Two Speeds of Gravity: Constraints and Waves in General Relativity

How fast does gravity propagate? The standard answer, "at the speed of light," is correct for gravitational waves but misses something important. Within linearized general relativity on a flat background (and analogously in the ADM initial-value formulation of the exact theory), the gravitational field decomposes into a constraint sector and a radiative sector. The constraint sector fixes the Newtonian-like binding potential at each moment on a chosen spacelike hypersurface. The radiative sector describes transverse-traceless gravitational waves, which carry energy on the light cone. We show that these two sectors can be cleanly decomposed in the linearized setting using a "velocity gauge," built in direct analogy with a well-known construction in electromagnetism [4]. In this framework, the apparent propagation speed of the gauge-variant binding potential Φ_v is a freely chosen gauge parameter (not a physical quantity), while gravitational waves always travel at c and all measurable quantities (tidal forces, geodesic deviation) remain the same regardless of the choice. The standard post-Newtonian (PN) approximation independently corroborates this picture: through 3PN order, conservative dynamics is built from instantaneous potentials, with wave-like propagation entering only at 2.5PN (radiation reaction) and 4PN (hereditary tails) [5]. The well-known absence of gravitational aberration in orbiting bodies is reinterpreted as a property of the constraint sector, not a measurement of wave speed. The 2017 neutron-star merger observation (GW170817) measured the speed of gravitational waves; it did not and could not measure the "speed" of gauge-variant binding potentials, because constraint equations are not radiative propagation laws. We work strictly within general relativity and introduce no new physics.