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Einstein‘s Theory of R elativity

Einstein‘s Theory of R elativity. 9. Einstein‘s Field E quations , Spherical Symmetric Fields,Schwarzschild Metric. Ulrich R.M.E. Geppert. b asic idea of Einstein‘s theory of gravitation :. c alculate for a given distribution of mass and energy ( mass energy )

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Einstein‘s Theory of R elativity

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  1. Einstein‘sTheoryofRelativity 9. Einstein‘s Field Equations, Spherical SymmetricFields,SchwarzschildMetric Ulrich R.M.E. Geppert U.R.M.E. Zielona Gora

  2. basicideaofEinstein‘stheoryofgravitation: calculatefor a givendistributionofmassandenergy (massenergy) thespace time metricandviceversa GR: gravitationishidden in themetricequ.ofmotionfor a pointlikeparticle in a gravitationalfieldarethegeodesicequations: parameter, e.g. pathlength, ifpathisparametrizedby proper time ⇒geodesicequation U.R.M.E. Zielona Gora

  3. searchingtherelativisticgeneralizationofNewton‘sequ. ofmotion potential energy starting withvariationalprinciple andLagrangian (kinetic – potentialrestenergy) reminder: gravitational potential Lagrangian explicitely: U.R.M.E. Zielona Gora

  4. U.R.M.E. Zielona Gora

  5. generalrelativisticvariationalprinciple: comparison ofbothintegrants: U.R.M.E. Zielona Gora

  6. wherethegravitationis in thesmallperturbation U.R.M.E. Zielona Gora

  7. inverse metric: because: U.R.M.E. Zielona Gora

  8. withmetric : Christoffel symbol the only non-constantmatrixelementsareand the metricisstatic, i.e. 0 0 since U.R.M.E. Zielona Gora

  9. Christoffel symbolcurvaturetensor (only) curvaturetensor curvaturetensor Ricci tensorby (contraction) U.R.M.E. Zielona Gora

  10. staticsphericalsymmetricmassdistribution (stars) sphericallydistributed total masshas a spherical symmetric potential Schwarzschild radius Karl Scwarzschild 1873-1916, foundfirstsolutionof Einstein equation, died due to a skindeseasecontagioned in thedugoutattheRussian front 1915 (Earth)9 mm (Sun)km U.R.M.E. Zielona Gora

  11. curvatureofspace (metric) due toweakfields theonlyremaining Christoffel symbol: the relatedcomponentsofthecurvaturetensor: for, staticgravitationalfield theonlyremainingcomponentofthe Ricci tensor: (fordetailsseeannex) U.R.M.E. Zielona Gora

  12. „derivation“ ofthefieldequations lookforgeneralizationof Newton: (Poisson) distributionofmassas sourceofthefield firstandsecondspatial derivatives ofthefield, second derivative onlylinear since problem: isonlyonecomponentof a tensor, i.e. theequation has NOT thecorrecttransformationproperties U.R.M.E. Zielona Gora

  13. wesearchfor a Riemann tensorequationofthe type „matter“ „field“ remember GR4: energy-momentumtensor since U.R.M.E. Zielona Gora

  14. thelatter in moredetail: since bothpressureandhydrodynamicalvelocitiesarenegligible andonlythe : U.R.M.E. Zielona Gora

  15. GR4: energyandmomentumconservation in a closedsystem in flat space in curvedspace ⇒ is not thecorrectfieldequationsince in general U.R.M.E. Zielona Gora

  16. requirements on the GR generalizationofNewton‘sPoissonequation: 1. fieldequationshouldbetensorequation (reflectstheindependenceofphysicallaws on CS) 2. l.h.s. shouldbe a symmetrictensorof rank 2 as 3. as all fieldequations in physicsitshouldbe a partial differential equationofmaxsecondorder, linear in highestderivation 4. l.h.sshouldbe (as ) solenoidal (divergencefree) 5. in Minkowskispace time: l.h.s. 6. smooth transitiontoNewton‘sformulaif only U.R.M.E. Zielona Gora

  17. wesearchverygenerallyfor an equationof type to bedetermined in accordancewithrequests: 1, 2, 4: existsonlyonetensorthatfulfillstheserequests: because also (GR5, slide 40 and GR7, slide17) and , : Ricci tensor, : curvaturescalar wesearchfor an equation andhavetodeterminefromtherequeststheconstants U.R.M.E. Zielona Gora

  18. request 4: sincealways (metricisalwayssolenoidal): can‘tbedeterminedfromthisequation ⇒determination of coefficient U.R.M.E. Zielona Gora

  19. Bianchi idendity: multiply Bianchi idenditywith symmetryproperties: exchangingtheindices in the 1stor 2nd pair ofthecurvaturetensorchangesthesignofthetensor: U.R.M.E. Zielona Gora

