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The document is a problem sheet for Advanced General Relativity, consisting of various problems related to the energy momentum tensor of particles, their velocities, and the calculation of metrics in Mathematica. It includes tasks such as deriving expressions for stationary and moving particles, calculating the average energy momentum tensor for a uniform density of particles, and programming in Mathematica to compute the Christoffel symbols and Riemann tensors. Additionally, it poses a question about the Levi-Civita tensor in spherical coordinates in Minkowski space.

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Dominic Wynes-Devlin
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48 views2 pages

Ps 1

The document is a problem sheet for Advanced General Relativity, consisting of various problems related to the energy momentum tensor of particles, their velocities, and the calculation of metrics in Mathematica. It includes tasks such as deriving expressions for stationary and moving particles, calculating the average energy momentum tensor for a uniform density of particles, and programming in Mathematica to compute the Christoffel symbols and Riemann tensors. Additionally, it poses a question about the Levi-Civita tensor in spherical coordinates in Minkowski space.

Uploaded by

Dominic Wynes-Devlin
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Advanced General Relativity: Problem Sheet 1

1. The energy momentum tensor of a stationary point particle of mass M at the


origin is
T00 = mc2 δ 3 (x) , Ti0 = Tij = 0 .
In the rest of the question you can set c = 1.

a) Write the four velocity of the particle uµ = dxµ /dτ where τ is the affine pa-
rameter along the world-line of the particle (which as stated is stationary).
b) Express T µν in terms of uµ .
c) Now assume the particle is not stationary, but has a constant velocity. What
is uµ and what would that give for T µν .
d) Can you get this answer by acting on the original T µν by a Lorentz transfor-
mation?
e) Now assume you have a uniform density of particles in some region. If they are
all stationary, What is the average energy momentum tensor in that region.
f) Finally, assume all particles have fixed velocity, but in an arbitrary direction.
What is T µν in that case?
For simplicity you can solve the question in a 2-dimensional spacetime.

2. Hard problem: Download Mathematica and get it to work....

3. Write a code where in the first line you define the metric as a matrix. For example
for S 2
coords = {Theta,Phi};
metric={{1,0},{0,Sin[Theta]^2}};
Then the code calculates Γ and the different R’s.
You may need to use the commands:

• Length[xxx] gives the length of the vector xxx.


• Inverse[matrix]
• D[f[x],x] (for derivative),
• Table[f[n],n,1,10] creates a length n vector. With two iterators it will
make a matrix, etc.
• Sum[a[[n,n]],n,1,10] where a[[n,n]] is the (n,n) entry in a matrix. This
sum calculate the trace of a 10d matrix.
If you are not experienced with Mathematica, you can look at the hint file on the
keats page.
There are other ways to program this not relying on Table, but on functions and
so on. You can try to be creative.

4. For Minkowski space in spherical coordinates ds2 = −dt2 +dr 2 +r 2 dθ2 +r 2 sin2 θ dφ2 ,
how many nonzero components are there to the Levi-Civita tensor?
You may rely on mathematica or do it by hand or both.

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