888

§¿ ÛÌÃß³ Î¬Ò¨ (Ê ) Á Ü 1 £¸ ¦§
¥À . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Ü 2 £ É ¥À . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Ü 3 £ÂÙ ( Æ ) ¥À . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Ü 4 £ÆÑ ¡ Killing É ¥À . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46 Ü 5 £¾ ×°Ó ¥À . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Ü 6 £Ä Í«Ñ¥À . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83 Ü 7 £ Í«ÑÑÍ¥À . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Ü 8 £ £ ÃÕ ¥À . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119 Ü 9 £ ¹ ¥À . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Ü 10 £´ Ñ¥À . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 (Dis)claimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .167 i

1 « 1 Ã¤ º ¦ ˜1. ÓÅ A −
B = A ∩ (X − B), ∀ A, B ⊂ X. Æ ÍÔÅ¢× ³ Ã (A) Î x ∈ A − B, x ∈ A, x ∈ B. · x ∈ A ² A ⊂ X ÍÔ x ∈ X. ² x ∈ B ÍÔ x ∈ X − B. Í· x ∈ A Æ x ∈ A ∩ (X − B). ° Ý ° A − B Õ¹§¤Ý° A ∩ (X − B), A − B ⊂ A ∩ (X − B). (B) Î x ∈ A ∩ (X − B), x ∈ A, x ∈ X − B. ÑÏ x ∈ X, x ∈ B. x ∈ B ²· x ∈ A ÍÔ x ∈ A − B. ° Ý° A ∩ (X − B) Õ¹§² Ý° A − B, A ∩ (X − B) ⊂ A − B. ˜2. ÓÅ X − (B − A) = (X − B) ∪ A, ∀ A, B ⊂ X. Æ ÍÔÅ¢× ³ Ã (A) Î x ∈ X − (B − A), x ∈ X, x ∈ B − A. ÑÏ x ∈ B Ð x ∈ A. x ∈ B ²· x ∈ X ÍÔ x ∈ X − B. È x ∈ A Ð x ∈ X − B, Ø x ∈ (X − B) ∪ A. ° Ý° X − (B − A) Õ¹§¤Ý° (X − B) ∪ A, X − (B − A) ⊂ (X − B) ∪ A. (B) Î x ∈ (X − B) ∪ A, x ∈ X − B Ð x ∈ A. · ÑÏ x ∈ X, x ∈ B. x ∈ B ² x ∈ A Ð Õ Í¯Ø x ∈ A Ð x ∈ B Ê x ∈ B − A. x ∈ X − (B − A). ° Ý° (X − B) ∪ A Õ¹§¤Ý° X − (B − A), (X − B) ∪ A ⊂ X − (B − A). ˜3. £«¥Ð£ ¥ Åº f : RR → RR ¦¦Õ ÒÉÕ f(x) = x3 ( ) ( ) f(x) = x2 ( ÄÖf(x) ℄) ( µÖf(x) Ý ℄) f(x) = ex ( ) ( 0 ÊµÖf(x) Ý ℄) f(x) = cos x ( |f(x)| ∈ [0, 1] ℄) ( |f(x)| ∈ (1, ∞) Ý ℄) f(x) = 5, ∀ x ∈ RR ( ℄) ( Ì f(x) = 5 Ý ℄) ˜4. ªÅ℄¡ Õ ¾§ÞÄ (a) Ã»Ä RR Ò RR Õ Ì § « x = nπ + π 2 (n Â ) Ü tan x. (b) « Ä RR Ò RR Õ Ì § « x ∈ (−∞, 0) Ü log x.

2 (c) (a, b] ⊂ RR Tu Ö § «
Tu Õ¹§ RR Õ ¦Ð ¦Ç¾¯ (a, b] ∈ Tu. Í Tu Õ¹§ Ö¯Æ (a, b] Tu Ö (d) [a, b] ⊂ RR Tu ±Ö § « [a, b] ⊂ RR, −[a, b] = RR − [a, b] = (−∞, a) ∪ (b, ∞) ∈ Tu, Æ » 6 ÆÓ Tu [a, b] ±Ö [ (c) Õ ÀÉ (d) ÕµÄÁÃ ÆÓ ° ¸ Tu ¯(a, b] ß Ö £ ±ÖÆ] ˜5. ¹¦ ÊÅ¢£ £ (RR, Tu) Õ Ê¿ ÒÖÇÆ ¥ Î ¦Ø On = (0, 1/n) ∈ Tu, (n = 1, 2, · · ·). Æ ∩∞ n=1 On = {0}. Þ Ö ( ¦) ¦¯Ø {0} ∈ Tu, £ Ê¿ ÒÖÇÆ ¥ É {0} ±Ö¯ −{0} = RR − {0} = (−∞, 0) ∪ (0, ∞) ¡ 1(c) ∈ Tu, » §1.2 6 ÞÖ {0} ±Ö ˜6. ÓÅ §1.2 Ê 5 Õ Ñ¸ Ø× 1 Õ 3 ¿ « Æ Ñ¸ Õ (1-2-2) S := {V ⊂ A| ∃ O ∈ T V = A ∩ O}. « 1: ²Ô ℄ O ⊃ A, V = A ∩ O = A, Ø A S Õ¹§ ºÁÀ O = ∅, V = A ∩ ∅ = ∅ £ S Õ¹§ « 2: ÁÀ Vi = A ∩ Oi ∈ S, ∩iVi = ∩i(A ∩ Oi) Î = A ∩ (∩iOi), Ç ∩iVi ⊂ A. º §1.2 1(b), ∩iOi ∈ T , Ñ¸ Õ Æ ∩iVi £ S Õ¹§ « 3: ÁÀ Vα = A ∩ Oα ∈ S ∀ α, ÖÆ » Å¢ ∪αVα = ∪α(A ∩ Oα) = A ∩ (∪αOα). Ò (A ∩ O1) ∪ (A ∩ O2) = A ∩ (O1 ∪ O2), Æ (A∩O1)∪(A∩O2)∪(A∩O3) = [A∩(O1 ∪O2)]∪(A∩O3) = A∩[(O1 ∪O2)∪O3] Î = A∩[O1 ∪O2 ∪O3]. µ Ô ∪α(A∩Oα) = A∩(∪αOα). ° ∪αVα = A∩(∪αOα), ∪αVα ⊂ A. º §1.2 1(c), ∪αOα ∈ T , Ñ¸ Õ Æ ∪αVα £ S Õ¹§ 7. ¹Ê¡¢ (RR3, Tu) ℄ ±ÕÒÖ § ¦Õ¦ Æ ÕÆ Þ× º ÞÝ°·ÒÖ A, ªÛ A ß £ ± Ý° Tu, Ø ¯°Á ÊÕ ÇÆÉ Ê Õ Ç¾ÔÒ ± −A = RR3 − A £ Ý° Tu, Í −A ∈ Tu ¯ A ±Õ ˜8. Ê Ì f : (X, T ) → (Y, S) Ï× Û

3 § Ï× Æ Ê Ì (§1.1 7) Ë ∀
x ∈ X, x → f(x) = y0 ∈ Y , Í Y Õ¦¿¹§ ( ¦Þ). « ∀ O ∈ S, » §1.1 5(2) ¬¯¢Û f−1[O] = ∅ ( y0 ∈ O ∈ S), ¢Û f−1[O] = X ( y0 ∈ O). ∅ É X ¤ T Õ¹§¯Æ f−1[O] ∈ T . » §1.2 3a, ¦ ÌÏ× ˜9. Î T Ö X ÉÕÃÆ¸ ¯ S Ö Y ÉÕ º¸ ¯ (a) Ê (X, T ) Ò (Y, S) Õ Ï× Ì (b) Ê (Y, S) Ò (X, T ) Õ Ï× Ì (X, T ) ÃÆ¸ ¦ (§1.2 Ê 1), T Õ¹§ X ÕÆ ÒÖ (Y, S) º¸ ¦ (§1.2 Ê 2), S Õ¹§Í 2 ¿ — ∅ É Y . ( ÁÀ X = Y = RR, T Õ¹§ RR ÕÆ ¦ÕÖÍ¯ Ñ ± S Õ¹§£Í 2 ¿ — ∅ É RR.) (a) (X, T ) Ò (Y, S) Õ Ë Ì¤ Ï×Õ Í¢ Ì¯µ ¯ ÕÖÍ¦ X ÕÒÖ¯ X ÕÒÖ¦ ÃÆ¸ T Õ¹ § » §1.2 3a, ¦ Ì Ï×Õ S º¸ ¯ 3a È Õ O Í ¯ ¿ ¦ O = ∅ ∈ S, f−1[∅] = ∅ ∈ T ; O = Y ∈ S, f−1[Y ] = X ∈ T . µÄÁÃ Ê¯Í¢ (X, T ) ÃÆ¸ ¦¯ (Y, S) Ë¸ ¦¯ ¦ Ñ ÀË ( Á À X = Y = RR, µ ¡ ÕÞÕÖÍ ÀÕ ¦ Ñ ±¤ RR ÕÒ Ö¯£µ T Õ¹§ ) (b) (Y, S) Ò (X, T ) Í Ê Ì Ï×Õ Ê ÌÕÏ×Ñ À 8 Õ Ñ È Ê¯Å ¢Å¢°¯ Ë Ì¤ Ï×Õ ÁÀ Ê Ì¯ ÎË ¿ x1, x2 ∈ X, x1 = x2. È X Õ ¹ §ÒÖ O1 = {x1 } É O2 = {x2 }. T ÃÆ¸ ¯ X ÕÆ ÒÖ ¤ Õ¹§¯Æ O1, O2 ∈ T . Å Õ f−1[O1] = f−1[{x1 }] É f−1[O2] = f−1[{x2 }], Y Õ ÒÖ ( Õ x1 É x2). ÖÆ Ò f−1[O1] ∩ f−1[O2] = ∅, Õ ²℄ º Þ y ∈ f−1[O1] ∩ f−1[O2], ¿ ±Õ x1 É x2, ² §1.1 ÌÕ 5 ± ÁÀ ¦ Ì Ï×Õ¯ » 3a, ¢ f−1[O1], f−1[O2] ∈ S. S º¸ ¯ ¹§Í Y , Ï× Ì¢ f−1[O1] = f−1[O2] = Y , Ç ² Õ ÍÆÑÚ¯ ¯ Ë °¦¿ Õ Ì ¾ Å¤ Ï×Õ ²· ¦¡¯Í¢ (Y, S) º¸ ¦¯ (X, T ) Ë¸ ¦¯ ¦ Ñ ÀË ˜10. ÓÅ §1.2 3a ² 3b Õ×£Ñ Æ (A) 3a Ò 3b, Ø¢Å¢ÁÀ 3a Õ ÌÏ×¯ 3b Õ Þ¤Ï× (B) 3b Ò 3a, Ø¢Å¢ÁÀ 3b Õ Þ¤Ï×¯ 3a Õ ÌÏ×

4 (A) ¦ÞÕ Ì x → f(x). ¸ S ¿¹§
G′ ÉG′′, Ø× f(x) ∈ G′′ ⊂ G′. Ì Ï×Õ¯Ï» 3a G ≡ f−1[G′′] ∈ T , x ∈ G. ÆÈ ∃ G ∈ T x ∈ G ¼ f[G] = G′′ ⊂ G′. » 3b, Ì Þ x ÎÏ× x ∈ X Õ (B) Õ¦¿ Ö O ∈ S, Î Õ¹§ yα, Ø∀ α, yα ∈ O. ÁÀ yα xα = f−1(yα) ¼ Ì Ï×Õ¯ » 3b, ¦ ℄ Ö Gα ∈ T xα ∈ Gα ¼ f[Gα] ⊂ O. Å¢ ∪αGα = f−1[O], 1(c) Æ f−1[O] ∈ T . ° » 3a, ÌÏ× Ú Å¢ ∪αGα = f−1[O], (i) Æ Ý° ∪αGα Õ¹§²Ý° f−1[O]; (ii) Æ Ý° ∪αGα Õ¹§² Ý ° f−1[O]. (i) Î x ∈ ∪αGα, f[∪αGα] = ∪αf[Gα] f[Gα]⊂O ⊂ O, f(x) ∈ f[∪αGα] ⊂ O, Ø f(x) ∈ O. » 5 ¬Õ f−1[O] = {x ∈ X| f(x) ∈ O}, Æ x ∈ f−1[O]. (ii) Î x ∈ ∪αGα, x ∈ ∩α(X − Gα) = ∩α(−Gα), f(x) ∈ f[∩α(−Gα)] = ∩αf[−Gα]. x ² Ý° f−1[O], Õ f(x) ∈ O, ² f(x) ∈ ∩αf[−Gα] Ú¯ Ú Í¢Å¢ÁÅ£ ∀ α, f[Gα] ⊂ O, O ∩ ( ∩α f[−Gα]) = ∅. Ò f[−Gα] = f[X − Gα] = Y − f[Gα], ∩αf[−Gα] = ∩α(Y − f[Gα]) DM = Y − ∪αf[Gα] = − ∪α f[Gα]. Aα ≡ f[Gα], Ø¢Å¢ ∀ α, Aα ⊂ O, O ∩ (− ∪α Aα) = ∅, Ç ¦ Ñ ÀËÆ » 3b ¦℄ ÔÒÁÅ Ä ÁÀ Æ Þ x ∈ X ÉÏ×¯ « ∀ Ø× f(x) ∈ G′ Õ G′ ∈ S, ∃ G ∈ T x ∈ G ¼ f[G] = G′. ² Õ » ° f[G] = G′ f[G] ⊂ G′. È Õ¦¿ Ö O ∈ S, Î Õ¹§ yα, Ø ∀ α, yα ∈ O. ÁÀ yα xα = f−1(yα) ¼ Ì Ï×Õ¯ » 3b µÊÕ Ä¯¦ ℄ Ö Gα ∈ T xα ∈ Gα ¼ f[Gα] = O. ° f−1[O] = Gα ∈ T , ° » 3a, ÌÏ× 11. ÓÅ ¦ ¦ (a, b) ⊂ RR ² RR ± Æ Ò Ì f : x → tan x (−π 2 , π 2 ) → (−∞, ∞) = RR Õ¦¦ÒÉ Ì y = αx + β, −π 2 = αa + β π 2 = αb + β Ô α = π b−a β = −π 2 a+b b−a , ° y = π b−a x − π 2 a+b b−a = π b−a (x − a+b 2 ). Ì f : x → tan [ π b−a (x − a+b 2 )], (a, b) → RR Õ¦¦ÒÉ Ì Å Å¢ f « ((a, b), T ′ u ) → (RR, Tu) Ï×× f−1 « (RR, Tu) → ((a, b), T ′ u ) Ï ×¯° T ′ u Tu Õ Ñ¸ » §1.2 3a, ´ É ¦ Ñ ÅÓ¢Õ » §1.2 4, ± Ì¯¸ Ò ¦ ((a, b), T ′ u ) ² ¸ ¦ (RR, Tu) ± 12. ÎX1 ÉX2 RR ÕÒÖ¯X1 ≡ (1, 2)∪(2, 3), X2 ≡ (1, 2)∪[2, 3). T1 ÉT2 » º RR Õ° ¸ X1 É X2 ÉÕ Ñ¸ ¸ ¦ (X1, T1)

5 É (X2, T2) Ï° È´ ÆÓ (X1, T1) Ï°
(X2, T2) Ï°¯Å Å¢ ¦ Ñ ÖÆ » Ñ¸ Õ [§1.1 Ê 5 (1-2-2)] ÆÓ T1 É T2 » Ã° X1 É X2 ÕÆ ¦× ¦Ç¾ÕÖÍ (A) (X1, T1) Õ Ï°Ñ ÖÆ (1, 2) É (2, 3) ¤ ¦¯Æ ¤ T1 Õ¹§¯Æ Õ ° ¯ −(1, 2) = X1 − (1, 2) = (2, 3) ∈ T1 Õ¯ Æ » §1.2 6, (1, 2) ±Õ ±¡ ¡¢ (2, 3) £ ±Õ ¸ ¦ (X1, T1) ( ÎË) 4 ¿ß ±ÕÒÖ ∅ ¡ X1 ¡ (1, 2) É (2, 3), Æ » §1.2 7, Ï°Õ ¯ Ã §1.2 Ê 9 Õ¦¿ Ê Å ¡¢Ë 4 ¿¯Ü °¯Õß ±ÕÒ ¦ Ý° X1 ÕÒ ¦ ( Ò ¦¡ÒÖ) Í 4 Á ¯Á (a) (1.6, 1.7) ¡(b) [1.6, 1.7] ¡(c) (1.6, 1.7] É (d) [1.6, 1.7). ÏÀÆ Ê¯° (a) ±Õ ( ÏÝ° T1 Ö Ý ° T1), (b) ± Õ ( ÖÝ° T1 Ï Ý° T1), (c) É (d) ß £ ±Õ ( Ï Ý° T1 Ö£ Ý° T1). Ü °¯Õß ± ÕÒÖ ( ÏÉ Ö¤Ý° T1). (B) (X2, T2) ÕÏ°Ñ ÖÆ Ò (1, 2) Õ [2, 3) Õ ( ° ±Õ¯©Å). [2, 3) Õ Ñ » Ñ¸ Õ ¯È Ò O ∈ Tu, ¯ [2, 3) = X2 ∩ O = [(1, 2) ∪ [2, 3)] ∩ O, Æ [2, 3) ∈ T2. ° ¯ −(1, 2) = X2 − (1, 2) = [2, 3) ∈ T2, Õ¯Æ » §1.2 6, (1, 2) ±Õ −[2, 3) = X2 − [2, 3) = (1, 2) ∈ T2, Æ » 6, [2, 3) ±Õ £µ ¡ (1, 2) ±Õ¯ [2, 3) ± Õ » (A) Ú Õ Ñ¯ ÆË ∅ É X2, Ü °¯Õß ±ÕÒÖ ( Ò ¦), » §1.2 7, Ï°Õ 13. ÖÍ X ÃÆ¸ T ÆÔÕ¸ ¦ Ï° Ï°¯Å¢ÁÅ Æ ÃÆ¸ Ë X ÕÆ ÒÖ¤ T Õ¹§ (§1.2 Ê 1). « Õ A ⊂ X, −A = X − A ∈ T . » §1.2 6, A ± Õ £µ ¡ X Õ ËÒÖ¤ ß ±Õ Æ » §1.2 7, ÃÆ¸ ¦ Ï°¯ £Ã £ÃÆ¥Õ £ Ê º¸ T ¦ Ï°Õ¯ Í ¿¹§ ∅ É X, ° À « Ë°¯Õ A ⊂ X, A ∈ T × −A = X − A ∈ T . » 6, A ß Õ£ ±Õ Í ∅ É X ß ±Õ¯ » 7, Ï ° £ £ º¥Õ ˜14. Î A ⊂ B, ÓÅ (a) ¯ A ⊂ ¯ B; A ⊂ B º¢ ¯ B Ã A Õ±Ö (b) i(A) ⊂ i(B). Æ ¿ Ñ ´ É Ç Õ

6 (a) §1.2 Ä 1-2-3(a) ¬¯ B ⊂ ¯ B,
A ⊂ B ⊂ ¯ B. » §1.2 8, ± ¯ A Ã A Õ Ú ±Ö¯£µ Ã A Õ Æ ±ÖÕÆÖ ß È A ⊂ ¯ B, » §1.2 Ä 1-2-3(a) «¯ ¯ B ±Ö¯Æ ¯ B £ 8 Õ Cα Ç¦ ¯ A ⊂ Cα ∀ α, ¯ A ⊂ ¯ B. É À 15 Õ ÀÅ¢ ∀ x ∈ ¯ A, x Õ ¶ N, » ⇒, Æ N ∩ A A ⊂ B, ÁÀ N ∩ A ¯² N ∩ B É ⇐, x ∈ ¯ B. ° Ý° ¯ A Õ¹§²Ý° ¯ B, » §1.1 Õ 1, ¯ A ⊂ ¯ B. (b) §1.2 Ä 1-2-3(b) ¬¯i(A) ⊂ A, i(A) ⊂ A ⊂ B. » §1.2 9, Æ i(B) Ã° B Õ Ú Ö¯£µ Ã° B Õ Æ ÖÕ¾Ö ß È i(A) ⊂ B, » §1.2 Ä 1-2-3(b) «¯ i(A) Ö¯Æ i(A) £ 9 Õ Oα Ç¦ Oα ⊂ i(B) ∀ α, i(A) ⊂ i(B). ˜15. ÓÅx ∈ ¯ A ⇔ x Õ ¦ ¶² A ÇÆÖ « ⇒ Å¢Õ Î O ∈ T ¼ O ∩ A = ∅, ÆÅ A ⊂ X − O, Å ( É ± ) ¯ A ⊂ X − O. Æ ¿ ¤°Á×£Õ £ Å¢ ⇒ Õ £ ºÞ ÁÀ ℄ x Õ ¶¯ ² A ÕÆÖ ¯ x ∈ ¯ A; ⇐ Õ £ ºÞ ÁÀ x ∈ ¯ A, ¦ ℄ x Õ ¶¯ ² A ÕÆÖ ⇒: ÖÆ¯ÁÀ O∩A = ∅, ªÛ x ∈ A ¯² x ∈ O. x ∈ X, » Ö Õ (§1.1 2) Æ ² x ∈ X −O. Ä O∩A = ∅, A ⊂ X −O. È ÎO ∈ T ¦ Ö¯ X −(X −O) Â 2 = (X −X)∪O = ∅∪O = O ∈ T . » §1.2 6 Æ X − O ±Ö ß A ⊂ X − O, X − O ±Ö¯ » §1.2 ± Õ 8: ¯ A Æ Ã A Õ±ÖÕÆÖ¯Ó X − O Õ Cα Ç¦¯Æ ¯ A ⊂ X − O. Î Å¢ « ¦ Ö O ∈ T , Á À O ∩ A = ∅, ¯ A ⊂ X − O. Å O x Õ ¶¯Ø x ∈ O. ² ¯ A ⊂ X − O ÍËØÆÓ x ∈ ¯ A. ÁÀ x ∈ ¯ A Õ ¯ ¯ A ⊂ X − O x ∈ X − O, Ø x ∈ O, ² x ∈ O Ú¯ ° Å¢ ÁÀ x ℄ ¦ ¿ ¶ O, ² A ÕÆÖ ¯ x ∈ ¯ A. µ ⇒ Õ £ Á À O Õ ¯ Ñ§ ÀË¯ Õ Õ¢£ ¥ ¯ÁÀ Õ² A ÕÆÖ ¯ Õ ² A ÕÆÖ ¾ ¡ ÕÎË ³ ¯ ÁÀ² A ÍÆ¯ªÛ Õ¢½Ã A ½ ( Å¢Á Ã Ë¢ Ò O ) ⇐: ÁÀ x ∈ ¯ A, » §1.2 Ä 1-2-3(a) ¬ A ⊂ ¯ A x ∈ A. ² ℄ x Õ¦¿ ¶ N, x ∈ N, Ô N ∩ A = ∅. Å¢ÁÅ x ∈ A, Æ x ∈ X − A. È X − A Æ x Õ¦¿ ¶¯Ø x ∈ N ⊂ X − A. ¡ÔÒ Õ N ²Ø× N ∩ A = ∅, N ⊂ X − A º¢ N Õ¹§² Ý° A [ Ý° N Õ¹§²Ý° X − A (§1.1 1), Ý° X − A Õ¹§² Ý° A (§1.1 2).], N ² A ÍÆ £ ÔÅ

7 16. ÓÅ RR ÏÕ Æ NN ºÓ Ö¯
{(−n, n)|n ∈ NN} RR Õ ²¹¯ Ü ÊÒ²¹ Æ C ¿ßÉ (X, T ) Þ¡ ¯ A ⊂ C A (X, T ) ¡ ¯ A Þ Æ C ⊂ X, ©° C = X, X Þ¯¾Ý £Î 1-3-3 Á ÕÆ C X Ô¡ ¯ £ÇÍÑÝ (?) Ä¡ B Ð¤ C ⊂ B ⊂ X. Ø× B − A Ä Æ X − (B − A) ¸2 = (X − B)∪ A. B Ä¯ X −B « A ¯ ℄ §1.2 £Î1-2-2(b) Ù(X −B)∪A Õ« X −(B −A) ¯B −A Ä ¦¹Æ {Oα } A §ËÄ ¯ {Oα, B − A} ÇÍÄ A ∪ (B − A) = B, {Oα } A, B − A B − A. ÎÝ C ⊂ B, « {Oα, B − A} É C Ë Ä C Þ ¯´ §µÄ ¦Ö ¡ ¯Æ {Oα1 , · · · , Oαn ; B − A}. A ∩ (B − A) = ∅, ³Í B − A Ý A, ´ÝÚ C − A. É {Oα1 , · · · , Oαn } A Ä {Oα } A §ËÄ ¯¦ ¡ {Oα1 , · · · , Oαn }, Á §1.3 £Ï 1 Ù A Þ « 2 Ã¤° Å £¦ ˜1. ÓÅ §2.1 Ê 2 Õ¸ ± Ì ψ± i O± i ÕÆ Æ ÉØ×ÍÀÑ «¯ Å S1 1 ÆÎ(x, y) RR2 ÕÓ Þ¸¯ » ( Ã ¨Þ) ÁÅ O+ i := {(x, y) ∈ S1| xi > 0}, O− i := {(x, y) ∈ S1| xi < 0}, i = x, y, »« Û É ÉÅ » O± i Ò RR Õ ¦ V = (−1, 1) Õ± Ì ψ± i ÁÅÕ² Ì ψ± x (x, y) = y, ψ± y (x, y) = x. Å Å¢ »Æ Ø× ÍÀÑ « µÜ¦ ÊÕÆ Ê¯ O+ x É O+ y ÕÆ ¯Ø O+ x ∩ O+ y = ∅. Í Õ Ì O+ x → V + x = ψ+ x [O+ x ]: ψ+ x : (x, y) → x1 = y , Ø x1 = ψ+ x (x, y) = ψ+ x 1 − y2, y = y , ° Ì V + x → O+ x = (ψ+ x )−1[V + x ]: (ψ+ x )−1 : x1 → (x, y) = 1 − (x1)2, x1 ,

8 Ø (x, y) = (ψ+ x )−1(x1) = 1
− (x1)2, x1 ; O+ y → V + y = ψ+ y [O+ y ]: ψ+ y : (x, y) → x′1 = x , Ø x′1 = ψ+ y (x, y) = ψ+ y (x, 1 − x2) = x . ° ³Í Ì ψ+ y ◦ (ψ+ x )−1 V + x → ψ+ x ∩ ψ+ y → V + y . » §1.1 8 Æ ψ+ y ◦ (ψ+ x )−1 : x1 → x′1 Ê³ÍÄ x′1 = (ψ+ y ◦ (ψ+ x )−1)(x1) = ψ+ y (ψ+ x )−1(x1) = ψ+ y 1 − (x1)2, x1 = 1 − (x1)2 . ° ÆÓ³Í Ì ψ+ y ◦ (ψ+ x )−1 Õ 1 ¿ 1 ¹Ä x′1 = φ1(x1) = 1 − (x1)2 , Ç Ë©Æ (|x1| < 1) Ê ¾¾Ï×¯Ø C∞ Õ¯ ± ¡ Å¢°¯Õ³Í Ì ψβ ◦ ψ−1 α £¤ Õ¯° ´ (O− 1 , ψ− 1 ) ²´ (O+ 2 , ψ+ 2 ) ÍÆÍÀ ( Ü Ê), ´ (O− 1 , ψ− 1 ) ²´ (O− 2 , ψ− 2 ) ÍÆÍÀ ( ÜÅ Ê), ´ (O+ 1 , ψ+ 1 ) ²´ (O− 2 , ψ− 2 ) ÍÆÍÀ ( Ü¥ Ê). »ÇÆÇÍØ× 3 S2 2 Á Æ (x, y, z) RR3 ¢¦¨ ℄¯£ÏÄÍ¡Õ ( Î² ) © O+ i := {(x, y, z) ∈ S2| xi > 0}, O− i := {(x, y, z) ∈ S2| xi < 0}, i = x, y, z, ª ¦ ¡ ¹¡¬ ÄÍ ¡Õ £Ï O± i Ý RR2 Ú Ä Ü D = {(x, y) ∈ RR2| x2 + y2 < 1} »Ý ψ± i © ¼ ψ± x (x, y, z) = (y, z), ψ± y (x, y, z) = (x, z), ψ± z (x, y, z) = (x, y). ÕØ×ÄÍ¡Õ ¢£Ð¤ ¨Âº Ó Í Ë© ¢£ ¯´ O+ x ¡O+ y ´ O+ z ¢ Î 3 Ê¨℄ ¢¯È Ø×´Ó℄℄ ¨ ÕÍ O+ x ∩ O+ y O+ x → D+ x = ψ+ x [O+ x ]: ψ+ x : (x, y, z) → (x1, x2) = (y, z) , (x1, x2) = ψ+ x (x, y, z) = ψ+ x 1 − y2 − z2, y, z = (y, z) , Û D+ x → O+ x = (ψ+ x )−1[D+ x ]: (ψ+ x )−1 : (x1, x2) → (x, y, z) = 1 − (x1)2 − (x2)2, x1, x2 , (x, y, z) = (ψ+ x )−1(x1, x2) = 1 − (x1)2 − (x2)2, x1, x2 ;

9 O+ y → D+ y = ψ+ y [O+
y ]: ψ+ y : (x, y, z) → (x′1, x′2) = (x, z) , (x′1, x′2) = ψ+ y (x, y, z) = ψ+ y x, 1 − x2 − z2, z = (x, z) . ¶ ψ+ y ◦ (ψ+ x )−1 D+ x → ψ+ x ∩ ψ+ y → D+ y . Á §1.1 £Ï 8 Ù ψ+ y ◦ (ψ+ x )−1 : (x1, x2) → (x′1, x′2) Ò ¶³ (x′1, x′2) = (ψ+ y ◦ (ψ+ x )−1)(x1, x2) = ψ+ y (ψ+ x )−1(x1, x2) = ψ+ y 1 − (x1)2 − (x2)2, x1, x2 = 1 − (x1)2 − (x2)2, x2 . ÓÙÞ ¶ ψ+ y ◦ (ψ+ x )−1 2 2 ³ x′1 = φ1(x1, x2) = 1 − (x1)2 − (x2)2 , x′2 = φ2(x1, x2) = x2 , ¦ Ú ÜÙ Ç Å¯ Ô º 2. ¡¢ n ¦ Ý n § n ¦ ¯ ¦¿Þ¸¶²¹Õ ¯Æ 3. Î X É Y ¸ ¦¯ f : X → Y ± Ä X ¿ ¯Ó Y ¦¿¾ f : X → Y ¾ ¾ ± § » §1.2 4, ÁÀ f ¸ ¦Ç¦Õ± Ì¯ªÛ ¦¦Ò ÉÕ¯℄ Ì f−1 : Y → X, ¼ f É f−1 ¤Ï× È X ¿ ¯ª Û Y °Á X Ô¾ « X Ì ψ Þ¸ ψ : X → RRn, « Y °Á Ì ψ ◦ f−1 : Y → RRn Þ¸ ˜4. Î{x, y} RR2 ÕÓ Þ¸¯C(t) Ë¯ º x = cos t, y = sin t, t ∈ (0, π). Ä p = C(π/3), Ê Ë p Õ» Ó Þ¸ÑÕ ¯¾ ´ºÊ· Ë×·» Ó Þ¸Å¯C(t) = (x(t), y(t)) = (cos t, sin t), d dt C(t) = (sin t, − cos t). p ÞÕ» v(p) = d dt C(t)|p = d dt C(t)|t=π/3 = (sin π 3 , − cos π 3 ) = ( √ 3 2 , −1 2 ), Ø p ÞÕ» Ó Þ¸ÑÕ vx = √ 3 2 , vy = − √ 1 2 . 5. Î Ë C(t) É C′(t) ≡ C(2t0 − t) C(t0) = C′(t0) ÞÕ» » v É v′, ÓÅ v + v′ = 0. Æ C′(t′) = C(2t0 − t′), » ¯ Ë C′(t′) Õ» v′(t′) = d dt′ C′(t′) = −C(1)(2t0 −t′), ÇC(1)(x) ºC(x) Õ¦ Ñ ° t′ = t0 Þ¯v′(t0) = −C(1)(t0). ¦ ¯ Ë C(t) Õ» v(t) = d dt C(t) = C(1)(t). ° t = t0 Þ¯v(t0) = C(1)(t0). ¦ÞÉ C(t0) = C′(t0), v(t0) + v′(t0) = 0.

10 ˜6. Î O Þ¸Ã {xµ} ÕÞ¸¶¯ p ∈ O,
v ∈ Vp, vµ v ÕÞ¸ ¯ Þ¸ xµ Ý O ÉÕ C∞ Ä ¯ÓÅ vµ = v(xµ). v = vνXν ³Ý ° Ä xµ. Æ Þ¸ xµ Ý O ÉÕ C∞ Ä ¯Ø (2-2-1’) Õ f, v = vνXν Ý É ÔÒ (Vp → RR): v(xµ) = vνXν(xµ). Ç v ∈ Vp p Þ Õ ¯ v(xµ) ∈ RR ¿ p ÞÕÞ¸Ñ Xν ∈ Vp £ ¯ Xν(xµ) ∈ RR £ ¿ vν ∈ RR ¿ º v ÕÞ¸ É (2-2-1’), Xν(xµ) = ∂ ∂xν xµ = δν µ = δµ ν = δµ ν , Ø v(xµ) = vνδµ ν = vµ. 7. Î M 2 ¯ (O, ψ) É (O′, ψ′) M ÉÕ ¿Þ¸Ã¯Þ¸ » {x, y} É {x′, y′}, O ∩ O′ ÉÕÞ¸¶ x′ = x, y′ = y − Ωx (Ω = ), Ó » ÊÞ¸Ñ ∂/∂x, ∂/∂y Þ¸Ñ ∂/∂x′, ∂/∂y′ Õ Þ¸Ñ Xµ = ∂ ∂xµ Õ¶ Ã (2-2-5) Xµ = ∂x′ν ∂xµ | p X′ ν . ÆÈ Xx = ∂x′ ∂x X′ x + ∂y′ ∂x X′ y = X′ x − ΩX′ y , Xy = ∂x′ ∂y X′ x + ∂y′ ∂y X′ y = X′ y , Ø ∂ ∂x = ∂ ∂x′ − Ω ∂ ∂y′ , ∂ ∂y = ∂ ∂y′ . £ ¡ÔÒ f(x, y) = f′(x′, y′), ∂ ∂x (f) = ∂f ∂x = ∂x′ ∂x ∂f′ ∂x′ + ∂y′ ∂x ∂f′ ∂y′ = ∂f′ ∂x′ − Ω ∂f′ ∂y′ = ∂ ∂x′ − Ω ∂ ∂y′ (f) , ØÉ ÕÜ¦¿ Ã ˜8. (a) ÓÅ (2-2-9) Õ [u, v] ÞÞØ× (§2.2 2) Õ «¯ Õ (b) Î u, v, w M ÉÕ ¯ÓÅ [[u, v], w] + [[w, u], v] + [[v, w], u] = 0 ( ¢ ª ) . Æ (a) [u, v] Ø× §2.2 2(a): [u, v](αf + βg) (2-2-9) = u(v(αf + βg)) − v(u(αf + βg)) ¡ 2(a) = u(αv(f) + βv(g)) − v(αu(f) + βu(g)) ¡ 2(a) = αu(v(f)) + βu(v(g)) − αv(u(f)) − βv(u(g)) = α[u(v(f)) − v(u(f))] + β[u(v(g)) − v(u(g))] (2-2-9) = α[u, v](f) + β[u, v](g) .

11 [u, v] Ø× §2.2 2(b): [u, v](fg) (2-2-9) =
u(v(fg)) − v(u(fg)) ¡ 2(b) = u(f|pv(g) + g|pv(f)) − v(f|pu(g) + g|pu(f)) ¡ 2(a) = f|pu(v(g)) + g|pu(v(f)) − f|pv(u(g)) − g|pv(u(f)) = f|p[u(v(g)) − v(u(g))] + g|p[u(v(f)) − v(u(f))] (2-2-9) = f|p[u, v](g) + g|p[u, v](f) . (b) Ý × [[u, v], w] + [[w, u], v] + [[v, w], u] = [uv − vu, w] + [wu − uw, v] + [vw − wv, u] = (uv − vu)w − w(uv − vu) + (wu − uw)v − v(wu − uw) +(vw − wv)u − u(vw − wv) = uvw − vuw − wuv + wvu + wuv − uwv − vwu + vuw +vwu − wvu − uvw + uwv = 0 . ˜9. Î {r, ϕ} RR2 ¦ Ö ( Þ¸¶) ÉÕÕÞ¸¯ {x, y} Ó Þ¸¯ (a) ÊÕÞ¸ÃÕÞ¸Ñ ∂/∂r É ∂/∂ϕ ( Ý Þ¸¶ÉÕ ) ∂/∂x, ∂/∂y Õº (b) [∂/∂r, ∂/∂x] ∂/∂x, ∂/∂y Õº (c) ˆ er ≡ ∂/∂r, ˆ eϕ ≡ r−1∂/∂ϕ, [ˆ er, ˆ eϕ] ∂/∂x, ∂/∂y Õº (a) x = r cos ϕ, y = r sin ϕ, r = x2 + y2, tan ϕ = y x É cos ϕ = x √ x2+y2 , sin ϕ = y √ x2+y2 . É Ñ Õ¶ Ã (2-2-5) eµ = ∂x′ν ∂xµ e′ ν , er = ∂x ∂r ex + ∂y ∂r ey = cos ϕ ex + sin ϕ ey , eϕ = ∂x ∂ϕ ex + ∂y ∂ϕ ey = −r sin ϕ ex + r cos ϕ ey , Ø ∂ ∂r = cos ϕ ∂ ∂x + sin ϕ ∂ ∂y = x x2 + y2 ∂ ∂x + y x2 + y2 ∂ ∂y , ∂ ∂ϕ = = −r sin ϕ ∂ ∂x + r cos ϕ ∂ ∂y = −y ∂ ∂x + x ∂ ∂y .

12 £ ÁÅµÊ frϕ(r, ϕ) = fxy(x, y), ∂ ∂r
(f) = ∂frϕ(r, ϕ) ∂r = ∂fxy(x, y) ∂r = ∂x ∂r ∂fxy(x, y) ∂x + ∂y ∂r ∂fxy(x, y) ∂y = cos ϕ ∂ ∂x + sin ϕ ∂ ∂y (f) , ∂ ∂ϕ (f) = ∂frϕ(r, ϕ) ∂ϕ = ∂fxy(x, y) ∂ϕ = ∂x ∂ϕ ∂fxy(x, y) ∂x + ∂y ∂ϕ ∂fxy(x, y) ∂y = − r sin ϕ ∂ ∂x + r cos ϕ ∂ ∂y (f) , ° · ±¡Õ À (b) É (a) Õ À¯ ∂ ∂r , ∂ ∂x = x x2 + y2 ∂ ∂x + y x2 + y2 ∂ ∂y , ∂ ∂x = x x2 + y2 ∂ ∂x + y x2 + y2 ∂ ∂y ∂ ∂x − ∂ ∂x x x2 + y2 ∂ ∂x + y x2 + y2 ∂ ∂y = − ∂ x √ x2+y2 ∂x ∂ ∂x − ∂ y √ x2+y2 ∂x ∂ ∂y = − y2 (x2 + y2)3/2 ∂ ∂x + xy (x2 + y2)3/2 ∂ ∂y . (c) É (a) Õ À¯ ˆ er ≡ er = x x2 + y2 ∂ ∂x + y x2 + y2 ∂ ∂y , ˆ eϕ ≡ 1 r eϕ = − y x2 + y2 ∂ ∂x + x x2 + y2 ∂ ∂y . ° [ˆ er, ˆ eϕ] = x x2 + y2 ∂ ∂x + y x2 + y2 ∂ ∂y , − y x2 + y2 ∂ ∂x + x x2 + y2 ∂ ∂y = x x2 + y2 ∂ ∂x + y x2 + y2 ∂ ∂y − y x2 + y2 ∂ ∂x + x x2 + y2 ∂ ∂y − − y x2 + y2 ∂ ∂x + x x2 + y2 ∂ ∂y x x2 + y2 ∂ ∂x + y x2 + y2 ∂ ∂y = − x x2 + y2 ∂ y √ x2+y2 ∂x − y x2 + y2 ∂ y √ x2+y2 ∂y ∂ ∂x + x x2 + y2 ∂ x √ x2+y2 ∂x + y x2 + y2 ∂ x √ x2+y2 ∂y ∂ ∂y − − y x2 + y2 ∂ x √ x2+y2 ∂x + x x2 + y2 ∂ x √ x2+y2 ∂y ∂ ∂x

13 − − y x2 + y2 ∂ y √
x2+y2 ∂x + x x2 + y2 ∂ y √ x2+y2 ∂y ∂ ∂y = x2y (x2 + y2)2 − x2y (x2 + y2)2 ∂ ∂x + xy2 (x2 + y2)2 − xy2 (x2 + y2)2 ∂ ∂y − − y3 (x2 + y2)2 − x2y (x2 + y2)2 ∂ ∂x − xy2 (x2 + y2)2 + x3 (x2 + y2)2 ∂ ∂y = y x2 + y2 ∂ ∂x − x x2 + y2 ∂ ∂y = y x2 + y2 ˆ ex − x x2 + y2 ˆ ey . × £ É Å Õ À Ô ˜10. Î u, v M ÉÕ ¯ÓÅ [u, v] ËÞ¸ÑÚÕ Ø× [u, v]µ = uν∂vµ/∂xν − vν∂uµ/∂xν . (2-2-3’) É (2-2-3). Æ (2-2-3’), [u, v] ÕÜ µ [u, v]µ [u, v] Ý ÒÄ xµ ÉÕ Ê¯Ø [u, v]µ = [u, v](xµ) (2-2-9’) = u(v(xµ)) − v(u(xµ)) (2-2-3’) = u(vµ) − v(uµ) (2-2-3) = uνXν(vµ) − vνXν(uµ) (2-2-1’) = uν ∂vµ ∂xν − vν ∂uµ ∂xν . [ ÁÔ Çß¬· (b) ´ (c) ° (b) ∂ ∂r = x √ x2+y2 ∂ ∂x + y √ x2+y2 ∂ ∂y , ∂ ∂x = ∂ ∂x , ∂ ∂r , ∂ ∂x x = ∂ ∂r x ∂( ∂ ∂x )x ∂x + ∂ ∂r y ∂( ∂ ∂x )x ∂y − ∂ ∂x x ∂( ∂ ∂r )x ∂x − ∂ ∂x y ∂( ∂ ∂r )x ∂y = x x2 + y2 ∂1 ∂x + y x2 + y2 ∂1 ∂y − 1 ∂ x √ x2+y2 ∂x − 0 ∂ x √ x2+y2 ∂y = − y2 (x2 + y2)3/2 , ∂ ∂r , ∂ ∂x y = ∂ ∂r x ∂( ∂ ∂x )y ∂x + ∂ ∂r y ∂( ∂ ∂x )y ∂y − ∂ ∂x x ∂( ∂ ∂r )y ∂x − ∂ ∂x y ∂( ∂ ∂r )y ∂y = x x2 + y2 ∂0 ∂x + y x2 + y2 ∂0 ∂y − 1 ∂ y √ x2+y2 ∂x − 0 ∂ y √ x2+y2 ∂y = xy (x2 + y2)3/2 ,

14 (b) ° ∂ ∂r , ∂ ∂x = −
y2 (x2 + y2)3/2 ∂ ∂x + xy (x2 + y2)3/2 ∂ ∂y . (c) ˆ er = x √ x2+y2 ex + y √ x2+y2 ey, ˆ eϕ = − y √ x2+y2 ex + x √ x2+y2 ey, [ˆ er, ˆ eϕ]x = (ˆ er)x ∂(ˆ eϕ)x ∂x + (ˆ er)y ∂(ˆ eϕ)x ∂y − (ˆ eϕ)x ∂(ˆ er)x ∂x − (ˆ eϕ)y ∂(ˆ er)x ∂y = x x2 + y2 ∂ − y √ x2+y2 ∂x + y x2 + y2 ∂ − y √ x2+y2 ∂y + y x2 + y2 ∂ x √ x2+y2 ∂x − x x2 + y2 ∂ x √ x2+y2 ∂y = x2y (x2 + y2)2 − x2y (x2 + y2)2 + y3 (x2 + y2)2 + x2y (x2 + y2)2 = y x2 + y2 , [ˆ er, ˆ eϕ]y = (ˆ er)x ∂(ˆ eϕ)y ∂x + (ˆ er)y ∂(ˆ eϕ)y ∂y − (ˆ eϕ)x ∂(ˆ er)y ∂x − (ˆ eϕ)y ∂(ˆ er)y ∂y = x x2 + y2 ∂ x √ x2+y2 ∂x + y x2 + y2 ∂ x √ x2+y2 ∂y + y x2 + y2 ∂ y √ x2+y2 ∂x − x x2 + y2 ∂ y √ x2+y2 ∂y = xy2 (x2 + y2)2 − xy2 (x2 + y2)2 − xy2 (x2 + y2)2 − x3 (x2 + y2)2 = − x x2 + y2 , (c) ° [ˆ er, ˆ eϕ] = y x2 + y2 ˆ ex − x x2 + y2 ˆ ey . Îª er ≡ ∂ ∂r = x √ x2+y2 ex + y √ x2+y2 ey, eϕ ≡ ∂ ∂ϕ = −y ex + x ey, ° [er, eϕ]x = (er)x ∂(eϕ)x ∂x + (er)y ∂(eϕ)x ∂y − (eϕ)x ∂(er)x ∂x − (eϕ)y ∂(er)x ∂y = x x2 + y2 ∂(−y) ∂x + y x2 + y2 ∂(−y) ∂y + y ∂ x √ x2+y2 ∂x − x ∂ x √ x2+y2 ∂y = 0 − y (x2 + y2)1/2 + y3 (x2 + y2)3/2 + x2y (x2 + y2)3/2 = 0 , [er, eϕ]y = (er)x ∂(eϕ)y ∂x + (er)y ∂(eϕ)y ∂y − (eϕ)x ∂(er)y ∂x − (eϕ)y ∂(er)y ∂y

15 = x x2 + y2 ∂(x) ∂x + y
x2 + y2 ∂(x) ∂y + y ∂ y √ x2+y2 ∂x − x ∂ y √ x2+y2 ∂y = x (x2 + y2)1/2 + 0 − xy2 (x2 + y2)3/2 − x3 (x2 + y2)3/2 = 0 , Ó× £Î 2-2-7 ³ÏØ ] ˜11. Î {eµ } V ÕÑÚ¯ {eµ∗} °« ÑÚ¯ v ∈ V , ω ∈ V ∗, ÓÅ ω = ω(eµ)eµ∗ , v = eµ∗(v)eµ . Æ ¬ ω = ω(eµ)eµ∗ Ý ° V Õ ¦Ñ eν, ÇÕ ω, eµ∗ ∈ V ∗ « É« ÕÑ ¯ ¤Ý É ÔÒ Æ ω(eµ) ∈ RR Û³ = ω(eν) (2-3-3) = ων , ³ = ω(eµ)eµ∗(eν) (2-3-2) = ω(eµ)δµ ν = ω(eν) , Ø ω É ω(eµ)eµ∗ Ý Ò ( ¦Ñ eν) ¤ÔÒ ων ≡ ω(eν). ¬« ( ¦« ÕÑ )eν∗ Ý Ò v = eµ∗(v)eµ, Çeµ∗(v) ∈ RR ¯ v, eµ ∈ V É ÕÑ ¯ Û³ = eν∗(v) , ³ = eµ∗(v)eν∗(eµ) (2-3-2) = eµ∗(v)δν µ = eν∗(v) , Ø« Õ ¦Ñ Ý Ò v É eµ∗(v)eµ ÔÒ±¦¿ eν∗(v). ˜12. ÓÅ ω′ ν = ∂xµ ∂x′ν ωµ ( Ä 2-3-4). Æ » Ñ Õ¶ Ã (2-2-5) eµ = ∂x′ν ∂xµ e′ ν É e′ µ = ∂xν ∂x′µ eν. « ω Ý Ò ¿ ¯ÔÒ ω(eµ) = ∂x′ν ∂xµ ω(e′ ν ) É ω(e′ µ ) = ∂xν ∂x′µ ω(eν), É (2-3-3), Ø ωµ = ∂x′ν ∂xµ ω′ ν É ω′ µ = ∂xν ∂x′µ ων. ˜13. ÓÅ (2-3-5) Õ Ì v → v∗∗ ± Ì É ËÑ Õ Ñ¯Ø± ¦Ç¦Õ¦¦ËÑ Ì²ÒÉ Æ ÁÀ (2-3-5) v∗∗(ω) = ω(v) ÀË¯Î v = vµeµ, ω(v) = ω(vµeµ) = vµω(eµ) (2-3-3) = vµωµ , ° v, eµ ∈ V , ω ∈ V ∗, vµ, ωu ∈ RR. ¦ ¯ ω (2-3-4) = ωµeµ∗, v∗∗(ω) = v∗∗(ωµeµ∗) = ωµv∗∗(eµ∗) = ωµv∗∗µ ,

16 ° v∗∗(eµ∗) ≡ v∗∗µ ∈ RR, ω, eµ∗ ∈
V ∗, v∗∗ ∈ V ∗∗. ω Õ¯ Ø n ¿ ωµ Õ¯¸ ·× ωµv∗∗µ = v∗∗µωµ = vµωµ ÀË¯² v∗∗µ = vµ, ¿Ó ± Õ ¦ Í V ∗∗ = V . ° ¿± ¦Ç¦ÕËÑ ÌÁÀ ¦¦Õ¯ªÛ² ÒÉÕ ℄ ¦¦ÒÉÕËÑ Ì¯µ Å Ç¦± ˜14. Î C1 1 T É (C1 1 T)′ » (2, 1) T ¿ÑÚ {eµ } É {e′ µ } Õ ¾¯ÓÅ (C1 1 T)′ = C1 1 T. Æ (C1 1 T)′ ¡ = T(e′µ∗, •; e′ µ ) ¡Å2-3-2 = T((A−1)µ ρ eρ∗, •; Aν µeν) ÌÒÒ = Aν µ(A−1)µ ρ T(eρ∗, •; eν ) ¸ÁÏÂ = (AA−1)ν ρ T(eρ∗, •; eν ) = δν ρT(eρ∗, •; eν ) = T(eν∗, •; eν ) ¡ = C1 1 T . Ò C1 1 T ¿ (1, 0) ¯Ø C1 1 T ∈ V , ¶ ¯ ¶Õ ¾ ¡¯ÁÀÉ • Â Í Õ ¯× ¾ ÀËÆ *˜15. Î g V Õ§ ¯ÓÅ g : V → V ∗ ± Ì ( ©Ü 13 Õ ). Æ§ g (0, 2) g(; •, •), « v ∈ V ÕÝ Êg(; v, •) Ðg(; •, v), ¤ « ¯ Ý ° u ∈ V Ê g(; v, u) Ðg(; u, v), ° g(; v, •) É g(; •, v) ¤Ý° V ∗. ¬ g À ¦¿ ¶À¦ ¿« Õ Ì g : V → V ∗, ¼ ËÑ Ì ¦ ¯ ° g · Õ¯ ¡Õ Ì² ¦¦Õ £µ ¡¯«° ¦ Þ ω ∈ V ∗, Í ¦Õº Þ v, Ø× g(; v, •) = ω ( Ð g(; •, v) = ω). Õ ² g Õ · ÑÚ¯ ÁÀ g(; v, •) = ω É g(; v′, •) = ω, ¼ v = v′, » g ÕËÑÑ¯ Í¨ g(; v − v′, •) = 0, g · Ú »ËÑ ¯ ¿± ¦ Õ¦¦ Ì² ÒÉ¯ ¦¦ÒÉÕËÑ Ì Å ± Ì ˜16. ÓÅË ² ËÕ Æ Î Ë C(t) Õ Ë C′(t′), Ø C(t) = C′(t′), t′ = α(t) ( © §2.2.1 4). C É t1 Ò t2 ©ÕË l = t2 t1 dC(τ) dτ dτ = C(t2) − C(t1), ªÛ C′ ÉÕÍ § l′ = C′(t′ 2 ) − C′(t′ 1 ) = C(t2) − C(t1) = l. 17. Î {x, y} 2 Ñ ¦ÕØ Þ¸Ã¯ÓÅ (2-5-14) Õ {x′, y′} £ Ø Ã Æ ° ( ° µ ¶ Ã Ä 2-4-2) δ ∂ ∂x′α , ∂ ∂x′β = δ ∂xµ ∂x′α ∂ ∂xµ , ∂xν ∂x′β ∂ ∂xν = ∂xµ ∂x′α ∂xν ∂x′β δ ∂ ∂xµ , ∂ ∂xν ,

