Conformal invariance and the conformal-traceless decomposition of the gravitational field

TL;DR

This paper explores conformal invariance in general relativity using a conformal-traceless decomposition, presenting a conformally invariant formulation of Einstein's equations and applying it to initial data construction.

Contribution

It introduces a conformally invariant formulation of Einstein's equations and applies it to the BSSN system and initial data methods.

Findings

Conformal invariance can be maintained or broken by specific evolution conditions.

The BSSN system can be expressed in conformally invariant form.

The approach clarifies the role of conformal flatness in initial data approximation.

Abstract

Einstein's theory of general relativity is written in terms of the variables obtained from a conformal--traceless decomposition of the spatial metric and extrinsic curvature. The determinant of the conformal metric is not restricted, so the action functional and equations of motion are invariant under conformal transformations. With this approach the conformal--traceless variables remain free of density weights. The conformal invariance of the equations of motion can be broken by imposing an evolution equation for the determinant of the conformal metric g. Two conditions are considered, one in which g is constant in time and one in which g is constant along the unit normal to the spacelike hypersurfaces. This approach is used to write the Baumgarte--Shapiro--Shibata--Nakamura system of evolution equations in conformally invariant form. The presentation includes a discussion of the conformal thin sandwich construction of gravitational initial data, and the conformal flatness condition as an approximation to the evolution equations.