Gravitational Wave Signatures of the Discrete Voxel Lattice

If spacetime is discrete at the Planck scale, the voxel lattice leaves imprints on gravitational waves propagating across cosmological distances. This paper derives three distinctive signatures that distinguish discrete spacetime from the continuum and assesses their detectability.

What this paper provides:

· Signature 1: Modified dispersion relation. From the discrete lattice wave equation, we derive \omega^2 = c^2 k^2 - \xi \ell_P^2 c^2 k^4 + O(k^6), where \xi > 0 is determined by the lattice geometry. High-frequency gravitational waves travel slightly slower than low-frequency ones, producing a frequency-dependent time delay. For LIGO frequencies (f \sim 100 Hz), the effect is \Delta t \sim 10^{-30} s over cosmological distances—far below detectable levels.
· Signature 2: Direction-dependent dispersion. The cubic lattice breaks continuous rotational symmetry. Gravitational waves propagating along a lattice axis versus a diagonal experience slightly different dispersion. The anisotropy is suppressed by (\ell_P/\lambda)^2 \sim 10^{-70} in the LIGO band and is undetectable. The stochastic gravitational wave background would exhibit a quadrupolar anisotropy pattern with amplitude \epsilon \sim 10^{-100} at pulsar timing array frequencies—completely negligible.
· Signature 3: Stochastic background from Planck-mass remnant mergers. The canvas model predicts dark matter consists of Planck-mass black hole remnants (M_{\text{rem}} \sim M_P). Their mergers throughout cosmic history produce a stochastic gravitational wave background with a characteristic spectrum \Omega_{\text{GW}}(f) \sim \alpha_0 (f/f_{\text{char}})^{2/3}, where f_{\text{char}} \sim 1/t_P \sim 10^{43} Hz and \alpha_0 \approx 1/140. The low-frequency tail extends into the pulsar timing array band (f \sim 10^{-8} Hz) but predicts \Omega_{\text{GW}} \sim 10^{-42}—33 orders of magnitude below the observed nanohertz background (NANOGrav, EPTA). The observed signal is astrophysical (supermassive black hole binaries), not cosmological.

The sobering conclusion:

All three signatures are suppressed by powers of \ell_P/\lambda, where \lambda is the gravitational wave wavelength. For astrophysical gravitational waves, this suppression is 10^{-38} to 10^{-55}. The Planck scale is 25 orders of magnitude below the electroweak scale and 35 orders of magnitude below the LIGO band. Direct detection of discrete spacetime signatures through gravitational wave observations is not feasible with current or foreseeable technology.

What this means for tests of quantum gravity:

This negative result is important. It establishes that the discrete spacetime hypothesis does not conflict with gravitational wave observations, and it directs experimental efforts toward signatures that are testable: the tensor-to-scalar ratio r \ll 0.01 (cosmology), the \pi/2 waveform asymmetry (laboratory wave-intersection experiments), and cosmological constraints on the dark energy equation of state.

The Planck scale remains hidden. But the physics it determines—the constants of nature, the dimensionality of space, the gauge group of the Standard Model—is visible everywhere. The challenge is not to see the lattice directly, but to recognize its imprint on the laws of physics.

Keywords: gravitational waves, discrete spacetime, voxel lattice, Planck scale, modified dispersion, Lorentz invariance violation, anisotropic dispersion, stochastic background, Planck-mass remnants, dark matter, pulsar timing arrays, LIGO, quantum gravity signatures