17 ÁÀ {xµ} Ø Ã¯ δ( ∂ ∂xµ , ∂
∂xν ) (2-5-12) = δµν, δ( ∂ ∂x′α , ∂ ∂x′β ) = ∂xµ ∂x′α ∂xµ ∂x′β . ( Ø ÃÜ « É ( ¾) ÉÅ¸Õ¢ Æ ) È ¯(2-5-14) Õ ¶ x = x′ cos α−y′ sin α, y = x′ sin α+y′ cos α, ∂x ∂x′ = cos α, ∂x ∂y′ = − sin α, ∂y ∂x′ = sin α, ∂y ∂y′ = cos α. δ ∂ ∂x′ , ∂ ∂x′ = ∂x ∂x′ ∂x ∂x′ + ∂y ∂x′ ∂y ∂x′ = (cos α)(cos α) + (sin α)(sin α) = 1 , δ ∂ ∂x′ , ∂ ∂y′ = ∂x ∂x′ ∂x ∂y′ + ∂y ∂x′ ∂y ∂y′ = (cos α)(− sin α) + (sin α)(cos α) = 0 , δ ∂ ∂y′ , ∂ ∂x′ = ∂x ∂y′ ∂x ∂x′ + ∂y ∂y′ ∂y ∂x′ = (− sin α)(cos α) + (cos α)(sin α) = 0 , δ ∂ ∂y′ , ∂ ∂y′ = ∂x ∂y′ ∂x ∂y′ + ∂y ∂y′ ∂y ∂y′ = (− sin α)(− sin α) + (cos α)(cos α) = 1 , Ø δ( ∂ ∂x′α , ∂ ∂x′β ) = δαβ . ° {x′, y′} £ Ø Ã ° ¦ Ñ µ Ò n Ø Ã Î {xµ} Ø Ã¯°ÁÃ Æ¶ ² ÃÍÎÃÕ ¦Þ¸Ã x′α = Aα µxµ £² Ø Ã Ç A n ÃÆ·À¯¼ Ñ A−1 = ˜ A, Ø (A−1)µ α = ˜ Aµ α = Aα µ. ¶ xµ = (A−1)µ α x′α = Aα µx′α, ∂xµ ∂x′α = (A−1)µ α = Aα µ, ° É Õ À δ ∂ ∂x′α , ∂ ∂x′β = ∂xµ ∂x′α ∂xµ ∂x′β = Aα µAβ µ = Aα µ(A−1)µ β = δα β = δαβ . ©§ δ « {x′µ} Ø× (2-5-12) ¯ £ Ø Ã 18. Î {t, x} 2 Ñ ¦ÕÕÐÑÞ¸Ã¯ÓÅ (2-5-20) Õ {t′, x′} £ ÕÐÑÃ Æ ° ( ° µ ¶ Ã Ä 2-4-2) η ∂ ∂x′α , ∂ ∂x′β = η ∂xµ ∂x′α ∂ ∂xµ , ∂xν ∂x′β ∂ ∂xν = ∂xµ ∂x′α ∂xν ∂x′β η ∂ ∂xµ , ∂ ∂xν , ÁÀ {xµ} ÕÐÑÃ¯ η( ∂ ∂xµ , ∂ ∂xν ) = ηµν, η( ∂ ∂x′α , ∂ ∂x′β ) = ηµν ∂xµ ∂x′α ∂xν ∂x′β . «° 2 Ñ ¦¯ ηxx = −ηtt = +1, ηtx = ηxt = 0. È ¯ (2-5-20) Õ ¶ t = t′ cosh λ − x′ sinh λ, x = −t′ sinh λ + x′ cosh λ, ∂t ∂t′ = cosh λ, ∂t ∂x′ = − sinh λ, ∂x ∂t′ = − sinh λ, ∂x ∂x′ = cosh λ. η ∂ ∂t′ , ∂ ∂t′ = ηtt ∂t ∂t′ ∂t ∂t′ + ηxx ∂x ∂t′ ∂x ∂t′ = (−1)(cosh λ)(cosh λ) + (+1)(− sinh λ)(− sinh λ) = −1 , η ∂ ∂t′ , ∂ ∂x′ = ηtt ∂t ∂t′ ∂t ∂x′ + ηxx ∂x ∂t′ ∂x ∂x′ = (−1)(cosh λ)(− sinh λ) + (+1)(− sinh λ)(cosh λ) = 0 , η ∂ ∂x′ , ∂ ∂t′ = ηtt ∂t ∂x′ ∂t ∂t′ + ηxx ∂x ∂x′ ∂x ∂t′ = (−1)(− sinh λ)(cosh λ) + (+1)(cosh λ)(− sinh λ) = 0 ,

18 η ∂ ∂x′ , ∂ ∂x′ = ηtt ∂t
∂x′ ∂t ∂x′ + ηxx ∂x ∂x′ ∂x ∂x′ = (−1)(− sinh λ)(− sinh λ) + (+1)(cosh λ)(cosh λ) = +1 , Ø ηx′x′ = −ηt′t′ = +1, ηt′x′ = ηx′t′ = 0. ° {t′, x′} £ 2 ÕÐÑÃ ° ¦ Ñ µ Ò n Ø Ã Î {xµ} ÕÐÑÃ¯°ÁÕ ÐÑ¶ ² ÃÍÎÃÕ ¦Þ¸Ã x′α = Λα µxµ £² ÕÐÑÃ Ç Λ n ÕÐÑ¶ ·À¯¼ Ñ ηαβ Λα µΛβ ν = ηµν, ηµνΛα µΛβ ν = ηαβ, × (Λ−1)µ α = Λα µ = ηαβ ηµνΛβ ν. ¶ xµ = (Λ−1)µ α x′α = ηαβ ηµνΛβ νx′α, ∂xµ ∂x′α = ηαβ ηµνΛβ ν, ° É Õ À η ∂ ∂x′α , ∂ ∂x′β = ηµν ∂xµ ∂x′α ∂xν ∂x′β = ηµν (ηαγηµρΛγ ρ) (ηβδ ηνσΛδ σ) = (ηµνηµρ) ηαγηβδ ηνσΛγ ρΛδ σ = δρ ν ηαγηβδ ηνσΛγ ρΛδ σ = ηαγηβδ (ηνσΛγ νΛδ σ) = ηαγ(ηβδ ηγδ) = ηαγδγ β = ηαβ . ©§ η « {x′µ} Ø× (2-5-18) ¯ £ ÕÐÑÃ ˜19. (a) ¶ Ê 3 Ñ§ Þ¸Ã Õ g′ µν . (b) ªÆ 4 Ñ§ g ÕÐÑÃ ÕË¹º ds2 = −dt2 + dx2 + dy2 + dz2, g ×° g−1 Þ¸Ã {t′, x′, y′, z′} Õ g′ µν × g′µν, · Þ¸ Ã ÁÅ t′ = t , z′ = z , x′ = (x2 + y2)1/2 cos(ϕ − ωt) , y′ = (x2 + y2)1/2 sin(ϕ − ωt) , ω = , ° ϕ Ø× cos ϕ = y(x2 + y2)−1/2, sin ϕ = x(x2 + y2)−1/2. Æ g′µν g′ µν . (a) Þ¸²È¯Þ¸Õ Ã x = r cos θ cos ϕ, y = r cos θ sin ϕ, z = r sin θ. ¶ ¯ Ä 2-4-2, g′ µν = ∂xα ∂x′µ ∂xβ ∂x′ν gαβ = ∂xα ∂x′µ ∂xα ∂x′ν , ° g′ rr = ∂x ∂r ∂x ∂r + ∂y ∂r ∂y ∂r + ∂z ∂r ∂z ∂r = (cos θ cos ϕ)2 + (cos θ sin ϕ)2 + (sin θ)2 = 1 , g′ θθ = ∂x ∂θ ∂x ∂θ + ∂y ∂θ ∂y ∂θ + ∂z ∂θ ∂z ∂θ = (−r sin θ cos ϕ)2 + (−r sin θ sin ϕ)2 + (r cos θ)2 = r2 , g′ ϕϕ = ∂x ∂ϕ ∂x ∂ϕ + ∂y ∂ϕ ∂y ∂ϕ + ∂z ∂ϕ ∂z ∂ϕ

19 = (−r cos θ sin ϕ)2 + (r cos
θ cos ϕ)2 + (0)2 = r2 sin2 θ , g′ rθ = ∂x ∂r ∂x ∂θ + ∂y ∂r ∂y ∂θ + ∂z ∂r ∂z ∂θ = (cos θ cos ϕ)(−r sin θ cos ϕ) + (cos θ sin ϕ)(−r sin θ sin ϕ) + (sin θ)(r cos θ) = 0 , g′ rϕ = ∂x ∂r ∂x ∂ϕ + ∂y ∂r ∂y ∂ϕ + ∂z ∂r ∂z ∂ϕ = (cos θ cos ϕ)(−r cos θ sin ϕ) + (cos θ sin ϕ)(r cos θ cos ϕ) + (sin θ)(0) = 0 , g′ θϕ = ∂x ∂θ ∂x ∂ϕ + ∂y ∂θ ∂y ∂ϕ + ∂z ∂θ ∂z ∂ϕ = (−r sin θ cos ϕ)(−r cos θ sin ϕ) + (−r sin θ sin ϕ)(r cos θ cos ϕ) + (r cos θ)(0) = 0 . Ô g′ rr = 1, g′ θθ = r2, g′ ϕϕ = r2 sin2 θ, «¯¹¤ (b) r ≡ (x2 + y2)1/2, Φ ≡ ϕ − ωt. cos ϕ = y r , we have − sin ϕ ∂ϕ ∂x = −xy r3 ⇒ x r ∂ϕ ∂x = xy r3 ⇒ ∂ϕ ∂x = y r2 . Á¦ Ô ∂ϕ ∂y = − x r2 . ° ∂ cos Φ ∂x = − sin Φ ∂ϕ ∂x = − y r2 sin Φ , ∂ cos Φ ∂y = − sin Φ ∂ϕ ∂y = x r2 sin Φ , ∂ sin Φ ∂x = cos Φ ∂ϕ ∂x = y r2 cos Φ , ∂ sin Φ ∂y = cos Φ ∂ϕ ∂y = − x r2 cos Φ . Å Ó¢ Ò ∂t′ ∂t = 1 , ∂t′ ∂x = ∂t′ ∂y = ∂t′ ∂z = 0 ; ∂x′ ∂t = (−r sin Φ)(−ω) = rω sin Φ , ∂x′ ∂x = x r cos Φ + r ∂ cos Φ ∂x = x r cos Φ + r − y r2 sin Φ = sin ϕ cos Φ − cos ϕ sin Φ = sin(ϕ − Φ) = sin ωt , ∂x′ ∂y = y r cos Φ + r ∂ cos Φ ∂y = y r cos Φ + r x r2 sin Φ = cos ϕ cos Φ + sin ϕ sin Φ = cos(ϕ − Φ) = cos ωt , ∂x′ ∂z = 0 ; ∂y′ ∂t = (r cos Φ)(−ω) = −rω cos Φ ,

20 ∂y′ ∂x = x r sin Φ + r
∂ sin Φ ∂x = x r sin Φ + r y r2 cos Φ = sin ϕ sin Φ + cos ϕ cos Φ = cos(ϕ − Φ) = cos ωt , ∂y′ ∂y = y r sin Φ + r ∂ sin Φ ∂y = y r sin Φ + r − x r2 cos Φ = cos ϕ sin Φ − sin ϕ cos Φ = − sin(ϕ − Φ) = − sin ωt , ∂y′ ∂z = 0 ; ∂z′ ∂t = ∂z′ ∂x = ∂z′ ∂y = 0 , ∂z′ ∂z = 1 . ¶ ¯ Ä 2-4-2, g′µν = ∂x′µ ∂xα ∂x′ν ∂xβ gαβ Ô g′tt = − ∂t′ ∂t ∂t′ ∂t + ∂t′ ∂x ∂t′ ∂x + ∂t′ ∂y ∂t′ ∂y + ∂t′ ∂z ∂t′ ∂z = −(1)2 + (0)2 + (0)2 + (0)2 = −1 , g′xx = − ∂x′ ∂t ∂x′ ∂t + ∂x′ ∂x ∂x′ ∂x + ∂x′ ∂y ∂x′ ∂y + ∂x′ ∂z ∂x′ ∂z = −(rω sin Φ)2 + (sin ωt)2 + (cos ωt)2 + (0)2 = 1 − r2ω2 sin2 Φ = 1 − (x2 + y2)ω2 sin2(ϕ − ωt) , g′yy = − ∂y′ ∂t ∂y′ ∂t + ∂y′ ∂x ∂y′ ∂x + ∂y′ ∂y ∂y′ ∂y + ∂y′ ∂z ∂y′ ∂z = −(−rω cos Φ)2 + (cos ωt)2 + (− sin ωt)2 + (0)2 = 1 − r2ω2 cos2 Φ = 1 − (x2 + y2)ω2 cos2(ϕ − ωt) , g′zz = − ∂z′ ∂t ∂z′ ∂t + ∂z′ ∂x ∂z′ ∂x + ∂z′ ∂y ∂z′ ∂y + ∂z′ ∂z ∂z′ ∂z = −(0)2 + (0)2 + (0)2 + (1)2 = 1 ; g′tx = − ∂t′ ∂t ∂x′ ∂t + ∂t′ ∂x ∂x′ ∂x + ∂t′ ∂y ∂x′ ∂y + ∂t′ ∂z ∂x′ ∂z = −(1)(rω sin Φ) + (0)(sin ωt) + (0)(cos ωt) + (0)(0) = −rω sin Φ = −(x2 + y2)1/2ω sin(ϕ − ωt) , g′ty = − ∂t′ ∂t ∂y′ ∂t + ∂t′ ∂x ∂y′ ∂x + ∂t′ ∂y ∂y′ ∂y + ∂t′ ∂z ∂y′ ∂z = −(1)(−rω cos Φ) + (0)(cos ωt) + (0)(− sin ωt) + (0)(0) = rω cos Φ = (x2 + y2)1/2ω cos(ϕ − ωt) , g′tz = − ∂t′ ∂t ∂z′ ∂t + ∂t′ ∂x ∂z′ ∂x + ∂t′ ∂y ∂z′ ∂y + ∂t′ ∂z ∂z′ ∂z = −(1)(0) + (0)(0) + (0)(0) + (0)(1) = 0 ; g′xy = − ∂x′ ∂t ∂y′ ∂t + ∂x′ ∂x ∂y′ ∂x + ∂x′ ∂y ∂y′ ∂y + ∂x′ ∂z ∂y′ ∂z = −(rω sin Φ)(−rω cos Φ) + (sin ωt)(cos ωt) + (cos ωt)(− sin ωt) + (0)(0)

21 = r2ω2 sin Φ cos Φ = (x2 +
y2)ω2 sin(ϕ − ωt) cos(ϕ − ωt) , g′xz = − ∂x′ ∂t ∂z′ ∂t + ∂x′ ∂x ∂z′ ∂x + ∂x′ ∂y ∂z′ ∂y + ∂x′ ∂z ∂z′ ∂z = −(rω sin Φ)(0) + (sin ωt)(0) + (cos ωt)(0) + (0)(1) = 0 ; g′yz = − ∂y′ ∂t ∂z′ ∂t + ∂y′ ∂x ∂z′ ∂x + ∂y′ ∂y ∂z′ ∂y + ∂y′ ∂z ∂z′ ∂z = −(−rω cos Φ)(0) + (cos ωt)(0) + (− sin ωt)(0) + (0)(1) = 0 . Ô ·À g′µν =        −1 −rω sin Φ rω cos Φ 0 −rω sin Φ 1 − r2ω2 sin2 Φ r2ω2 sin Φ cos Φ 0 rω cos Φ r2ω2 sin Φ cos Φ 1 − r2ω2 cos2 Φ 0 0 0 0 1        . g′ µν g′νρ = δρ µ , Æ g′ µν Õ ·À É·ÀÕ ·À g′ µν =        −1 + r2ω2 −rω sin Φ rω cos Φ 0 −rω sin Φ 1 0 0 rω cos Φ 0 1 0 0 0 0 1        , ° r = (x2 + y2)1/2, Φ = ϕ − ωt. ˜20. ÓÅ 3 Ñ ¦ Þ¸Ñ ∂/∂r, ∂/∂θ, ∂/∂ϕ Õ §§ 1, r, r sin θ. Æ É (a) ª° Ô Þ¸Õ grr = 1, gθθ = r2, gϕϕ = r2 sin2 θ, Þ¸Ñ Õ §§ 1, r, r sin θ. ˜21. È Ë¸ÞÇÅ¢ T′µ ν = ∂x′µ ∂xρ ∂xσ ∂x′ν Tρ σ. Æ É È Ë¸ÞÇ¯ Ta b = Tµ ν(eµ)a(eν)b [ © (2-6-1) ], Tµ ν = Ta b (eµ)a(eν)b [ © (2-6-2) ], ° T′µ ν = Ta b (e′µ)a(e′ ν )b = Tρ σ(eρ)a(eσ)b (e′µ)a(e′ ν )b = Tρ σ(eρ)a(e′µ)a(eσ)b (e′ ν )b , ° (eρ)a(e′µ)a = ∂ ∂xρ a (dx′µ)a = ∂x′µ ∂xρ , (eσ)b (e′ ν )b = (dxσ)b ∂ ∂x′ν b = ∂xσ ∂x′ν . Ô T′µ ν = ∂x′µ ∂xρ ∂xσ ∂x′ν Tρ σ .

22 22. g É g′ » º§ gab Þ¸Ã
{xµ} É {x′µ} Õ gµν É g′ µν Ù ÀÕ ¿ n × n ·ÀÐ℄ ¯ÓÅ g′ = |∂xρ/∂x′σ|2g, ° |∂xρ/∂x′σ| Þ ¸¶ {xµ} → {x′µ} ÕÝ Ð℄ ¯Ø ∂xρ/∂x′σ ÙÀÕ n × n Ð℄ º¢§ ÕÐ℄ Þ¸¶ Å ¶ × g′ ρσ = (∂xµ/∂x′ρ)(∂xν/∂x′σ)gµν ÕÐ℄ Æ ¬ (0, 2) Õ gµν É g′ µν À·À¹¯° µ É ν » ÐÉ℄ Ë¸ ±¡¯¬ (1, 1) Õ Aρσ ≡ ∂xρ/∂x′σ £ À·À¹¯° ρ É σ » ÐÉ℄Ë¸ ° ¯¶ Ã g′ ρσ = (∂xµ/∂x′ρ)(∂xν/∂x′σ)gµν À g′ ρσ = AµρAνσgµν = ˜ AρµgµνAνσ = ( ˜ AgA)ρσ, ° ˜ A A Õ ·À¯ Í Õ·À× g′ = ˜ AgA. ³ Ð℄ det g′ = det ˜ A det g det A = (det A)2 det g = |∂xρ/∂x′σ|2 det g, Ã ¢ÅÕ Ã È¯Þ¸Ò Þ¸ Ê¯ · ª°ÆÓÐ℄ |∂xρ/∂x′σ| = r2 sin θ, det g ß¹ = r4 sin2 θ det g É ß¹ = r4 sin2 θ, det g É ß¹ = 1. ˜23. Î {xµ} ÉÕ ¦¶¶Þ¸Ã¯Ó ªÅ℄× Õ (1) (∂/∂xµ)a(∂/∂xν )a = gµν, ° (∂/∂xν)a ≡ gab (∂/∂xν)b; (2) (dxµ)a(dxν)a = gµν, ° (dxµ)a ≡ gab(dxµ)b ; (3) (∂/∂xµ)a = (dxµ)a; (4) (dxµ)a = (∂/∂xµ)a; (5) vµωµ = vµωµ; (6) gµνTνρSρ σ = TµρSρσ; (7) vaub = vbua; (8) vaub = ubva. § (1) (2) (3) (4) (5) (6) (7) (8) Á° (6) gµνTνρSρ σ = Tµ ρSρ σ = Tµτ gρτ Sρ σ = Tµτ Sτσ = TµρSρσ. 24. Î Tab ¦ V ÉÕ (0, 2) ¯ÓÅ Tab vavb = 0, ∀ va ∈ V ⇒ Tab = T[ab] . va º ¿ ua É wa ÇÉ © Ò ª. Æ Å²°×£Õ Õ£ ÁÀ Tµνvµvν = 0 ∀ vµ, Tµν = T[µν] , Ç µ, ν = 1, · · · , n. ÖÆ¯ vµ = (v, 0, · · · , 0), Ø v1 = v ∈ RR, °¯ µ = 1 Õ ¤ × ¶ T11v2 = 0 ⇒ T11 = 0. ±¡ ÆÓÆ «¯¹ § Tµµ = 0. Å vµ = (v, v, 0, · · · , 0), Ø v1 = v2 = v, °¯ µ = 1 É 2 Õ ¤ × ¶ (T11 + T22 + T12 + T21)v2 = 0. ªÆ T11 = T22 = 0, Æ² T12 + T21 = 0. Á¦ Å µ = ν ¯ Tµν + Tνµ = 0, Tµν = −Tνµ = T[µν] , £ ÔÅ

23 25. ÓÅ Tabcd = Ta[bc]d = Tab[cd] ⇒ Tabcd
= Ta[bcd] . É (1) µ Î¦ Õ Ñ T···a···b···c··· = T···[a···b]···c··· = T···a···[b···c]··· ⇒ T···a···b···c··· = T···[a···b···c]··· . É Õ· Í ¿×Ç¯ ª T···[a···b]···c··· É T···a···[b···c]··· ÕË¸ b ¤ ÇÆ (2) · É Ñ Õ Ç¸ » Ç¯ µ · Õ£ ÀË Æ ÁÀ Tabcd = Ta[bc]d = 1 2 (Tabcd − Tacbd ) É Tabcd = Tab[cd] = 1 2 (Tabcd − Tabdc ), Tacbd = −Tabcd É Tabdc = −Tabcd , ØÆ ¦ ¿Ë¸ÉÆ Ú ¿Ë ¸¤ ¶ ¦´Ç ° Ta[bcd] = 1 6 (Tabcd − Tabdc + Tacdb − Tacbd + Tadbc − Tadcb ) = 1 6 Tabcd [1 − (−1) + (−1)2 − (−1) + (−1)2 − (−1)3] = Tabcd . ¦ ÑÏÀÆµ ¯ [a · · · b] É [b · · · c] ÆÕ ÑÏ [a · · · b · · · c] ÆÕ « 3 Ã¤ ´ (·Ë) ¾²Å ¦ ˜1. ´ ∇a Õ Ñ « (e), (1) ÓÅ℄ Tc ab ( ¶±Ä ) ∇a ∇b f − ∇b ∇af = −Tc ab ∇cf , ∀ f ∈ F . ˜ ∇a ª±¯¤ Ä 3-1-4 Å¢ ÕµÑ (2) ÓÅ Tc ab uavb = ua∇avc − va∇auc − [u, v]c ∀ ua, va ∈ F(1, 0). Æ (1) (3-1-2), ωb = ∇b f = ˜ ∇b f, ° ˜ ∇b Ñ ª± » Ä 3-1-3 (3-1-6): ∇aωb = ˜ ∇aωb − Cc ab ωc, ∇a ∇b f = ˜ ∇a ˜ ∇b f − Cc ab ∇cf. Æ Ë¸Ô ∇b ∇af = ˜ ∇b ˜ ∇af − Cc ba ∇cf. Í¨¾É ˜ ∇a Õ Ñ¯ ∇a ∇b f−∇b ∇af = −(Cc ab −Cc ba )∇cf = −Tc ab ∇cf, ° ª Cc ab −Cc ba ≡ Tc ab (2) ¬«ÆÒÝ °¸ f ∈ FM (0, 0) [u, v](f) (2-2-9) = u(v(f)) − v(u(f)) ¡ 1(d) = ua∇a(vb∇b f) − va∇a(ub∇b f)

24 = ua(∇avb)(∇b f) + uavb(∇a ∇b f) − va(∇aub)(∇b
f) − vaub(∇a ∇b f) = [ua(∇avb) − va(∇aub)](∇b f) + uavb[∇a ∇b f − ∇b ∇af] = [ua∇avc − va∇auc](∇cf) + uavb[−Tc ab ∇cf] = ua∇avc − va∇auc − Tc ab uavb ∇cf . ¦ ¯ [u, v] ∈ FM (1, 0) Ï ¯Ý ° f » 1(d) [u, v](f) = [u, v]c∇cf, Ô ua∇avc − va∇auc − Tc ab uavb = [u, v]c ˜2. Î va ¯ vν É v′ν va Þ¸Ã {xν} É {x′ν} Õ ¯ Aν µ ≡ ∂vν/∂xµ, A′ν µ ≡ ∂v′ν/∂x′µ, ÓÅ Aν µ É A′ν µ Õ Ã¦ Ø× ¶ É vν ² v′ν Ç¦Õ¶ Æ va É (1, 1) Ta b Þ¸Ã¶ ÅØ×Õ¶ Ã » v′µ = va(e′µ)a = vρ(eρ)a(e′µ)a = vρ(∂/∂xρ)a(dx′µ)a = vρ ∂x′µ ∂xρ , T′µ ν = Ta b (e′µ)a(e′ ν )b = Tρ σ(eρ)a(eσ)b (e′µ)a(e′ ν )b = Tρ σ(∂/∂xρ)a(dx′µ)a(dxσ)b (∂/∂x′ν )b = Tρ σ ∂x′µ ∂xρ ∂xσ ∂x′ν , Ø Ä 2-4-2 Õ¶ È » A′µ ν = ∂ ∂x′ν v′µ = ∂xσ ∂x′ν ∂ ∂xσ vρ ∂x′µ ∂xρ = ∂xσ ∂x′ν ∂vρ ∂xσ ∂x′µ ∂xρ + vρ ∂2x′µ ∂xσ∂xρ = Aρ σ ∂xσ ∂x′ν ∂x′µ ∂xρ + vρ ∂xσ ∂x′ν ∂2x′µ ∂xσ∂xρ , ² Õ¶ ¯Ç ³Ü ¶ [ ÁÀ¶ ËÑÕ ( ÁÕÐÑ¶ ), ªÛÜ Æ ] ˜3. ÓÅ Ä 3-1-7. Æ Ä 3-1-5 (3-1-7) É 2: ∇avb = ∂avb + Γb acvc, vν ;µ = ∇avb(eν)b (eµ)a = (∂avb + Γb acvc)(eν)b (eµ)a = ∂avb(eν)b (eµ)a + Γb acvc(eν)b (eµ)a = ∂vν ∂xµ + Γν µσvσ = vν ,µ + Γν µσvσ .

25 Ä 3-1-3 (3-1-6) É 2: ∇aωb = ∂aωb −
Γc ab ωc, ων;µ = ∇aωb (eν)b(eµ)a = (∂aωb − Γc ab ωc)(eν )b(eµ)a = ∂aωb (eν)b(eµ)a − Γc ab ωc(eν)b(eµ)a = ∂ων ∂xµ − Γσ µνωσ = ων,µ − Γσ µνωσ . Ø Ä 3-1-7 Õ (3-1-11) Õ 4. Å Γσ µν: ( ∂ ∂xν )b∇b ( ∂ ∂xµ )a = Γσ µν( ∂ ∂xσ )a, ÓÅ (a) Γσ µν = Γσ νµ ( É ∇a Õ ÑÉÞ¸Ñ ¦Õ«ÆÑ ); (b) vν ;µ = vν ,µ + Γν µβ vβ ( ° Ñ±Õ×£ ). Æ § ³©¯Å¤ Ñ ±Çº Þ¸ÑÚÕÑ ¯Ø ÕÑ ( ∂ ∂xµ )a ≡ (eµ)a, « ÕÑ (dxµ)a ≡ (eµ)a. ° (eν)b∇b (eµ)a = Γσ µν(eσ)a . (a) » Ä 3-1-9 (3-1-13), «° ¾ ª± ∇a ÀË [u, v]a = ub∇b va − vb∇b ua . «°Þ¸ÑÚÕÑ {(eµ)a}, Í«Æ¯ 0 = [eµ, eν]a = (eµ)b∇b (eν)a − (eν)b∇b (eµ)a , Ø (eµ)b∇b (eν)a = (eν)b∇b (eµ)a. « Ñ (eρ)a Ý (eρ)a(eν)b∇b (eµ)a = Γσ µν(eρ)a(eσ)a = Γσ µνδρ σ = Γρ µν . É «Æ Ã¯° Γρ µν = (eρ)a(eν)b∇b (eµ)a = (eρ)a(eµ)b∇b (eν)a = Γρ νµ (b) ³Ý (eν)c ° (eν)c(eν)b[∇b (eµ)a] = δb c[∇b (eµ)a] = ∇c(eµ)a = (eν)cΓσ µν(eσ)a, Ø ∇a(eµ)b = Γσ µν(eν)a(eσ)b vν ;µ = (eµ)a(eν)b ∇avb = (eµ)a(eν)b ∇a[vρ(eρ)b] = (eµ)a(eν)b (∇avρ)(eρ)b + vρ[∇a(eρ)b] (3-1-1) = (eµ)a(eν)b (dvρ)a(eρ)b + vρ[Γσ ρλ (eλ)a(eσ)b] = (eµ)a(dvρ)a(eν)b (eρ)b + vρΓσ ρλ (eµ)a(eλ)a(eν)b (eσ)b = (eµ)a(dvρ)aδν ρ + vρΓσ ρλ δλ µδν σ = (eµ)a(dvν)a + vρΓν ρµ

26 ° (eµ)a(dvν)a = dvν(eµ) = dvν( ∂ ∂xµ )
= ∂vν ∂xµ = vν ,µ. É (a) Õ ÀÔ vν ;µ = vν ,µ + Γν µρvρ . ˜5. ª (1) ∇a(dxµ)b = 0 ; (2) vν ;µ = (∇avb)(∂/∂xµ)a(dxν)b ; (3) vν ,µ = (∂avb)(∂/∂xµ)a(dxν)b ; (4) vν ;µ = (∂/∂xµ)a∇avν ; (5) vν ,µ = (∂/∂xµ)a∇avν . § (1) ©É (a) Õ À¯« Ñ ª± ∇a(dxµ)b = ∇b (dxµ)a. (2) « (3) « £ (4) ∇avν = ∇a[vb(dxν)b ] = (∇avb)(dxν)b +vb[∇a(dxν)b ], Æ ³ (∂/∂xµ)a(∇avb)(dxν)b +(∂/∂xµ)avb[∇a(dxν)b ] = vν ;µ+(∂/∂xµ)avb[∇a(dxν)b ] = vν ;µ. º (5) Õ ÀÆ ³° vν ,µ, ¦ ×° vν ;µ. (5) « ÁÀ vµ À¸ Ä ¯ (3-1-2) Æ ∇avν = ∂avν = (dvν)a. ° (∂/∂xµ)a∇avν = (∂/∂xµ)a∂avν = (∂/∂xµ)a(dvν)a = (dvν )(∂/∂xµ) (2-3-7) = ∂vν/∂xµ = vν ,µ. £ ¡ ∇a vν = ∂avν = ∂a[vb(dxν)b ] = (∂avb)(dxν)b + vb[∂a(dxν)b ] (3-1-10) = (∂avb)(dxν)b , ° ³(∂/∂xµ)a∇avν = (∂/∂xµ)a(∂avb)(dxν)b , » µ vν ,µ [ © (3)]. ˜6. Î C(t) {xµ} ÕÞ¸¶ÆÕ Ë¯ xµ(t) C(t) ·ÃÕ º ¯ va C(t) ÉÕ ¯ Dvµ/dt ≡ (dxµ)a(∂/∂t)b∇b va, ÓÅ Dvµ/dt ≡ dvµ/dt + Γµ νσvσdxν(t)/dt . Æ Dvµ dt = (dxµ)a Dva dt (3-2-13) = (dxµ)aTb∇b va = (dxµ)a ∂ ∂t b ∇b va (3-1-7) = (dxµ)a ∂ ∂t b (∂b va + Γa bc vc) (3-1-10) = ∂ ∂t b ∂b [(dxµ)ava] + (dxµ)aΓa bc vc = ∂ ∂t b ∂b [vµ] + Γµ bσ vσ = ∂ ∂t ν ∂νvµ + Γµ νσvσ ,

27 ° ( ∂ ∂t )ν ËÕ» ( ∂ ∂t
)b ÕÞ¸ ( ∂ ∂t )ν (2-2-7) = dxν (t) dt , Ð ( ∂ ∂t )ν = ( ∂ ∂t )a(dxν)a = dxν( ∂ ∂t ) (2-3-7) = ∂xν ∂t . ° Dvµ dt = dxν(t) dt ∂vµ ∂xν + Γµ νσvσ = dvµ dt + Γµ νσvσ dxν(t) dt . ˜7. Ê 3 Ñ ¦ Þ¸ÃÕ Γσ µν. Æ » (3-2-10’) Γσ µν = 1 2 gσρ(gρµ,ν + gρν,µ − gµν,ρ). «° Þ¸¯Í [ © ·¦ À 19(a)] grr = 1 , gθθ = r2 , gϕϕ = r2 sin2 θ . Æ grr = 1 , gθθ = r−2 , gϕϕ = r−2 sin−2 θ . Ñ Ô grr,r = grr,θ = grr,ϕ = 0 ; gθθ,r = 2r , gθθ,θ = gθθ,ϕ = 0 ; gϕϕ,r = 2r sin2 θ , gϕϕ,θ = r2 sin 2θ , gϕϕ,ϕ = 0 . Â ÔÒ Γσ rr = 1 2 gσρ(gρr,r + gρr,r − grr,ρ) = 1 2 gσr(grr,r + grr,r − grr,r) = 0 , Γσ rθ = 1 2 gσρ(gρr,θ + gρθ,r − grθ,ρ ) = 1 2 gσθgθθ,r = 1 2 (δσθr−2)(2r) = δσθr−1 , Γσ rϕ = 1 2 gσρ(gρr,ϕ + gρϕ,r − grϕ,ρ) = 1 2 gσϕgϕϕ,r = 1 2 (δσϕr−2 sin−2 θ)(2r sin2 θ) = δσϕr−1 ; Γσ θθ = 1 2 gσρ(gρθ,θ + gρθ,θ − gθθ,ρ ) = − 1 2 gσrgθθ,r = − 1 2 (δσr)(2r) = −δσrr , Γσ θϕ = 1 2 gσρ(gρθ,ϕ + gρϕ,θ − gθϕ,ρ ) = 1 2 gσϕgϕϕ,θ = 1 2 (δσϕr−2 sin−2 θ)(r2 sin 2θ) = δσϕ sin−1 θ cos θ = δσϕ cot θ ; Γσ ϕϕ = 1 2 gσρ(gρϕ,ϕ + gρϕ,ϕ − gϕϕ,ρ) = − 1 2 gσrgϕϕ,r − 1 2 gσθgϕϕ,θ = − 1 2 (δσr)(2r sin2 θ) − 1 2 (δσθr−2)(r2 sin 2θ) = −δσrr sin2 θ − δσθ sin θ cos θ .

28 Ô 3 Ñ ¦ Þ¸ÃÕ Ñ±ÁÅ Γθ rθ =
Γθ θr = r−1 , Γϕ rϕ = Γϕ ϕr = r−1 , Γr θθ = −r , Γϕ θϕ = Γϕ ϕθ = cot θ = cos θ sin θ , Γr ϕϕ = −r sin2 θ , Γθ ϕϕ = − sin θ cos θ . 8. Î I RR Õ¦¿ ¦¯ C : I → M (M, ∇a) Õ Ë¯ÓÅ ∀ s, t ∈ I, ¨ Ì ψ : VC(s) → VC(t) ( ©´ 3-2) ± Ì Æ C(s) → C(t) ¦¦ÒÉÕËÑ Ì¯Æ va(s) ¨Ò ˜ va(t) £ ¦¦ÒÉÕËÑ Ì¯ ψ : VC(s) → VC(t) ± Ì ˜9. ÓÅ Ä 3-3-2 ¡ 3-3-3 É 3-3-5. Æ (1) Ä 3-3-2 ÕÅ¢ Î T′a Ë γ′(t′) [= γ(t)] Õ» ¯ Ã T′a = ∂ ∂t′ a = dt dt′ ∂ ∂t a = dt dt′ Ta . ¢ γ′(t′) ÛË¯ T′a ²ÔØ× 0 = T′b∇b T′a = dt dt′ Tb∇b dt dt′ Ta = dt dt′ TaTb∇b dt dt′ + dt dt′ 2 Tb∇b Ta = dt dt′ Ta ∂ ∂t b d dt dt′ b + dt dt′ 2 αTa = dt dt′ d dt dt dt′ + α dt dt′ 2 Ta = d2t dt′2 + α dt dt′ 2 Ta , ° ¢ d2t dt′2 + α dt dt′ 2 = 0 . « t Õ¾ Ã¯ « t′ Õ¾ Ã¯ d2t dt′2 = d dt′ dt dt′ = dt dt′ d dt dt′ dt −1 = − dt′ dt −1 dt′ dt −2 d2t′ dt2 = − dt′ dt −3 d2t′ dt2 , − dt′ dt −3 d2t′ dt2 + α dt′ dt −2 = 0 ,

29 Ø d2t′ dt2 = α(t) dt′ dt . µ
t′ = t′(t) Ø×Õ¾ Ã¯ Ê t′, ªÛµ Ò ÛË γ′(t′) [= γ(t)]. (2) Ä 3-3-3 ÕÅ¢ «²¢Ñ Ä t ÛË γ(t) Õ Ì ¯ Ä 3-3-2 Õ α = 0, t′ Ø×Õ Ã¶ d2t′ dt2 = 0, °° ² t′ = at + b. t′ ±¦ ÛË γ′(t′) Õ Ì ¬Ç Ñ Ä t′ = at + b ÛË γ′(t′) Õ Ì ¯ªÛ Ä 3-3-2 Õ α(t) = (dt′ dt )−1 d2t′ dt2 = 0, ° Tb∇b Ta = 0, Ø t ÛË γ(t) [= γ′(t′)] Õ Ì (3) Ä 3-3-5 ÕÅ¢ Î γ(t) Ì t Õ ÛË¯ γ(t) Õ » Ta(t) ≡ Ta(γ(t)), ° § ( Õ ) T2 = g(T, T) = TaTbgab . Ta ÛËÕ» ¯ÆØ× Tc∇cTa = 0, ¦ § gab ²Ñ ª± ∇a ÍÐ ¯ ∇cgab = 0. ° Tc∇cT2 = Tc∇c(TaTbgab ) = gab TbTc∇cTa + gab TaTc∇cTb + TaTbTc∇cgab = 0, ÛË» Õ § ÛË |T| = C. ÛËÕË (2-5-3) Ê l = t t0 |T(t′)| dt′ = C(t − t0). ªÛ ± ¦ ÛË£ Õ γ′(l) Þ¯ l Ë Ú » Ä 3-3-3 Õ À¯ÁÀ t γ(t) Õ Ì ¯ªÛ l ² γ′(l) [= γ(t)] Õ Ì ˜10. (a) Ê § ds2 = R2(dθ2 +sin2 θdϕ2) (R ) Õ ÛË Ã (b) Å ¦ » ( Ð ) Ø× ÛË Ã Ù Þ¸Ã {θ, ϕ} Æ » ÆÓÕ¦ ¯¾ ϕ Ì Æ(a) Õ§ gθθ = R2, gϕϕ = R2 sin2 θ. É À 7 ÔÕ Þ ¸ÃÕ Ñ±¯ªÛÈ Í Γθ ϕϕ = − sin θ cos θ ÉΓϕ ϕθ = Γϕ θϕ = cos θ sin θ . ° ÛËÕ Ã (3-3-1) 0 = d2θ dt2 + Γθ ϕϕ dϕ dt dϕ dt = d2θ dt2 − sin θ cos θ dϕ dt 2 , 0 = d2ϕ dt2 + Γϕ ϕθ dϕ dt dθ dt + Γϕ θϕ dθ dt dϕ dt = d2ϕ dt2 + 2 cos θ sin θ dθ dt dϕ dt , Ø ÛË Ã θ,tt − sin θ cos θ ϕ2 ,t = 0 , ϕ,tt + 2 cot θ ϕ,t θ,t = 0 . (b) ÆÜ Þ¸ÃÕØ ¶ Ü¦ ¯ O ÃÕx Ø α ¯§ÔO′ Ã¯ ÃÇ¦ÕÞ¸ Ã        x′ = x , y′ = y cos α + z sin α , z′ = −y sin α + z cos α .        x = x′ , y = y′ cos α − z′ sin α , z = y′ sin α + z′ cos α .

30 O′ ÃÕ z′ Ø β ¯§Ô O′′ Ã¯ ÃÇ¦ÕÞ¸
Ã        x′′ = x′ cos β + y′ sin β , y′′ = −x′ sin β + y′ cos β , z′′ = z′ .        x′ = x′′ cos β − y′′ sin β , y′ = x′′ sin β + y′′ cos β , z′ = z′′ . Ô O Ã² O′′ ÃÕÞ¸ Ã x′′ =     x′′ y′′ z′′     =     cos β cos α sin β sin α sin β − sin β cos α cos β sin α cos β 0 − sin α cos α         x y z     = Rx . x = R−1x′′ = RT x′′ =     x y z     =     cos β − sin β 0 cos α sin β cos α cos β − sin α sin α sin β sin α cos β cos α         x′′ y′′ z′′     =     cos β − sin β 0 cos α sin β cos α cos β − sin α sin α sin β sin α cos β cos α         R sin θ′′ cos ϕ′′ R sin θ′′ sin ϕ′′ R cos θ′′     = R     sin θ cos ϕ sin θ sin ϕ cos θ     = R     sin θ′′ cos(ϕ′′ + β) sin θ′′ sin(ϕ′′ + β) cos α − cos θ′′ sin α sin θ′′ sin(ϕ′′ + β) sin α + cos θ′′ cos α     , cos θ = sin θ′′ sin(ϕ′′ + β) sin α + cos θ′′ cos α , tan ϕ = tan(ϕ′′ + β) cos α − cot θ′′ sin α cos(ϕ′′ + β) . O ÃÕ Ë » ( ©) ¤ O′′ ÃÕ (i) ÆÓË ( ©) Ð (ii) °Ë ( ©) Þ ÆÓË ( ©) θ′′ = π 2 , φ′′ = at + b; °Ë ( ©) θ′′ = at + b, φ′′ = c. (i) ÁÀ O′′ ÕÆÓË ( ©), θ′′ = π 2 , φ′′ = at + b: cos θ = sin(at + b + β) sin α = sin α sin φ(t) , tan ϕ = tan(at + b + β) cos α = cos α tan φ(t) , ° φ(t) = at + b + β, Ø θ(t) = arccos[sin α sin φ(t)] , ϕ(t) = arctan[cos α tan φ(t)] .

31 « t Ñ Ô θ,t = −a sin α
cos φ (1 − sin2 α sin2 φ)−1/2 = − a sin α cos φ (1 − sin2 α sin2 φ)1/2 , θ,tt = a2 sin α sin φ (1 − sin2 α sin2 φ)−1/2 −a sin α cos φ − 1 2 (1 − sin2 α sin2 φ)−3/2[−2a sin2 α sin φ cos φ] = a2 sin α sin φ (1 − sin2 α sin2 φ)−1/2 −a2 sin3 α cos2 φ sin φ (1 − sin2 α sin2 φ)−3/2 = a2 sin α sin φ (1 − sin2 α sin2 φ)−3/2 (1 − sin2 α sin2 φ) − sin2 α cos2 φ = a2 sin α sin φ (1 − sin2 α sin2 φ)−3/2 1 − sin2 α = a2 cos2 α sin α sin φ (1 − sin2 α sin2 φ)−3/2 = a2 cos2 α sin α sin φ (1 − sin2 α sin2 φ)3/2 ; ϕ,t = a cos α sec2 φ (1 + cos2 α tan2 φ)−1 = a cos α sec2 φ 1 + cos2 α tan2 φ , ϕ,tt = a cos α 2a sec2 φ tan φ (1 + cos2 α tan2 φ)−1 −a cos α sec2 φ (1 + cos2 α tan2 φ)−2 cos2 α 2a tan φ sec2 φ = 2a2 cos α sec2 φ tan φ (1 + cos2 α tan2 φ)−1 −2a2 cos3 α sec4 φ tan φ (1 + cos2 α tan2 φ)−2 = 2a2 cos α sec2 φ tan φ (1 + cos2 α tan2 φ)−2 × (1 + cos2 α tan2 φ) − cos2 α sec2 φ = 2a2 cos α sec2 φ tan φ (1 + cos2 α tan2 φ)−2 1 − cos2 α = 2a2 cos α sin2 α sec2 φ tan φ (1 + cos2 α tan2 φ)2 . ÛË Ã θ,tt − sin θ cos θ ϕ2 ,t = 0 , ϕ,tt + 2 cot θ ϕ,t θ,t = 0 . Â Å¯Ü¦¿ Ã θ,tt − sin θ cos θ ϕ2 ,t = a2 cos2 α sin α sin φ (1 − sin2 α sin2 φ)3/2 − sin θ cos θ a cos α sec2 φ 1 + cos2 α tan2 φ 2 = a2 cos2 α sin α sin φ (1 − sin2 α sin2 φ)3/2 − (1 − sin2 α sin2 φ)1/2 sin α sin φ a2 cos2 α sec4 φ (1 + cos2 α tan2 φ)2

32 = a2 cos2 α sin α sin φ (1
− sin2 α sin2 φ)3/2 − (1 − sin2 α sin2 φ)1/2 a2 cos2 α sin α sin φ sec4 φ (1 + cos2 α tan2 φ)2 = a2 cos2 α sin α sin φ (1 − sin2 α sin2 φ)3/2 1 − (1 − sin2 α sin2 φ)2 sec4 φ (1 + cos2 α tan2 φ)2 , ° (1 − sin2 α sin2 φ) sec2 φ (1 + cos2 α tan2 φ) = sec2 φ − sin2 α tan2 φ 1 + cos2 α tan2 φ = sec2 φ − tan2 φ + cos2 α tan2 φ 1 + cos2 α tan2 φ = 1 , Ü¦¿ ÃÀË Ü ¿ Ã ϕ,tt + 2 cot θ ϕ,t θ,t = 2a2 cos α sin2 α sec2 φ tan φ (1 + cos2 α tan2 φ)2 +2 cos θ sin θ a cos α sec2 φ 1 + cos2 α tan2 φ − a sin α cos φ (1 − sin2 α sin2 φ)1/2 = 2a2 cos α sin2 α sec2 φ tan φ (1 + cos2 α tan2 φ)2 − cos θ sin θ 2a2 cos α sin α sec φ (1 + cos2 α tan2 φ)(1 − sin2 α sin2 φ)1/2 = 2a2 cos α sin2 α sec2 φ tan φ (1 + cos2 α tan2 φ)2 − sin α sin φ (1 − sin2 α sin2 φ)1/2 2a2 cos α sin α sec φ (1 + cos2 α tan2 φ)(1 − sin2 α sin2 φ)1/2 = 2a2 cos α sin2 α sin φ sec3 φ (1 + cos2 α tan2 φ)2 − 2a2 cos α sin2 α sin φ sec φ (1 + cos2 α tan2 φ)(1 − sin2 α sin2 φ) = 2a2 cos α sin2 α sin φ sec3 φ (1 + cos2 α tan2 φ)2 1 − (1 + cos2 α tan2 φ) cos2 φ (1 − sin2 α sin2 φ) , ° (1 + cos2 α tan2 φ) cos2 φ (1 − sin2 α sin2 φ) = cos2 φ + cos2 α sin2 φ 1 − sin2 α sin2 φ = 1 , Ü ¿ Ã£ÀË (ii) ÁÀ O′′ Õ°Ë ( ©), θ′′ = at + b, φ′′ = c: cos θ = sin(at + b) sin(c + β) sin α + cos(at + b) cos α , tan ϕ = tan(c + β) cos α − cot(at + b) sin α cos(c + β) . φ(t) ≡ at + b, A ≡ sin(c + β), B ≡ sin α, cos θ = AB sin φ(t) + 1 − B2 cos φ(t) , tan ϕ = A √ 1 − B2 √ 1 − A2 − B √ 1 − A2 cot φ(t) .

33 ° θ(t) = arccos AB sin φ(t) + 1
− B2 cos φ(t) , ϕ(t) = arctan A √ 1 − B2 √ 1 − A2 − B √ 1 − A2 cot φ(t) . Mathematica È Å¯ ¿ º £Ø× ÛË Ã θ,tt − sin θ cos θ ϕ2 ,t = 0 , ϕ,tt + 2 cot θ ϕ,t θ,t = 0 . ˜11. ÓÅ Ä 3-4-2. Î ωc, ω′ c ∈ F(0, 1) ¼ ω′ c |p = ωc |p, [(∇a ∇b − ∇b ∇a)ω′ c ]|p = [(∇a ∇b − ∇b ∇a)ωc]|p . Æ ÙÞ¸Ã {xµ} °Þ¸¶Ã p Þ¯·Þ¸ÃÕ« ÑÚ ωc = ωµ(dxµ)c , ω′ c = ω′ µ (dxµ)c . ° [(∇a ∇b − ∇b ∇a)ωc]|p = (∇a ∇b − ∇b ∇a)[ωµ(dxµ)c] p = ωµ(∇a ∇b − ∇b ∇a)(dxµ)c p = ωµ |p [(∇a ∇b − ∇b ∇a)(dxµ)c]|p , [(∇a ∇b − ∇b ∇a)ω′ c ]|p = ω′ µ |p [(∇a ∇b − ∇b ∇a)(dxµ)c]|p . Ç Ò Ä 3-4-1 [ ωµ É ω′ µ Ý f, (dxµ)c Ý ωc]. ω′ c |p = ωc |p Å ω′ µ |p = ωµ |p, £ ÔÅ *12. ÓÅ (3-4-10). Æ »ÂÙ ÕÛ × Ñ (3-4-7) 0 = R[abc] e (2-6-14) = 1 6 Rabc e + Rbca e + Rcab e − Rbac e − Racb e − Rcba e (3-4-6) = 1 3 Rabc e + Rbca e + Rcab e . gde Ý É ¯ Rabcd = gde Rabc e Ô 1 3 Rabcd + Rbcad + Rcabd = 0 . µ R[abc]d = 0. Û ¥¿Ë¸¾Í Ô 0 = 3(R[abc]d + R[bcd]a + R[cda]b + R[dab]c )

34 = (Rabcd + Rbcad + Rcabd ) + (Rbcda
+ Rcdba + Rdbca ) +(Rcdab + Rdacb + Racdb ) + (Rdabc + Rabdc + Rbdac ) = (Rabcd + Rbcad − Racbd ) + (−Rbcad − Rcdab + Rbdac ) +(Rcdab + Radbc − Racbd ) + (−Radbc − Rabcd + Rbdac ) = −2Racbd + 2Rbdac , ° Ò Ñ (3-4-6) É (3-4-9). Racbd = Rbdac , Ø¼ « « Ñ (pair interchange symmetry) Õ (3-4-10) Rabcd = Rcdab . ˜13. Ê § ( © 10) ÕÂÙ Þ¸Ã {θ, ϕ} Õ Õ§ gθθ = R2 , gϕϕ = R2 sin2 θ ; gθθ = R−2 , gϕϕ = R−2 sin−2 θ . É À 7 ÔÕ Þ¸ÃÕ Ñ±¯ªÛÈ Í Γθ ϕϕ = − sin θ cos θ , Γϕ ϕθ = Γϕ θϕ = cos θ sin θ ; × Γθ ϕϕ,θ = sin2 θ − cos2 θ = − cos 2θ , Γϕ ϕθ,θ = Γϕ θϕ,θ = − sin−2 θ = − 1 sin2 θ . É ÝªÂÙ Õ (3-4-20’): Rµνσ ρ = Γρ µσ,ν − Γρ νσ,µ + Γλ σµΓρ νλ − Γλ σνΓρ µλ . «· ¿Ë¸ « ¯ÆÍÔª µ = ν ¾ ¯Ø Rθϕσ ρ = −Γρ ϕσ,θ + Γλ σθ Γρ ϕλ − Γλ σϕΓρ θλ = −Γρ ϕσ,θ + Γλ σθ Γρ ϕλ − Γλ σϕΓρ θλ . ° Rθϕθ θ = −Γθ ϕθ,θ + Γλ θθ Γθ ϕλ − Γλ θϕ Γθ θλ = −0 + 0 − Γϕ θϕ Γθ θϕ = 0 , Rθϕθ ϕ = −Γϕ ϕθ,θ + Γλ θθ Γϕ ϕλ − Γλ θϕ Γϕ θλ = −Γϕ ϕθ,θ + 0 − Γϕ θϕ Γϕ θϕ

35 = − − 1 sin2 θ − cos θ
sin θ 2 = 1 , Rθϕϕ θ = −Γθ ϕϕ,θ + Γλ ϕθ Γθ ϕλ − Γλ ϕϕΓθ θλ = −Γθ ϕϕ,θ + Γϕ ϕθ Γθ ϕϕ − 0 = −(− cos 2θ) + cos θ sin θ (− sin θ cos θ) = − sin2 θ , Rθϕϕ ϕ = −Γϕ ϕϕ,θ + Γλ ϕθ Γϕ ϕλ − Γλ ϕϕΓϕ θλ = −0 + Γϕ ϕθ Γϕ ϕϕ − Γθ ϕϕΓϕ θθ = 0 . ÔÕ ÂÙ Rθϕθ ϕ = −Rϕθθ ϕ = 1 , Rθϕϕ θ = −Rϕθϕ θ = − sin2 θ . Ò Rθϕθ ϕ = gϕϕRθϕθϕ (3-4-9) = −gϕϕRθϕϕθ = −gϕϕgθθ Rθϕϕ θ = −(R−2)(sin−2 θR2)Rθϕϕ θ = − sin−2 θ Rθϕϕ θ , Ç É Õ ÀØ× ¦ Ã É¯ ° Rθϕθϕ = gϕϕRθϕθ ϕ = (R2 sin2 θ)(+1) = R2 sin2 θ , Rθϕϕθ = gθθ Rθϕϕ θ = (R2)(− sin2 θ) = −R2 sin2 θ , Rθϕθϕ = −Rϕθθϕ = −Rθϕϕθ = Rϕθϕθ = R2 sin2 θ , Ê Ø× (3-4-9) É (3-4-10) Õ Ã È n = dim M = 2, Rabc d Õ¥Ë Õ¿ N = n2(n2 − 1)/12 = 1. 14. § ds2 = Ω2(t, x)(−dt2 + dx2) ÕÂÙ {t, x} ÃÕ ( À ˙ Ω É Ω′ » ºÄ Ω « t É x Õ Ñ ). ¦ÕÞ¸ÑÚÕ§ gtt = −Ω2 , gxx = Ω2 ; gtt = −Ω−2 , gxx = Ω−2 . ° gtt,t = −2Ω ˙ Ω , gtt,x = −2ΩΩ′ ; gxx,t = 2Ω ˙ Ω , gxx,x = 2ΩΩ′ . ÆÉ (3-2-10’) Γσ µν = 1 2 gσρ(gρµ,ν + gρν,µ − gµν,ρ) ,

36 Ýª Ñ± Γt tt = 1 2 gtt(gtt,t +
gtt,t − gtt,t) = 1 2 (−Ω−2)(−2Ω ˙ Ω) = Ω−1 ˙ Ω ≡ A , Γx xx = 1 2 gxx(gxx,x + gxx,x − gxx,x) = 1 2 (Ω−2)(2ΩΩ′) = Ω−1Ω′ ≡ B , Γx tt = 1 2 gxx(gxt,t + gxt,t − gtt,x) = 1 2 (Ω−2)(2ΩΩ′) = Ω−1Ω′ = B , Γt xx = 1 2 gtt(gtx,x + gtx,x − gxx,t) = 1 2 (−Ω−2)(−2Ω ˙ Ω) = Ω−1 ˙ Ω = A ; Γt tx = Γt xt = 1 2 gtt(gtt,x + gtx,t − gtx,t) = 1 2 (−Ω−2)(−2ΩΩ′) = Ω−1Ω′ = B , Γx tx = Γx xt = 1 2 gxx(gxt,x + gxx,t − gtx,x) = 1 2 (Ω−2)(2Ω ˙ Ω) = Ω−1 ˙ Ω = A . Í ÕÑ Γt tt,t = A,t = Ω−1 ¨ Ω − Ω−2 ˙ Ω2 ≡ T , Γt tt,x = A,x = Ω−1 ˙ Ω′ − Ω−2 ˙ ΩΩ′ ≡ U , Γx xx,t = B,t = Ω−1 ˙ Ω′ − Ω−2 ˙ ΩΩ′ = U , Γx xx,x = B,x = Ω−1Ω′′ − Ω−2Ω′2 ≡ X ; Γx tt,t = B,t = U , Γx tt,x = B,x = X , Γt xx,t = A,t = T , Γt xx,x = A,x = U ; Γt tx,t = Γt xt,t = B,t = U , Γt tx,x = Γt xt,x = B,x = X , Γx tx,t = Γx xt,t = A,t = T , Γx tx,x = Γx xt,x = A,x = U . É (3-4-20’) Rµνσ ρ = Γρ µσ,ν − Γρ νσ,µ + Γλ σµΓρ νλ − Γλ σνΓρ µλ ÝªÂÙ «· ¿Ë¸ « ¯ÆÍÔª µ = ν ¾ ¯Ø Rtxσ ρ = Γρ tσ,x − Γρ xσ,t + Γλ σtΓρ xλ − Γλ σxΓρ tλ . ° Rtxt t = Γt tt,x − Γt xt,t + Γλ ttΓt xλ − Γλ txΓt tλ = Γt tt,x − Γt xt,t + Γt ttΓt xt + Γx ttΓt xx − Γt txΓt tt − Γx txΓt tx = U − U + AB + BA − BA − AB

37 = 0 , Rtxt x = Γx tt,x −
Γx xt,t + Γλ ttΓx xλ − Γλ txΓx tλ = Γx tt,x − Γx xt,t + Γt ttΓx xt + Γx ttΓx xx − Γt txΓx tt − Γx txΓx tx = X − T + AA + BB − BB − AA = X − T , Rtxx t = Γt tx,x − Γt xx,t + Γλ xtΓt xλ − Γλ xxΓt tλ = Γt tx,x − Γt xx,t + Γt xtΓt xt + Γx xtΓt xx − Γt xxΓt tt − Γx xxΓt tx = X − T + BB + AA − AA − BB = X − T , Rtxx x = Γx tx,x − Γx xx,t + Γλ xtΓx xλ − Γλ xxΓx tλ = Γx tx,x − Γx xx,t + Γt xtΓx xt + Γx xtΓx xx − Γt xxΓx tt − Γx xxΓx tx = U − U + BA + AB − AB − BA = 0 . Ô ÕÂÙ Rtxt x = −Rxtt x = Rtxx t = −Rxtx t = X − T = Ω−1Ω′′ − Ω−2Ω′2 − Ω−1 ¨ Ω + Ω−2 ˙ Ω2 = Ω′′ − ¨ Ω Ω + ˙ Ω2 − Ω′2 Ω2 . È n = dim M = 2, Rabc d Õ¥Ë Õ¿ N = n2(n2 − 1)/12 = 1. º Rtxtx = gxxRtxt x = Ω2(X − T) , Rtxxt = gttRtxx t = −Ω2(X − T) , Rtxtx = −Rxttx = −Rtxxt = Rxtxt = Ω2(X − T) = (Ω′′ − ¨ Ω)Ω + ˙ Ω2 − Ω′2 , © Ø× (3-4-9) É (3-4-10) Õ Ã 15. § ds2 = z−1/2(−dt2 + dz2) + z(dx2 + dy2) ÕÂÙ {t, x, y, z} ÃÕ ¦ÕÞ¸ÑÚÕ§ gtt = −gzz = −z−1/2 , gxx = gyy = z ; gtt = −gzz = −z1/2 , gxx = gyy = z−1 .

