Gravity / Waves
Gravitational waves travel on the light cone.
Long waves on the lattice move at the speed of light in every direction, with corrections of order (kb)², where b is the lattice spacing. At the frequencies gravitational-wave detectors see, the correction is more than thirty orders of magnitude below the bound measured with GW170817; along one direction the paper computes its exact leading term, fifty orders below.
On the light cone, corrections (kb)²: provedκ = 1: identified, not derived
From the lattice to a detector
Live · computed from Corollary 5.3 and Theorem 5.5
106 PLANCK LENGTHS
5,000 HZ
At these values
- Wavelength
- –lattice spacings
- k b
- –2πfb/c
- Plus
- –slower than light, (59/24)(kb)²: – orders below the GW170817 bound
- Cross
- –faster than light, (5/24)(kb)²
Over the Hubble distance, 1.4 × 10²⁶ m
- Drift
- –radians of phase, plus mode
- Split
- –radians between the two polarizations
Along the paired direction, with the exact leading terms of Theorem 5.5. Light here is the null cone of the lattice metric: comparing with GW170817 assumes photons travel on the same cone, an identification about the photon sector that the paper does not derive.
With JavaScript on, set a lattice spacing and a frequency. At 10⁶ Planck lengths and 5,000 Hz the plus mode along the paired direction differs from light by about 7 × 10⁻⁶⁶, and every mode in every direction by at most 2 × 10⁻⁵⁰.
The numbers
How close to light.
Every direction
2×10−50
Every non-gauge mode up to 5,000 Hz, for a spacing up to 10⁶ Planck lengths, has a phase speed this close to light’s; over the Hubble distance its phase drifts by at most 3 × 10⁻²⁸ rad.
Earlier paper, Corollary 5.3
Plus, paired direction
7×10−66
(59/24)(kphb)², exact at leading order: the plus mode’s phase drifts by about 10⁻⁴³ rad over the same distance.
Earlier paper, Corollary 5.3 and Theorem 5.5
Two polarizations
1.2×10−43
Radians: the most the plus and cross modes can part along the paired direction over 1.4 × 10²⁶ m, at b ≤ 10⁶ Planck lengths and f ≤ 5,000 Hz.
Earlier paper, Theorem 5.5
GW170817
7×10−16
The measured bound on the speed of gravity the paper compares with: −3 × 10⁻¹⁵ ≤ v/c − 1 ≤ 7 × 10⁻¹⁶.
Earlier paper, after Corollary 5.3
The lattice scale
Short waves show the lattice.
Along the paired direction u = (z+t, z+t, z−t, z−t) the wave number is |k| = 2t and the frequency ω = 2z/√3, and each polarization satisfies a printed polynomial equation. Solved exactly here, they leave the light cone only near the lattice scale.
- Light cone
- Plus
- Cross
- Cross, returning root
Live · solved from the polynomials of Appendix F
0.400
Wavelength 15.7 lattice units.
| Crest speed | Packet speed | Leading term | |
|---|---|---|---|
| Plus | – | – | – |
| Cross | – | – | – |
Speeds in units of light. The leading terms are Theorem 5.5’s near-cone crest speeds, 1 − (59/24)|k|² and 1 + (5/24)|k|², shown up to |k| = 0.7.
How it works
Why long waves keep to the cone.
1 · Every direction
The tree-level cone holds in every direction.
The free waves of the flat κ = 1 lattice travel on the light cone in every direction, up to a relative correction of order (kphb)², where kph is the physical wave number. Near the origin of the Brillouin zone there is no slow branch, no fast branch and no extra low-frequency mode. The constants are rigorous and crude: products of row and entry bounds of exact rational tables.
Earlier paper, Theorem 5.1 and Corollary 5.2
2 · Birefringence
Along the paired direction, plus is slower and cross faster.
Near the cone every zero mode is a vertex move plus a plus-polarized mode plus a cross-polarized mode, and each exists. Their crest speeds are 1 − (59/24)|k|² and 1 + (5/24)|k|² and their packet speeds 1 − (59/8)|k|² and 1 + (5/8)|k|² at leading order: the lattice graviton is birefringent along this direction. The cross mode’s speed differs from 1 by exactly (5/24)|k|² at leading order, so no bound smaller than order |k|² holds in every direction.
Earlier paper, Theorem 5.5 and Appendix F
3 · What is open
Every direction’s exact coefficients are open.
Whether every direction has a (kphb)² law with definite coefficients, beyond the bound of Corollary 5.2, is open; so is whether a bound as small as the paired direction’s holds in every direction. And light here is the null cone of the lattice metric: that photons travel on the same cone is a question about the photon sector, which the paper does not address.
Earlier paper, Section 5
Sources: the earlier paper is Linearized Einstein gravity from Lorentzian Regge calculus on the Kuhn lattice; the quantum gravity paper is Quantum Gravity from the Recognition Ledger, version 1.2.
Words on this page, in plain terms: Kuhn lattice · light cone · κ (kappa) · Brillouin zone.
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