Why Light and Gravity Travel at the Same Speed
In any medium, different kinds of waves travel at different speeds. Sound waves in steel travel at 6,000 m/s; shear waves at 3,200 m/s. Water waves travel at one speed on the surface, pressure waves at another through the bulk. The dc1 substrate is no different — it supports multiple wave types, and they do not all travel at the same speed.
What is remarkable is that the two wave types we actually observe — photons and gravitational waves — both propagate at c, and they do so for the same underlying reason.
The quasiparticle speed: why c is c
The speed of light in the substrate is not imposed by fiat. It emerges from the BEC quasiparticle spectrum. In the strong-coupling regime where the gap energy dominates over the Fermi energy (\Delta_0 \gg E_F), the low-energy excitations of a superfluid condensate automatically obey a Dirac-like dispersion:
E^2 = \mu^2 + c^2 p^2 \qquad \text{with} \qquad c = \frac{\hbar}{m_1\,\xi}
This spectrum is isotropic — the same speed in all directions — and it governs all low-energy excitations of the condensate, regardless of their polarisation or spin structure. Phonon-like excitations (scalar), modon-like excitations (vector), and metric perturbations (tensor) all inherit the same characteristic speed c from the BEC medium.
The substrate’s logarithmic equation of state sharpens why this c is robust. The logarithm makes the sound speed density-independent (dc^2/d\rho \equiv 0), so c cannot drift as the substrate dilutes with cosmic expansion, and it singles out a marginal background density at which the low-momentum mode is exactly massless. Exact Lorentz invariance is therefore structural — and the late-time attractor a diluting substrate self-organizes toward — rather than a condition that must be held in place by some external scaffold (see Substrate Particles § The Marginal Point).
This is the deep reason photons and gravitational waves travel at the same speed. They are different excitations of the same medium, and the medium has a single characteristic velocity. GW170817 confirmed this to extraordinary precision: the gravitational wave and its electromagnetic counterpart arrived within 1.7 seconds of each other after travelling 130 million light-years, constraining |c_\text{GW}/c_\text{EM} - 1| < 6 \times 10^{-15}.
In the substrate framework, this is not a surprise. It would be surprising if they didn’t match.
What travels at c — and what doesn’t
The substrate supports at least three distinct wave types:
Modons (photons) are nonlinear vortex dipoles — counter-rotating pairs that self-advect through the substrate. Their speed is set by the BEC quasiparticle dispersion: c = \hbar/(m_1\,\xi). This is constraint C1 in its Volovik form. The modon’s internal structure (the Bessel function matching, the L-R boundary conditions) determines its shape, not its speed — much as the shape of a water wave doesn’t determine the speed of sound.
Gravitational waves are perturbations of the effective acoustic metric. In the Barceló-Liberati-Visser framework, the acoustic geometry of a flowing superfluid gives an effective metric whose perturbations propagate at the sound speed. Since the substrate’s sound speed is c (from the same BEC dispersion), gravitational waves automatically propagate at c. The vortex lattice provides the tensor structure — the spin-2 polarisation that makes these modes gravitational rather than merely acoustic — but the speed comes from the medium, not the lattice.
Tkachenko waves are shear oscillations of the vortex lattice itself — slow, elastic modes where the vortices wobble about their equilibrium positions. In the stiff (incompressible) limit, their speed is [@baym2003]:
c_T = \sqrt{\frac{\hbar\,\Omega}{4\,m_1}} \approx 9 \;\text{km/s} \approx 3 \times 10^{-5}\,c
where \Omega = \kappa_q/(2\xi^2) is the effective 2D rotation rate (the Feynman relation applied to the lattice cell). This is five orders of magnitude below the speed of light — comparable to sound in metals. The 4 in the denominator comes from the 2D triangular lattice shear modulus (the 8\pi of Baym’s formula) — a completely different geometric factor from the 4\pi in SC2.
In superfluid helium-4, the Tkachenko speed is approximately 10^{-4} m/s — even slower relative to the helium sound speed of 238 m/s [@andronikashvili1966; @coddington2003]. The substrate’s Tkachenko waves are slow for the same reason: the lattice shear modulus C_2 \propto \Omega is tiny compared to the bulk modulus that sets the sound speed.
The Tkachenko modes are a prediction of the substrate model: they are a very low-frequency (\sim 3{,}700 Hz) oscillation of the vortex lattice with no counterpart in standard physics. Whether they have observable consequences — kHz modulation of dark matter density, second-order CMB imprints, or laboratory detection via precision interferometry at \sim 100\;\mum scales — is an open question (see Open Problems).
What SC2 actually constrains
The structural condition SC2,
\kappa_q \cdot \Omega_v = 4\pi\,c^2
is not a statement that Tkachenko waves travel at c. It is a condition on the background lattice configuration — the requirement that the vortex lattice’s circulation and density, combined with the acoustic metric, produce the correct tensor structure for linearised Einstein equations.
