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

7.2.1  Special Relativity Problems

Problem 4:  Moving mirror and generalized Snell's Law.

Consider a mirror at rest in reference frame S'.  As observed in S'. the mirror's rest frame, an incident beam of light makes an angle θm relative to the normal and, according to Snell's Law, reflects at the same angle.  the frequency of the light in S' is fm and is the same fro the incident and reflected beams.  The laboratory frame S is moving with a velocity β along the normal.  In the laboratory frame, the incident light has a frequency fi and makes an angle θi with the normal to the mirror.  The reflected beam in S has a frequency fr and makes an angle θr with the normal.

Part (a)  Show that the generalized Doppler formula is:
		 fi = fr (1-β^2) / (1 + β^2 + 2βCos[θi])

Ki = fi{1, Cos[θi],Sin[θi],0}

Kr = fr{1,-Cos[θr],Sin[θr],0}

Kmi = fm{1, Cos[θm],Sin[θm],0}

Kmr = fm{1,-Cos[θm],Sin[θm],0}

eq1 = (Ki == boostx[γ,β].Kmi)//Thread;

eq1//ColumnForm

eq2 = (Kr == boostx[γ,β].Kmr)//Thread;

eq2//ColumnForm

eq3 = {eq1,eq2,{fr!=0,fi!=0,γ!=0,β!=0}}//Flatten;

eq4 = Eliminate[eq3,{Cos[θr],Sin[θr],Cos[θm],Sin[θm]}]

sol1 = Solve[eq4,{fi,fm}]//Flatten

Part (b)  Show that the generalized Snell's Law is:
		 Cot[θi/2] Tan[θr/2] = (1-β) / (1 + β)

eq5 = Eliminate[eq3,{Cos[θm],Sin[θm]}]

eq6 = eq5/.sol1//Simplify

sol2 = Solve[eq6,{Sin[θi],Cos[θr]}]//Flatten

sol3 = (Solve[Cos[θr] == (Cos[θr]/.sol2),β][[1]]
		//Simplify
		//PowerExpand
		//Simplify)

temp=(1-β)/(1+β)

temp==(temp/.sol3)//Simplify