Mirror symmetry and conformal flatness in general Relativity

Using symmetry arguments only, we show that every spacetime with mirror-symmetric spatial sections is necessarily conformally flat. The general form of the Ricci tensor of such spacetimes is also determined. 1. Introduction. It is well known that the curvature tensor of any four-dimensional differentiable manifold has only 20 algebraically independent components. Ten out of these 20 components can be associated with its Weyl tensor, the remaining ten making up its Ricci tensor. When the four-dimensional manifold corresponds to an empty spacetime, its Ricci tensor becomes identically zero. The Weyl tensor can thus be seen as describing that part of the curvature of the spacetime which is not due to the presence of matter. The spacetime is said to be conformally flat when its Weyl tensor is identically zero (see, e.g., [4, Chapter 8]). In this note, we are interested in the conditions on the curvature tensor R of a space-time ᏹ 4 which follow from the assumption that ᏹ 4 has mirror-symmetric spatial sections. We will show that any such ᏹ 4 is conformally flat. We will also obtain the general form of the corresponding Ricci tensor.