148
10 Black Holes and Gravitational Collapse
Thus the spherical surface is spacelike outside the Schwarzschild radius, and
becomes null on it. By the above geometric arguments we see that the surface of
a Schwarzschild black hole is a null surface, and we therefore expect that nothing
from the interior could pass through it and emerge outside, neither a particle nor
light. The name black hole is thus appropriate. On the other hand objects may fall
onto it in terms of their proper time or approach it asymptotically in terms of the
external Schwarzschild time, as we discussed in the preceding section.
The above comments are based on classical relativity. If quantum effects are
considered however the situation changes in an interesting way: a black hole may
emit radiation as if it were a black body. We will study this later in Sect. 10.7.
For the Schwarzschild case that we have discussed the surface at r = 2m is both
an infinite redshift surface and a null surface. In the more general case of a black
hole which is rotating these two surfaces are not the same, and we will discuss this
further in Sect. 10.5 on the Kerr metric.
10.3 Stellar Evolution, Very Briefly
A typical star is born when a gas cloud, mainly of hydrogen, collapses under the
influence of gravity. As it collapses it heats up as gravitational potential energy is
converted into heat energy of the gas. When the temperature has risen sufficiently
high a number of thermonuclear processes begin, such as the fusion of protons via the
overall process 4p → He + 2e
+
+ 2ν + radiation. These release energy as heat and
radiation, and the pressure due to the increasing temperature and radiation pressure
stabilize the star against further collapse. It then becomes a stable energy emitting
star for a relatively long time. We can discuss briefly and superficially the behavior
of some stars at the end of their stable lifetime. See also the material on the death of
stars in the free online textbook of Fraknoi (2016).
A low mass star like the sun emits radiation for billions of years until its hydrogen
is depleted and the radiation pressure can no longer stabilize it. It then collapses into
a denser and denser state, and may emit material from its surface as it does, called a
planetary nebula. It finally becomes small and dense, about the size of the earth with
a density roughly 10
6 times water. This is called a white dwarf. It is prevented from
further collapse by the pressure of the degenerate electron gas in its dense interior,
much as the electrons in a metal make the metal highly incompressible. Such white
dwarfs are stable only if their mass is less than about 1.4 solar masses, which is called
the Chandrasekhar limit.
A medium mass star will also emit radiation, but for a shorter time, until its
hydrogen is depleted and radiation pressure can no longer stabilize it. Unlike a low
mass star it may then eject large amounts of material and huge amounts of energy
in a spectacular and complicated supernova explosion. The remnant left behind in
such an explosion may have a mass greater than the Chandrasekhar limit of 1.4 solar
masses; if that is the case it cannot be a white dwarf. In such a remnant the electrons
may be absorbed by protons via inverse beta decay to form neutrons, and a neutron
10 Black Holes and Gravitational Collapse
Thus the spherical surface is spacelike outside the Schwarzschild radius, and
becomes null on it. By the above geometric arguments we see that the surface of
a Schwarzschild black hole is a null surface, and we therefore expect that nothing
from the interior could pass through it and emerge outside, neither a particle nor
light. The name black hole is thus appropriate. On the other hand objects may fall
onto it in terms of their proper time or approach it asymptotically in terms of the
external Schwarzschild time, as we discussed in the preceding section.
The above comments are based on classical relativity. If quantum effects are
considered however the situation changes in an interesting way: a black hole may
emit radiation as if it were a black body. We will study this later in Sect. 10.7.
For the Schwarzschild case that we have discussed the surface at r = 2m is both
an infinite redshift surface and a null surface. In the more general case of a black
hole which is rotating these two surfaces are not the same, and we will discuss this
further in Sect. 10.5 on the Kerr metric.
10.3 Stellar Evolution, Very Briefly
A typical star is born when a gas cloud, mainly of hydrogen, collapses under the
influence of gravity. As it collapses it heats up as gravitational potential energy is
converted into heat energy of the gas. When the temperature has risen sufficiently
high a number of thermonuclear processes begin, such as the fusion of protons via the
overall process 4p → He + 2e
+
+ 2ν + radiation. These release energy as heat and
radiation, and the pressure due to the increasing temperature and radiation pressure
stabilize the star against further collapse. It then becomes a stable energy emitting
star for a relatively long time. We can discuss briefly and superficially the behavior
of some stars at the end of their stable lifetime. See also the material on the death of
stars in the free online textbook of Fraknoi (2016).
A low mass star like the sun emits radiation for billions of years until its hydrogen
is depleted and the radiation pressure can no longer stabilize it. It then collapses into
a denser and denser state, and may emit material from its surface as it does, called a
planetary nebula. It finally becomes small and dense, about the size of the earth with
a density roughly 10
6 times water. This is called a white dwarf. It is prevented from
further collapse by the pressure of the degenerate electron gas in its dense interior,
much as the electrons in a metal make the metal highly incompressible. Such white
dwarfs are stable only if their mass is less than about 1.4 solar masses, which is called
the Chandrasekhar limit.
A medium mass star will also emit radiation, but for a shorter time, until its
hydrogen is depleted and radiation pressure can no longer stabilize it. Unlike a low
mass star it may then eject large amounts of material and huge amounts of energy
in a spectacular and complicated supernova explosion. The remnant left behind in
such an explosion may have a mass greater than the Chandrasekhar limit of 1.4 solar
masses; if that is the case it cannot be a white dwarf. In such a remnant the electrons
may be absorbed by protons via inverse beta decay to form neutrons, and a neutron