38 ° gtt,z = −gzz,z = z−3/2/2 , gxx,z =
gyy,z = 1 . ÆÉ (3-2-10’) Γσ µν = 1 2 gσρ(gρµ,ν + gρν,µ − gµν,ρ) , Ýª Ñ± ( µ, ν, ρ ²Ô z ): Γt µν = 1 2 gtt(gtµ,ν + gtν,µ − gµν,t) = 1 2 gtt(gtµ,ν + gtν,µ) ⇒ Γt tz = Γt zt = 1 2 gtt(gtt,z) = 1 2 (−z1/2)(z−3/2/2) = − 1 4z ; Γz µν = 1 2 gzz(gzµ,ν + gzν,µ − gµν,z) ⇒ Γz tt = 1 2 gzz(−gtt,z) = 1 2 (z1/2)(−z−3/2/2) = − 1 4z , Γz zz = 1 2 gzz(gzz,z) = 1 2 (z1/2)(−z−3/2/2) = − 1 4z , Γz xx = 1 2 gzz(−gxx,z) = 1 2 (z1/2)(−1) = − z1/2 2 = Γz yy ; Γx µν = 1 2 gxx(gxµ,ν + gxν,µ − gµν,x) = 1 2 gxx(gxµ,ν + gxν,µ) ⇒ Γx xz = Γx zx = 1 2 gxx(gxx,z) = 1 2 (z−1)(1) = 1 2z = Γy yz = Γy zy . Õ Ñ±ÁÅ Γt tz = Γt zt = Γz tt = Γz zz = − 1 4z , Γz xx = Γz yy = − z1/2 2 , Γx xz = Γx zx = Γy yz = Γy zy = 1 2z . Ñ Γt tz,z = Γt zt,z = Γz tt,z = Γz zz,z = 1 4z2 , Γz xx,z = Γz yy,z = − 1 4z1/2 , Γx xz,z = Γx zx,z = Γy yz,z = Γy zy,z = − 1 2z2 . Ú É (3-4-20) Rµνσ ρ = Γρ µσ,ν − Γρ νσ,µ + Γλ σµΓρ νλ − Γλ σνΓρ µλ

39 ÝªÂÙ «· ¿Ë¸ « ¯ÆÍÔÝª µ = ν
Õ¾ Å t, z, x, y Õ ÖÝª¯ ¼ x É y « Õ Rtzσ ρ = Γρ tσ,z − Γρ zσ,t + Γλ σtΓρ zλ − Γλ σzΓρ tλ = Γρ tσ,z + Γλ σtΓρ zλ − Γλ σzΓρ tλ , Rtxσ ρ = Γρ tσ,x − Γρ xσ,t + Γλ σtΓρ xλ − Γλ σxΓρ tλ = Γλ σtΓρ xλ − Γλ σxΓρ tλ = Rtyσ ρ , Rzxσ ρ = Γρ zσ,x − Γρ xσ,z + Γλ σzΓρ xλ − Γλ σxΓρ zλ = −Γρ xσ,z + Γλ σzΓρ xλ − Γλ σxΓρ zλ = Rzyσ ρ , Rxyσ ρ = Γρ xσ,y − Γρ yσ,x + Γλ σxΓρ yλ − Γλ σyΓρ xλ = Γλ σxΓρ yλ − Γλ σyΓρ xλ = −Ryxσ ρ . « x É y « t É z ¡ « Õ¯¾ ÐÜ Rxyσ ρ. ° Rtzt ρ = Γρ tt,z + Γλ ttΓρ zλ − Γλ tzΓρ tλ ⇒ Rtzt t = Γt tt,z + Γλ ttΓt zλ − Γλ tzΓt tλ = 0 , Rtzt z = Γz tt,z + Γλ ttΓz zλ − Γλ tzΓz tλ = Γz tt,z + Γz ttΓz zz − Γt tzΓz tt = 1 4z2 + − 1 4z − 1 4z − − 1 4z − 1 4z = 1 4z2 , Rtzt x = Γx tt,z + Γλ ttΓx zλ − Γλ tzΓx tλ = 0 = Rtzt y ; Rtzz ρ = Γρ tz,z + Γλ ztΓρ zλ − Γλ zzΓρ tλ ⇒ Rtzz t = Γt tz,z + Γλ ztΓt zλ − Γλ zzΓt tλ = Γt tz,z + Γt ztΓt zt − Γz zzΓt tz = 1 4z2 + − 1 4z − 1 4z − − 1 4z − 1 4z = 1 4z2 , Rtzz z = Γz tz,z + Γλ ztΓz zλ − Γλ zzΓz tλ = 0 , Rtzz x = Γx tz,z + Γλ ztΓx zλ − Γλ zzΓx tλ = 0 = Rtzz y ; Rtzx ρ = Γρ tx,z + Γλ xtΓρ zλ − Γλ xzΓρ tλ = 0 = Rtzy ρ .

40 Rtxt ρ = Γλ ttΓρ xλ − Γλ txΓρ
tλ = Γλ ttΓρ xλ ⇒ Rtxt t = Γλ ttΓt xλ = 0 = Rtyt t , Rtxt z = Γλ ttΓz xλ = 0 = Rtyt z , Rtxt x = Γλ ttΓx xλ = Γz ttΓx xz = − 1 4z 1 2z = − 1 8z2 = Rtyt y , Rtxt y = Γλ ttΓy xλ = 0 = Rtyt x ; Rtxz ρ = Γλ ztΓρ xλ − Γλ zxΓρ tλ = 0 = Rtyz ρ ; Rtxx ρ = Γλ xtΓρ xλ − Γλ xxΓρ tλ = −Γλ xxΓρ tλ = −Γz xxΓρ tz ⇒ Rtxx t = −Γz xxΓt tz = − − z1/2 2 − 1 4z = − 1 8z1/2 = Rtyy t ; Rtxy ρ = Γλ ytΓρ xλ − Γλ yxΓρ tλ = 0 = Rtyx ρ . Rzxt ρ = −Γρ xt,z + Γλ tzΓρ xλ − Γλ txΓρ zλ = 0 = Rzyt ρ , Rzxz ρ = −Γρ xz,z + Γλ zzΓρ xλ − Γλ zxΓρ zλ ⇒ Rzxz t = −Γt xz,z + Γλ zzΓt xλ − Γλ zxΓt zλ = 0 = Rzyz t , Rzxz z = −Γz xz,z + Γλ zzΓz xλ − Γλ zxΓz zλ = Γz zzΓz xz − Γx zxΓz zx = 0 = Rzyz z , Rzxz x = −Γx xz,z + Γλ zzΓx xλ − Γλ zxΓx zλ = −Γx xz,z + Γz zzΓx xz − Γx zxΓx zx = − − 1 2z2 + − 1 4z 1 2z − 1 2z 1 2z = 1 8z2 = Rzyz y , Rzxz y = −Γy xz,z + Γλ zzΓy xλ − Γλ zxΓy zλ

41 = 0 = Rzyz x ; Rzxx ρ =
−Γρ xx,z + Γλ xzΓρ xλ − Γλ xxΓρ zλ ⇒ Rzxx t = −Γt xx,z + Γλ xzΓt xλ − Γλ xxΓt zλ = Γx xzΓt xx − Γz xxΓt zz = 0 = Rzyy t , Rzxx z = −Γz xx,z + Γλ xzΓz xλ − Γλ xxΓz zλ = −Γz xx,z + Γx xzΓz xx − Γz xxΓz zz = − − 1 4z1/2 + 1 2z − z1/2 2 − − z1/2 2 − 1 4z = − 1 8z1/2 = Rzyy z , Rzxx x = −Γx xx,z + Γλ xzΓx xλ − Γλ xxΓx zλ = Γx xzΓx xx − Γz xxΓx zz = 0 = Rzyy y , Rzxx y = −Γy xx,z + Γλ xzΓy xλ − Γλ xxΓy zλ = Γx xzΓy xx − Γz xxΓy zz = 0 = Rzyy x ; Rzxy ρ = −Γρ xy,z + Γλ yzΓρ xλ − Γλ yxΓρ zλ = Γy yzΓρ xy = 0 = Rzyx ρ ; Rxyx y = Γλ xxΓy yλ − Γλ xyΓy xλ = Γz xxΓy yz = − z1/2 2 1 2z = − 1 4z1/2 = −Ryxx y , Rxyy x = Γλ yxΓx yλ − Γλ yyΓx xλ = −Γz yyΓx xz = − − z1/2 2 1 2z = 1 4z1/2 = −Ryxy x . Ú℄ Ô ÕÂÙ Rtzt z = −Rztt z = Rtzz t = −Rztz t = 1 4z2 , Rxtt x = −Rtxt x = Rytt y = −Rtyt y = Rzxz x = −Rxzz x = Rzyz y = −Ryzz y = 1 8z2 , Rxtx t = −Rtxx t = Ryty t = −Rtyy t = Rxzx z = −Rzxx z = Ryzy z = −Rzyy z = 1 8z1/2 , Rxyy x = −Ryxy x = −Rxyx y = Ryxx y = 1 4z1/2 . Rtztz = gzzRtzt z = z−1/2 1 4z2 = 1 4z5/2 ,

42 Rtzzt = gttRtzz t = −z−1/2 1 4z2 =
− 1 4z5/2 ; Rxttx = gxxRxtt x = z 1 8z2 = 1 8z , Rzxzx = gxxRzxz x = z 1 8z2 = 1 8z ; Rxtxt = gttRxtx t = −z−1/2 1 8z1/2 = − 1 8z , Rxzxz = gzzRxzx z = z−1/2 1 8z1/2 = 1 8z ; Rxyyx = gxxRxyy x = z 1 4z1/2 = z1/2 4 . ÆÂÙ 1 4z5/2 = Rtztz = −Rzttz = −Rtzzt = Rztzt , 1 8z = Rxttx = −Rtxtx = Rytty = −Rtyty = Rzxzx = −Rxzzx = Rzyzy = −Ryzzy = −Rxtxt = Rtxxt = −Rytyt = Rtyyt = Rxzxz = −Rzxxz = Ryzyz = −Rzyyz , z1/2 4 = Rxyyx = −Ryxyx = −Rxyxy = Ryxxy . ÏÀÆ Ê Ø× (3-4-9) É (3-4-10) Õ Ã 16. Î α(z), β(z), γ(z) Ä ¯ h = t + α(z)x + β(z)y + γ(z), § ds2 = −dt2 + dx2 + dy2 + h2dz2 ÕÂÙ {t, x, y, z} ÃÕ § gtt = −gxx = −gyy = −1 , gzz = h2 ; gtt = −gxx = −gyy = −1 , gzz = h−2 . ° gzz,t = 2h , gzz,z = 2h(α′x + β′y + γ′) ≡ 2hh′ , gzz,x = 2hα , gzz,y = 2hβ . ÆÉ (3-2-10’) Γσ µν = 1 2 gσρ(gρµ,ν + gρν,µ − gµν,ρ)

43 Ýª Ñ± Γt µν = 1 2 gtt(gtµ,ν +
gtν,µ − gµν,t) = − 1 2 gttgµν,t ⇒ Γt zz = − 1 2 gttgzz,t = − 1 2 (−1)(2h) = h ; Γz µν = 1 2 gzz(gzµ,ν + gzν,µ − gµν,z) ⇒ Γz zz = 1 2 gzzgzz,z = 1 2 (h−2)(2hh′) = h−1h′ , Γz zt = 1 2 gzz(gzz,t + gzt,z − gzt,z) = 1 2 gzzgzz,t = 1 2 (h−2)(2h) = h−1 , Γz zx = 1 2 gzz(gzz,x + gzx,z − gzx,z) = 1 2 gzzgzz,x = 1 2 (h−2)(2hα) = h−1α , Γz zy = 1 2 gzz(gzz,y + gzy,z − gzy,z) = 1 2 gzzgzz,y = 1 2 (h−2)(2hβ) = h−1β ; Γx µν = 1 2 gxx(gxµ,ν + gxν,µ − gµν,x) = − 1 2 gxxgµν,x ⇒ Γx zz = − 1 2 gxxgzz,x = − 1 2 (1)(2hα) = −hα , Γy zz = − 1 2 gyygzz,y = − 1 2 (1)(2hβ) = −hβ . Õ Ñ±ÁÅ Γt zz = h , Γz zz = h−1(α′x + β′y + γ′) ≡ h−1h′ , Γz zt = Γz tz = h−1 , Γz zx = Γz xz = h−1α , Γz zy = Γz yz = h−1β , Γx zz = −hα , Γy zz = −hβ . Ñ Γt zz,t = 1 , Γz zz,t = −h−2h′ , Γz zt,t = Γz tz,t = −h−2 , Γz zx,t = Γz xz,t = −h−2α , Γz zy,t = Γz yz,t = −h−2β , Γx zz,t = −α , Γy zz,t = −β ; Γt zz,z = h′ ,

44 Γz zz,z = −h−2h′2 + h−1h′′ , Γz zt,z
= Γz tz,z = −h−2h′ , Γz zx,z = Γz xz,z = −h−2h′α + h−1α′ , Γz zy,z = Γz yz,z = −h−2h′β + h−1β′ , Γx zz,z = −h′α − hα′ , Γy zz,z = −h′β − hβ′ ; Γt zz,x = α , Γz zz,x = −h−2αh′ + h−1α′ , Γz zt,x = Γz tz,x = −h−2α , Γz zx,x = Γz xz,x = −h−2α2 , Γz zy,x = Γz yz,x = −h−2αβ , Γx zz,x = −α2 , Γy zz,x = −αβ ; Γt zz,y = β , Γz zz,y = −h−2βh′ + h−1β′ , Γz zt,y = Γz tz,y = −h−2β , Γz zx,y = Γz xz,y = −h−2αβ , Γz zy,y = Γz yz,y = −h−2β2 , Γx zz,y = −αβ , Γy zz,y = −β2 . Ú É (3-4-20’) Rµνσ ρ = Γρ µσ,ν − Γρ νσ,µ + Γλ σµΓρ νλ − Γλ σνΓρ µλ ÝªÂÙ «· ¿Ë¸ « ¯ÆÍÔÝª µ = ν Õ¾ Å t, z, x, y Õ ÖÝª Rtzt t = Γt tt,z − Γt zt,t + Γλ ttΓt zλ − Γλ tzΓt tλ = 0 , Rtzt z = Γz tt,z − Γz zt,t + Γλ ttΓz zλ − Γλ tzΓz tλ = −Γz zt,t − Γz tzΓz tz = −(−h−2) − (h−1)2 = 0 , Rtzt x = Γx tt,z − Γx zt,t + Γλ ttΓx zλ − Γλ tzΓx tλ = 0 = Rtzt y ; Rtzz t = Γt tz,z − Γt zz,t + Γλ ztΓt zλ − Γλ zzΓt tλ

45 = −Γt zz,t + Γz ztΓt zz = −(1)
+ (h−1)(h) = 0 , Rtzz z = Γz tz,z − Γz zz,t + Γλ ztΓz zλ − Γλ zzΓz tλ = Γz tz,z − Γz zz,t + Γz ztΓz zz − Γz zzΓz tz = (−h−2h′) − (−h−2h′) = 0 , Rtzz x = Γx tz,z − Γx zz,t + Γλ ztΓx zλ − Γλ zzΓx tλ = −Γx zz,t + Γz ztΓx zz = −(−α) + (h−1)(−hα) = 0 = Rtzz y . Mathematica ´Ã ÅÂÙ ÕÆ ·Þ¸ÃÅ Æ ªÛ » Ä 3-4-9, ¦ ℄ ( ¶¶) È§ ¯Ø§ Õ Æ 17. ÓÅ 2 ÂÙ ¦Õ £ 2 ÂÙ ¦Õ ÂÙ Í ¦¿¥Ë Æ 2 ÂÙ ¦ÕÂÙ Í 22(22−1) 12 = 1 ¿¥Ë ¯Ø Ã R1212 = R2121 = −R1221 = −R2112 = a . Ç± Rab = gcdRacbd Õ R11 = gcdR1c1d = g22R1212 = ag22 , R12 = gcdR1c2d = g21R1221 = −ag21 , R21 = gcdR2c1d = g12R2112 = −ag12 , R22 = gcdR2c2d = g11R2121 = ag11 . ¸ R = gabRab Õ R = g11R11 + g12R12 + g21R21 + g22R22 = g11(ag22) + g12(−ag21) + g21(−ag12) + g22(ag11) = 2a(g11g22 − g12g21) . ° £ Gab = Rab − 1 2 Rgab Õ G11 = R11 − 1 2 Rg11 = ag22 − 1 2 2a(g11g22 − g12g21)g11 = ag22 − ag11g11g22 + ag12g21g11

46 = ag22 − a(δ1 1 − g12g21)g22 + ag12g21g11
= ag12g21g22 + ag12g21g11 = ag12(g21g11 + g22g21) = ag12δ2 1 = 0 , G22 = R22 − 1 2 Rg22 = ag11 − 1 2 2a(g11g22 − g12g21)g22 = ag11 − ag11g22g22 + ag12g21g22 = ag11 − ag11(δ2 2 − g21g12) + ag12g21g22 = ag11g21g12 + ag12g21g22 = ag21(g11g12 + g12g22) = ag21δ1 2 = 0 , G12 = R12 − 1 2 Rg12 = −ag21 − 1 2 2a(g11g22 − g12g21)g12 = −ag21 − ag11g12g22 + ag12g12g21 = −ag21 − ag11g12g22 + a(δ1 1 − g11g11)g21 = −ag11g12g22 − ag11g11g21 = −ag11(g22g21 + g21g11) = −ag11δ2 1 = 0 , G21 = R21 − 1 2 Rg21 = −ag12 − 1 2 2a(g11g22 − g12g21)g21 = −ag12 − ag11g22g21 + ag12g21g21 = −ag12 − ag11g22g21 + ag12(δ2 2 − g22g22) = −ag11g22g21 − ag12g22g22 = −ag22(g11g12 + g12g22) = −ag22δ1 2 = 0 . £ ÔÅ « 4 Ã¤ ¨ ¢ Killing £ ¤¾µ¦ ˜1. ÓÅ (4-1-1) Õ (φ∗v)a Ø× §2.2 2 « Õ ¿¢ ¯ φ(p) ÞÕ Æ (a) ËÑÑ (φ∗v)(αf + βg) (4-1-2) = v[φ∗(αf + βg)] (4-1-1)(1) = v[α(φ∗f) + β(φ∗g)] §2.2 ¡ 2(a) = α v(φ∗f) + β v(φ∗g) (4-1-2) = α (φ∗v)(f) + β (φ∗v)(g) . (b) (φ∗v)(fg) (4-1-2) = v[φ∗(fg)]

47 (4-1-1)(2) = v[(φ∗f)(φ∗g)] §2.2 ¡ 2(b) = (φ∗f)|p v(φ∗g)
+ (φ∗g)|p v(φ∗f) (4-1-2) = (φ∗f)|p (φ∗v)(g) + (φ∗g)|p (φ∗v)(f) ¡ 1 = f|φ(p) (φ∗v)(g) + g|φ(p) (φ∗v)(f) . φ∗v Þ φ(p) ∈ N Õ¦¿ ¯ φ∗v ∈ Vφ(p) , ∀ f, g ∈ FN , Ø× §2.2 2 ¢ Õ ÕËÑÑÉ ˜2. ÓÅ Ä 4-1-1 ¡ 4-1-2 É 4-1-3. Æ (a) Ä 4-1-1 ÕÅ¢ ∀ f ∈ FN [φ∗(αua + βva)](f) (4-1-2) = (αu + βv)(φ∗f) = α u(φ∗f) + β v(φ∗f) (4-1-2) = α (φ∗ua)(f) + β (φ∗va)(f) . ( u É v Õ É¸ a £ ) φ∗ : Vp → Vφ(p) ËÑ Ì¯Ø × φ∗(αua + βva) = α φ∗ua + β φ∗va . (b) Ä 4-1-2 ÕÅ¢ p ≡ C(t0) ∈ M, φ(p) ≡ φ(C(t0)) ∈ N. ∀ f ∈ FN [(φ∗Ta)(f)]|φ(C(t0 )) = [(φ∗Ta)(f)]|φ(p) ¡ 2 = [T(φ∗f)]|p = [T(φ∗f)]|C(t0 ) = [T(φ∗f|C(t) )]|t=t0 ¡ 1 = [T(f|φ(C(t)) )]|t=t0 = T(f(φ(C(t)))|t=t0 = d(f ◦ φ(C(t))) dt t=t0 . » §2.2 6 (2-2-6), × ³ Ê Ë φ(C(t)) Õ» M É Õ Ë C(t) Õ» Ta N ÉÕ φ∗Ta, M ÉÕ Ë C(t) N ÉÕ φ(C(t)) Õ» (c) Ä 4-1-3 ÕÅ¢ Å¢ Ä·¯ÔÆÅ¢ (4-1-4) É (4-1-5). » Ä 4-1-2: Ë» Õµ· ×° Ëµ· Õ» M É q ÞÕ¦ ¿¶ Þ¸Ã {x′µ}, φ ÌÒ N É φ(q) ÞÕ¦¿¶ Þ¸Ã {yµ}, Ø Ø×x′µ(q) = yµ(φ(p)). {x′µ} ÃÕÞ¸Ë Ì {yµ} ÃÕÞ¸Ë¯ Ò ÙÞ¸Ë q É φ(q) ÞÕ» » (∂/∂x′µ)a|q É (∂/∂yµ)a|φ(q) , Ä 4-1-2 Æ φ∗[(∂/∂x′µ)a|q] = (∂/∂yµ)a|φ(q) , Ø (4-1-4). ¦ ¯ δµ ν = φ∗[δµ ν] = φ∗[(dx′µ)a |q(∂/∂x′ν )a|q] (4-1-10) = φ∗[(dx′µ)a |q]φ∗[(∂/∂x′ν )a|q] (4-1-4) = φ∗[(dx′µ)a |q](∂/∂yν )a|φ(q) ,

48 ³Ý (dyν)b |φ(q) Ô δµ ν(dyν)b |φ(q) = φ∗[(dx′µ)a
|q](∂/∂yν)a|φ(q) (dyν)b |φ(q) = φ∗[(dx′µ)a |q]δa b , Ø (4-1-5) φ∗[(dx′µ)b |q] = (dyµ)b |φ(q) . ¿ Ã£ ×£Û À φ∗[(∂/∂yµ)a|φ(q) ] = (∂/∂x′µ)a|q , φ∗[(dyµ)a |φ(q) ] = (dx′µ)a |q . «° p ∈ M ÞÕ T ∈ FM (k, l), °Á¾ ± Õµ· Ì ¶ φ(p) ∈ N Þ Õ φ∗T ∈ FN (k, l), ° Þ¸Ã À (φ∗T)µ1 ···µk ν1 ···νl |φ(p) = (φ∗T)a1 ···ak b1 ···bl (dyµ1 )a1 |φ(p) · · · (dyµk )ak |φ(p) ∂ ∂yν1 b1 φ(p) · · · ∂ ∂yνl bl φ(p) = Ta1 ···ak b1 ···bl φ∗[(dyµ1 )a1 |φ(p) ] · · · φ∗[(dyµk )ak |φ(p) ] φ∗ ∂ ∂yν1 b1 φ(p) · · · φ∗ ∂ ∂yνl bl φ(p) = Ta1 ···ak b1 ···bl (dx′µ1 )a1 |p · · · (dx′µk )ak |p ∂ ∂x′ν1 b1 p · · · ∂ ∂x′νl bl p = T′µ1 ···µk ν1 ···νl |p . © Û³ Þ φ(p) Õ φ∗T ( À) Þ¸Ã {yµ} Õ ¯ ³ ÀÞ p ÕÀ T Þ¸Ã {x′µ} Õ 3. Î φ : M → N Ì¯ p ∈ M, {yµ} φ(p) Þ¦ ¶ÉÕÞ¸¯ÓÅ (φ∗v)a = v(φ∗yµ)(∂/∂yµ)a , ∀ va ∈ Vp . Æ (φ∗v)a ∈ Vφ(p) , Þ¸Ã {yµ} (φ∗v)a|φ(p) = (φ∗v)µ|φ(p) ∂ ∂yµ a φ(p) . ¦ ¯ (φ∗v)µ|φ(p) (4-1-6) = v′µ|p = va(dx′µ)a |p = vaφ∗[(dyµ)a |φ(p) ] = va[d(φ∗yµ)]a = v(φ∗yµ) . 4. Î M, N ¯ φ : M → N ¾ ± ¯ p ∈ M, q ≡ φ(p), ÓÅµ· Ì φ∗ : Vp → Vq ± Ì Æ ¾ ± Ì ¦¦ÒÉÕ Ì¯Æµ· Ì£ ¦¦ÒÉÕ ¿ ¦¦¦ÒÉÕËÑ ÌØ ± Ì

49 5. Î M, N, Q ¯ φ : M
→ N É ψ : N → Q Ì (a) ÓÅ (ψ ◦ φ)∗f = (φ∗ ◦ ψ∗)f, ∀ f ∈ FQ. (b) ÓÅ (ψ ◦ φ)∗va = ψ∗(φ∗va), ∀ p ∈ M, va ∈ Vp. (c) (ψ ◦ φ)∗ É φ∗ ◦ ψ∗ ¤ Ý FQ(0, l) Ò FM (0, l) Õ Ì¯ÓÅ (ψ ◦ φ)∗ = φ∗ ◦ ψ∗ . Æ «°³Í Ì ψ ◦ φ : p → q = φ(p) → r = ψ(q) = ψ(φ(p)), ° p ∈ M, q ∈ N, r ∈ Q. (a) ∀ f ∈ FQ, » 1, Ì (ψ◦φ)∗f|p = f|ψ(φ(p)) . ¦ ¯ψ∗f|q = f|ψ(q) = g|q É φ∗g|p = g|φ(p) , Æ (φ∗ ◦ ψ∗)f|p = φ∗g|p = g|φ(p) = f|ψ(φ(p)) . (ψ ◦ φ)∗f|p = (φ∗ ◦ ψ∗)f|p = f|ψ(φ(p)) . Ð É Ã φ∗f = f ◦ φ, È (ψ ◦ φ)∗f = f ◦ (ψ ◦ φ) = (f ◦ ψ) ◦ φ = (ψ∗f) ◦ φ = φ∗ ◦ (ψ∗f) = (φ∗ ◦ ψ∗)f . (b) » 2, ∀ f ∈ FQ, µ· Ì [(ψ ◦ φ)∗va](f)|ψ(φ(p)) = v[(ψ ◦ φ)∗f]|p = v(f)|ψ(φ(p)) , Ú ¦ É (a) Õ À ¦ ¯±¡ » 2: [ψ∗(φ∗va)](f)|ψ(φ(p)) = (φ∗va)(ψ∗f)|φ(p) = v(φ∗(ψ∗f))|p = v((φ∗ ◦ ψ∗)f)|p = v(f)|ψ(φ(p)) . [(ψ ◦ φ)∗va](f) = [ψ∗(φ∗va)](f), ÑÏ (ψ ◦ φ)∗va = ψ∗(φ∗va) ÀË (c) ∀ T ∈ FQ(0, l), ψ∗T ∈ FN (0, l), φ∗(ψ∗T) = (φ∗ ◦ ψ∗)T ∈ FM (0, l). » 3 (4-1-3) [(ψ ◦ φ)∗T]a1 ···al |p(v1)a1 · · · (vl )al = Ta1 ···al |ψ(φ(p)) [(ψ ◦ φ)∗v1]a1 · · · [(ψ ◦ φ)∗vl ]al (b) = Ta1 ···al |ψ(φ(p)) [ψ∗(φ∗v1)]a1 · · · [ψ∗(φ∗vl )]al , ° v1, · · · , vl ∈ Vp. ¦ ¯±¡ » 3 (4-1-3), [(φ∗ ◦ ψ∗)T]a1 ···al |p(v1)a1 · · · (vl )al = [φ∗(ψ∗T)]a1 ···al |p(v1)a1 · · · (vl )al = (ψ∗T)a1 ···al |φ(p) (φ∗v1)a1 · · · (φ∗vl )al = Ta1 ···al |ψ(φ(p)) [ψ∗(φ∗v1)]a1 · · · [ψ∗(φ∗vl )]al . [(ψ ◦ φ)∗T]a1 ···al |p(v1)a1 · · · (vl )al = [(φ∗ ◦ ψ∗)T]a1 ···al |p(v1)a1 · · · (vl )al , ÑÏ (ψ ◦ φ)∗ = φ∗ ◦ ψ∗.

50 6. Î φ : M → N ¾ ±
¯ va, ua M É ¯ÓÅ φ∗([v, u]a) = [φ∗v, φ∗u]a, ° [v, u]a º«ÆÒ Æ 2 φ∗[v, u](f) = [v, u](φ∗f) = v(u(φ∗f)) − u(v(φ∗f)) . [φ∗v, φ∗u](f) = φ∗v(φ∗u(f)) − φ∗u(φ∗v(f)) (4-1-9) = φ∗(vu)(f) − φ∗(uv)(f) (4-1-2) = (vu)(φ∗f) − (uv)(φ∗f) = v(u(φ∗f)) − u(v(φ∗f)) . φ∗[v, u](f) = [φ∗v, φ∗u](f), ∀ f ∈ FN , Ê φ∗[v, u] = [φ∗v, φ∗u] . ˜7. ÓÅ Ä 4-2-4. Æ ÖÆ¯ ÆÑ Ø× ¯ Lv(ωava) = (Lvωa)va + ωa(Lvva) = vaLvωa , ° É Ä 4-2-3 (4-2-6) Lvva = [v, v]a = 0. ¦ ¯ ωava ¸ ¯ Ä 4-2-1 (4-2-2) Lv(ωava) = v(ωava) §3.1 ¡ 1(d) = vb∇b (ωava) = vb[(∇b ωa)va + ωa(∇b va)] = vavb∇b ωa + vbωa ∇b va = va[vb∇b ωa + ωb ∇avb] . É ¯ Ò va Õ Ñ¯Æ Lvωa = vb∇b ωa + ωb ∇avb . ÇÕ ∇a ¦ Ñ ª± ˜8. Î va ∈ FM (1, 0), ωa ∈ FM (0, 1), ÓÅ« ¦Þ¸Ã {xµ} (Lvω)µ = vν∂ωµ/∂xν + ων∂vν/∂xµ . (4-2-7) ¾ ° ∇a ∂a . Æ Þ¸ÑÚ (Lvω)µ = (∂/∂xµ)a(Lvω)a (4-2-7) = (∂/∂xµ)a(vb∂b ωa + ωb ∂avb) (3-1-10) = vb∂b [(∂/∂xµ)aωa] + ωb [(∂/∂xµ)a∂a]vb = vb∂b ωµ + ωb ∂µvb (3-1-10) = vν∂νωµ + ων∂µvν = vν∂ωµ/∂xν + ων∂vν/∂xµ .

51 ˜9. Î ua, va ∈ FM (1, 0), Å
Ý ° ¤ÀË [Lv, Lu] = L[v,u] ( ° [Lv, Lu] ≡ Lv Lu − Lu Lv) . ÓµÝ « f ∈ FM É wa ∈ FM (1, 0) Õ¾ ÊÅ¢ Ý « wa Ý × ( Ü 2 À 8). Æ (a) Ý °¸ f: [Lv, Lu](f) = (Lv Lu − Lu Lv)(f) = Lv Lu(f) − Lu Lv(f) = Lv(Luf) − Lu(Lvf) (4-2-2) = Lv(u(f)) − Lu(v(f)) (4-2-2) = v(u(f)) − u(v(f)) (2-2-9) = [v, u](f) (4-2-2) = L[v,u] (f) . (b) Ý ° wa: [Lv, Lu](wa) = (Lv Lu − Lu Lv)(wa) = Lv Lu(wa) − Lu Lv(wa) = Lv(Luwa) − Lu(Lvwa) (4-2-6) = Lv([u, w]a) − Lu([v, w]a) (4-2-6) = [v, [u, w]]a − [u, [v, w]]a = ([v, [u, w]]a + [u, [w, v]]a + [w, [v, u]]a) − [w, [v, u]]a = −[w, [v, u]]a = [[v, u], w]a (4-2-6) = L[v,u] (wa) , ° Ð Ü¥ Ý × (c) Ý °« ωa. Ò Lv Lu(ωa) = Lv(Luωa) (4-2-7) = Lv(ub∇b ωa + ωb ∇aub) (4-2-7) = vc∇c(ub∇b ωa + ωb ∇aub) + (ub∇b ωc + ωb ∇cub)∇avc = vc∇cub∇b ωa + vcub∇c ∇b ωa + vc∇cωb ∇aub + vcωb ∇c ∇aub +ub∇b ωc ∇avc + ωb ∇cub∇avc = vc∇cub∇b ωa + vcub∇c ∇b ωa + vc∇aub∇cωb + vcωb ∇c ∇aub

52 +ub∇avc∇b ωc + ωb ∇cub∇avc = vcub∇c ∇b ωa
+ vc∇cub∇b ωa + vc∇aub∇cωb + ub∇avc∇b ωc +vcωb ∇c ∇aub + ωb ∇cub∇avc , ° Lu Lv(ωa) = ucvb∇c ∇b ωa + uc∇cvb∇b ωa + uc∇avb∇cωb + vb∇auc∇b ωc +ucωb ∇c ∇avb + ωb ∇cvb∇auc = vcub∇b ∇cωa + uc∇cvb∇b ωa + ub∇avc∇b ωc + vc∇aub∇cωb +ucωb ∇c ∇avb + ωb ∇cvb∇auc . É Í¨¯Ü¦¡Å¡¥ Í Ù ¯Ô [Lv, Lu](ωa) = Lv Lu(ωa) − Lu Lv(ωa) = (vc∇cub − uc∇cvb)∇b ωa +ωb (vc∇c ∇aub − uc∇c ∇avb) + ωb (∇avc∇cub − ∇auc∇cvb) = (vc∇cub − uc∇cvb)∇b ωa + ωb ∇a(vc∇cub − uc∇cvb) (4-2-6’) = [v, u]b∇b ωa + ωb ∇a[v, u]b (4-2-7) = L[v,u] (ωa) . 10. ÎFab 4 Ñ ¦ÉÕ « ¯° ÕÐÑÞ¸Ã {t, x, y, z} Õ F01 = −F13 = xρ−1, F02 = −F23 = yρ−1, F03 = F12 = 0, ° ρ ≡ (x2 + y2)1/2. ÓÅFab Ø « Ñ¯ Ø LvFab = 0, ° va = −y(∂/∂x)a+x(∂/∂y)a. Æ » Ä 4-2-5 (4-2-8), LvFab = vc∇cFab + Fcb ∇avc + Fac ∇b vc . ÇÕÑ ª± ∇a ¯ÆÙ °Ñ ∂a, ° É Õ LvFµν = vσ∂σFµν + Fσν ∂µvσ + Fµσ∂νvσ . ° LvF01 = vσ∂σF01 + Fσ1∂0vσ + F0σ∂1vσ = (v1∂1F01 + v2∂2F01) + 0 + F02∂1v2 = (−y) ∂ ∂x (xρ−1) + x ∂ ∂y (xρ−1) + (yρ−1) ∂ ∂x (x) = −yρ−1 − yx − 1 2 ρ−3/22x + x2 − 1 2 ρ−3/22y + yρ−1 = −yρ−1 + yx2ρ−3/2 − x2yρ−3/2 + yρ−1 = 0 ,

53 LvF02 = vσ∂σF02 + Fσ2∂0vσ + F0σ∂2vσ = (v1∂1F02
+ v2∂2F02) + 0 + F01∂2v1 = (−y) ∂ ∂x (yρ−1) + x ∂ ∂y (yρ−1) + (xρ−1) ∂ ∂y (−y) = −y2 − 1 2 ρ−3/22x + xρ−1 + xy − 1 2 ρ−3/22y − xρ−1 = xy2ρ−3/2 + xρ−1 − xy2ρ−3/2 − xρ−1 = 0 , LvF03 = Lv(0) = 0 , LvF12 = Lv(0) = 0 , LvF13 = vσ∂σF13 + Fσ3∂1vσ + F1σ∂3vσ = (v1∂1F13 + v2∂2F13) + F23∂1v2 + 0 = −(v1∂1F01 + v2∂2F01) − F02∂1v2 = −LvF01 = 0 , LvF23 = vσ∂σF23 + Fσ3∂2vσ + F2σ∂3vσ = (v1∂1F23 + v2∂2F23) + F13∂2v1 + 0 = −(v1∂1F02 + v2∂2F02) − F01∂2v1 = −LvF02 = 0 . « va = −y(∂/∂x)a + x(∂/∂y)a LvFab = 0. 11. Î ξa (M, gab ) Õ Killing ¯ ∇a ² gab Ð ¯ÓÅ ∇aξa = 0. Æ Ò ∇aξa = ∇a(gabξb ) = ξb ∇agab + gab∇aξb . ° Ü¦ ∇a ² gab ÕÐ Ñ ∇agbc = 0 ( © §3.2.2 Ê 1 ·ÕÅ¢), ° ∇agab = 0. Ü gab∇aξb = g(ab)∇aξb ¡Å2-6-2(a) = g(ab)∇(a ξb) ¡Å4-3-1 = 0. ∇aξa = 0. 12. Î ξa (M, gab ) Õ Killing ¯ φ : M → N © M ª ×§ Ì¯ÓÅ φ∗ξa £ (M, gab ) Õ Killing É À 5(c) Õ Ñ Æ Î (M, gab ) ÉÕ Killing ξa ÊÕ ¾ ± ψ, ° ¹ ψt, t ξa ÕÓ ËÉÕ » Killing Õ (§4.3 2) ×ÆÑ Õ (§4.2 1), Lξ gab = lim t→0 1 t (ψ∗ t gab − gab ) = 0 . È ¢ÅÕ ÁÀ φ : M → M ×§ Ì¯ØØ× φ∗gab = gab (§4.3 1), ªÛ Lφ∗ ξ gab = 0, Øφ∗ξa £ Killing ÖÆ Å¢ φ∗ξa ÊÕ ¾ ± φ ◦ ψt. » Ä 4-2-1 (4-2-2), Lφ∗ ξ = (φ∗ξ)(f),

54 ∀ f ∈ FM . µ· ÌÕ (4-1-2) (φ∗ξ)(f)
= ξ(φ∗f) = Lξ (φ∗f). ¬ À ÆÑ Õ (4-2-1) Ô Lφ∗ ξ f = Lξ (φ∗f) = lim t→0 1 t [ψ∗ t (φ∗f) − f] = lim t→0 1 t [(ψ∗ t ◦ φ∗)f − f] = lim t→0 1 t [(φ ◦ ψt)∗f − f] , ° Ú ¦ É À 5(c) Õ Ñ (ψ∗ t ◦ φ∗) = (φ ◦ ψt)∗. f Ä ¯° Ê φ∗ξ ÀÕ ¾ ± ¹ φ ◦ ψt. Ú Lφ∗ ξ gab = lim t→0 1 t [(φ ◦ ψt)∗gab − gab ] = lim t→0 1 t [(ψ∗ t ◦ φ∗)gab − gab ] = = lim t→0 1 t [ψ∗ t (φ∗gab ) − gab ] = lim t→0 1 t [ψ∗ t (gab ) − gab ] = Lξ gab = 0 , ° É ×§ Ì φ∗gab = gab É ξ Õ Killing Ñ Lξ gab = 0. 13. Î ξa, ηa (M, gab ) Õ Killing ¯ÓÅ°«ÆÒ [ξ, η]a £ Killing Ñ Ô M É Killing ÕÖÍ ¦¯ ¼ Æ ( ©Å ¶ G). Æ »À 9 Õ À¯ L[ξ,η] gab = [Lξ , Lη]gab = Lξ Lηgab − Lη Lξ gab ¡ 2 = Lξ (0) − Lη(0) = 0 , 2 Æ¯ [ξ, η]a £ Killing Ð È Å Ã (4-3-1): ∇a[ξ, η]b + ∇b [ξ, η]a = ∇a[ξc∇cηb − ηc∇cξb ] + ∇b [ξc∇cηa − ηc∇cξa] (4-3-1) = ∇a[−ξc∇b ηc + ηc∇b ξc] + ∇b [−ξc∇aηc + ηc∇aξc] = ∇aηc∇b ξc + ηc∇a ∇b ξc − ∇aξc∇b ηc − ξc∇a ∇b ηc +∇b ηc∇aξc + ηc∇b ∇aξc − ∇b ξc∇aηc − ξc∇b ∇aηc = (∇aηc∇b ξc − ∇b ξc∇aηc) + (∇b ηc∇aξc − ∇aξc∇b ηc) +ηc(∇a ∇b + ∇b ∇a)ξc − ξc(∇a ∇b + ∇b ∇a)ηc = (∇aηc∇b ξc − ∇b ξc ∇aηc) + (∇b ηc∇aξc − ∇aξc ∇b ηc) +ηc(∇a ∇b + ∇b ∇a)ξc − ξc(∇a ∇b + ∇b ∇a)ηc = ηc(∇a ∇b + ∇b ∇a)ξc − ξc(∇a ∇b + ∇b ∇a)ηc . É Killing ÑÔ (∇a ∇b + ∇b ∇a)ξc = −∇a ∇cξb − ∇b ∇cξa = −∇c ∇aξb − Racbd ξd − ∇c ∇b ξa − Rbcad ξd

55 = −∇c ∇aξb − Racbd ξd − ∇c ∇b
ξa − Rbcad ξd = −∇c(∇aξb + ∇b ξa) − Racbd ξd − Rbcad ξd = −Racbd ξd − Rbcad ξd , (∇a ∇b + ∇b ∇a)ηc = −Racbd ηd − Rbcad ηd . ° ∇a[ξ, η]b + ∇b [ξ, η]a = ηc(−Racbd ξd − Rbcad ξd) − ξc(−Racbd ηd − Rbcad ηd) = −Racbd ηcξd − Rbcad ηcξd + Racbd ξcηd + Rbcad ξcηd = −Radbc ηdξc − Rbdac ηdξc + Racbd ξcηd + Rbcad ξcηd = (Rbcad − Radbc )ξcηd + (Racbd − Rbdac )ξcηd (3-4-10) = 0 . 14. Î ξa ÂÙ ¦ (M, gab ) Õ Killing ¯ Rabc d gab ÕÂÙ (a) ÓÅ ∇a ∇b ξc = −Rbca dξd . «Å¢ Ä 4-3-4 ¢ Î Rabc d Õ × Killing Ã (4-3-1) Æ ∇a ∇b ξc + ∇b ∇cξa = Rabc dξd . Ü¦ ÝË¸ a → b, b → c, c → a ÔÜ ¯ ¦ ÔÜ Å Ü¦¡ ÇÉ¨ÜÅ ¾É (3-4-7) µÔÅ (b) É (a) Õ ÀÅ¢ ∇a∇aξc = −Rcd ξd, ° Rcd Ç± Æ (a) »ÂÙ Õ (3-4-3) É Killing Ø×Õ Ã (4-3-1) (∇a ∇b − ∇b ∇a)ξc = ∇a ∇b ξc − ∇b ∇aξc = ∇a ∇b ξc + ∇b ∇cξa = Rabc dξd . ÝË¸ a → b, b → c, c → a Ô ∇b ∇cξa + ∇c ∇aξb = Rbca dξd , ¦ Ô ∇c ∇aξb + ∇a ∇b ξc = Rcab dξd . Ü¦¡ ÇÉ¨ÜÅ Ô ∇a ∇b ξc + ∇b ∇cξa + ∇b ∇cξa + ∇c ∇aξb − ∇c ∇aξb − ∇a ∇b ξc = 2∇b ∇cξa = Rabc dξd + Rbca dξd − Rcab dξd = (Rabc d + Rbca d + Rcab d)ξd − 2Rcab dξd = −2Rcab dξd , Ú ¦ Ò ÂÙ ÕÛ × (3-4-7). ° ∇b ∇cξa = −Rcab dξd , Ø ∇a ∇b ξc = −Rbca dξd .

56 (b) É ³Ý gab: gab∇a ∇b ξc = ∇b∇b
ξc = −gabRbca dξd = −gabRbcad ξd = −Rcd ξd , ° Ú ¦ Ò ÂÙ Ñ (3-4-6) ¡ (3-4-9) ÉÇ± Õ gabRbcad = gabRcbda = Rcd . ∇a∇aξc = −Rcd ξd. ˜15. Å (4-3-3) Õ (∂/∂η)a ÕØ× Killing Ã (4-3-1). Æ ¸Å ξa = (∂/∂η)a = t(∂/∂x)a + x(∂/∂t)a Ø× Killing Ã (4-3-1): ∇aξb + ∇b ξa = 0, Ò² Ñ§ ηab ÍÐ ÕÑ ª± °Ñ ∂a, ÍÔÅ ∂aξb + ∂b ξa = 0. °Í Õ Ã ∂µξν + ∂νξµ = 0, Ç µ, ν = 0, 1 º t, x. Ç ξ0 = x É ξ1 = t. ° ¯ (µ, ν) = (0, 0) : ∂0ξ0 + ∂0ξ0 = 2∂0ξ0 = 2∂0(η0ρξρ) = 2∂0(−ξ0) = −2∂0ξ0 = −2 ∂ ∂t (x) = 0 , (µ, ν) = (0, 1) Ð(1, 0) : ∂0ξ1 + ∂1ξ0 = ∂0(η1ρξρ) + ∂1(η0ρξρ) = ∂0(ξ1) + ∂1(−ξ0) = ∂ ∂t (t) − ∂ ∂x (x) = 0 , (µ, ν) = (1, 1) : ∂1ξ1 + ∂1ξ1 = 2∂1ξ1 = 2∂1(η1ρξρ) = 2∂1(ξ1) = 2 ∂ ∂x (t) = 0 Æ ∂aξb + ∂b ξa = 0 ÀË ˜16. Ê 2 Ñ ¦ Ra = x(∂/∂y)a − y(∂/∂x)a ÊÕ ×§ Õ ¦¹§ φα ÑÕÞ¸¶ Ra = x(∂/∂y)a − y(∂/∂x)a ÕÓ ËÕ Ã dxµ(t) dt = Rµ (µ = 1, 2 = x, y), Ø dx1(t) dt = dx(t) dt = R1 = −y(t) , dx2(t) dt = dy(t) dt = R2 = x(t) . ∀ p ∈ RR2, Î C(t) Ø× p = C(0) ÕÓ Ë¯Ø x(0) = xp, y(0) = yp, ÀÆ ÊÉ ÃÕ [ Ø·ËÕ ] x(t) = xp cos t − yp sin t , y(t) = xp sin t + yp cos t . Î q ≡ φα(p), q µ C(t) É Ê t = α ÕÞ¯Ø q = C(α), φα ÑÕ Þ¸ x′ É y′ Ø× x′ p ≡ xq = xp cos α − yp sin α , y′ p ≡ yq = xp sin α + yp cos α .

57 p Þ ¯ Å¸ p À x′ = x
cos α − y sin α , y′ = x sin α + y cos α . ØÜÆÕ Ø ( ÃÆ) ¶ *17. Î (M, gab ) Õ φ[S] ÉÞÞ¤ Á » Á » ( £» ¥Ë»° φ[S]), ÓÅ ² Á «Å¢²Á ta Ã ÆÕ ²Á [ ÙÃÆ ¦ÑÚ {(eµ)a} (e0)a = ta ]; ¬Å¢Á ÉÞÞ¤ Á » É ÞÅ¢ £ Æ«ÖÆ Å¢²Á ÃÆÕ ²Á Î ta Á ¯ gab tatb < 0. ÁÀ va ² ta ÃÆ¯ØØ× gab vatb = 0, ² gab vavb > 0. Î n M Õ¦¦ÃÆ ¦ÑÚ {(eµ)a}. ¦ Ñ Ù (e0)a = ta, ° gab (e0)a(e0)b = g00 < 0. va ·ÑÚÕ va = vµ(eµ)a, ÃÆÑ Ê gab vatb = gab vµ(eµ)a(e0)b = gµ0vµ = g00v0 = 0 . ÆÓ v0 = 0. gab vavb = gab vµ(eµ)avν(eν)b = gµνvµvν = g00v0v0 + gµ0vµv0 + g0µv0vµ + gijvivj = gijvivj = gii(vi)2 > 0 , ° É n − 1 Õ ¦ § ÕÃ Ñ ÔÒ Ñ Å ¢£ ÆÓ²Á ÃÆÕ ²¦ Á ¯°º ° ¦Í ¦ ¦ ¼°¦ ¯ Á Ç¦ Í ÃÆ ¬° Å¢Á ÉÞÞ¤ Á » Æ Á ¯ » 4 Õ ÎÎÁ ÎÁ φ[S] É q ÞÕ» ¦ Wq. ÕÁ na ∈ Wq, Wq ÕÑÚ ² ¦¿Á ¯ µ Õ Á » ÁÀ φ[S] ÉÞÞ¤ Á » Á » ¯ªÛ ß ¯ Á £ ¯ Á ¯ Í ¯ Á ÁÀ Á ¯ªÛ »¬¯ ÞÞ¤ Á » ¯ ² ÎÞÞ¤ Á » ± ÁÀ Á ¯ªÛ ÞÞÕ ¤ Á Õ¯ Õ Ñ ½© Õ» ¤² ÃÆ¯Æ «Õ Ñ ÆÓ » ¤ Á Õ Ü Á ÕÉÁ Õ¯ Î Á » ¯Æ£ Ú ¯ ¦Õ ¯Ñµ · Á

58 « 5 Ã¤ » ¦ ˜1. Ä 5-1-3 Å¢
Å {(e1)a ∧ (e2)b , (e2)a ∧ (e3)b , (e3)a ∧ (e1)b } ËÑ¥Ë Æ ¢Î ËÑ¥Ë¯ ² Õ a, b, c Ø× a (e1)a ∧ (e2)b + b (e2)a ∧ (e3)b + c (e3)a ∧ (e1)b = 0 . (e1)a(e2)b Ý É ¯ÆÔ (e1)a(e2)b(e1)a ∧ (e2)b = (e1)a(e2)b[(e1)a(e2)b − (e2)a(e1)b ] = 1 , (e1)a(e2)b(e2)a ∧ (e3)b = (e1)a(e2)b[(e2)a(e3)b − (e3)a(e2)b ] = 0 , (e1)a(e2)b(e3)a ∧ (e1)b = (e1)a(e2)b[(e3)a(e1)b − (e1)a(e3)b ] = 0 , É ¶ a = 0. ±Ä Æ b = c = 0. ° ËÑ¥Ë ˜2. Î V ¦¯ {(e1)a, (e2)a, (e3)a, (e4)a } V ∗ ÕÑÚ¯ Ê ωa ∈ Λ(1), ωabc ∈ Λ(3) É ωabcd ∈ Λ(4) ÑÚÕ ¯¡¢ Ã ( Á ω12) Õ » C1 4 = 4, C3 4 = 4, C4 4 = 1 ωa = ω1(e1)a + ω2(e2)a + ω3(e3)a + ω4(e4)a , ωabc = ω123(e1)a ∧ (e2)b ∧ (e3)c + ω124(e1)a ∧ (e2)b ∧ (e4)c +ω134(e1)a ∧ (e3)b ∧ (e4)c + ω234(e2)a ∧ (e3)b ∧ (e4)c , ωabcd = ω1234(e1)a ∧ (e2)b ∧ (e3)c ∧ (e4)d . Ã Õ Á ω134 = ωabc (e1)a(e3)b(e4)c . ˜3. Ú « Å¢ (ω1)a1 ∧· · ·∧(ωl)al = l!(ω1)[a1 · · · (ωl)al] , ° (ω1)a, · · · , (ωl)a « Æ Î« l ÀË (ω1)a1 ∧ · · · ∧ (ωl)al = l!(ω1)[a1 · · · (ωl)al] , ¿ l ¯ ×° F1···l a1 ···al , ° » ÓÕ 2 (5-1-2) É Í (ω1)a1 ∧ · · · ∧ (ωl)al ∧ (ωl+1)al+1 = F1···l a1 ···al ∧ (ωl+1)al+1 (5-1-2) = (l + 1)! l! 1! F1···l [a1 ···al (ωl+1)al+1 ] = (l + 1)l!(ω1)[[a1 · · · (ωl)al] (ωl+1)al+1 ] = (l + 1)!(ω1)[a1 · · · (ωl)al (ωl+1)al+1 ] , Ú ¦ Ò Ä 2-6-2(b)— ÇÆÕ± Ò Ç ¬ Ç « l + 1 ÀË É 2 (5-1-2) È Ê (ω1)a1 ∧ · · · ∧ (ωl)al = (1 + · · · + 1)! 1! · · · 1! (ω1)[a1 · · · (ωl)al] = l!(ω1)[a1 · · · (ωl)al] .

