General Relativity (Advanced Physics)

General Relativity 2022 Hand in problem 2 The Tolman IV perfect fluid solution to Einstein's equations: ds' = (7 + A' ) 2r2 + 42 R2 (R' -r? )(12 + A ) R'd - r de' + sin' edo ] , where A and R are constants, is static and spherically symmetric. a) Determine the energy density, pressure and 4-velocity of the fluid. 1.5p b) Assume that the solution describes the interior of a spherically symmetric star. The exterior should then be described by the Schwarzschild metric. Determine the radius r, of the star and the mass parameter m of the Schwarzschild metric in terms of the constants A and R. 1.5p Hint: Which physical condition defines the surface of a star surrounded by vacuum? What should hold for the two metrics on the surface r= r; ? c) Plot the density and pressure as functions of r for some choice of A and R, to see if the curves look reasonable. 0.5p d) Show that the radius of the star must satisfy r, > 3m for pressure and density to be regular and positive. 0.5p Note 1: In general one has to take into account for that the coordinates could be defined differently in the two regions, but in this case you may assume that they are already adopted to each other, i.e., t, r, and are continuous across the surface r=