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The Einstein field equations (EFE) may be written in the form: [5] [1] + = EFE on a wall in Leiden, Netherlands. where is the Einstein tensor, is the metric tensor, is the stress–energy tensor, is the cosmological constant and is the Einstein gravitational constant.
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 exact form of the metric g μν depends on the gravitating mass, momentum and energy, as described by the Einstein field equations. Einstein developed those field equations to match the then known laws of Nature; however, they predicted never-before-seen phenomena (such as the bending of light by gravity) that were confirmed later.
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 Einstein field equations are a system of coupled, nonlinear partial differential equations. In general, this makes them hard to solve. In general, this makes them hard to solve. Nonetheless, several effective techniques for obtaining exact solutions have been established.
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.
Non-exact solutions in general relativity are solutions of Albert Einstein's field equations of general relativity which hold only approximately. These solutions are typically found by treating the gravitational field, , as a background space-time, , (which is usually an exact solution) plus some small perturbation, .
The metric tensor is a central object in general relativity that describes the local geometry of spacetime (as a result of solving the Einstein field equations). Using the weak-field approximation, the metric tensor can also be thought of as representing the 'gravitational potential'. The metric tensor is often just called 'the metric'.