59 ˜4. ÓÅ Ä 5-1-4. Æ ωa1 ···al = C
ωµ1 ···µl (dxµ1 )a1 ∧ · · · ∧ (dxµl )al , ° Ã ωµ1 ···µl 0 ¯ Ñ (dxµ1 )a1 ∧ · · · ∧ (dxµl )al l º¾ ª± 3 (5-1-11) Æ db ωa1 ···al = (dω)ba1 ···al = (l + 1)∇[b ωa1 ···al] . Õ ∇b °Ñ ∂b , Ò (3-1-10) Õ À ∂b (dxµ)a = 0, ∂b [(dxµ1 )a1 ∧ · · · ∧ (dxµl )al ] = 0. ° ∇b ωa1 ···al = ∂b ωa1 ···al = C ∂b ωµ1 ···µl (dxµ1 )a1 ∧ · · · ∧ (dxµl )al = C (∂b ωµ1 ···µl )(dxµ1 )a1 ∧ · · · ∧ (dxµl )al (3-1-2) = C (dωµ1 ···µl )b (dxµ1 )a1 ∧ · · · ∧ (dxµl )al . (dωµ1 ···µl )b 1 (dxµ1 )a1 ∧ · · · ∧ (dxµl )al l ° » ÓÕ 2 (5-1-2) C (dωµ1 ···µl )b ∧ (dxµ1 )a1 ∧ · · · ∧ (dxµl )al = C (1 + l)! 1!l! (dωµ1 ···µl )[b (dxµ1 )a1 ∧ · · · ∧ (dxµl )al] = (l + 1) C (dωµ1 ···µl )[b (dxµ1 )a1 ∧ · · · ∧ (dxµl )al] = (l + 1)∇[b ωa1 ···al] . Íº¾ ª±Õ 3 (5-1-11) ØÔ Ä 5-1-4 (5-1-12): (dω)ba1 ···al = (l + 1)∇[b ωa1 ···al] = C (dωµ1 ···µl )b ∧ (dxµ1 )a1 ∧ · · · ∧ (dxµl )al . 5. Îω 1 ¯u, v ¯ÓÅ dω(u, v) = u(ω(v))−v(ω(u))−ω([u, v]). × Û³ º dω « u, v ÕÝ À¯Ø (dω)ab uavb. Æ 3 (5-1-11): (dω)ab = 2∇[a ωb] = ∇aωb − ∇b ωa. ° Û³= (∇aωb − ∇b ωa)uavb = uavb∇aωb − uavb∇b ωa . ³ u(ω(v)) = u(ωb vb) = ua∇a(ωb vb) = uavb∇aωb + uaωb ∇avb , v(ω(u)) = v(ωb ub) = va∇a(ωb ub) = vaub∇aωb + vaωb ∇aub , ω([u, v]) = ωb (ua∇avb − va∇aub) = uaωb ∇avb − vaωb ∇aub . ° ³= uavb∇aωb − vaub∇aωb = uavb∇aωb − uavb∇b ωa = Û³.

60 ˜6. Î vb É ωa1 ···al » M ÉÕ
É l ¯ÓÅ (a) Lvωa1 ···al = da1 (vbωba2 ···al ) + (dω)ba1 ···al vb. µa2 ···al ≡ vbωba2 ···al , da1 µa2 ···al Ë (dµ)a1 a2 ···al . (b) Lvdω = dLvω ( Ïµ ¦¿Ï Õ£ ). (1) Å(a) ÆÅ l = 2 Õ Ê¯ Òº¿ ¬µ Î¦ ¾ (2) É (a) Õ À¬ (b) ÕÅ¢¶Ô℄ § Æ (a) Æ l = 2 Õ Ê ¸Å× ÕÛ³ » Ä 4-2-5 (4-2-8) Lvωa1 a2 = vb∇b ωa1 a2 + ωba2 ∇a1 vb + ωa1 b ∇a2 vb . ³Ü¦ da1 (vbωba2 ) (5-1-11) = 2∇[a1 (vbω|b|a2 ] ) = ∇a1 (vbωba2 ) − ∇a2 (vbωba1 ) (5-1-1) = ∇a1 (vbωba2 ) + ∇a2 (vbωa1 b ) = ωba2 ∇a1 vb + vb∇a1 ωba2 + ωa1 b ∇a2 vb + vb∇a2 ωa1 b (5-1-1) = ωba2 ∇a1 vb + ωa1 b ∇a2 vb + vb∇a1 ωba2 − vb∇a2 ωba1 = ωba2 ∇a1 vb + ωa1 b ∇a2 vb + 2vb∇[a1 ω|b|a2 ] , ³Ü (dω)ba1 a2 vb (5-1-11) = 3∇[b ωa1 a2 ] vb = 3vb∇[b ωa1 a2 ] = 3 1 3! vb ∇b ωa1 a2 + ∇a1 ωa2 b + ∇a2 ωba1 − ∇b ωa2 a1 − ∇a2 ωa1 b − ∇a1 ωba2 (5-1-1) = 1 2 vb 2∇b ωa1 a2 − 2∇a1 ωba2 + 2∇a2 ωba1 = vb∇b ωa1 a2 − 2vb∇[a1 ω|b|a2 ] . Í ³= ωba2 ∇a1 vb + ωa1 b ∇a2 vb + vb∇b ωa1 a2 = Û³. Å ¦ l ¾ ÖÆÛ³ » Ä 4-2-5 (4-2-8) Lvωa1 ···al = vb∇b ωa1 ···al + l j=1 ωa1 ···aj−1 baj+1 ···a2 ∇aj vb . ³Ü¦ da1 (vbωba2 ···al ) (5-1-11) = l ∇[a1 (vbω|b|a2 ···al] ) . Ò [a1a2 · · · al ] = 1 l! π δπaπ(1) aπ(2) · · · aπ(l)

61 = 1 l a1[a2a3 · · · al ]
− a2[a1a3 · · · al ] + · · · + (−1)l−1al [a1a2 · · · al−1 ] = 1 l l j=1 (−1)j−1aj[a1 · · · aj−1aj+1 · · · al ] , Æ da1 (vbωba2 ···al ) = l ∇[a1 (vbω|b|a2 ···al] ) = l 1 l l j=1 (−1)j−1∇aj (vbωb[a1 ···aj−1 aj+1 ···al] ) = l j=1 (−1)j−1∇aj (vbω[ba1 ···aj−1 aj+1 ···al] ) (5-1-1) = l j=1 ∇aj (vbω[a1 ···aj−1 baj+1 ···al] ) = l j=1 ∇aj (vbωa1 ···aj−1 baj+1 ···al ) = l j=1 ωa1 ···aj−1 baj+1 ···al ∇aj vb + vb l j=1 ∇aj ωa1 ···aj−1 baj+1 ···al . ³Ü (dω)ba1 ···al vb (5-1-11) = (l + 1)∇[b ωa1 a2 ···al] vb = vb(l + 1)∇[b ωa1 a2 ···al] . ²· Á¦¯È Ò [ba1a2 · · · al ] = 1 l + 1 b[a1a2 · · · al ] − a1[ba2 · · · al ] + · · · + (−1)lal [ba1a2 · · · al−1 ] = 1 l + 1 b[a1a2 · · · al ] − 1 l + 1 a1[ba2 · · · al ] − · · · − (−1)lal [ba1a2 · · · al−1 ] = 1 l + 1 b[a1a2 · · · al ] − 1 l + 1 l j=1 (−1)j−1aj[ba1 · · · aj−1aj+1 · · · al ] = 1 l + 1 b[a1a2 · · · al ] − 1 l + 1 l j=1 aj[a1 · · · aj−1baj+1 · · · al ] , Æ (dω)ba1 ···al vb = vb(l + 1)∇[b ωa1 a2 ···al] = vb∇b ω[a1 a2 ···al] − vb l j=1 ∇aj ω[a1 ···aj−1 baj+1 ···al] = vb∇b ωa1 a2 ···al − vb l j=1 ∇aj ωa1 ···aj−1 baj+1 ···al .

62 Å Õ ³ = da1 (vbωba2 ···al ) +
(dω)ba1 ···al vb = l j=1 ωa1 ···aj−1 baj+1 ···al ∇aj vb + vb l j=1 ∇aj ωa1 ···aj−1 baj+1 ···al +vb∇b ωa1 a2 ···al − vb l j=1 ∇aj ωa1 ···aj−1 baj+1 ···al = l j=1 ωa1 ···aj−1 baj+1 ···al ∇aj vb + vb∇b ωa1 a2 ···al = Û³. (b) ¸Å ÕÛ³ Lvdω = Lv(dω)ba1 ···al (a) = db [vc(dω)ca1 ···al ] + [d(dω)]cba1 ···al vc = db [vc(dω)ca1 ···al ] , Ú ¦ Ä 5-1-5 Õ À d ◦ d = 0. ¸Å Õ ³ dLvω (a) = db [da1 (vcωca2 ···al ) + (dω)ca1 ···al vc] = db [vc(dω)ca1 ···al ] , Ç±¡ Ò d ◦ d = 0. Ô Lvdω = dLvω = db [vc(dω)ca1 ···al ] . 7. Î O n M ÉÞ¸Ã {xµ} ÕÞ¸¶ ( ¼ O ± ° RRn), ωa O ÉÕ 1 ¯ÓÅ ∂ωµ/∂xν = ∂ων/∂xµ (µ, ν = 1, · · · , n) ¼ ℄ f : O → RR ∇af = ωa . §5.1 µÑ 5-1-6 ÕÅ¢ Æ (A) [ Ç Ñ] ÁÀ 1 ωa = ∇af = daf ¶ Õ¯ªÛ » Ä 5-1-5 ² ±Õ¯ 0 = db ωa = 2∇[b ωa] = ∇b ωa − ∇aωb , Ø ∇b ωa = ∇aωb . ∇a °Ñ ∂a, ∂b ωa = ∂aωb . Þ¸ÃÕ º µ ∂νωµ = ∂µων, Ø ∂ωµ/∂xν = ∂ων/∂xµ. (B) [ ²¢Ñ] ÁÀ∂νωµ = ∂µων, ° ∂b ωa = ∂aωb Édb ωa = ∇b ωa −∇aωb = 0, Ø ωa ±Õ « RRn Ä 5-1-5 Õ ÄÀË¯Æ ² ¶ Õ¯ Ø º ωa = daf = ∇af. 8. Î {x, y, z} É {r, θ, ϕ} » 3 Ñ ¦ÕØ Þ¸ÃÉ Þ¸Ã¯ Ê dr ∧ dθ ∧ dϕ dx ∧ dy ∧ dz Õº

63 x = r sin θ cos ϕ, y =
r sin θ sin ϕ, z = r cos θ, Ô dx = sin θ cos ϕ dr + r cos θ cos ϕ dθ − r sin θ sin ϕ dϕ , dy = sin θ sin ϕ dr + r cos θ sin ϕ dθ + r sin θ cos ϕ dϕ , dz = cos θ dr − r sin θ dθ . ° dx ∧ dy = sin θ cos ϕ dr ∧ [r cos θ sin ϕ dθ + r sin θ cos ϕ dϕ] +r cos θ cos ϕ dθ ∧ [sin θ sin ϕ dr + r sin θ cos ϕ dϕ] −r sin θ sin ϕ dϕ ∧ [sin θ sin ϕ dr + r cos θ sin ϕ dθ] = r sin θ cos θ sin ϕ cos ϕ dr ∧ dθ + r sin2 θ cos2 ϕ dr ∧ dϕ +r sin θ cos θ sin ϕ cos ϕ dθ ∧ dr + r2 sin θ cos θ cos2 ϕ dθ ∧ dϕ −r sin2 θ sin2 ϕ dϕ ∧ dr − r2 sin θ cos θ sin2 ϕ dϕ ∧ dθ = r sin2 θ dr ∧ dϕ + r2 sin θ cos θ dθ ∧ dϕ , × dx ∧ dy ∧ dz = [r sin2 θ dr ∧ dϕ + r2 sin θ cos θ dθ ∧ dϕ] ∧ [cos θ dr − r sin θ dθ] = −r2 sin3 θ dr ∧ dϕ ∧ dθ + r2 sin θ cos2 θ dθ ∧ dϕ ∧ dr = r2 sin θ dr ∧ dθ ∧ dϕ . Á µ dr ∧ dθ ∧ dϕ = 1 r2 sin θ dx ∧ dy ∧ dz = 1 x2 + y2 + z2 x2 + y2 dx ∧ dy ∧ dz . ˜9. Ï° M ÕÐÑÇ Õ§ gab (spacetime). Î Fab 4 Õ 2 ( Ü 6 ¬ Òß Fab µ ¦¿ 2 ), ÓÅ 1 2 (FacFb c + ∗Fac ∗Fb c) = FacFb c − 1 4 gab Fcd Fcd , ° ∗Fac ≡ (∗F)ac, ∗Fb c = gac ∗Fba ( «Þ³ß ). Æ ÖÆ¯«° 4 ( Ñ ), 2 Fab Õ¾ « 2 ∗Fac (5-6-1) = 1 2 Fdeεdeac , ° ∗Ffc = 1 2 Fgh εghfc É ∗Fb c = gbf ∗Ffc = 1 2 gbf Fgh εghfc .

64 Ýª× Û³ÕÜ ∗Fac ∗Fb c = 1 2 Fdeεdeac
1 2 gbf Fgh εghfc = 1 4 gbf FdeFgh [εghfcεdeac ] (5-4-10) = 1 4 gbf FdeFgh [(−1)1(4 − 3)!3! δ[g d δh eδf] a] = − 3 2 FdeFgh gfb δ[g d δh eδf] a (2-6-19) = − 3 2 FdeF[gh gf]b δg d δh eδf a = − 3 2 FdeF[de ga]b = − 3 2 Fde 1 3 (F[de] gab + F[ad] geb + F[ea] gdb ) (2-6-20) = − 1 2 Fde(Fde gab + Fad geb + Feagdb ) = − 1 2 gab Fde Fde − 1 2 Fad Fd b − 1 2 FeaFb e = − 1 2 gab Fde Fde + 1 2 Fad Fb d + 1 2 FaeFb e = − 1 2 gab Fcd Fcd + FacFb c . ¬ À Â Å× ÕÛ³ 1 2 FacFb c − 1 2 gab Fcd Fcd + FacFb c = FacFb c − 1 4 gab Fcd Fcd . × ÀË *10. ÓÅ ˆ εa1 ···an−1 ≡ ±nbˆ εba1 ···an−1 © εba1 ···an−1 ª ∂N É² Ñ§ hab ÍÐ Õ ¹ Æ Ø¢Å¢ ˆ εa1 ···an−1 = ±nbεba1 ···an−1 Ø× (5-5-5): ˆ εa1 ···an−1 ˆ εa1 ···an−1 = ha1 b1 · · · han−1 bn−1 ˆ εb1 ···bn−1 ˆ εa1 ···an−1 = (−1)ˆ s(n − 1)! , ° ˆ s N ÉÕ§ gab ∂N ÉÕ Ñ§ hab = gab ∓ nanb Õ« ¯¹Õ´ Õ¿ ÖÆ ha1 b1 · · · han−1 bn−1 ˆ εb1 ···bn−1 ˆ εa1 ···an−1 = ha1 b1 · · · han−1 bn−1 (±ndεdb1 ···bn−1 )(±ncεca1 ···an−1 ) = ncndha1 b1 · · · han−1 bn−1 εdb1 ···bn−1 εca1 ···an−1 = ncnd(ga1 b1 ∓ na1 nb1 ) · · · (gan−1 bn−1 ∓ nan−1 nbn−1 )εdb1 ···bn−1 εca1 ···an−1 . ÏÀÆ ÊÉ Õ (gaj bj ∓ naj nbj ) Õ naj nbj Ü Ë É¯ Á ² Ò ncndnaj nbj εdb1 ···bn−1 εca1 ···an−1 = ncnaj εcaj ··· ndnbj εdbj ···

65 = n(cnaj)ε[caj···an−1 ] n(dnbj)ε[dbj ···bn−1 ] (2-6-19) = ncnaj
ε[(caj )···an−1 ] ndnbj ε[(dbj )···bn−1 ] (2-6-21) = 0 . ° ha1 b1 · · · han−1 bn−1 ˆ εb1 ···bn−1 ˆ εa1 ···an−1 = ncndga1 b1 · · · gan−1 bn−1 εdb1 ···bn−1 εca1 ···an−1 = ncnd εda1 ···an−1 εca1 ···an−1 (5-4-10) = ncnd (−1)s(n − 1)!δd c = ncnc(−1)s(n − 1)! . (−1)ˆ s ≡ ncnc(−1)s. ÁÀ ∂N Á ¯ ncnc = +1, ˆ s = s; ÁÀ ∂N Á ¯ ncnc = −1, ˆ s = s − 1 ( © §4.4 3 ÕÊÒ). ÔÒ ˆ εa1 ···an−1 ˆ εa1 ···an−1 = (−1)ˆ s(n − 1)! , ÉÕ ¹² ÉÕ Ñ§ ÍÐ 11. ÓÅ Ä 5-6-1 É 5-6-2. Æ Ä 5-6-1 ÕÅ¢ §5.6 « ¾ Õ 1 (5-6-1), « l ω = ωa1 ···al , ∗∗ωa1 ···al = 1 (n − l)! ∗ωb1 ···bn−l εb1 ···bn−la1 ···al = 1 (n − l)! 1 l! ωc1 ···cl εc1 ···clb1 ···bn−l εb1 ···bn−la1 ···al = 1 (n − l)! l! [εc1 ···clb1 ···bn−l εb1 ···bn−la1 ···al ]ωc1 ···cl = 1 (n − l)! l! (−1)l(n−l)[εb1 ···bn−lc1 ···clεb1 ···bn−la1 ···al ]ωc1 ···cl (5-4-10) = 1 (n − l)! l! (−1)l(n−l)[(−1)sl!(n − l)!δ[c1 a1 · · · δcl] al ]ωc1 ···cl = (−1)s+l(n−l) δ[c1 a1 · · · δcl] al ωc1 ···cl (2-6-19) = (−1)s+l(n−l) δc1 a1 · · · δcl al ω[c1 ···cl] = (−1)s+l(n−l) ωa1 ···al , Ø Ä 5-6-1 (5-6-2) ∗∗ω = (−1)s+l(n−l)ω . Ä 5-6-2 ÕÅ¢ Î f É A 3 Ñ ¦ÕÄ É ¯ (a) grad f = df , (b) curl A = ∗dA , (c) div A = ∗d(∗ A) .

66 (a) f 0 ¯ df 1 ¯ (df)a =
∇af = ∂af = (∇f)a = (grad f)a . (b) A 1 ¯ dA 2 ¯ ∗dA 1 º¾ (5-1-11) (dA)ba = db Aa = 2∇[b Aa] = 2∂[b Aa] , ×« ¾ (5-6-1) ∗(dA)c = 1 2! (dA)ba εbac = 1 2 2∂[bAa] εbac = εabc ∂[aAb] = ε[abc] ∂[aAb] (2-6-19) = ε[[ab]c] ∂aAb (2-6-20) = ε[abc] ∂aAb = εabc ∂aAb (5-6-5)(c) = (∇ × A )c = (curl A )c . (c) A 1 ¯ ∗ A 2 ¯ d(∗ A) 3 ¯ ∗d(∗ A) 0 ( ¸ ). ÖÆ Aa Õ« ¾ ∗Abc (5-6-1) = Aaεabc , Õº¾ d(∗A)dbc (5-1-11) = 3∂[d ∗Abc] = 3∂[d Aaε|a|bc] , « ¾ ∗d(∗A) (5-6-1) = 1 3! d(∗A)dbc εdbc = 1 3! 3∂[dAaε|a|bc] εdbc = 1 2 ∂[dAaε|a|bc] εdbc (5-4-6) = 1 2 ∂[d Aaε|a|bc]εdbc (2-6-19) = 1 2 ∂d Aaεabcε[dbc] = 1 2 ∂d Aaεabcεdbc (5-4-10) = 1 2 ∂d Aa2δa d = ∂aAa = ∂aAa (5-6-5)(b) = ∇ · A = div A . ˜12. Î x, y, z 3 Ñ ¦ÕØ Þ¸¯ÓÅ (a) ∗dx = dy ∧ dz ; (b) ∗(dx ∧ dy ∧ dz) = 1 . Æ (a) 1 (dx)a Õ« ¾ ∗(dx)bc (5-6-1) = (∂/∂x)aεabc ,

67 3 Ø Þ¸ÃÕÐ ¹ ( Õ) εabc (5-4-4) =
(dx)a ∧ (dy)b ∧ (dz)c = (dx ∧ dy ∧ dz)abc , ° ∗(dx)bc = (∂/∂x)a(dx ∧ dy ∧ dz)abc (5-1-2) = (∂/∂x)a3!(dx)[a (dy)b (dz)c] . [abc] = 1 3 (a[bc] + b[ca] + c[ab]) , É (∂/∂x)a(dx)a = 1 É (∂/∂x)a(dy)a = (∂/∂x)a(dz)a = 0 Ô ∗(dx)bc = 2(∂/∂x)a (dx)a(dy)[b (dz)c] + (dx)b (dy)[c (dz)a] + (dx)c(dy)[a (dz)b] = 2(dy)[b (dz)c] (5-1-2) = (dy)b ∧ (dz)c = (dy ∧ dz)bc , Ø ∗dx = dy ∧ dz . (b) 3 (dx ∧ dy ∧ dz)abc = (dx)a ∧ (dy)b ∧ (dz)c, ° µ 3 Ø Þ ¸ÃÕÐ Õ ¹ εabc [ © (5-4-4) ]. °« ¾ 0 ∗(dx ∧ dy ∧ dz) (5-6-1) = 1 3! (∂/∂x)a ∧ (∂/∂y)b ∧ (∂/∂z)cεabc (5-4-4) = 1 3! [(∂/∂x)a ∧ (∂/∂y)b ∧ (∂/∂z)c][(dx)a ∧ (dy)b ∧ (dz)c] (5-1-2) = 3![(∂/∂x)[a(∂/∂y)b(∂/∂z)c]][(dx)[a (dy)b (dz)c] ] (2-6-19) = 3![(∂/∂x)a(∂/∂y)b(∂/∂z)c][(dx)[a (dy)b (dz)c] ] = 3![(∂/∂y)b(∂/∂z)c] 1 3 [(dy)[b (dz)c] ] = 2[(∂/∂z)c] 1 2 [(dz)c] = 1 . ° É (5-4-3) Õ ÀÈ ÔÒ¯ «°ÃÆ ¦ÕØ Ã (∂/∂x)a ∧ (∂/∂y)b ∧ (∂/∂z)c = εabc, ∗(dx ∧ dy ∧ dz) = 1 3! εabcεabc (5-4-3) = 1 3! 3! = 1 . 13. Î {r, θ, ϕ} 3 Ñ ¦Õ Þ¸Ã¯ÓÅ ∗dr = (r2 sin θ) dθ ∧ dϕ . Æ ÖÆ¯ » (5-4-4), 3 Ñ ¦ Þ¸ÃÕ ÕÐ ¹ ( g = det(gµν) = r4 sin2 θ) εabc = r2 sin θ (dr)a ∧ (dθ)b ∧ (dϕ)c = r2 sin θ (dr ∧ dθ ∧ dϕ)abc .

68 1 (dr)a Õ« ¾ 2: ∗(dr)bc (5-6-1) = (∂/∂r)a
εabc = (∂/∂r)a r2 sin θ (dr)a ∧ (dθ)b ∧ (dϕ)c = r2 sin θ (dθ)b ∧ (dϕ)c = r2 sin θ (dθ ∧ dϕ)bc , Ø ∗dr = (r2 sin θ) dθ ∧ dϕ . 14. Î A, B RR3 ÉÕ ¯∇ RR3 É² Ñ§ ÍÐ ÕÑ ª±¯ ÓÅ ∇ × (A × B ) = (B · ∇)A + (∇ · B)A − (A · ∇)B − (∇ · A)B . Æ É (A × B)c = εcabAaBb É (∇ × B)c = εcab∂aBb , É Û³ [∇ × (A × B )]c = εcab∂a(A × B )b = εcab∂a(εbde AdBe) (5-4-6) = εcabεbde ∂a(AdBe) = εbcaεbde ∂a(AdBe) (5-4-10) = 2δ[c d δa] e∂a(AdBe) = (δc d δa e − δa d δc e)∂a(AdBe) = ∂a(AcBa) − ∂a(AaBc) = Ba∂aAc + (∂aBa)Ac − Aa∂aBc − (∂aAa)Bc = (B · ∇)Ac + (∇ · B)Ac − (A · ∇)Bc − (∇ · A)Bc = [(B · ∇)A + (∇ · B)A − (A · ∇)B − (∇ · A)B]c Ø É ³ 15. ¾ Å¢ 3 Ñ ¦ Ñ ¾ ÆÅÕÅ℄ÜÆ£ (1) Ø ² º § (2) Æ ² º Ø§ ( © §5.6 ¥). Æ(1) Ä5-6-2 Õcurl A = ∗dA Æ¯ÁÀ curl A = 0, ∗dA = 0, dA = 0, Ø1 A ±Õ «° RR3, ± ²¶ ( Ø Ä 5-1-5 Õ ÀË), A ² ¦ 0 ( ¸ ) φ Õº¾ ¯Ø Aa = Aa = daφ (5-1-11) = ∇aφ = ∂aφ , Ø A = ∇φ. (2) Ä 5-6-2 Õ div A = ∗d(∗ A) Æ¯ÁÀ div A = 0, ∗d(∗ A) = 0, d(∗ A) = 0, Ø 2 ∗ A ±Õ «° RR3, ± ²¶ ( Ø Ä 5-1-5 Õ ÀË), ∗ A ² ¦ 1 Ba Õº¾ ¯Ø ∗Abc = db Bc (5-1-11) = 2∇[b Bc] = 2∂[b Bc] ,

69 ° ∗∗Aa (5-6-1) = 1 2 ∗Abc εbca =
∂[b Bc] εbca (2-6-19) = ∂b Bcε[bc]a (2-6-20) = ∂b Bcεbca (5-6-5)(c) = (∇ × B )a . Ú ¯ Ä 5-6-1 (5-6-2) Æ ∗∗Aa = (−1)0+1(3−1)Aa = Aa , Õ A = ∇ × B . 16. Î ∇a ÂÙ ¦ (M, gab ) ÉÕÐ Ñ ª± ( Ø ∇agbc = 0), ε Ð ¹ ( Ø ∇aεb1 ···bn = 0), va M É ¯ va ≡ gab vb va Í Õ 1 ¯ ∗v va Õ« ¯ÓÅ (∇ava)ε = d∗v. ¿ Ñ ÜÁÅµ Î Fa1 ···ak k (k ≤ n), §ÞÝ F , k − 1 ∇ak Fa1 ···ak ÞÝ div F , ∗(div F ) = d∗F . ß Õ×Ñ Ã [ (12-6-2)] µ ¦Ê Æ 1 va Õ« ¾ [(n − 1) ] ∗vb1 ···bn−1 (5-6-1) = vcεcb1 ···bn−1 . Õº¾ (n ) da ∗vb1 ···bn−1 = (d ∗v)ab1 ···bn−1 (5-1-11) = n∇[a ∗vb1 ···bn−1 ] = n∇[a (vcε|c|b1 ···bn−1 ] ) . n Õ n ÕÖÍ 1 ¦¯ ² εab1 ···bn−1 ·Í ¦¿ Ò ( Î h), Ø n∇[a (vcε|c|b1 ···bn−1 ] ) = hεab1 ···bn−1 . εab1 ···bn−1 ¾ ¯ ³Ô h(−1)sn! [ Ð ¹ÕÑ (5-4-3)], Û³ nεab1 ···bn−1 ∇[a (vcε|c|b1 ···bn−1 ] ) (2-6-19) = nε[ab1 ···bn−1 ]∇a(vcεcb1 ···bn−1 ) = nεab1 ···bn−1 ∇a(vcεcb1 ···bn−1 ) (5-4-6) = nεab1 ···bn−1 εcb1 ···bn−1 ∇avc (5-4-10) = n[(−1)s(n − 1)!δa c]∇avc = (−1)sn!∇cvc . Æ h = ∇cvc. Â· Õ À da ∗vb1 ···bn−1 = (∇cvc)εab1 ···bn−1 ,

70 §ÞÝ d∗v = (∇ava)ε. µ Ò k F =
Fa1 ···ak , °« ¾ (n − k) ∗Fb1 ···bn−k (5-6-1) = 1 k! Fc1 ···ck εc1 ···ckb1 ···bn−k . °º¾ (n − k + 1) (d∗F )ab1 ···bn−k = (d∗F)ab1 ···bn−k = da ∗Fb1 ···bn−k (5-1-11) = (n − k + 1)∇[a ∗Fb1 ···bn−k] = (n − k + 1)∇[a 1 k! Fc1 ···ck ε|c1 ···ck|b1 ···bn−k] = (n − k + 1) k! ∇[a (Fc1 ···ck ε|c1 ···ck|b1 ···bn−k] ) . ¦ ¯(k−1) ∇ck Fc1 ···ck−1 ck = div F Õ« ¾ (n−k+1) ∗(div F )ab1 ···bn−k (5-6-1) = 1 (k − 1)! (∇ck Fc1 ···ck−1 ck )εc1 ···ck−1 ab1 ···bn−k . ¸Å ∗(div F ) = d∗F ØÅ (n − k + 1) k! ∇[a (Fc1 ···ck ε|c1 ···ck|b1 ···bn−k] ) = 1 (k − 1)! (∇ck Fc1 ···ck−1 ck )εc1 ···ck−1 ab1 ···bn−k , Ø (n − k + 1)∇[a (Fc1 ···ck ε|c1 ···ck|b1 ···bn−k] ) = k(∇ck Fc1 ···ck−1 ck )εc1 ···ck−1 ab1 ···bn−k . ¯Ø (n − k + 1)∇[a (Fc1 ···ck εb1 ···bn−k]c1 ···ck ) = k(∇ck Fc1 ···ck−1 ck )εb1 ···bn−kc1 ···ck−1 a . ³Ý εab1 ···bn−kd1 ···dk−1 (n − k + 1)∇[a (Fc1 ···ck εb1 ···bn−k]c1 ···ck εab1 ···bn−kd1 ···dk−1 ) = (n − k + 1)∇a(Fc1 ···ck εb1 ···bn−kc1 ···ck ε[ab1 ···bn−k]d1 ···dk−1 ) = (n − k + 1)∇a(Fc1 ···ck εb1 ···bn−kc1 ···ck εab1 ···bn−kd1 ···dk−1 ) = k(∇ck Fc1 ···ck−1 ck )εb1 ···bn−kc1 ···ck−1 a εab1 ···bn−kd1 ···dk−1 . Ø (n − k + 1)∇a(Fc1 ···ck εb1 ···bn−kc1 ···ck εb1 ···bn−kd1 ···dk−1 a) = k(∇ck Fc1 ···ck−1 ck )εb1 ···bn−kc1 ···ck−1 a εb1 ···bn−kd1 ···dk−1 a .

71 É (5-4-10) εb1 ···bn−kd1 ···dk−1 aεb1 ···bn−kc1 ···ck =
(−1)sk!(n − k)!δ[d1 c1 · · · δdk−1 ck−1 δa] ck , εb1 ···bn−kd1 ···dk−1 aεb1 ···bn−kc1 ···ck−1 a = εab1 ···bn−kd1 ···dk−1 εab1 ···bn−kc1 ···ck−1 = (−1)s(k − 1)!(n − k + 1)!δ[d1 c1 · · · δdk−1 ] ck−1 . Å Û³ (n − k + 1)∇a Fc1 ···ck (−1)sk!(n − k)!δ[d1 c1 · · · δdk−1 ck−1 δa] ck = (−1)sk!(n − k + 1)!δ[d1 c1 · · · δdk−1 ck−1 δa] ck ∇aFc1 ···ck , Å ³ k(∇ck Fc1 ···ck−1 ck )(−1)s(k − 1)!(n − k + 1)!δ[d1 c1 · · · δdk−1 ] ck−1 = (−1)sk!(n − k + 1)!δ[d1 c1 · · · δdk−1 ] ck−1 ∇ck Fc1 ···ck−1 ck . ° Å ¶ δ[d1 c1 · · · δdk−1 ck−1 δa] ck ∇aFc1 ···ck = δ[d1 c1 · · · δdk−1 ] ck−1 ∇ck Fc1 ···ck−1 ck . ¿× Ç Õ¯ Û³ = δd1 [c1 · · · δdk−1 ck−1 δa ck] ∇aFc1 ···ck (2-6-19) = δd1 c1 · · · δdk−1 ck−1 δa ck ∇aF[c1 ···ck] = δd1 c1 · · · δdk−1 ck−1 δa ck ∇aFc1 ···ck = ∇aFd1 ···dk−1 a , ³ = δd1 [c1 · · · δdk−1 ck−1 ] ∇ck Fc1 ···ck−1 ck (2-6-19) = δd1 c1 · · · δdk−1 ck−1 ∇ck F[c1 ···ck−1 ]ck = δd1 c1 · · · δdk−1 ck−1 ∇ck Fc1 ···ck−1 ck = ∇ck Fd1 ···dk−1 ck . £ ÔÅ 17. ÓÅ (5-7-2) Õ Γσ µτ Ã §3.1 Õ Ñ± Γc ab (5-7-2) Í× ÕÞ¸ÑÚÕ Æ Ñ±Õ §3.1 2: ∇aωb = ∂b ωa − Γc ab ωc [ © (3-1-6)]. ÇÕ ωa « Þ¸Ñ (dxν)a, ∇a(dxν)b = ∂b (dxν)a − Γc ab (dxν)c . Ñ (∂/∂xµ)a(∂/∂xν)d ÛÝ (∂/∂xµ)a(∂/∂xν)d∇a(dxν)b = (∂/∂xµ)a(∂/∂xν)d∂b (dxν)a − (∂/∂xµ)a(∂/∂xν)dΓc ab (dxν)c (3-1-10) = (∂/∂xµ)a∂b [(∂/∂xν )d(dxν)a] − (∂/∂xµ)a[(∂/∂xν )d(dxν)c]Γc ab = (∂/∂xµ)a∂b δd a − (∂/∂xµ)aδd cΓc ab = −(∂/∂xµ)aΓd ab ,

72 ÔÒ [ ° » (3-1-10), È ∂b (dxν)a =
0] (∂/∂xµ)aΓd ab = −(∂/∂xµ)a(∂/∂xν)d∇a(dxν)b . Ò 0 = ∇aδd b = ∇a[(∂/∂xν )d(dxν)b ] = (∂/∂xν )d∇a(dxν)b + (dxν)b ∇a(∂/∂xν)d , Æ (∂/∂xµ)aΓd ab = (∂/∂xµ)a(dxν)b ∇a(∂/∂xν )d . Ñ±ÕÞ¸ Γσ µτ = (∂/∂xτ )b(dxσ)d (∂/∂xµ)aΓd ab = (∂/∂xτ )b(dxσ)d (∂/∂xµ)a(dxν)b ∇a(∂/∂xν )d = (dxσ)d (∂/∂xµ)aδν τ ∇a(∂/∂xν)d = (dxσ)d (∂/∂xµ)a∇a(∂/∂xτ )d . ÛÝ (∂/∂xσ)b: (∂/∂xσ)bΓσ µτ = (∂/∂xσ)b(dxσ)d (∂/∂xµ)a∇a(∂/∂xτ )d = δb d (∂/∂xµ)a∇a(∂/∂xτ )d = (∂/∂xµ)a∇a(∂/∂xτ )b . Ø (5-7-2) Õ ( Γσ µτ = Γσ τµ). ©Ü 3 À 4. *18. ÃÆ ¦¸¤ » Ü 3 À 14∼16 Æ § Õ Õ ¸¤ ¯ ¾ ÅÆÔ À² Þ¸ÑÚ ÔÕ Í± ² Rabc d ÕÞ ¸ Rµνσ ρ »¯ Ô Rabc d Õ ¸¤ ©¸ ±Ç R(µ)(ν)(σ) (ρ) º¸¤ (A) À 14. (a) ÙÃÆ ¦¸¤ Ë¹ ds2 = Ω2(t, x)(−dt2 + dx2), ¦Þ¸ÑÚÕ § gtt = −Ω2(t, x) , gxx = Ω2(t, x) ; gtt = −Ω−2(t, x) , gxx = Ω−2(t, x) . § gab = gtt(dt)a(dt)b + gxx(dx)a(dx)b = η00(e0)a(e0)b + η11(e1)a(e1)b , gab = gtt(∂t)a(∂t)b + gxx(∂x)a(∂x)b = η00(e0)a(e0)b + η11(e1)a(e1)b ,

73 ° {(eµ)a} É {(eµ)a } (µ = 0, 1)
ÃÆ ¦ÕÑÚÉ« ÑÚ¯Ø§ −η00 = −η00 = η11 = η11 = 1. Ô (e0)a = Ω−1 (∂t)a , (e1)a = Ω−1 (∂x)a ; (e0)a = Ω (dt)a , (e1)a = Ω (dx)a . gab (eµ)b Ð ηµν (eν)a, Á (e0)a = gab (e0)b = η00(e0)a(e0)b (e0)b = −(e0)a = −Ω (dt)a, Ð (e0)a = η0ν(eν)a = −(e0)a = −Ω (dt)a. (e0)a = −Ω (dt)a , (e1)a = Ω (dx)a . (b) (5-7-19) Ýª Λµνρ É (5-7-20) Ýª ωµνρ. Ò Ã Λµνρ = −Λρνµ, È Í 2 ¿¥Ë Λ001 = −Λ100 É Λ011 = −Λ110. (e0)λ = (e0)a(∂λ )a = −Ω (dt)a(∂λ )a = −Ω δ0 λ , (e1)λ = (e1)a(∂λ )a = Ω (dx)a(∂λ )a = Ω δ1 λ , (e0)λ,τ = ∂τ (−Ω δ0 λ ) = −δ0 λ δ0 τ ˙ Ω − δ0 λ δ1 τ Ω′ , (e1)λ,τ = ∂τ (Ω δ1 λ ) = δ1 λ δ0 τ ˙ Ω + δ1 λ δ1 τ Ω′ . Â (5-7-19) Λµνρ = [(eν)λ,τ − (eν)τ,λ ](eµ)λ(eρ)τ : Λµ0ρ = [(e0)λ,τ − (e0)τ,λ ](eµ)λ(eρ)τ = [−δ0 λ δ0 τ ˙ Ω − δ0 λ δ1 τ Ω′ + δ0 τ δ0 λ ˙ Ω + δ0 τ δ1 λ Ω′](eµ)λ(eρ)τ = − ˙ Ω (eµ)0(eρ)0 − Ω′ (eµ)0(eρ)1 + ˙ Ω (eµ)0(eρ)0 + Ω′ (eµ)1(eρ)0 = −Ω′ (eµ)0(eρ)1 + Ω′ (eµ)1(eρ)0 , Λµ1ρ = [(e1)λ,τ − (e1)τ,λ ](eµ)λ(eρ)τ = [δ1 λ δ0 τ ˙ Ω + δ1 λ δ1 τ Ω′ − δ1 τ δ0 λ ˙ Ω − δ1 τ δ1 λ Ω′](eµ)λ(eρ)τ = ˙ Ω(eµ)1(eρ)0 + Ω′(eµ)1(eρ)1 − ˙ Ω(eµ)0(eρ)1 − Ω′(eµ)1(eρ)1 = ˙ Ω(eµ)1(eρ)0 − ˙ Ω(eµ)0(eρ)1 . ÔÒ Õ Λµνρ Λ001 = −Λ100 = −Ω′ (e0)0(e1)1 = −Ω′ Ω−1Ω−1 = −Ω′ Ω−2 , Λ011 = −Λ110 = − ˙ Ω (e0)0(e1)1 = − ˙ Ω Ω−1Ω−1 = − ˙ Ω Ω−2 . Â (5-7-20) ωµνρ = 1 2 (Λµνρ + Λρµν − Λνρµ) Ô Õ ωµνρ ( Ã¯ µ = ν): ω010 = 1 2 (Λ010 + Λ001 − Λ100) = Λ001 = −Ω′ Ω−2 = −ω100 , ω011 = 1 2 (Λ011 + Λ101 − Λ110) = Λ011 = − ˙ Ω Ω−2 = −ω101 .

74 ÎÖ 1 ωµν = ωµνa = ωµνλ (eλ)a =
ωµνλeλ: ω01 = ω010e0 + ω011e1 = −Ω′ Ω−2e0 − ˙ Ω Ω−2e1 = −Ω′ Ω−2Ω dt − ˙ Ω Ω−2Ω dx = −Ω′ Ω−1 dt − ˙ Ω Ω−1 dx , Í ¦¿ Õ¥Ë¹§ (c) Ü Ã 2 ωµ ν = ηνσωµσ Æ ω0 1 = ω01. Â (5-7-8): Rµ ν = dωµ ν + ωµ λ ∧ ωλ ν, ÔÂÙ 2 R0 1 = dω0 1 + ω0 λ ∧ ωλ 1 = dω01 + 0 = d(−Ω′ Ω−1 dt − ˙ Ω Ω−1 dx) (5-1-12) = − ˙ Ω′Ω − Ω′ ˙ Ω Ω2 dt − Ω′′Ω − Ω′2 Ω2 dx ∧ dt + − ¨ ΩΩ − ˙ Ω2 Ω2 dt − ˙ Ω′Ω − ˙ ΩΩ′ Ω2 dx ∧ dx = − Ω′′Ω − Ω′2 Ω2 dx ∧ dt − ¨ ΩΩ − ˙ Ω2 Ω2 dt ∧ dx = Ω′′Ω − Ω′2 − ¨ ΩΩ + ˙ Ω2 Ω2 dt ∧ dx = Ω′′Ω − Ω′2 − ¨ ΩΩ + ˙ Ω2 Ω4 (Ω dt) ∧ (Ω dx) = Ω′′ − ¨ Ω Ω3 + ˙ Ω2 − Ω′2 Ω4 e0 ∧ e1 ≡ R e0 ∧ e1 . Ø (R0 1)ab = R (e0)a ∧ (e1)b = 2R (e0)[a (e1)b] . ÂÙ ÃÆ ¦¸¤ÑÚÕ Rµν0 1 = (R0 1)ab (eµ)a(eν)b = 2R (e0)[a (e1)b] (eµ)a(eν)b = 2R δ0 [µ δ1 ν] = R (δ0 µδ1 ν − δ0 νδ1 µ) , ÔÂÙ R010 1 = −R100 1 = R = Ω′′ − ¨ Ω Ω3 + ˙ Ω2 − Ω′2 Ω4 . ²Ü 3 À 14 Õ À Rtxt x Õ Ã Rtxt x = Rabc d(∂t)a(∂x)b(∂t)c(dx)d = Rµνσ τ (eµ)a(eν)b (eσ)c(eτ )d(∂t)a(∂x)b(∂t)c(dx)d

75 = Rµνσ τ (eµ)a(eν)b (eσ)c(eτ )dΩ(e0)aΩ(e1)bΩ(e0)cΩ−1(e1)d = Rµνσ τ
Ω2(eµ)a(e0)a(eν)b (e1)b(eσ)c(e0)c(eτ )d(e1)d = Rµνσ τ Ω2δµ 0δν 1δσ 0δ1 τ = R010 1Ω2 = Ω′′ − ¨ Ω Ω + ˙ Ω2 − Ω′2 Ω2 , Ø · °ÁÞ¸ÑÚ Õ§ ÝªÕ À (B) À 15. (a) ÙÃÆ ¦¸¤ Ë¹ ds2 = z−1/2(−dt2 + dz2) + z(dx2 + dy2), ¦ Þ¸ÑÚÕ§ gtt = −z−1/2 , gzz = z−1/2 , gxx = z , gyy = z ; gtt = −z1/2 , gzz = z1/2 , gxx = z−1 , gyy = z−1 . § gab = gtt(dt)a(dt)b + gzz(dz)a(dz)b + gxx(dx)a(dx)b + gyy(dy)a(dy)b = η00(e0)a(e0)b + η33(e3)a(e3)b + η11(e1)a(e1)b + η22(e2)a(e2)b , gab = gtt(∂t)a(∂t)b + gzz(∂z)a(∂z)b + gxx(∂x)a(∂x)b + gyy(∂y)a(∂y)b = η00(e0)a(e0)b + η33(e3)a(e3)b + η11(e1)a(e1)b + η22(e2)a(e2)b , ° {(eµ)a} É {(eµ)a } (µ = 0, 1, 2, 3) ÃÆ ¦ÕÑÚÉ« ÑÚ¯Ø§ ÕÐÑ§ −η00 = −η00 = η11 = η11 = η22 = η22 = η33 = η33 = 1. Ô (e0)a = z1/4 (∂t)a , (e3)a = z1/4 (∂z)a , (e1)a = z−1/2 (∂x)a , (e2)a = z−1/2 (∂y)a ; (e0)a = z−1/4 (dt)a , (e3)a = z−1/4 (dz)a , (e1)a = z1/2 (dx)a , (e2)a = z1/2 (dy)a . gab (eµ)b Ð ηµν (eν)a, Á (e0)a = gab (e0)b = η00(e0)a(e0)b (e0)b = −(e0)a = −z−1/4 (dt)a, Ð (e0)a = η0ν(eν)a = −(e0)a = −z−1/4 (dt)a. (e0)a = −z−1/4 (dt)a, (e3)a = z−1/4 (dz)a, (e1)a = z1/2 (dx)a, (e2)a = z1/2 (dy)a . (b) (5-7-19) Ýª Λµνρ É (5-7-20) Ýª ωµνρ. Ò Ã Λµνρ = −Λρνµ, ÍÔÝª µ = ρ ¾ (e0)λ = (e0)a(∂λ )a = −z−1/4 (dt)a(∂λ )a = −z−1/4 δ0 λ , (e3)λ = (e3)a(∂λ )a = z−1/4 (dz)a(∂λ )a = z−1/4 δ3 λ , (e1)λ = (e1)a(∂λ )a = z1/2 (dx)a(∂λ )a = z1/2 δ1 λ , (e2)λ = (e2)a(∂λ )a = z1/2 (dy)a(∂λ )a = z1/2 δ2 λ ,

76 (e0)λ,τ = ∂τ (−z−1/4 δ0 λ ) = δ0
λ δ3 τ 1 4 z−5/4 , (e3)λ,τ = ∂τ (z−1/4 δ3 λ ) = −δ3 λ δ3 τ 1 4 z−5/4 , (e1)λ,τ = ∂τ (z1/2 δ1 λ ) = δ1 λ δ3 τ 1 2 z−1/2 , (e2)λ,τ = ∂τ (z1/2 δ2 λ ) = δ2 λ δ3 τ 1 2 z−1/2 . Â (5-7-19) Λµνρ = [(eν)λ,τ − (eν)τ,λ ](eµ)λ(eρ)τ : Λµ0ρ = [(e0)λ,τ − (e0)τ,λ ](eµ)λ(eρ)τ = 1 4 z−5/4[δ0 λ δ3 τ − δ0 τ δ3 λ ](eµ)λ(eρ)τ = 1 4 z−5/4[(eµ)0(eρ)3 − (eµ)3(eρ)0] , Λµ3ρ = [(e3)λ,τ − (e3)τ,λ ](eµ)λ(eρ)τ = 1 4 z−5/4[−δ3 λ δ3 τ + δ3 τ δ3 λ ](eµ)λ(eρ)τ = 0 , Λµ1ρ = [(e1)λ,τ − (e1)τ,λ ](eµ)λ(eρ)τ = 1 2 z−1/2[δ1 λ δ3 τ − δ1 τ δ3 λ ](eµ)λ(eρ)τ = 1 2 z−1/2[(eµ)1(eρ)3 − (eµ)3(eρ)1] , Λµ2ρ = [(e2)λ,τ − (e2)τ,λ ](eµ)λ(eρ)τ = 1 2 z−1/2[δ2 λ δ3 τ − δ2 τ δ3 λ ](eµ)λ(eρ)τ = 1 2 z−1/2[(eµ)2(eρ)3 − (eµ)3(eρ)2] . ÔÒ Õ Λµνρ Λ003 = −Λ300 = 1 4 z−5/4 (e0)0(e3)3 = 1 4 z−5/4 z1/4z1/4 = 1 4 z−3/4 , Λµ3ρ = 0 , Λ113 = −Λ311 = 1 2 z−1/2 (e1)1(e3)3 = 1 2 z−1/2 z−1/2z1/4 = 1 2 z−3/4 Λ223 = −Λ322 = 1 2 z−1/2 (e2)2(e3)3 = 1 2 z−1/2 z−1/2z1/4 = 1 2 z−3/4 . Â (5-7-20) ωµνρ = 1 2 (Λµνρ + Λρµν − Λνρµ) Ô Õ ωµνρ ( Ã¯ µ = ν). ÀÆ Ê (µ, ν) = (0, 1) Ü ¯(0, 2) Ü ¯(0, 3) , (1, 2) Ü ¯ (1, 3) , (2, 3) : ω030 = 1 2 (Λ030 + Λ003 − Λ300) = Λ003 = 1 4 z−3/4 = −ω300 ,

77 ω131 = 1 2 (Λ131 + Λ113 − Λ311)
= Λ113 = 1 2 z−3/4 = −ω311 ω232 = 1 2 (Λ232 + Λ223 − Λ322) = Λ223 = 1 2 z−3/4 = −ω322 . ÎÖ 1 ωµν = ωµνa = ωµνλ (eλ)a = ωµνλeλ: ω03 = ω030 e0 = 1 4 z−3/4 e0 = 1 4 z−3/4 z−1/4 dt = 1 4 z−1 dt , ω13 = ω131 e1 = 1 2 z−3/4 e1 = 1 2 z−3/4 z1/2 dx = 1 2 z−1/4 dx , ω23 = ω232 e2 = 1 2 z−3/4 e2 = 1 2 z−3/4 z1/2 dy = 1 2 z−1/4 dy , 3 ¿ Õ¥Ë¹§ (c) Ü Ã 2 ωµ ν = ηνσωµσ Æ ω0 i = ω0i, ωi 0 = −ωi0 = ω0i × ωi j = ωij. Â (5-7-8): Rµ ν = dωµ ν + ωµ λ ∧ ωλ ν, ÔÂÙ 2 Ç Ä 3-4-6 Ñ (4) (3-4-9): Rµν = −Rνµ. Å¢ ÁÅ Rµν = dωµν + ωµ λ ∧ ωλν = −dωνµ − ωλν ∧ ωµ λ = −dωνµ + ωνλ ∧ ωµ λ = −dωνµ + ων λ ∧ ωµλ = −dωνµ − ων λ ∧ ωλµ = −Rνµ. R0 1 = dω0 1 + ω0 λ ∧ ωλ 1 = 0 + ω03 ∧ ω31 = 1 4 z−3/4 e0 ∧ − 1 2 z−3/4 e1 = − 1 8 z−3/2 e0 ∧ e1 , R0 2 = dω0 2 + ω0 λ ∧ ωλ 2 = 0 + ω03 ∧ ω32 = 1 4 z−3/4 e0 ∧ − 1 2 z−3/4 e2 = − 1 8 z−3/2 e0 ∧ e2 , R0 3 = dω0 3 + ω0 λ ∧ ωλ 3 = dω03 + 0 = d 1 4 z−1 dt (5-1-12) = − 1 4 z−2 dz ∧ dt = 1 4 z−2 dt ∧ dz = 1 4 z−2z1/4z1/4 (z−1/4 dt) ∧ (z−1/4 dz) = 1 4 z−3/2 e0 ∧ e3 , R1 0 = dω1 0 + ω1 λ ∧ ωλ 0 = 0 + ω13 ∧ ω03 = 1 2 z−3/4 e1 ∧ 1 4 z−3/4 e0

78 = 1 8 z−3/2 e1 ∧ e0 = −
1 8 z−3/2 e0 ∧ e1 = R0 1 , R1 2 = dω1 2 + ω1 λ ∧ ωλ 2 = 0 + ω13 ∧ ω32 = 1 2 z−3/4 e1 ∧ − 1 2 z−3/4 e2 = − 1 4 z−3/2 e1 ∧ e2 , R1 3 = dω1 3 + ω1 λ ∧ ωλ 3 = dω13 + 0 = d 1 2 z−1/4 dx (5-1-12) = − 1 8 z−5/4 dz ∧ dx = 1 8 z−5/4 dx ∧ dz = 1 8 z−5/4z−1/2z1/4 (z1/2 dx) ∧ (z−1/4 dz) = 1 8 z−3/2 e1 ∧ e3 , R2 0 = dω2 0 + ω2 λ ∧ ωλ 0 = 0 + ω23 ∧ ω03 = 1 2 z−3/4 e2 ∧ 1 4 z−3/4 e0 = 1 8 z−3/2 e2 ∧ e0 = − 1 8 z−3/2 e0 ∧ e2 = R0 2 , R2 1 = dω2 1 + ω2 λ ∧ ωλ 1 = 0 + ω23 ∧ ω31 = 1 2 z−3/4 e2 ∧ − 1 2 z−3/4 e1 = − 1 4 z−3/2 e2 ∧ e1 , R2 3 = dω2 3 + ω2 λ ∧ ωλ 3 = dω23 + 0 = d 1 2 z−1/4 dy (5-1-12) = − 1 8 z−5/4 dz ∧ dy = 1 8 z−5/4 dy ∧ dz = 1 8 z−5/4z−1/2z1/4 (z1/2 dy) ∧ (z−1/4 dz) = 1 8 z−3/2 e2 ∧ e3 R3 0 = dω3 0 + ω3 λ ∧ ωλ 0 = dω03 + 0 = R0 3 , R3 1 = −R1 3 , R3 2 = −R2 3 . Ò (Rσ τ )ab = R(σ, τ) (eσ)a ∧ (eτ )b = 2R(σ, τ) (eσ)[a (eτ )b] . ÂÙ ÃÆ ¦¸¤ÑÚÕ Rµνσ τ = (Rσ τ )ab (eµ)a(eν)b

79 = 2R(σ, τ) (eσ)[a (eτ )b] (eµ)a(eν)b = 2R(σ,
τ) δσ [µ δτ ν] = R(σ, τ) (δσ µδτ ν − δσ νδτ µ) , ° Ô ÂÙ R010 1 = −R100 1 = R(0, 1) = − 1 8 z−3/2 , R020 2 = −R200 2 = R(0, 2) = − 1 8 z−3/2 , R030 3 = −R300 3 = R(0, 3) = 1 4 z−3/2 , R121 2 = −R211 2 = R(1, 2) = − 1 4 z−3/2 , R131 3 = −R311 3 = R(1, 3) = 1 8 z−3/2 , R232 3 = −R322 3 = R(2, 3) = 1 8 z−3/2 . ²Ü 3 À 15 Õ À R(µ)(ν)(σ) (τ) Õ Ã Rtxt x = Rabc d(∂t)a(∂x)b(∂t)c(dx)d = Rµνσ τ (eµ)a(eν)b (eσ)c(eτ )d(∂t)a(∂x)b(∂t)c(dx)d = Rµνσ τ (eµ)a(eν)b (eσ)c(eτ )dz−1/4(e0)az1/2(e1)bz−1/4(e0)cz−1/2(e1)d = Rµνσ τ z−1/2(eµ)a(e0)a(eν)b (e1)b(eσ)c(e0)c(eτ )d(e1)d = Rµνσ τ z−1/2δµ 0δν 1δσ 0δ1 τ = R010 1z−1/2 = − 1 8 z−3/2z−1/2 = − 1 8z2 , Rtyt y = Rabc d(∂t)a(∂y)b(∂t)c(dy)d = Rµνσ τ (eµ)a(eν)b (eσ)c(eτ )d(∂t)a(∂y)b(∂t)c(dy)d = Rµνσ τ (eµ)a(eν)b (eσ)c(eτ )dz−1/4(e0)az1/2(e2)bz−1/4(e0)cz−1/2(e2)d = Rµνσ τ z−1/2(eµ)a(e0)a(eν)b (e2)b(eσ)c(e0)c(eτ )d(e2)d = Rµνσ τ z−1/2δµ 0δν 2δσ 0δ2 τ = R020 2z−1/2 = − 1 8 z−3/2z−1/2 = − 1 8z2 , Rtzt z = Rabc d(∂t)a(∂z)b(∂t)c(dz)d = Rµνσ τ (eµ)a(eν)b (eσ)c(eτ )d(∂t)a(∂z)b(∂t)c(dz)d = Rµνσ τ (eµ)a(eν)b (eσ)c(eτ )dz−1/4(e0)az−1/4(e3)bz−1/4(e0)cz1/4(e3)d = Rµνσ τ z−1/2(eµ)a(e0)a(eν)b (e3)b(eσ)c(e0)c(eτ )d(e3)d = Rµνσ τ z−1/2δµ 0δν 3δσ 0δ3 τ = R030 3z−1/2 = 1 4 z−3/2z−1/2 = 1 4z2 ,

80 Rxyx y = Rabc d(∂x)a(∂y)b(∂x)c(dy)d = Rµνσ τ (eµ)a(eν)b
(eσ)c(eτ )d(∂x)a(∂y)b(∂x)c(dy)d = Rµνσ τ (eµ)a(eν)b (eσ)c(eτ )dz1/2(e1)az1/2(e2)bz1/2(e1)cz−1/2(e2)d = Rµνσ τ z(eµ)a(e1)a(eν)b (e2)b(eσ)c(e1)c(eτ )d(e2)d = Rµνσ τ zδµ 1δν 2δσ 1δ2 τ = R121 2z = − 1 4 z−3/2z = − 1 4z1/2 , Rxzx z = Rabc d(∂x)a(∂z)b(∂x)c(dz)d = Rµνσ τ (eµ)a(eν)b (eσ)c(eτ )d(∂x)a(∂z)b(∂x)c(dz)d = Rµνσ τ (eµ)a(eν)b (eσ)c(eτ )dz1/2(e1)az−1/4(e3)bz1/2(e1)cz1/4(e3)d = Rµνσ τ z(eµ)a(e1)a(eν)b (e3)b(eσ)c(e1)c(eτ )d(e3)d = Rµνσ τ zδµ 1δν 3δσ 1δ3 τ = R131 3z = 1 8 z−3/2z = 1 8z1/2 , Ryzy z = Rabc d(∂y)a(∂z)b(∂y)c(dz)d = Rµνσ τ (eµ)a(eν)b (eσ)c(eτ )d(∂y)a(∂z)b(∂y)c(dz)d = Rµνσ τ (eµ)a(eν)b (eσ)c(eτ )dz1/2(e2)az−1/4(e3)bz1/2(e2)cz1/4(e3)d = Rµνσ τ z(eµ)a(e2)a(eν)b (e3)b(eσ)c(e2)c(eτ )d(e3)d = Rµνσ τ zδµ 2δν 3δσ 2δ3 τ = R232 3z = 1 8 z−3/2z = 1 8z1/2 . Ã · °ÁÞ¸ÑÚ Õ§ ÝªÕ À (C) À 16. (a) ÙÃÆ ¦¸¤ Ë¹ ds2 = −dt2 +dx2 +dy2 +h2dz2, ° h = t+α(z)x+ β(z)y + γ(z), ¦Þ¸ÑÚÕ§ gtt = −1 , gzz = h2 , gxx = 1 , gyy = 1 ; gtt = −1 , gzz = h−2 , gxx = 1 , gyy = 1 . § gab = gtt(dt)a(dt)b + gzz(dz)a(dz)b + gxx(dx)a(dx)b + gyy(dy)a(dy)b = η00(e0)a(e0)b + η33(e3)a(e3)b + η11(e1)a(e1)b + η22(e2)a(e2)b , gab = gtt(∂t)a(∂t)b + gzz(∂z)a(∂z)b + gxx(∂x)a(∂x)b + gyy(∂y)a(∂y)b = η00(e0)a(e0)b + η33(e3)a(e3)b + η11(e1)a(e1)b + η22(e2)a(e2)b , ° {(eµ)a} É {(eµ)a } (µ = 0, 1, 2, 3) ÃÆ ¦ÕÑÚÉ« ÑÚ¯Ø§ ÕÐÑ§ −η00 = −η00 = η11 = η11 = η22 = η22 = η33 = η33 = 1.

