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The Einstein tensor is a tensor of order 2 defined over pseudo-Riemannian manifolds.In index-free notation it is defined as =, where is the Ricci tensor, is the metric tensor and is the scalar curvature, which is computed as the trace of the Ricci tensor by = .
The Einstein field equations (EFE) may be written in the form: [5] [1] + = EFE on the wall of the Rijksmuseum Boerhaave 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.
where is the Einstein tensor, is the cosmological constant (sometimes taken to be zero for simplicity), is the metric tensor, is a constant, and is the stress–energy tensor. The Einstein field equations relate the Einstein tensor to the stress–energy tensor, which represents the distribution of energy, momentum and stress in the spacetime ...
Einstein's field equations: = where the Ricci curvature tensor = and the scalar curvature = relate the metric (and the associated curvature tensors) to the stress–energy tensor. This tensor equation is a complicated set of nonlinear partial differential equations for the metric components.
Since giving the Einstein tensor does not fully determine the Riemann tensor, but leaves the Weyl tensor unspecified (see the Ricci decomposition), the Einstein equation may be considered a kind of compatibility condition: the spacetime geometry must be consistent with the amount and motion of any matter or non-gravitational fields, in the ...
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'.
A metric tensor is a (symmetric) (0, 2)-tensor; it is thus possible to contract an upper index of a tensor with one of the lower indices of the metric tensor in the product. This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index.
In general relativity, the stress–energy tensor is studied in the context of the Einstein field equations which are often written as + =, where = is the Einstein tensor, is the Ricci tensor, = is the scalar curvature, is the metric tensor, Λ is the cosmological constant (negligible at the scale of a galaxy or smaller), and = / is the ...