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

Understand

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.