Consistency Problems of Conformal Killing Gravity

TL;DR

This paper discusses the consistency issues in Conformal Killing Gravity, a rank-3 tensor extension of General Relativity, highlighting problems in defining conserved quantities like energy and angular momentum.

Contribution

It identifies fundamental consistency problems in Conformal Killing Gravity related to the interpretation of integration constants as conserved quantities.

Findings

Integration constants in the theory lack a clear conserved quantity interpretation.

The theory's extension does not provide a straightforward way to define energy or angular momentum.

Challenges in formulating gravitational wave formulas in the weak field limit.

Abstract

We show that gravity field equations based on a tensor with rank greater than 2 have consistency problems in the sense that integration constants in the solutions, such as the parameter $m$ in the Schwarzschild metric, do not allow for an interpretation in terms of conserved quantities in the theory. The recently introduced Conformal Killing Gravity, an interesting extension of General Relativity that inherits all the solutions of the latter, and defined with a rank-3 tensor field equation that does not arise from a diffeomorphism-invariant action, is plagued with this problem. In this theory, it is not clear at all how one can define the energy and angular momentum for black hole solutions, or define the analogues of the formulas, such as the quadrupole formula, in the weak field limit for gravitational waves emitted by compact sources.