Do the Ricci and energy-momentum tensors have "duality'' in the context of their Lie symmetries?

TL;DR

This paper investigates whether the Lie symmetry algebras of Ricci and energy-momentum tensors are dual, finding that they are not identical nor subsets of each other despite their algebraic similarities.

Contribution

The study demonstrates that Ricci and energy-momentum tensors do not share identical Lie symmetry algebras in cylindrically symmetric static spacetimes, challenging the expectation of their duality.

Findings

Lie symmetry algebras of the tensors are not identical

Neither algebra is a subset of the other

Duality does not extend to their Lie symmetries

Abstract

The Ricci and energy-momentum tensors have the same algebraic symmetries. In the Einstein equations they look ``dual'' to each other, in that interchanging them and inverting the gravitational coupling leaves the equations invariant. It may then be expected that their differential symmetry Lie algebras would also be identical. Using cylindrically symmetric static spacetimes it is shown that they are not identical and neither algebra is a subset of the other.