On the Weyl Tensor for a Curved Spacetime Endowed with a Torsionful Affinity

TL;DR

This paper extends the Weyl tensor formalism to curved spacetimes with torsion within Einstein-Cartan theory, revealing that conformal flatness cannot generally be achieved even with vanishing graviton wave functions.

Contribution

It reformulates the Weyl tensor in a torsionful framework, showing the limitations of conformal flatness in Einstein-Cartan spacetimes.

Findings

Conformal flatness is not generally attainable in torsionful spacetimes.

The formalism adapts the { extepsilon}-formalism to include torsion effects.

Discussion on the special case of general relativity as a limit.

Abstract

We transcribe into the framework of the torsionful version of the {\epsilon}-formalism of Infeld and van der Waerden the world definition of the Weyl tensor for a curved spacetime that occurs in the realm of Einstein-Cartan's theory. The resulting expression shows us that it is not possible to attain any general condition for conformal flatness in such a spacetime even if wave functions for gravitons are eventually taken to vanish identically. A short discussion on the situation concerning the limiting case of general relativity is presented thereafter.