Deconstructing scaling virial identities in General Relativity: spherical symmetry and beyond

TL;DR

This paper reinterprets Derrick-type virial identities in General Relativity as integrals of equations of motion, providing a unified framework for spherical and axisymmetric solutions, applicable to black holes and solitons.

Contribution

It offers a new derivation of virial identities in relativistic gravity, extending their applicability beyond spherical symmetry to stationary, axisymmetric solutions in a gauge-independent manner.

Findings

Derived a master form of virial identity for spherical symmetry.

Extended virial identities to stationary, axisymmetric solutions.

Connected virial identities to energy-momentum balance conditions.

Abstract

Derrick-type virial identities, obtained via dilatation (scaling) arguments, have a variety of applications in field theories. We deconstruct such virial identities in relativistic gravity showing how they can be recast as self-evident integrals of appropriate combinations of the equations of motion. In spherical symmetry, the appropriate combination and gauge choice guarantee the geometric part can be integrated out to yield a master form of the virial identity as a non-trivial energy-momentum balance condition, valid for both asymptotically flat black holes and self-gravitating solitons, for any matter model. Specifying the matter model we recover previous results obtained via the scaling procedure. We then discuss the more general case of stationary, axi-symmetric, asymptotically flat black hole or solitonic solutions in General Relativity, for which a master form for their virial identity is proposed, in a specific gauge but regardless of the matter content. In the flat spacetime limit, the master virial identity for both the spherical and axial cases reduces to a balance condition for the principal pressures, discussed by Deser.