636 18. Einstein’s Equations
If K>0, these are orbits for the Kepler problem, that is, for the two-body
planetary motion problem. If K<0, these are orbits for the Coulomb problem,
for the motion of charged particles with like charges, hence for motion under a
repulsive force. Repulsive gravitational fields have not been observed.
Note that if we take K>0in (2.61), then the formula is degenerate at r D K.
Only on fr>Kg is @=@t a timelike vector. It is this region that is properly said to
carry the Schwarzschild metric.
It follows from the fact that the sectional curvature of the plane spanned by
@=@t and @=@r is S
U
that the Schwarzschild metric (2.61) is singular at r D 0.
On the other hand, the apparent singularity in the Schwarzschild metric at r D K
actually arises from a coordinate singularity, which can be removed as follows.
First, set
(2.63) v D t C
Z
1
K
r
1
dr D t C r C K log.r K/:
Using coordinates .v;r;;'/, the metric tensor (2.61) takes the form
(2.64) ds
2
D
1
K
r
dv
2
C 2 dv dr Cr
2
d!
2
:
These coordinates are called Eddington–Finkelstein coordinates. The region
fr>Kg in .t;r;;'/-coordinates corresponds to the region fr>Kg in the new
coordinate system, but the metric (2.64) is smooth and nondegenerate on the
larger region fr>0g. Note that if '; ,andv are held constant and r & K,then
t %1.
The shell † Dfr D Kg is a null surface for the metric (2.64); that is, the
restriction of the metric to † is everywhere degenerate. Thus, for each p 2 †,
the light cone formed by null geodesics through p is tangent to † at p. Figure 2.1
depicts the extended Schwarzschild metric, in Eddington–Finkelstein coordinates.
The function v arises from considering null geodesics in the Schwarzschild
spacetime for which ! 2 S
2
is constant or, equivalently, considering null
geodesics in the two-dimensional spacetime
(2.65) ds
2
D
1
K
r
dt
2
C
1
K
r
1
dr
2
:
On the region r>K, there are a family of null geodesics given by v D const. and
a family of null geodesics given by u D const., where
(2.66) u D t r K log.r K/:
The coordinates .u;r;;'/ are called outgoing Eddington–Finkelstein coordi-
nates (the ones above then being called incoming), and in this coordinate system
the Schwarzschild metric takes the form