Group velocity of gravitational wave packet on curved background

Question

Within the framework of General Relativity, is there a well-defined way of determining how the curvature affects the group velocity of a wave packet, more specifically a localised gravitational wave? I know there is such a thing as gravitational lensing of waves, but what I'm aiming for is an equation (similar to the geodesic equation?) that somehow relates the group velocity to the connection or the curvature. More specifically I'm talking about the case where the metric of the space is of the form: $$\gamma_{\mu \nu} \approx g_{\mu \nu} +h_{\mu \nu}$$ where $g$ is the metric of the curved background, and $h$ describes the wave packet and is sufficiently small. So the intended equation would probably involve only first-order terms in $h$.

When I searched online for answers, I found some papers on accelerated wave packets (i.e their centre of mass velocity, or group velocity changes with time) but they only treated particular cases, for example, a wave packet in a spherical shell, and the acceleration is only given explicitly.

I suppose defining such an equation may be problematic for several reasons: the group velocity is usually defined w.r.t to an absolute time coordinate, and I don't know how one would go about defining a "group 4-velocity". Also, due to dispersion, the equation may not always hold, and if it does it will only approximately hold for the short spacetime interval where the wave packet is localised. If such an equation exists, can it be derived from the wave equation in curved space-time (derived from the sourceless field equations)? How would one go about defining such as an equation?

Any help or further clarification on the subject would be greatly appreciated!

Answer 1

Analyzing Group Velocity of Localized Gravitational Waves in a Curved Background

To explore how curvature affects the group velocity of a localized gravitational wave in General Relativity, we can consider the dynamics of wave packets within a perturbed metric. Given the form $$ \gamma_{\mu \nu} \approx g_{\mu \nu} + h_{\mu \nu} $$, where $$ g_{\mu \nu} $$ represents the background metric and $$ h_{\mu \nu} $$ denotes the wave packet perturbation (assumed small), we can analyze the propagation and evolution of this wave packet through curved spacetime. While gravitational lensing influences the trajectory of waves, as you pointed out, our focus here is specifically on the group velocity, a measure of the wave packet's "center of mass" motion through spacetime.

For small perturbations $$ h_{\mu \nu} $$, the wave packet's evolution can be studied using a weak-field approximation to the Einstein field equations. In this regime, the propagation is influenced by the effective potential created by the curvature, which can be treated by examining the perturbed wave equation derived from the linearized Einstein equations. In essence, this approach treats the wave packet like a collection of quasi-particles, with each component following trajectories approximated by null or nearly-null geodesics, depending on the precise structure of $$ h_{\mu \nu} $$ and the dispersion relation in the background $$ g_{\mu \nu} $$.

One can approximate the group velocity $$ v_g $$ using a framework similar to the geodesic equation, but accounting for the wave vector $$ k^\mu $$ evolving according to the connection associated with the background curvature. The phase velocity in curved space-time generally follows the null geodesic condition $$ g^{\mu \nu} k_\mu k_\nu = 0 $$, while the group velocity requires accounting for the perturbation's influence. In first-order approximation, the perturbed wave equation indeed implies an effective geodesic equation for $$ k^\mu $$, where $$ \nabla_\nu k^\nu = 0 $$ governs the transport of the wave vector under curvature. This setup describes the deviation of the wave packet’s trajectory, analogous to gravitational lensing but here specifically adapted to the packet's group motion, enabling us to express the group velocity’s evolution.

Although defining a “group 4-velocity” is non-trivial in General Relativity, since it lacks a clear absolute time, one can interpret the group velocity in terms of the locally measured propagation speed. Within the limits of the locality of the wave packet, the dispersion effects remain minimal, allowing the wave packet’s group velocity to approximate the local speed of propagation—modulated by spacetime curvature. To derive an explicit equation, we start from the wave equation for $$h_{\mu \nu}$$ in a curved background and apply the eikonal approximation, yielding an evolution equation for $$k^\mu$$ (the wave packet's central wave vector) under curvature, which relates directly to the group velocity and thus indirectly to the connection and curvature tensor.