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 6:  Circular geodesics for the Schwartzschild metric.

Consider the timelike circular geodesics for the Schwartzschild metric with four-velocity given by v = {t'[s],r'[s],θ'[s],ϕ'[s]}.  Assume that the geodesics are in the plane θ[s] = π/2 and that the radius is a constant with r[s] = r0.

Part (a)  Solve for t[s] and ϕ[s] with the initial conditions that t[0[=0 and ϕ[0]=0.

circleRule = {r->(((r0))&),θ->(((π/2))&)}

eq1 = (geodesic[ssMet/.ssComp,{t,r,θ,ϕ}]/.circleRule)//Simplify

tRule = DSolve[{eq1[[1]],t[0]==0},{t},s]/.C[2]->c1

ϕRule = DSolve[{eq1[[4]],ϕ[0]==0},{ϕ},s]/.C[2]->c2

eq2 = eq1[[2]]/.tRule/.ϕRule//Flatten

vel = {t'[s],r'[s],θ'[s],ϕ'[s]}

met = (ssMet/.ssComp/.{θ->θ[s],r->r[s]})

eq3 = 1 == vel.met.vel/.circleRule/.tRule/.ϕRule//Flatten

constRule = ((Solve[{eq3,eq2}/.m->temp r0//Flatten,
					{c1,c2}][[4]]//ExpandAll)/.
					temp->m/r0//PowerExpand)/.
					1/Sqrt[a_] -> -I/Sqrt[-a]

eq4 = CoordT == (t[s]/.tRule/.constRule)[[1]]

eq5 = Φ == (ϕ[s]/.ϕRule/.constRule)[[1]]

Part (b)  Find the coordinate and proper periods for the circular orbit.

coordPeriod = CoordT/.Solve[{eq4,eq5},CoordT,s][[1]]/.Φ->2π

properPeriod = MapAt[Apart,s/.Solve[eq5,s][[1]]/.Φ->2π,{5,1}]