Physics 523, Introduction to Relativity
Homework 5
Due Thursday, 17
th
November 2011
Hans Bantilan
TOV Star in 2+1 Dimensions
Consider a static circularly symmetric spacetime in 2+1 dimensions, so that the metric takes the form
g
ij
dx
i
dx
j
=

exp (2
α
)
dt
2
+ exp (2
β
)
dr
2
+
r
2
dθ
2
where
α
=
α
(
r
) and
β
=
β
(
r
), and include perfect fluid matter
1
with energymomentum tensor compo
nents
T
ij
= (
ρ
+
P
)
U
i
U
j
+
Pg
ij
.
a.
We are to write down the TOV equation in 2+1 dimensions. We begin by computing the Einstein
tensor, which gives us
G
ij
dx
i
dx
j
=
1
r
exp (2
α

2
β
)
dβ
dr
dt
2
+
1
r
dα
dr
dr
2
+
r
2
dα
dr
2

dα
dr
dβ
dr
+
d
2
α
dr
2
exp (

2
β
)
dθ
2
.
This calculation, along with the perfect fluid energy momentum tensor
T
ij
dx
i
dx
j
=
ρ
exp (2
α
)
dt
2
+
P
exp (2
β
)
dr
2
+
r
2
Pdθ
2
implies that
tt
and
rr
components of the Einstein equations
G
ij
= 8
πT
ij
read,
dm
dr
= 2
πrρ
(
r
)
and
dα
dr
=
8
πrP
(
r
)
1

8
m
(
r
)
respectively, where
m
(
r
) = (1

exp (

2
β
))
/
8 is such that exp (2
β
) = 1
/
(1

8
m
(
r
)).
Instead of in
specting the
θθ
component, we equivalently apply the second Bianchi identity
∇
R
= 0 to the Einstein
equations, giving
0
=
(
∇
i
T
)
ij
=
(
∇
i
T
)
ir
=
∂T
rr
∂r
+
(
Γ
t
tr
T
rr
+ Γ
r
rr
T
rr
+ Γ
θ
θr
T
rr
)
+
(
Γ
r
tt
T
t
+ Γ
r
rr
T
rr
+ Γ
r
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 '09
 FRANSPRETORIUS
 Work, General Relativity, Tensor, DR1, Einstein field equations, dxi dxj

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