10.7 Hawking Radiation from a Black Hole
157
smaller than the Planck distance are not physically meaningful, so it is possible that
black holes might not evaporate entirely away and could leave a remnant of order
the Planck size. Such remnants could be a candidate for the dark matter particles
that will be discussed in Chap. 16; we will come back to this also in Appendix 3 in
Chap. 19 (Adler 2001).
Exercises
10.1 Calculate the geometric mass m of the sun in km. What is the radius of the sun
in km? How much does the Schwarzschild metric component g 00 = 1−2m/r
differ from 1 near the surface of the sun? Repeat the calculations for the earth.
Do you see why it is difficult to do interesting general relativity experiments
in the solar system and especially on the surface of the Earth?
10.2 What is the area of the black hole surface? This area plays an interesting role
in the quantum mechanical properties of black holes, as we have discussed
in Sect. 10.7 on Hawking radiation.
10.3 Evaluate the Riemann curvature tensor to see if it is singular or zero at the
Schwarzschild radius. (You may instead simply look up the Riemann tensor
for the Schwarzschild metric.)
10.4 Is the Ricci tensor singular at the Schwarzschild radius? Is the Riemann scalar
singular at the Schwarzschild radius? Hint: this requires no calculation.
10.5 Verify that a photon falling onto a black hole approaches it exponentially;
that is verify (10.6).
10.6 Verify that a particle falling onto a black hole approaches it exponentially;
that is verify (10.13).
10.7 Consider a particle (or photon) falling onto a black hole surface. Take the
geometric mass to be about 5 km and the initial position to be about 5 km.
Calculate approximately how far away the particle is after, 10
−9 , 10
−5 , 1 s,
1 year. Does it really make physical sense to say that the particle never quite
reaches the surface, or that a collapsing dust star never quite becomes a black
hole? See Sect. 19.6 for comments on small distances.
10.8 In Sect. 10.1 of the text we refer to an outside observer situated far from the
black hole,. Repeat the discussion of the red shift using an observer at a finite
fixed distance outside the black hole using laboratory time intervals t ob =
√
1 − 2m/r ob t. Do any of the important qualitative conclusion concerning
the fall of a particle to the surface change significantly?
10.9 Using some reference on the Kruskal Szekeres coordinates, such as Adler
(1975), show how the region inside an empty theoretical black hole surface
is only relevant to outside observers for t > ∞; that is we in the exterior
simply cannot communicate with that interior region.
10.10 Make a qualitative sketch of the trajectories of light traveling radially inward
and radially outward in the exterior of a black hole with Schwarzschild geometry. In the sketch draw the local light cones for such radial motion, and notice
that they degenerate at the Schwarzschild radius and lie along the surface.
157
smaller than the Planck distance are not physically meaningful, so it is possible that
black holes might not evaporate entirely away and could leave a remnant of order
the Planck size. Such remnants could be a candidate for the dark matter particles
that will be discussed in Chap. 16; we will come back to this also in Appendix 3 in
Chap. 19 (Adler 2001).
Exercises
10.1 Calculate the geometric mass m of the sun in km. What is the radius of the sun
in km? How much does the Schwarzschild metric component g 00 = 1−2m/r
differ from 1 near the surface of the sun? Repeat the calculations for the earth.
Do you see why it is difficult to do interesting general relativity experiments
in the solar system and especially on the surface of the Earth?
10.2 What is the area of the black hole surface? This area plays an interesting role
in the quantum mechanical properties of black holes, as we have discussed
in Sect. 10.7 on Hawking radiation.
10.3 Evaluate the Riemann curvature tensor to see if it is singular or zero at the
Schwarzschild radius. (You may instead simply look up the Riemann tensor
for the Schwarzschild metric.)
10.4 Is the Ricci tensor singular at the Schwarzschild radius? Is the Riemann scalar
singular at the Schwarzschild radius? Hint: this requires no calculation.
10.5 Verify that a photon falling onto a black hole approaches it exponentially;
that is verify (10.6).
10.6 Verify that a particle falling onto a black hole approaches it exponentially;
that is verify (10.13).
10.7 Consider a particle (or photon) falling onto a black hole surface. Take the
geometric mass to be about 5 km and the initial position to be about 5 km.
Calculate approximately how far away the particle is after, 10
−9 , 10
−5 , 1 s,
1 year. Does it really make physical sense to say that the particle never quite
reaches the surface, or that a collapsing dust star never quite becomes a black
hole? See Sect. 19.6 for comments on small distances.
10.8 In Sect. 10.1 of the text we refer to an outside observer situated far from the
black hole,. Repeat the discussion of the red shift using an observer at a finite
fixed distance outside the black hole using laboratory time intervals t ob =
√
1 − 2m/r ob t. Do any of the important qualitative conclusion concerning
the fall of a particle to the surface change significantly?
10.9 Using some reference on the Kruskal Szekeres coordinates, such as Adler
(1975), show how the region inside an empty theoretical black hole surface
is only relevant to outside observers for t > ∞; that is we in the exterior
simply cannot communicate with that interior region.
10.10 Make a qualitative sketch of the trajectories of light traveling radially inward
and radially outward in the exterior of a black hole with Schwarzschild geometry. In the sketch draw the local light cones for such radial motion, and notice
that they degenerate at the Schwarzschild radius and lie along the surface.