Chapter 10
Black Holes and Gravitational Collapse
Abstract Black holes are one of the strangest predictions of relativity theory. In this
chapter we study some properties of black holes and discuss how they are expected
to be the end result of the collapse of some types of stars in the real universe. One
extraordinary theoretical property of black holes is that they should radiate energy like
a classical black body; this profound prediction connects classical general relativity
with quantum theory, although the radiation has not yet been observed.
10.1 Schwarzschild Black Hole
A typical star like the sun is roughly spherically symmetric and has a geometric mass
of about 1 km and a radius of about 10
6 km. Thus the Schwarzschild radius is deep
within such a star as shown in Fig. 10.1.
Assuming it is approximately spherically symmetric we know the metric is the
Schwarzschild metric for the exterior of such a star—but only the exterior. We have
not yet studied the interior, which is an entirely different problem. For a typical star
such as the sun the metric function 1 − 2m/r differs from 1 by less than about a part
in 10
6 , so gravity is indeed weak. For a dense star it may become significantly less
than 1 and gravity is strong. We will study the exterior of such a dense star in this
section but will not discuss the interior until later.
Consider first the gravitational redshift of light from the surface of a small dense
star, with its radius near to 2m. The gravitational redshift of light from the stellar
surface is given by (7.25), which we rewrite in terms of the frequency as
ν ob
ν s
=
√
g 00 (s)
√
g 00 (ob)
=
√
1 − 2m/r s
√
1 − 2m/r ob
.
(10.1)
Here s refers to the stellar surface and ob refers to the observer, typically at a large
distance from the surface. We see that for r s → 2m the observed frequency goes
to zero. Thus a photon emitted from a body at the surface loses all of its energy
as it travels outwards. This means that light does not actually escape. Such a star
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. J. Adler, General Relativity and Cosmology, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-61574-1_10
141
Black Holes and Gravitational Collapse
Abstract Black holes are one of the strangest predictions of relativity theory. In this
chapter we study some properties of black holes and discuss how they are expected
to be the end result of the collapse of some types of stars in the real universe. One
extraordinary theoretical property of black holes is that they should radiate energy like
a classical black body; this profound prediction connects classical general relativity
with quantum theory, although the radiation has not yet been observed.
10.1 Schwarzschild Black Hole
A typical star like the sun is roughly spherically symmetric and has a geometric mass
of about 1 km and a radius of about 10
6 km. Thus the Schwarzschild radius is deep
within such a star as shown in Fig. 10.1.
Assuming it is approximately spherically symmetric we know the metric is the
Schwarzschild metric for the exterior of such a star—but only the exterior. We have
not yet studied the interior, which is an entirely different problem. For a typical star
such as the sun the metric function 1 − 2m/r differs from 1 by less than about a part
in 10
6 , so gravity is indeed weak. For a dense star it may become significantly less
than 1 and gravity is strong. We will study the exterior of such a dense star in this
section but will not discuss the interior until later.
Consider first the gravitational redshift of light from the surface of a small dense
star, with its radius near to 2m. The gravitational redshift of light from the stellar
surface is given by (7.25), which we rewrite in terms of the frequency as
ν ob
ν s
=
√
g 00 (s)
√
g 00 (ob)
=
√
1 − 2m/r s
√
1 − 2m/r ob
.
(10.1)
Here s refers to the stellar surface and ob refers to the observer, typically at a large
distance from the surface. We see that for r s → 2m the observed frequency goes
to zero. Thus a photon emitted from a body at the surface loses all of its energy
as it travels outwards. This means that light does not actually escape. Such a star
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. J. Adler, General Relativity and Cosmology, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-61574-1_10
141