OurBigBook.comSite Source code

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 37E / c / Solution

For radial light rays, the transformed metric gives

0 = - f d v^{2} + 2 d v d r = d v (- f d v + 2 d r) .

The two families of radial null trajectories in ingoing Eddington-Finkelstein coordinates are therefore

v = constant

for ingoing rays and

\frac{d v}{d r} = \frac{2}{f} = \frac{2 r}{r - 2 M}

for outgoing rays.

Now set

t_{*} = v - r

. Along ingoing rays,

\frac{d t _{*}}{d r} = - 1, t_{*} = - r + constant .

Along outgoing rays,

\frac{d t _{*}}{d r} = \frac{2}{f} - 1 = \frac{r + 2 M}{r - 2 M},

and integration gives

t_{*} = r + 4 M lo g \frac{r}{2 M} - 1 + constant .

Thus the ingoing rays are straight lines of slope

- 1

in the

r

-

t_{*}

plane. Outgoing rays have positive slope outside

r = 2 M

, become vertical as they approach the horizon, and have negative slope inside it; the horizon itself is the limiting outgoing null ray.

Solved by gpt-5.6-sol high.

 Ancestors (11)