| Date | November 2014 | Marks available | 3 | Reference code | 14N.3.HL.TZ0.20 |
| Level | Higher level | Paper | Paper 3 | Time zone | Time zone 0 |
| Command term | Calculate | Question number | 20 | Adapted from | N/A |
Question
This question is about black holes.
A space probe is stationary in the gravitational field of a black hole.
The mass of the black hole is \(4.5 \times {10^{31}}{\text{ kg}}\). The space probe is emitting a pulse of blue light at a time interval of 1.0 seconds as measured on the space probe. The light is received by an observer on a distant space station that is stationary with respect to the space probe.
Explain why the light reaching the space station will be red-shifted.
The time between the pulses as measured by the observer on the distant space station is found to be 1.5 s. Calculate the distance of the space probe from the black hole.
Markscheme
the principle of equivalence predicts photon energy decreases as it moves against a \(g\) field;
this energy is given by \(E = hf\);
hence as \(E\) decreases, \(f\) must also decrease;
or
photon is subject to extreme warping of spacetime;
under these conditions a distant observer observes dilation of the photon period / OWTTE;
increase in time period is equal to decrease in photon frequency;
\({R_{\text{S}}} = \frac{{2GM}}{{{c^2}}} = \frac{{2 \times 6.67 \times {{10}^{ - 11}} \times 4.5 \times {{10}^{31}}}}{{9 \times {{10}^{16}}}} = 66700{\text{ m}}\);
\(\Delta t = \frac{{\Delta {t_0}}}{{\sqrt {1 - \frac{{{R_{\text{S}}}}}{r}} }} = 1.5 = \frac{{1.0}}{{\sqrt {1 - \frac{{66700}}{r}} }}\);
\(r = 1.2 \times {10^5}{\text{ m}}\);
Examiners report
The stronger candidates clearly explained gravitational red-shift in a black hole gravitational field.
The stronger candidates clearly explained gravitational red-shift in a black hole gravitational field. In (b), some candidates applied formulae from the Data Booklet; the best candidates also explained the formula.
