The metric (i) - (6-2m) 1 b(r2 dr-2 (d2 sin2 do dt2-2b(r)dt dr- ds2 given in coordinates (t, r, 0. ) is a solution of einstein equation gab = spherically symmetric spacetime. call it the b metric show that the b metric is equivalent to the standard schwarzschild metric for any function b(r). recall that the standard schwarzschild (d02sin2 do2), 0 in vacuum for a 2m) dt2-dr metric is written as ds2(1 1-- 27 where (t, r, 0, ) are the standard schwarzschild coordinates. hint: try a transformation on the standard schwarzschild metric of the form t(t, r)tp(r). (ii) (a) impose that grr =- in the b metric and obtain the metric in this case. the metric in this form is called the painlevé-gullstrand metric. show that the painlevé-gullstrand metric is non-singular for all positive radius r, including r = 2m, i. e., show that the metric coefficients are bounded and the determinant is non-zero for all positive radius r. (1.3 points) 2m? (b) compute the function p(r) in this case. is it singular at r = why? (1.2 points) note: d 2/ in -1 (iii (a) returning to the b metric what is the appropriate choice of b to obtain the metric in eddington-finkelstein coordinates? (1.0 points) check that indeed in this case t is the (b) find p(r) in this case advanced null coordinate v of eddington-finkelstein. (1.0 points (c) draw a finkelstein coordinates, denoting and justifying with care the null curves and other important features that appear on the diagram. diagram of the whole spacetime in advanced