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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 allows the Einstein field equations to be written in the concise form: + =, where is the cosmological constant and is the Einstein gravitational constant. From the explicit form of the Einstein tensor , the Einstein tensor is a nonlinear function of the metric tensor, but is linear in the second partial derivatives of the ...
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.
The Einstein field equations (EFE) are the core of general relativity theory. The EFE describe how mass and energy (as represented in the stress–energy tensor) are related to the curvature of space-time (as represented in the Einstein tensor).
The Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero: = = = = The Ricci tensor, the Einstein tensor, and the traceless Ricci tensor are symmetric 2-tensors:
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 ...
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 ...
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 ...