Solutions of vacuum fieldequationhab = Aabeikeakak° =0Null wavevector0?hab = 0Transverse wavekAab = 0nabrab = 0Extra degree of freedom comesfrom gaugetransformationα'α(P) = αa(P) + ≤℃(P)A°= 0Tracelesshab = hab - Sa,b - Sb,aaAab=0ot2c~ = 0
Solutions of vacuum field equation Null wave vector Transverse wave Traceless Extra degree of freedom comes from gauge transformation
Constrains on AAA02A01A03000AabA11A12=0AA1310otA21A22LA2320k~ Aab = 0DU0A11 = -A22A° = 0h+hx0hx-h+0000
Constrains on A
Geometrical interpretationh+00etwt-h+000001+h+eiwt00Aa1-hteiwt0Ay0A=(AAzAy000AzGraviataionalwaveofGR(TT)@KJLee
Geometrical interpretation + x
GenerationofGW22w22hii=3RhμM2/3,2/3R-1(M+m2uaa3mMum+Mh ~ M5/3w2/3 R-1RInGWastronomy,wearedetectingh~1/R.Increasingsensitivitybyafactorof10,thenumberofsourceis 1000times.Forotherdetectortheydependonenergyflux,whichgoesas1/R^2,numberofsourceincreaseas100times
Generation of GW a R In GW astronomy, we are detecting h~1/R. Increasing sensitivity by a factor of 10, the number of source is 1000 times. For other detector, they depend on energy flux, which goes as 1/R^2, number of source increase as 100 times
Inhistorytischen Klssevom 22.Junf 1916Minteilungvom 31.Jotiua154Geamtsitzungvom14.Februar1918,MITSCIRUCRLIGnAATAS.A.3IlUber GravitationswellenNaherungsweise IntegrationderFeldgleichungenVon A.EINSTEIN.der GravitationVon A.EnsTEIN.(Vorgelegtam31,Januar1918[s.oben.79].)ForGR,onlytwo polarizationmodesexist.EinsteinderivethewaveequationsoonafterformulatedtheGR.However,in 1916,hemademistakeand statethattherearethree polarizationmodes.The mistakewascorrectedin1918
In history ⚫For GR, only two polarization modes exist. Einstein derive the wave equation soon after formulated the GR. However, in 1916, he made mistake and state that there are three polarization modes. The mistake was corrected in 1918