Question

please write clearly and explain

172 14. EVENT HORIZON BOX 14.1 Finite Distance to r = 2GM ysical distance ds cor According to the Schwarzschild metric, the differential physical distan responding to a purely radial differential displacement dr is ds dr 1 - 20 Mir As - J71- 2GMI in most integral tables (and This is not an integral that one can look up directly in most integral table Wolfram Alpha gave me something really ugly). But let's define NEVI 2GM , r 2GM Exercise 14.1.1. Verify that the last equality is correct. correct. With this substitution, the total distance from radial coordinate r = 2GM to rais As = 4GM/_du (14.9) Exercise 14.1.2. Check that this is correct.

Answer 1

Given in AS dr 2Gm us Vincome out = 1-2m 1- 2GM 1-20m - r = 20m 1-42 to prove . To prove ! Asa 4 GM s du 2 AS = 4 GM du (-u22 Proof! - AS a = in dom Differentiating. Gm. 2u du 22 a dr AS = Putting in as term asos as= ( 2ure du 1-2 GM u As = 2 ur du u. 2GM (2 G m du AS 2 GM 1-4272 AS As = UGM s dry UGM