AY10/11
PC4248 RELATIVITY EXAM SOLUTIONS
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Question 1
∆ = 2 ∞
1
2
1
− 1 − = ⇒ = − ∆ = 2 = 2
2
1 −
1 −
1 − 1 − 2 = 2
1 − 2 1
2 − 1−
≈ 2
1 + 2 − 1 + = 2
1 + − − 1+ ≈ 2
1 − − 1 +
≈ 2
1
2 = 2
+
1 − − 1 − − 4
1+ −
2 = 2
+
1 − − 1 − − 1
1 − − −
2
1 − −
!
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AY10/11
PC4248 RELATIVITY EXAM SOLUTIONS
2
&
1 −
= 2 "sin −
+
'
≈ ( − 2 sin& −
2 2
+
4
=(+
≈(+
2
+
4 4.
=
/ In the case of grazing the edge of the sun,
4.⊙
*+,- = / 1⊙
∴ *+,- = ∆ − ( =
Question 2
2
6
51 + 7
= ±4 − 1 −
2
Proper time after the particle crossed the horizon,
2 = −
4 − 1 −
2
6
+
1
For maximum proper time, 4 = 0, 6 = 0:
2=
2 − 1
= 9 ,
= 299
2=
√
=
√
299
2
−1
9
29
√2 − 9
9
= − <29=2 − 9 >
=
√
√
2=2 − 9 9
+
√
2=2 − 9 9
2
AY10/11
PC4248 RELATIVITY EXAM SOLUTIONS
9 = √2 sin ? ,
9 = √2 cos ? ?
B
2 = 2 =2 − 2 sin ? √2 cos ? ?
B
= 2 2 cos ? ?
= 4 B
1
2
+
cos 2?
?
2
B
? sin 2? = 4 " +
'
2
4 (
= 4 = (
∴ 2CDE = (
Question 3 (a)
F = −29G + H I9JK + I9JL 2
F 9 G
K
M=
= − = 2
− H N
N
N
N N
−
L
N
OM
OM
=
O9 N O 9
N
K L 1 N
G 1 N
− 2
′ =
P−2HH′ Q
2
N
N
2 2 2 N 2 2
G
K
′
=
−HH
2 2
OM
OM
=
OG N O G N
9 1 N
0=
2
2 N 2 2
9
I2J
= 0,
2 L
− 2
′
,
I1J
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AY10/11
PC4248 RELATIVITY EXAM SOLUTIONS
OM
OM
=
OK N O K N
K 1 N
0=
"−2H
'
N
N 2 2
K
=
−H
2
2
9 K
K
= −2HH′
− H 2 2
2
′
2H 9 K
K
I3J
=−
,
H 2 2
2
By symmetry,
L
2
′ 9 L
=−
,
2 2 2
I4J
So the non-zero Christoffel symbols are
ΓSEE = HH′
ΓSTT = ′
H′
H
′
=
E
E
ΓUE
= ΓEU
=
T
T
ΓUT
= ΓTU
Question 3 (b)
X
X
X
X
1VW = OX ΓVW
− OW ΓVX
+ ΓVW
ΓXZ − ΓVZ
ΓWX
Z
Z
E
E E
1UU = −OU ΓUE
− OU ΓUT − ΓUE
ΓUE − ΓUT ΓUT
T
T
T
H[
H[[
[
[[
H′
′
=5 − 7+5 − 7−5 7 −5 7
H
H
H
= −5
1SS = 0,
H[[ [[
+ 7
H
1EE = 0,
1US = 1SU = 0,
1UE = 1EU = 0,
1SE = 1ES = 0,
1TT = 0
1ET = 1TE = 0
1UT = 1TU = 0
1ST = 1TS = 0
∴ The only non-zero Ricci curvature tensor,
1UU
H[[ [[
= −5 + 7
H
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AY10/11
PC4248 RELATIVITY EXAM SOLUTIONS
Question 3 (c)
1VW = 0
1UU = − 5
H[[ [[
1 H 1 + 7=−
−
=0
H
H 9 9
1 H 1 −
=0
H 9 9
1 H 1 +
=0
H 9 9
H
H H
+
=0
H 9
9
H
+H =0
9
9
IH
J = 0
9
IH
J = \
9
−
∴ HI9J
I9J = \ 9 + \
where \ , \ are constants.
Question 3 (d)
Solutions provided by:
Dr Wang Qinghai (Q1-2)
John Soo (Q3a-c)
☺ 2012, NUS Physics Society.
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