The Einstein tensor is built up from the metric tensor and its partial derivatives; thus, the EFE are a system of ten partial differential equations to be solved for the metric.
12.
In the above field equations, G ^ { \ alpha \ beta } is the Einstein tensor, computed uniquely from the vacuum regions ", which contain no matter or nongravitational fields.
13.
Adding a " constant " to the right-hand side of the Einstein equations will effect a change in the Einstein tensor, and thus also in the curvature properties of space-time.
14.
But in the Brans Dicke theory, the Einstein tensor is determined partly by the immediate presence of mass-energy and momentum, and partly by the long-range scalar field \ phi \,.
15.
Where "'\ kappa "'is a constant, and the Einstein tensor on the left side of the equation is equated to the stress energy tensor representing the energy and momentum present in the spacetime.
16.
First published by Einstein in 1915 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 ).
17.
The special case of n = 4 dimensions in physics ( 3 space, 1 time ) gives G \,, the trace of the Einstein tensor, as the negative of R \,, the Ricci tensor's trace.
18.
Then we need only ensure that the divergences vanish ( i . e . that the second Maxwell equation is satisfied for a " source-free " field ) and that the electromagnetic stress-energy matches the Einstein tensor.
19.
It is a matter of pure mathematics that, in any metric theory, the Riemann tensor can always be written as the sum of the Weyl curvature ( or " conformal curvature tensor " ) plus a piece constructed from the Einstein tensor.
20.
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.
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