81 Ô (e0)a = (∂t)a , (e3)a = h−1 (∂z)a
, (e1)a = (∂x)a , (e2)a = (∂y)a ; (e0)a = (dt)a , (e3)a = h (dz)a , (e1)a = (dx)a , (e2)a = (dy)a . gab (eµ)b Ð ηµν (eν)a, Á (e0)a = gab (e0)b = η00(e0)a(e0)b (e0)b = −(e0)a = −(dt)a, Ð (e0)a = η0ν(eν)a = −(e0)a = −(dt)a. (e0)a = −(dt)a, (e3)a = h (dz)a, (e1)a = (dx)a, (e2)a = (dy)a . (b) (5-7-19) Ýª Λµνρ É (5-7-20) Ýª ωµνρ. Ò Ã Λµνρ = −Λρνµ, ÍÔÝª µ = ρ ¾ (e0)λ = (e0)a(∂λ )a = −(dt)a(∂λ )a = −δ0 λ , (e3)λ = (e3)a(∂λ )a = h (dz)a(∂λ )a = h δ3 λ , (e1)λ = (e1)a(∂λ )a = (dx)a(∂λ )a = δ1 λ , (e2)λ = (e2)a(∂λ )a = (dy)a(∂λ )a = δ2 λ , (e0)λ,τ = ∂τ (δ0 λ ) = 0 , (e3)λ,τ = ∂τ (h δ3 λ ) = δ3 λ (δ0 τ ht + δ3 τ hz + δ1 τ hx + δ2 τ hy) = hσ δ3 λ δσ τ , (e1)λ,τ = ∂τ (δ1 λ ) = 0 , (e2)λ,τ = ∂τ (δ2 λ ) = 0 , ° h = t + α(z)x + β(z)y + γ(z), ht ≡ h0 = 1 , hz ≡ h3 = α′(z)x + β′(z)y + γ′(z) , hx ≡ h1 = α(z) , hy ≡ h2 = β(z) . Â (5-7-19) Λµνρ = [(eν)λ,τ − (eν)τ,λ ](eµ)λ(eρ)τ : Λµ0ρ = [(e0)λ,τ − (e0)τ,λ ](eµ)λ(eρ)τ = 0 , Λµ3ρ = [(e3)λ,τ − (e3)τ,λ ](eµ)λ(eρ)τ = hσ [δ3 λ δσ τ − δ3 τ δσ λ ](eµ)λ(eρ)τ = hσ [(eµ)3(eρ)σ − (eµ)σ(eρ)3] , Λµ1ρ = [(e1)λ,τ − (e1)τ,λ ](eµ)λ(eρ)τ = 0 , Λµ2ρ = [(e2)λ,τ − (e2)τ,λ ](eµ)λ(eρ)τ = 0 .

82 Í Λµ3ρ Λ33σ = −Λσ33 = hσ (e3)3(eσ)σ =
hσ h−1(eσ)σ = h−1hσ (eσ)σ , (σ É, σ = 3) Ø Λ330 = −Λ033 = h−1h0 (e0)0 = h−1ht , Λ331 = −Λ133 = h−1h1 (e1)1 = h−1hx , Λ332 = −Λ133 = h−1h2 (e2)2 = h−1hy . Â (5-7-20) ωµνρ = 1 2 (Λµνρ + Λρµν − Λνρµ) Ô Õ ωµνρ ( Ã¯ µ = ν). ÀÆ Ê (µ, ν) = (0, 1) Ü ¯(0, 2) Ü ¯(0, 3) ¯ (1, 2) Ü ¯ (1, 3) ¯ (2, 3) ω033 = 1 2 (Λ033 + Λ303 − Λ330) = Λ033 = −h−1ht = −ω330 , ω133 = 1 2 (Λ133 + Λ313 − Λ331) = Λ133 = −h−1hx = −ω331 ω233 = 1 2 (Λ233 + Λ323 − Λ332) = Λ233 = −h−1hy = −ω322 . ÎÖ 1 ωµν = ωµνa = ωµνλ (eλ)a = ωµνλeλ: ω03 = ω033 e3 = −h−1ht e3 = −h−1ht h dz = −dz , ω13 = ω133 e3 = −h−1hx e3 = −h−1hx h dz = −α(z) dz , ω23 = ω233 e3 = −h−1hy e3 = −h−1hy h dz = −β(z) dz , 3 ¿ Õ¥Ë¹§ (c) Ü Ã 2 ωµ ν = ηνσωµσ Æ ω0 i = ω0i, ωi 0 = −ωi0 = ω0i × ωi j = ωij. Â (5-7-8): Rµ ν = dωµ ν + ωµ λ ∧ ωλ ν, ÔÂÙ 2 Ç Ä 3-4-6 Ñ (4) (3-4-9): Rµν = −Rνµ. Å¢ ÁÅ Rµν = dωµν + ωµ λ ∧ ωλν = −dωνµ − ωλν ∧ ωµ λ = −dωνµ + ωνλ ∧ ωµ λ = −dωνµ + ων λ ∧ ωµλ = −dωνµ − ων λ ∧ ωλµ = −Rνµ. R0 1 = dω0 1 + ω0 λ ∧ ωλ 1 = 0 + ω03 ∧ ω31 = (−dz) ∧ (α dz) = 0 ,

83 R0 2 = dω0 2 + ω0 λ ∧
ωλ 2 = 0 + ω03 ∧ ω32 = (−dz) ∧ (β dz) = 0 , R0 3 = dω0 3 + ω0 λ ∧ ωλ 3 = dω03 + 0 = d(−dz) = 0 , ( Ä 5-1-5) R1 2 = dω1 2 + ω1 λ ∧ ωλ 2 = 0 + ω13 ∧ ω32 = (−α dz) ∧ (β dz) = 0 , R1 3 = dω1 3 + ω1 λ ∧ ωλ 3 = dω13 + 0 = d(−α dz) (5-1-12) = −α′ dz ∧ dz = 0 , R2 3 = dω2 3 + ω2 λ ∧ ωλ 3 = dω23 + 0 = d(−β dz) (5-1-12) = −β′ dz ∧ dz = 0 . ÆÓ Æ²· °ÁÞ¸ÑÚ Õ§ ÝªÕ À Í± « 6 Ã¤ Æ³¦ ˜1. Ñ G É G′ Í«¨ u = 0.6c, Í· ¦ ¤ ´ Ñ (a) G ÆÝÕ Ñ Ã ( °± ª), G ¦ 5µs ¯G′ Õ¦ Ë (b) G ¦ 5µs ¯¯ © G′ Õ¦ Ë γ = (1 − u2)−1/2 = 1.25. ´ ÕÚË ÃÆ (a) lob = l2 oa − l2 ab = l2 oa − (uloa)2 = √ 1 − u2 loa = γ−1loa = 5/1.25 = 4µs. G ¦ 5µs ¯ G′ Õ¦ 4µs. (b) loc = γ−1lod = γ−1(loa − lad ) = γ−1(loa − lcd ) = γ−1(loa − ulod ) = γ−1(loa − uγloc). Ô loc = (1 + u)−1γ−1loa = 1−u 1+u loa = 1−0.6 1+0.6 × 5 = 2.5µs. G ¦ 5µs ¯¯ © G′ Õ¦ 2.5µs.

84 ˜2. ½ 0.8c Õ¨ ( ¿¨ÈËÛ) Ã ¯
Ô °Ì ÕÈ 5 ¥Õ ¶ ´ É ÔÕÈ »É (b) Õ À ÆÓ lob = (1 + v)−1γ−1loa, lod = (1 + v)−1γ−1lob , Í¨ØÔ ∆t′ = lbd = lod − lob = (1+ v)−1γ−1(loc − loa) = (1+ v)−1γ−1lac = (1 + v)−1γ−1∆t = 1−u 1+u ∆t = 1−0.8 1+0.8 × 5 = 5 3 ¥ ÇÕ Ò (1 + u)−1γ−1 = 1−u 1+u £ °ÁÕÐÑ¶ ÁÅ Ô Î ºÞ Ã ( É ) Í Õ ¦ t ¯ ½Ã vt, ÉÕ ÖÁ γ−1t. Ò ¦ §² vt/c = vt ¦Ç Õ ÖÁ t + vt = (1 + v)t £ Ò¥ Õ ÖÁ γ−1t. ¾ ¡ Õ ÖÁ t £ Ò¥ ÉÕ ÖÁ (1 + u)−1γ−1t. ˜3. ´ 6-20 Õ oa ©É oe ©Ë »ÞÝ τ É τ′. (a) ÕÍ«¨ u º Ê τ′/τ; (b) u = 0.6c É u = 0.8c ¾ Å Ê τ′/τ Õ Ê (a) loa = lob − lab = lob − lbe = lob − ulob = (1 − u)lob = (1 − u)γloe = (1 + u)−1/2loe, Ø τ = (1 + u)−1/2τ′, τ′/τ = (1 + u)1/2. (b) u = 0.6c ¯ τ′/τ = (1 + 0.6)1/2 = 1.265; u = 0.8c ¯ τ′/τ = (1 + 0.8)1/2 = 1.342. ¿ Ê ¿ÕÊ √ 2 = 1.414. 4. Ñ ÞA, B, C À¦ÈË¾ ËÍ« £ ( ©´6-42), Í«¨ uBA = 0.6c, uCA = 0.8c, A, B Æ ÑÃ RA É RB. Î RB Ã ( Ô) C Ö 60m, Ê ´¾ RA ( Ô) ¦ÁÃÕ ¦ 1. À´ ÕÚËµ ¯ Õ ªÆ loa, Ê lod = lfb . uB ≡ uBA, uC ≡ uCA. ÖÆ Ã lof = lfb uC, Ø log + lgf = (lfe + leb )uC, Æ loaγB + lab γBuB = (loaγBuB + lab γB)uC , Ô Ã loa lab = uC − uB 1 − uCuB . ° µ Í«Ñ¨§ ° lod = lfb = lfe + leb = loaγBuB + lab γB = γB(loauB + lab ) = γB loauB + loa 1 − uCuB uC − uB = loaγB 1 − u2 B uC − uB = loa 1 − u2 B uC − uB . 2. A É B Ç¦ÕÍ«¨ vA = uBA = uB, B É C Ç¦ÕÍ«¨ vC = uCA−uBA 1−uCAuBA = uC−uB 1−uC uB . Õ ªÆ loa, Ê lod . ÖÆ

85 Ã ldf = lof vA = lod γAvA, lof
= loc + lce + lef = lab + lcb vA + ldf vA = loa/vC + loavA + (lod γAvA)vA = lod γA , Ô lod = loaγA 1 vC + vA = loaγB 1 − uCuB uC − uB + uB = loa 1 − u2 B uC − uB . ² 1 Õ ÀÍ± ¯Ú Õ RA Ô ¦ÁÃÕ ¦ 60 × √ 1−0.62 0.8−0.6 = 240m/c . ˜5. A, B ±¦ ÑÃÕ ¿ Ñ ¯¯ Í Ì Ò¯Þ¦ ÒÍ« ¨ 0.6c Ã Ò¸ Î B Ô B ¸Õ Ò Ì 104s−1 ( ØÞ¡ 104 ¿), A Æ Ò ( » ÒÓÜÕ¸ ) ÔÕ B ¸Õ Ò Ì ( ¢ ´ ). ´ Ê loa É lob Ç¦Õ Ã loc = loaγA , loe = lob γB ; ldc = lcavA = (locvA)vA = loaγAv2 A , led = lbe vA = (loevB)vA = lob γBvAvB . loc = loe + led + ldc = lob γB + lob γBvAvB + loaγAv2 A = loaγA , Ô loa = lob γB(1 + vAvB) γA(1 − v2 A ) = lob γAγB(1 + vAvB) . «° vA = vB = v = 0.6c, γA = γB = γ = 1.25, loa = lob 1+v2 1−v2 . A Æ Ò ÔÕ B ¸Õ Ò Ì 104 × 1−0.62 1+0.62 = 4.71×103 s−1. ÆÔ ÀÕ« ÆÓ¯B Æ Ò ÔÕ A ¸Õ Ò Ì £ 4.71 × 103 s−1. ˜6. ±Ì µ ÒÕ Ø£ τ0 = 2 × 10−6s. ´ Ë Õ µ ÒÍ«°Û 0.995c Õ¨ ¿¨ÈËÅÔ¯ ´ Û ÔÕ (a) µ ÒÕ Ø £ (b) µ Ò ° Ø£ÆÆÖÁÕ½Ã lob = loaγ, loc = lcav = lob v = loaγv. (a) µ ÒÕ Ø£ τ = lob = loaγ = τ0γ = 2 × 10−6 × (1 − 0.9952)−1/2 = 2.00 × 10−5s. (b) µ Ò ° Ø£ ÆÆÖÁÕ½Ã loc = loaγv = τv = 1.99 × 10−5s ×c.

86 7. ÑÃ R ( ¯ Ô), °¦Û A Õ
¸ ¡«Ë ¨ v = 0.6c ¦±Ü¿¨ÈË £ Ë 1s Ò ¦Û B. Ò B ËØ¨ v A Û¿¨ ¯« B Û 1s ( ¯Õ ) ¨ v A Û¿¨ ¼ ¦È A Û¯¼ ¡«Ã A Û £Ë ¯(a) Ê ¡«¡¼Õ Ë (b) « A Û Å Õ¦ τ ¡, τ ¬ É τ ½. γ = 1.25. « Õ¦ τ ¬ = loa + lac + lcd = 1 + 1 + 1 = 3s. Õ¦ τ ¡ = loa + lab + lbd = loa + lab + lac = 1 + 1 + 1 = 3s. ¼ Õ¦ τ ½ = lod = loe + lef + lfd = loaγ + lac + lcd γ = 2 × 1.25 + 1 = 3.5 s . ˜8. ( Ù ) Ò A, B ±Ì°¦ ÑÃ R Õ±¦ ¦ÞÉ A ¦ ( A, B Í×) ¢¨ u Í«° R ÃÜ Ñ £¯¦© ¦ B ¨ v > u ¢ É A, Í· A Õ (1) B ¯ (2) B ¯ (3) ² B × (1) B A Õ ¦ loa, B Õ ¦ lob + lba . Á Ë ÛË ( ÈË) Ú ¯ loa > lob + lba . Å Å¢ ¦ × Ã Ò lca = locu = loaγuu, lba = lbc /γv = lca/vγv = loaγuu/vγv. º lob = loc − lbc = loaγu − lca/v = loaγu − loaγuu/v. ° lob + lba = loaγu − loaγuu/v + loaγuu/vγv = loaγu(1 − u/v + u/vγv) Å¢ v > u γu(1 − u/v + u/vγv) < 1. Ø 1 − u v (1 − 1 − v2 ) < 1 − u2 . Û³ Ã¯ Ô 1 + u2 v2 (1 − 1 − v2 )2 − 2 u v (1 − 1 − v2 ) < 1 − u2 , Ø 1 + u2 v2 (2 − v2 − 2 1 − v2) − 2 u v (1 − 1 − v2 ) = 1 + 2u2 v2 − u2 − 2u2 v2 1 − v2 − 2u v + 2u v 1 − v2 = (1 − u2) − 2u v 1 − u v − 2u v 1 − u v 1 − v2 < (1 − u2) . ©· × v > u ÀË (v = u ¶ × ).

87 ˜9. ¸ A, B ±Ì°¦ ÑÃ Õ±¦ ¦ÞÉ A
¦ ¨ u = 0.6c ¿¨ÈË Ê¯2s( » A ) u = 0.6c ¿¨ÈË Å ªÆ Õ Ë (1) Õ¦ (2) A Ë 3s A © B Ë Ë γ = 1.25. (1) lob = loc + lcb = loaγ + lab γ, A Õ¦ loa + lab = 2 + 2 = 4 s, B Õ¦ lob = 2 × 1.25 + 2 × 1.25 = 5 s. (2) A 3s (d Þ) Ò B Ë Õ lof , Ê lof = lob − leb − lfe = lob − ldb γ − led = lob − leb − led = lob − ldb γ − ldb γu = (loa + lab )γ − 1 2 lab γ(1 + u) = 4 × 1.25 − 1 2 × 2 × 1.25 × 1.6 = 3 s . A Ë 3s A © B £»ÆË 3s. ˜10. Û Ó Ë¨ ÆÓÇÊ¾ Þ 1600km. ¡« ÆÓÉÕ¦« Ò Á ÒÞ 1600km Õ¨ ÆÓ Ð¦ ²« ( Û É Í Õ Ü 7 Æ ÍÕ℄ « ° Õ » ). (a) ÊÛ º Õ É ¡«Õ Ë ( Í«°Û Õ £ Ù Û Ó Õ ¯Æ Ñ ); (b) ²« ½ (c) Ë ( ¾ 10−7s.) ª° 1971 ¼À¯ « « ºÒ © Hafele and Keating (1972). « ½¯ ÑÃ¯ Ë ÛË¯ ßÈÕ ¦Ë¯ « Õ Ë ÓØË¯°Ë¹ ds = dt/γ. Æ τ ¬ = τ ¡/γ. Õ τ ¡(1 − 1/γ). v c = 1.6 × 106/(60 × 60) 3 × 108 = 1.48 × 10−6 ≪ 1 , Æ 1 − 1/γ = 1 − [1 − (v/c)2]1/2 ≈ v2 2c2 = 1.097 × 10−12 . Ô τ ¡ v2 2c2 ≈ 24 × 60 × 60 × 1.097 × 10−12 = 9.48 × 10−8 s . « ( ±Ì°ÆÓÉ) ( ÆÓ Ð) ½¾ 10−7 s. ˜11. ± l = 5m Õµ u = 0.6c Õ¨ ¿¨ ¯ ¥ ¹ § ¯¢ ³ ¹Õ ¨Ï¿¯ Ï ¦Þ Ò ËØ Å (a) Î¤ Ô ³ ¹Õ± Õ CW Ë ¯ £ ¥ ³ ¹

88 ¦ CW Õ¦ (b) ¼ Å Õ± ˆ l;
(c) u ºÊ ´± ˆ l/l. © ´ 6-23. (a) ¤ ³ ¹ Þ c, ÉÕ CW Ë Õ Ë lcf , ÈÒ Þ f ÔÒ ³ ¹ ÊÕ Ç lgc = lgf u = lcf γu É log = lgf = lcf γ, log + lgc = loc = l/γ, lcf γ + lcf γu = lcf γ(1 + u) = l/γ. Ô £ ¥ ³ ¹ ¦ CW Õ¦ lcf = l(1 + u)−1γ−2 = l(1 − u) = 5 × (1 − 0.6) 3 × 108 = 6.67 × 10−9 s . (b) ¼ Å Õ± lfh = log = lgf = lcf γ, Ø ˆ l = l(1 − u)γ = l 1 − u 1 + u . (c) Õ ´± ˆ l l = 1 − u 1 + u = 1 − 0.6 1 + 0.6 = 0.5 . 12. ÓÅ£ 6-3-4. Æ 0 = Ub∂b (−1) = Ub∂b (UaUa) = UbUa∂b Ua + UbUa∂b Ua, ° UbUa∂b Ua = UbUa∂b (ηacUc) = UbUaηac∂b Uc = UbUc∂b Uc = UbUb ∂b Ub, ÇÉ ∂a ² ηab ÕÐ Ñ ∂b ηac = 0. ° 0 = 2UbUa∂b Ua = 2UaAa. £ ÔÅ É¯ ¦ À£ Ua É Aa Õ 3 + 1 Õ (6-3-30) É (6-3-37) Ê Uµ = (γ, γu) , Aµ = (γ4u · a, γ2a + γ4(u · a)u) , UaAa = UµAµ = −γ γ4u · a + γu · [γ2a + γ4(u · a)u] = −γ5u · a + γ3u · a + u2γ5(u · a) = [−γ5(1 − u2) + γ3]u · a = [−γ3 + γ3]u · a = 0 . ˜13. Î Ë t∼x ÆÕ Ë G ( ©´ 6-43), ´ K Ê ªÆ¯Aa Õ 4 ¨¯ AaAa ( Ñ AaAa ¯ G ¿ ¨ £ ¿ ËÕ 4 ¨ )

89 Ë Ã x2 − t2 = K2. « t
Ñ 2xu − 2t = 0 Ô 3 ¨ u = t/x. ÑÔ 3 ¨ a = x − tu x2 = x − t2/x x2 = x2 − t2 x3 = K2 x3 . γ = (1 − u2)−1/2 = (1 − t2/x2)−1/2 = x/K. (6-3-37) Ô 4 ¨ · ÑÕÐÑ ÃÉÕ A0 = x K 4 t x K2 x3 = t K2 , A1 = x K 2 K2 x3 + x K 4 t x K2 x3 t x = 1 x + t2 K2x = 1 x + x2 − K2 K2x = x K2 , A2 = A3 = 0 . AaAa = AµAµ = −(A0)2 + (A1)2 = − t K2 2 + x K2 2 = x2 − t2 K4 = 1 K2 . · ËÕ Õ 4 ¨ ¿ ( Ë Rindler Ë¯ Rindler ) ˜14. ÓÅ£ 6-6-2. Æ ß Ø× ¶ Ã ¶ ÃÆ E′ 1 = F′ 10 = Fab (e′ 1 )a(e′ 0 )b = Fµν(eµ)a(eν)b (e′ 1 )a(e′ 0 )b = Fµν[(dxµ)a(∂/∂x′)a][(dxν)b (∂/∂t′)b] = Fµν(∂xµ/∂x′)(∂xν/∂t′) = F01(∂t/∂x′)(∂x/∂t′) + F10(∂x/∂x′)(∂t/∂t′) = F01(γv)(γv) + F10(γ)(γ) = F10γ2(1 − v2) = F10 = E1 , E′ 2 = F′ 20 = Fab (e′ 2 )a(e′ 0 )b = Fµν(eµ)a(eν)b (e′ 2 )a(e′ 0 )b = Fµν[(dxµ)a(∂/∂y′)a][(dxν)b (∂/∂t′)b] = Fµν(∂xµ/∂y′)(∂xν/∂t′) = F20(∂y/∂y′)(∂t/∂t′) + F21(∂y/∂y′)(∂x/∂t′) = F20(1)(γ) + F21(1)(γv) = γF20 − γvF12 = γ(E2 − vB3) , E′ 3 = F′ 30 = Fab (e′ 3 )a(e′ 0 )b = Fµν(eµ)a(eν)b (e′ 3 )a(e′ 0 )b = Fµν[(dxµ)a(∂/∂z′)a][(dxν )b (∂/∂t′)b] = Fµν(∂xµ/∂z′)(∂xν/∂t′) = F30(∂z/∂z′)(∂t/∂t′) + F31(∂z/∂z′)(∂x/∂t′) = F30(1)(γ) + F31(1)(γv) = γF30 + γvF31 = γ(E3 + vB2) . £ É Ù¦ 6-6-1 Õ Ã (6-6-6): (e′ 0 )a = γ(e0)a + γv(e1)a, É (6-6-7): (e′ 1 )a = γv(e0)a + γ(e1)a. B′ 1 = F′ 23 = Fab (e′ 2 )a(e′ 3 )b = Fab (e2)a(e3)b = F23 = B1 ,

90 B′ 2 = F′ 31 = Fab (e′ 3
)a(e′ 1 )b = Fab (e3)a[γv(e0)b + γ(e1)b] = γvFab (e3)a(e0)b + γFab (e3)a(e1)b = γvF30 + γF31 = γ(B2 + vE3) , B′ 3 = F′ 12 = Fab (e′ 1 )a(e′ 2 )b = Fab [γv(e0)a + γ(e1)a](e2)b = γvFab (e0)a(e2)b + γFab (e1)a(e2)b = γvF02 + γF12 = γ(B3 − vE2) . *15. Î Fab ÆÔß É » Ea É Ba ( £ÞÝ E É B), ÓÅ (a) Fab Fab = 2(B2 − E2) , (b) Fab ∗Fab = 4E ·B . ÑÞ¸ÑÚ Fab ∗Fab À º º¢¯« E É B ¤ § Õ¯ B2 − E2 É E · B ± É¯ Fab ¯ Õ¥ËÕ ¶ Í ¿ Æ ß Éß (2 ) ÑÞ¸ÑÚÕ Ã Ei = Ei = Fi0 = −F0i = −Fi0 = F0i = 1 2 ˆ εijk ∗Fjk = 1 2 ˆ εijk ∗Fjk , Bi = Bi = − ∗Fi0 = ∗F0i = ∗Fi0 = − ∗F0i = 1 2 ˆ εijk Fjk = 1 2 ˆ εijkFjk , ∗Fij = ˆ εijkEk , ∗Fij = ˆ εijk Ek , Fij = ˆ εijkBk , Fij = ˆ εijk Bk . ° (a) Fab Fab = Fµν Fµν = F0iF0i + Fi0Fi0 + FijFij = (−Ei)(Ei) + (Ei)(−Ei) + (ˆ εijk Bk)(ˆ εijlBl ) (5-4-10) = −2EiEi + (−1)0(3 − 2)!2!δl k BkBl = −2EiEi + 2BiBi = 2(B2 − E2) . (b) Fab ∗Fab = Fµν ∗Fµν = F0i ∗F0i + Fi0 ∗Fi0 + Fij ∗Fij = (−Ei)(−Bi) + (Ei)(Bi) + (ˆ εijk Bk)(ˆ εijlEl ) = 2EiBi + (−1)0(3 − 2)!2!δl k BkEl = 2EiBi + 2EiBi = 4E · B . (c) ∗Fab ∗Fab = ∗Fµν ∗Fµν = ∗F0i ∗F0i + ∗Fi0 ∗Fi0 + ∗Fij ∗Fij = (−Bi)(−Bi) + (−Bi)(Bi) + (ˆ εijk Ek)(ˆ εijlEl ) = −2BiBi + (−1)0(3 − 2)!2!δl k EkEl = −2BiBi + 2EiEi = 2(E2 − B2) = −Fab Fab . ´Ç °« Õ E, B

91 ˜16. ÓÅ£ 6-6-5 ( ÍÔÅ ¿×Ñ Ã). Æ δab
ºÆÙ ÑÃÕ× t ÉÕ ( Ñ) Ñ§ ¯ˆ ∂a É∂a » º ² δab É ηab Ð ÕÑ ª±¯ Za ≡ (∂/∂t)a. ×Ñ Ã (6-6-12) Õ (c): Ò ¦ Ba Ø× B0 = 0, µ ∇ · B = ˆ ∂aBa = ∂Bi/∂xi = ∂aBa = ∂a(− ∗Fab Zb) = −Zb∂a ∗Fab = −Zb∂a 1 2 εabcd Fcd = − 1 2 Zbεabcd ∂aFcd = 1 2 ˆ εacd ∂aFcd = 1 2 ˆ ε[acd] ∂aFcd (2-6-19) = 1 2 ˆ εacd ∂[aFcd] (6-6-11) = 0 , ° ˆ εacd ≡ Zbεbacd × t É² δab Ð Õ ¹ ×Ñ Ã (6-6-12) Õ (d): (5-6-5c) Æ (∇ × B )c = ˆ εabc ˆ ∂aBb , ° Õ ˆ ∂aBb º [ » (3-1-9)] ˆ ∂aBb = (dxi)a(dxj)b ˆ ∂iBj = (dxi)a(dxj)b ∂iBj , B0 = 0 ÑÏ ∂aBb = (dxµ)a(dxj)b ∂µBj = (dx0)a(dxj)b ∂0Bj + (dxi)a(dxj)b ∂iBj , ¬É ² Ò× t ¯ Ò (dx0)a Õ² ¯(dxi)a Õ² ×°ÓÏ¯ ²· Ô ˆ ∂aBb = ha dhb e ∂d Be . Ò ˆ εabc = Zd εdabc ¦ ¯°² ×°ÓÏ ha dhb e ˆ εabc = (δa d + ZaZd)(δb e + Zb Ze)Zf εfabc = (δa d + ZaZd)Zf εfaec = Zf εfdec = ˆ εdec , Â (∇ × B )c, µÔ (∇ × B )c = ˆ εabc ha dhb e ∂d Be = ˆ εdec ∂d Be , ° (∇ × B )c = ˆ εabc ∂aBb = ˆ εabc ∂a(− ∗Fbd Zd) = −Zdˆ εabc ∂a ∗Fbd = −Zdˆ εabc ∂a 1 2 εbdef Fef = − 1 2 Zdεbdef ˆ εabc ∂aFef = 1 2 ˆ εbef ˆ εabc ∂aFef = − 1 2 ˆ εbef ˆ εbac ∂aFef (5-4-10) = − 1 2 (−1)0(3 − 1)!1!δ[a eδc] f ∂aFef = −∂aFac (6-6-10) = 4πJc .

92 ¿ À ÕÆÃÕÜ (∇ × B )c = −
1 2 Zdεbdef ˆ εabc ∂aFef = − 1 2 Zdεbdef Zgεgabc ∂aFef = − 1 2 ZdZgεbdef εbgac ∂aFef (5-4-10) = − 1 2 ZdZg(−1)1(4 − 1)!1!δ[g d δa eδc] f ∂aFef = 3Zgδg [d δa eδc f] ∂a(ZdFef ) (2-6-19) = 3Zgδg d δa eδc f ∂a(Z[dFef]) = 3Zg∂a(Z[gFac]) = Zg∂a(ZgFac + ZcFga + ZaFcg) = ZgZg∂aFac − Zc∂a(FagZg) + Za∂a(FcgZg) = −∂aFac − Zc∂aEa + Za∂aEc (6-6-10) = 4πJc − Zc∂aEa + Za∂aEc . ° (∇ × B )i = (dxi)c(∇ × B )c = (dxi)c(4πJc − Zc∂aEa + Za∂aEc) = 4πJi − 0 + Za∂aEi = 4πji + ∂ ∂t a ∂aEi = 4πji + ∂Ei ∂t , ° É (dxi)cZc = (dxi)c(∂/∂t)c = ∂xi/∂t = 0. µ ×Ñ Ã (6-6-12) Õ (d). Û ¯ Zdεbdef ˆ εabc = −ˆ εbef ˆ εabc = ˆ εbef ˆ εbac (5-4-10) = (−1)0(3 − 1)!1!δ[a eδc] f = 2δ[a eδc] f ? Û³ Zdεbdef Zgεgabc = ZdZgεbdef εbgac (5-4-10) = ZdZg(−1)1(4 − 1)!1!δ[g d δa eδc] f = −6ZdZgδ[g d δa eδc] f = −2ZdZg(δg d δ[a eδc] f + δc d δ[g eδa] f + δa d δ[c eδg] f ) = −2(ZdZd δ[a eδc] f + ZcZgδg [e δa f] + ZaZgδc [e δg f] ) = −2(−δ[a eδc] f + ZcZ[e δa f] + Zaδc [e Zf] ) = 2δ[a eδc] f − 2(ZcZ[e δa f] + Zaδc [e Zf] ) . Ç · ˆ εbef ˆ εbac Ýª ¯Ë ÆÍ Ë¸Ê 3 ¦ ¯ ˆ εbef ˆ εbac ¯ÔÒÃ À ˜17. ÓÅ ÔÕß ¯ §É 3 £ § » T00 = (E2 + B2)/8π É wi = −Ti0 = (E × B )i/4π, i = 1, 2, 3. Fab Õ« º (6-6-28’) § T00 ÕÝª

93 Æ ß Éß (2 ) ÑÞ¸ÑÚÕ Ã Ei
= Ei = Fi0 = −F0i = −Fi0 = F0i = 1 2 ˆ εijk ∗Fjk = 1 2 ˆ εijk ∗Fjk , Bi = Bi = − ∗Fi0 = ∗F0i = ∗Fi0 = − ∗F0i = 1 2 ˆ εijk Fjk = 1 2 ˆ εijkFjk , ∗Fij = ˆ εijkEk , ∗Fij = ˆ εijk Ek , Fij = ˆ εijkBk , Fij = ˆ εijk Bk . É ß ¯£ Õº (6-6-28’) Tab = 1 8π (FacFb c +∗Fac ∗Fb c) Æß ¯ § T00 = 1 8π (F0iF0 i +∗F0i ∗F0 i) = − 1 8π (F0iF0i +∗F0i ∗F0i) = − 1 8π [(−Ei)(Ei) + (Bi)(−Bi)] = 1 8π (E2 + B2) . £ É º (6-6-28) Tab = 1 4π (FacFb c − 1 4 ηab Fcd Fcd), T00 = 1 4π F0iF0 i − 1 4 η00Fcd Fcd = 1 4π − F0iF0i + 1 4 Fcd Fcd = 1 4π − (−Ei)(Ei) + 1 4 2(B2 − E2) = 1 8π (E2 + B2) . É (6-6-28) Ýªß Õ¯ § ( ) É£ § (c = 1 Í×) wi = −Ti0: wi = −Ti0 = − 1 4π FijF0 j − 1 4 ηi0Fcd Fcd = 1 4π FijF0j = 1 4π ˆ εijk BkEj = 1 4π (E × B )i . ©¥¹ Ó Ñ ±¤ Ô¨ (3 ) Ð ¯ É Æ§ 6-4-1 ¶ ± 3 Ð Tij = Tij = 1 4π (FicFj c− 1 4 ηijFcd Fcd), ηij = δij, Fcd Fcd = 2(B2−E2), « FicFj c = FiµFj µ = Fi0Fj 0 + Fik Fj k = Fi0Fj0 + Fik Fjk = (Ei)(−Ej) + (ˆ εikl Bl)(ˆ εjkmBm) = −EiEj + 2δ[j iδm] l BlBm = −EiEj + (δj iδm l − δm iδj l )BlBm = −EiEj + δj iBlBl − BjBi = −EiEj + δijB2 − BiBj . Ø¾¬ ß Tij = 1 4π − EiEj + δijB2 − BiBj − 1 4 δij2(B2 − E2) = 1 4π − EiEj − BiBj + δij 1 2 (E2 + B2) = − 1 4π (EiEj + BiBj) + δij 1 8π (E2 + B2) .

94 Ó× Ó ¤ Ô¨ (3 ) Ð [ ¸
(1995) 220 Ê (7.5) ↔ T ]. ª 18. (a) ÓÅ 4 ß § Ja Õß Fab Õ¯£ Tab Ø× ∂aTab = −Fbc Jc ( Æ Ja = 0 ∂aTab = 0); *(b) ÓÅÉ ÑÞ¸Ã Õ ¦ ¯ × ¯Ø ¢ (1995) 40 ¤ (6.2); ¦ 3 £ × ¯Ø Û 220 ¤ (7.6). 4 ÕÐÑÍº (6-6-20) Fbc Jc ¸ ÕÐÑÍ § Æ (a) ¯£ Tab = 1 4π (FacFb c − 1 4 ηab Fcd Fcd). ∂a(FacFb c) = Fb c∂aFac + Fac∂aFb c (6-6-10) = Fb c(−4πJc) + Fac∂aFbc = −4πFbc Jc + Fac∂aFbc , ∂a(ηab Fcd Fcd) = ∂b (Fcd Fcd) = Fcd∂b Fcd + Fcd ∂b Fcd = 2Fcd∂b Fcd (6-6-11) = 2Fcd(−∂d Fbc − ∂cFdb ) = 2Fcd(∂d Fcb − ∂cFdb ) = 4Fcd∂[d Fc]b (2-6-19) = −4F[dc]∂d Fcb = −4Fdc∂d Fcb = 4Fac∂aFbc . Â¯£ Õº Ô ∂aTab = 1 4π [−4πFbc Jc + Fac∂aFbc ] − 1 4 [4Fac∂aFbc ] = −Fbc Jc . ¼ Jc = 0 ¯ ∂aTab = 0. (b) ÕÐÑ 4 Í §Õ ˜ fa = Fa b Jb , Ð ˜ fa = Fab Jb . ¬ 4 ß §Ü 3 + 1 Jb = ρZb + jb = ρZb + ρub, ˜ fa = Fab (ρZb + jb) = ρFab Zb + Fab jb = ρEa + Fab jb . ° ( Ea É jb Ü ¦ ) ˜ f0 = F0iji = −Eiji = −E · j , ˜ fi = ρEi + Fijjj = ρEi + ˆ εijk Bkjj = ρEi + (j × B )i . Ü Ø ˜ f = ρE + j × B. ° j = ρu, ˜ f · u = ρu · E = j · E, ˜ f0 = − ˜ f · u Ð ˜ f0 = ˜ f · u.

95 £ (6-6-27), Ýª (j × B )c = ˆ
εcab jaBb = ˆ εcab ja(− ∗FbdZd ) = ˆ εcab ja − 1 2 εbdef Fef Zd = − 1 2 Zd ˆ εcab εbdef jaFef = − 1 2 Zd Zgεgcab εbdef jaFef = 1 2 Zd Zgεbgca εbdef jaFef (5-4-10) = 1 2 Zd Zg(−1)1(4 − 1)!1!δ[d gδe cδf] ajaFef = −3Zgδ[d gδe cδf] aZd Fef ja (2-6-19) = −3Zgδd gδe cδf aZ[d Fef] ja = −3ZgZ[g Fca] ja = −Zg(ZgFca + ZaFgc + ZcFag)ja = −(ZgZgFca + ZcFagZg)ja = −(−Fca + ZcEa)ja = Fcaja − ZcEaja , ° Ò ja Í ¦ ¯ Zaja = 0. Ô ˜ fa = Fab Jb = Fab (ρZb + jb) = ρFab Zb + Fab jb = ρEa + ZaEb jb + (j × B )a . Æ±¡ ˜ f0 = (e0)a ˜ fa = Za ˜ fa = ρZaEa + ZaZaEb jb + Za(j × B )a = 0 − Eb jb + 0 = −E · j , ˜ fi = (ei)a ˜ fa = (∂/∂xi)a ˜ fa = ρEi + 0 + (j × B )i = ρEi + (j × B )i . ÕÐÑ 4 Í § ˜ fa = Fab Jb Õ ÁÅ ¦ ˜ f = ρE + j × B , ¦ ˜ f0 = j · E = ˜ f · u . ÀÅ ¯ × É£ × ¯É ∂aTab = −Fbc Jc = − ˜ fb , Ø ∂aTab = −FbcJc = − ˜ fb Õ ∂µTµν = − ˜ fν. (i) ¯ × ν = 0, ∂µTµ0 = ∂0T00 +∂iTi0 = − ˜ f0. É À 17 Õ À¯° ∂0T00 = ∂0T00 = ∂ ∂t w, Â w ≡ T00 = 1 8π (E2 + B2) µ ÛÕß ¯ § ∂iTi0 = ∂i(−Ti0) = ∂iwi = ∂iwi = ∇ · w = ∇ · S, Â S ≡ w = 1 4π E × B µ ÛÕß ¯ § ( ). ÔÒ ∂w ∂t + ∇ · S = −j · E = − ˜ f · u . Ã ºß ¯ × Ñ Õ ¢ Û (1995) 40 ¤Õ (6.2). (ii) £ × ν = i, ∂µTµi = ∂0T0i + ∂jTji = − ˜ fi. É À 17 Õ À¯ ° ∂0T0i = ∂0(−T0i) = ∂0(−Ti0) = ∂ ∂t wi = ∂ ∂t Si = ∂ ∂t gi = ∂gi ∂t , Â 3 £

96 § gi, ×°¯ § (c = 1 ); ∂jTji
= ∂jTji = (∇· ↔ T )i = (∇· ↔ T )i, ° Tji £ § (3 ) ( ©À 17 Õ ÇÝª). ÔÒ ∂gi ∂t + (∇· ↔ T )i = − ˜ fi , Ø ºß £ × Ñ Õ ¢ Û (1995) 220 ¤Õ 3 (7.6): ˜ f + ∂g ∂t = −∇· ↔ T . 19. ÓÅ (6-6-29) Õ aa É φ Ø× B = ∇ × a É E = −∇φ − ∂a/∂t, Õ ß£ÍÚ Õ 3 É¸ Æ É 4 Aa ÑÃ {t, xi} Õ ¯ (6-6-29): Aa = φZa + aa = φ(∂/∂t)a + aa , Ð Aa = φZa + aa = −φ(dt)a + aa , (∇ × a)c (5-6-5c) = ˆ εcab ∂aab = ˆ εcab ∂a(Ab − φZb) = ˆ εcab ∂aAb = ˆ εc[ab] ∂aAb (2-6-19) = ˆ εcab ∂[aAb] = 1 2 ˆ εcab Fab = 1 2 Zdεdcab Fab = Zd ∗Fdc = − ∗Fcd Zd = Bc , ° É ˆ εcab Õ ¦Ñ¯ ˆ εcab Zb = Zdεdcab Zb = Z(dZb)ε[d|ca|b] = 0. Ø Ã B = ∇ × a. ¦ ¯ Fab = ∂aAb − ∂bAa = ∂a(φZb + ab) − ∂b(φZa + aa) (3-1-10) = Zb∂aφ + ∂aab − Za∂bφ − ∂baa , Ô Ea = FabZb = Zb (Zb∂aφ + ∂aab − Za∂bφ − ∂baa) = Zb Zb∂aφ + ∂a(Zb ab) − ZaZb ∂bφ − Zb ∂baa = −∂aφ − ZaZb ∂bφ − Zb ∂baa , Ú ¦ Ò ab Õ ¦Ñ Zb ab = 0. É Õ ¦ Ei = (dxi)aEa = (dxi)a(−∂aφ − ZaZb ∂bφ − Zb ∂baa) = −(dxi)a∂aφ − (dxi)aZaZb ∂bφ − Zb ∂b[(dxi)aaa] = −∂iφ − 0 − Zb∂b ai = −∂iφ − (∂/∂t)b∂b ai = − ∂φ ∂xi − ∂ai ∂t ,

97 ° Ò Za Õ ¦Ñ (dxi)aZa = (dxi)a(∂/∂t)a =
∂xi/∂t = 0. Ø Ã E = −∇φ − ∂a/∂t. Ú ÅÉ Õ Ea Õ ÕÜ ¦ E0 = (dt)aEa = −Za(−∂aφ − ZaZb ∂bφ − Zb ∂baa) = −Za∂aφ − ZaZaZb ∂bφ − Zb ∂b(Zaaa) = −Za∂aφ + Zb ∂bφ − Zb ∂b(0) = 0 . 20. Ù¦ 6-1-1 ¯(a) ÓÅ ∇a(dt)b = 0, ° t « ¦¯ ∇a Æ ÕÑ ª± [ (5-7-2) Ê ]; (b) Î wa ¦ ( Ø»° « ± Õ ), va ¦ 4 ¯ÓÅ va∇awb ¦ [ ∇at «± Õ ± ]. Æ (a) » (5-7-2): ∂ ∂xν a ∇a ∂ ∂xµ b = Γσ µν ∂ ∂xσ b . Æ ¦Õ Ñ± Γi 00, É ¶ ∂ ∂t a ∇a ∂ ∂t b = Γi 00 ∂ ∂xi b . ³Ý (dt)b , ³ ¯ Û³×° (dt)b ∂ ∂t a ∇a ∂ ∂t b = ∂ ∂t a ∇a (dt)b ∂ ∂t b − ∂ ∂t a ∂ ∂t b ∇a(dt)b = ∂ ∂t a ∇a[1] − ∂ ∂t a ∂ ∂t b ∇a(dt)b = − ∂ ∂t a ∂ ∂t b ∇a(dt)b . ∇a(dt)b = 0. (b) ∇at (3-1-2) = (dt)a «± Õ ± ¯Æ ² ¦ wa Ã Æ (∇at)wa = (dt)awa = 0. Å¢ va∇awb ¦ ¯ÍÔÅ¢ £² ∇b t = (dt)b ÃÆ É (a) Õ À¯ (dt)b va∇awb = va∇a[(dt)b wb] = va∇a[0] = 0 . « 4 va, va∇awb ¦ ¾Ü§ÎÃª 4 Fa §Î℄ Â Ú ¼ Æ ℄ Â R ´ R′, ´Ó 4 ° Ua = ( ∂ ∂t )a ´ Ua = ( ∂ ∂t′ )a, Æ ¨℄ {xµ} ´ {x′µ}. ÓÈÑÒ§Î 4 Fa Ó℄ ¨℄ Ú ¼ ∂ ∂x′µ a = ∂xν ∂x′µ ∂ ∂xν a .

98 ª ¨ ηab = ηµν(dxµ)a(dxν)b = ηµν(dx′µ)a(dx′ν)b , ηµν
= ηab ∂ ∂x′µ a ∂ ∂x′ν b = ∂xλ ∂x′µ ∂xρ ∂x′ν ηab ∂ ∂xλ a ∂ ∂xρ b = ∂xλ ∂x′µ ∂xρ ∂x′ν ηλρ . Û¦¯Ó £Î 2-4-2 Ð ¼ ∂x0 ∂x′0 = ∂t ∂t′ ≡ γ , ∂xi ∂x′0 = ∂xi ∂t′ = ∂t ∂t′ dxi dt ≡ γui , ∂x0 ∂x′i = ∂t ∂x′i ≡ αi , ∂xi ∂x′j = βi j , ui R′ ª R ¤ 3 ° ¨℄ {xµ} ¬ −1 = η00 = ∂x0 ∂x′0 ∂x0 ∂x′0 η00 + ∂xi ∂x′0 ∂xi ∂x′0 ηii = (γ)(γ)(−1) + (γui)(γui)(δii) = −γ2(1 − u2) , 0 = η0i = ∂x0 ∂x′0 ∂x0 ∂x′i η00 + ∂xj ∂x′0 ∂xj ∂x′i ηjj = (γ)(αi)(−1) + (γuj)(βj i )(δii) = −γαi + γujβji , δij = ηij = ∂x0 ∂x′i ∂x0 ∂x′j η00 + ∂xk ∂x′i ∂xk ∂x′j ηkk = (αi)(αj)(−1) + (βk i )(βk j )(δkk ) = −αiαj + βki βkj . ℄ Ë Æ ß γ = (1 − u2)−1/2, ¬ Æ ß αi = ujβji = βijuj , ÎÝÓÏ βij = βji ª Øª « Æ ß δij = −βik uk ul βlj + βik βkj . Õ αi ´ ui Å ¯ βij Å ÀÕ¯ ¬℄ αT = uT β , ¿ α = βu , I = −βuuT β + β2 .

