P.K.Townsend - Black Holes

Acknowledgements

These notes were written to accompany a course taught in Part III of the Cambridge University Mathematical Tripos. There are occasional references to questions on four 'example sheets', which can be found in the Appendix. The writing of these course notes has greatly bene tted from discussions with Gary Gibbons and Stephen Hawking. The organisation of the course was based on unpublished notes of Gary Gibbons and owes much to the 1972 Les Houches and 1986 Cargese lecture notes of Brandon Carter, and to the 1972 lecture notes of Stephen Hawking. Finally, I am very grateful to Tim Perkins for typing the notes in LATEX, producing the diagrams, and putting it all together.

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Contents

1.1The Chandrasekhar Limit . . . . . . . . . . . . . . . . . . . . 6

1.2Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1Test particles: geodesics and a ne parameterization . . . . . 11

2.2Symmetries and Killing Vectors . . . . . . . . . . . . . . . . . 13

2.3Spherically-Symmetric Pressure Free Collapse . . . . . . . . . 15

2.3.1Black Holes and White Holes . . . . . . . . . . . . . . 18

2.3.2Kruskal-Szekeres Coordinates . . . . . . . . . . . . . . 20

2.3.3 Eternal Black Holes . . . . . . . . . . . . . . . . . . . 24

2.3.4Time translation in the Kruskal Manifold . . . . . . . 26

2.3.5Null Hypersurfaces . . . . . . . . . . . . . . . . . . . . 27

2.3.9Tolman Law - Unruh Temperature . . . . . . . . . . . 39

2.4 Carter-Penrose Diagrams . . . . . . . . . . . . . . . . . . . . 40

2.4.1 Conformal Compacti cation . . . . . . . . . . . . . . . 40

2.5Asymptopia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.6The Event Horizon . . . . . . . . . . . . . . . . . . . . . . . . 49

2.7Black Holes vs. Naked Singularities . . . . . . . . . . . . . . . 53

3.2Pressure-Free Collapse to RN . . . . . . . . . . . . . . . . . . 65

3.3Cauchy Horizons . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.4 Isotropic Coordinates for RN . . . . . . . . . . . . . . . . . . 70

3.4.1Nature of Internal 1 in Extreme RN . . . . . . . . . . 74

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4.1.1Spacetime Symmetries . . . . . . . . . . . . . . . . . . 76

4.3The Ergosphere . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.4The Penrose Process . . . . . . . . . . . . . . . . . . . . . . . 88

4.4.1Limits to Energy Extraction . . . . . . . . . . . . . . . 89

4.4.2Super-radiance . . . . . . . . . . . . . . . . . . . . . . 90

5.3.1 Angular Momentum in Axisymmetric Spacetimes . . . 98

5.4Energy Conditions . . . . . . . . . . . . . . . . . . . . . . . . 99

6.1Geodesic Congruences . . . . . . . . . . . . . . . . . . . . . . 101 6.1.1 Expansion and Shear . . . . . . . . . . . . . . . . . . . 106

6.2The Laws of Black Hole Mechanics . . . . . . . . . . . . . . . 109

6.2.1 Zeroth law . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2.2Smarr's Formula . . . . . . . . . . . . . . . . . . . . . 110

6.2.3First Law . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.1Quantization of the Free Scalar Field . . . . . . . . . . . . . . 119

7.2 Particle Production in Non-Stationary Spacetimes . . . . . . 123

7.3Hawking Radiation . . . . . . . . . . . . . . . . . . . . . . . . 125

7.4 Black Holes and Thermodynamics . . . . . . . . . . . . . . . 129

7.4.1The Information Problem . . . . . . . . . . . . . . . . 130

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A.4 Example Sheet 4 . . . . . . . . . . . . . . . . . . . . . . . . . 141

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Chapter 1

Gravitational Collapse

1.1The Chandrasekhar Limit

A Star is a self-gravitating ball of hydrogen atoms supported by thermal pressure P nkT where n is the number density of atoms. In equilibrium,

where hEi is average kinetic energy of atoms. Eventually, fusion at the core must stop, after which the star cools and contracts. Consider the possible nal state of a star at T = 0. The pressure P does not go to zero as T ! 0 because of degeneracy pressure. Since me mp the electrons become degenerate rst, at a number density of one electron in a cube of side Compton wavelength.

Can electron degeneracy pressure support a star from collapse at T = 0?

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So, since n = ne,

E .

~2M 1=3

Rmin Gmem5p=3

.Rmin

The collapse of the star is therefore prevented. It becomes a White Dwarf or a cold, dead star supported by electron degeneracy pressure.

At equilibrium

But the validity of non-relativistic approximation requires that hpei mec, i.e.

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For su ciently large M the electrons would have to be relativistic, in

For smaller M, R must increase until electrons become non-relativistic, in which case the star is supported by electron degeneracy pressure, as we just saw. For larger M, R must continue to decrease, so electron degeneracy pressure cannot support the star. There is therefore a critical mass MC

above which a star cannot end as a White Dwarf. This is the Chandrasekhar limit. Detailed calculation gives MC ' 1:4M .

1.2Neutron Stars

The electron energies available in a White Dwarf are of the order of the Fermi

energy. Necessarily E < m c2 since the electrons are otherwise relativistic

F e

and cannot support the star. A White Dwarf is therefore stable against inverse -decay

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since the reaction needs energy of at least ( mn)c2 where mn is the neutron-proton mass di erence. Clearly m > me ( -decay would otherwise be impossible) and in fact m 3me. So we need energies of order of 3mec2 for inverse -decay. This is not available in White Dwarf stars but for M > MC the star must continue to contract until EF ( mn)c2. At this point inverse -decay can occur. The reaction cannot come to equilibrium with the reverse reaction

cannot occur because all electron energy levels below E < ( mn)c2 arelled when E > ( mn)c2. Since inverse -decay removes the electron degeneracy pressure the star will undergo a catastrophic collapse to nuclear matter density, at which point we must take neutron-degeneracy pressure into account.

Can neutron-degeneracy pressure support the star against collapse?

The ideal gas approximation would give same result as before but with me ! mp. The critical mass MC is independent of me and so is una ected,

which is the Schwarzschild radius, so the neglect of GR e ects was not justi ed. Also, at nuclear matter densities the ideal gas approximation is not justi ed. A perfect uid approximation is reasonable (since viscosity can't help). Assume that P ( ) ( = density of uid) satis es

More massive stars must continue to collapse either to an unknown new ultra-high density state of matter or to a black hole. The latter is more

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likely. In any case, there must be some mass at which gravitational collapse to a black hole is unavoidable because the density at the Schwarzschild radius decreases as the total mass increases. In the limit of very large mass the collapse is well-approximated by assuming the collapsing material to be a pressure-free ball of uid. We shall consider this case shortly.

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