Einstein field equation
imported>Pythagoras0님의 2012년 10월 28일 (일) 15:26 판 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로)
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
- cosmological constant
- differential geometry and topology
- Yang-Mills Theory(Non-Abelian gauge theory)
- string theory and Einstein equations
encyclopedia==
\(ds^2=-dt^2+e^{2b(t,r)}dr^2+R(t,r)d\phi^2\)
where b, R are unknown functions
- Einstein-Hilbert action
\(S= - {1 \over 2\kappa}\int R \sqrt{-g} \, d^4x \\)
\(\kappa = {8 \pi G \over c^4} \)
- Einstein
- eventually led to the graviton idea\