State the principle of equivalence.

The gravitational field strength near the surface of a neutron star is 1.2 ×10¹³Nkg^–1. A light ray is emitted from a stationary probe at a height of 250 m above the surface. The frequency of the light measured in the probe is 4.8×10¹⁴ Hz.

(i) Determine the frequency of the light received at the surface of the star according to an observer at the surface.

(ii) Describe how gravitational red-shift leads to the concept of gravitational time dilation.

General relativity predicts the existence of black holes.

(i) State what is meant by a black hole.

(ii) Suggest two ways in which a black hole may be detected.

a frame of reference that is freely falling in a gravitational field is equivalent to an inertial frame of reference far from all masses;

or

inertial and gravitational effects are indistinguishable;

or

a frame of reference accelerating in empty space is equivalent to a frame of reference at rest in a gravitational field;

(i) $\left( {\frac{{\Delta f}}{{{f_0}}} = \frac{{g\Delta h}}{{{c^2}}}{\rm{and so}}} \right)\Delta f = \left( {\frac{{1.2 \times {{10}^{13}} \times 250 \times 4.8 \times {{10}^{14}}}}{{9 \times {{10}^{16}}}} = } \right)0.16 \times {10^{14}}{\rm{Hz}}$;
$f = \left( {4.8 \times {{10}^{14}} + 0.16 \times {{10}^{14}} = } \right)5.0 \times {10^{14}}{\rm{Hz}}$;
ECF applies only if the wrong value of Δf is added to f₀.

(ii) as light travels away from a massive body it is red-shifted/frequency decreases/time period increases;
hence closer to the body the time period is shorter / time runs slower (than higher up) – this is (equivalent to) time dilation;

(i) a point of infinite curvature / a point of infinite space / a singularity of spacetime / a region from which nothing can escape / escape velocity ≥c;

(ii) matter falling into the black hole radiates;
the (gravitational) influence on other objects;
by observing its gravitational lensing effect;
emission of Hawking radiation;