 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-1/37e/b/solution

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

Because

τ

is proper time, the four-velocity has normalization

g_{μν} \overset{x}{˙}^{μ} \overset{x}{˙}^{ν} = - 1.

t = 0

, the initial data therefore give

- \dot{t} (0)^{2} + v_{0}^{2} = - 1.

Both

t

and

τ

are future oriented, so the positive root must be chosen:

\dot{t} (0) = 1 + v_{0}^{2} .

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