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16 Physics Formulary by ir. J.C.A. Wevers
• r > 2 m :











u =
r
2 m
- 1 exp
r
4 m
cosh
t
4 m
v =
r
2 m
- 1 exp
r
4 m
sinh
t
4 m
• r < 2 m :











u = 1 -
r
2 m
exp
r
4 m
sinh
t
4 m
v = 1 -
r
2 m
exp
r
4 m
cosh
t
4 m
• r = 2 m : here, the Kruskal coordinates are singular, which is necessary to eliminate the coordinate
singularity there.
The line element in these coordinates is given by:
ds
2
= -
32 m
3
r
e
- r/ 2 m
( dv
2
- du
2
) + r
2
d Ω
2
The line r = 2 m corresponds to u = v = 0 , the limit x
0
→ ∞ with u = v and x
0
→ -∞ with u = - v . The
Kruskal coordinates are only singular on the hyperbole v
2
- u
2
= 1 , this corresponds with r = 0 . On the line
dv = ± du holds d θ = d ϕ = ds = 0 .
For the metric outside a rotating, charged spherical mass the Newman metric applies:
ds
2
= 1 -
2 mr - e
2
r 2 + a 2 cos 2 θ
c
2
dt
2
-
r
2
+ a
2
cos
2
θ
r 2 - 2 mr + a 2 - e 2
dr
2
- ( r
2
+ a
2
cos
2
θ ) d θ
2
-
r
2
+ a
2
+
(2 mr - e
2
) a
2
sin
2
θ
r 2 + a 2 cos 2 θ
sin
2
θ d ϕ
2
+
2 a (2 mr - e
2
)
r 2 + a 2 cos 2 θ
sin
2
θ ( d ϕ