Einstein field equation

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

\(S= - {1 \over 2\kappa}\int R \sqrt{-g} \, d^4x \\)

\(\kappa = {8 \pi G \over c^4} \)

 

 

equation of motion

• http://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action#Derivation_of_Einstein.27s_field_equations
  

 

 

 

 

 

solutions example : gravitational wave

• Einstein
  
• eventually led to the graviton idea\
  

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items

• cosmological constant
• differential geometry and topology
• Yang-Mills Theory(Non-Abelian gauge theory)
• string theory and Einstein equations

 

 

encyclopedia

• [1]http://en.wikipedia.org/wiki/Einstein_field_equations
• http://en.wikipedia.org/wiki/Riemann_curvature_tensor
• http://en.wikipedia.org/wiki/Ricci_curvature
• http://en.wikipedia.org/wiki/Scalar_curvature