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HOMEWORK SET 7, PHYS. 131
Instructor:
Anthony Zee
Email:
[email protected]
TA:
Kevin Moore
Email:
[email protected]
12.4.
Consider the spacetime specifed by the line element
ds
2
=

±
1

M
r
²
2
dt
2
+
±
1

M
r
²

2
dr
2
+
r
2
(
dθ
2
+ sin
2
θdφ
2
)
except For
r
=
M
, the coordinate
t
is always timelike and the coordinate
r
is
spacelike.
(a)
±ind a transFormation to new coordinates (
v, r, θ, φ
) analagous to (12.1)
that sets
g
rr
= 0 and shows that the geometry is not singular at
r
=
M
.
Let
t
=
f
(
r, v
)
while
r, θ,
and
φ
are unchanged. We are attempting to fnd a coordinate trans
Formation where the metric has
g
±
rr
= 0, or
0=
g
μν
∂x
μ
∂r
∂x
ν
∂r
=

±
1

M
r
²
2
±
∂t
∂r
²
2
+
±
1

M
r
²

2
which leaves
∂t
∂r
=
±
1

M
r
²

2
Integrating, and choosing
v
as the constant oF integration, gives
t
=
r
+
M
1

r/M
+2
M
log(
r

M
)+
v
and
ds
2
=

±
1

M
r
²
2
dv
2
+2
drdv
+
r
2
dθ
2
+
r
2
sin
2
θdφ
2
which is maniFestly regular at
r
=
M
.
(b)
Sketch a (
±
t, r
) diagram analagous to ±igure 12.1 showing the world lines oF
ingoing and outgoing light rays and light cones.
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This note was uploaded on 07/15/2008 for the course PHYS 131 taught by Professor Kevin during the Spring '08 term at UCSB.
 Spring '08
 KEVIN
 Physics, Work, General Relativity

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