The 3D form \kappa_q \cdot n_1\omega_0 = 4\pi c^2 has a dimensional mismatch: LHS is [m⁻¹s⁻²], RHS is [m²s⁻²], off by [m³]. The root cause: n_1 is a 3D number density [m⁻³] while the Feynman relation operates on a 2D areal vortex density [m⁻²]. The substrate’s vortex lattice is organized into chirality-coherent 2D sheets (same-chirality orbital systems clustering through the Mexican hat mechanism; see Higgs Field), and the correct 3D→2D projection involves the inter-sheet spacing — a quantity determined by the chirality ordering thermodynamics that has not yet been computed from first principles.
The formula gives correct numerical values in SI units (verified to 0.27%) but is not a valid physical equation (confirmed by CGS cross-check: the bridge equation values diverge by 10^4 between unit systems). The naive 2D repair (using n_v^{(2D)} = 1/(\pi\xi^2)) is dimensionally correct but gives \xi \approx 82 fm — the effective quantum’s Compton wavelength, not the lattice spacing — because it misses the layered stacking structure.
The dimensionless packing-fraction form of the bridge equation, f = \rho_\text{DM}c\xi^4/\hbar = 4\pi/(K\sqrt{2}), encodes the same content with no dimensional ambiguity (both sides [1], verified to 0.18%). See the open problems WIP-15 for the path to full dimensional repair, which is connected to the open problem of deriving the Higgs VEV from substrate parameters.
Physically: for the effective spacetime geometry to include a proper spin-2 gravitational sector with the right coupling strength, the vortex density and the circulation quantum \kappa_q = h/m_\text{eff} must satisfy this specific relationship. This is a constraint on how the lattice is arranged, not on how fast its perturbations propagate.
Combined with the modon existence condition (old C1: n_1\omega_0\xi^3 = Kc, where K = j_{11}^2 + 1 = 15.682) and the superfluid relation \kappa_q = 2\pi\hbar/m_\text{eff} (from C2), SC2 uniquely determines the coherence length:
\xi_\text{SC2}^3 = \frac{\hbar\,K\,\alpha_{mf}}{2\,m_e\,c} \qquad \text{⚠️ NUMERICAL RECIPE: LHS [m³], RHS [m]}
Numerically: \xi_\text{SC2} = 96.9\;\mu\text{m}. This formula gives the correct value in SI but has a dimensional deficit of [m²] — the combined effect of the dimensional mismatches in old C1 (off by [m]) and SC2 (off by [m³]). It should be treated as a numerical recipe for computing \xi_\text{SC2} in meters. Read honestly, the n_1\omega_0 common to old C1 and SC2 cancels in this ratio, so \xi_\text{SC2} never depended on \omega_0: each condition on its own is an identity — old C1 collapses to the Volovik speed c = \hbar/(m_1\xi), and SC2 to the effective-quantum Compton clock \Omega_v = 2m_\text{eff}c^2/\hbar — and the outer rotation \omega_0 is fixed separately, in the gravity sector (see WIP-15).
This is the particle physics route to the coherence length. It contains only \hbar, m_e, c, \alpha_{mf} (determined by the Weinberg angle), and a Bessel zero j_{11}. No free parameters. The coherence length is determined by the intersection of two conditions:
- Modon structure (Bessel matching): the soliton boundary requires n_1\omega_0\xi^3 = Kc
- Metric structure (SC2): the effective Einstein equations require \kappa_q \cdot \Omega_v = 4\pi c^2
There is a completely independent cosmological route: the Volovik quasiparticle relation c = \hbar/(m_1\xi) combined with n_1 m_1 = \rho_\text{DM} and close-packing gives \xi_\text{CP} = (\hbar/(\rho_\text{DM}\,c))^{1/4} \approx 111.8\;\mu\text{m}. The two routes share no parameters beyond \hbar and c — one uses electroweak physics, the other uses the dark matter density — yet they agree through the bridge equation: n_1\xi_\text{SC2}^3 = 4\pi/(K\sqrt{2}) = 0.5666, verified to 0.18%.
The result is mesoscopic: ~100 \mum is far larger than atoms and far smaller than everyday objects. It sits in the range of far-infrared wavelengths and fine biological structures.
The helium analogy, corrected
The previous version of this section noted that helium-4 has a six-order-of-magnitude gap between its sound speed and its Tkachenko speed, and argued that the substrate must close this gap. The opposite is true: the substrate also has a large gap between its sound speed (c) and its Tkachenko speed (\sim 10 km/s) — five orders of magnitude. This is expected for any superfluid vortex lattice where the rotation rate is slow compared to phonon frequencies.
