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Massachusetts Institute of Technology Department of Physics 8.962 Spring 2006 Problem Set 4 Post date: Thursday, March 9th Due date: Thursday, March 16th 1. Connection in Rindler spacetime The spacetime for an accelerated observer that we derived on Pset 2, ¯2 + dx ds2 = −(1 + gx ¯)2 dt ¯2 + dy ¯2 + dz ¯2 (1)
is known as “Rindler spacetime”. Compute all non-zero Christoﬀel symbols for this spacetime. (Carroll problem 3.3 will help you quite a bit here.) 2. Relativistic Euler equation � ⊗U � + P

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Massachusetts Institute of Technology Department of Physics 8.962Spring2006 ProblemSet4
Post date: Thursday, March 9th Duedate:Thursday,March16th 1. Connection in Rindler spacetime The spacetime for an accelerated observer that we derived on Pset 2,
ds
2
=
−
(1+
gx
¯)
2
dt
¯
2
+
dx
¯
2
+
dy
¯
2
+
dz
¯
2
(1) is known as “Rindler spacetime”. Compute all nonzero Christoﬀel symbols for this spacetime. (Carroll problem 3.3will help you quite a bit here.) 2. Relativistic Euler equation (a)Startingfromthe stressenergy tensorforaperfect ﬂuid,
T
=
ρU �
⊗
U �
+
P
h
, where
h
=
g
−
1
+
U �
⊗
U �
, using local energy momentum conservation,
∇·
T
=0,derivethe relativistic Euler equation, (
ρ
+
P
)
∇
U �
U �
=
−
h
·∇
P.
(2) (Note:Becauseboth
T
and
h
are symmetric tensors, there is no ambiguity in the dot products that appear in this problem.) (b)Foranonrelativistic ﬂuid(
ρ
�
P
,
v
t
�
v
i
)and a cartesian basis, show that this equation reduces to the Euler equation,
∂v
i
1 +
v
k
∂
k
v
i
=
−
∂
i
P.
(3)
∂t ρ
(
i
,
k
are spatial indices running from 1 to 3.) What extra terms are present if the connectionisnonzero(e.g., spherical coordinates)? (c)Applythe relativistic Euler equation to Rindler spacetime for hydrostatic equilib- rium. Hydrostatic equilibrium means that the ﬂuid is at rest in the ¯
x
coordinates,i.e.
U
x
¯
=0. Supposethattheequationof state(relationbetweenpressureanddensity) is
P
=
wρ
where
w
isapositiveconstant. Find thegeneral solution
ρ
(¯
x
)with
ρ
(0)=
ρ
0
. (d)Suppose now instead that
w
=
w
0
/
(1+
gx
¯)where
w
0
is a constant. Show that the solutionis
ρ
(¯
x
)=
ρ
0
exp(
−
¯Find
L
, the density scale height, in terms of
g
and
x/L
.
w
0
. Convert to “normal” units by inserting appropriate factors of
c
—
L
should be a length. (e) Compare your solution to the density proﬁle of a nonrelativistic, planeparallel, isothermal atmosphere(for which
P
=
ρkT/µ
, where
T
is temperature and
µ
is the meanmolecularweight)inaconstantgravitational ﬁeld. [UsethenonrelativisticEuler equation with gravity: add a term
−
∂
i
Φ=
g
i
, where Φ is Newtonian gravitational potential and
g
i
is Newtonian gravitational acceleration, to the right hand side of Eq. (3).] Why does hydrostatic equilibrium in Rindler spacetime — where there is no gravity — give such similar results to hydrostatic equilibrium in a gravitational ﬁeld?
�
3. Spherical hydrostatic equilibrium As we shall derive later in the course, the line element for a spherically symmetric static spacetime may be written
−
1
ds
2
=
−
e
2Φ(
r
)
dt
2
+1
−
2
GM
(
r
)
dr
2
+
r
2
(
dθ
2
+ sin
2
θdφ
2
)
, r
whereΦ(
r
)and
M
(
r
)are somegivenfunctions. Inhydrostaticequilibrium,
U
i
=0for
i
∈
[
r,θ,φ
]. Using the relativistic Euler equation, show that in hydrostatic equilibrium
p
=
p
(
r
)with
∂p ∂
Φ =
−
(
ρ
+
P
)
. ∂r ∂r
4. Converting from nonaﬃne to aﬃne parameterization Suppose
v
α
=
dx
α
/dλ
∗
obeys the geodesic equation in the form
Dv
αα
=
κ
(
λ
∗
)
v. dλ
∗
Clearly
λ
∗
is not an aﬃne parameter. Showthat
u
α
=
dx
α
/dλ
obeys the geodesic equation in the form
Du
α
=0
dλ
provided that
dλ
=exp
κ
(
λ
∗
)
dλ
∗
. dλ
∗
5. Conserved quantities with charge A particle with electric charge
e
moves with 4velocity
u
α
in a spacetime with metric
g
αβ
in the presence of a vector potential
A
µ
. The equation describing this particle’s motion can be written
u
β
∇
β
u
α
=
eF
αβ
u
β
,
where
F
αβ
=
∇
α
A
β
−∇
β
A
α
.
The spacetime admits a Killing vector ﬁeld
ξ
α
such that
L
ξ�
g
αβ
=0
,
L
ξ�
A
α
=0
.
Showthatthequantity(
u
α
+
eA
α
)
ξ
α
is constant along the worldline of the particle.

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