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Concept
Einstein tensor
Summary
In 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. 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 . In component form, the previous equation reads as
The Einstein tensor is symmetric
and, like the on shell stress–energy tensor, has zero divergence:
The Ricci tensor depends only on the metric tensor, so the Einstein tensor can be defined directly with just the metric tensor. However, this expression is complex and rarely quoted in textbooks. The complexity of this expression can be shown using the formula for the Ricci tensor in terms of Christoffel symbols:
where is the Kronecker tensor and the Christoffel symbol is defined as
and terms of the form represent its partial derivative in the μ-direction, i.e.:
Before cancellations, this formula results in individual terms. Cancellations bring this number down somewhat.
In the special case of a locally inertial reference frame near a point, the first derivatives of the metric tensor vanish and the component form of the Einstein tensor is considerably simplified:
where square brackets conventionally denote antisymmetrization over bracketed indices, i.e.
The trace of the Einstein tensor can be computed by contracting the equation in the definition with the metric tensor . In dimensions (of arbitrary signature):
Therefore, in the special case of n = 4 dimensions, . That is, the trace of the Einstein tensor is the negative of the Ricci tensor's trace. Thus, another name for the Einstein tensor is the trace-reversed Ricci tensor.
Official source
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