# Einstein space

## Einstein space

[¦īn¦stīn ′spās]
(mathematics)
A Riemannian space in which the contracted curvature tensor is proportional to the metric tensor.
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an Einstein space which is Ricci-flat and consequently has zero scalar curvature, too.
They adequately generalize the Einstein space endowed with the cosmological constant [LAMBDA] defined as:
The background space is not one an Einstein space (where [R.sub.[alpha][beta]] = k[g.sub.[alpha][beta]], k = const) due to the fact that Einstein's equations have k=0 in the background space.
If [T.sub.[alpha][beta]] [varies][g.sub.[alpha][beta]], the associated space is called an Einstein space. Thus, Einstein spaces include those described by partially degenerate metrics of this form.
Metric (1a) and Euclidean-like structure (1b) are complementary to each other in the Einstein space. These space-time coordinates form not just a mathematical coordinate system since a light speed ([ds.sup.2] = 0) is defined in terms of dx/dt, dy/dt, and dz/dt .
From the purely geometrical perspective, an Einstein space  is described by any metric obtained from
in which case the space, where the gravitational field is located, is called an Einstein space. If [kappa] = 0, we have an empty space (without gravitating matter).
Equation (3.4) is the Einstein equation for Einstein spaces in differential geometry [1,2];
In  the result for odd dimensional Einstein spaces was obtained by Dumitru.
The topics are notations and prerequisites from analysis, curves in Rn, the local theory of surfaces, the intrinsic geometry of surfaces, Riemannian manifolds, spaces of constant curvature, and Einstein spaces. This third English edition incorporates changes in all six German editions, the most recent published in 2013.
Besides, Schwarzschild space is only a very particular case of Einstein spaces of type I.
In a previous paper the writer treated of particular classes of cosmological solutions for certain Einstein spaces and claimed that no such solutions exist in relation thereto.

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