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

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- Spring '10
- Ye
- Physics, General Relativity, a2 cos2 θ, Kruskal coordinates, exp cosh, perihelion shift, exp sinh
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