  20. contractionofthetensorsofthe Bianchi – idendity: ⇒ multiplywith : ⇒ rename U.R.M.E. Zielona Gora

  21. multiplywith and insert a Delta-function: raise index foristhel.h.softhefieldequationssolenoidal U.R.M.E. Zielona Gora

  22. determinationofthecoefficientsand: or useresultsofthe non-relativisticlimit (forand) andfor matter atrest (for): insertinto: U.R.M.E. Zielona Gora

  23. request 5: since in theMinkowskilimit (thel.h.s. → 0 ⇒ with (slide 12) verysmall!! U.R.M.E. Zielona Gora

  24. Einstein‘sfieldequations (1915): or, bycontraction replaceby U.R.M.E. Zielona Gora

  25. Einstein‘sfieldequations andtheequationsofmotion arethebasicequationsofgeneralrelativity. U.R.M.E. Zielona Gora

  26. remark on thecosmologicalconstant Riemann scalar thisequationfulfillsconditions 1,2,3,4, but not 5 and 6, l.h.s.=0 in the Minkowskilimitandthe smooth transitiontoNewton‘slimit second derivatives wrtcoordinates⇒hasthedimension HOWEVER Newton‘stheoryworksverywellwithinour solar system: must beverysmall length must bemuch larger thenthediameterofour solar system, i.e. ly U.R.M.E. Zielona Gora

  27. cosmologicalconstantimportantforthevery large-scaleevolutionoftheuniverse „Cosmology“ state oftheartmodelsprefer a non-vanishingcosmlogicalconstantly rememberslide 23: forcontractionther.h.sbecomes thetermr.h.s. andconsideredas a contributiontothemassdensity g cm-3 forcm-2 , 1ly cm one proton in onecubicmeter U.R.M.E. Zielona Gora

  28. External Schwarzschild metric:(e.g. stellar radius) considerstaticsphericalsymmetricmassdistribution: star surroundedbyvacuum: rememberslide 23: forand i.e. Einstein‘sfieldequationssimplifyto U.R.M.E. Zielona Gora

  29. generalansatzfor a sphericalsymmetricmetric: becauseof time reversalinvariance metric static⇒ all coefficientsindependent on bysuitablecoordinatetransformation, thenrename as usualforsphericalcoordinateswith U.R.M.E. Zielona Gora

  30. metrictensor inverse metrictensor U.R.M.E. Zielona Gora

  31. withthe Christoffel symbolscanbecalculated: , (a) , (b) , (c) , (d) , (e) U.R.M.E. Zielona Gora

  32. , (f) , (g) , (h) Christoffel symbols⇒ curvaturetensor ⇒Ricci tensor forfieldequationsonly Ricci tensorisnecessary U.R.M.E. Zielona Gora

  33. remember: in all detail andsimilarily: (littleexercise?) U.R.M.E. Zielona Gora

  34. if if if if if if if U.R.M.E. Zielona Gora

  35. U.R.M.E. Zielona Gora

  36. Einstein equations outside themassdistribution: isfulfilledforandsince⇒ becauseof (D) fromremains ⇒ ⇒ for , i.e. farawayfromthemassdistribution, themetricbecomesMinkowski i.e. hence U.R.M.E. Zielona Gora

  37. ⇒ insertthisinto (B) and (C): integration: ourchoice: ⇒ outside themassdistribution (forstars Schwarzschild metric: U.R.M.E. Zielona Gora

  38. determinationoftheconstantofintegrationbyuseofNewtonianlimit:determinationoftheconstantofintegrationbyuseofNewtonianlimit: „Schwarzschildradius“ U.R.M.E. Zielona Gora

  39. not a real but a coordinatesingularity singularityat in sphericalcoordinatesissingularatthenorthpole – however, spacetherehasthe same propertiesasanywhereelse, e.g. velocityof a freefallingparticlehasnothingspecialat but radiusisphysicallyspecial: a clockatrestatshowsthe time remember GR1, GR2: is time shownby a clockatrestatinfinity (coordinate time) divergesif i.e. a photonemittedathas an infinite redshift ⇒ is not a suitable time coordinatefor starwitradius: black hole, from ist surfacecannophotonreachregion U.R.M.E. Zielona Gora

  40. Annex U.R.M.E. Zielona Gora

  41. calculationofthe Christoffel symbol general: the inverse of sincegravitational potential isstatic, onlythespacial derivatives are different fromzero, i.e., (not ), but onlycomponentsaredependent on U.R.M.E. Zielona Gora

  42. calculationofthecurvaturetensor general: sinceonly is , and U.R.M.E. Zielona Gora

  43. calculationofthecurvaturescalar U.R.M.E. Zielona Gora

  44. considerthe 3rdterm: rename and find: U.R.M.E. Zielona Gora

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