General Relativity: An Introduction for Physicists · Problem 9.28

The Schwarzschild geometry — Hobson, Efstathiou & Lasenby

Physics · static-axisymmetric-spacetime-line-element

Problem

Consider a static axisymmetric spacetime that is invariant under translations and reflections along the axis of symmetry. Show that, in general, the line element for such a spacetime can be written in the form ds2=A(ρ) dt2−dρ2−B(ρ) dϕ2−C(ρ) dz2,ds^2 = A(\rho)\,dt^2 - d\rho^2 - B(\rho)\,d\phi^2 - C(\rho)\,dz^2, for arbitrary functions AA, BB and CC. Show that the non-zero connection coefficients for this line element are given by Γ001=A′2A,Γ122=−B′2,Γ133=−C′2,\Gamma^0{}_{01} = \frac{A'}{2A}, \quad \Gamma^1{}_{22} = -\frac{B'}{2}, \quad \Gamma^1{}_{33} = -\frac{C'}{2}, Γ100=A′2,Γ221=B′2B,Γ331=C′2C,\Gamma^1{}_{00} = \frac{A'}{2}, \quad \Gamma^2{}_{21} = \frac{B'}{2B}, \quad \Gamma^3{}_{31} = \frac{C'}{2C}, where the primes denote d/dρd/d\rho. Hence show that the non-zero components of the Ricci tensor are given by R00=−A′′2+A′2(A′2A−B′2B−C′2C),R_{00} = -\frac{A''}{2} + \frac{A'}{2}\left(\frac{A'}{2A} - \frac{B'}{2B} - \frac{C'}{2C}\right), R11=A′′2A−(A′)24A2+B′′2B−(B′)24B2+C′′2C−(C′)24C2,R_{11} = \frac{A''}{2A} - \frac{(A')^2}{4A^2} + \frac{B''}{2B} - \frac{(B')^2}{4B^2} + \frac{C''}{2C} - \frac{(C')^2}{4C^2}, R22=B′′2−B′2(B′2B−A′2A−C′2C),R_{22} = \frac{B''}{2} - \frac{B'}{2}\left(\frac{B'}{2B} - \frac{A'}{2A} - \frac{C'}{2C}\right), R33=C′′2−C′2(C′2C−A′2A−B′2B).R_{33} = \frac{C''}{2} - \frac{C'}{2}\left(\frac{C'}{2C} - \frac{A'}{2A} - \frac{B'}{2B}\right).

Browse the problem library