TH = /8π kM and entropy kA/4 . This analogy had been noticed as soon as the area theorem was discovered (see Bekenstein 1973, 1974, and Misner et al. 1973, Box 33.4), but at that time it was thought to be an incomplete analogy because black holes did not have a true temperature. The Hawking radiation fits the missing piece into the puzzle.

But the Hawking radiation has raised other questions. One of them concerns information. The emission of radiation from the black hole raises the possibility that the radiation could carry information that, in the classical picture, is effaced by the formation of the horizon. If the radiation is perfectly thermal, then it contains no information. But it is possible that the outgoing photons and gravitons have a thermal black-body spectrum but also have weak correlations that contain the information. This information would not have come from inside the hole, but from the virtual photons and gravitons just outside the horizon, which are affected by the details of the collapsing matter that passes through them on its the way to forming the black hole, and which then become real photons and gravitons by the process described above. Whether this picture is indeed correct, and what kind of information can in principle be recovered from the outgoing radiation, are still matters of considerable debate among physicists.

The Hawking radiation has also become a touchstone for the development of full theories of quantum gravity. Physicists test new quantization methods by showing that they can predict the Hawking radiation and associated physics, such as the entropy of the black hole. This is not sufficient to prove that a method will work in general, but it is regarded as necessary.

11.6 F u r t h e r re a d i n g

The story of Karl Schwarzschild’s discovery of the solution named after him is an extraordinary one. See the on-line biography of Schwarzschild by J. J. O’Connor and E. F. Robertson in the MacTutor History of Mathematics archive, at the URL www

-history.mcs.st-andrews.ac.uk/Biographies/Schwarzschild.html

(cited April 2008).

The perihelion shift and deflection of light are the two classical tests of GR. Other theories predict different results: see Will (1993, 2006). A short, entertaining account of the observation of the deflection of light and its impact on Einstein’s fame is in McCrea (1979). To learn more about gravitational lensing, see Wambsganss (1998), Schneider et al (1992), Schneider (2006), or Perlick (2004).

The Kerr metric has less symmetry than the Schwarzschild metric, so it might be expected that particle orbits would have fewer conserved quantities and therefore be harder to calculate. This is, quite remarkably, false: even orbits out of the equator have three conserved quantities: energy, angular momentum, and a difficult-to-interpret quantity associated with the θ motion. The same remarkable property carries over to the wave equations that govern electromagnetic fields and gravitational waves in the Kerr metric: these equations separate completely in certain coordinate systems. See Teukolsky (1972) for the first general proof of this and Chandrasekhar (1983) for full discussions.

Black-hole thermodynamics is treated thoroughly in Carter (1979), while the related theory of quantum fields in curved spacetimes is reviewed by Wald (1994). This relates to

numerical relativity section of the online journal Living Reviews in Relativity at http://relativity.livingreviews.org/ or, at a more popular level, the Einstein Online website http://www.einstein-online.info/en/.

11.7E xe rc i s e s

1Consider a particle or photon in an orbit in the Schwarzschild metric with a certain E and L, at a radius r M. Show that if spacetime were really flat, the particle would travel on a straight line which would pass a distance b := L/[E2 − m2]1/2 from the center of coordinates r = 0. This ratio b is called the impact parameter. Show also that

photon orbits that follow from Eq. (11.12) depend only on b.

2Prove Eqs. (11.17) and (11.18).

˜ 2 ˜ 2 = 2 ˜ 2 = 2 ˜ 2 = 2

3 Plot V against r/M for the three cases L 25 M , L 12 M , L 9 M and verify the qualitative correctness of Figs. 11.1 and 11.3.

4What kind of orbits are possible outside a star of radius (a) 2.5 M, (b) 4 M, (c) 10 M?

5The centers of active galaxies and quasars contain black holes of mass 106 − 109 M or more.

(a)Find the radius R0.01 at which −g00 differs from the ‘Newtonian’ value 1 − 2 M/R by only 1%. (We may think of this as a kind of limit on the region in which relativistic effects are important.)

(b)A ‘normal’ star may have a radius of 1010 m. Approximately how many such stars could occupy the volume of space between the horizon R = 2 M and R0.01?

6Compute the wavelength of light that gets to a distant observer from the following sources.

(a)Light emitted with wavelength 6563 Å (Hα line) by a source at rest where = −10−6. (Typical star.)

(b)Same as (a) for = −6 × 10−5 (value for the white dwarf 40 Eridani B).

(c) Same as (a) for a source at rest at radius r = 2.2 M outside a black hole of mass

M = 1 M = 1.47 × 105 cm.

(d)Same as (c) for r = 2.02 M.

7A clock is in a circular orbit at r = 10 M in a Schwarzschild metric.

(a)How much time elapses on the clock during one orbit? (Integrate the proper time dτ = |ds2|1/2 over an orbit.)

(b)It sends out a signal to a distant observer once each orbit. What time interval does the distant observer measure between receiving any two signals?

(c)A second clock is located at rest at r = 10 M next to the orbit of the first clock. (Rockets keep it there.) How much time elapses on it between successive passes of the orbiting clock?

(d)Calculate (b) again in seconds for an orbit at r = 6 M where M = 14 M . This is the minimum fluctuation time we expect in the X-ray spectrum of Cyg X-1: why?

