Search for a command to run...
We present a rigorous derivation of the gravitational wave speed in the framework of Causal Field Theory (CFT), a modified gravity theory rooted in the Principle of Causal Optimality. In previous work, CFT was shown to successfully account for galactic rotation curves via the saturated causal field $\phi_{\max} = 0.429$ (Paper I), as well as dark energy and the Hubble tension through the vacuum attractor $\phi_{\text{vac}} = 0.3476$ (Paper II). However, the propagation speed of gravitational waves—a critical test for any theory of gravity—remained to be derived from first principles. Starting from the modified Einstein equations, we perform a complete linear perturbation analysis around a homogeneous and isotropic background. Working in the transverse-traceless gauge, we derive the wave equation for the metric perturbations $h_{ij}^{\text{TT}}$ and obtain a Klein–Gordon equation with a mass term $\mu^2 = 16\pi G \rho_{\Lambda}$, where $\rho_{\Lambda}$ is the dark energy density. The corresponding dispersion relation $\omega^2 = k^2 - \mu^2$ yields a group velocity $v_g = c / \sqrt{1 - \mu^2/k^2}$. For LIGO-band frequencies ($f \sim 100$ Hz), the deviation from the speed of light is $\sim 2 \times 10^{-24}$; for LISA-band frequencies ($f \sim 10^{-3}$ Hz), it is $\sim 2 \times 10^{-14}$. These values are orders of magnitude below the bound $|v_{\text{GW}} - c|/c < 10^{-15}$ set by the multimessenger event GW170817. The tachyonic instability for $k < \mu$ affects only ultra-low frequencies ($f < 2 \times 10^{-10}$ Hz) that are not accessible to current or near-future detectors. This work resolves the potential tension between Causal Field Theory and multimessenger observations, thereby completing the classical formulation of the theory and establishing its consistency with all existing observational constraints. The results lay the foundation for subsequent quantization of the theory.