Page 2 SECTION A Answer ALL questions in Section A [Expect to use one single-sided A4 page for each Section-A sub question.] Question A1 SPA6308 (2024) Consider Minkowski spacetime in Cartesian coordinates th = (t, x, y, z), such that ds² = dt² + dx² + dy² + dz². (a) Consider the vector with components V" = (1,-1,0,0). Determine V and V. V. (b) Consider now the coordinate system x' (u, v, y, z) such that u =t-x, v=t+x. [2 marks] Write down the line element, the metric, the Christoffel symbols and the Riemann curvature tensor in the new coordinates. [See the Appendix of this document.] [5 marks] (c) Determine V", that is, write the object in question A1.a in the coordinate system x'. Verify explicitly that V. V is invariant under the coordinate transformation. Question A2 [5 marks] Suppose that A, is a covector field, and consider the object Fv=AAμ. (a) Show explicitly that F is a tensor, that is, show that it transforms appropriately under a coordinate transformation. [5 marks] (b) Show that the definition F = VμA₁ - V₁₂Aμ, which uses the covariant derivative, is equiv- alent to the definition above. [5 marks] Question A3 In special relativity, we introduced the momentum of a particle in a given Lorentz frame as pH = (E,p'), where E is the energy of the particle and p² is the relativistic 3-momentum. Explain why, in general relativity, the energy of a particle measured by an observer is given, irrespective of the coordinate system used, by -p "obs, where pt is the momentum of the particle and ubs is the velocity of the observer in those coordinates. [5 marks] SPA6308 (2024) Question A4 Any two-dimensional metric satisfies R Πλρμν (9μ9pv - vpμ). 2 Page 3 Show that the vacuum Einstein equations (with zero cosmological constant) are satisfied for any 2D metric. [6 marks] Question A5 State the value of the Ricci tensor R₁ on the event horizon of a Schwarzschild black hole, and justify your answer. Question A6 Consider the energy-momentum tensor T= (p+p) + P9μv [5 marks] applied to the matter/energy distribution in the universe on large scales, and assume an equation of state of the form p = wp, with wa constant. Determine the type of matter/energy dominating the universe if the energy-momentum tensor is traceless, that is, T" = 0. Question A7 Consider the following line element, ds² = = −dt² + a²(t) (dx² + dy²) + b²(t) dz², [6 marks] where a(t) and b(t) are distinct functions. State whether or not this line element obeys the Cosmological Principle, if applied to describe the universe on large scales. Justify your answer. [6 marks]