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The value of the Einstein convention is that it applies to other vector spaces built from V using the tensor product and duality. For example, V ⊗ V, the tensor product of V with itself, has a basis consisting of tensors of the form e ij = e i ⊗ e j. Any tensor T in V ⊗ V can be written as: =.
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 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.
On a manifold, a tensor field will typically have multiple, upper and lower indices, where Einstein notation is widely used. When the manifold is equipped with a metric, covariant and contravariant indices become very closely related to one another. Contravariant indices can be turned into covariant indices by contracting with the metric tensor ...
The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci. [1] An equivalent definition of a tensor uses the representations of the general linear group. There is an action of the general linear group on the set of all ordered bases of an n-dimensional vector space.
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.
First published by Einstein in 1915 [7] as a tensor equation, the EFE equate local spacetime curvature (expressed by the Einstein tensor) with the local energy and momentum within that spacetime (expressed by the stress–energy tensor). [8] The Einstein field equations can be written as =, where G μν is the Einstein tensor and T μν is the ...
The stress–energy tensor involves the use of superscripted variables (not exponents; see Tensor index notation and Einstein summation notation).If Cartesian coordinates in SI units are used, then the components of the position four-vector x are given by: [ x 0, x 1, x 2, x 3].