You may use the following values of the fundamental constants for this ex-
amination where appropriate.

c= 3.00 x 10⁸ ms^-1
G = 6.67 x 10^-11 N rn² kg^-2
h = 6.63 x 10^-34 Js
ε_o = 8.85 x 10^-12 Fm^-1
ε_oμ_o= c^-2
Mass of the sun = 1.99 x 10³⁰ kg
Mass of the earth = 5.98 x 10²⁴ kg
Mass of the electron = 0.511 MeV//c²
Mass of the proton = 938 MeV/c²

Part 1:

1. Inertial observer 0' moves at speed v in the negative z-direction with respect to inertial observer 0 . Write down the relationship between the sets of coordinates used by the two different observers.

2. A rocket moves at speed 4c/5 towards the earth. It fires a missile at a speed of 3c/5 opposite to its direction of motion as measured on the rocket. With what speed does the missile strike the earth?

3. If A = (λ, 1, 2, 2) is a future-pointing timelike 4-vector, what is the maximum information you know about λ?

4. A particle is measured to have a kinetic energy that is four times its rest mass energy. How fast is the particle moving? Express the speed as a fraction of c.

5. Write down the transformations that relate the electric field E^→ and the magnetic field B^→ in two different inertial frames. Use your result to show that E^→ • B^→ is the same for all inertial observers. You may choose the motion to be in the x-direction if you wish.

6. Define the Schwarzschild radius of a black hole and give a brief descrip-tion of its significance.

7. Give a very brief description of two observational tests of general rela-tivity.

8. An inertial observer sees a particle moving at a speed 3c/5. It then is seen to decay into two photons. Draw an accurate spacetime diagram to illustrate the situation before and after the decay takes place.

9. What energies expressed in terms of the rest mass energy of the initial particle, would the observer in question A8 measure for the two photons?

SECTION B

1. (a) Define what is meant by a 4-vector A.

(b) If A and B are both 4-vectors, prove that A • B is invariant under a Lorentz transformation.

(c) If T is a timelike 4-vector, prove that it is possible to perform a Lorentz transformation to a new inertial frame so that T has only a time component in this new frame.

(d) If T is a timelike 4-vector and N is a null 4-vector, is it ever possible for T + N to be a null 4-vector?

2. (a) What is meant by the threshold energy in a particle reaction? If the threshold energy in a particle reaction holds, what can you say about the motion of the particles in the centre of momentum frame? What does this imply about the motion of the particles in the original laboratory frame?

(b) Two photons of energies E1 and E2 collide with their incident direc-tions of motion making an angle θ with respect to each other. The collision annihilates the two photons and produces two new particles of equal rest mass m. Show that at threshold the condition

E1 E2 sin2(θ/2) = m²c⁴

holds.

3. (a) Write down the line element that describes the spacetime geometry outside the event horizon of a spherically symmetric black hole of mass M.

(b) What equation describes the radial infall of a light ray into a spher-ically symmetric black hole?

(c) At time t = 0 a light signal is sent from a fixed position at r = 4rs radially outwards to a distant fixed observer outside a Schwarzschild black hole where r_s is the Schwarzschild radius. A second light signal is sent from r = 2r_s radially outwards towards the same observer also at the time t = 0. If the difference in travel times for these two signals is measured to be 3.05 x 10^-2s by the observer, deduce the mass of the black hole in terms of the mass of the sun.