Question A5 Consider the line element ds² = du dv+dx² + dy². State whether it corresponds to a Riemannian manifold or to a Lorentzian manifold, and justify your answer. In addition, state whether the manifold is curved or not, and justify your answer. Question A6 [6 marks] (a) Write down the equation of motion obeyed by the trajectory in spacetime of a free massive particle, parametrised by the proper time T. [3 marks] (b) Suppose that, instead of the proper time, we want to parametrise the trajectory by σ, with σ a function of T, and determine the equation of motion in terms of the new parameter σ. [3 marks] (c) Finally, restrict to the case where σ(T) = a + b, where a and b are constants, and write down the equation of motion in this case. Question A7 The Schwarzschild black hole is described by ds²=- · (1 2GM dr² 1- dt² + 2GM +2(do² + sin² Odo²). 1 [2 marks] Without calculation, state the value of the Ricci tensor R at the event horizon, and justify your answer. [5 marks] Question A8 Suppose that you are in a spaceship static outside of a non-rotating black hole described by the Schwarzschild metric. Your position is at r = 8GM. Your enemy in a spaceship static but far away from the black hole shoots a laser gun at you. The laser gun emission frequency is ve. Your specially reinforced spaceship was designed to withstand this type of laser attack up to laser frequency ve. Determine whether your spaceship can keep you safe from the attack. [6 marks] Question B3 Consider the following FLRW spacetime: t2 ds² = -dt² + (dx² + dy²+ dz²), t2 where t is a constant. a) State whether this universe is spatially open, closed or flat. [2 marks] b) Determine the Hubble factor H(t), and represent it in a (roughly drawn) plot as a function of time t, starting at t = 0. [3 marks] c) Taking galaxy A to be located at (x, y, z) = (0,0,0), determine the proper distance to galaxy B located at (x, y, z) = (L, 0, 0). Determine the recessional velocity of galaxy B with respect to galaxy A. d) The Friedmann equations are 2 k 8πG а 4πG + a² (p+3p). 3 a 3 [5 marks] Use these equations to determine the energy density p(t) and the pressure p(t) for the FLRW spacetime specified at the top of the page. [5 marks] e) Given the result of question B3.d, state whether the FLRW universe in question is (i) radiation-dominated, (ii) matter-dominated, (iii) cosmological-constant-dominated, or (iv) none of the previous. Justify your answer. f) [5 marks] A conformally flat spacetime is one that admits coordinates such that ds² = N(x) dxdx", where (x) is any function of the coordinates x", and 7 is the usual Minkowski metric. Determine whether the FLRW spacetime in question is conformally flat. Hint: Consider a transformation of the coordinate t. [5 marks]