Coordinate conditionsIn general relativity, the laws of physics can be expressed in a generally covariant form. In other words, the description of the world as given by the laws of physics does not depend on our choice of coordinate systems. However, it is often useful to fix upon a particular coordinate system, in order to solve actual problems or make actual predictions. A coordinate condition selects such coordinate system(s). The Einstein field equations do not determine the metric uniquely, even if one knows what the metric tensor equals everywhere at an initial time.
Expansion of the universeThe expansion of the universe is the increase in distance between gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion; the universe does not expand "into" anything and does not require space to exist "outside" it. To any observer in the universe, it appears that all but the nearest galaxies (which are bound by gravity) recede at speeds that are proportional to their distance from the observer, on average.
Einstein tensorIn differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field equations for gravitation that describe spacetime curvature in a manner that is consistent with conservation of energy and momentum. The Einstein tensor is a tensor of order 2 defined over pseudo-Riemannian manifolds.
Exact solutions in general relativityIn general relativity, an exact solution is a solution of the Einstein field equations whose derivation does not invoke simplifying assumptions, though the starting point for that derivation may be an idealized case like a perfectly spherical shape of matter. Mathematically, finding an exact solution means finding a Lorentzian manifold equipped with tensor fields modeling states of ordinary matter, such as a fluid, or classical non-gravitational fields such as the electromagnetic field.
