- #1

kderakhshani

- 13

- 0

As you know, the Einstein field equation

[itex]R_{μ\nu} - 1/2Rg_{μ\nu} =κT_{μ\nu} [/itex]

implies that at any point with vanishing energy-momentum tensor the Ricci curvature also vanishes:

[itex]T_{μ\nu} = 0 \Rightarrow R_{μ\nu} = 0 [/itex]

hence a Ricci-flat space-time (the vacuum solutions of the field equation).

However, [itex]R_{μ\nu}[/itex] includes just 10 elements of the Riemann-Christofel curvature tensor [itex]R_{μ\nuλρ}[/itex]. The remaining 10 elements are coded in the Weyl tensor (the conformal tensor) [itex]C_{μ\nuλρ}[/itex], which has all symmetries of [itex]R_{μ\nuλρ}[/itex] but is totally traceless, and does not vanish only for [itex]n≥4[/itex].

http://en.wikipedia.org/wiki/Weyl_tensor

The Weyl tensor is often associated with tidal forces or gravitational waves.

Thus, in an absolute (classical) vacuum (with no mass, energy, momentum densities), [itex]R_{μ\nuλρ}[/itex] can still be nonzero.

Does it mean a curved space-time (and a gravitation) without mass-energy?