Causality in gravitational theories with second order equations of motion

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 factorizes 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” polarization, whereas the latter generically involves a mixture of gravitational and scalar field polarizations. 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 nonsingular 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.

Figure 1

Slowness surface when $W^{\mu \nu \rho \sigma }\ne 0$. The figures show the intersection of the slowness surface (in ${\mathbb{R}}^3$ with coordinates ${\xi }_i$) with a plane passing through the origin and two of the four points (the green dots) corresponding to principal null directions (PNDs). The dashed black curve shows the sphere corresponding to the null cone of $(C^{-1}{)}^{\mu \nu }$. The solid blue curve corresponds to the quartic surface. The left sketch shows the generic behavior, where the quartic surface is nonsingular and the PNDs correspond to double points where the quadratic surface touches the quartic surface smoothly at the green points. The right sketch shows nongeneric behavior where one of the PNDs corresponds to a triple point at which the quartic surface is singular. (The figures are exaggerated for clarity; for a weakly coupled theory, all surfaces will be close to each other and to the ellipsoid corresponding to the null cone of the physical metric.)