The Volume of the Past Light-Cone and the Paneitz Operator

TL;DR

This paper investigates a conjecture linking the invariant volume of the past light-cone to a conformal anomaly operator, confirming its validity in homogeneous isotropic spacetimes and exploring perturbative violations in flat spacetime.

Contribution

It extends the conjecture to homogeneous isotropic geometries and analyzes perturbative violations, providing insights into quantum gravity and alternative gravity models.

Findings

Conjecture holds for homogeneous isotropic geometries.

Perturbation theory shows violations in flat spacetime.

Violations vanish for vacuum Einstein solutions.

Abstract

We study a conjecture involving the invariant volume of the past light-cone from an arbitrary observation point back to a fixed initial value surface. The conjecture is that a 4th order differential operator which occurs in the theory of conformal anomalies gives $8\pi$ when acted upon the invariant volume of the past light-cone. We show that an extended version of the conjecture is valid for an arbitrary homogeneous and isotropic geometry. First order perturbation theory about flat spacetime reveals a violation of the conjecture which, however, vanishes for any vacuum solution of the Einstein equation. These results may be significant for constructing quantum gravitational observables, for quantifying the back-reaction on spacetime expansion and for alternate gravity models which feature a timelike vector field.