6.31 This final exercise consists of a long calculation which will be needed later in the book. If we take coordinates xa=(x0,x1,x2,x3)=(t,r,θ,ϕ), then the four-dimensional spherically symmetric line element is ds2=ev dt2−eλdr2−r2 dθ2−r2sin2θdϕ2, where v=v(t,r) and λ=λ(t,r) are arbitrary functions of t and r. (i) Find gab,g, and gab (see Exercise 6.13). (ii) Use the expressions in (i) to calculate Γbca. [Hint: remember Γbca=Γcba⋅] (iii) Calculate Rabcd. [Hint: use the symmetry relations (6.81).] (iv) Calculate Rab,R, and Gab. (v) Calculate Gab(=gacGcb=Gba).
