  Two universal predictions of general relativity are the propagation of
gravitational waves of large momentum along null geodesics and the
isospectrality of quasinormal modes in many families of black holes. In
extensions of general relativity, these properties are typically lost:
quasinormal modes are no longer isospectral and gravitational wave propagation
is no longer geodesic and it exhibits birefringence -- polarization-dependent
speed. We study these effects in an effective-field-theory extension of general
relativity with up to eight-derivative terms and show that there is a unique
Lagrangian that has a non-birefringent dispersion relation for gravitational
waves and isospectral quasinormal modes in the eikonal limit. We argue that
both properties are related through a generalized correspondence between
eikonal quasinormal modes and unstable photon-sphere orbits. These properties
define a special class of theories that we denote as isospectral effective
field theories. We note that the lowest-order isospectral correction to general
relativity coincides with the quartic-curvature correction arising from type II
string theory, suggesting that isospectrality might be a key feature of quantum
gravity.
