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1、TheenergymomentumtensorThisisalsoalittleexerciseofinsertingcatthecorrectplaces.Weputcequal1forconvenienceandre-insertitattheend.RecalltheEulerequationsforanidealfluidwithdensityρ(xi,t)andvelocityvi(xj,t):∂ρ∂(ρvi)+=0∂t∂xi∂pj∂(pjvi)∂pj+=(Forcedensity)=−∂t∂xi∂xjwherepj≡ρvjisthej-componentofthemo
2、mentumdensityandpisthepressure.Thefirstequation(thecontinuityequation)expressestheconservationofmass,thenextequationexpressesthatthechangeof(acomponentof)momentumpervolumeaccordingtoNewton’ssecondlawisequaltotheforcecomponentpervolume,i.e.minusthegradientofthepressure.Usingthecontinuityequati
3、oninthemomentumequation,thislattercanbewrittenas∂v1+(v·∇)v=−∇p,(1)∂tρwhilethecontinuityequationcanbewrittenasacurrentconservation:µµ∂µj=0,j=(ρ,ρv)(2)Foranidealfluidwehave:µνµνµνµνT=pη+(p+ρ)UU,∂νT=0,(3)whereUµ=γ(v)(1,vi)isthefour-velocity(UUµ=−1).µ(4)Writeouttheequationsexplicitlyforµ=0a
4、ndforµ=i,andshow(usingbothequations)thattheoneforµ=icanbewrittenas∂v1−v2∂p+(v·∇)v=−∇p+v.∂tp+ρ∂t(5)Showthatthisequationreducesto(1)inthenon-relativisticlimitandthattheequationforµ=0likewisereducesto(2)inthenon-relativisticlimit.2Problemset2constantacceleration,partIConsidertheequation
5、ofmotioninSRforapointparticle:dp1v=F,p=m0γv,γ=,β=.dt1−β2cConsiderthesituationwhereFpointsalongthex-axisandhastheconstantvalueF=m0g.Assumethatthevelocityiszeroattimet=0.(1)Showthatthemotionoftheparticleishyperbolic:2222c02cx+−(x)=,(4)ggwherex0=ctandy=z=0.(2)LetIdenotetheinitialsyste
6、mwheretheparticleisatrestatt=0.Showthatthepropertime(timesc)ofaclockfollowingtheacceleratedparticleisgivenby:c2gx0−1τ=sinh,(5)gc2i.e.20cgτx=sinh.(6)gc2(3)ShowthatthetransformationfromaninertialsystemI’,wheretheacceleratedparticleisatrestatpropertimeτatx=0(andy=z=0)withx0=0toIisgivenb
7、y:2cgτgτ0gτx=cosh−1+xcosh+xsinh(7)gc2c2c220cgτgτ0gτx=sinh+xsinh+xcosh(8)gc2c2c23DerivationofthegeodesicequationDefinethefollowingquantity:λ2dxS[x(λ)]=dλL(x(λ),x˙(λ)),x˙(λ)≡.(30)λ1dλSisafunctionalofthesetofpathsx(λ).AssumenowthatL
