Black-hole event horizons
277
As r ranges from 2M to 00, ,* ranges from -00 to 00. Thus, d(1 ± ,*) = 0
on radial nulI geodesics and the ingoing radial null (Eddington-Finkelstein)
coordinate is defined as
v == I + ,* = t + r + 2M In 12~ - I1 - 00 < v < 00. (10.6)
Using v as a coordinate instead of I gives
ds 2 = (I - 2~) dv 2 _ 2dv dr - r2 de 2 - ,2 sin 2 (J dtfJ2. (10.7)
This metric is defined initially for r > 2M, since the relation v = , + ,*(r) is
only defined for, > 2M. However, it can now be analytically continued to alI
r > 0 and, in these coordinates, there is no singularity at, = 2M. (There is
a singularity at r = 0 where the curvature becomes infinite. Such a singularity
cannot, of course, be removed by a coordinate transformation.) At large values of
r, the light cones are almost Minkowskian and they alIow a particle (or photon)
to move outwards or inwards on a timelike (or null) worldline. However, as ,
decreases the lightcones gradually tilt over. When, :5 2M, 2dr dv :5 0 for alI
non-spacelike (i.e. timelike or null) world lines. Since dv > 0 for future directed
world lines, it follows that dr :5 0, with equality for radial null geodesics when
r = 2M. When, < 2M, all non-spacelike curves necessarily move inwards
and hit the singularity at r = O. Thus, if the massive body emits light from its
(spherical) surface at r = rB < 2M, the light never escapes to an observer in
the region, > 2M. Such an observer might infer the presence of the body from
its gravitational field but she could not see it. The hypersurface traced out in
spacetime by the spherical surface r = 2M is calIed the 'event horizon' of the
(Schwarzchild) black hole. The area of a two-dimensional section of the event
horizon is
An = 4tr(2M)2 = 16trM2.
(10.8)
Let S(x) be a smooth function of the spacetime coordinates x# and
consider a family of hypersurfaces S(x) = constant. The vectors normal to the
hypersurfaces are given by
- as
(10.9)
1# = f(x) ax#
where j is an arbitrary non-zero function. If 12 = 0 for a particular hypersurface
.N in the family, then .N is said to be a 'null' hypersurface. So for the spherical
hypersurfaces S == r = constant, with the black-hole metric (10.7),
P = g#v1/L1v = grr p = _ (I _ 2~) p.
(10.10)
Thus, the event horizon r = 2M is a null hypersurface and (exercise 2)
1#lr=2M = g# V1 vlr=2M = - j~~.
(10.11 )
277
As r ranges from 2M to 00, ,* ranges from -00 to 00. Thus, d(1 ± ,*) = 0
on radial nulI geodesics and the ingoing radial null (Eddington-Finkelstein)
coordinate is defined as
v == I + ,* = t + r + 2M In 12~ - I1 - 00 < v < 00. (10.6)
Using v as a coordinate instead of I gives
ds 2 = (I - 2~) dv 2 _ 2dv dr - r2 de 2 - ,2 sin 2 (J dtfJ2. (10.7)
This metric is defined initially for r > 2M, since the relation v = , + ,*(r) is
only defined for, > 2M. However, it can now be analytically continued to alI
r > 0 and, in these coordinates, there is no singularity at, = 2M. (There is
a singularity at r = 0 where the curvature becomes infinite. Such a singularity
cannot, of course, be removed by a coordinate transformation.) At large values of
r, the light cones are almost Minkowskian and they alIow a particle (or photon)
to move outwards or inwards on a timelike (or null) worldline. However, as ,
decreases the lightcones gradually tilt over. When, :5 2M, 2dr dv :5 0 for alI
non-spacelike (i.e. timelike or null) world lines. Since dv > 0 for future directed
world lines, it follows that dr :5 0, with equality for radial null geodesics when
r = 2M. When, < 2M, all non-spacelike curves necessarily move inwards
and hit the singularity at r = O. Thus, if the massive body emits light from its
(spherical) surface at r = rB < 2M, the light never escapes to an observer in
the region, > 2M. Such an observer might infer the presence of the body from
its gravitational field but she could not see it. The hypersurface traced out in
spacetime by the spherical surface r = 2M is calIed the 'event horizon' of the
(Schwarzchild) black hole. The area of a two-dimensional section of the event
horizon is
An = 4tr(2M)2 = 16trM2.
(10.8)
Let S(x) be a smooth function of the spacetime coordinates x# and
consider a family of hypersurfaces S(x) = constant. The vectors normal to the
hypersurfaces are given by
- as
(10.9)
1# = f(x) ax#
where j is an arbitrary non-zero function. If 12 = 0 for a particular hypersurface
.N in the family, then .N is said to be a 'null' hypersurface. So for the spherical
hypersurfaces S == r = constant, with the black-hole metric (10.7),
P = g#v1/L1v = grr p = _ (I _ 2~) p.
(10.10)
Thus, the event horizon r = 2M is a null hypersurface and (exercise 2)
1#lr=2M = g# V1 vlr=2M = - j~~.
(10.11 )