  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.
