7. Relativity and Cosmology

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Setup

\(Clear["Global`*"]; 
Off[General::spell]; 
Off[General::spell1]; 
    \)

Needs["Calculus`VectorAnalysis`"]

Needs["Graphics`PlotField3D`"]

<< "D:\Mma\Inits\ch7init.m"

Clear["Global`*"];

7.2 Problems

7.2.2  General Relativity and Cosmology Problems

Problem 1:  Schwartzschild solution in null coordinates

Consider a spherically symmetric metric expressed in null coordinates {u,r,θ,ϕ}.  The metric is non-diagonal and is of the form:
	{{g00[r]	-1	 0	 0		}
	  {-1		 0	 0	 0		}
	  { 0		 0	-r^2	 0		}
	  { 0		 0	 0	-r^2Sin[θ]^2	}}  .

Part (a)  Find the solution of Einstein's vacuum equations with the boundary conditions that g00[r] -> (1 - 2m/r) as r goes to infinity.

varNull = {u,r,θ,ϕ}

nullMet = {	{g00[r],-1,0,0},
	  		{-1,0,0,0},
	  		{0,0,-r^2,0},
	  		{0,0,0,-(r Sin[θ])^2} }

nullChrist = christ[nullMet,varNull]

nullCur = curvR[nullChrist,varNull]//Simplify;

Short[nullCur,7]

nullRicci = ricciR[nullCur]//Simplify

vacuumEq =
	Select[(nullRicci//Flatten)==0//Thread,FreeQ[#,
		True]&]//Together//Numerator

nullRule = DSolve[vacuumEq[[4]],g00,r]//Flatten

vacuumEq/.nullRule

nullMet/.nullRule/.{C[1]->-2 m}