Light-Like Shockwaves in Scalar-Tensor Theories

Bence Racskó,László Gergely

doi:10.3390/universe4030044

Abstract

Both electromagnetic shock-waves and gravitational waves propagate with the speed of light. If they carry significant energy-momentum, this will change the properties of the space-time they propagate through. This can be described in terms of the junction conditions between space-time regions separated by a singular, null hypersurface. We derived generic junction conditions for Brans-Dicke theory in the Jordan frame, exploring a formalism based on a transverse vector, rather than normal, which can be applied to any type of hypersurfaces. In the particular case of a non-null hypersurface we obtain a generalised Lanczos equation, in which the jump of the extrinsic curvature is sourced by both the distributional energy-momentum tensor and by the jump in the transverse derivative of the scalar. In the case of null hypersurfaces, the distributional source is decomposed into surface density, current and pressure. The latter however ought to vanish by virtue of the scalar junction condition.

Highlights

General relativity (GR) has withstood the confrontation with observations both in Solar System tests and in strong field regimes, the latter of which had been the experimental detection of gravitational waves by the LIGO Scientific Collaboration and Virgo Collaboration [1,2,3,4,5,6]
Brans-Dicke theory in the Jordan frame, exploring a formalism based on a transverse vector, rather than normal, which can be applied to any type of hypersurfaces
In the case of null hypersurfaces, the distributional source is decomposed into surface density, current and pressure

Summary

Introduction

General relativity (GR) has withstood the confrontation with observations both in Solar System tests and in strong field regimes, the latter of which had been the experimental detection of gravitational waves by the LIGO Scientific Collaboration and Virgo Collaboration [1,2,3,4,5,6]. In the Einstein frame, the matter stress-energy tensor rather than obeying a continuity equation is subject to ∇μ T μν = − T ∇ν ln Ω [11,14] Both in GR and in modified gravity theories it is of special interest to match spacetime regions with different matter sources or even different set of symmetries. We opt for the Jordan frame, motivated by the desire to keep the generic form of the function G4 in the Hordeski Lagrangian Another motivation for assuming that the sources couple to gravity only via the metric tensor, and not via the scalar field would be to avoid any non-gravitational interaction of the scalar field with the baryonic matter fields ψi , to be able to describe the dark sector with φ.

Junction Conditions across Arbitrary Hypersurfaces

Brans-Dicke Theory

The Null Case

Conclusions