99 β Á I + AuuT , A Ù£ ¯Øª
¬ I = −(I + AuuT )uuT (I + AuuT ) + (I + AuuT )2 = −(uuT + 2AuuT uuT + A2uuT uuT uuT ) + (I + 2AuuT + A2uuT uuT ) = −uuT − 2Au2uuT − A2u4uuT + I + 2AuuT + A2u2uuT = I − uuT + 2A(1 − u2)uuT + A2u2(1 − u2)uuT = I − uuT + 2Aγ−2uuT + A2u2γ−2uuT = I + γ−2(A2u2 + 2A − γ2)uuT , uT u = u2. Ô A Ð¤ u2A2 + 2A − γ2 = 0 . A± = −2 ± 4 + 4u2γ2 2u2 = −1 ± γ u2 = ±(γ ∓ 1)u−2 . ¥ × ©Þ Ò Þ α Ã γu, ©° −γu »¯ ¼ Ì × ª¯ ßÝ β = I + (γ − 1)u−2uuT , α = βu = [I + (γ − 1)u−2uuT ]u = u + (γ − 1)u−2uuT u = u + (γ − 1)u−2u2u = γu . ¾ Á αi = γui , βi j = δi j + (γ − 1)u−2uiuj . Æ Fa §Î 4 ¯ Ãª ¯´ ÇÍ R Ä ( {xµ} ¨℄) É Ç R′ Ä ( {x′µ} ¨℄): Fa = f0 ∂ ∂t a + fi ∂ ∂xi a = f′0 ∂ ∂t′ a + f′i ∂ ∂x′i a , « ∂ ∂t′ a = ∂t ∂t′ ∂ ∂t a + ∂xi ∂t′ ∂ ∂xi a = γ ∂ ∂t a + γui ∂ ∂xi a , ∂ ∂x′i a = ∂t ∂x′i ∂ ∂t a + ∂xj ∂x′i ∂ ∂xj a = αi ∂ ∂t a + βj i ∂ ∂xj a , ³Í f′0 ∂ ∂t′ a + f′i ∂ ∂x′i a = f′0 γ ∂ ∂t a + γui ∂ ∂xi a + f′i αi ∂ ∂t a + βj i ∂ ∂xj a = (f′0γ + f′iαi) ∂ ∂t a + (f′0γui + f′jβi j ) ∂ ∂xi a = f0 ∂ ∂t a + fi ∂ ∂xi a ,

100 ßÝ ¼ f0 = f′0γ + f′iαi = f′0γ
+ f′iγui = γ(f′0 + f′iui) , fi = f′0γui + f′jβi j = f′0γui + f′j[δi j + (γ − 1)u−2uiuj] = f′0γui + f′i + (γ − 1)u−2uif′juj , f0 = γ(f′0 + f′ · u) , f = f′ + γuf′0 + (γ − 1)u−2u(f′ · u) . Ó Û ¼¯ × ¼ f′0 = γ(f0 − f · u) , f′ = f − γuf0 + (γ − 1)u−2u(f · u) . ÎÓÏ u Â R′ ª Â R 3 °¨¯ {xµ} ¨℄Ö R′ °¨ « 7 Ã¤ Æ³ ¦¦ ˜1. ÓÅ» ×Ñ Ã ∇aFab = −4πJb ÃßÈ× ¯Ø ∇aJa = 0. ∇aFab = −4πJb ×£° (7-2-8) (7-2-9), º¢ (7-2-8) (7-2-9) µÊßÈ× Æ Ã ∇aFab = −4πJb Æ −4π∇b Jb = ∇b ∇aFab (3-4-5) = ∇a ∇b Fab + Rabc aFcb + Rabc bFac = ∇a ∇b Fab − Rbc Fcb + RacFac = ∇a ∇b Fab − Rcb Fcb + RacFac = ∇a ∇b Fab , ° É Ç± Õ« Ñ Rac = Rca = R(ac) . −4π∇b Jb = ∇(b ∇a) Fab = ∇(b ∇a) F[ab] = 0 . £ ÔÅ £ ÕÐÑ «Õ (7-2-8) ∇a ∇aAb − RbdAd = −4πJb ÔÒß È× −4π∇b Jb = ∇b ∇a ∇aAb − ∇b (RbdAd )

101 (3-4-5) = ∇a ∇b ∇aAb + Rabc a∇cAb +
Rabc b∇aAc − ∇b(Rbd Ad) = ∇a ∇b ∇aAb − Rcb ∇cAb + Rac ∇aAc − ∇b(Rbd Ad) = ∇a ∇b ∇aAb − ∇b(Rbd Ad) = ∇a∇b ∇aAb − ∇b(Rbd Ad) = ∇a(∇a ∇b Ab + Rad Ad) − ∇b(Rbd Ad) = 0 , Ú ¦ Ò ÕÐÑ « ∇b Ab = 0. ²ÔÕ¯ ÔÒ (7-2-8) ª° Á ¦ « ° Ê (7-2-9) µ ÊßÈ× ˜2. ÓÅ DF ωa dτ = Dωa dτ + (Aa ∧ Zb )ωb ∀ ωa ∈ FG(0, 1). Æ É DF gab dτ = 0 É Dgab dτ = 0, DF ωa dτ = DF (gab ωb) dτ = gab DF ωb dτ = gab Dωb dτ + (Ab ∧ Zc)ωc = D(gab ωb) dτ + gab (Ab ∧ Zc)ωc = Dωa dτ + (Aa ∧ Zc)ωc . ˜3. ÓÅ Ñ Ñ 3. Æ Ä wa G(τ) ÉÕ ¦ ¯ waZa = 0. » Ñ Õ DF wa dτ = Dwa dτ + (AaZb − ZaAb)wb = Dwa dτ − ZaAbwb . ¦ ¯ °² Ì ha b = δa b + ZaZb , ha b Dwb dτ = (δa b + ZaZb ) Dwb dτ = Dwa dτ + ZaZb Zc∇cwb = Dwa dτ + ZaZc(Zb ∇cwb) = Dwa dτ − ZaZc(wb∇cZb ) = Dwa dτ − Za(Zc∇cZb )wb = Dwa dτ − ZaAb wb = Dwa dτ − ZaAbwb , ° Ò 0 = ∇c(0) = ∇c(wbZb ) = Zb ∇cwb + wb∇cZb . DF wa dτ = ha b Dwb dτ . 4. ÓÅÁ Ë G(τ) É § ¶ ( ¼ ) Õ va ²°Ù £ ua ≡ Dva/dτ, uava = 0. ÆÅ Ñ vava ² ¯Õ G(τ) É v′a v′ a va = 1. Å va °Ù Ωab ≡ 2v′ [a ub] ¯¨§ 2 Õ £

102 Æ va G(τ) § ¶¯ 0 = D(vava) dτ
= 2va Dva dτ = 2vaua, ° ua ≡ Dva dτ . Õ Ò G(τ) ÉÕ v′a Ø× v′ a va = 1. ° Ã Dva dτ = ua = ua(1) − v′a(0) = ua(v′bbvb ) − v′a(ubvb ) = −(v′aub − uav′b)vb = −2v′[aub]vb = −Ωabvb , ° Ωab ≡ 2v′[aub], Õ¯¨§ 2 Ωab ≡ 2v′ [a ub] = v′ a ∧ ub . © va Á Ë G(τ) ¯¨§ Ωab Ü £ 5. Î {T, X, Y, Z} Ñ ÕÕÐÑÞ¸Ã¯ Ë G(τ) Õ º T = A−1 sinh Aτ , X = A−1 cosh Aτ , Y = Z = 0 , ( ° A ) (a) ÓÅ G(τ) Á Ë ( Ø´ 6-43 Õ G), τ ¯ A G(τ) Õ 4 ¨ Aa Õ § *(b) ÓÅ {T, X, Y, Z} ÃºÞ o Ê Õ² G(τ) ÆÕ ¦ ÈË µ(s) ¤ ² G(τ) ÃÆ *(c) Î (b) Õ µ(s) Õ s µ ÕË ¯¬Ð µ(s) ·Æ o Ê ¾ ² G(τ) ÆÕ ÈË¯µÔ G(τ) ÉÕ¦¿ ¦ wa ≡ (∂/∂s)a, ÓÅ wa G(τ) ¨ *(d) Za ≡ (∂/∂τ)a, Ù {Za, wa, (∂/∂Y )a, (∂/∂Z)a} G(τ) ÉÕÃÆ ¦ 4 ¸¤ ¯ Ê G(τ) Õ Þ¸Ã {t, x, y, z} ¾ËÊ°Þ¸¶ T = (A−1 + x) sinh At , X = (A−1 + x) cosh At , Y = y Z = z . (e) Ê Ñ§ ÉÞ Þ¸Ã ÕË¹º Ýª Ñ§ ·Ã Õ Ñ±¯ Å Ø× Ä 7-4-3, Ø (7-4-10). (a) ËG(τ) ÑÕÐÑÃÕ Þ¸(A−1 sinh Aτ, A−1 cosh Aτ, 0, 0) Ø X2 − T2 = A−2 Þ¯ Õ» (4 ¨) Za = ( ∂ ∂τ )a, ÕÐÑÃÕ Zµ(τ) = Za(dXµ)a = ∂ ∂τ a (dXµ)a = ∂Xµ ∂τ = ∂ ∂τ (T, X, Y, Z) = (cosh Aτ, sinh Aτ, 0, 0) . ZaZa = Zµ(τ)Zµ(τ) = ηµνZµ(τ)Zν(τ) = − cosh2 Aτ + sinh2 Aτ = −1 , Æ Á Ë G(τ) Õ 4 ¨ Aa = Zb∇b Za, ² Ñ Õ§ ηab ÍÐ ÕÑ ª± °Ñ ∂a, Aa = Zb∂b Za, ° ÕÐÑÞ¸ÃÕ

103 Aµ(τ) = (dXµ)aZb∂b Za = Zb∂b [(dXµ)aZa] = ∂
∂τ Zµ(τ) = ∂ ∂τ (cosh Aτ, sinh Aτ, 0, 0) = (A sinh Aτ, A cosh Aτ, 0, 0) . 4 ¨ Aa Õ § ( ) AaAa = AµAµ = ηµνAµAν = −(A sinh Aτ)2 + (A cosh Aτ)2 = A2 , Ø A Aa Õ §¯Æ G Ü¿ ¨ £ ( ©Ü 6 À 13). (b) ºÞ o Ê Õ² G(τ) ÆÕ ¦ ÈË µ(s) ¤ Ñ ÕÁ ÛË¯ °ÕÐÑÞ¸Õ º (sA−1 sinh Aτ, sA−1 cosh Aτ, 0, 0), ° s Ì ÁÀ¢ s µ Ë ¯ Þ¸ (s sinh Aτ, s cosh Aτ, 0, 0). µ(s) Õ» wa = ( ∂ ∂s )a, ÕÐÑÃÕ wµ(s) = wa(dXµ)a = ∂ ∂s a (dXµ)a = ∂Xµ ∂s = ∂ ∂s (s sinh Aτ, s cosh Aτ, 0, 0) = (sinh Aτ, cosh Aτ, 0, 0) . © wawa = wµwµ = − sinh2 Aτ + cosh2 Aτ = 1 , wa Á º¯Ç waZa = wµZµ = − sinh Aτ cosh Aτ + cosh Aτ sinh Aτ = 0 , Ø wa ² Za ÃÆ¯£µ µ(s) ² G(τ) ÃÆ¯ÆÞ p ÕÕÐÑÞ¸ (s sinh Aτ, s cosh Aτ, 0, 0) = (A−1 sinh Aτ, A−1 cosh Aτ, 0, 0) Ø Ì ( o ÒÆÞÕË ) sp = A−1. (c) wa G(τ) Õ Ñ DF wa dτ = Dwa dτ + (AaZb − ZaAb)wb = Dwa dτ − ZaAbwb , Í ÕÕÐÑÃÕ DF wµ dτ = Dwµ dτ − ZµAνwν , ° Dwµ dτ = dwµ dτ = d dτ (sinh Aτ, cosh Aτ, 0, 0) = (A cosh Aτ, A sinh Aτ, 0, 0) , Aνwν = −(A sinh Aτ) sinh Aτ + (A cosh Aτ) cosh Aτ = A , ZµAνwν = (cosh Aτ, sinh Aτ, 0, 0)A = (A cosh Aτ, A sinh Aτ, 0, 0) .

104 DF wµ dτ = 0 , Ø wa G(τ)
¨ (d) Ù{Za, wa, (∂/∂Y )a, (∂/∂Z)a} G(τ) ÉÕÃÆ ¦ 4 ¸¤ Î µ(s) ² G(τ) Æ° p, µ(s) É ¦Þ q ¸¤ÉÕ (t(q), x(q), y(q), z(q)). » (7-4-1) Õ ¯t(q) = τp, τp ×°G(τ) ËÕp ÞÒÒ ¦Þ ( Áp ²X ÆÞ) ÕË x(q) ×°µ(s) ÉÕÈË©pq ÕË ¯ Øx(q) = sq −sp = sq − A−1. Ò q ÞÕÕÐÑÞ¸ (sq sinh Aτq, sq cosh Aτq, 0, 0), Ô Ò Ã T(q) = sq sinh Aτq = [A−1 + x(q)] sinh At(q) , X(q) = sq cosh Aτq = [A−1 + x(q)] cosh At(q) , Y (q) = Z(q) = 0 . p G(τ) ÉÕ Þ¯q µ(s) ÉÕ Þ¯° Ò G(τ) Õ Þ¸Ã {t, x, y, z} ÉÕÐÑÞ¸Ã {T, X, Y, Z} Õ Ã T = (A−1 + x) sinh At , X = (A−1 + x) cosh At , Y = y , Z = z . Ê t, y, z ¤ ´ ÒÃ ¯ X ≥ 0, Æ x ÕÞ¸¶ [−A−1, +∞) (e) dT = (1 + Ax) cosh At dt + sinh At dx , dX = (1 + Ax) sinh At dt + cosh At dx , dY = dy , dZ = dz , Ô Ñ§ ÕË¹ ds2 = −dT2 + dX2 + dY 2 + dZ2 = −[(1 + Ax) cosh At dt + sinh At dx]2 +[(1 + Ax) sinh At dt + cosh At dx]2 + dy2 + dz2 = −(1 + Ax)2dt2 + dx2 + dy2 + dz2 .

105 Ñ§ G Õ Þ¸ÃÕ g00 = −(1 + Ax)2
, g11 = g22 = g33 = 1 , Ð g00 = −(1 + Ax)−2 , g11 = g22 = g33 = 1 . Þ¸ÑÚ« ÃÆ ¦ ÔÒÃÆ ¦ÑÚ¯ § gab = gµν(dxµ)a(dxν)b = ηµν(eµ)a(eν)b « ÔÊ« ÑÚ (e0)a = (1 + Ax)(dt)a , (e1)a = (dx)a , (e2)a = (dy)a , (e3)a = (dz)a . (eµ)a = ηµνgab(eν)b = ηµνgσρ(∂/∂xσ)a(∂/∂xρ)b(eν)b ÆÑÚ (e0)a = η00gσρ(∂/∂xσ)a(∂/∂xρ)b(e0)b = −gσρ(∂/∂xσ)a(∂/∂xρ)b(1 + Ax)(dt)b = −g00(∂/∂t)a(∂/∂t)b(1 + Ax)(dt)b = (1 + Ax)−1(∂/∂t)a , (e1)a = η11gσρ(∂/∂xσ)a(∂/∂xρ)b(e1)b = gσρ(∂/∂xσ)a(∂/∂xρ)b(dx)b = g11(∂/∂x)a(∂/∂x)b(dx)b = (∂/∂x)a , (e2)a = η22gσρ(∂/∂xσ)a(∂/∂xρ)b(e2)b = gσρ(∂/∂xσ)a(∂/∂xρ)b(dy)b = g22(∂/∂y)a(∂/∂y)b(dy)b = (∂/∂y)a , (e3)a = η33gσρ(∂/∂xσ)a(∂/∂xρ)b(e3)b = gσρ(∂/∂xσ)a(∂/∂xρ)b(dz)b = g33(∂/∂z)a(∂/∂z)b(dz)b = (∂/∂z)a , Ø (e0)a = (1 + Ax)−1(∂t)a , (e1)a = (∂x)a , (e2)a = (∂y)a , (e3)a = (∂z)a . ¦ Ã£ § gab = gµν(∂/∂xµ)a(∂/∂xν )b = ηµν (eµ)a(eν)b È « ÔÊ º [ Á (e0)a = η0ν(eν)a = η00(e0)a = −(e0)a] (e0)a = −(1 + Ax)(dt)a , (e1)a = (dx)a , (e2)a = (dy)a , (e3)a = (dz)a . Å É (5-7-19) Ýª Λαβγ É (5-7-20) Ýª ωαβγ . Ò Ã Λαβγ = −Λγβα , ÍÔÝª α = γ ¾ (e0)λ = (e0)a(∂/∂xλ)a = −(1 + Ax)(dt)a(∂/∂xλ)a = −(1 + Ax)δ0 λ , (e1)λ = (e1)a(∂/∂xλ)a = (dx)a(∂/∂xλ)a = δ1 λ , (e2)λ = (e2)a(∂/∂xλ)a = (dy)a(∂/∂xλ)a = δ2 λ , (e3)λ = (e3)a(∂/∂xλ)a = (dz)a(∂/∂xλ)a = δ3 λ .

106 (e1)λ,τ = (e2)λ,τ = (e3)λ,τ = 0 , (e0)λ,τ
= ∂ ∂xτ [−(1 + Ax)δ0 λ ] = −Aδ1 τ δ0 λ . Â (5-7-19) Λαβγ = [(eβ )λ,τ − (eβ )τ,λ ](eα)λ(eγ)τ ( ÇÕ α, β ¸¤ Ë¸ 0, 1, 2, 3, λ, τ Þ¸ÃË¸ 0, 1, 2, 3 !): Λα0γ = [(e0)λ,τ − (e0)τ,λ ](eα)λ(eγ)τ = [−Aδ1 τ δ0 λ + Aδ1 λ δ0 τ ](eα)λ(eγ)τ = −A(eα)0(eγ)1 + A(eα)1(eγ)0 = −A(1 + Ax)−1δ0 αδ1 γ + A(1 + Ax)−1δ1 αδ0 γ . ÔÒ Õ Λαβγ : Λ001 = −Λ100 = −A(1 + Ax)−1 . Â (5-7-20) ωαβγ = 1 2 (Λαβγ + Λγαβ − Λβγα ) Ô Õ ωαβγ ( Ã¯ α = β): ω010 = 1 2 (Λ010 + Λ001 − Λ100) = Λ001 = −A(1 + Ax)−1 = −ω100 , ÎÖ 1 ωαβ = ωαβa = ωαβγ (eγ)a = ωαβγeγ: −ω10 = ω01 = ω010e0 = −A(1 + Ax)−1e0 = −A(dt)a , Ø ω1 0 = ω0 1 = −A(1 + Ax)−1e0 = −A(dt)a . Ü Ã (5-7-8) ÏÀÆ ÊÈ ÕÂÙ R1 0 = dω1 0 + ω1 γ ∧ ωγ 0 = −Ad(dt) = 0, Ñ Õ ÈÑ ÃÆ ¦¸¤ÉÕÎÖ (5-7-1) Ê (eβ )b∇b (eα)a = γγ αβ (eγ)a , Ø γγ αβ = (eγ)a(eβ )b∇b (eα)a = −(eγ)a(eβ )bωα δ b (eδ )a = −ωα γ b (eβ )b , ¯ µ (5-7-4). ° ÕÎÖÍ γ1 0β = −ω0 1 b (eβ )b = A(1 + Ax)−1(e0)b (eβ )b = A(1 + Ax)−1δ0 β , γ0 1β = −ω1 0 b (eβ )b = A(1 + Ax)−1(e0)b (eβ )b = A(1 + Ax)−1δ0 β ,

107 Ø γ0 01 = γ0 10 = γ1 00
= A(1 + Ax)−1 . £ ÁÅ Ô § G(τ) Õ Þ¸ÃÕ Ñ± (3-2-10’) Γσ µν = 1 2 gσρ(gρµ,ν + gρν,µ − gµν,ρ) Ê¯ ÆÝª g00,1 = −2A(1 + Ax) . ÖÆÀÆ Ê¯Í µ, ν ÎË ¦¿ 0 ¯ Ñ± ¯° Γσ ij = 0. Ç Γ0 00 = 1 2 g0ρ(gρ0,0 + gρ0,0 − g00,ρ) = 1 2 g00(g00,0 + g00,0 − g00,0) = 0 . Γ0 0i = Γ0 i0 = 1 2 g0ρ(gρ0,i + gρi,0 − g0i,ρ) = 1 2 g00(g00,i + g0i,0 − g0i,0) = 1 2 g00(g00,i) = 1 2 [−(1 + Ax)−2][−2A(1 + Ax)]δi1 = A(1 + Ax)−1δi1 , Γi 00 = 1 2 giρ(gρ0,0 + gρ0,0 − g00,ρ) = 1 2 gii(gi0,0 + gi0,0 − g00,i) = − 1 2 giig00,i = − 1 2 [1][−2A(1 + Ax)]δi1 = A(1 + Ax)δi1 , Γi 0j = Γi j0 = 1 2 giρ(gρ0,j + gρj,0 − g0j,ρ) = 1 2 gii(gi0,j + gij,0 − g0j,i) = 0 . ÒÃÆ ¦¸¤ÉÕ Ñ± ( ÎÖ): γ0 0i = γc ab (e0)c(e0)a(ei)b = γc ab [(1 + Ax)(dt)c][(1 + Ax)−1(∂t)a](∂i)b = γc ab (dt)c(∂t)a(∂i)b = Γ0 0i = A(1 + Ax)−1δi1 , γi 00 = γc ab (ei)c(e0)a(e0)b = γc ab (dxi)c[(1 + Ax)−1(∂t)a][(1 + Ax)−1(∂t)b] = (1 + Ax)−2γc ab (dxi)c(∂t)a(∂t)b = (1 + Ax)−2Γi 00 = A(1 + Ax)−1δi1 . ÃÆ ¦¸¤ÉÕ Ñ± ( ÎÖ¯ Þ¸ÑÚ) γ0 01 = γ0 10 = γ1 00 = A(1 + Ax)−1 . ²· Õ ÀÍ± Ú G Õ 4 ¨ Aa G Õ Þ¸Ã {xµ} ÉÕ º ¯ ˆ Aµ, Ø ˆ Aµ = (dxµ)aAa = (dxµ)aAν ∂ ∂Xν a = ∂xµ ∂Xν Aν ,

108 ° Aν = (A sinh At, A cosh At,
0, 0) ÕÐÑÃÕ ¯ª (a) ÔÒ Å Æ ·À ∂Xν ∂xµ , Õ ·ÀØ ∂xµ ∂Xν . ÏÀÆ Ê ∂Xν ∂xµ =        (1 + Ax) cosh At sinh At 0 0 (1 + Ax) sinh At cosh At 0 0 0 0 1 0 0 0 0 1        , Ô ∂xµ ∂Xν = ∂Xν ∂xµ −1 =        (1 + Ax)−1 cosh At −(1 + Ax)−1 sinh At 0 0 − sinh At cosh At 0 0 0 0 1 0 0 0 0 1        . Ú Ô ˆ Aµ =        ˆ A0 ˆ A1 ˆ A2 ˆ A3        =        (1 + Ax)−1 cosh At −(1 + Ax)−1 sinh At 0 0 − sinh At cosh At 0 0 0 0 1 0 0 0 0 1               A sinh At A cosh At 0 0        =        0 A 0 0        . G Õ 4 ¨ Aa G Õ Þ¸ÃÉÍ x É ¦¿ ¯ 4 ¨Õ A, Ø ˆ A1 = A. Ä 7-4-3 « G ËÉ (x = 0) ÀË¯ Ç È Γ0 01 |x=0 = Γ0 10 |x=0 = Γ1 00 |x=0 = ˆ A1 . 6. Î G Þ L Þ p ∈ L Õ ±Ì Ó ÅÔ ( Ø G Õ 4 ¨ Za ² L Õ 4 ¨ Ua p ÞÍ»), Aa L p ÞÕ 4 ¨¯ aa L p ÞÍ«° G Õ 3 ¨ [ (7-4-3) ], ÓÅ aa = Aa. £ Ò £ 6-3-6 » Õµ Æ ¦¿ ¦ Õ¾ Î G(t) ( ¦ Ó ÅÔ ¼ Ó ) ¯° 4 ¨ Za = ( ∂ ∂t )a, Õ Þ¸Ã {t, xi} Õ ( ¦©) Þ¸¶Æ ¿ £Õ Þ L(τL), ° 4 ¨ Ua = ( ∂ ∂τL )a. Þ L Õ £ G Õ Þ¸Õ º Õ Ë {xµ(t)} = {t, xi(t)} Þ¯ªÛ Þ L Í«° G Õ 3 ¨ ua = dxi(t) dt ∂ ∂xi a ,

109 3 ¨ aa = d2xi(t) dt2 ∂ ∂xi a
. G É L ¤ £¯Æ ¤ 4 ¨¯ » ˆ A = Zb∇b Za É A = Ub∇b Ua. G Ó Õ ¦ £¯¨§ ωa. ¨ ¢Å ¢Õ G Ó (ωa = 0) ¼Ó ÅÔ ( ˆ Aa = 0) ¯Ä ¦¦ Þ p ÃÆÉ Þ L Í«±Ì ( Õ Ë p ÞÍ»¼ 4 ¨Í×), Ø Z|p = U|p, ªÛ Þ L Õ 4 ¨ Aa ×° G ÒÕ ÞÕ 3 ¨ aa ( Ø ÞÍ«° Õ 3 ¨). ÖÆ« Þ L Õ 4 ¨ Ua Ü 3 + 1 Ua = ∂ ∂τL a = dt dτL ∂ ∂t a + dxi dτL ∂ ∂xi a = γ ∂ ∂t a + γ dxi dt ∂ ∂xi a = γ(e0)a + γui(ei)a = γZa + γua , ° γ ≡ dt dτL , (eµ)a ≡ ( ∂ ∂xµ )a Þ¸Ñ Þ L Õ 4 ¨ Aa £ Ü 3 + 1 Aa = Ub∇b Ua = Ub∇b [γ(e0)a + γui(ei)a] = Ub (e0)a∇b γ + γ∇b (e0)a + (ei)a∇b (γui) + γui∇b (ei)a = (e0)a dγ dτL + γ[γ(e0)b + γuj(ej)b]∇b (e0)a +(ei)a d(γui) dτL + γui[γ(e0)b + γuj(ej)b]∇b (ei)a = (e0)aγ dγ dt + γ2[(e0)b∇b (e0)a + uj(ej)b∇b (e0)a] +(ei)aγ d(γui) dt + γ2ui[(e0)b∇b (ei)a + uj(ej)b∇b (ei)a] (7-4-13) = (e0)aγ dγ dt + γ2[Γσ 00(eσ)a + ujΓσ 0j(eσ)a] +(ei)aγ d(γui) dt + γ2ui[Γσ i0(eσ)a + ujΓσ ij(eσ)a] (7-4-10) = (e0)aγ dγ dt + γ2[Γi 00(ei)a + ujΓ0 0j(e0)a + ujΓi 0j(ei)a] +(ei)aγ d(γui) dt + γ2ui[Γ0 i0(e0)a + Γj i0(ej)a + 0] (7-4-10) = (e0)aγ dγ dt + γ2[ ˆ Ai(ei)a + uj ˆ Aj(e0)a + uj(−ωk εki j)(ei)a] +(ei)aγ d(γui) dt + γ2ui[ ˆ Ai(e0)a + (−ωk εkj i)(ej)a] = (e0)aγ dγ dt + (ei)aγ d(γui) dt + γ2 ˆ Ai(ei)a + 2γ2(uj ˆ Aj)(e0)a + 2γ2εi kj ωkuj(ei)a = (e0)a γ dγ dt + 2γ2(uj ˆ Aj) + (ei)a γ d(γui) dt + γ2 ˆ Ai + 2γ2εi kj ωkuj

110 = (e0)a γ dγ dt + 2γ2(uj ˆ Aj)
+ (ei)a γ2ai + uiγ dγ dt + γ2 ˆ Ai + 2γ2εi kj ωkuj . [ Ä7-4-3 (7-4-10) «G ËÉÕÞÀË ] Æ Ú Ú¾ «ÁÀ G Ó Ó ( ˆ Aa = 0, ωa = 0), ªÛ Aa = (e0)a γ dγ dt + (ei)a γ2ai + uiγ dγ dt . ¦ Ò G ÑÃ ( ÈÕ Ñ ) µ· Ò£ 6-3-5 Õ Ñ [ (6-3-37)], dγ dt = γ3uiai = γ3u · a. ¬ÁÀ L Ó ÅÔ Þ¯ G Õ £ ¯ L(τL) Á ÛË¯ ÞÕ 4 ¨ Aa = 0, ° Ã γ dγ dt + 2γ2(uj ˆ Aj) = 0 , Ã £ 7-4-2 ÕÅ¢ÁÃ ÒÕ¦¿× ( ©Ù¦ 7-4-1 Ú ¦¿ Ò), × γ2ai + uiγ dγ dt + γ2 ˆ Ai + 2γ2εi kj ωkuj = 0 , É É Õ× Ô ai − 2ui(uj ˆ Aj) + ˆ Ai + 2εi kj ωkuj = 0 , ¤ ¦ ¯Ü ¦ ¯¸ È Ë¸Ø aa − 2ua(ub ˆ Ab ) + ˆ Aa + 2εa bc ωbuc = 0 , Ã £ 7-4-2 (7-4-7) Õ À Ú ÒÀ Ï ÁÀ G Ó ¯ Aa = (e0)a γ dγ dt + 2γ2(uj ˆ Aj) + (ei)a γ2ai + uiγ dγ dt + γ2 ˆ Ai . Å¢ Z|p = U|p ui|p = 0 ( Ø ua|p = 0) × γ|p = 1, dγ dt |p = 0. Ö Æ Ub = γZb + γub, gab ZaUb = gab Za(γZb + γub). ° p ÞÉÛ³ gab |pZa|pUb|p = ηab Za|pZb|p = −1, ³ gab |pZa|p(γ|pZb|p + γ|pub|p) = γ|pgab |pZa|pZb|p = −γ, Ô γ|p = 1. Ua|p = γ|pZa|p + γ|pua|p = Za|p + ua|p, ua|p = Ua|p − Za|p = 0. ©©µØ× dγ dt |p = 0? ª ¬ À É ¯ Aa|p = (ei)a ai|p + ˆ Ai|p = aa|p + ˆ Aa|p . G Ó ÅÔ Aa|p = aa|p. ˜7. § gab ¼¹Ç Õ¯Ä gab ÕÇ± ÓÅ gab £ ÃÕ ÕÇ¢ « gab Ç± ÈÕ

111 Æ £ Ã Gab = Rab − 1 2
gab R = 0. ÁÀ gab Ç± ÈÕ¯ Ç± Rab = 0, ° ¸ R = gabRab = 0, Ç £ Ã Õ ÁÀ Rab £ ÃÕ ¯Ø× Rab − 1 2 gab R = 0, § gab Ý ¯ gabRab − 1 2 gabgab R = R − 1 2 δa aR = R − 2R = −R = 0, ÃØ Rab = 0, ÆÇ± È ˜8. Î(M, gab ) Ç± È ( ©É ), ξa ° Õ¦¿ Killing ¯ ÓÅFab := (dξ)ab Ø×(M, gab ) Õ ¼ (Ja = 0) ×Ñ Ã É Killing ξa Ø×Õ ∇aξa = 0 ( Ü 4 À 11 Õ À). Æ ¼×Ñ Ã ∇aFab = 0 É ∇[a Fbc] = 0. È Fab = (dξ)ab (5-1-11) = 2∇[a ξb] = ∇aξb − ∇b ξa . Ü¦¿ ÃÓÅ∇a∇aξb −∇a∇b ξa = 0. ξa Killing ¯ Ø×∇aξb = −∇b ξa, ° É ¶ −2∇a∇b ξa = 0. °§ ²Ñ ª±Ð ¯ Ø¢Å ∇a ∇b ξa = 0. Ò ∇a ∇b ξa (3-4-4) = ∇b ∇aξa − Rabc aξc (3-4-6) = ∇b ∇aξa + Rbac aξc = ∇b ∇aξa + Rbc ξc . Ç± È¯Æ Rbc = 0; Ü 4 À 11 Õ ÀÆ« Killing ∇aξa = 0, ∇a ∇b ξa = 0, ØÜ¦¿×Ñ Ã ∇aFab = 0 ÀË ° ∇a ∇b ξc (3-4-3) = ∇b ∇aξc + Rabc dξd , «Ü ¿ Ã ∇[a Fbc] = 2∇[a ∇[b ξc]] (2-6-20) = 2∇[a ∇b ξc] = 2∇[b ∇aξc] + 2R[abc] dξd (3-4-7) = 2∇[b ∇aξc] = −2∇[a ∇b ξc] , ∇[a Fbc] = 0, Ü ¿ Ã£ÀË 9. Î ξµ (µ = 0, 1, 2, 3) Ã ∂b∂b ξµ = 0 É « (7-9-10)∼(7-9-13) ÅÕ ¯ ÓÅ ξa = ξµ(dxµ)a ×γab (7-9-8) Õγ′ ab ¼ ßØ×ÕÐÑ « ∂a¯ γ′ ab = 0 Ø× γ′ = 0 É γ′ 0i = 0 (i = 1, 2, 3). (1) » Õ ¦Ñ Ä¯ÍÔÅ¢ γ′ = 0 É γ′ 0i = 0 » Ã ∂c∂cγ′ = 0 É∂c∂cγ′ 0i = 0 ÕØ×É « γ′|Σ0 = 0, ∂γ′/∂t|Σ0 = 0, γ′ 0i |Σ0 = 0 É ∂γ′ 0i /∂t|Σ0 = 0 Õ (2) ∂b∂b ξµ = 0 Ô ∂2ξµ/∂t2 = ∇2ξµ. Æ ÖÆ¯ÁÀ γab ( Ø ¯ γab ) ÕÐÑ « (∂a¯ γab = 0) ÅÕËÑ £ Ã (∂c∂c¯ γab = −16πTab ) Õ ¯ªÛ ( ¦ ) °Á ¶ (7-9-8), ¾Ø × « ∂b∂b ξa = 0, ÔÒÕ γ′ ab = γab + ∂aξb + ∂b ξa ,

112 £ ¦ ÃÕ ¯ØØ× ∂a¯ γ′ ab = 0.
Ç Õ¯ ¶ ¸¶ ÂÙ ¯ ¸¶ Ã¯Ð È Å Ò γ′ = ηabγ′ ab = ηab(γab + ∂aξb + ∂b ξa) = γ + 2∂cξc , ¯ γ′ ab = γ′ ab − 1 2 ηab γ′ = (γab + ∂aξb + ∂b ξa) − 1 2 ηab (γ + 2∂cξc) = ¯ γab + ∂aξb + ∂b ξa − ηab ∂cξc , ÁÀ ∂c∂cξa = 0, Ç ∂c∂c¯ γ′ ab = ∂c∂c¯ γab = −16πTab , © ¯ γ′ ab £Ø× ( ÕÐ Ñ «ÅÕ) ËÑ £ Ã¯ÆÓ¢ ÅÍ ÕÕÐÑ « ∂a¯ γ′ ab = 0 Ø× Ç ∂a¯ γ′ ab = ∂a(¯ γab + ∂aξb + ∂b ξa − ηab ∂cξc) = ∂a¯ γab + ∂a∂aξb + ∂b ∂aξa − ∂b ∂cξc = ∂a¯ γab + ∂a∂aξb = 0 , ° Ò ÕÐÑ « ∂a¯ γab = 0 É ∂a∂aξb = 0. Å Ø×ÕÐÑ «Õ γab ( Ø ¯ γab ) Ê ¯°Á ¦ ¶ Ô ¼ γ′ = 0, γ′ 0i = 0. ÖÆ ° ξa Ø× Ã ∂c∂cξa = 0, Þ¸ Å ∂ν∂νξµ = 0, Ø¢ ξµ Ø× − ∂2 ∂t2 + ∇2 ξµ = 0 , Ø ∂2ξµ ∂t2 = ∇2ξµ . ¼ ¯ ¯ γ′ ab Ø× ∂c∂c¯ γ′ ab = ∂c∂c(γ′ ab − 1 2 ηab γ′) = 0, ηab Ý Ô ∂c∂c(γ′ − 1 2 4γ′) = −∂c∂cγ′ = 0. Î Σ0 t = t0 Õ ¯ÁÀ γ|Σ0 Ø × (7-9-10) × ∂γ ∂t |Σ0 Ø× (7-9-11), γ′ = γ + 2∂cξc = γ + 2∂µξµ = γ − 2∂0ξ0 + 2∂iξi = γ + 2(∇ · ξ − ∂ξ0 ∂t ) γ′|Σ0 = γ|Σ0 + 2 ∇ · ξ − ∂ξ0 ∂t Σ0 (7-9-10) = 0 , ∂γ′ ∂t Σ0 = ∂γ ∂t Σ0 + 2 ∇ · ∂ξ ∂t − ∂2ξ0 ∂t2 Σ0 = ∂γ ∂t Σ0 + 2 ∇ · ∂ξ ∂t − ∇2ξ0 Σ0 (7-9-11) = 0 , ° É ∂2ξ0 ∂t2 = ∇2ξ0. Ã ∂c∂cγ′ = 0 É ¿ÉÊ «¯ ¦ γ′ = 0. ° È ËÑ £ Ã¶ ∂c∂c¯ γ′ ab = ∂c∂c(γ′ ab − 1 2 ηab γ′) = ∂c∂cγ′ ab = 0. Å γ′ 0i . γ′ 0i = γ0i + ∂0ξi + ∂iξ0 = γ0i + ∂ξi ∂t + ∂ξ0 ∂xi , (7-9-12) É (7-9-13) γ′ 0i |Σ0 = γ0i |Σ0 + ∂ξi ∂t + ∂ξ0 ∂xi Σ0 (7-9-12) = 0 , ∂γ′ 0i ∂t Σ0 = ∂γ0i ∂t Σ0 + ∂2ξi ∂t2 + ∂ ∂t ∂ξ0 ∂xi Σ0 = ∂γ0i ∂t Σ0 + ∇2ξi + ∂ ∂xi ∂ξ0 ∂t Σ0 (7-9-13) = 0 ,

113 ° É ∂2ξi ∂t2 = ∇2ξi. Ã ∂c∂cγ′ 0i
= 0 É ¿ÉÊ «¯ ¦ γ′ 0i = 0. ²ß Õ¾ Þ ±¯ªÇ [ Ã(7-9-6) É(7-9-7)] Í Ò É Ê A0 |Σ0 É a|Σ0 , ¾Ü Ò ∂A0 ∂t |Σ0 , °ÁÕÐÑ «¶ ∇ · a|Σ0 , ÆÈ £ ·É ÕÐÑ « ∂a¯ γab = 0 (7-9-11) É (7-9-13) ³ Õ ∂γ ∂t |Σ0 É ∂γ0i ∂t |Σ0 , ¦℄Í γ00 = 0 ¯ÜÒÆ ÕÐÑ « 0 = ∂µ¯ γµν = ∂µ γµν − 1 2 ηµνγ = ∂µγµν − 1 2 ∂νγ , ∂γ ∂t Σ0 = ∂0γ|Σ0 = 2∂µγµ0 |Σ0 = 2∂iγi0 |Σ0 = 2 ∂γ0i ∂xi Σ0 , ÆÍÓÆÓ γ0i |Σ0 , µ ∂γ0i ∂xi |Σ0 ( Ó¢« ¦ Ñ¯Æ Ó¢ÆÓ Σ0 ºÕÐ ) É ∂γ ∂t |Σ0 . Ú ∂γ0i ∂t Σ0 = ∂0γ0i |Σ0 = −∂0γ0i |Σ0 = ∂jγji − 1 2 ∂iγ Σ0 = ∂γij ∂xj − 1 2 ∂γ ∂xi Σ0 , ¡Í¢ÆÓ γ0i |Σ0 É γij |Σ0 , µ ∂γ0i ∂t |Σ0 . 10. Î γab Ø× (a) ∂a¯ γab = 0; (b) γ = 0; (c) γ0i = 0 (i = 1, 2, 3); (d) γ00 = Ó Ê¦¿£ Ê ¥ ξa ˜ γab ≡ γab + ∂aξb + ∂b ξa Ø× (a) ∂a¯ ˜ γab = 0 ; (b) ˜ γ = 0 ; (c) ˜ γ0i = 0 (i = 1, 2, 3) ; (d) ˜ γ00 = 0 . (a) · ÆÓ ¶ Å ÄÕÐÑ «¯¢ ∂b∂b ξa = 0, Ø − ∂2 ∂t2 + ∇2 ξµ = 0 . (b) ˜ γ = ηab˜ γab = γ + 2∂aξa = 2∂aξa = 0, ¢ ∂0ξ0 = ∂iξi, Ø ∂ξ0 ∂t = ∂ξi ∂xi = ∇ · ξ . (c) ¢ ˜ γ0i = γ0i + ∂0ξi + ∂iξ0 = ∂0ξi + ∂iξ0 = 0, Ø ∂ξi ∂t + ∂ξ0 ∂xi = 0 . (d) ¢ ˜ γ00 = γ00 + 2∂0ξ0 = 0, γ00 ¯Ø¢ ∂ξ0 ∂t = − γ00 2 .

114 Ø×É (a)–(d) Õ¦¿Ç Æ©Õ ξ0 = − γ00 2
t = − γ00 2 x0 , ξi = − γ00 6 xi ( Ø ξ = − γ00 6 x ) . ÅÁÅ (a) « ËÑ§ ¯° Ñ ¤ (b) ∂ξ0 ∂t = ∂ ∂t (−γ00 2 t) = −γ00 2 , ∇ · ξ = ∇ · (−γ00 6 x) = −γ00 6 × 3 = −γ00 2 . (c) ∂ξi ∂t = ∂ ∂t (−γ00 6 xi) = 0, ∂ξ0 ∂xi = ∂ ∂xi (−γ00 2 t) = 0. (d) Æ© 11. ÓÅ£ 7-9-2. Æ ª° ÊÂÙ (7-9-32) Rabc d = [f(e1)a ∧ (e4)b + g(e2)a ∧ (e4)b ][(e1)c(e3)d + (e4)c(e1)d] +[g(e1)a ∧ (e4)b − f(e2)a ∧ (e4)b ][(e2)c(e3)d + (e4)c(e2)d] , Æ Rabcd = gde Rabc e = [f(e1)a ∧ (e4)b + g(e2)a ∧ (e4)b ][(e1)cgde (e3)e + (e4)cgde (e1)e] +[g(e1)a ∧ (e4)b − f(e2)a ∧ (e4)b ][(e2)cgde (e3)e + (e4)cgde (e2)e] , ° gde (e1)e = (e1)d = g1β (eβ)d = g11(e1)d = (e1)d , gde (e2)e = (e2)d = g2β (eβ)d = g22(e2)d = (e1)d , gde (e3)e = (e3)d = g3β (eβ)d = g34(e4)d = −(e4)d . Ã£ § gab Õ (7-9-23) ª¯Á gde (e3)e = ηde + 2P[(dt)d − (dz)d ][(dt)e − (dz)e] [(∂/∂t)e + (∂/∂z)e] = −(dt)d + (dz)d + 2P[(dt)d − (dz)d ][1 − 1] = −(du)d = −(e4)d . Ô Rabcd = [f(e1)a ∧ (e4)b + g(e2)a ∧ (e4)b ][−(e1)c(e4)d + (e4)c(e1)d ] +[g(e1)a ∧ (e4)b − f(e2)a ∧ (e4)b ][−(e2)c(e4)d + (e4)c(e2)d ] = [f(e1)a ∧ (e4)b + g(e2)a ∧ (e4)b ](e4)c ∧ (e1)d +[g(e1)a ∧ (e4)b − f(e2)a ∧ (e4)b ](e4)c ∧ (e2)d , Ø£ 7-9-2 Õ À 12. Å (7-9-41) Õ (1)∼(3).

115 Æ « {(Ei)a} ÕÃÆ ¦Ñ (7-9-41) (E1)a = (∂/∂x)a
+ E−1Z1Ka = (e1)a + E−1Z1(e3)a , (E2)a = (∂/∂y)a + E−1Z2Ka = (e2)a + E−1Z2(e3)a , (E3)a = E−1Ka − Za = E−1(e3)a − Za , ° E = −gab ZaKb = −gab Za(e3)b , Z1 = gab Za(∂/∂x)b = gab Za(e1)b , Z2 = gab Za(∂/∂y)b = gab Za(e2)b . Ç ¦ gab (E1)a(E1)b = gab [(e1)a + E−1Z1(e3)a][(e1)b + E−1Z1(e3)b] = g11 + g13E−1Z1 + g31E−1Z1 + g33E−2Z2 1 = 1 + 0 + 0 + 0 = 1 , gab (E2)a(E2)b = gab [(e2)a + E−1Z2(e3)a][(e2)b + E−1Z2(e3)b] = g22 + g23E−1Z2 + g32E−1Z2 + g33E−2Z2 1 = 1 + 0 + 0 + 0 = 1 , gab (E3)a(E3)b = gab [E−1(e3)a − Za][E−1(e3)b − Zb] = g33E−2 − gab (e3)aZbE−1 − gab Za(e3)bE−1 + gab ZaZb = 0 − (−E)E−1 − (−E)E−1 + (−1) = 1 , ¼ÃÆ gab (E1)a(E2)b = gab [(e1)a + E−1Z1(e3)a][(e2)b + E−1Z2(e3)b] = g12 + g13E−1Z2 + g32E−1Z1 + g33E−2Z2 1 = 0 + 0 + 0 + 0 = 0 , gab (E1)a(E3)b = gab [(e1)a + E−1Z1(e3)a][E−1(e3)b − Zb] = g13E−1 − gab (e1)aZb + gab (e3)a(e3)bE−2Z1 − gab (e3)aZbE−1Z1 = 0 − Z1 + 0 − (−E)E−1Z1 = 0 , gab (E2)a(E3)b = gab [(e2)a + E−1Z2(e3)a][E−1(e3)b − Zb] = g23E−1 − gab (e2)aZb + gab (e3)a(e3)bE−2Z2 − gab (e3)aZbE−1Z2 = 0 − Z2 + 0 − (−E)E−1Z2 = 0 . ¬² Za ÃÆÕ² ª± ha b = δa b + ZaZb ¬ p ÞÕ 4 ² Ò Wp 3 ¦Ò ¦¯° Ka Wp ÉÕ² ha b Kb = (δa b + ZaZb )Kb = Ka + ZaZb Zb = Ka + Za(−E) = Ka − EZa ,

116 Õ£ §¥Õ gab (Ka − EZa)(Kb − EZb) =
gab KaKb − gab KaZbE − gab EaKbE + gab ZaZbE2 = 0 − (−E)E − (−E)E + (−1)E2 = E2 , ·² ¦ E−1(Ka − EZa) = E−1Ka − Za = (E3)a . ÖÆÅ¢ (E3)a ÛË γ(τ) ¨¯Ø Zc∇c(E3)a = 0. Ò Zc∇cE = Zc∇c(−gab ZaKb) = −gab KbZc∇cZa − gab ZaZc∇cKb = 0 , ° É γ(τ) Õ ÛÑ Zc∇cZa = 0 É Ka Õ Killing Ñ ∇cKb = 0. ° Zc∇c(E3)a = Zc∇c(E−1Ka − Za) = Zc∇c(E−1Ka) − Zc∇cZa = KaZc∇cE−1 + E−1Zc∇cKa − Zc∇cZa = 0 . Å¢(E1)a ÛË ¨¯ É (5-7-5): ωα β a = (eα)c∇a(eβ)c = −(eβ)c ∇a(eα)c, ³Ý (eβ )b: (eβ )bωα β a = −(eβ )b(eβ)c ∇a(eα)c = −∇a(eα)b, Ø ∇a(eα)b = −ωα β a (eβ )b. È ∇a(e1)b = −ω1 β a (eβ )b (7-9-30) = −ω1 3 a (e3)b , ° ω1 3 a » (7-9-30) ω1 3 a = (fx + gy)(du)a. ° Za∇a(E1)b = Za∇a[(e1)b + E−1Z1(e3)b] = Za∇a(e1)b + Z1(e3)bZa∇aE−1 + E−1(e3)bZa∇aZ1 + E−1Z1Za∇a(e3)b = Za∇a(e1)b + 0 + E−1(e3)bZa∇a[gcd Zc(e1)d] + 0 = Za∇a(e1)b + E−1gcd Zc(e3)bZa∇a(e1)d = Za[−ω1 3 a (e3)b] + E−1gcd Zc(e3)bZa[−ω1 3 a (e3)d] = −ω1 3 a Za(e3)b − ω1 3 a E−1[gcd Zc(e3)d]Za(e3)b = −ω1 3 a Za(e3)b − ω1 3 a E−1[−E]Za(e3)b = 0 . ±¡ Å (E2)a ÛË ¨¯Ø Za∇a(E2)b = 0.

117 13. ÓÅ (7-9-43). Æ (7-9-33) Â (7-9-42), ψi j
= −Rabcd Za(Ej)bZc(Ei)d = −[f(e1)a ∧ (e4)b + g(e2)a ∧ (e4)b ](e4)c ∧ (e1)d Za(Ej)bZc(Ei)d −[g(e1)a ∧ (e4)b − g(e2)a ∧ (e4)b ](e4)c ∧ (e2)d Za(Ej)bZc(Ei)d = −[f(e1)a ∧ (e4)b Za(Ej)b + g(e2)a ∧ (e4)b Za(Ej)b](e4)c ∧ (e1)d Zc(Ei)d −[g(e1)a ∧ (e4)b Za(Ej)b − f(e2)a ∧ (e4)b Za(Ej)b](e4)c ∧ (e2)d Zc(Ei)d , ° (e1)a ∧ (e4)b Za(Ej)b = [(e1)a(e4)b − (e4)a(e1)b ]Za(Ej)b = [(e1)aZa][(e4)b (Ej)b] − [(e4)aZa][(e1)b (Ej)b] , (e2)a ∧ (e4)b Za(Ej)b = [(e2)a(e4)b − (e4)a(e2)b ]Za(Ej)b = [(e2)aZa][(e4)b (Ej)b] − [(e4)aZa][(e2)b (Ej)b] , (e4)c ∧ (e1)d Zc(Ei)d = [(e4)c(e1)d − (e1)c(e4)d ]Zc(Ei)d = [(e4)cZc][(e1)d (Ei)d] − [(e1)cZc][(e4)d (Ei)d] , (e4)c ∧ (e2)d Zc(Ei)d = [(e4)c(e2)d − (e2)c(e4)d ]Zc(Ei)d = [(e4)cZc][(e2)d (Ei)d] − [(e2)cZc][(e4)d (Ei)d] . (e1)aZa = g11(e1)aZa = gab (e1)aZb = Z1 , (e2)aZa = g22(e2)aZa = gab (e2)aZb = Z2 , (e4)aZa = g43(e3)aZa = −gab (e3)aZb = E , (e1)a(Ei)a = g11(e1)a δ1 i[(e1)a + E−1Z1(e3)a] + δ2 i[(e2)a + E−1Z2(e3)a] +δ3 i[E−1(e3)a − Za] = δ1 i[g11 + g13E−1Z1] + δ2 i[g12 + g13E−1Z2] +δ3 i[g13E−1 − (e1)aZa] = δ1 i − δ3 iZ1 , (e2)a(Ei)a = g22(e2)a δ1 i[(e1)a + E−1Z1(e3)a] + δ2 i[(e2)a + E−1Z2(e3)a] +δ3 i[E−1(e3)a − Za] = δ1 i[g21 + g23E−1Z1] + δ2 i[g22 + g23E−1Z2] +δ3 i[g23E−1 − (e2)aZa] = δ2 i − δ3 iZ2 , (e4)a(Ei)a = g43(e3)a δ1 i[(e1)a + E−1Z1(e3)a] + δ2 i[(e2)a + E−1Z2(e3)a]

118 +δ3 i[E−1(e3)a − Za] = −δ1 i[g31 + g33E−1Z1]
− δ2 i[g32 + g33E−1Z2] −δ3 i[g33E−1 − (e3)aZa] = δ3 i(e3)aZa = −δ3 iE . ° (e1)a ∧ (e4)b Za(Ej)b = [Z1][−δ3 jE] − [E][δ1 j − δ3 jZ1] = −δ1 jE , (e2)a ∧ (e4)b Za(Ej)b = [Z2][−δ3 jE] − [E][δ2 j − δ3 jZ2] = −δ2 jE , (e4)c ∧ (e1)d Zc(Ei)d = [E][δ1 i − δ3 iZ1] − [Z1][−δ3 iE] = δ1 iE , (e4)c ∧ (e2)d Zc(Ei)d = [E][δ2 i − δ3 iZ2] − [Z2][−δ3 iE] = δ2 iE . Â ψi j Õº ψi j = −[f(−δ1 jE) + g(−δ2 jE)](δ1 iE) − [g(−δ1 jE) − f(−δ2 jE)](δ2 iE) = E2fδ1 iδ1 j + E2gδ1 iδ2 j + E2gδ2 iδ1 j − E2fδ2 iδ2 j = αδ1 iδ1 j + βδ1 iδ2 j + βδ2 iδ1 j − αδ2 iδ2 j , ° α = E2f, β = E2g. À·À (ψi j ) =     α β 0 β −α 0 0 0 0     , Ø (7-9-43). 14. ÓÅ (7-9-36), Ø ∇a∇aP = (∂2P/∂x2) + (∂2P/∂y2) . Æ ÖÆ ∇aP = ∇aP(x, y, u) (3-1-2) = (dP)a = Px(dx)a + Py(dy)a + Pu(du)a (7-9-28) = Px(e1)a + Py(e2)a + Pu(e4)a , ° Þ Px ≡ ∂P/∂x × ∇aP = Px(e1)a + Py(e2)a + Pu(e4)a = Pxg11(e1)a + Pyg22(e2)a + Pug43(e3)a = Px(e1)a + Py(e2)a − Pu(e3)a . ° ∇a ∇aP = ∇a[Px(e1)a] + ∇a[Py(e2)a] − ∇a[Pu(e3)a]

119 = (e1)a∇aPx + Px ∇a(e1)a +(e2)a∇aPy + Py ∇a(e2)a
−(e3)a∇aPu − Pu ∇a(e3)a = (e1)a[Pxx(dx)a + Pxy(dy)a + Pxu(du)a] + Px ∇a(e1)a +(e2)a[Pyx(dx)a + Pyy(dy)a + Pyu(du)a] + Py ∇a(e2)a −(e3)a[Pux(dx)a + Puy(dy)a + Puu(du)a] − Pu ∇a(e3)a = (e1)a[Pxx(e1)a + Pxy(e2)a − Pxu(e3)a] + Px ∇a(e1)a +(e2)a[Pyx(e1)a + Pyy(e2)a + Pyu(e3)a] + Py ∇a(e2)a −(e3)a[Pux(e1)a + Puy(e2)a + Puu(e3)a] − Pu ∇a(e3)a = g11Pxx + g12Pxy − g13Pxu + Px ∇a(e1)a +g21Pyx + g22Pyy + g23Pyu + Py ∇a(e2)a −g31Pux − g32Puy − g33Puu − Pu ∇a(e3)a = Pxx + Pyy + Px ∇a(e1)a + Py ∇a(e2)a − Pu ∇a(e3)a . É ∇a(eα)b = −ωα β a (eβ )b ¾ (7-9-30), ∇a(e1)a = −ω1 β a (eβ )a = −ω1 3 a (e3)a = −(fx + gy)(du)a(e3)a = (fx + gy)(e3)a(e3)a = g33(fx + gy) = 0 , ∇a(e2)a = −ω2 β a (eβ )a = −ω2 3 a (e3)a = −(gx − fy)(du)a(e3)a = (gx − fy)(e3)a(e3)a = g33(gx − fy) = 0 , ∇a(e3)a = −ω3 β a (eβ )a = 0 . Ú ÔÒ ∇a ∇aP = Pxx + Pyy = ∂2P ∂x2 + ∂2P ∂y2 . « 8 Ã¤¡Á ¯¥©½ ¦ ˜1. ÓÅ£ 8-1-1. Æ Killing ξa = (∂/∂t)a Õ ×§ ¹ φt, ¬ÎÎ² ξa ÃÆÕ Σ0 = {p ∈ M| t(p) = 0} ÉÕÞ p ÌÒ Σt1 = {q ∈ M| t(q) = t1 } ÉÕ q = φt1 (p) Þ [ p = φ−1 t1 (p) = φ−t1 (p)]. Î q ÞÕ» ¦ Vq, ² Σt1 Í»ÕÒ ¦ Wq ( Ø£ ¥ Σt1 ÆÕ ÀÕ ¦). Å ¢Å¢Ý° Wq Õ wa|q ¤² ξa|q ÃÆ¯Ø (gab ξawb)|q = gab |qξa|qwb|q = 0 .

