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
   

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2. Gravitational collapse: Two views
   Falling with collapsing star
   Far away
   

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3. Gravitational collapse: Two views
   Falling with collapsing star
   Far away
   

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4. Gravitational collapse: View from
   inside
   Sending light rays at regular intervals
   

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5. GOTO Mathematica
   Remind behavior of light rays emitted in Schwarzschild spacetime
   light-rays near black holes, Eddington-Finkelstein coords.nb
   

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6. Hartle
   ˜
   t
   r
   Spacetime
   diagram of
   gravitational
   collapse
   

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7. View from inside
   

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8. Hartle
   ˜
   t
   r
   

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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
   

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10. Hartle
    ˜
    t
    r
    distant observer
    Singularity
    hidden
    from
    observers
    outside
    black hole
    

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11. View from inside: Conclusion
    Death crushed at singularity (r=0)
    In a finite proper time
    

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12. View from outside
    

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13. Hartle
    ˜
    t
    r
    distant observer
    period (λ) of
    received signal
    

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14. Hartle
    ˜
    t
    r
    

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15. Hartle
    ˜
    t
    r
    distant observer
    period (λ) of
    received signal
    light from star
    increasingly
    redshifted
    

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16. Misner, Thorne & Wheeler
    open
    gravitational
    collapse in
    Eddington-
    Finkelstein
    coords, Misner.tif
    

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17. Hartle
    ˜
    t
    r
    distant observer
    distant observer never
    sees star cross r=2M
    redshift→∞ as r →2M
    

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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)
    

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19. GOTO Mathematica
    Remind behavior of light rays emitted in Schwarzschild spacetime
    visualizing gravitational collapse.nb
    

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20. http://hubblesite.org/explore_astronomy/black_holes/encyc_mod3_q15.html
    

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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
    

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22. Growth of black holes
    If particles fall into the black hole
    M increases
    Schwarzschild radius rS = 2M increases
    surface area increases
    

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23. RS
    =
    2GM
    c2
    

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24. RS
    =
    2GM
    c2
    

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25. RS
    =
    2GM
    c2
    

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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
    

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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
    

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28. Types of black holes
    Mass M
    Spin a
    Charge Q
    Schwarzschild spacetime
    Kerr spacetime
    Reissner–Nordström spacetime
    

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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
    

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30. Spin
    angular
    velocity
    Period
    Velocity
    A Kerr black hole is spinning
    

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31. event
    horizon
    singularity
    “ergosphere”
    Event horizon radius
    decreases with spin:
    for a/M = 1 (maximal spin)
    Structure of a Kerr black hole
    

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32. Black hole spin generates spacetime whirlwind
    (non-Newtonian effect)
    Huge energy stored in rotating spacetime
    black hole
    Credit: Thorne
    

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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)
    

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34. https://en.wikipedia.org/wiki/File:Orbit_um_ein_rotierendes_schwarzes_Loch_(Animation).gif#file
    

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35. Thorne; W. W. Norton & Company
    Buraco negro Planeta
    

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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)
    

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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
    

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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
    

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39. Hawking radiation
    If BHs have a temperature, then they emit blackbody
    radiation
    F = σT4
    Stefan-Boltzmann law
    

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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
    

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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
    

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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.
    

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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)
    

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44. Hawking radiation
    If BHs have a temperature, then they emit blackbody
    radiation
    ∴ BHs lose mass
    Credit: BBC
    

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45. Hawking (2001); Moonrunner Design Ltd UK and The Book Laboratory TM Inc.
    

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46. AGA0319
    Rodrigo Nemmen
    Black Holes in
    Astrophysics
    Astrophysical Applications of GR III
    

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47. Begin at slide 22
    black hole primer for
    undergrads physics
    

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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
    

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