Overview
This article completes the elliptic spectral-geometry framework by constructing a Lorentzian, causal dynamics for perturbations of the effective metric. Starting from the projective spectral entropy functional $S_\Pi[g]=\tfrac12\log\det' A_g$, the paper introduces a normally hyperbolic operator $D_g$ whose spatial restriction reproduces the elliptic operator $A_g$ on constant-time slices.
The Lorentzian spectral action is defined as $S_\Pi^{(L)}[g]=\tfrac12\,\mathrm{Re}\,\log\det' D_g$, equivalently via a Schwinger–Keldysh prescription, so that the inverse entering $\delta^2 S_\Pi^{(L)}$ is the retarded Green operator $G_R$. This ensures a real quadratic kernel and causal propagation.
Core contributions
- Lorentzian completion: a hyperbolic operator $D_g$ whose spatial block reproduces the elliptic $A_g$ in a quasi-static $3+1$ reduction.
- Causal quadratic kernel: $S_\Pi^{(L)}=\tfrac12\,\mathrm{Re}\,\log\det' D_g$ (or Schwinger–Keldysh) implies a retarded inverse $G_R$ in $\delta^2 S_\Pi^{(L)}$.
- Infrared propagation theorem: for $R\ell_\chi^2\ll1$ and $\omega\ll\ell_\chi^{-1}$, exactly two TT tensor modes of helicity $\pm2$ propagate; scalar and vector sectors remain non-dynamical.
- Ultraviolet pole separation: higher-derivative terms generate an extra pole at $k^2\sim\ell_\chi^{-2}$ that decouples from the observational infrared window.
- Spectrally fixed dispersion: $\omega^2=c^2k^2-\gamma\ell_\chi^2k^4+O(k^6)$ with $\gamma=1/(180\zeta)$ and $\zeta=O(1)$; in the natural spectral scheme, $\gamma=1/30$.
Lorentzian spectral action
The Lorentzian completion is defined by a causal prescription: $S_\Pi^{(L)}[g]=\tfrac12\,\mathrm{Re}\,\log\det' D_g$, equivalently formulated on a Schwinger–Keldysh contour, so that functional variations involve the retarded Green operator $G_R$. This choice makes the quadratic kernel real and enforces causal support $G_R(x,y)=0$ when $x\notin J^+(y)$.
In a $3+1$ decomposition, $\Box_g$ reduces to $-\partial_t^2+\Delta_\gamma$ in a quasi-static regime, and the elliptic operator of the Riemannian theory is recovered as the spatial block of the Lorentzian operator.
Infrared spectrum and polarizations
Linearizing $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ and imposing harmonic gauge, the SVT decomposition shows that scalar and vector components remain constrained in the infrared. The TT sector obeys a factorized wave operator of the form $c_{EH}\Box+\beta\Box^2$, yielding a standard massless graviton pole and an additional pole at $k^2\sim\ell_\chi^{-2}$. For $\omega\ll\ell_\chi^{-1}$, only the massless pole is accessible, so the polarization content matches general relativity (two helicity-$\pm2$ modes).
Dispersion and gravitational-wave constraints
The Weyl-squared sector fixed by the Seeley–DeWitt coefficient $a_4$ induces a universal $k^4$ correction to the dispersion relation: $\omega^2=c^2k^2-\gamma\ell_\chi^2k^4+O(k^6)$ with $\gamma=1/(180\zeta)$ and $\zeta=O(1)$ encoding scheme-dependent normalizations. This maps directly to the LVK parametrization $\omega^2=c^2k^2+A_4k^4$, giving the bound $\ell_\chi\lesssim\sqrt{180\,\zeta\,A_4^{\mathrm{obs}}}$, and $\ell_\chi\lesssim\sqrt{30\,A_4^{\mathrm{obs}}}$ in the natural scheme.
The characteristic operator remains luminal at leading order, while group-velocity deviations are suppressed by $(k\ell_\chi)^2$ in the observational regime.
Relation to the Cosmochrony program
This article is formulated as a self-contained completion of the spectral entropy approach. It connects to the broader Cosmochrony program through the pre-geometric scale $\ell_\chi$ and the infrared Einstein response, but it is intended to be evaluable independently of the full substrate construction.
References
Jérôme Beau. Causal Propagation and Gravitational Waves from Projective Spectral Dynamics. Preprint. 10.5281/zenodo.18826644