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.
is a tensor of order 2 defined over pseudo-Riemannian manifolds.
is the scalar curvature, which is computed as the trace of the Ricci tensor
In component form, the previous equation reads as
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:
is the Kronecker tensor and the Christoffel symbol
represent partial derivatives in the μ-direction, e.g.:
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:
γ [ β , μ ] α
α [ μ , β ] γ
ϵ [ μ , σ ] γ
γ [ σ , μ ] ϵ
ϵ [ μ , σ ] γ
γ [ σ , μ ] ϵ
α [ β , γ ] ϵ
α β , γ ϵ
α γ , β ϵ
The trace of the Einstein tensor can be computed by contracting the equation in the definition with the metric tensor
case is especially relevant in the theory of general relativity.
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 metric.
It follows that the Einstein field equations are a set of 10 quasilinear second-order partial differential equations for the metric tensor.
The contracted Bianchi identities can also be easily expressed with the aid of the Einstein tensor:
The (contracted) Bianchi identities automatically ensure the covariant conservation of the stress–energy tensor in curved spacetimes:
The physical significance of the Einstein tensor is highlighted by this identity.
In terms of the densitized stress tensor contracted on a Killing vector
, an ordinary conservation law holds:
David Lovelock has shown that, in a four-dimensional differentiable manifold, the Einstein tensor is the only tensorial and divergence-free function of the
[1][2][3][4][5] However, the Einstein field equation is not the only equation which satisfies the three conditions:[6] Many alternative theories have been proposed, such as the Einstein–Cartan theory, that also satisfy the above conditions.