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ISBN 0-19-853209-1. Archived from the original on 2007-06-10. The definitive resource on colliding plane waves, but also useful to anyone interested in other exact solutions. Hoenselaers, C.; Dietz, W. (1985). Solutions of Einstein's Equations: Techniques and Results. Springer. ISBN 3-540-13366-6. Ehlers, Jürgen; Kundt, Wolfgang (1962).
These generally covariant theories describes a spacetime endowed with both a metric and a unit timelike vector field named the aether. The aether in this theory is "a Lorentz-violating vector field" [1] unrelated to older luminiferous aether theories; the "Einstein" in the theory's name comes from its use of Einstein's general relativity ...
This diagram gives the route to find the Schwarzschild solution by using the weak field approximation. The equality on the second row gives g 44 = −c 2 + 2GM/r, assuming the desired solution degenerates to Minkowski metric when the motion happens far away from the blackhole (r approaches to positive infinity).
As historians such as John Stachel argue, Einstein's views on the "new aether" are not in conflict with his abandonment of the aether in 1905. As Einstein himself pointed out, no "substance" and no state of motion can be attributed to that new aether. [10] Einstein's use of the word "aether" found little support in the scientific community, and ...
[5] The Socolar–Taylor tile was proposed in 2010 as a solution to the einstein problem, but this tile is not a connected set. In 1996, Petra Gummelt constructed a decorated decagonal tile and showed that when two kinds of overlaps between pairs of tiles are allowed, the tiles can cover the plane, but only non-periodically. [6]
But if the exact solution is required or a solution describing strong fields, the evolution of the metric and the stress–energy tensor must be solved for together. To obtain solutions, the relevant equations are the above quoted EFE (in either form) plus the continuity equation (to determine evolution of the stress–energy tensor):
Einstein showed how the velocity of light in a moving medium is calculated, in the velocity-addition formula of special relativity. Einstein's theory of general relativity provides the solution to the other light-dragging effects, whereby the velocity of light is modified by the motion or the rotation of nearby masses.
The Ellis drainhole is the earliest-known complete mathematical model of a traversable wormhole.It is a static, spherically symmetric solution of the Einstein vacuum field equations augmented by inclusion of a scalar field minimally coupled to the geometry of space-time with coupling polarity opposite to the orthodox polarity (negative instead of positive):