7. Relativity and Cosmology

Legal Notice, Disclaimer, and Book Ordering Information:

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 2:  The horizons and surfaces of infinite redshift.

The Kerr metric kMet is defined in the introduction, and its components follow from the user-defined rule kComp.  The surfaces of infinite redshift are defined by the equation g00 = 0, and the event horizons are definrd by the inverse metric component equations g^11 = 0.

Part (a) Assume that m^2 > a^2 and find the surfaces of infinite redshift and the horizons.  Plot the surfaces.

eq1 = (g00[r,θ]/.kComp)==0//FullSimplify

redShift = Solve[eq1,r]//Simplify

eq2 = Inverse[kMet][[2,2]]==0//Simplify

horizon = Solve[eq2/.kComp,r]

Part (b) Plot the surfaces of infinite redshift and the horizons.

list = {{r/.redShift},{r/.horizon}}//Flatten

pt = SphericalPlot3D[Evaluate[#/.m->1/.a->0.995]
		,{θ,0,π},{ϕ,-π/2,π/2}
		,AspectRatio->Automatic
		,Boxed->False
		,Axes->False
		,DisplayFunction->Identity]&/@list;

Show[pt, ViewPoint->{-1,-0.6,0.4},DisplayFunction->$DisplayFunction];