Problem 5. (15 pts) (8) Consider the Schwarzschild metric denoted by ds² == (1 — ²^²) dt² + (1 - ²^ )¯¹ dr² +r² (d0² + s

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Problem 5 15 Pts 8 Consider The Schwarzschild Metric Denoted By Ds 1 Dt 1 Dr R D0 S 1 (199.93 KiB) Viewed 30 times

Problem 5. (15 pts) (8) Consider the Schwarzschild metric denoted by ds² == (1 — ²^²) dt² + (1 - ²^ )¯¹ dr² +r² (d0² + sin² 0dø²), - (t, r, 0, 0). where s is the Schwarzschild radius, and we have chosen units c = = 1, with the coordinates x (a) (5pts) Consider a timelike geodesic xª(7), where 7 is the proper time, lying on the plane 0 = d0/dT that = π π/2 with = 0. Use the Lagrangian L = 9µv*¹*' to derive the equations governing the geodesic, first showing r 1 - 7² ) t = E, r²6 = L, where E, L are constants. (b) (5pts) Using the results in (a) and eq.(8), show that 2 dr (²) ² - 5 - (1 + ²) (1 - ² ) = . dT (c) (5pts) Consider an observer who falls radially inward to a Schwarzschild black hole with Schwarzschild radius rs, starting with zero kinetic energy at r = 5rg. Please calculate the observer's total proper time elapse from r = 5r, to r = 0. END OF QUESTION 5