Calculate the number of ticks detected in 10 ks by the distant observer.

Observer A now sends a beam of light initially parallel to the surface of the planet.

Explain why the path of the light is curved.

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

Suggest, whether your answer to (a) underestimates or overestimates the correction required to the time signal.

The probe is stationary above the event horizon of the black hole in (a). The probe sends a radio pulse every 1.0 seconds (as measured by clocks on the probe). The spacecraft receives the pulses every 2.0 seconds (as measured by clocks on the spacecraft). Determine the distance of the probe from the centre of the black hole.

Calculate the shift in frequency observed by A in terms of Δf.

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

Explain whether the detected frequency would be greater or less than the emitted frequency.

The probe emits 20 short pulses of these radio waves every minute, according to a clock in the probe. Calculate the time between pulses as measured by the observer.

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.

Outline why the clock near the black hole runs slowly compared to a clock close to the distant observer.

Star S-2 is in an elliptical orbit around a black hole. The distance of S-2 from the centre of the black hole varies between a few light-hours and several light-days. A periodic event on S-2 occurs every 5.0 s.

Discuss how the time for the periodic event as measured by an observer on the Earth changes with the orbital position of S-2.

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

Explain the cause of the frequency shift for the gamma rays in your answer in (a) if the tower and detector were accelerating towards the gamma rays in free space.

Explain the cause of the frequency shift for the gamma rays in your answer in (a) in the Earth’s gravitational field.

Calculate the fractional change in frequency of the gamma rays at the detector.

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

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

The mass of the black hole is 4.0 × 10³⁶kg. Calculate the Schwarzschild radius of the black hole.

Explain why the frequency of the radio waves detected by the observer is lower than $f$.

The gravitational field strength at 20 Mm above the surface of the Earth is about 0.6 N kg^–1. Estimate the time correction per day needed to the time signals, due to the gravitational redshift.

Calculate the expected shift in frequency between the emitted and the detected gamma rays.

Calculate the distance of the event horizon of the black hole from its centre.

                                                                   Mass of Sun = 2 × 10³⁰ kg