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Caltech Tutorial on Relativity — A simple introduction to Einstein's Field Equations. The Meaning of Einstein's Equation — An explanation of Einstein's field equation, its derivation, and some of its consequences; Video Lecture on Einstein's Field Equations by MIT Physics Professor Edmund Bertschinger. Arch and scaffold: How Einstein found ...
Next, notice that only 10 of the original 14 equations are independent, because the continuity equation ; = is a consequence of Einstein's equations. This reflects the fact that the system is gauge invariant (in general, absent some symmetry, any choice of a curvilinear coordinate net on the same system would correspond to a numerically ...
The initial value formulation of general relativity is a reformulation of Albert Einstein's theory of general relativity that describes a universe evolving over time.. Each solution of the Einstein field equations encompasses the whole history of a universe – it is not just some snapshot of how things are, but a whole spacetime: a statement encompassing the state of matter and geometry ...
The static assumption is unneeded, as Birkhoff's theorem states that any spherically symmetric vacuum solution of Einstein's field equations is stationary; the Schwarzschild solution thus follows. Birkhoff's theorem has the consequence that any pulsating star that remains spherically symmetric does not generate gravitational waves , as the ...
The Einstein field equations are nonlinear and considered difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But in 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. This solution laid the ...
However, perturbation expansions are generally not reliable for questions of long-term existence and stability, in the case of nonlinear equations. The full field equation is highly nonlinear, so we really want to prove that the Minkowski vacuum is stable under small perturbations which are treated using the fully nonlinear field equation.
The field of numerical relativity emerged from the desire to construct and study more general solutions to the field equations by approximately solving the Einstein equations numerically. A necessary precursor to such attempts was a decomposition of spacetime back into separated space and time.
A rotating black hole is a solution of Einstein's field equation. There are two known exact solutions, the Kerr metric and the Kerr–Newman metric , which are believed to be representative of all rotating black hole solutions, in the exterior region.