The sky invariant: A new conformal invariant for Schwarzschild spacetime

TL;DR

This paper introduces a new conformal invariant called sky curvature for Schwarzschild spacetime, based on the geometry of light rays and skies, with an algorithm for its computation.

Contribution

It develops a novel conformal invariant framework using sky geometry and provides an explicit computational algorithm for Schwarzschild spacetime.

Findings

Defined the sky invariant and sky curvature in conformal geometry.

Established an algorithm for computing sky curvature.

Illustrated the computation in Schwarzschild spacetime.

Abstract

A new class of conformal invariants for a given spacetime $M$ is introduced exploiting the conformal geometry of any light ray $\Gamma$. Each congruence of light rays passing through a given point $p$ defines the sky $S(p)$ of such point. The new conformal invariants are defined on the bundle of skies of the spacetime $M$, being called sky invariants accordingly. The natural conformal covariant derivative defined on a light ray and its associated covariant calculus allows us show the existence of a natural conformal invariant differential of arc that, together with the restriction of the curvature of the conformal covariant derivative, can be used to construct a sky invariant that will be called the sky curvature. An algorithm, that can be implemented on any symbolic manipulation software system, to compute the sky curvature will be discussed and the main ideas and the explicit computation of the sky curvature are illustrated in Schwarzschild spacetime.