10.1 Schwarzschild Black Hole
145
metric function 1 − 2m/r , which goes to zero at the surface. This is a profound
difference characteristic of black holes!
As seen from the outside, where physicists live, light and particles falling onto a
black hole would take an infinite amount of time to reach the surface; however, if
one were to fall onto the surface of the black hole carrying a clock he would reach
the surface in the finite time (10.15). For the falling observer the entire history of the
external world would thus be seen to pass during his fall. This remarkable behavior
has never been directly tested by a falling physicist, but observations of material
falling onto a black hole from an accreting disk of matter are consistent with it.
As noted earlier we have not discussed the interior region, r < 2m. This is
because the Schwarzschild coordinates simply do not work there. Indeed the signature
changes from (1, −1, −1, −1) to (−1, 1, −1, −1) at the Schwarzschild radius, so
t cannot be thought of as a time coordinate and r cannot be thought of as a radial
coordinate (see Exercise 9.10). Moreover the point r = 0 should not be thought of
as the “center” of the black hole, a fact which is not always appreciated. To study the
interior one must use other coordinates that should be consistent with Schwarzschild
coordinates outside but remain well-behaved inside the Schwarzschild radius. The
best-known coordinates of this type are called the Kruskal Szekeres coordinates and
serve their purpose quite well. See Exercise 10.9 and Kruskal (1960).
Another thing we should note about the Schwarzschild line element is that at
the Schwarzschild radius the time term goes to zero while the radial term becomes
infinite, but the product of the two in the determinant remains finite. Thus the 4-space
volume element is well behaved at the Schwarzschild radius but the 3-space volume
element is not.
We also emphasize that we have not yet discussed how a Schwarzschild black
hole could form in the real universe that we observe, for example from a collapsing
star. We will discuss this in more detail when we study the collapse of model stars
in Sect. 10.4.
10.2 Null Surfaces
We have seen in the previous section that the black hole surface has some interesting
properties. In particular the behavior of both light and particles is quite peculiar
as they approach the surface from outside. In terms of the time used by external
observers, such as physicists, neither light nor particles can reach the surface. Indeed
the surface is special in a way that is independent of the choice of coordinates; the
surface is called a null surface and we will see that it acts like a one-way membrane
or horizon. In this section we will study the relation of a general surface in spacetime
to the local light cone and obtain some elegant geometric results characterizing a
null surface that are relevant to black holes.
Consider a smooth surface S in spacetime defined by
S: f (x
α
) = C = const.
(10.16)
145
metric function 1 − 2m/r , which goes to zero at the surface. This is a profound
difference characteristic of black holes!
As seen from the outside, where physicists live, light and particles falling onto a
black hole would take an infinite amount of time to reach the surface; however, if
one were to fall onto the surface of the black hole carrying a clock he would reach
the surface in the finite time (10.15). For the falling observer the entire history of the
external world would thus be seen to pass during his fall. This remarkable behavior
has never been directly tested by a falling physicist, but observations of material
falling onto a black hole from an accreting disk of matter are consistent with it.
As noted earlier we have not discussed the interior region, r < 2m. This is
because the Schwarzschild coordinates simply do not work there. Indeed the signature
changes from (1, −1, −1, −1) to (−1, 1, −1, −1) at the Schwarzschild radius, so
t cannot be thought of as a time coordinate and r cannot be thought of as a radial
coordinate (see Exercise 9.10). Moreover the point r = 0 should not be thought of
as the “center” of the black hole, a fact which is not always appreciated. To study the
interior one must use other coordinates that should be consistent with Schwarzschild
coordinates outside but remain well-behaved inside the Schwarzschild radius. The
best-known coordinates of this type are called the Kruskal Szekeres coordinates and
serve their purpose quite well. See Exercise 10.9 and Kruskal (1960).
Another thing we should note about the Schwarzschild line element is that at
the Schwarzschild radius the time term goes to zero while the radial term becomes
infinite, but the product of the two in the determinant remains finite. Thus the 4-space
volume element is well behaved at the Schwarzschild radius but the 3-space volume
element is not.
We also emphasize that we have not yet discussed how a Schwarzschild black
hole could form in the real universe that we observe, for example from a collapsing
star. We will discuss this in more detail when we study the collapse of model stars
in Sect. 10.4.
10.2 Null Surfaces
We have seen in the previous section that the black hole surface has some interesting
properties. In particular the behavior of both light and particles is quite peculiar
as they approach the surface from outside. In terms of the time used by external
observers, such as physicists, neither light nor particles can reach the surface. Indeed
the surface is special in a way that is independent of the choice of coordinates; the
surface is called a null surface and we will see that it acts like a one-way membrane
or horizon. In this section we will study the relation of a general surface in spacetime
to the local light cone and obtain some elegant geometric results characterizing a
null surface that are relevant to black holes.
Consider a smooth surface S in spacetime defined by
S: f (x
α
) = C = const.
(10.16)