general relativity (225a) fall 2013 assignment 1 -...

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University of California at San Diego – Department of Physics – Prof. John McGreevy General Relativity (225A) Fall 2013 Assignment 1 Posted September 29, 2013 Due Monday, Oct 7, 2013 1. Show that lmn R li R mj R nk = det R ijk for any 3 × 3 matrix R. (Recall that the determinant is a scalar function of square matrices which is odd under interchange of rows and columns and has det 1 = 1. And ijk is the completely antisymmetric collection of numbers with 123 = 1.) 2. Show that the Maxwell equations (in Minkowski space) ijk j E k + 1 c t B i =0, i H i =0 ijk j B k - 1 c t E i = 4π c J i , i E i =4πρ. are invariant under the following set of transformations: x i 7˜ x i R ij x j , R O(3) E i (x) 7˜ E i x)= R ij E j (x) B i (x) 7˜ B i x) = det RR ij B j (x) ρ(x) 7˜ ρx)= ρ(x), J i (x) 7˜ J i x)= R ij J j (x). (1) That is, ~ E is a polar vector and ~ B is an axial vector. Recall that an O(3) matrix satisfies R T R =1 . 3. Prove the identity ijk lmn = det δ il δ im δ in δ jl δ jm δ jn δ kl δ km δ kn . Use this identity to show that ij k lmk = δ il δ jm - δ im δ jl . 4. Lorentz contraction exercise [from Brandenberger] (a) Suppose frame S 0 moves with velocity v relative to frame S . A projectile in frame S 0 is fired with velocity v 0 at an angle θ 0 with respect to the forward direction of motion ( ~v). What is this angle θ measured in S ? What if the projectile is a photon? 1

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Page 1: General Relativity (225A) Fall 2013 Assignment 1 - …mcgreevy.physics.ucsd.edu/f13/225A-pset01.pdf · General Relativity (225A) Fall 2013 Assignment 1 Posted September 29, 2013 Due

University of California at San Diego – Department of Physics – Prof. John McGreevy

General Relativity (225A) Fall 2013Assignment 1

Posted September 29, 2013 Due Monday, Oct 7, 2013

1. Show thatεlmnRliRmjRnk = detRεijk

for any 3× 3 matrix R.

(Recall that the determinant is a scalar function of square matrices which is oddunder interchange of rows and columns and has det 1 = 1. And εijk is the completelyantisymmetric collection of numbers with ε123 = 1.)

2. Show that the Maxwell equations (in Minkowski space)

εijk∂jEk +1

c∂tBi = 0, ∂iHi = 0

εijk∂jBk −1

c∂tEi =

cJi, ∂iEi = 4πρ.

are invariant under the following set of transformations:

xi 7→ xi ≡ Rijxj, R ∈ O(3)

Ei(x) 7→ Ei(x) = RijEj(x)Bi(x) 7→ Bi(x) = detRRijBj(x)ρ(x) 7→ ρ(x) = ρ(x), Ji(x) 7→ Ji(x) = RijJj(x). (1)

That is, ~E is a polar vector and ~B is an axial vector. Recall that an O(3) matrixsatisfies RTR = 1.

3. Prove the identity

εijkεlmn = det

δil δim δinδjl δjm δjnδkl δkm δkn

.

Use this identity to show that

εijkεlmk = δilδjm − δimδjl.

4. Lorentz contraction exercise [from Brandenberger]

(a) Suppose frame S ′ moves with velocity v relative to frame S. A projectile in frameS ′ is fired with velocity v′ at an angle θ′ with respect to the forward directionof motion (~v). What is this angle θ measured in S? What if the projectile is aphoton?

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Page 2: General Relativity (225A) Fall 2013 Assignment 1 - …mcgreevy.physics.ucsd.edu/f13/225A-pset01.pdf · General Relativity (225A) Fall 2013 Assignment 1 Posted September 29, 2013 Due

(b) An observer A at rest relative to the fixed distant stars sees an isotropic distri-bution of stars in a galaxy which occupies some region of her sky. The number ofstars seen within an element of solid angle dΩ is

PdΩ =N

4πdΩ

where N is the total number of stars that A can see. Another observer B movesuniformly along the z axis relative to A with velocity v. Letting θ′ and ϕ′ berespectively the polar (with respect to z) and azimuthal angle in the inertialframe of B, what is the distribution function P ′(θ′, ϕ′) such that P ′(θ′, ϕ′)dΩ′ isthe number of stars seen by B in the solid angle dΩ′ = sin θ′dθ′dϕ′.

(c) Check that when integrating the distribution function over the sphere in thecoordinates of B you obtain N ! Discuss the behavior of the distribution P ′ in thelimiting cases when the velocity v goes to 0 or to 1.

5. Show that the half of the Maxwell equations

0 = εµνρσ∂νFρσ

is invariant under the general coordinate transformation,

xµ 7→ xµ = fµ(x), Fµν(x) 7→ Fµν(x) =∂xρ

∂xµ∂xσ

∂xνFρσ(x)

for an arbitrary fµ(x) with non-zero Jacobian.

The following problems are optional.

6. Eotvos

What is the optimal latitude at which to perform the Eotvos experiment?

7. Poincare group

Show that the Poincare group satisfies all the properties of a group. (That is: it hasan identity, it is closed under the group law, it is associative, and every element hasan inverse in the group.)

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