Black hole physics

Lecture of the course "General Relativity With Astrophysical Applications", taught by Prof. Rodrigo Nemmen (USP).

• Gravitational collapse: View from observer collapsing with the star
• Collapse: View from distant observer
• Growth of black holes
• Black holes have no hair
• Kerr black holes
• Black hole thermodynamics
• Hawking radiation

Credit for the slides/figures belongs to Rodrigo Nemmen, unless otherwise stated.

https://rodrigonemmen.com/teaching/relatividade-geral-e-aplicacoes-astrofisicas/

Transcript

1. AGA0319 Rodrigo Nemmen Black Holes
2. Gravitational collapse: Two views Falling with collapsing star Far away
3. Gravitational collapse: Two views Falling with collapsing star Far away
4. Gravitational collapse: View from inside Sending light rays at regular intervals
5. GOTO Mathematica Remind behavior of light rays emitted in Schwarzschild spacetime light-rays near black holes, Eddington-Finkelstein coords.nb
6. Hartle ˜ t r Spacetime diagram of gravitational collapse
7. View from inside
8. Hartle ˜ t r
9. Hartle ˜ t r distant observer once inside rS=2M, falling observer cannot communicate with distant one anymore Collapse to singularity at r=0 is inevitable
10. Hartle ˜ t r distant observer Singularity hidden from observers outside black hole
11. View from inside: Conclusion Death crushed at singularity (r=0) In a finite proper time
12. View from outside
13. Hartle ˜ t r distant observer period (λ) of received signal
14. Hartle ˜ t r
15. Hartle ˜ t r distant observer period (λ) of received signal light from star increasingly redshifted
16. Misner, Thorne & Wheeler open gravitational collapse in Eddington- Finkelstein coords, Misner.tif
17. Hartle ˜ t r distant observer distant observer never sees star cross r=2M redshift→∞ as r →2M
18. View from outside: Conclusion Distant observer sees gravitational collapse: slow down get redshifted darken All records of star’s history and its properties will be erased from the exterior geometry Reduced to one number: mass M (Schwarzschild spacetime)
19. GOTO Mathematica Remind behavior of light rays emitted in Schwarzschild spacetime visualizing gravitational collapse.nb
20. http://hubblesite.org/explore_astronomy/black_holes/encyc_mod3_q15.html
21. Generic features of gravitational collapse Formation of spacetime singularity: unavoidable once star crosses r=2M (singularity theorems) point where theory breaks down Formation of event horizon: singularities inside event horizon, hidden from external observers Cosmic censorship conjecture: singularities always hidden inside event horizon even for nonspherical collapse Area increase: if mass falls in a black hole, its area will increase analogous to entropy in thermodynamic → laws of black hole thermodynamics
22. Growth of black holes If particles fall into the black hole M increases Schwarzschild radius rS = 2M increases surface area increases
23. RS = 2GM c2
24. RS = 2GM c2
25. RS = 2GM c2
26. Growth of black holes If particles fall into the black hole M increases Schwarzschild radius rS = 2M increases surface area increases There is no limit to how big a BH can grow. From astrophysics: Mmin = 3.6 MSun Mmax ~ 1010 MSun
27. A black hole has no hair All black hole solutions of Einstein’s equation completely characterized by only three externally observable classical parameters: Mass M Spin: angular momentum J Charge Q J ≡ a GM2 c −1 ≤ a ≤ 1 spin parameter No-hair theorem All other information (“hair”=metaphor) disappears behind the event horizon, therefore permanently inaccessible to external observers
28. Types of black holes Mass M Spin a Charge Q Schwarzschild spacetime Kerr spacetime Reissner–Nordström spacetime
29. Kerr black hole Conservation of angular momentum leads to spinning black holes Rotational energy deforms spacetime → Kerr spacetime Kerr metric considerably more complex than Schwarzschild ds2 = −(1 − 2Mr ρ2 ) dt2 − 4Mar sin2 θ ρ2 dϕdt + ρ2 Δ dr2 + ρ2dθ2 + ( r2 + a2 + 2Mra2 sin2 θ ρ2 ) sin2 θdϕ2 a ≡ J/M, ρ2 ≡ r2 + a2 cos2 θ, Δ ≡ r2 − 2Mr + a2
