Spacetime and Geometry · Problem 2

More General Black Holes — Sean Carroll

Physics — General Relativity · tov-equation-lower-dimensions

Problem

Consider a perfect fluid in a static, circularly symmetric (2+1)(2+1)-dimensional spacetime, equivalently, a cylindrical configuration in (3+1)(3+1) dimensions with perfect rotational symmetry. (a) Derive the analogue of the Tolman--Oppenheimer--Volkov (TOV) equation for (2+1)(2+1) dimensions. (b) Show that the vacuum solution can be written as ds2=−dt2+11−8GMdr2+r2dθ2ds^2 = -dt^2 + \frac{1}{1-8GM}dr^2 + r^2 d\theta^2 Here MM is a constant. (c) Show that another way to write the same solution is ds2=−dτ2+dξ2+ξ2dϕ2ds^2 = -d\tau^2 + d\xi^2 + \xi^2 d\phi^2 where ϕ∈[0,2π(1−8GM)1/2]\phi \in [0, 2\pi(1-8GM)^{1/2}]. (d) Solve the (2+1)(2+1) TOV equation for a constant density star. Find p(r)p(r) and solve for the metric. (e) Solve the (2+1)(2+1) TOV equation for a star with equation of state p=κρ3/2p = \kappa\rho^{3/2}. Find p(r)p(r) and solve for the metric. (f) Find the mass M(R)=∫02π∫0Rρ dr dθM(R) = \int_0^{2\pi}\int_0^R \rho\, dr\, d\theta and the proper mass M~(R)=∫02π∫0Rρ−g dr dθ\tilde{M}(R) = \int_0^{2\pi}\int_0^R \rho\sqrt{-g}\, dr\, d\theta for the solutions in parts (d) and (e).

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