General Relativity Homework 1

Lecturer: Dr. Eugene A. Lim

2017 Year 3 Semester 2 Theoretical Physics

Week 1+2

1. In the lectures, we argue that Lorentz transformations in 3+1D are represented by 4 × 4 matrices

given by Λµ′µ, which obey the equation

Λµ′

µηµ′ν′Λν′

ν = ηµν . (1)

Suppose Λµ′µ and Λµ

′µ are two different Lorentz transforms. Show that the product of these two

transforms must also obey Eq. (1). Using these results, discuss whether a rotation in the x − y plane

followed by a boost along the z direction is also a Lorentz transformation.

2. Consider a timelike vector V µ

ηµνVµV ν = −1. (2)

Show that, under Lorentz transformation V µ′

= Λµ′µV

µ, the vector remains timelike ηµ′ν′V µ′V ν

′= −1.

What about spacelike and null-like vectors?

3. The Maxwell equations are

∇×B = µ0J + µ0ε0∂tE, (3)

∇×E = −∂tB, (4)

∇ ·E =ρcε0, (5)

∇ ·B = 0. (6)

As we discussed in class, we can define the vector potential Aµ = (V,A) and the 4-current jµ = (ρ,J), and

the electromagnetic tensor Fµν = ∂µAν − ∂νAµ. Show that the covariant form of the Maxwell equations

∂µFνρ + ∂ρFµν + ∂νFρµ = 0 , ∂µFµν = jν (7)

is equivalent to the Maxwell equations written above in the usual 3-vector notation.

4. In class, we discussed the Pound-Rebka experiment. We argue that even though gravitational redshift

is observed in a general relativistic interpretation of gravity, the effect is also present in a purely Newtonian

interpretation of gravity, as long as the WEP is obeyed. In the Pound-Rebka experiment, Alice and Bob

are at rest in a fixed gravitational field g = ∇Φ, or assuming spherical symmetry g = ∂Φ/∂r. Let h be

the distance between Alice and Bob. Now, the WEP tells us that this system is equivalent to Alice and

Bob falling with acceleration g with respect to a free-falling frame, so their trajectories are given as usual

by

rA(t) = h+1

2gt2 , rB(t) =

1

2gt2, (8)

and both have non-relativistic velocities v = gt � c. Alice emits the first photon (with velocity c) at

time t = t1, and the second photon at time t = t1 + ∆tA.

(i) Show that Bob receives the first photon at time t = T1 which is given by the implicit formula

h+1

2gt21 − c(T1 − t1) =

1

2gT 2

1 . (9)

1

Alice

Bob

r t

rAliceBob

Photon 2

Photon 1

t1

t1 +dt

t2

t2 +dtgravity

Figure 1: The Pound-Rebka Experiment.

(ii) Suppose that Bob receives the second photon at time t = T1 + ∆tB . By neglecting all terms of order

O(∆t)2, show that

c(∆tA −∆tB) = g(∆tBT1 −∆tAt1). (10)

Discuss why we can neglect the terms of order O(∆t)2.

(iii) Hence, by neglecting terms of O(g2T 21 /c

2), show that

∆tB ≈(

1− g(T1 − t1)

c

)∆tA. (11)

Argue that this is approximately equal to the result we obtained in the GR treatment we discussed in

class (restoring c’s)

∆tB ≈(

1 +(Φ(rB)− Φ(rA)

c2

)∆tA. (12)

5. Show that the function

f(x) = e−1/x2

(13)

converges to 0 as x→ 0. Is this function analytic? Why (or why not)?

6. (Spherical coordinates basis). Let (r, θ, φ) be the spherical coordinate system related to the

Cartesian coordinate system (x, y, z) by the coordinate transforms

x = r sinφ sin θ , y = r cosφ sin θ , z = r cos θ (14)

The metric for flat Euclidean space in 3D in Cartesian basis is given by

ds2 = dx2 + dy2 + dz2. (15)

(i) The Cartesian basis is given by e(i) = ∂i = (∂x, ∂y, ∂z). As discussed in class, the norm of any vector

A is given by g(A,A). The norm of the basis vector e(i) is then g(e(i), e(i)). Show that the norm for all

3 basis vectors of the Cartesian basis is unity, and the basis is orthogonal g(e(i), e(j)) = δij .

(ii) By executing the coordinate transforms above, derive the metric for flat 3D Euclidean space in the

spherical coordinate basis.

(iii) As discussed in class, the coordinate basis of any coordinate system xi is given by e(i) = ∂/∂xi, so

the spherical coordinate basis is e(i) = ∂i = (∂r, ∂θ, ∂φ). Show that the norms for this basis is given by

1, r2 and r2 sin2 θ for i = r, θ, φ respectively.

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(iv**) Consider the non-coordinate basis e(i) = (∂r, r−1∂θ, r

−1 sin−1 θ∂φ). Show that this basis is or-

thonormal, i.e. it has unity norm and is orthogonal to each other g(e(i), e(j)) = δij .

(7) The energy-momentum of cold dust is a perfect fluid with ρ > 0 and zero pressure P = 0. In its rest

frame, with coordinates xµ,

Tµν =

ρ

0

0

0

. (16)

Consider a Lorentz boost in the x direction into the new frame xµ′, given by

Λµ′

ν =

γ −γβ/c2

−γβ γ

1

1

(17)

Calculate the energy-momentum tensor of cold dust in this primed frame Tµ′ν′

.

(8) The energy-momentum tensor for electromagnetic waves is given by

TµνEM = ηαβFµαF νβ − 1

4ηµν |F 2| , |F 2| ≡ ηµαηνβFµνFαβ . (18)

Using the covariant form of the Maxwell equations, show that this equation obey the conservation

equation

∂µTµνEM = 0. (19)

(9) (Schulz) The trajectory of a particle is described by the equations

x(t) = at+ b sinωt , y(t) = b cosωt , z(t) = 0 , |bω| < 1 (20)

in some inertial frame with coordinates (t, x, y, z). Describe the motion with an illustration, and compute

the components of the four-velocity and four-acceleration.

(10) The trajectory of a particle is parameterized by a monotonic parameter λ as

t(λ) = a sinh(λ/a) , x(λ) =a

2cosh(λ/a) , y(λ) =

a

2cosh(λ/a) , z(λ) = b (21)

where a and b are constants. Compute the four-velocity and four-acceleration of this particle. If the

metric of the spacetime is Minkowski, is the trajectory spacelike or timelike?

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