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HOMEWORK SET 4, PHYS. 131
Instructor:
Anthony Zee
Email:
[email protected]
TA:
Kevin Moore
Email:
[email protected]
7.2.
The following line element corresponds to Fat spacetime:
dS
2
=

dt
2
+2
dxdt
+
dy
2
+
dz
2
±ind a coordinate transformation that puts the line element in the usual Fat
space form.
Here we notice that only the
x
and
t
coordinates are mixed, so we can take
our coordinate transformation to only involve these coordinates.
Aside: You may recognize the form of this problem from classical mechanics.
What we’re doing is similar to doing a coordinate transformation to uncouple a
system of coupled harmonic oscillators
So our general coordinate transformations look like:
t
±
=
t
±
(
t, x
)
x
±
=
x
±
(
t, x
)
y
±
=
yz
±
=
z
Now, we want the metric written in the new coordinates to look like:
dS
2
=

dt
±
2
+
dx
±
2
+
dy
±
2
+
dz
±
2
So we use the general tensor transformation formula to get
g
αβ
=
∂x
α
±
α
β
±
β
g
α
±
β
±
We can write down 3 relevant independent equations from this:
g
xx
= 0 =

±
∂t
±
²
2
+
±
±
²
2
g
xt
= 1 =

±
±
±
²
+
±
±
±
²
g
tt
=

1=

±
±
²
2
+
±
±
²
2
These equations can basically be solved by inspection, giving the solution:
t
±
=
t

x,
x
±
=
x
1
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View Full Document 8.2.
In usual spherical coordinates the metric on a twodimensional sphere is
dS
2
=
a
2
(
dθ
2
+ sin
2
θdφ
2
)
where
a
is constant.
(a)
Calculate the Christofel symbols “by hand”.
The metric is given by
g
AB
=
a
2
±
10
0
sin
2
θ
²
The only element oF
g
AB
that has nonvanishing derivatives is
g
φφ
, and it de
pends only on
θ
. The only nonvanishing Christofel symbols will be given by
g
φA
Γ
A
φθ
=
g
φφ
Γ
φ
θφ
=
1
2
±
∂g
φφ
∂θ
+
φθ
∂φ

θφ
²
=
a
2
sin
θ
cos
θ
g
θB
Γ
B
φφ
=
g
θθ
Γ
θ
φφ
=
1
2
±
θφ
+
θφ

φφ
²
=

a
2
sin
θ
cos
θ
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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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