State what is meant by the event horizon of a black hole.

Show that the surface area A of the sphere corresponding to the event horizon is given by

\(A = \frac{{16\pi {G^2}{M^2}}}{{{c^4}}}\).

Suggest why the surface area of the event horizon can never decrease.

The diagram shows a box that is falling freely in the gravitational field of a planet.

A photon of frequency f is emitted from the floor of the box and is received at the ceiling. State and explain the frequency of the photon measured at the ceiling.

the surface at which the escape speed is the speed for light
OR
the surface from which nothing/not even light can escape to the outside
OR
the surface of a sphere whose radius is the Schwarzschild radius

Accept distance as alternative to surface.

[1 mark]

use of \(A = 4\pi {R^2}\) and \(R = \frac{{2GM}}{{{c^2}}}\)

«to get \(A = \frac{{16\pi {G^2}{M^2}}}{{{c^4}}}\)»

[1 mark]

since mass and energy can never leave a black hole and \(A = \frac{{16\pi {G^2}{M^2}}}{{{c^4}}}\)

OR

some statement that area is increasing with mass

«the area cannot decrease»

[1 mark]

ALTERNATIVE 1 — (student/planet frame):

photon energy/frequency decreases with height
OR
there is a gravitational redshift

detector in ceiling is approaching photons so Doppler blue shift

two effects cancel/frequency unchanged

ALTERNATIVE 2 – (box frame):

by equivalence principle box is an inertial frame

so no force on photons

so no redshift/frequency unchanged

[3 marks]