30. Spin angular velocity Period Velocity A Kerr black hole is spinning
31. event horizon singularity “ergosphere” Event horizon radius decreases with spin: for a/M = 1 (maximal spin) Structure of a Kerr black hole
32. Black hole spin generates spacetime whirlwind (non-Newtonian effect) Huge energy stored in rotating spacetime black hole Credit: Thorne
33. spinning BH https://www.youtube.com/watch?v=9MHuhcFQsBg Frame-dragging effect Penrose effect: rotational energy can be extracted (more about this in next lecture)
34. https://en.wikipedia.org/wiki/File:Orbit_um_ein_rotierendes_schwarzes_Loch_(Animation).gif#file
35. Thorne; W. W. Norton & Company Buraco negro Planeta
36. Laws of thermodynamics 0th law: If two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other (define temperature) 1st law: 2nd law: S cannot decrease for isolated system 3rd law: dE = TdS − PdV lim T→0 S = Smin (typically close to zero)
37. Laws of black hole thermodynamics 0th law: A non-rotating BH has uniform gravity at its event horizon 㲗 a BH is at thermal equlibrium 1st law: 2nd law: A cannot decrease 3rd law: Extreme BHs (i.e. maximum spin or charge) have minimum entropy Simpler way to understand complex interactions between BHs. The rules first devised to describe thermodynamics also apply to BHs dE = κ 8π dA + ΩdJ + ΦdQ
38. Laws of black hole thermodynamics BH temperature T = ℏc3 8πkB GM BH entropy Hawking temperature S = kB A 4ℏ Bekenstein-Hawking formula Credit: BBC
39. Hawking radiation If BHs have a temperature, then they emit blackbody radiation F = σT4 Stefan-Boltzmann law
40. 0 10 20 30 40 -10 0 10 20 30 log(T / K) log(M / g) Hawking temperature of a BH T = ℏc3 8πkB GM = 6 × 10−8 ( M M⊙ ) −1 K astrophysical masses
41. 0 10 20 30 40 -10 0 10 20 30 log(T / K) log(M / g) Hawking temperature of a BH T = ℏc3 8πkB GM = 6 × 10−8 ( M M⊙ ) −1 K astrophysical masses M = Planck mass T=2.3 K T=5700 K Do BHs form for M<2 Msun? Nobody knows, but probably not Hawking radiation does not seem to be astrophysically relevant
42. Nature of Hawking radiation 1 19 (FIG. 4.17) Above; Virtual particles appearing and annihilating one another; close to the event horizon of a black hole. One of the pair falls into the black hole while its twin is free to escape. From outside the event horizon it appears that the black hole is radiating the parti- cles that escape. (FIG. 4.16) Left: In empty space particle pairs appear lead a brief existence, and then annihilate one another (FIG. 4.17) Above; Virtual particles appearing and annihilating one another; close to the event horizon of a black hole. One of the pair falls into the black hole while its twin is free to escape. From outside the event horizon it appears that the black hole is radiating the parti- cles that escape. (FIG. 4.16) Left: In empty space particle pairs appear lead a brief existence, and then annihilate one another Hawking (2001); Moonrunner Design Ltd UK and The Book Laboratory TM Inc.
43. Nature of Hawking radiation (FIG. 4.17) Above; Virtual particles appearing and annihilating one another; close to the event horizon of a black hole. One of the pair falls into the black hole while its twin is free to escape. From outside the event horizon it appears that the black hole is radiating the parti- cles that escape. Hawking (1988)
44. Hawking radiation If BHs have a temperature, then they emit blackbody radiation ∴ BHs lose mass Credit: BBC
45. Hawking (2001); Moonrunner Design Ltd UK and The Book Laboratory TM Inc.
46. AGA0319 Rodrigo Nemmen Black Holes in Astrophysics Astrophysical Applications of GR III
47. Begin at slide 22 black hole primer for undergrads physics
48. Supermassive 106-1010 solar masses one in every galactic nucleus 5-30 solar masses ~107 per galaxy Stellar black holes ~1 Mpc ~100 kpc Active galactic nuclei Quasars Radio galaxies black holes Gamma-ray bursts Microquasars 1 pc = 3×1013 km