introduction to tensor calculus for general relativity - mit

《introduction to tensor calculus for general relativity - mit》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、MassachusettsInstituteofechnologyTDepartmentofPhysicsPhysics8.962Spring2000troInductiontoTensorCalculusforGeneralyRelativitc2000EdmundBertschinger.1troInductionTherearethreeessentialideasunderlyinggeneralrelativity(GR).Therstisthatspace-timemaybedescribedasacurved,four-dimensionalmathemat

2、icalstructurecalledapseudo-Riemannianmanifold.Inbrief,timeandspacetogethercompriseacurvedfour-dimensionalnon-Euclidean.geometryConsequently,thepractitionerofGRmustbefamiliarwiththefundamentalgeometricalpropertiesofcurvedspacetime.Inparticu-lar,thelawsofphysicsmustbeexpressedinaformthatisal

3、idvindependentlyofanycoordinatesystemusedtolabelpointsinspacetime.ThesecondessentialideaunderlyingGRisthatateveryspacetimepointthereexistlocallyinertialreferenceframes,correspondingtolocally atcoordinatescarriedbyfreelyfallingobservers,inwhichthephysicsofGRislocallyindistinguishablefromtha

4、tofspecial.relativityThisisEinstein'sfamousstrongalenceequivprincipleanditmakesgeneralrelativityanextensionofspecialrelativitytoacurvedspacetime.Thethirdkeyideaisthatmass(aswellasmassandmomentum ux)curvesspacetimeinamannerdescribedbythetensoreldequationsofEinstein.Thesethreeideasareexempl

5、iedbycontrastingGRwithNewtonian.gravityIntheNewtonianview,gravityisaforceacceleratingparticlesthroughEuclideanspace,whiletimeisabsolute.romFtheviewpointofGRasatheoryofcurvedspacetime,thereisnogravitationalforce.Rather,intheabsenceofelectromagneticandotherforces,particlesfollowthestraighte

6、stpossiblepaths(geodesics)throughaspacetimecurvedbymass.reelyFfallingparticlesdenelocallyinertialreferenceframes.Timeandspacearenotabsolutebutarecombinedintothefour-dimensionalmanifoldcalledspacetime.Inspecialrelativitythereexistglobalinertialframes.Thisisnolongertrueinthepresenceofgravit

7、.yHowever,thereareloalcinertialframesinGR,suchthatwithina1suitablysmallspacetimevolumearoundanevent(justhowsmallisdiscussede.g.inMTWChapter1),onemaychoosecoordinatescorrespondingtoanearly- atspacetime.Thus,theloalcpropertiesofspecialrelativitycarryovertoGR.The