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Date May 2017 Marks available 1 Reference code 17M.3.HL.TZ2.7
Level Higher level Paper Paper 3 Time zone Time zone 2
Command term State Question number 7 Adapted from N/A

Question

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

[1]
a.i.

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

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

[1]
a.ii.

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

[1]
a.iii.

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.

[3]
b.

Markscheme

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]

a.i.

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]

a.ii.

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]

a.iii.

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]

b.

Examiners report

[N/A]
a.i.
[N/A]
a.ii.
[N/A]
a.iii.
[N/A]
b.

Syllabus sections

Option A: Relativity » Option A: Relativity (Additional higher level option topics) » A.5 – General relativity (HL only)
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