GR
ALEXANDER J. WEAVER
Last time we talked about the Schwartzchild Metric, which and the form
rs 2
dr2
+ r2 dΩ2
ds2 = − 1 −
dt +
r
1 − rrs
where we noted there was a coordinate singularity at r = rs so to get around it we transformed
to the coordinates or analytically continued them.
r
r
t
r
2r
s
T =
−∞<T <∞
− 1 e sinh
rs
2rs
r
r
t
r
2r
s
− 1 e cosh
X>0
X=
rs
2rs
Where me make X the negative of this guy in the coordinate range. We also used a second
transformation for region 2 from his drawing on the board
r
t
r 2rr
T = 1 − e s sinh
−∞<T <∞
rs
2rs
r
r 2rr
t
X>0
X = 1 − e s cosh
rs
2rs
while in the fourth region we make T the minus of it’s value. This set of coordinates are called
the kruetzal coordinates. These are the only coordinates that are good everywhere, in terms of
the coordinates we can right the metric as
4r3
ds2 = − s −dT 2 + dX 2 + r2 dΩ2
r
r
Where we have in all four regimes that T 2 − X 2 = − rrs − 1 e rs . We see we have a true
singularity at r=0 so that we have a black hole, which we see because any light like line entering
region 2 must end at r = 0 and nothing inside region 2 can leave to regions 1 or 3. Similarly in
region 4 we have a white hole, all light rays exit region 4.
Date: 10/28.
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