121 Ü 3 À 9 ( Ä 3-3-2 ÕÅ¢) Õ
À¯Ä Ã r′ = r′(r) Ø× ¾ Ã d2r′(r) dr2 = α(r) dr′(r) dr . γ′(r′) Õ» T′a = ( ∂ ∂r′ )a Ø× ÛË Ã T′b∇b T′a = 0, r′ Ì ˜3. Î ξa ÕÁ Killing ¯ χ ≡ (−gab ξaξb)1/2. (a) ÓÅ χ ξa ÕÓ ËÉ (b) ÓÅ Õ 4 ¨ Aa = ∇a(ln χ). É Killing Ã ∇(aξb) = 0 É (a) Õ À Æ χ ≡ (−ξaξa)1/2 = (−gab ξaξb)1/2 = (−g00)1/2. (a) χ ξa ÕÓ ËÉÕ¶ ξb∇b χ = ξb∇b (−ξaξa)1/2 = 1 2 (−ξaξa)−1/2 − ξbξa ∇b ξa − ξbξa∇b ξa = −χ−1ξbξa∇b ξa (4-3-1) = −χ−1ξ(bξa)∇[b ξa] = 0 , ° É ξa Õ Killing Ñ ¦ À¡¢ ξa ÕÓ ËÉ ξa Õ £ §¥ ¶ (b) Î τ Õ ¯° Ë² ξa ÕÓ Ë Í Õ 4 ¨ Za = ( ∂ ∂τ )a, −1 = ZaZa = gab ZaZb = gab ∂ ∂τ a ∂ ∂τ b = gab dt dτ 2 ∂ ∂t a ∂ ∂t b = g00 dt dτ 2 , Ô dτ dt = (−g00)1/2 = χ . ° Ã Za = χ−1ξa. Õ 4 ¨ Aa = Zb∇b Za = χ−1ξb∇b (χ−1ξa) = χ−2ξb∇b ξa , Ú ¦ É (a) Õ À ¦ ¯ ∇a ln χ = χ−1∇aχ = χ−1∇a(−ξb ξb)1/2 = χ−1 1 2 (−ξb ξb)−1/2 − 2ξb ∇aξb = −χ−2ξb ∇aξb (4-3-1) = χ−2ξb ∇bξa = χ−2ξb∇b ξa . Aa = ∇a ln χ.

122 ˜4. ÓÅ (a) ß ¯£ ÕÔ ¯Ø T ≡
gabTab = 0; (b) ß Õ¸ R = 0. Æ (a) ß Õ¯£ (8-4-1) Tab = 1 4π FacFb c − 1 4 gab Fcd Fcd , ÕÔ gabTab = Ta a = 1 4π FacFac − 1 4 ga aFcd Fcd = 1 4π FacFac − Fcd Fcd = 0 , ° É ga a = δa a = 4. (b) ß Ø×Õ £ Ã Gab = Rab − 1 2 Rgab = 8πTab , Ô Ra a − 1 2 Rga a = R − 2R = Ta a = 0 , ° ¸ R = 0. ˜5. ÓÅ (8-4-7) É (8-4-28). Æ (8-4-7) ÕÅ¢ (8-4-5) Õ Σab = Fab + i ∗Fab , Æ Σab = Fab + i ∗Fab, ° Σab Σab = (Fab + i ∗Fab )(Fab + i ∗Fab) = Fab Fab − ∗Fab ∗Fab + iFab ∗Fab + i ∗Fab Fab = Fab Fab − ∗Fab ∗Fab + 2iFab ∗Fab , ° ∗Fab ∗Fab (5-6-1) = 1 2 Fcdεcdab 1 2 Fef εefab = 1 4 FcdFef εcdab εefab (5-4-10) = 1 4 FcdFef (−1)12!2!δ[e cδf] d = −FcdF[ef] δe cδf d = −FcdFcd . Ô Σab Σab = 2(Fab Fab + iFab ∗Fab) . Ø (8-4-7). (8-4-28) ÕÅ¢ ²Ü 3 (b) Á¦¯ »Ë¹ (8-4-23), ± G Õ 4 ¨ Za ² Killing ξa Õ Ã Za = ∂ ∂τ a = (−g00)−1/2 ∂ ∂t a = (−g00)−1/2ξa .

123 È g00 = − 1 + Q2 r2 +
C r ≡ −f , Za = f−1/2ξa = f−1/2(∂/∂t)a. G ÕÃÆ ¦ 4 ¸¤Õ« ÑÚ Ë¹ (8-4-23) Ê (e0)a = f1/2(dt)a , (e1)a = f−1/2(dr)a , (e2)a = r(dθ)a , (e3)a = r sin θ(dϕ)a , Í ÕÑÚ (e0)a = f−1/2(∂t)a, (e1)a = f1/2(∂r)a, (e2)a = r−1(∂θ )a, (e3)a = (r sin θ)−1(∂ϕ)a. ± G ÔÕß É » Ea = Fab Zb É Ba = − ∗Fab Zb, ° Zb = (e0)b. Ò (8-4-27) Fab = − Q r2 [f−1/2(e0)a] ∧ [f1/2(e1)b ] = − Q r2 (e0)a ∧ (e1)b = − Q r2 [(e0)a(e1)b − (e0)b (e1)a] , Fab = − Q r2 [(e0)a(e1)b − (e0)b(e1)a] = Q r2 [(e0)a(e1)b − (e0)b(e1)a] . ° Ea = Fab Zb = − Q r2 [(e0)a(e1)b − (e0)b (e1)a](e0)b = Q r2 (e1)a , Ba = − ∗Fab Zb = − 1 2 Fcdεcdab Zb = − 1 2 Q r2 [(e0)c(e1)d − (e0)d(e1)c]εcdab (e0)b = − Q r2 (e0)c(e1)dεcdab (e0)b = Q r2 (e0)b(e0)c(e1)dεbcda = Q r2 (e0)(b(e0)c)(e1)dε[bc]da = 0 , É Ea = Q r2 (e1)a , Ba = 0 , [ ° (e1)a = f1/2(∂r)a = f1/2(∂/∂r)a ] . Ø (8-4-28). 6. Î Fab Õ 2 ¯ ∗Fab Fab Õ« 2 ¯ α ∈ [0, 2π] ¯ F′ ab ≡ Fab cos α − ∗Fab sin α Fab Õ¡¯§ α Õ¦¿ ¸ Ê¬ (duality rotation). (a) ÓÅ Fab ¼ß ¼ F′ ab ¼ß [ Å¢ÏÆ Ä × Ñ ÃÕº¾ º (7-2-4’) É (7-2-5’) Î¦½µÆ ].

124 (b) ÓÅß Fab É F′ ab Í±¯£ Tab Õ«
º (6-6-28’) § Å¢ (c) M ≡ 2Fab Fab, N ≡ 2Fab ∗Fab, M′ ≡ 2F′ ab F′ab, N′ ≡ 2F′ ab ∗F′ab, ÓÅ M′ = M cos 2α − N sin 2α , N′ = M sin 2α + N cos 2α . (d) Σab ≡ Fab + i ∗Fab , Σ′ ab ≡ F′ ab + i ∗F′ ab , K ≡ Σab Σab É K′ ≡ Σ′ ab Σ′ab ³¸ ¯ Þ¦ ÞÕ K É K′ Í °³ ÉÕ ¿ Ó (c) Õ ÀÅ¢ K′ K 2α ¯Õ À ( Ø |K| = |K′|, K′ ² K Õ°¯ 2α.). (e) Î (E, B ) É (E′, B′) » Fab É F′ ab ÆÔÕß É ¯ ÓÅ E′ = E cos α + B sin α , B′ = −E sin α + B cos α , . « £Õ ¦ Ä © ÛÅ × Jackson (1975). Æ (a) ÖÆ ° ∗∗Fab (5-6-2) = (−1)1+2(4−2)Fab = −Fab , » F′ ab ≡ Fab cos α − ∗Fab sin α, ∗F′ ab = ∗Fab cos α − ∗∗Fab sin α = Fab sin α + ∗Fab cos α , ° ¶ Fab = F′ ab cos α + ∗F′ ab sin α . ∗Fab = −F′ ab sin α + ∗F′ ab cos α . ×Ñ ÃÕº¾ º (7-2-4’) É (7-2-5’) Ê d ∗ F = 4π ∗ J , dF = 0 , ¼ ² É ÕËÑ¶ ÃËØÆÓ Fab ¼ß ¼ F′ ab ¼ß (b) ß ¯£ Tab Õ« º (6-6-28’): Tab = 1 8π (FacFb c +∗Fac ∗Fb c) . Ç T′ ab = 1 8π (F′ ac F′ b c +∗F′ ac ∗F′ b c) = 1 8π (Fac cos α − ∗Fac sin α)(Fb c cos α − ∗Fb c sin α) +(Fac sin α + ∗Fac cos α)(Fb c sin α + ∗Fb c cos α) = 1 8π (FacFb c +∗Fac ∗Fb c) = Tab .

125 (c) Ü 5 Ü¦ÕÅ¢ ÔÒÕ¦¿ Ã ∗Fab ∗Fab =
−Fab Fab. ° M′ = 2F′ ab F′ab = 2(Fab cos α − ∗Fab sin α)(Fab cos α − ∗Fab sin α) = 2Fab Fab cos2 α − 2Fab ∗Fab sin α cos α −2 ∗Fab Fab sin α cos α + 2 ∗Fab ∗Fab sin2 α = 2Fab Fab(cos2 α − sin2 α) − 4Fab ∗Fab sin α cos α = M cos 2α − N sin 2α ; N′ = 2F′ ab ∗F′ab = 2(Fab cos α − ∗Fab sin α)(Fab sin α + ∗Fab cos α) = 2Fab Fab sin α cos α + 2Fab ∗Fab cos2 α −2 ∗Fab Fab sin2 α − 2 ∗Fab ∗Fab sin α cos α = 4Fab Fab sin α cos α + 2Fab ∗Fab(cos2 α − sin2 α) = M sin 2α + N cos 2α . (d) » K′ = Σ′ ab Σ′ab = (F′ ab + i ∗F′ ab )(F′ab + i ∗F′ab) = F′ ab F′ab + iF′ ab ∗F′ab + i ∗F′ ab F′ab − ∗F′ ab ∗F′ab = 2F′ ab F′ab + 2iF′ ab ∗F′ab = M′ + iN′ . (c) Õ ÀÆ³ ÉÕ K′ K 2α ¯Õ À (e) ± ÔÕß É » Ea = Fab Zb É Ba = − ∗Fab Zb, E′ a = F′ ab Zb = (Fab cos α − ∗Fab sin α)Zb = Fab Zb cos α − ∗Fab Zb sin α = Ea cos α + Ba sin α ; B′ a = − ∗F′ ab Zb = −(Fab sin α + ∗Fab cos α)Zb = −Fab Zb sin α − ∗Fab Zb cos α = −Ea sin α + Ba cos α . Ø E′ = E cos α + B sin α , B′ = −E sin α + B cos α , . 7. n À , Ä Rab = Rgab /2, ° gab , Rab É R » § ¡Ç± É¸ ÓÅß ( ° ß ) £ Ü 3 À 17 Æ 2 ² £

126 Æ £ Ø £ Gab = 0 Õ
¯ ß Õ £ Ã Gab = Rab − 1 2 Rgab = 8πTab , ° Tab ß Õ¯£ ß Tab ¦ ¯ Gab ¦ £ ¯Æß £ Ü3 À 17 Õ Ñ½© 2 ¦Ð2 Õ £ ¤ ¯ 2 ² £ ¯ Á £µ ¡ 2 Õß ¯ £ ²Ô «° 1+1 ¯ ¾ ¾ Á ß Õ¯£ Tab = 1 4π (FacFb c− 1 4 gab Fcd Fcd), ° Tµν = 1 4π (FµσFν σ − 1 4 gµνFσρFσρ). É º Õ Ñ¯È Í ¦¿¥Ë F01 = −F10. ÒFσρFσρ = F01F01 + F10F10 = 2F01F01, ° T00 = 1 4π F0σF0 σ − 1 4 g002F01F01 = 1 4π F01F0 1 − 1 2 g00F01F01 = 1 4π g0σF01Fσ1 − 1 2 g00F01F01 = 1 8π g00F01F01 , T01 = 1 4π F0σF1 σ − 1 4 g012F01F01 = 1 4π F01F1 1 − 1 2 g01F01F01 = 1 4π g1σF01Fσ1 − 1 2 g01F01F01 = 1 8π g01F01F01 , T11 = 1 4π F1σF1 σ − 1 4 g112F01F01 = 1 4π F10F1 0 − 1 2 g11F01F01 = 1 4π g1σF10Fσ0 − 1 2 g11F01F01 = 1 8π g11F01F01 . © Tµν = 0, ² £ Gµν = 0 ÍÀÆ 1 + 3 Õß ¯£ Ð ° 1 + 1 ¾ 8. Taub Õ « (8-6-1’). (a) Ê± Õ 4 ¨ Þ¸Ñ Õº (b) Î ± Õ ¦Þ¸ » (x, y, z1) É (x, y, z2), ¯ ¦Õ ¦ ½Ã (a) Taub « ds2 = z−1/2(−dt2 + dz2) + z(dx2 + dy2) Æ g00 = −z−1/2. Î± Õ τ, dτ = √ −g00 dt, ° ± Õ 4 ¨ Za = ∂ ∂τ a = (−g00)−1/2 ∂ ∂t a = z1/4 ∂ ∂t a = z1/4ξa . (b) ° (x, y, z1) É (x, y, z2) Õ ± Õ ¦½Ã ( Î z2 > z1 > 0) l = z2 z1 z−1/2dz2 = z2 z1 z−1/4dz = 4 3 (z3/4 2 − z3/4 1 ) .

127 Î¯ ÕÞ¸½Ã lc = z2 − z1. z1 <
1 ¯«° Õ lc l > lc, «° Õ lc l < lc; Í× Õ lc Ã 4 3 [(lc + z1)3/4 − z3/4 1 ] = lc z1 > 1 ¯Õ l < lc. 9. ÓÅ (eq8-6-5) Õ Fab « Ñ¯Ø Lξi Fab = 0 (i = 1, 2, 3), ° ξa 1 ≡ (∂/∂x)a, ξa 2 ≡ (∂/∂y)a, ξa 3 ≡ −y(∂/∂x)a + x(∂/∂y)a § (8-6-3) « ÑÕ Killing Æ Ç Æ©Õ¯ ξa 1 Éξa 2 ÕÓ Ë » x Éy Þ¸Ë¯ ξa 3 ÕÓ Ë ϕ Þ¸Ë¯° ϕ Ø× cos ϕ = x/ x2 + y2 ( Ð sin ϕ = y/ x2 + y2). Ò Fab Õ [ (8-6-5)] ¤Í z ÕÄ ¯ » Ä 4-2-2 (4-2-3) (Lξ1 F)µν = ∂Fµν ∂x = 0 , (Lξ2 F)µν = ∂Fµν ∂y = 0 , (Lξ3 F)µν = ∂Fµν ∂ϕ = 0 , ¯Ø Lξi Fab = 0. Ð É Ä 4-2-5 Õ (4-2-8) Ýª Lξi Fab = ξc i ∂cFab + Fcb ∂aξc i + Fac∂b ξc i . ³ÕÜ¦ ξc i ∂cFab = (dxµ)a(dxν)b ξσ i ∂σFµν(z) = (dxµ)a(dxν)b ξσ i δ3 σ ∂ ∂z Fµν(z) = (dxµ)a(dxν)b ξ3 i ∂ ∂z Fµν(z) = 0 . Ú ¦ ξ3 i = 0 ( Õ Killing Í ξ1 1 = 1, ξ2 2 = 1, ξ1 3 = −y É ξ2 3 = x). ³Õ Õ ∂aξc i » (3-1-10) ∂aξc 1 = ∂a(∂/∂x)c = ∂aξc 2 = ∂a(∂/∂y)c = 0 , Ç « ξa 1 É ξa 2 ÑÀË « ∂aξc 3 ∂aξc 3 = (dxµ)∂µ − y ∂ ∂x c + x ∂ ∂y c = (dy)a − ∂ ∂x c + (dx)a ∂ ∂y c = −(dx2)a ∂ ∂x1 c + (dx1)a ∂ ∂x2 c , ÑÏ Fcb ∂aξc 3 + Fac∂b ξc 3 = Fcb − (dx2)a ∂ ∂x1 c + (dx1)a ∂ ∂x2 c

128 +Fac − (dx2)b ∂ ∂x1 c + (dx1)b ∂
∂x2 c = −(dx2)aF1b + (dx1)aF2b − (dx2)b Fa1 + (dx1)b Fa2 = −(dx2)a(dxµ)b F1µ + (dx1)a(dxµ)b F2µ − (dx2)b (dxµ)aFµ1 + (dx1)b (dxµ)aFµ2 (8-6-5) = −(dx2)a(dx2)b F12 + (dx1)a(dx1)b F21 − (dx2)b (dx2)aF21 + (dx1)b (dx1)aF12 = 0 . « ξa 3 Ñ£ÀË £ ÔÅ *10. µÊ ¼×Ñ Ã NP Õº (8-8-3a)–(8-8-3d) ÕÞ ³ ¦ ¯ § −4πJ4, −4πJ2, −4πJ1, −4πJ3 © 2π! ª(J1, J2, J3, J4 Ja Á ¸¤Õ ). (8-8-3a) ÕµÑ Ü¦¿ Ã 2DΦ1 = kc∇c[Fab (kalb + ¯ mamb)] = Fab kakc∇clb + Fab lbkc∇cka + kalbkc∇cFab +Fab ¯ makc∇cmb + Fab mbkc∇c ¯ ma + ¯ mambkc∇cFab , ° Fab kakc∇clb = F4ν(εν)b (ε4)c∇c(ε3)b = F4νgνµωµ34 = F41g12ω234 + F42g21ω134 + F43g34ω434 = F41ω234 + F42ω134 + F43ω344 , Fab lbkc∇cka = Fν3(εν)a(ε4)c∇c(ε4)a = Fν3gνµωµ44 = F13g12ω244 + F23g21ω144 + F43g43ω344 = F13ω244 + F23ω144 − F43ω344 , Fab ¯ makc∇cmb = F2ν(εν)b (ε4)c∇c(ε1)b = F2νgνµωµ14 = F21g12ω214 + F23g34ω414 + F24g43ω314 = −F21ω124 + F23ω144 − F42ω134 , Fab mbkc∇c ¯ ma = Fν1(εν)a(ε4)c∇c(ε2)a = Fν1gνµωµ24 = F21g21ω124 + F31g34ω424 + F41g43ω324 = F21ω124 − F13ω244 + F41ω234 , Ø Fab kakc∇clb + Fab lbkc∇cka + Fab ¯ makc∇cmb + Fab mbkc∇c ¯ ma = 2F41ω234 + 2F23ω144 = 2(πΦ0 − κΦ2) . Ô 2DΦ1 = 2(πΦ0 − κΦ2) + kalbkc∇cFab + ¯ mambkc∇cFab .

129 Á¦Û¯ ¯ δΦ0 = ¯ mc∇c[Fab kamb] = Fab
ka ¯ mc∇cmb + Fab mb ¯ mc∇cka + kamb ¯ mc∇cFab , ° Fab ka ¯ mc∇cmb = F4ν(εν)b (ε2)c∇c(ε1)b = F4νgνµωµ12 = F41g12ω212 + F42g21ω112 + F43g34ω412 = −F41ω122 + F43ω142 , Fab mb ¯ mc∇cka = Fν1(εν)a(ε2)c∇c(ε4)a = Fν1gνµωµ42 = F21g21ω142 + F31g34ω442 + F41g43ω342 = F21ω142 − F41ω342 , Ø Fab ka ¯ mc∇cmb + Fab mb ¯ mc∇cka = −F41(ω122 + ω342) + (F43 + F21)ω142 = −Φ0(−2α) + 2Φ1(−ρ) = 2(αΦ0 − ρΦ1) . Ô ¯ δΦ0 = 2(αΦ0 − ρΦ1) + kamb ¯ mc∇cFab . ° DΦ1 − ¯ δΦ0 = (πΦ0 − κΦ2) + 1 2 (kalbkc + ¯ mambkc)∇cFab − 2(αΦ0 − ρΦ1) − kamb ¯ mc∇cFab = (π − 2α)Φ0 + 2ρΦ1 − κΦ2 + 1 2 (kalbkc + ¯ mambkc − 2kamb ¯ mc)∇cFab . Ü ¿ Ã DΦ2 = kc∇c[Fab ¯ malb] = Fab ¯ makc∇clb + Fab lbkc∇c ¯ ma + ¯ malbkc∇cFab , ° Fab ¯ makc∇clb = F2ν(εν)b (ε4)c∇c(ε3)b = F2νgνµωµ34 = F21g12ω234 + F23g34ω434 + F24g43ω334 = F21ω234 + F23ω344 , Fab lbkc∇c ¯ ma = Fν3(εν)a(ε4)c∇c(ε2)a = Fν3gνµωµ23 = F13g12ω224 + F23g21ω124 + F43g43ω324 = F23ω124 + F43ω234 ,

130 Ø Fab ¯ makc∇clb + Fab lbkc∇c ¯ ma
= F23(ω124 + ω344) + (F21 + F43)ω234 = Φ2(−2ε) + 2Φ1π = 2(πΦ1 − εΦ2) . Ô DΦ2 = 2(πΦ1 − εΦ2) + ¯ malbkc∇cFab . Á¦Û¯ 2¯ δΦ1 = ¯ mc∇c[Fab (kalb + ¯ mamb)] = Fab ka ¯ mc∇clb + Fab lb ¯ mc∇cka + kalb ¯ mc∇cFab +Fab ¯ ma ¯ mc∇cmb + Fab mb ¯ mc∇c ¯ ma + ¯ mamb ¯ mc∇cFab , ° Fab ka ¯ mc∇clb = F4ν(εν)b (ε2)c∇c(ε3)b = F4νgνµωµ32 = F41g12ω232 + F42g21ω132 + F43g34ω432 = F41ω232 + F42ω132 + F43ω342 , Fab lb ¯ mc∇cka = Fν3(εν)a(ε2)c∇c(ε4)a = Fν3gνµωµ42 = F13g12ω242 + F23g21ω142 + F43g43ω342 = F13ω242 + F23ω142 − F43ω342 , Fab ¯ ma ¯ mc∇cmb = F2ν(εν)b (ε2)c∇c(ε1)b = F2νgνµωµ14 = F21g12ω212 + F23g34ω412 + F24g43ω312 = −F21ω122 + F23ω142 − F42ω132 , Fab mb ¯ mc∇c ¯ ma = Fν1(εν)a(ε2)c∇c(ε2)a = Fν1gνµωµ22 = F21g21ω122 + F31g34ω422 + F41g43ω322 = F21ω122 − F13ω242 + F41ω232 , Ø Fab ka ¯ mc∇clb + Fab lb ¯ mc∇cka + Fab ¯ ma ¯ mc∇cmb + Fab mb ¯ mc∇c ¯ ma = 2F41ω232 + 2F23ω142 = 2(λΦ0 − ρΦ2) . Ô 2¯ δΦ1 = 2(λΦ0 − ρΦ2) + kalb ¯ mc∇cFab + ¯ mamb ¯ mc∇cFab . ° DΦ2 − ¯ δΦ1 = 2(πΦ1 − εΦ2) + ¯ malbkc∇cFab − (λΦ0 − ρΦ2) − 1 2 (kalb ¯ mc + ¯ mamb ¯ mc)∇cFab = −λΦ0 + 2πΦ1 + (ρ − 2ε)Φ2 + 1 2 (−kalb ¯ mc − ¯ mamb ¯ mc + 2 ¯ malbkc)∇cFab .

131 ÜÅ¿ Ã 2δΦ1 = mc∇c[Fab (kalb + ¯ mamb)]
= Fab kamc∇clb + Fab lbmc∇cka + kalbmc∇cFab +Fab ¯ mamc∇cmb + Fab mbmc∇c ¯ ma + ¯ mambmc∇cFab , ° Fab kamc∇clb = F4ν(εν)b (ε1)c∇c(ε3)b = F4νgνµωµ31 = F41g12ω231 + F42g21ω131 + F43g34ω431 = F41ω231 + F42ω131 + F43ω341 , Fab lbmc∇cka = Fν3(εν)a(ε1)c∇c(ε4)a = Fν3gνµωµ41 = F13g12ω241 + F23g21ω141 + F43g43ω341 = F13ω241 + F23ω141 − F43ω341 , Fab ¯ mamc∇cmb = F2ν(εν)b (ε1)c∇c(ε1)b = F2νgνµωµ11 = F21g12ω211 + F23g34ω411 + F24g43ω311 = −F21ω121 + F23ω141 − F42ω131 , Fab mbmc∇c ¯ ma = Fν1(εν)a(ε1)c∇c(ε2)a = Fν1gνµωµ21 = F21g21ω121 + F31g34ω421 + F41g43ω321 = F21ω121 − F13ω241 + F41ω231 , Ø Fab kamc∇clb + Fab lbmc∇cka + Fab ¯ mamc∇cmb + Fab mbmc∇c ¯ ma = 2F41ω231 + 2F23ω141 = 2(µΦ0 − σΦ2) . Ô 2δΦ1 = 2(µΦ0 − σΦ2) + kalbmc∇cFab + ¯ mambmc∇cFab . Á¦Û¯ ∆Φ0 = lc∇c[Fab kamb] = Fab kalc∇cmb + Fab mblc∇cka + kamblc∇cFab , ° Fab kalc∇cmb = F4ν(εν)b (ε3)c∇c(ε1)b = F4νgνµωµ13 = F41g12ω213 + F42g21ω113 + F43g34ω413 = −F41ω123 + F43ω143 , Fab mblc∇cka = Fν1(εν)a(ε3)c∇c(ε4)a = Fν1gνµωµ43 = F21g21ω143 + F31g34ω443 + F41g43ω343 = F21ω143 − F41ω343 ,

132 Ø Fab kalc∇cmb + Fab mblc∇cka = −F41(ω123 +
ω343) + (F43 + F21)ω143 = −Φ0(−2γ) + 2Φ1(−τ) = 2(γΦ0 − τΦ1) . Ô ∆Φ0 = 2(γΦ0 − τΦ1) + kamblc∇cFab . ° δΦ1 − ∆Φ0 = (µΦ0 − σΦ2) + 1 2 (kalbmc + ¯ mambmc)∇cFab − 2(γΦ0 − τΦ1) − kamblc∇cFab = (µ − 2γ)Φ0 + 2τΦ1 − σΦ2 + 1 2 (kalbmc + ¯ mambmc − 2kamblc)∇cFab . Ü¥¿ Ã δΦ2 = mc∇c[Fab ¯ malb] = Fab ¯ mamc∇clb + Fab lbmc∇c ¯ ma + ¯ malbmc∇cFab , ° Fab ¯ mamc∇clb = F2ν(εν)b (ε1)c∇c(ε3)b = F2νgνµωµ31 = F21g12ω231 + F23g34ω431 + F24g43ω331 = F21ω231 + F23ω341 , Fab lbmc∇c ¯ ma = Fν3(εν)a(ε1)c∇c(ε2)a = Fν3gνµωµ21 = F13g12ω221 + F23g21ω121 + F43g43ω321 = F23ω121 + F43ω231 , Ø Fab ¯ mamc∇clb + Fab lbmc∇c ¯ ma = F23(ω121 + ω341) + (F21 + F43)ω231 = Φ2(−2β) + 2Φ1µ = 2(µΦ1 − βΦ2) . Ô δΦ2 = 2(µΦ1 − βΦ2) + ¯ malbmc∇cFab . Á¦Û¯ 2∆Φ1 = lc∇c[Fab (kalb + ¯ mamb)] = Fab kalc∇clb + Fab lblc∇cka + kalblc∇cFab +Fab ¯ malc∇cmb + Fab mblc∇c ¯ ma + ¯ mamblc∇cFab ,

133 ° Fab kalc∇clb = F4ν(εν)b (ε3)c∇c(ε3)b = F4νgνµωµ33 =
F41g12ω233 + F42g21ω133 + F43g34ω433 = F41ω233 + F42ω133 + F43ω343 , Fab lblc∇cka = Fν3(εν)a(ε3)c∇c(ε4)a = Fν3gνµωµ43 = F13g12ω243 + F23g21ω143 + F43g43ω343 = F13ω243 + F23ω143 − F43ω343 , Fab ¯ malc∇cmb = F2ν(εν)b (ε3)c∇c(ε1)b = F2νgνµωµ13 = F21g12ω213 + F23g34ω413 + F24g43ω313 = −F21ω123 + F23ω143 − F42ω133 , Fab mblc∇c ¯ ma = Fν1(εν)a(ε3)c∇c(ε2)a = Fν1gνµωµ23 = F21g21ω123 + F31g34ω423 + F41g43ω323 = F21ω123 − F13ω243 + F41ω233 , Ø Fab kalc∇clb + Fab lblc∇cka + Fab ¯ malc∇cmb + Fab mblc∇c ¯ ma = 2F41ω233 + 2F23ω143 = 2(νΦ0 − τΦ2) . Ô 2∆Φ1 = 2(νΦ0 − τΦ2) + kalblc∇cFab + ¯ mamblc∇cFab . ° δΦ2 − ∆Φ1 = 2(µΦ1 − βΦ2) + ¯ malbmc∇cFab − (νΦ0 − τΦ2) − 1 2 (kalblc + ¯ mamblc)∇cFab = −νΦ0 + 2µΦ1 + (τ − 2β)Φ2 + 1 2 (−kalblc − ¯ mamblc + 2 ¯ malbmc)∇cFab . G1 = 1 2 (kalbkc + ¯ mambkc − 2kamb ¯ mc)∇cFab , G2 = 1 2 (−kalb ¯ mc − ¯ mamb ¯ mc + 2 ¯ malbkc)∇cFab , G3 = 1 2 (kalbmc + ¯ mambmc − 2kamblc)∇cFab , G4 = 1 2 (−kalblc − ¯ mamblc + 2 ¯ malbmc)∇cFab ,

134 ¥¿ Ã¶ DΦ1 − ¯ δΦ0 = (π −
2α)Φ0 + 2ρΦ1 − κΦ2 + G1 , DΦ2 − ¯ δΦ1 = −λΦ0 + 2πΦ1 + (ρ − 2ε)Φ2 + G2 , δΦ1 − ∆Φ0 = (µ − 2γ)Φ0 + 2τΦ1 − σΦ2 + G3 , δΦ2 − ∆Φ1 = −νΦ0 + 2µΦ1 + (τ − 2β)Φ2 + G4 . (8-7-3) Æ gac = ma ¯ mc + ¯ mamc − lakc − kalc , ¼×Ñ Ã ∇aFab = −4πJb º (ma ¯ mc + ¯ mamc − lakc − kalc)∇cFab = −4πJb . ² kb ¾Ô −4πJ4 = (makb ¯ mc + ¯ makbmc − lakbkc − kakblc)∇cFab = [makb ¯ mc + ¯ makbmc − (lakbkc + kakblc)]∇cFab = [makb ¯ mc − (ma ¯ mbkc + kamb ¯ mc) + kalbkc]∇cFab = [−mbka ¯ mc − (−mb ¯ makc + kamb ¯ mc) + kalbkc)]∇cFab = (kalbkc + ¯ mambkc − 2kamb ¯ mc)∇cFab = 2G1 , ° ÜÅ ∇[c Fab] = 0 ÑÏ ¯ m[akbmc]∇cFab = 0 É l[akbkc]∇cFab = 0. ² ¯ mb ¾Ô −4πJ2 = (ma ¯ mb ¯ mc + ¯ ma ¯ mbmc − la ¯ mbkc − ka ¯ mblc)∇cFab = [(ma ¯ mb ¯ mc + ¯ ma ¯ mbmc) − la ¯ mbkc − ka ¯ mblc]∇cFab = [− ¯ mamb ¯ mc − la ¯ mbkc + ( ¯ malbkc + lakb ¯ mc)]∇cFab = [− ¯ mamb ¯ mc + lb ¯ makc + ( ¯ malbkc − lbka ¯ mc)]∇cFab = (kalbmc + ¯ mambmc − 2kamblc)∇cFab = 2G3 , ° ÜÅ ∇[c Fab] = 0 ÑÏ m[a ¯ mb ¯ mc]∇cFab = 0 É k[a ¯ mblc]∇cFab = 0. ² mb ¾Ô −4πJ1 = (mamb ¯ mc + ¯ mambmc − lambkc − kamblc)∇cFab = [(mamb ¯ mc + ¯ mambmc) − lambkc − kamblc]∇cFab = [−ma ¯ mbmc + (kalbmc + makblc) − kamblc]∇cFab = [mb ¯ mamc + (kalbmc − mbkalc) − kamblc]∇cFab = (kalbmc + ¯ mambmc − 2kamblc)∇cFab = 2G3 ,

135 ° ÜÅ ∇[c Fab] = 0 ÑÏ m[amb ¯
mc]∇cFab = 0 É l[ambkc]∇cFab = 0. ² lb ¾Ô −4πJ3 = (malb ¯ mc + ¯ malbmc − lalbkc − kalblc)∇cFab = [malb ¯ mc + ¯ malbmc − (lalbkc + kalblc)]∇cFab = [−( ¯ mamblc + la ¯ mbmc) + ¯ malbmc + lakblc]∇cFab = [−( ¯ mamblc − lb ¯ mamc) + ¯ malbmc − lbkalc]∇cFab = (−kalblc − ¯ mamblc + 2 ¯ malbmc)∇cFab = 2G4 . ° ÜÅ ∇[c Fab] = 0 ÑÏ m[alb ¯ mc]∇cFab = 0 É l[albkc]∇cFab = 0. Í· Õ À¯ Ú℄µÔ ¼×Ñ ÃÕ NP DΦ1 − ¯ δΦ0 = (π − 2α)Φ0 + 2ρΦ1 − κΦ2 − 2πJ4 , DΦ2 − ¯ δΦ1 = −λΦ0 + 2πΦ1 + (ρ − 2ε)Φ2 − 2πJ2 , δΦ1 − ∆Φ0 = (µ − 2γ)Φ0 + 2τΦ1 − σΦ2 − 2πJ1 , δΦ2 − ∆Φ1 = −νΦ0 + 2µΦ1 + (τ − 2β)Φ2 − 2πJ3 . ¥¿ Ã Ã (8-8-3a)–(8-8-3d) ¼ Õµ *11. ÓÅ (8-8-7) É (8-8-10). Æ (8-8-7) ÕÅ¢ ß Õ¯£ (7-2-6) Tµν = 1 4π FµσFν σ − 1 4 gµνFρσFρσ . ÖÆ FρσFρσ = 2(F43F43 + F42F42 + F41F41 + F32F32 + F31F31 + F21F21) = 2(F43F34 − F42F31 − F41F32 − F32F41 − F31F42 + F21F12) = 2(−F2 43 + F42F13 + F41F23 + F23F41 + F13F42 − F2 21 ) = 2(−F2 43 + 2F42F13 + 2F41F23 − F2 21 ) = −2(F2 43 + F2 21 ) + 4(F41F23 + F42F13) = −2(F2 43 + F2 21 ) + 4(F41F23 + ¯ F41 ¯ F23) . É (8-8-1b), Φ1 = 1 2 (F43 + F21), ¯ Φ1 = 1 2 (F43 + F12) = 1 2 (F43 − F21), Φ2 1 + ¯ Φ2 1 = 1 2 (F2 43 + F2 21 ) , Φ2 1 − ¯ Φ2 1 = F43F21 , Φ1 ¯ Φ1 = 1 4 (F2 43 − F2 21 ) . ° Í (8-8-1a) É (8-8-1c) Ô FρσFρσ = −4(Φ2 1 + ¯ Φ2 1 ) + 4(Φ0Φ2 + ¯ Φ0 ¯ Φ2) .

136 ß Õ¯£ Á ¸¤Õ T11 = 1 4π F1σF1
σ − 1 4 g11FρσFρσ = 1 4π F12F1 2 + F13F1 3 + F14F1 4 − 0 = 1 4π F12F11 − F13F14 − F14F13 = 1 2π F41F13 = 1 2π Φ0 ¯ Φ2 , T12 = 1 4π F1σF2 σ − 1 4 g12FρσFρσ = 1 4π F12F2 2 + F13F2 3 + F14F2 4 − 1 4 FρσFρσ = 1 4π F12F21 − F13F24 − F14F23 − 1 4 FρσFρσ = 1 4π − F2 21 + F13F42 + F41F23 − 1 4 [−2(F2 43 + F2 21 ) + 4(F41F23 + ¯ F41 ¯ F23)] = 1 4π − F2 21 + ¯ F41 ¯ F23 + F41F23 + 1 2 (F2 43 + F2 21 ) − (F41F23 + ¯ F41 ¯ F23) = 1 8π (F2 43 − F2 21 ) = 1 2π Φ1 ¯ Φ1 , T13 = 1 4π F1σF3 σ − 1 4 g13FρσFρσ = 1 4π F12F3 2 + F13F1 3 + F14F3 4 − 0 = 1 4π F12F31 − F13F34 − F14F33 = 1 4π F13(F21 + F43) = 1 2π ¯ Φ2Φ1 , T14 = 1 4π F1σF4 σ − 1 4 g14FρσFρσ = 1 4π F12F4 2 + F13F4 3 + F14F4 4 − 0 = 1 4π F12F41 − F13F44 − F14F43 = 1 4π F41(F43 − F21) = 1 2π Φ0 ¯ Φ1 , T22 = 1 4π F2σF2 σ − 1 4 g22FρσFρσ = 1 4π F21F2 1 + F23F2 3 + F24F2 4 − 0 = 1 4π F21F22 − F23F24 − F24F23 = 1 2π F23F42 = 1 2π Φ2 ¯ Φ0 , T23 = 1 4π F2σF3 σ − 1 4 g23FρσFρσ = 1 4π F21F3 1 + F23F3 3 + F24F3 4 − 0

137 = 1 4π F21F32 − F23F34 − F24F33 =
1 4π F23(F43 − F21) = 1 2π Φ2 ¯ Φ1 , T24 = 1 4π F2σF4 σ − 1 4 g24FρσFρσ = 1 4π F21F4 1 + F23F4 3 + F24F4 4 − 0 = 1 4π F21F42 − F23F44 − F24F43 = 1 4π F42(F43 + F21) = 1 2π ¯ Φ0Φ1 , T33 = 1 4π F3σF3 σ − 1 4 g33FρσFρσ = 1 4π F31F3 1 + F32F3 2 + F34F3 4 − 0 = 1 4π F31F32 + F32F31 − F34F33 = 1 2π F23F13 = 1 2π Φ2 ¯ Φ2 , T34 = 1 4π F3σF4 σ − 1 4 g34FρσFρσ = 1 4π F31F4 1 + F32F4 2 + F34F4 4 + 1 4 FρσFρσ = 1 4π F31F42 + F32F41 − F34F43 + 1 4 FρσFρσ = 1 4π − F13F42 − F23F41 + F2 43 + 1 4 [−2(F2 43 + F2 21 ) + 4(F41F23 + ¯ F41 ¯ F23)] = 1 4π − ¯ F23 ¯ F41 − F23F41 + F2 43 − 1 2 (F2 43 + F2 21 ) + (F41F23 + ¯ F41 ¯ F23) = 1 8π (F2 43 − F2 21 ) = 1 2π Φ1 ¯ Φ1 , T44 = 1 4π F4σF4 σ − 1 4 g44FρσFρσ = 1 4π F41F4 1 + F42F4 2 + F43F4 3 − 0 = 1 4π F41F42 + F42F41 − F43F44 = 1 2π F41F42 = 1 2π Φ0 ¯ Φ0 . Ø (8-8-7) (8-8-10) ÕÅ¢ (8-4-7) Æ Σab Σab = 2(Fab Fab + iFab ∗Fab) . É ª°Å¢ Fab Fab = FµνFµν = −4(Φ2 1 + ¯ Φ2 1 ) + 4(Φ0Φ2 + ¯ Φ0 ¯ Φ2) .

138 Å Fab ∗Fab. »« ¾ Õ Fab ∗Fab =
Fab 1 2 Fcd εcdab = 1 2 FabFcdεcdab = 1 2 FµνFρσεµνρσ = F12Fρσε12ρσ + F13Fρσε13ρσ + F14Fρσε14ρσ +F23Fρσε23ρσ + F24Fρσε24ρσ + F34Fρσε34ρσ = 2F12F34ε1234 + 2F13F24ε1324 + 2F14F23ε1423 +2F23F14ε2314 + 2F24F13ε2413 + 2F34F12ε3412) = ε1234(2F12F34 − 2F13F24 + 2F14F23 + 2F23F14 − 2F24F13 + 2F34F12) = ε1234(4F12F34 − 4F13F24 + 4F14F23) = 4ε1234(F21F43 − F24F13 + F23F14) = 4ε1234(F43F21 + F42F13 − F41F23) = 4ε1234(Φ2 1 − ¯ Φ2 1 + ¯ Φ0 ¯ Φ2 − Φ0Φ2) . È ÝªÁ ¸¤ Õ ¹ ε1234, · ÀÃÆ ¦¸¤ ÕÍ ε1234 = εabcd (ε1)a(ε2)b(ε3)c(ε4)d (8-7-1) = εabcd 1 4 [(e1)a − i(e2)a][(e1)b + i(e2)b][(e0)c − (e3)c][(e0)d + (e3)d] = 1 4 εabcd i[(e1)a(e2)b − (e2)a(e1)b][(e0)c(e3)d − (e3)c(e0)d] = iεabcd (e1)a(e2)b(e0)c(e3)d = iε1203 = iε0123 = i . Fab ∗Fab = Fµν ∗Fµν = 4i(Φ2 1 − ¯ Φ2 1 + ¯ Φ0 ¯ Φ2 − Φ0Φ2) . Ú Ô Σab Σab = ΣµνΣµν = 2(Fµν Fµν + iFµν ∗Fµν) = 2 − 4(Φ2 1 + ¯ Φ2 1 ) + 4(Φ0Φ2 + ¯ Φ0 ¯ Φ2) − 4(Φ2 1 − ¯ Φ2 1 + ¯ Φ0 ¯ Φ2 − Φ0Φ2) = 16(Φ0Φ2 − Φ2 1 ) . Ø (8-8-10). « 9 Ã¤¿ ¦ ˜1. Taub Õ « ± ¯°Ë¹ (8-6-1’), Ó Killing ÊÁ ÛË γ(τ) Õ º t(τ), x(τ), y(τ), z(τ) ÆØ×Õ Ã ( §9.1).

139 Taub « ± ÕË¹ ds2 = z−1/2(−dt2 + dz2)
+ z(dx2 + dy2) , Ë¹ ÏÀÆ Ê§ g00 = −g33 = −z−1/2 , g11 = g22 = z , °Í Õ Ñ±ª Ü 3 À 15 Ô Γ0 03 = Γ0 30 = Γ3 00 = Γ3 33 = − 1 4z , Γ3 11 = Γ3 22 = − z1/2 2 , Γ1 13 = Γ1 31 = Γ2 23 = Γ2 23 = 1 2z . § Í z ÕÄ Æ « ± Õ¥Ë Killing 4 ¿ — 1 ¿ ¦ ¨« ÑÕÁ Killing ξa 0 = ( ∂ ∂t )a; 2 ¿ ¦ ¨« Ñ ÕÁ Killing ξa 1 = ( ∂ ∂x )a É ξa 2 = ( ∂ ∂y )a; 1 ¿ ¦ £« ÑÕÁ Killing ξa 3 = −y( ∂ ∂x )a + x( ∂ ∂y )a. É Ä 4-3-3 ÛË γ(τ) ÉÕ 3 ¿ E = −gab ∂ ∂t a ∂ ∂τ b = −g00(dt)b ∂ ∂τ b = −g00 dt dτ = z−1/2 dt dτ , Px = gab ∂ ∂x a ∂ ∂τ b = g11(dx)b ∂ ∂τ b = g11 dx dτ = z dx dτ , Py = gab ∂ ∂y a ∂ ∂τ b = g22(dy)b ∂ ∂τ b = g22 dx dτ = z dy dτ , º » ÛËÕÁ Ñ (κ = 1) ÐÁ Ñ (κ = 0) κ := −gab UaUb = −gab ∂ ∂τ a ∂ ∂τ b = −g00 dt dτ 2 − g33 dz dτ 2 − g11 dx dτ 2 − g22 dy dτ 2 = z−1/2 dt dτ 2 − z−1/2 dz dτ 2 − z dx dτ 2 − z dy dτ 2 . 3 ¿ ÂÔ κ = z1/2E2 − z−1/2 dz dτ 2 − z−1P2 x − z−1P2 y , Ø dz dτ 2 = E2z − (P2 x + P2 y )z−1/2 − κz1/2 = E2z − P2z−1/2 − κz1/2 , ° P2 ≡ P2 x + P2 y . Æ dz dτ = ± E2z − P2z−1/2 − κz1/2 ,

140 Ê z = z(τ), Â º 3 ¿¾ Ã
dt dτ = Ez1/2 , dx dτ = Pxz−1 , dy dτ = Pyz−1 , Ø Ô ÛË γ(τ) Õ º xµ = xµ(τ). Å ÅÉ 4 ¿ ÃÕ² ÛË ÃÙ d2xµ dτ2 + Γµ νσ dxν dτ dxσ dτ = 0 , µ = 0, 1, 2, 3 ¦Ï µ = 0 : d2t dτ2 + 2 − 1 4z dt dτ dz dτ = 1 2 Ez−1/2 dz dτ − 1 2z dt dτ dz dτ = 1 2 Ez−1/2 dz dτ − 1 2z (Ez1/2) dz dτ = 0 , µ = 1 : d2x dτ2 + 2 1 2z dx dτ dz dτ = −Pxz−2 dz dτ + 1 z dx dτ dz dτ = −Pxz−2 dz dτ + 1 z (Pxz−1) dz dτ = 0 , µ = 2 : d2y dτ2 + 2 1 2z dy dτ dz dτ = −Pyz−2 dz dτ + 1 z dy dτ dz dτ = −Pyz−2 dz dτ + 1 z (Pyz−1) dz dτ = 0 , µ = 3 : d2z dτ2 + − 1 4z dt dτ 2 + dz dτ 2 + − z1/2 2 dx dτ 2 + dy dτ 2 = d2z dτ2 − 1 4z (E2z + E2z − P2z−1/2 − κz1/2) − z1/2 2 (P2 x z−2 + P2 y z−2) = d2z dτ2 − 1 2 E2 + 1 4 P2z−3/2 + 1 4 κz−1/2 − 1 2 P2z−3/2 = d2z dτ2 − 1 2 E2 − 1 4 P2z−3/2 + 1 4 κz−1/2 = 0 , Ú ¦ ° (dz dτ )2 = E2z − P2z−1/2 − κz1/2, ³« z ÑÔ 2 dz dτ d2z dτ2 = E2 + 1 2 P2z−3/2 − 1 2 κz−1/2 dz dτ , ° d2z dτ2 = 1 2 E2 + 1 4 P2z−3/2 − 1 4 κz−1/2 . ©· Õ 4 ¿· Ã² ÛË ÃÕ 4 ¿ Ã¦Ï Ú 3 ¿ ÛËÕ E ¡ Px É Py Õ Ä E = z−1/2 dt dτ = z−1/2γ = z−1/2 m γm = z−1/2 m E , Px = z dx dτ = zγ dx dt = zγvx = z m γmvx = z m px , Py = z dy dτ = zγ dy dt = zγvy = z m γmvy = z m py ,

141 Ç× ³Õ E ¡px É py » Û Õ
ÞÕ¯ É Ð x É y Õ£ ¯ × Û³ Í ÕÕ 2. Æ ÍÑ ´ 9-8 È µÊ (9-3-18). ´ 9-8 Æ¯ ¦ ¹ dV = drdS ÆÕ ρdV = ρdrdS, ÙÒ Õ ÆÕ Í m(r)(ρdrdS) r2 , ° m(r) ² r ÆÕ Õ ¯ (9-3-8) Ê º¯ ℄ Üº §¯· ¹ ÙÒ ºÕÜÍ¯ [p − (p + dp)]dS = −dpdS, Ô Ã m(r)(ρdrdS) r2 = −dpdS, ° Ø Ã (9-3-18): dp dr = −ρm(r) r2 . ˜3. ÓÅ OV ±ÍÚ Ã (9-3-17) ¸ 1 − 2m(r) r 1/2 dp dr = −(ρ + p)g , (9-4-60) ° g º ÞÕ 4 ¨ Ub∇b Ua Õ É Æ ¦Å [1 − 2m(r)/r]1/2 ∼ = 1, p ∼ = 0, (9-4-60) À dp/dr ∼ = −ρg. g ∼ = m(r)/r2, Ô (9-3-18), Ø dp/dr ∼ = −ρm(r)/r2. Æ É Ü 8 À 3 Õ Ñ ÞÕ 4 ¨ Aa = Ub∇b Ua = ∇a ln χ, ° χ = (−ξaξa)1/2 = (−g00)1/2 = [e2A(r)]1/2 = eA(r). ° 4 ¨Õ ( Õ ) g2 = AaAa = gab(∇a ln χ)(∇b ln χ) = χ−2gab(∇aχ)(∇b χ) = e−2A(r) eA(r) dA(r) dr 2 gab(dr)a(dr)b (9-3-11) = m(r) + 4πpr3 r[r − 2m(r)] 2 g11 = m(r) + 4πpr3 r[r − 2m(r)] 2 1 − 2m(r) r . Æ ¦Å¯ g ∼ = m(r) r2 , Ø Í ¨§ (9-4-60) dp dr = −(ρ + p) 1 − 2m(r) r −1/2 g = −(ρ + p) 1 − 2m(r) r −1/2 m(r) + 4πpr3 r[r − 2m(r)] 1 − 2m(r) r 1/2 = −(ρ + p) m(r) + 4πpr3 r[r − 2m(r)] . Ø OV ±ÍÚ Ã (9-3-17). ˜4. ÓÅ R ≫ M (9-3-26) ¦ Ò Æ ÍÑÕ (9-3-23).

142 Æ ¿ § Õ ¹ Æ (9-3-25): p(r) =
ρ (1 − 2M/R)1/2 − (1 − 2Mr2/R3)1/2 (1 − 2Mr2/R3)1/2 − 3(1 − 2M/R)1/2 , Üº (9-3-26): p0 = ρ 1 − (1 − 2M/R)1/2 3(1 − 2M/R)1/2 − 1 . R ≫ M ¯ » Ò Æ ÍÑÕ (9-3-24) É (9-3-23): p(r) ≈ ρ (1 − M/R) − (1 − Mr2/R3) (1 − Mr2/R3) − 3(1 − M/R) ≈ ρ Mr2/R3 − M/R −2 = ρ M 2R3 (R2 − r2) = ρ 4πR3ρ/3 2R3 (R2 − r2) = 2 3 πρ2(R2 − r2) , p0 ≈ ρ 1 − (1 − M/R) 3(1 − M/R) − 1 ≈ ρ M/R 2 = ρ 4πR3/3ρ 2R = 2 3 πρ2R2 . ˜5. Ñ Rindler Þ¸ t, x ²ÕÐÑÞ¸ T, X Õ Ã Ã (9-4-16) ¡(9-4-11) ¡(9-4-12) ¡(9-4-6) ÔÕÐÑÞ¸² Rindler Þ¸Õ Ã T = 1 2 (V + U) = 1 2 (ev − e−u) = 1 2 (eln x+t − eln x−t) = x sinh t , X = 1 2 (V − U) = 1 2 (ev + e−u) = 1 2 (eln x+t + eln x−t) = x cosh t . ° dT = x cosh t dt + sinh t dx , dX = x sinh t dt + cosh t dx , Ë¹ ds2 = −dT2 + dX2 = −(x cosh t dt + sinh t dx)2 + (x sinh t dt + cosh t dx)2 = −x2dt2 + dx2 . ˜6. Rindler ÕÁ Killing (∂/∂t)a Ñ Õ©¿ Killing

143 É Õ À T = x sinh t, X
= x cosh t Æ ∂ ∂t a = ∂ ∂T a ∂T ∂t + ∂ ∂X a ∂X ∂t = ∂ ∂T a x cosh t + ∂ ∂X a x sinh t = X ∂ ∂T a + T ∂ ∂X a , º 2 Ñ £Õ Killing ˜7. ¹ ± Õ 4 ¨Õ § A ≡ (AaAa)1/2. Ü 8 À 3 Õ Ñ¯Ø Aa = ∇a ln χ. »Ü 8 À 3 Õ Ñ¯Î ξa = ( ∂ ∂t )a ¹ ÕÁ Killing ¯ ± Õ 4 ¨ Aa = ∇a ln χ = ∇a ln(−ξaξa)1/2 = ∇a ln(−g00)1/2 = ∇a ln 1 − 2M r 1/2 = 1 − 2M r −1 M r2 (dr)a , Ô A = (AaAa)1/2 = (gabAaAb )1/2 = (g11A1A1)1/2 = 1 − 2M r 1 − 2M r −1 M r2 2 1/2 = 1 − 2M r −1/2 M r2 . Æ ¦Å A ∼ = M r2 = g, Ø Í ¨§ ˜8. ´ 9-13(a) Õ N1 ( Ð N2) Æ ºÕ² Á ÛË§ N1 ( Ð N2), Ó Å (1) Þ¸ V ( Ð U) Á ÛË N1 ( Ð N2) Õ Ì (2) Þ¸ r Ë N1 É N2 ºÕ² Á ÛËÕ Ì Æ Î η(λ) ¦² Á ÛË¯° t = t(λ), r = r(λ), θ = ¯ ϕ = °» ∂ ∂λ a = ∂ ∂t a dt(λ) dλ + ∂ ∂r a dr(λ) dλ Ø× 0 = gab ∂ ∂λ a ∂ ∂λ b = g00 dt(λ) dλ 2 + g11 dr(λ) dλ 2 = − 1 − 2M r dt(λ) dλ 2 + 1 − 2M r −1 dr(λ) dλ 2 , Ø dt = ± 1 − 2M r −1 dr = ±dr∗ .

