Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 354 / 2 / b

Table of contents
- Solution b

Solution

The horizon equation gives

2 M = \frac{r _{+}^{3}}{L ^{2}} + \frac{Q ^{2}}{r _{+}} .

With

Q = r_{+}^{2} μ / L

g (z) = 1 - (1 + μ^{2}) z^{3} + μ^{2} z^{4} .

The coordinate changes in the question give

d s^{2} = \frac{L ^{2}}{z ^{2}} [- g (z) d τ^{2} + \frac{d z ^{2}}{g ( z )} + d X^{2} + d Y^{2}],

A = μL (1 - z) d τ .

The horizon is at

z = 1

, and all nontrivial dimensionless dependence is through

μ

;

L

remains only as the overall curvature scale.

Solved by gpt-5.6-sol high.

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