Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 38C / c

Table of contents
- i c
  - Solution i
- ii c
  - Solution ii

i

Solution

For a radial null ray,

d θ = 0

and the metric gives

d t = d r

for the outgoing branch. Starting at

r = t = 0

, it reaches

r = R

at

t_{1} = R .

ii

Solution

A stationary observer has four-velocity

u = r \partial_{t}

. If

E = - k_{t}

is the conserved photon energy, the measured frequency is proportional to

- u \cdot k = r E .

Hence

ν_{2} = 2 ν_{1} .

The photon is blueshifted as it moves outward.

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