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 4:  Potential analysis for timelike geodesics.

Consider the Schwartzschild metric and assume that the timelike geodesics are confined to the plane θ=π/2.  Use the first two integrals that follow from Killing vectors: E = t'[s](1 - 2m/r) and ℓ = r^2 ϕ'[s].

Part (a)  Consider the timelike condition 1 = v . η . v , where v is the geodesic four-velocity and η is the metric.  If the effective potential V is defined by the equation r'[s] == -(1 - E^2) -V, show that V = (-2ℓ^2 m)/r^3 + ℓ^2/r^2 - (2m)/r .  The first term is the relativistic correction, the second term is the centrifugal barrier, and the third term is the gravitational potential.

planeRule = {θ -> ((π/2)&)}

SSconst = {t'[s] -> en/(1-2 m/r[s])
		  ,t''[s]-> D[en/(1-2 m/r[s]),s]
		  ,ϕ'[s] -> ℓ/r[s]^2
		  ,ϕ''[s]-> D[ℓ/r[s]^2,s]}

vel = {t'[s],r'[s],θ'[s],ϕ'[s]}

eq1 = 1 == ((ssMet/.ssComp/.r->r[s]/.θ->θ[s]).vel.vel)/.planeRule/.SSconst

eq2 = r'[s]^2 == -(1-en^2)-V

eq3 = (Solve[{eq1,eq2},V,r'[s]][[1,1]])/.{r[s]->r}//ExpandAll

Part (b)  Show that the potential has two extrema if ℓ^2 > 12 m^2 .

{rmax,rmin} = r/.Solve[D[V/.eq3,r]==0,r]//Simplify

{Vmax,Vmin} = {V/.eq3/.r->rmin,V/.eq3/.r->rmax}//Together//Simplify

Part (c)  Graph the three parts of the effective potential for ℓ^2 > 12 m^2 .  Also, graph the effective potential for the three cases: ℓ^2 > 12 m^2, ℓ^2 = 12 m^2 .ℓ^2 < 12 m^2 .

list = {eq3[[2]],List @@ eq3[[2]]}//Flatten

potPlot[L_] :=
	Plot[list//.{m->1,ℓ->L}//Evaluate
		,{r,2.1,50}
		,Prolog->Graphics[{PointSize[0.02]
			,Map[Point,{{rmax,Vmin},{rmin,Vmax}}
			]/.m->1/.ℓ->(1.3 Sqrt[12])}][[1]]
		,Axes->False
		,Frame->True
		,PlotLabel-> eq3[[2]]
		,FrameLabel->{"Radius","Potential","Radius","Potential"}
		,FrameTicks->Automatic
		,PlotStyle->{Thickness[0.006]
					,Dashing[{0.01}]
					,Dashing[{0.02}]
					,Dashing[{0.03}]	}
		,DisplayFunction->Identity];

label =
	Graphics[
		{Text[FontForm[" Unstable ",{"Times-Roman",10}],{4,.16}]
		,Text[FontForm[" Stable   ",{"Times-Roman",10}],{17,-.08}]			,Text[FontForm[  eq3[[2,2]],{"Times-Roman",10}],{36,-.1}]			,Text[FontForm[  eq3[[2,3]],{"Times-Roman",10}],{20,.11}]			,Text[FontForm[  eq3[[2,1]],{"Times-Roman",10}],{14,.03}]
		}];

Show[potPlot[1.3 Sqrt[12]],label
			,PlotRange->{0.2,-0.2}
			,DisplayFunction->$DisplayFunction];

Plot[Evaluate[(eq3[[2]]/.m->1/.
		ℓ->{.9 Sqrt[12],Sqrt[12],1.1 Sqrt[12]})]
		,{r,2.1,15}
		,Axes->False
		,Frame->True
		,PlotLabel->"Potential with Different ℓ"
		,FrameLabel->{"Radius","Potential","Radius","Potential"}
		,FrameTicks->Automatic
		,PlotStyle->{Thickness[0.006]
					,Dashing[{0.01}]
					,Dashing[{0.02}]
					,Dashing[{0.03}]	}
		,PlotRange->{0,-0.2}
		,Epilog->Graphics[
			{Text[FontForm[ℓ^2 > 12m^2//StandardForm,{"Times-Roman",10}]
						,{4.5,-0.03}]
			,Text[FontForm[ℓ^2 == 12m^2//StandardForm,{"Times-Roman",10}]
						,{7,-0.095}]
			,Text[FontForm[ℓ^2 < 12m^2//StandardForm,{"Times-Roman",10}]
						,{11,-0.14}]}][[1]]];