 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-3/38e/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Hold

p = mγ

fixed. At a point with

x \neq = t

A = \frac{p}{2 γ γ ^{2} ( x - v t ) ^{2} + y ^{2} + z ^{2}} ⟶ 0,

but

γ^{2} A ⟶ \frac{p}{2∣ x - t ∣} .

The small-

A

expansions are

f (A) = 1 + 4 A + O (A^{2}), g (A) = 8 A + O (A^{2}) .

Consequently, with

H = 4 p /∣ x - t ∣

d s^{2} = - d t^{2} + d x^{2} + d y^{2} + d z^{2} + H (d t - d x)^{2}, x \neq = t .

Equivalently,

g_{tt} = - 1 + H, g_{t x} = g_{x t} = - H, g_{xx} = 1 + H, g_{yy} = g_{zz} = 1,

with all other components zero.

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