1 PHYS3002 - General Relativity Assignment 3 Question 1 Let V be an n-dimensional vector space and g a metric on V . (i) Show that there always exists an orthonormal basis {v1 , ..., vn } of V , i.e., a basis of V such that g(vi , vj ) = ij (use induction). (10 marks) (ii) (Bonus Question) Show that the signature of g is independent of the choice of orthonormal basis. (3 marks) Question 2 Let g be a Lorentzian metric on M (n-dimensional manifold). If w1 , w2 Vp such that g(w1 , w1 ) 0, g(w2 , w2 ) 0 and w1 and w2 lie in the same null cone at p, show that: (i) g(w1 , w2 ) 0. (5 marks) (ii) g(w1 , w2 ) = 0 only if w1 and w2 are parallel null vectors, i.e., if w1 = w2 , R+ , g(w1 , w1 ) = 0. (5 marks) \f\f