Einstein field equation
imported>Pythagoras0님의 2014년 5월 13일 (화) 18:54 판
introduction
- prerequiste : Riemann tensor, the Ricci tensor, and the Ricci scalar
- relativistic matter field equation
\[R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}\] where \(R_{\mu \nu}\) is the Ricci curvature tensor, \(R\) the Ricci scalar curvature, \(\Lambda\) is the cosmological constant, \(T_{\mu \nu}\) momentum-energy tensor
steps to solve an Einstein equation
- consider a metric, for example\(ds^2=-dt^2+e^{2b(t,r)}dr^2+R(t,r)d\phi^2\) where b, R are unknown functions
- find the components of the curvature tensor
- find the components of the Einstein tensor
Einstein-Hilbert action
- Einstein-Hilbert action
\[S= - {1 \over 2\kappa}\int R \sqrt{-g} \, d^4x \] \[\kappa = {8 \pi G \over c^4} \]
equation of motion
solutions example : Schwarzschild black hole
solutions example : gravitational wave
- Einstein
- eventually led to the graviton idea\
history
- Einstein metrics and Ricci solitons
- cosmological constant
- differential geometry and topology
- Yang-Mills Theory(Non-Abelian gauge theory)
- string theory and Einstein equations