Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 37E (General Relativity)

Table of contents
- a 37E
  - Solution a
- b 37E
  - i b
    - Solution i
  - ii b
    - Solution ii
  - iii b
    - Solution iii

a

Solution

For a slowly moving particle in the weak-field approximation,

d t / d τ = 1 + O (Φ, v^{2})

and terms containing two spatial velocities may be neglected. The spatial geodesic equation reduces to

\frac{d ^{2} x ^{i}}{d t ^{2}} = - Γ^{i}_{00} .

For

g_{00} = - (1 + 2Φ)

, the Levi-Civita connection is

Γ^{i}_{00} = \frac{1}{2} g^{ij} (- \partial_{j} g_{00}) = \partial_{i} Φ + O (Φ^{2}) .

Hence the Newtonian limit of general relativity is

\frac{d ^{2} x}{d t ^{2}} = - \nablaΦ .

b

i

Solution

Along the radial curve at fixed

t, θ, φ

, the spatial line element is

d S = 1 + 2 M / r d r

. To first order,

S = \int_{r_{1}}^{r_{2}} (1 + \frac{M}{r}) d r,

so

S = (r_{2} - r_{1}) + M lo g \frac{r _{2}}{r _{1}} .

ii

Solution

A radial null geodesic satisfies

0 = - (1 - \frac{2 M}{r}) d t^{2} + (1 + \frac{2 M}{r}) d r^{2},

so for the outgoing ray

\frac{d t}{d r} = 1 + \frac{2 M}{r} + O (M^{2} / r^{2}) .

The coordinate-time interval is therefore

Δ t = (r_{2} - r_{1}) + 2 M lo g \frac{r _{2}}{r _{1}} .

The stationary observer at

r_{2}

measures proper time

d T = 1 - 2 M / r_{2} d t

. Hence

T = (r_{2} - r_{1}) + 2 M lo g \frac{r _{2}}{r _{1}} - \frac{M}{r _{2}} (r_{2} - r_{1}) .

iii

Solution

Put

Δ r = r_{2} - r_{1}

. Expanding the quotient

S / T

consistently to first order gives

v_{eff} = 1 + \frac{M}{r _{2}} - \frac{M}{Δ r} lo g \frac{r _{2}}{r _{1}} .

Thus

v_{eff} = 1 + \frac{M}{r _{2}} - \frac{M}{r _{2} - r _{1}} lo g \frac{r _{2}}{r _{1}} .

Because

lo g (r_{2} / r_{1}) / (r_{2} - r_{1}) \to 1/ r_{2}

r_{1} \to r_{2}

,

r_{1} \to r_{2} lim v_{eff} = 1 .

This local limit is the expected speed of a massless particle measured in a freely falling local inertial frame; the finite-interval effective velocity combines nonlocal length and clock comparisons.

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