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Physics 570
Homework No. 4
due Wednesday, 17 February, 2010
1.
Recall the gravitational ﬁeld of a static, sphericallysymmetric mass by Φ, exterior to that mass,
that was discussed in the previous problem set:
g
=
ds
2
=
1
H
2
dr
2
+
r
2
(
dθ
2
+ sin
2
θ dϕ
2
)
 H
2
dt
2
,
H ≡
√
1 + 2Φ
,
Φ
≡ 
M
r
.
During that problem you a) evaluated the metriccompatible connection forms known as Christ¨oﬀel
symbols and wrote down the geodesic equations for a timelike path, and b) created an orthonor
mal set of basis 1forms.
I would now like you to determine the connection 1forms Γ
∼
ˆ
α
ˆ
β
associated with this orthonormal
basis, and again write down the geodesic equations, but of course this time for the 4 components
of the 4velocity vector relative to the orthonormal basis, i.e., determine the 4 equations
d
dτ
u
ˆ
α
+
u
β
Γ
∼
ˆ
α
ˆ
β
(
e
u
) = 0
that determine geodesic, timelike worldlines in these coordinates, where
τ
is the parameter along
these curves with tangent vector
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 Spring '10
 DavidS.King

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