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Problem Sheet 1

This document is a problem sheet for a course on General Theory of Relativity, specifically focusing on Special Relativity for the Monsoon 2025-26 semester. It includes five problems related to velocity addition of moving carts, properties of 4-velocity, angle transformations between different frames, star distribution for moving observers, and the relationship between imaginary coordinates and Lorentz boosts. The submission date for the problem sheet is August 14, 2025.

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458 views1 page

Problem Sheet 1

This document is a problem sheet for a course on General Theory of Relativity, specifically focusing on Special Relativity for the Monsoon 2025-26 semester. It includes five problems related to velocity addition of moving carts, properties of 4-velocity, angle transformations between different frames, star distribution for moving observers, and the relationship between imaginary coordinates and Lorentz boosts. The submission date for the problem sheet is August 14, 2025.

Uploaded by

anudev301
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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PH7320E GENERAL THEORY OF RELATIVITY

Monsoon 2025-26
Problem sheet 1- Special relativity

(Submission date: August 14, 2025)

1. velocity addition
A cart rolls on a long table with velocity v. A smaller cart rolls on the first cart in the same direction with
velocity v relative to the first cart. A third cart rolls on the second cart in the same direction with velocity v
relative to the second cart and so on upto n carts. What is the velocity vn of the n th cart relative to the table.
what does vn tend to as n → ∞.
α
 α
2. Consider the quantity uα = dx ds , α = 0, 1, 2, 3 (4-velocity). Show that uα
u α = 1 and u α
du
ds =0

3. Frame S ′ moves with velocity ⃗v with respect to frame S. A rod in frame S ′ makes an angle θ′ with respect to
the forward direction of motion. What is the angle θ as measured in S?
4. Suppose that an observer at rest with respect to the fixed distant stars sees an isotropic distribution of stars.
That is, in any solid angle dΩ he sees dN = N dΩ 4π stars, where N is the total number of stars he can see.
Suppose now that another observer (whose rest frame is S ′ ) is moving at a relativistic velocity β in the ex
direction. What is the distribution of stars seen by this observer? Specifically what is the distribution function
P (θ′ , ϕ′ ) such
R that′ the number of stars seen by this observer in his solid angle dΩ′ is P (θ′ , ϕ′ )dΩ′ ? Check to
see that P (θ , ϕ′ )dΩ′ = N and check that P (θ′ , ϕ′ ) → 4π
N
as β → 0. Where will the observer see the stars
sphere
bunch up?
5. Define an imaginary coordinate w = it. Show that a rotation of angle θ in the xi , w plane (i=1,2,3) where θ is a
pure imaginary number, corresponds to a pure lorentz boost in t, x, y, z coordinates. How is the boost velocity
v related to the angle θ?

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