Scribd
Upload
67 views2 pages

HW 1

1. This document outlines the goals and problems for a homework assignment on special and general relativity. The homework includes calculations involving photon rocket trips, particle decays, covariance equations, gravitational time dilation between twins, light deflection by the sun, gravitational wave emission, and deriving the electromagnetic field energy-momentum tensor. 2. Students are asked to calculate the trip time and initial fuel mass for a photon rocket traveling to and from the center of the galaxy, as well as the energy of a decaying particle's photon daughter in different frames of reference. 3. The homework also tests which equations from a list are covariant and asks students to determine which twin sibling will outlive the other based on their separation and

Uploaded by

thai pham cong
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
67 views2 pages

HW 1

1. This document outlines the goals and problems for a homework assignment on special and general relativity. The homework includes calculations involving photon rocket trips, particle decays, covariance equations, gravitational time dilation between twins, light deflection by the sun, gravitational wave emission, and deriving the electromagnetic field energy-momentum tensor. 2. Students are asked to calculate the trip time and initial fuel mass for a photon rocket traveling to and from the center of the galaxy, as well as the energy of a decaying particle's photon daughter in different frames of reference. 3. The homework also tests which equations from a list are covariant and asks students to determine which twin sibling will outlive the other based on their separation and

Uploaded by

thai pham cong
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
You are on page 1/ 2

GR HW 1

Due Apr 19 at 11am, in class

Goals: Special Relativity, Gravitational Redshift, Geodesics, Gravitational Waves, Variation


Principle

1. An ideal photon spaceship (transforming fuel mass to energy of ideally collimated photons
with 100% efficiency) goes to the center of the galaxy (10 kpc from Earth) and back. It starts
from rest and finishes at rest. The accelerations and decelerations are equal to a = 10m/s2 .
Calculate:
a. The trip time by Earth and onboard clocks.
b. The initial mass of the fuel M . (The mass of the ship without fuel plus the mass of the
astronauts is m = 1ton.)

2. The neutral Sigma baryon, Σ0 , with mass mΣ , decays into a Lambda baryon, Λ, with mass
mΛ , and a massless photon.
(a) Find the energy of the photon in the frame in which the Σ0 is at rest.
(b) Find the energy of the photon in the frame in which the Λ is at rest.
Hint: it simplifies the algebra to use four-vectors.

3. φ, Aµ , T µν are scalar, vector and tensor. Which of the following equations are covariant
a. φ = A0
b. φ = Aµ Aµ
c. φ = A0 A0
d. φ = Tµν T µν
e. Tµν = T νµ
f. Tµν = Tνµ
g. T µν = Aµ + Aν
h. Tµν = −Tνµ
i. Tνµ = −Tµν
j. T µν = Aµ Aν
k. φ = det T µν
l. φ = det Tνµ

4. Twin brothers are separated at birth: one stays at sea level the other lives at a space station,
freely orbiting at radius 20,000km. Which one will outlive his brother (will see him die) and
by how much? (Earth mass 5.97 × 1027 gr.)

5. We will see that in Einstein gravity test particles moving in the field of a spherically symmetric
star of mass M follow the geodisics of the Schwarzschild metric:

rg 2 dr2
ds2 = −(1 − )dt + 2
r + r dΩ
2
(1)
r 1 − rg

1
where rg = 2GM/c2 .
Calculate the small angle scattering (scattering angle χ in lowest order in 1/b, b is the impact
parameter) for an ultrarelativistic particle.
In particular, calculate light deviation by the Sun, for a light ray just grazing the surface
of the Sun (solar mass M = 1.99 × 1033 gr, radius b = 700, 000 km, give your answer in
arcminutes).

6. Estimate how long it would take the Earth to fall down on the Sun due to emission of
gravitational waves.

7. Find the energy momentum tensor of electromagnetic field by varying the action
Z
1
S=− d4 xF 2 , Fµν = Aν;µ − Aµ;ν (2)
4
with respect to the metric.

8. When calculating the energy momentum tensor of electromagnetic field by varying with re-
spect to the metric, what should you keep fixed Aµ or Aµ ?

You might also like