ABSTRACT
We now apply the same technique to the Schwarzschild geometry that we
used in Chapter 4 to extend the Rindler geometry. Consider a radial light
beam, so that dφ = 0 (and as usual θ = π/2). Then the line element
becomes
ds2 = − ( 1− 2m
r
) dt2 +
dr2
1− 2mr = −
( 1− 2m
r
)( dt2 − dr
)2 ) ,
(5.1)
which we can factor as
ds2 = − ( 1− 2m
r
)( dt− dr
)( dt+
dr
) . (5.2)
This motivates the definition
du = dt− dr 1− 2mr
, (5.3)
dv = dt+ dr
1− 2mr , (5.4)
which we can integrate to obtain
v − u 2
=
∫ dr
1− 2mr =
∫ ( 1 +
1 r
2m − 1 )
dr
= r + 2m ln ( r 2m
− 1 ) . (5.5)
Expression (5.5) relating r to u and v is badly behaved at r = 2m. How-
like their
in Rindler geometry, as shown in Figure 5.1 (compare Figure 4.4). Fur-
thermore, we can extend the geometry across the apparent singularity at
r = 2m if we exponentiate, which yields
e(v−u)/4m = er/2m ( r 2m
− 1 ) = er/2m
r
2m
( 1− 2m
r
) . (5.6)
Inserting (5.6) back into the factored line element (5.2) leads to
ds2 = −2m r e−r/2me(v−u)/4m du dv
= −32m 3
r e−r/2m
( e−u/4m
du
4m
)( ev/4m
dv
4m
)
= −32m 3
r e−r/2m dU dV, (5.7)
where
U = −e−u/4m, (5.8) V = ev/4m. (5.9)
The line element (5.7) is perfectly well behaved at r = 2m! The coordinates
{U, V, θ, φ} are known as (double-null) Kruskal-Szekeres coordinates. The radial coordinate r can be expressed (implicitly) in terms of U and V via
UV = −e(v−u)/4m = er/2m ( 1− r
2m
) . (5.10)
Finally, orthogonal coordinates {T,X} can be introduced if desired by writing
U = T −X, (5.11) V = T +X. (5.12)
5.2 KRUSKAL GEOMETRY Since Kruskal-Szekeres coordinates U and V are well-defined at r = 2m,
we can use them to extend Schwarzschild geometry to r < 2m. The max-
imally extended Schwarzschild geometry, also called the Kruskal geometry,
is obtained by extending the domains of U , V as much as possible, and is
shown in Figure 5.2.