(e)If the orbiting ‘clock’ is the twin Artemis, in the orbit in (d), how much does she

age during the time her twin Diana lives 40 years far from the black hole and at rest with respect to it?

8(a) Derive Eqs. (11.20) and (11.24).

(b)Derive Eqs. (11.26) and (11.28).

9(This problem requires access to a computer.)

(a)Integrate numerically Eq. (11.26) or Eq. (11.28) for the orbit of a particle (i.e.

11The right-hand side of Eq. (11.28) is a polynomial in u. Trace the u3 term back through the derivation and show that it would not be present if we had started with the Newtonian version of Eq. (11.9). Interpret this term as a redshift effect on the orbital kinetic energy. Show that it is responsible for the maximum in the curve in Fig. 11.1.

12(a) Prove that Eq. (11.32) solves Eq. (11.31).

(b)Derive Eq. (11.33) from Eq. (11.32) and show that it describes an ellipse by transforming to Cartesian coordinates.

13(a) Derive Eq. (11.34) in the approximation that y is small. What must it be small compared to?

(b)Derive Eqs. (11.35) and (11.36) from (11.34).

(c)Verify the remark after Eq. (11.36) that y = 0 is not the correct circular orbit for

˜ ˜

the given E and L by using Eqs. (11.20) and (11.21) to find the correct value of y and comparing it to y0 in Eq. (11.36).

(d) Show from Eq. (11.13) that a particle which has an inner turning point in the ‘New-

˜

tonian’ regime, i.e. for r M, has a value L M. Use this to justify the step from Eq. (11.37) to Eq. (11.38).

14Compute the perihelion shift per orbit and per year for the following planets, given their distance from the Sun and their orbital period: Venus (1.1 × 1011 m, 1.9 × 107 s); Earth (1.5 × 1011 m, 3.2 × 107 s); Mars (2.3 × 1011 m, 5.9 × 107 s).

15(a) Derive Eq. (11.51) from (11.49), and show that it describes a straight line passing a distance b from the origin.

28(a) Prove Eq. (11.87).

(b)Derive Eq. (11.89).

29In the Kerr metric, show (or argue on symmetry grounds) that a geodesic which passes through a point in the equatorial ‘plane’ (θ = π/2) and whose tangent there is tangent to the plane (pθ = 0) remains always in the plane.

30Derive Eqs. (11.91) and (11.92).

31Show that a ZAMO has four-velocity components U0 = A, Uφ = ωA, Ur = Uθ = 0, A2 = gφφ /(−D), where D is defined in Eq. (11.87).

32Show, as argued in the text, that the Penrose process decreases the angular momentum of the hole.

33Derive Eq. (11.101) from Eq. (11.100).

34(a) Use the area theorem to calculate the maximum energy released when two Schwarzschild black holes of mass M collide to form a Schwarzschild hole.

(b)Do the same for holes of mass m1 and m2, and express the result as a percentage of m1 when m1 → 0 for fixed m2.

35The Sun rotates with a period of approximately 25 days.

(a)Idealize it as a solid sphere rotating uniformly. Its moment of inertia is 25 M R2 , where M = 2 × 1030 kg and R = 7 × 108 m. In SI units compute J .

(b)Convert this to geometrized units.

(c)If the entire Sun suddenly collapsed into a black hole, it would form a Kerr hole

of mass M and angular momentum J . What would be the Kerr parameter, a = J /M , in m? What is the ratio a /M ? Physicists expect that a Kerr hole will never be formed with a > M, because centrifugal forces will halt the collapse

or create a rotational instability. The result of this exercise is that even a quite ordinary star like the sun needs to get rid of angular momentum before forming a black hole.

(d)Does an electron have too much angular momentum to form a Kerr hole with a < M? (Neglect its charge.)

36(a) For a Kerr black hole, prove that for fixed M, the largest area is obtained for a = 0 (Schwarzschild).

(b)Conversely, prove that for fixed area, the smallest mass is obtained for a = 0.

37(a) An observer sits at constant r, φ in the equatorial plane of the Kerr metric (θ = π/2) outside the ergoregion. He uses mirrors to cause a photon to circle the hole along a circular path of constant r in the equatorial plane. Its world line is thus a null line with dr = dθ = 0, but it is not, of course, a geodesic. How much coordinate time t elapses between the emission of a photon in the direction of increasing φ and its receipt after it has circled the hole once? Answer the same for a photon sent off in the direction of decreasing φ, and show that this is a different amount of time. Does the photon return redshifted from its original frequency?

(b)A different observer rotates about the hole on an orbit of r = const. and angular velocity given by Eq. (11.77). Using the same arrangement of mirrors, he measures the coordinate time that elapses between his emission and his receipt of a photon sent in either direction. Show that in this case the two terms are equal. (This is a ZAMO, as defined in the text.)

38 Consider equatorial motion of particles with m =0 in the Kerr metric. Find the analogs

˜ ˜ ˜

of Eqs. (11.91)–(11.95) using E and L as defined in Eqs. (11.5) and (11.6). Plot V± for

= ˜ =

a 0.5 M and L/M 20, 12, and 6. Discuss the qualitative features of the trajectories.

˜ ˜

For arbitrary a determine the relations among E, L, and r for circular orbits with either sense of rotation. What is the minimum radius of a stable circular orbit? What happens to circular orbits in the ergosphere?