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 3:  Killing vectors and constants of motion

Consider the following problems for the Schwartzschild and Kerr metrics on killing vectors.

Part (a)  Consider the following four contravariant vectors:
		kt = {1,0,0,0}
		kz = {0,0,0,-1}
		kx = {0,0,-Sin[ϕ],-Cos[ϕ]Cot[θ]}
		ky = {0,0,Cos[ϕ],-Sin[ϕ]Cot[θ]}
Verify that they satisfy Killing's equations for the Schwartzschild metric.

kt = {1,0,0,0}

kz = {0,0,0,-1}

kx = {0,0,-Sin[ϕ],-Cos[ϕ]Cot[θ]}

ky = {0,0,Cos[ϕ],-Sin[ϕ]Cot[θ]}

ke = killingEq[#,ssMet,ssChrist]&/@{kt,kz,ky,kx}

ke/.ssComp//Simplify

Part (b)  let V = {t'[s],r'[s],θ'[s],ϕ'[s]} be a geodesic four-vector velocity.  Constants of motion follow from equating the scalar dot product V.η.K to a constant, where K is a Killing vector and η is the metric.  Express the components of the four-vector velocity V in terms of the killing constants.

velocity = {t'[s],r'[s],θ'[s],ϕ'[s]}

kconst = {	en == ssMet.velocity.kt,
			jx == ssMet.velocity.kx,
			jy == ssMet.velocity.ky,
			jz == ssMet.velocity.kz};

kconst//ColumnForm

tRule = (Solve[kconst[[1]],t'[s]]//Flatten)/.ssComp

ϕRule = (Solve[kconst[[4]],ϕ'[s]]//Flatten)

jRule = (Solve[kconst[[{2,3}]],{ϕ'[s],θ'[s]}]//Flatten)//Simplify

Part (c)  Verify that the two contravariant vectors kt = {1,0,0,0} and kz = {0,0,0,-1} satisfy Killing's equations for the Kerr metric.

kt = {1,0,0,0}

kz = {0,0,0,-1}

killingEq[#,kMet,kChrist]&/@{kt,kz}

Part (d)  Express the two components of the four-velocity {t'[s],r'[s]} for the Kerr metric in terms of the Killing constants.  Write the explicit expressions when θ=π/2.

kconst = {	en == kMet.velocity.kt,
			jz == kMet.velocity.kz}

eq1 = Solve[kconst,{t'[s],ϕ'[s]}]//Flatten//Simplify

(eq1/.kComp/.{θ->π/2})//Simplify