144 ° ² Á ÛË« ° v = t +
r∗ = 0 Ð u = t − r∗ = 0. ξa = ( ∂ ∂t )a Killing ¯ » Ä 4-3-3 Å Õ E ÛË η(λ) E := −gab ∂ ∂t a ∂ ∂λ b = −g00(dt)b ∂ ∂λ b = −g00 dt dλ = 1 − 2M r dt dλ . É Ã (1 − 2M r )dt = ±dr, Ô² ÛËÉ Ã dλ = ±dr E , Ø λ = ± r E + c , c = . λ Á ÛËÕ Ì ¯ » Ä 3-3-3, r £ ¦ ÛËÕ Ì É À Ð ° N1 Ð N2 Æ ºÕÁ ÛË¯ N1 Ð N2 É¯ r = 2M t = ±∞, É Ã ÀË » (9-4-26)–(9-4-28), dˆ s2 = 32M3 r e−r/2M (−dT2 + dX2) = − 32M3 r e−r/2M dV dU , Æ N1 U = T − X = 0 Þ¯ V Þ¸Ë¯» ( ∂ ∂V )a, Æ Õ Ì µ V ; Á¦Û¯ N2 V = T + X = 0 Þ¯ U Þ¸ Ë¯» ( ∂ ∂U )a, Æ Õ Ì µ U. ¤ Á ÛË¯Ø× gab ( ∂ ∂V )a( ∂ ∂V )b = gab ( ∂ ∂U )a( ∂ ∂U )b = 0. Õ Ó¶ Ì¯ γ(τ), t = t(τ), r = r(τ), θ = ¯ ϕ = ℄Æ (9-1-6) (κ = 1, L = 0) Ù −1 = − 1 − 2M r −1 E2 + 1 − 2M r −1 dr dτ 2 , dr dτ = ± E2 − (1 − 2M/r) , ¹Çß³ τ = τ(r). r Ü r = 0, τ É Ü τ(0), Ì¯ γ(τ) É Õ½ 9-13a ËÅ γ(τ) ¦ ÂÐ ˜9. Â² Kruskal Þ¸Á¦ÕÞ¸ ËÅ℄Ë¹ÕÞ¸±Ñ r = R: ds2 = −(1 − r2/R2)dt2 + (1 − r2/R2)−1dr2 + r2(dθ2 + sin2 θ dϕ2) , R = . ÔÒ Kruskal Þ¸ÕÁÃ¯ dˆ s2 = −(1 − r2/R2)dt2 + (1 − r2/R2)−1dr2 = (1 − r2/R2)[−dt2 + (1 − r2/R2)−2dr2] = (1 − r2/R2)(−dt2 + dr2 ∗ ) ,

145 ° dr∗ = (1 − r2/R2)−1dr = d[R arctanh(r/R)]
= d R 2 ln 1 + r/R 1 − r/R , « 0 < r < R, r∗ := R 2 ln 1 + r/R 1 − r/R . v := t + r∗ , u := t − r∗ Ø t = (v + u)/2 , r∗ = (v − u)/2 , v É u Õ Ê −∞ < v, u < ∞ . −dt2 + dr2 ∗ = −dvdu, Ô dˆ s2 = −(1 − r2/R2)dvdu . V := eβv , U := −e−βu (β ) , V É U Õ Ê 0 < V < ∞ , −∞ < U < 0 , ¼ dvdu = β−2eβ(u−v)dV dU , dˆ s2 = −β−2(1 − r2/R2)eβ(u−v)dV dU = −β−2(1 − r2/R2)e−2βr∗ dV dU = −β−2(1 − r2/R2)e−βR ln 1+r/R 1−r/R dV dU = −β−2(1 − r/R)(1 + r/R) 1 − r/R 1 + r/R βR dV dU . ËÉ r = R ÎÕ±Ñ¯ Ù βR = −1, Ø β = −1/R . ° dˆ s2 = −R2(1 + r/R)2dV dU = −(r + R)2dV dU . É º¢§ r = R Î ± ¯ V, U Õ Ê ß¸Î V ≤ 0 É U ≥ 0 Õ ¶ r = 0 ¯ ( ¿Ë¸ θ É ϕ Õ) ±Þ¯ Æ« V É U Õ ÊÕÊ ²ÔØ× r > 0 Õ « T := 1 2 (V + U) , X := 1 2 (V − U) ,

146 ¾ É ¯µÔ Þ¸Ã {T, X, θ, ϕ} ÕË¹º
ds2 = (r + R)2(−dT2 + dX2) + r2(dθ2 + sin2 θ dϕ2) . ² Kruskal « ¹ ÆÜÕß¸Á¦¯È £ 4 ¿ ( Kruskal Õ¸Þ). A (X > |T|, 0 < r < R, r∗ = R 2 ln 1+r/R 1−r/R ): V = e−v/R = e−(t+r∗ )/R = 1 − r/R 1 + r/R 1/2 e−t/R , U = −eu/R = −e(t−r∗ )/R = − 1 − r/R 1 + r/R 1/2 et/R , T = 1 2 (V + U) = − 1 − r/R 1 + r/R 1/2 sinh(t/R) , X = 1 2 (V − U) = 1 − r/R 1 + r/R 1/2 cosh(t/R) ; B (T > |X|, r > R, r∗ = R 2 ln r/R+1 r/R−1 ): V = e−v/R = e−(t+r∗ )/R = r/R − 1 r/R + 1 1/2 e−t/R , U = eu/R = e(t−r∗ )/R = r/R − 1 r/R + 1 1/2 et/R , T = 1 2 (V + U) = r/R − 1 r/R + 1 1/2 cosh(t/R) , X = 1 2 (V − U) = − r/R − 1 r/R + 1 1/2 sinh(t/R) ; W (T < −|X|, r > R, r∗ = R 2 ln r/R+1 r/R−1 ): V = −e−v/R = −e−(t+r∗ )/R = − r/R − 1 r/R + 1 1/2 e−t/R , U = −eu/R = −e(t−r∗ )/R = − r/R − 1 r/R + 1 1/2 et/R , T = 1 2 (V + U) = − r/R − 1 r/R + 1 1/2 cosh(t/R) , X = 1 2 (V − U) = r/R − 1 r/R + 1 1/2 sinh(t/R) ; A’ (X < −|T|, 0 < r < R, r∗ = R 2 ln 1+r/R 1−r/R ): V = −e−v/R = −e−(t+r∗ )/R = − 1 − r/R 1 + r/R 1/2 e−t/R , U = eu/R = e(t−r∗ )/R = 1 − r/R 1 + r/R 1/2 et/R ,

147 T = 1 2 (V + U) = 1
− r/R 1 + r/R 1/2 sinh(t/R) , X = 1 2 (V − U) = − 1 − r/R 1 + r/R 1/2 cosh(t/R) . ¶ A, B, W, A’ 1 − r/R 1 + r/R = X2 − T2 , A, A’ t/R = − arctanh(T/X) , B, W t/R = − arctanh(X/T) . Ë ¶¤ (r + R)2(−dT2 + dX2) = −(1 − r2/R2)dt2 + (1 − r2/R2)−1dr2 , ° r + R º 2R(X2 − T2 + 1)−1. ¢ Õ È Õ r = 0 « Õ X2 − T2 = 1, ° A É A’ ¯ B É W Ê r = 0 Î ℄ ±Ñ¯ ¬Ë¹¸ ds2 = (r + R)2(−dT2 + dX2) − dr2 + [dr2 + r2(dθ2 + sin2 θ dϕ2)] , ° Ç 3 È Ñ ¦ÕË¹¯ÆÜ ±Ñ 1−r/R 1+r/R = X2 − T2 Ô dr = R−1(r + R)2(TdT − XdX), dr2 = R−2(r + R)4(T2dT2 + X2dX2 − 2TXdTdX) , Ê r = 0 £Í Þ¸Ã {T, X, θ, ϕ} ÕÞ¸±Ñ r R É 0 ÎÕÞ¸±Ñ£ °ÁÝª Õ¸ (s.p.) Ô Õ Æ Ô ( Mathematica) ²É Þ¸Í Õ Ñ±ÁÅ Γ0 01 = Γ0 10 = −(1 − r2/R2)−1r/R2 , Γ1 00 = −(1 − r2/R2)r/R2 , Γ1 11 = (1 − r2/R2)−1r/R2 , Γ1 22 = −(1 − r2/R2)r , Γ1 33 = −(1 − r2/R2)r sin2 θ , Γ2 12 = Γ2 21 = 1/r , Γ2 33 = − sin θ cos θ , Γ3 13 = Γ3 31 = 1/r , Γ3 23 = Γ3 32 = cot θ ,

148 ÔÂÙ É Þ¸ÅÕ ÁÅ R0101 = −1/R2 , R0202
= (1 − r2/R2)r4/R4 , R0303 = (1 − r2/R2)(r4/R4) sin2 θ , R1212 = (1 − r2/R2)−1r2/R2 , R1313 = (1 − r2/R2)−1(r2/R2) sin2 θ , R2323 = (r4/R2) sin2 θ . ° ÔÇ± Õ R00 = −3R−2(1 − r2/R2) , R11 = 3R−2(1 − r2/R2)−1 , R22 = 3R−2r2 , R33 = 3R−2r2 sin2 θ . Å Ýª s.p.: «¸ R: R = gµνRµν = g00R00 + g11R11 + g22R22 + g33R33 = 12R−2 . ¬ Rab Rab Rab Rab = gµµ′ gνν′ RµνRµ′ν′ = 36R−4 . Rabcd Rabcd Rabcd Rabcd = gµµ′ gνν′ gσσ′ gρρ′ RµνσρRµ′ν′σ′ρ′ = 24R−4 . ÎË s.p. ¤Ü Ë±Ñ 10. ÓÅÚ ß¸ ¹ s.p. ±Ñ É (8-3-21). Æ É ¹ ÕÂÙ º (8-3-21), ÂÙ ÕÑ (3-4- 6) ¡ (3-4-9) É (3-4-10), ( £ Mathematica § ªÔ): Rabcd Rabcd = gµµ′ gνν′ gσσ′ gρρ′ RµνσρRµ′ν′σ′ρ′ = 4[(g00g11R0101)2 + (g00g22R0202)2 + (g00g33R0303)2 +(g11g22R1212)2 + (g11g33R1313)2 + (g22g33R2323)2] = 4 − 1 − 2M r −1 1 − 2M r − 2M r3 2 + − 1 − 2M r −1 1 r2 M r 1 − 2M r 2

149 + − 1 − 2M r −1 1 r2
sin2 θ M r 1 − 2M r sin2 θ 2 + 1 − 2M r 1 r2 − M r 1 − 2M r −1 2 + 1 − 2M r 1 r2 sin2 θ − M r 1 − 2M r −1 sin2 θ 2 + 1 r2 1 r2 sin2 θ 2Mr sin2 θ 2 = 4 4M2 r6 + M2 r6 + M2 r6 + M2 r6 + M2 r6 + 4M2 r6 = 48M2 r6 . r → 2M Ê r → 0 Æ¯ © r → 0 s.p. ±Ñ ¹ ÕÇ± Rab ¯ ¸ R É Rab Rab ¤ 11. ÓÅ´ 9-13(a) Õ N1 Á ÍÔÅ¢° na Á ¿ N1 Õ Ã U = 0, ° ± na = ∇aU. Æ N1 Ã U = 0 ¯° ± na = ∇aU = (dU)a. Ë¹ (9-4-28) É (9-4-26) Æ ds2 = 32M3 r e−r/2M (−dT2 + dX2) + r2(dθ2 + sin2 θ dϕ2) = − 32M3 r e−r/2M dV dU + r2(dθ2 + sin2 θ dϕ2) , ° nana = gabnanb = gab(dU)a(dU)b = gUU = 0 . N1 Á 12. Ó (9-4-50) µÊ (9-4-51), µÊ (9-4-54). (9-4-50) dˆ s2 = −(1 − 2M/r)dt2 + (1 − 2M/r)−1dr2 , v := t + r∗ ¾É (1 − 2M/r)−1dr = dr∗, dˆ s2 = (1 − 2M/r)[−dt2 + (1 − 2M/r)−2dr2] = (1 − 2M/r)[−dt2 + dr2 ∗ ] = (1 − 2M/r)d(r∗ + t)d(r∗ − t) = (1 − 2M/r)dvd(2r∗ − v) = 2(1 − 2M/r)dvdr∗ − (1 − 2M/r)dv2 = −(1 − 2M/r)dv2 + 2dvdr ,

150 Ø (9-4-51). ˜ t := v − r, dv2
= (d˜ t + dr)2 = d˜ t2 + dr2 + 2d˜ tdr , dvdr = (d˜ t + dr)dr = d˜ tdr + dr2 , ° dˆ s2 = −(1 − 2M/r)(d˜ t2 + dr2 + 2d˜ tdr) + 2(d˜ tdr + dr2) = −(1 − 2M/r)d˜ t2 + (1 + 2M/r)dr2 + (4M/r)d˜ tdr , Ø (9-4-54). ˜13. Ê ¹ § º Eddington Þ¸Ã {u, r, θ, ϕ} (u ≡ t − r∗) ÕË¹º ²É Á¦¯ dˆ s2 = (1 − 2M/r)[−dt2 + (1 − 2M/r)−2dr2] = (1 − 2M/r)[−dt2 + dr2 ∗ ] = (1 − 2M/r)d(r∗ + t)d(r∗ − t) = (1 − 2M/r)d(2r∗ + u)(−du) = −2(1 − 2M/r)dudr∗ − (1 − 2M/r)du2 = −(1 − 2M/r)du2 − 2dudr , µ ¹ § º Eddington Þ¸Ã {u, r, θ, ϕ} ÕË¹º *14. ÓÅ (∂/∂V )a É (∂/∂U)a Õ ξa [ © (9-4-40)] N1 É N2 É Á Killing Æ (9-4-26) Æ gV U = −16M3 r e−r/2M , gab ξaξb = gab 1 (4M)2 V ∂ ∂V a − U ∂ ∂U a V ∂ ∂V b − U ∂ ∂U b = − V U (4M)2 gab ∂ ∂V a ∂ ∂U b + gab ∂ ∂U a ∂ ∂V b = − V U (4M)2 2gV U = V U 2M r e−r/2M . N1 É U = 0, N2 É V = 0, Æ N1 É N2 É ξa Á Killing *15. ´ 9-21 ¸ ´ 9-23. Ó°ÁÝª´ Õ ∆τ′/∆τ Ê (9-4-58) Õ ¦ µÑ (1) U ≡ −e(r∗ −t)/4M Þ º Á ÛËÉ Æ º ± ËÉ Ó ÅÔ Ë Ô±¦ dU Õ ¿º

151 ( »Ã dτ′ É dτ), Ç¦ ×ÇµÔ (9-4-58). (2)
Ê dτ ºÊ dU Õ Ò ¢ Ò¯ E º dt/dτ É dr/dτ Õ ¯ §9.1 ÕÕ Ô ÖÆ ÒÅÕ Ñ¤Ê°² £¯ Í× θ É ϕ ( Ø dθ = dϕ = 0), ÍÔ · ÁÀ G(τ) º± Õ Ë ( T∼X ´ ºÈ A Õ ËØ), Õ 4 ¨ Za = ( ∂ ∂τ )a. ²Á Killing ξa = ( ∂ ∂t )a ÕÓ Ë Í¯ 4 ¨ ¦ÑÔ −1 = ZaZa = gab ∂ ∂τ a ∂ ∂τ b = g00(dt)a(dt)b ∂ ∂τ a ∂ ∂τ b = g00 dt dτ 2 , Ô dt dτ = (−g00)−1/2 = 1 − 2M r −1/2 ≡ χ−1(r) , Ø Ã ξa = χZa. ÁÀ p É p′ Á ÛË ( Ø× U Ë) ÎÃÕ ¿ º± ËÉÕ Þ¯ªÛµ Ã ω′ ω = χ χ′ Ð λ′ λ = χ′ χ , Ø º Í ¨Õ (9-2-2). ∆τ É ∆τ′ » × U Ë G(τ) (p Þ) É G′(τ′) (p′ Þ) ÔÕË ( Ë©¯©´), ªÛ Ã ∆τ′ ∆τ = χ′ χ = χ(r(p′)) χ(r(p)) = [1 − 2M/r(p′)]1/2 [1 − 2M/r(p)]1/2 . Å p À º ± G(˜ τ) ÕÞ p′ À º± Õ Þ ( ©´), É Õ ∆τ À ∆˜ τ: ∆τ′ ∆˜ τ = χ(r(p′)) χ(r(p)) = χ′ χ . ° µ (9-4-55). «° Üº ( ) « ¯ º ÞÞÕ Ë ² ( Æ ) Á ÛË γ(τ), ° 4 ¨ ( ∂ ∂τ )a. Ä 4-3-3 Æ· ÛËÕ¯ E = −gab ∂ ∂t a ∂ ∂τ b = −g00(dt)b ∂ ∂τ b = −g00 dt dτ = 1 − 2M r dt dτ = χ2(r) dt dτ , Ø dt dτ = E χ2(r) ,

152 ° r = r(p), º ¦¦ Õ ² [
Ø (9-1-4)]. º 4 ¨Õ ¦ «Ô −1 = gab ∂ ∂τ a ∂ ∂τ b = g00(dt)a(dt)b + g11(dr)a(dr)b ∂ ∂τ a ∂ ∂τ b = g00 dt dτ 2 + g11 dr dτ 2 = −χ2(χ−2E)2 + χ−2 dr dτ 2 , Ø dr dτ 2 = E2 − χ2(r) . ° γ(τ) Æ ¯ dr dτ = − E2 − χ2 . ° «° ÛË Ã¯ dr dt = − E2 − χ2 χ−2E = − χ2 E2 − χ2 E . Ú Ò U = −e(r∗ −t)/4M , Æ dU = U 1 4M (dr∗ − dt) = U 4M 1 − 2M r −1 dr − dt = U 4M [χ−2dr − dt] = U 4M χ−2 dr dt − 1 dt = U 4M − E2 − χ2 E − 1 dt = − U 4M E2 − χ2 + E E dt = − U 4M E2 − χ2(r(p)) + E E E χ2(r(p)) dτ = − U 4M E2 − χ2(r(p)) + E χ2(r(p)) dτ . ¦ ¯«° º± Õ Ë G′(τ′), ° r = ¯ dU = U 1 4M (−dt) = − U 4M [χ−1(r(p′))dτ′] . ²· Í×Ô dτ′ dτ = χ(r(p′))( E2 − χ2(r(p)) + E) χ2(r(p)) = χ′( E2 − χ2 + E) χ2 , Ø (9-4-59), ° χ = [1 − 2M/r(p)]1/2, χ′ = [1 − 2M/r(p′)]1/2. ª ÌÀ ±À¹ Õ¢℄µ² ÒÕ ± (r = 2M) ¹Ý¡ª (r = 0) ¯ ( ª M = 3M⊙ ·¥). ℄¬· ¢ ± ÙÞ¯À¹ Õ¢℄µ² Ò γ(τ), dr dτ = − E2 − χ2(r) ,

153 E Ö Ú ¯ ¯« χ(r) = (1 −
2M/r)1/2. ¬ ß ∆τ = − 0 r=2M dr E2 − 1 + 2M/r = M (E2 − 1)3/2 2E(E2 − 1)1/2 − ln 2E(E2 − 1)1/2 + 2E2 − 1 . Ô©°ÙÞÚ Û E, ÇÙ ¯ ∆τ. § ±¯ÇÆ E ¥ Ä ¥ Û 1 ( ª ª ¯ µ²° ¥ ). ∆τ = lim E→1 M (E2 − 1)3/2 2E(E2 − 1)1/2 − ln 2E(E2 − 1)1/2 + 2E2 − 1 = 4M 3 . ©°¥ M = 3M⊙, ∆τ = 4M⊙. ½ Ú ß¯ Ó ∆τ = 4GM⊙ c2 c = 4 × 6.67 × 10−11 × 1.99 × 1030 (3 × 108)3 = 1.97 × 10−5 s . Ô §9.4.6 Ë©Ø Æ « 10 Ã¤ÂÈ³¦ ˜1. Ó Å§ (10-1-12) Õ (3)Rabc d Ø× (3)Rab cd = 2 ¯ R−2δa [cδb d]. Æ 3 Ë¹ dl2 = ¯ R2[dψ2 + sin2 ψ(dθ2 + sin2 θ dϕ2)] Æ§ g11 ≡ gψψ = ¯ R2 , g22 ≡ gθθ = ( ¯ R sin ψ)2 , g33 ≡ gϕϕ = ( ¯ R sin ψ sin θ)2 ; × g11 = ¯ R−2 , g22 = ( ¯ R sin ψ)−2 , g33 = ( ¯ R sin ψ sin θ)−2 . ° ¯ g22,1 = ¯ R2 sin 2ψ , g33,1 = ¯ R2 sin 2ψ sin2 θ , g33,2 = ¯ R2 sin2 ψ sin 2θ . Â (3-4-19) Ô Ñ± (Mathematica): Γ1 22 = − sin ψ cos ψ , Γ1 33 = − sin ψ cos ψ sin2 θ , Γ2 12 = Γ2 21 = cot ψ , Γ2 33 = − sin θ cos θ , Γ3 13 = Γ3 31 = cot ψ , Γ3 23 = Γ3 32 = cot θ .

154 Â (3-4-20’) Ô ÂÙ (Mathematica): R121 2 = 1
, R122 1 = − sin2 ψ , R131 3 = 1 , R133 1 = − sin2 ψ sin2 θ , R211 2 = −1 , R212 1 = sin2 ψ , R232 3 = sin2 ψ , R233 2 = − sin2 ψ sin2 θ , R311 3 = −1 , R313 1 = sin2 ψ sin2 θ , R322 3 = − sin2 ψ , R323 2 = sin2 ψ sin2 θ , °« R1212 = −R1221 = −R2112 = R2121 = ¯ R2 sin2 ψ , R1313 = −R1331 = −R3113 = R3131 = ¯ R2 sin2 ψ sin2 θ , R2323 = −R2332 = −R3223 = R3232 = ¯ R2 sin4 ψ sin2 θ . ° R12 12 = g11R121 2 = ¯ R−2 = ¯ R−2(δ1 1δ2 2 − δ1 2δ2 1) , R12 21 = g22R122 1 = − ¯ R−2 = ¯ R−2(δ1 2δ2 1 − δ1 1δ2 2) , R13 13 = g11R131 3 = ¯ R−2 = ¯ R−2(δ1 1δ3 3 − δ1 3δ3 1) , R13 31 = g33R133 1 = − ¯ R−2 = ¯ R−2(δ1 3δ3 1 − δ1 1δ3 3) , R21 12 = g11R211 2 = − ¯ R−2 = ¯ R−2(δ2 1δ1 2 − δ2 2δ1 1) , R21 21 = g22R212 1 = ¯ R−2 = ¯ R−2(δ2 2δ1 1 − δ2 1δ1 2) , R23 23 = g22R232 3 = ¯ R−2 = ¯ R−2(δ2 2δ3 3 − δ2 3δ3 2) , R23 32 = g33R233 2 = − ¯ R−2 = ¯ R−2(δ2 3δ3 2 − δ2 2δ3 3) , R31 13 = g11R311 3 = − ¯ R−2 = ¯ R−2(δ3 1δ1 3 − δ3 3δ1 1) , R31 31 = g33R313 1 = ¯ R−2 = ¯ R−2(δ3 3δ1 1 − δ3 1δ1 3) , R32 23 = g22R322 3 = − ¯ R−2 = ¯ R−2(δ3 2δ2 3 − δ3 3δ2 2) , R32 32 = g33R323 2 = ¯ R−2 = ¯ R−2(δ3 3δ2 2 − δ3 2δ2 3) . Ø Rab cd = 2 ¯ R−2δa [cδb d], 3 ¦¯ K = ¯ R−2 > 0.

155 ℄ 3 ¤Õ dl2 = ¯ ξ2[dψ2 + sinh2
ψ(dθ2 + sin2 θ dϕ2)] Ù¨ g11 ≡ gψψ = ¯ ξ2 , g22 ≡ gθθ = (¯ ξ sinh ψ)2 , g33 ≡ gϕϕ = (¯ ξ sinh ψ sin θ)2 ; Í g11 = ¯ ξ−2 , g22 = (¯ ξ sinh ψ)−2 , g33 = (¯ ξ sinh ψ sin θ)−2 . ¯ g22,1 = ¯ ξ2 sinh 2ψ , g33,1 = ¯ ξ2 sinh 2ψ sin2 θ , g33,2 = ¯ ξ2 sinh2 ψ sin 2θ . Øª (3-4-19) ß¯ È (Mathematica): Γ1 22 = − sinh ψ cosh ψ , Γ1 33 = − sinh ψ cosh ψ sin2 θ , Γ2 12 = Γ2 21 = coth ψ , Γ2 33 = − sin θ cos θ , Γ3 13 = Γ3 31 = coth ψ , Γ3 23 = Γ3 32 = cot θ . Øª (3-4-20’) ß¯ ÍÑÐ (Mathematica): R121 2 = −1 , R122 1 = sinh2 ψ , R131 3 = −1 , R133 1 = sinh2 ψ sin2 θ , R211 2 = 1 , R212 1 = − sinh2 ψ , R232 3 = − sinh2 ψ , R233 2 = sinh2 ψ sin2 θ , R311 3 = 1 , R313 1 = − sinh2 ψ sin2 θ , R322 3 = sinh2 ψ , R323 2 = − sinh2 ψ sin2 θ , ª Á R1212 = −R1221 = −R2112 = R2121 = −¯ ξ2 sinh2 ψ , R1313 = −R1331 = −R3113 = R3131 = −¯ ξ2 sinh2 ψ sin2 θ , R2323 = −R2332 = −R3223 = R3232 = −¯ ξ2 sinh4 ψ sin2 θ .

156 R12 12 = g11R121 2 = −¯ ξ−2 =
−¯ ξ−2(δ1 1δ2 2 − δ1 2δ2 1) , R12 21 = g22R122 1 = ¯ ξ−2 = −¯ ξ−2(δ1 2δ2 1 − δ1 1δ2 2) , R13 13 = g11R131 3 = −¯ ξ−2 = −¯ ξ−2(δ1 1δ3 3 − δ1 3δ3 1) , R13 31 = g33R133 1 = ¯ ξ−2 = −¯ ξ−2(δ1 3δ3 1 − δ1 1δ3 3) , R21 12 = g11R211 2 = ¯ ξ−2 = −¯ ξ−2(δ2 1δ1 2 − δ2 2δ1 1) , R21 21 = g22R212 1 = −¯ ξ−2 = −¯ ξ−2(δ2 2δ1 1 − δ2 1δ1 2) , R23 23 = g22R232 3 = −¯ ξ−2 = −¯ ξ−2(δ2 2δ3 3 − δ2 3δ3 2) , R23 32 = g33R233 2 = ¯ ξ−2 = −¯ ξ−2(δ2 3δ3 2 − δ2 2δ3 3) , R31 13 = g11R311 3 = ¯ ξ−2 = −¯ ξ−2(δ3 1δ1 3 − δ3 3δ1 1) , R31 31 = g33R313 1 = −¯ ξ−2 = −¯ ξ−2(δ3 3δ1 1 − δ3 1δ1 3) , R32 23 = g22R322 3 = ¯ ξ−2 = −¯ ξ−2(δ3 2δ2 3 − δ3 3δ2 2) , R32 32 = g33R323 2 = −¯ ξ−2 = −¯ ξ−2(δ3 3δ2 2 − δ3 2δ2 3) . Ô Rab cd = −2¯ ξ−2δa [cδb d], É ¤ 3 É ¯ K = −¯ ξ−2 < 0. 2. ÓÅ ±Ñ Õ Ë ÛË É (10-2-5) Õ Ñ±º × (5-7-2) Ú℄¦½ Æ Æ ±Ñ Õ Ë² ¦Þ¸Ë Í¯°» (4 ¨) Za = (∂/∂t)a. » §3.3 1, « ÛË Ø× Zb∇b Za = 0. » (5-7-2): ∂ ∂xν b ∇b ∂ ∂xµ a = Γσ µν ∂ ∂xσ a , Zb∇b Za = ∂ ∂t b ∇b ∂ ∂t a = Γσ 00 ∂ ∂xσ a = 0 , Ú ¦ (10-2-5) Õ Ñ±Æ Γσ 00 = 0. ±Ñ Õ Ë ÛË 3. Ó ÁÅ ÑÊ´ Ú ¨ (10-2-8): (a) Å¢ ¦Á ÛË η(β) (β Ì ) dω/dβ = −KaKb∇aZb , ° Ka ≡ (∂/∂β)a , Za ≡ (∂/∂t)a , ω ≡ −gab ZaKb . (b) Å¢ ∇aZb = (˙ a/a)hab , ° hab gab ¿ ÉÕ Ñ§ ¯ ˙ a ≡ da/dt. ÆÅ¢ ∇aZb ¦ ¯Ø Za∇aZb = 0 = Zb∇aZb , Å¢ Å × ³Ý ° (∂/∂xi)a(∂/∂xj)b (i, j = 1, 2, 3) ÔÍ± À

157 (c) É (a) ¡ (b) Õ ÀµÊ dω/ω =
−da/a, Ô (10-2-8). (a) ω ¸ Ä ¯ dω dβ = Kc∇cω = −Kc∇c(gab ZaKb) = −Kc∇c(Zb Kb) = −KcKb∇cZb − KcZb ∇cKb = −KaKb∇aZb , Ú ¦ É Á ÛËÕÑ Kc∇cKb = 0. (b) É É¦ Õ ÀÆÓ ±Ñ Õ Ë ÛË¯ Õ» Ø× Za∇aZc = 0 ( ©É ÕÅ¢). Ð § gbc Ý Ô gbc Za∇aZc = Za∇aZb = 0. ¦ ¯ 4 ¨ Za Õ ¦Ñ ZaZa = −1 Ó 0 = ∇a(Zb Zb) = Zb∇aZb + Zb ∇aZb = 2Zb∇aZb ( Ò § ²¾ ª±ÕÐ Ñ). ß Za∇aZb = 0 = Zb∇aZb , Æ ∇aZb ² Za ÃÆ¯ ¦ ¯ Ø Õ Í ¦Ë¸ É Ã ∇aZb = ∂aZb − Γc ab Zc , Ô ∂ ∂xi a ∂ ∂xj b ∇aZb = ∂ ∂xi a ∂ ∂xj b ∂aZb − ∂ ∂xi a ∂ ∂xj b Γc ab Zc = ∂iZj − Γσ ijZσ . ° Zσ = ( ∂ ∂xσ )aZa = ( ∂ ∂xσ )a[−(dt)a] = −δ0 σ, Zj = 0, Ô ∂ ∂xi a ∂ ∂xj b ∇aZb = Γσ ijδ0 σ = Γ0 ij = a˙ a(1 − kr2)−1δ1 iδ1 j + a˙ ar2δ2 iδ2 j + a˙ ar2 sin2 θδ3 iδ3 j = a˙ a[a−2g11δ1 iδ1 j + a−2g22δ2 iδ2 j + a−2g33δ3 iδ3 j] = a−1 ˙ agij[δ1 iδ1 j + δ2 iδ2 j + δ3 iδ3 j] = a−1 ˙ ahij[δ1 iδ1 j + δ2 iδ2 j + δ3 iδ3 j] = a−1 ˙ ahab ∂ ∂xi a ∂ ∂xj b , ∇aZb = a−1 ˙ ahab . (c) Í (a) É (b) Õ À¯ ( Î Ka ² Á ÛË) dω dβ = −KaKb∇aZb = −KaKb(a−1 ˙ ahab ) = −a−1 ˙ ahab ∂ ∂t a dt dβ + ∂ ∂r a dr dβ ∂ ∂t b dt dβ + ∂ ∂r b dr dβ = −a−1 ˙ a h00 dt dβ 2 + h11 dr dβ 2 (h00 = 0) = −a−1 ˙ ah11 dr dβ 2 = −a−1 ˙ ag11 dr dβ 2

158 = − a˙ a 1 − kr2 dr dβ
2 (10-2-7) = − ˙ a a dt dβ 2 = − 1 a da dt dt dβ ω = − ω a da dβ , Ø dω/ω = −da/a, Ô ω = ω0a−1. 4. ´ ´ a = 0 Î a0 ≡ a(t0) ÆÓÕ ¦ ¦ a Ê ¤ ×´ ÕÅ§ Ò Î·ÊÆÓÕ ¦¯ · a ÊÍ Õ´ ¯ t Ý a ÕÄ (a) (10-2-29a)–(10-2-29c) É (10-2-25) Ê Å¢ Λ = 0 Õ ´ Õ Ä ÅÅ Ê « Ω0 = 1, t = 2 3 H−1 0 a a0 3/2 , « Ω0 > 1, t = H−1 0 Ω0 2(Ω0 − 1)3/2 cos−1 1−2(1−Ω−1 0 ) a a0 − 1 Ω0 − 1 Ω0 a a0 −(Ω0 −1) a a0 2 1/2 , « Ω0 < 1, t = H−1 0 −Ω0 2(1 − Ω0)3/2 cosh−1 1−2(1−Ω−1 0 ) a a0 + 1 1 − Ω0 Ω0 a a0 +(1−Ω0) a a0 2 1/2 . (b) ÉÅ ÑÊ Ω0 = 1, Ω0 > 1 É Ω0 < 1 Å ¾ Å ´ t0 Õº (a) ÂÂ H(t) = ˙ a(t)/a(t) É § Ω(t), (10-3-8) É(10-3-12) Ê H2 = 8πρ 3 − k a2 , Ω = 8πρ 3H2 . Ê ρ É a: k = 0 ¯ Ω = 1, ρ = 3H2 8π ; k = ±1 ¯ Ω > < 1, ρ = 3H2Ω 8π , a = 1 |Ω − 1|1/2|H| . º (10-2-25) Æ« ( ) ´ 8πρa3/3 = A ¯° « k = 0 (Ω = 1): A = 8π 3 3H2 8π a3 = H2a3 = H2 0 a3 0 ;

159 « k = ±1 (Ω > < 1): A
= 8π 3 3H2Ω 8π 1 |Ω − 1|3/2|H|3 = Ω |Ω − 1|3/2|H| = Ω0 |Ω0 − 1|3/2|H0 | = Ω0a0 |Ω0 − 1| , Ú ¦ É a0 = 1 |Ω0 −1|1/2|H0 | . Ò ´ ¯ H0 > 0, |H0 | = H0. É Ã Â ´ Õ ¯ (10-2-29a)–(10-2-29c). « k = 0 (Ω0 = 1), (10-2-29b) Ê t = 4 9A 1/2 a3/2 = 2 3A1/2 a3/2 = 2 3(H2 0 a3 0 )1/2 a3/2 = 2 3 H−1 0 a a0 3/2 . « k = +1 (Ω0 > 1), (10-2-29a) Ü¦ Ê ˆ t = arccos 1 − 2a A = arccos 1 − 2a Ω0a0/(Ω0 − 1) = arccos 1 − 2(1 − Ω−1 0 ) a a0 , sin ˆ t = (1 − cos2 ˆ t)1/2 = 1 − 1 − 2(1 − Ω−1 0 ) a a0 2 1/2 = 4(1 − Ω−1 0 ) a a0 − 4(1 − Ω−1 0 )2 a a0 2 1/2 = 2(Ω0 − 1)1/2 Ω0 Ω0 a a0 − (Ω0 − 1) a a0 2 1/2 , ÂÜ Ô t = Ω0 2(Ω0 − 1)3/2H0 arccos 1 − 2(1 − Ω−1 0 ) a a0 − 2(Ω0 − 1)1/2 Ω0 Ω0 a a0 − (Ω0 − 1) a a0 2 1/2 = H−1 0 Ω0 2(Ω0 − 1)3/2 arccos 1 − 2(1 − Ω−1 0 ) a a0 − 1 Ω0 − 1 Ω0 a a0 − (Ω0 − 1) a a0 2 1/2 . « k = −1 (Ω0 < 1), (10-2-29c) Ü¦ Ê ˆ t = arccosh 1 + 2a A = arccosh 1 + 2a Ω0a0/(1 − Ω0) = arccosh 1 − 2(1 − Ω−1 0 ) a a0 ,

160 sinh ˆ t = (cosh2 ˆ t − 1)1/2
= 1 − 2(1 − Ω−1 0 ) a a0 2 − 1 1/2 = − 4(1 − Ω−1 0 ) a a0 + 4(1 − Ω−1 0 )2 a a0 2 1/2 = 2(1 − Ω0)1/2 Ω0 Ω0 a a0 + (1 − Ω0) a a0 2 1/2 , ÂÜ Ô t = Ω0 2(1 − Ω0)3/2H0 − arccosh 1 − 2(1 − Ω−1 0 ) a a0 + 2(1 − Ω0)1/2 Ω0 Ω0 a a0 + (1 − Ω0) a a0 2 1/2 = H−1 0 − Ω0 2(1 − Ω0)3/2 arccosh 1 − 2(1 − Ω−1 0 ) a a0 + 1 1 − Ω0 Ω0 a a0 + (1 − Ω0) a a0 2 1/2 = H−1 0 Ω0 2(Ω0 − 1)3/2 arccosh 1 − 2(1 − Ω−1 0 ) a a0 − 1 Ω0 − 1 Ω0 a a0 − (Ω0 − 1) a a0 2 1/2 , ©ÍÔ ·¦ ¾ Õ arccos À arccosh, °¯ ¶ (b) «° ´ ¯ É a = a0, Ô Ä Ω0 = 1, t0 = 2 3 H−1 0 ; Ä Ω0 > 1, t0 = H−1 0 Ω0 2(Ω0 − 1)3/2 arccos 1 − 2(1 − Ω−1 0 ) − 1 Ω0 − 1 Ω0 − (Ω0 − 1) 1/2 = H−1 0 Ω0 2(Ω0 − 1)3/2 arccos(2Ω−1 0 − 1) − 1 Ω0 − 1 ; Ä Ω0 < 1, t0 = H−1 0 − Ω0 2(1 − Ω0)3/2 arccosh 1 − 2(1 − Ω−1 0 ) + 1 1 − Ω0 Ω0 + (1 − Ω0) 1/2 = H−1 0 − Ω0 2(1 − Ω0)3/2 arccosh (2Ω−1 0 − 1) + 1 1 − Ω0 . Å¢ «° Ω0 > < 1 ¾ ¯ÁÀ ÕÊ Ω0 → 1, ¤ Ò t0 = 2 3 H−1 0 . ÁÀ t0 º À t0 = H−1 0 f(Ω0) ,

161 ° f(x) =      
   x 2(x−1)3/2 arccos(2x−1 − 1) − 1 x−1 , x < 1 , 2 3 , x = 1 , x 2(x−1)3/2 arccosh(2x−1 − 1) − 1 x−1 , x > 1 . f(x) Ý¨Ä ¯ f(0) = 1 f(2) = π 2 − 1 = 0.5708. ˜5. ÓÅÃ Λ Õ £ ÃØ (Tab = 0) £ Õ È§ Ã Λ Õ £ ÃÊ Ô R ² T Õ Ã¯ Ã Õ R, µ È Tab = 0 Rab ¯ Æ Ã Λ Õ £ Ã Gab + Λgab = Rab − 1 2 Rgab + Λgab = 8πTab . ³Ý ( Ô) gab Ô R − 1 2 Rδa a + Λδa a = 8πTa a = 8πT , ° T ≡ Ta a ¯£ ÕÔ 4 Õ δa a = 4, 4Λ − R = 8πT , Ø R = 4Λ − 8πT. £ ÃÔ Rab − 1 2 (4Λ − 8πT)gab + Λgab = Rab − Λgab + 4πTgab = 8πTab , ° ÆÇ± Ø× Rab = Λgab + 8πTab − 4πTgab . ©Ø (Tab = 0, T = 0), Rab = Λgab = 0, £ È 6. ÓÅ k = −1 É k = +1 Õ RW § £ ( ¶ ) ÈÕ (10-4-2) ˆ t, (10-1-23a) É (10-1-23c) ÕË¹¸ Þ¸ ˆ t, ψ, θ, ϕ ºÊ¯ »« k = −1 É k = +1 Õ¾ ÜÁÅÞ¸¶ (ˆ t, ψ) → (˜ t, ˜ r): « k = −1, ˜ t = eˆ t cosh ψ , ˜ r = eˆ t sinh ψ , « k = +1, ˜ t = tan 1 2 (ˆ t + ψ) + tan 1 2 (ˆ t − ψ) , ˜ r = tan 1 2 (ˆ t + ψ) − tan 1 2 (ˆ t − ψ) , Ë¹ » ÁÅÕ¢Ç È

162 « k = −1, ds2 = a2(t(ˆ t))e−2ˆ t[−d˜
t2 + d˜ r2 + ˜ r2(dθ2 + sin2 θ dϕ2)] , « k = +1, ds2 = a2(t(ˆ t)) 4 (cos ˆ t + cos ψ)2[−d˜ t2 + d˜ r2 + ˜ r2(dθ2 + sin2 θ dϕ2)] . Æ Â Þ¸ ˆ t(t) ≡ t 0 dt′/a(t′) , dˆ t = dt/a(t) Ð a(ˆ t)dˆ t = dt, a2(ˆ t)dˆ t2 = dt2. ° Ë¹ (10-1-23a) É (10-1-23c) » ds2 = a2(ˆ t)[−dˆ t2 + dψ2 + sin2 ψ(dθ2 + sin2 θ dϕ2)] , [ «k = +1 ] , ds2 = a2(ˆ t)[−dˆ t2 + dψ2 + sinh2 ψ(dθ2 + sin2 θ dϕ2)] , [ «k = −1 ] . « k = +1, ˜ t = tan 1 2 (ˆ t + ψ) + tan 1 2 (ˆ t − ψ) , ˜ r = tan 1 2 (ˆ t + ψ) − tan 1 2 (ˆ t − ψ) , ° ¶ ˆ t = arctan 1 2 (˜ t + ˜ r) + arctan 1 2 (˜ t − ˜ r) , ψ = arctan 1 2 (˜ t + ˜ r) − arctan 1 2 (˜ t − ˜ r) . ° dˆ t = (d˜ t + d˜ r)/2 1 + [1 2 (˜ t + ˜ r)]2 + (d˜ t − d˜ r)/2 1 + [1 2 (˜ t − ˜ r)]2 = (d˜ t + d˜ r)/2 1 + tan2 1 2 (ˆ t + ψ) + (d˜ t − d˜ r)/2 1 + tan2 1 2 (ˆ t − ψ) = cos2 1 2 (ˆ t + ψ) 2 (d˜ t + d˜ r) + cos2 1 2 (ˆ t − ψ) 2 (d˜ t − d˜ r) , dψ = (d˜ t + d˜ r)/2 1 + [1 2 (˜ t + ˜ r)]2 − (d˜ t − d˜ r)/2 1 + [1 2 (˜ t − ˜ r)]2 = (d˜ t + d˜ r)/2 1 + tan2 1 2 (ˆ t + ψ) − (d˜ t − d˜ r)/2 1 + tan2 1 2 (ˆ t − ψ) = cos2 1 2 (ˆ t + ψ) 2 (d˜ t + d˜ r) − cos2 1 2 (ˆ t − ψ) 2 (d˜ t − d˜ r) , Ô −dˆ t2 + dψ2 = −(dˆ t + dψ)(dˆ t − dψ) = − cos2[(ˆ t + ψ)/2](d˜ t + d˜ r) cos2[(ˆ t − ψ)/2](d˜ t − d˜ r) = {cos[(ˆ t + ψ)/2] cos[(ˆ t − ψ)/2]}2(−d˜ t2 + d˜ r2) = 1 4 (cos ˆ t + cos ψ)2(−d˜ t2 + d˜ r2) .

163 ¦ ¯ sin2 ψ = tan2 ψ 1+tan2 ψ
, tan ψ = tan arctan 1 2 (˜ t + ˜ r) − arctan 1 2 (˜ t − ˜ r) = 1 2 (˜ t + ˜ r) − 1 2 (˜ t − ˜ r) 1 + 1 4 (˜ t + ˜ r)(˜ t − ˜ r) = ˜ r 1 + 1 4 (˜ t + ˜ r)(˜ t − ˜ r) , Ô sin2 ψ = ˜ r2 ˜ r2 + [1 + 1 4 (˜ t + ˜ r)(˜ t − ˜ r)]2 , ° § ˜ r2 + 1 + 1 4 (˜ t + ˜ r)(˜ t − ˜ r) 2 = tan 1 2 (ˆ t + ψ) − tan 1 2 (ˆ t − ψ) 2 + 1 + tan 1 2 (ˆ t + ψ) tan 1 2 (ˆ t − ψ) 2 = 4 (cos ˆ t + cos ψ)2 . sin2 ψ = 1 4 (cos ˆ t + cos ψ)2 ˜ r2 . ¬É À k = +1 ÕË¹º ds2 = a2 1 4 (cos ˆ t + cos ψ)2(−d˜ t2 + d˜ r2) + 1 4 (cos ˆ t + cos ψ)2 ˜ r2(dθ2 + sin2 θ dϕ2) = a2 4 (cos ˆ t + cos ψ)2[−d˜ t2 + d˜ r2 + ˜ r2(dθ2 + sin2 θ dϕ2)] . « k = −1, ˜ t = eˆ t cosh ψ , ˜ r = eˆ t sinh ψ , ° ¶ ˆ t = 1 2 ln(˜ t2 − ˜ r2) , ψ = arctanh(˜ r/˜ t) . ° dˆ t = ˜ td˜ t − ˜ rd˜ r ˜ t2 − ˜ r2 = e−2ˆ t(˜ td˜ t − ˜ rd˜ r) , dψ = 1 1 − (˜ r/˜ t)2 ˜ td˜ r − ˜ rd˜ t ˜ t2 = ˜ td˜ r − ˜ rd˜ t ˜ t2 − ˜ r2 = e−2ˆ t(˜ td˜ r − ˜ rd˜ t) , Ô −dˆ t2 + dψ2 = −e−4ˆ t(˜ td˜ t − ˜ rd˜ r)2 + e−4ˆ t(˜ td˜ r − ˜ rd˜ t)2 = e−4ˆ t(˜ t2 − ˜ r2)(−d˜ t2 + d˜ r2) = e−2ˆ t(−d˜ t2 + d˜ r2)

164 ¦ ¯ sinh2 ψ = tanh2 ψ 1−tanh2 ψ
, tanh ψ = ˜ r/˜ t, Ô sinh2 ψ = ˜ r2 ˜ t2 − ˜ r2 = e−2ˆ t ˜ r2 . ¬É À k = −1 ÕË¹º ds2 = a2[e−2ˆ t(−d˜ t2 + d˜ r2) + e−2ˆ t ˜ r2(dθ2 + sin2 θ dϕ2)] = a2e−2ˆ t[−d˜ t2 + d˜ r2 + ˜ r2(dθ2 + sin2 θ dϕ2)] . ÔÉ Ñ¯« k = +1 É k = −1, RW § £ ( ¶ ) ÈÕ¯ Î ÃÕÃ Ä » a2 4 (cos ˆ t + cos ψ)2 É a2e−2ˆ t. ˜7. Î p ±Ñ G ËÉÕ¦Þ¯ÓÅ G tp ÕÒ ½ÃØ × (10-4-5). É (10-1-28) É (10-2-7). Æ × É ÞÕ½Ã (10-4-5) Ê DAB(t) = a(t) rB rA dr √ 1 − kr2 . A ±Ñ G ËÉÕ¦Þ p, B Ò ³ ¯² Þ¸ rB, DH (tp) = a(tp) rB 0 dr √ 1 − kr2 . ¦ ¯ ¦² Á ÛËÕ {t(β), r(β)} Ø× Ã (10-2-7): dt dβ 2 = a2 1 − kr2 dr dβ 2 , Ø«Æ Á ÛË Ã dr dt = − √ 1 − kr2 a(t) . Ò ½ÃÕ ( ©´ 10-16) ÆÁ ÛË {0, rB } ÏÒ {tp, 0}, ¯ DH(tp) = a(tp) 0 tp 1 √ 1 − kr2 − √ 1 − kr2 a(t) dt = a(tp) tp 0 dt a(t) . Ø (10-4-5).

165 *˜8. (a) Î η(β) ² (dθ/dβ = dϕ/dβ =
0) Á ÛË¯ p1 = (t1, ψ1, θ, ϕ) É p2 = (t2, ψ2, θ, ϕ) η É Þ¯ÓÅ« k = 1, 0, −1 Å ¾ ¤ ψ2 − ψ1 = t2 t1 dt/a(t) . (b) « k = 1 Õ´ ¯ ±Þ ÊÕ ¦² Ë ÐÕ 3 É » · ÓÅ (b1) « ´ ¯· Ë 3 ÎÚ » Ö¼ ¿ »¯ 3 ¦Þ ( ÙÜ) »Ö¼¦¿ » ¯ ÎÚ ¦ ±Ñ Í¢ ¿ ¯Õ¯ Ò ¦ ±ÑÌÒ Õ ¯º¢¯ÕÌÒÒ ÎÚ [ © Wald (1984) P.106]. (b2) «°Ì´ ¯· Ë 3 ¦Þ ( Ù Ü) »»Ö¼ ¿ » ¯ ¦ ±Ñ Õ ¦ ¤℄ ÌÒ Ò Æ (a) ÒÁ ÛËØ×Õ « (10-2-7), Õº dr dt = √ 1 − kr2 a(t) . É Ã (10-1-24), « k = +1, ψ2 − ψ1 = arcsin r2 − arcsin r1 = r2 r1 dr √ 1 − r2 = t2 t1 dt a(t) ; « k = 0, ψ2 − ψ1 = r2 − r1 = r2 r1 dr = t2 t1 dt a(t) ; « k = −1, ψ2 − ψ1 = arcsinh r2 − arcsinh r1 = r2 r1 dr √ 1 + r2 = t2 t1 dt a(t) . ¯ÔÍ ψ2 − ψ1 = t2 t1 dt a(t) . (b1) « k = 1 Õ ´ ¯Å§ ÒÕ (10-2-29a): a = A(1 − cos ˆ t)/2 , t = A(ˆ t − sin ˆ t)/2 . Æ 3 ÎÚ ˆ t = π, t = Aπ/2. Þ¸ (0, 0, θ, ϕ) Ê Õ Ò ÖÒ (Aπ/2, ψ, θ, ϕ), ° ψ Þ¸ » (a) Õ À ψ = Aπ/2 0 dt a(t) = π 0 dˆ t = π ,

166 ÃÆ 3 Õ ¿ » ° 3 Ô (
ÙÜ) Ò¦Þ ˆ t = 2π, t = Aπ, ÒÕ ψ Þ¸ π ¯Ò ÙÜÕ¦Þ ÕÞ¸ (Aπ, 0, θ, ϕ), »ÆÖ¼ 3 Õ¦¿ » º 3 Õ Ó (10-1-29) ÆÓ¯ ψ < π ¯²Ç« Õ 3 Õ Ó ÓÕ ψ 0 sin2 ψ′dψ′ π 0 sin2 ψ′dψ′ = 1 π (ψ − sin ψ cos ψ) . ° ¬ ´ ÎÚ Ç·¯Ø ˆ t < π (t < Aπ/2) ¯ ψ = ˆ t Æ¯ÌÒÒ ¦Õ Ó¬ ¦Õ¶ Ã η(t) = 1 π (ˆ t − sin ˆ t cos ˆ t) , t = A(ˆ t − sin ˆ t)/2 . 0 Ò 1, ´ Ô¯ÌÒÒ ¦ ¼ÀÆª Ô¯ t ( Ø ˆ t) Ï ¯ ˆ t = 12 A 1/3 t1/3 + 1 5A t + O(t5/2) , η(t) = 8 Aπ t − 311041/3 5A5/3π t5/3 + O(t7/3) , ÌÒÒ É ¬ ¦ËÑ t Aπ/2 ( Ø ˆ t π) ¯ ¯ ˆ t ≡ π −ˆ t, ¯ t ≡ Aπ 2 − t, ÉÕ Ã¶ η(¯ t) = 1 − 1 π (¯ ˆ t − sin¯ ˆ t cos¯ ˆ t) , ¯ t = A(¯ ˆ t + sin¯ ˆ t)/2 . ÀÆªÔ ¯ ˆ t = 1 A ¯ t + 1 12A3 ¯ t3 + O(¯ t5) , η(¯ t) = 1 − 2 3A3π ¯ t3 − 1 30A5π ¯ t5 + O(¯ t7) . ´ ÎÚ ·¯ η(t) Å ° É η(t) = 1 − 2 3A3π Aπ 2 − t 3 − 1 30A5π Aπ 2 − t 5 + · · · . (b2) « k = 1 Õ°Ì´ ¯Å§ ÒÕ (10-2-24a): a = 2Bt − t2 . Æ 3 ÎÚ t = B. Þ¸ (0, 0, θ, ϕ) Ê Õ Ò ÖÒ (B, ψ, θ, ϕ), ° ψ Þ¸ » (a) Õ À ψ = B 0 dt a(t) = B 0 dt √ 2Bt − t2 = π 2 ,

167 ÃÆ 3 Õ 1/4 ¿ » ° 3 Ô
( ÙÜ) Ò¦Þ t = 2B, ÒÕ ψ Þ¸ π/2 Î π, Ò ÙÜÕ¦Þ ÕÞ¸ (2B, π, θ, ϕ), »ÆÖ¼ 3 Õ ¿ » Á¦Û£ Ê°Ì´ Ò ÙÜÕÂ¿È ÌÒÒ Â¿ ¦Õ ÊÄ η(t) Õ ψ(t) = t 0 dt′ √ 2Bt′ − t′2 = 2 arctan t 2B − t , Ô η(t) = 1 π (ψ − sin ψ cos ψ) = 1 π 2 arctan t 2B − t − (B − t) √ 2Bt − t2 B2 . (t = 0) ¡Ú ´ (t = B) É ÙÜ (t = 2B) ¶ ÕÐ » η(t) = 4 √ 2 3B3/2π t3/2 − √ 2 5B5/2π t5/2 + O(t7/2) , η(t) = 1 2 + 2 Bπ (t − B) − 1 3B3π (t − B)3 + O((t − B)5) , η(t) = 1 − 4 √ 2 3B3/2π (2B − t)3/2 − √ 2 5B5/2π (2B − t)5/2 + O((2B − t)7/2) . Ç É ÙÜ Þ « Õ¯ Õ« µ Ú ´ ¯ ÀË Ã η(t) + η(2B − t) = 1 . {(Dis)claimer: Since I thank no one for helping me in solving these problems, all errors are deﬁnitely my own. — 639 }

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