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Einstein’s Equations

  • Michael E. Taylor
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 117)

Abstract

In this chapter we discuss Einstein’s gravitational equations, which state that the presence of matter and energy creates curvature in spacetime, via
$${G}_{jk} = 8\pi \kappa {T}_{jk},$$
(0.1)
where G jk ={ Ric} jk −(1∕2)Sg jk is the Einstein tensor, T jk is the stress-energy tensor due to the presence of matter, and κ is a positive constant. In 1 we introduce this equation and relate it to previous discussions of stress-energy tensors and their relation to equations of motion. We recall various stationary action principles that give rise to equations of motion and show that (0.1) itself results from adding a term proportional to the scalar curvature of spacetime to standard Lagrangians and considering variations of the metric tensor.

Keywords

Scalar Curvature Fundamental Form Integral Curve Lorentz Manifold Spacelike Hypersurface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of North CarolinaChapel HillUSA

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