What makes the substrate special is not that the gap closes, but that the sound speed is c — determined by the BEC quasiparticle spectrum — and that the vortex lattice has exactly the right configuration (SC2) to produce a proper spin-2 gravitational sector. In helium, neither condition holds: the sound speed is 238 m/s (not a fundamental speed), and the vortex lattice has no reason to satisfy a gravitational consistency condition. The 4\pi in SC2 — from the Gauss’s law solid-angle factor — is distinct from the 8\pi in Baym’s Tkachenko formula, which comes from the shear modulus of the 2D triangular lattice. The bridge equation makes this distinction precise: the ratio 4\pi/8\pi = 1/2 lifts \xi_\text{SC2} above \xi_\text{Baym} by the factor 2^{1/3}, producing the structural 2.7% gap between the gravitational and elastic coherence lengths.
What the CMB tells us
The cosmic microwave background constrains the substrate through the properties of the primordial plasma, not through direct measurement of substrate wave speeds.
The baryon-photon sound speed is measured through the spacing of the CMB acoustic peaks. At recombination (z \approx 1100), c_s \approx c/\sqrt{3(1+R)} \approx 0.45c, where R = 3\rho_b/(4\rho_\gamma) \approx 0.63 is the baryon loading. This is standard photon-baryon physics — modons (photons) scattering off charged baryons in a tightly coupled plasma. The substrate determines c (the bare photon speed) and \rho_{DM} (which affects the gravitational potential wells the plasma falls into), but the plasma sound speed itself is set by radiation pressure and baryon inertia.
The sound horizon r_s = 147.09 \pm 0.26 Mpc (Planck 2018) integrates the baryon-photon sound speed from the Big Bang to recombination. The substrate contributes through the expansion history (which depends on \rho_{DM}), not through a modification of c_s.
The dark matter density \Omega_c h^2 = 0.1200 \pm 0.0012 constrains \rho_{DM} to about 1% precision. In the substrate model, this directly constrains m_1 and n_1 through \rho_{DM} = n_1 m_1. Combined with c = \hbar/(m_1\xi), the Planck measurement of \rho_{DM} provides the primary observational input to the bridge equation — the zero-parameter relation n_1\xi_\text{SC2}^3 = 4\pi/(K\sqrt{2}) that connects the cosmological \rho_\text{DM} to the particle physics parameters \sin^2\theta_W and m_e. If exact, this relation reduces the independent parameter count of SM + ΛCDM by one: \rho_\text{DM} is determined by electroweak physics.
Gravitational wave speed from GW170817 constrains |c_\text{GW}/c - 1| < 6 \times 10^{-15}. In the substrate model, both speeds are set by the same BEC dispersion relation, so exact equality is predicted. This constraint is automatically satisfied — it would require fine-tuning to violate it.
None of these observations constrain the Tkachenko speed or require it to equal c. The CMB is consistent with a substrate that has a single fast mode (c, carrying both photons and gravitational waves) and a separate slow mode (c_T \sim 10 km/s, the vortex lattice shear). The slow mode has no direct CMB signature because it couples to neither the photon field nor the gravitational wave sector at leading order.
Summary of substrate wave modes
| Mode | Speed | Mechanism | Observable as |
|---|---|---|---|
| Sound / quasiparticles | c | BEC dispersion: c = \hbar/(m_1\xi) | Sound in the primordial plasma |
| Modons (photons) | c | Nonlinear vortex dipole self-advection | Electromagnetic radiation |
| Metric perturbations (GWs) | c | Acoustic metric perturbations | Gravitational waves |
| Tkachenko (lattice shear) | \sim 9 km/s (3 \times 10^{-5}c) | Vortex lattice elasticity (8\pi shear) | No known counterpart (WIP-13) |
| Outer rotation | \sim 800 km/s (0.003c) | Lattice-scale vorticity (\omega_0 \xi) | CDM-MOND transition? |
The first three share the speed c because they are all excitations of the same BEC medium. The last two are internal substrate modes with no direct observational counterpart in standard physics.
The derivation of \xi_\text{SC2} from C1 + SC2 is unchanged. What has changed is the physical interpretation: SC2 is a background configuration condition (for the effective metric), not a wave-speed matching condition. The 4\pi in SC2 (vs the 8\pi in Baym’s Tkachenko formula) reflects its origin in the gravitational coupling of the effective metric, not in fluid shear mechanics. The Tkachenko wave speed in the substrate is \sim 10 km/s, five orders of magnitude below c, consistent with the substrate being deep in the stiff (incompressible) regime of vortex lattice dynamics. The coincidence of photon and gravitational wave speeds is automatic from the BEC quasiparticle spectrum — no additional speed matching is required.
The bridge equation connects these results to cosmology: the same \xi determined here by particle physics (SC2 + modon matching) is independently determined by the dark matter density (Volovik + close-packing), and the two routes agree through f = 4\pi/(K\sqrt{2}) = 0.5666 — a zero-parameter relation verified to 0.18%. Each factor in f traces to a distinct physical origin: 4\pi from Gauss’s law, K from Bessel matching, and 1/\sqrt{2} from the GP kinetic energy. This is the substrate framework’s most concrete cross-domain result.