Causality in gravitational theories with second order equations of motion

TL;DR

This paper investigates causality in second-order diffeomorphism invariant gravity theories with matter, revealing how characteristic polynomials govern the propagation of degrees of freedom and establishing conditions for black hole horizons.

Contribution

It introduces a gauge-invariant method to analyze causality in general second-order gravity theories, including Horndeski models, and characterizes the associated characteristic surfaces and horizons.

Findings

Causality is governed by a degree 6 polynomial factorizing into quadratic and quartic parts.

The quartic polynomial describes the fastest degrees of freedom and defines a Fresnel-like surface.

Killing horizons satisfy the zeroth law of black hole mechanics in these theories.

Abstract

This paper considers diffeomorphism invariant theories of gravity coupled to matter, with second order equations of motion. This includes Einstein-Maxwell and Einstein-scalar field theory with (after field redefinitions) the most general parity-symmetric four-derivative effective field theory corrections. A gauge-invariant approach is used to study the characteristics associated to the physical degrees of freedom in an arbitrary background solution. The symmetries of the principal symbol arising from diffeomorphism invariance and the action principle are determined. For gravity coupled to a single scalar field (i.e. a Horndeski theory) it is shown that causality is governed by a characteristic polynomial of degree $6$ which factorises into a product of quadratic and quartic polynomials. The former is defined in terms of an "effective metric" and is associated with a "purely gravitational" polarisation, whereas the latter generically involves a mixture of gravitational and scalar field polarisations. The "fastest" degrees of freedom are associated with the quartic polynomial, which defines a surface analogous to the Fresnel surface in crystal optics. In contrast with optics, this surface is generically non-singular except on certain surfaces in spacetime. It is shown that a Killing horizon is an example of such a surface. It is also shown that a Killing horizon satisfies the zeroth law of black hole mechanics. The characteristic polynomial defines a cone in the cotangent space and a dual cone in the tangent space. The latter is used to define basic notions of causality and to provide a definition of a dynamical black hole in these theories.