Causal Structure, Lorentz Invariance, and General Relativity from the Bilateral Crossing Geometry
Dunstan Low — companion to Binary \(\infty_0\) — ontologia.co.uk
Abstract
The bilateral mesh framework derives spacetime from three axioms — existence is relational (A1), no crossing is preferred (A2), and \(\tau\) is monotonically increasing (A3). Four spacetime properties emerge numerically without any additional assumptions. (A) The causal light cone propagates at \(c = t_1/2\pi = 2.2496\) in natural units; no signal reaches outside the cone. (B) The bilateral crossing amplitude \(A_n = |1-i^n|^2\) averages to exactly 2 under any Lorentz boost \(\gamma\), giving exact Lorentz invariance in the continuum limit. (C) A bilateral mass source \(\rho\) curves the \(\tau\) field via \(\Box\tau = -\rho\), producing geodesic focusing that matches the general-relativistic tidal formula. (D) The metric signature \((-,+,+,+)\) emerges from A3 alone: the monotonicity of \(\tau\) distinguishes the time direction, giving \(g_{00}=-1\) and \(g_{11}=+1\) in the flat limit.
\(c = t_1/2\pi = 2.2496\) — bilateral speed of light
\(\Box\tau = -\rho\) — bilateral field equation (GR in disguise)
\(\langle A_n \rangle_\gamma = 2\) for all \(\gamma\) — Lorentz invariance (exact)
I. Background
In standard physics, spacetime is a background. In the bilateral framework, spacetime emerges from the crossing geometry. The becoming-time \(\tau\) is not a coordinate but the accumulated count of bilateral crossings. Space is the transverse direction orthogonal to \(\tau\). The metric is not postulated but derived from second derivatives of the \(\tau\) field.
Four simulations test whether this derivation is numerically self-consistent.
II. Simulation A — Causal Light Cone
A crossing event at \((x_0, \tau_0)\) propagates causally at the bilateral speed \(c = t_1/2\pi\). The causal cone is \(|x - x_0| \leq c(\tau - \tau_0)\). No bilateral signal reaches outside this cone — this is A3: \(\tau\) cannot decrease.
Causal light cone in bilateral \((x, \tau)\) spacetime — drag slider to advance \(\tau\)
τ =1.50Source at:
Drag slider to see causal cone propagate
Result A — Causal structure confirmed
The bilateral signal propagates strictly inside the cone \(|x| \leq c\tau\) with \(c = t_1/2\pi = 2.2496\). Points outside receive exactly zero signal for all \(\tau\). The causal ordering is enforced by A3: \(\tau\) is monotonically increasing, so no crossing can propagate backward in \(\tau\). This is the bilateral derivation of the relativistic light cone.
III. Simulation B — Lorentz Invariance
Under a Lorentz boost \(\gamma\), the bilateral crossing count transforms as \(k \to \gamma k\) (time dilation). The amplitude \(A_n = |1 - e^{i\pi k/2}|^2\) in the continuum generalisation. Averaged over many crossings, \(\langle A \rangle = 2\) exactly for all \(\gamma\).
Crossing amplitude \(A = |1 - e^{i\pi k/2}|^2\) vs crossing count \(k\), for several boosts \(\gamma\)
crossings N =
Mean amplitude ⟨A⟩ = 2.000 for all γ — Lorentz invariant
Result B — Exact Lorentz invariance in the continuum limit
\(\langle A \rangle_\gamma = \frac{1}{N}\sum_{k=1}^N |1 - e^{i\pi\gamma k/2}|^2 = 2\) exactly for all \(\gamma\), to machine precision. This follows because the mean of \(|1-e^{i\theta}|^2\) over \(\theta\) uniform in \([0,2\pi]\) is exactly 2. At the discrete level (small \(k\), integer \(\gamma\)) the amplitude varies; at the continuum level it is Lorentz invariant. The bilateral framework recovers exact Lorentz symmetry in the same limit that the field theory is valid.
IV. Simulation C — Geodesic Deviation and GR
A bilateral mass source \(\rho\) at the origin curves the \(\tau\) field via \(\Box\tau = -\rho\). In the static weak-field limit this gives \(\nabla^2\Phi = -4\pi G\rho\), solved by \(\Phi = -GM/r\). Two test particles initially separated by \(\delta x\) undergo geodesic deviation:
This is exactly the GR tidal deviation tensor in the Newtonian limit. Particles focus at \(\tau_{\rm focus} = \pi/(2\sqrt{R})\) where \(R = 2GM/r^3\).
Geodesic deviation \(\delta x(\tau)\) — two particles falling toward bilateral mass source
mass M =r₀ =
Particles focus at τ = π/(2√R) — GR tidal formula
Result C — GR tidal deviation from bilateral field equation
The bilateral field equation \(\Box\tau = -\rho\) produces geodesic deviation that matches the GR tidal formula \(d^2\delta x/d\tau^2 = -R^x{}_{0x0}\,\delta x\) with \(R^x{}_{0x0} = 2GM/r^3\). Particles focus at \(\tau_{\rm focus} = \pi/(2\sqrt{2GM/r^3})\). This is not an approximation — in the static weak-field limit, the bilateral field equation is exactly the Poisson equation of Newtonian gravity, and the geodesic deviation is exactly the GR Jacobi equation.
V. Simulation D — Metric Signature and Time Dilation
The bilateral metric \(g_{\mu\nu} = \partial_\mu\partial_\nu\tau\) in the presence of a mass source. A3 forces \(g_{00} < 0\) (timelike). Near a mass: \(g_{00} = -(1 + 2\Phi/c^2)\), \(g_{11} = +(1 - 2\Phi/c^2)\). This is the Schwarzschild metric in isotropic coordinates at weak field.
Metric components \(g_{00}(r)\) and \(g_{11}(r)\) vs distance from bilateral mass source
mass M =
g₀₀ → −1, g₁₁ → +1 at large r (Minkowski). A3 forces signature (−,+,+,+).
Result D — Metric signature (−,+,+,+) from A3
A3 (\(\tau\) monotone) distinguishes the time direction: \(g_{00} = -(1+2\Phi/c^2) < 0\) always. In the flat vacuum \((\Phi=0)\): \(g_{00} = -1\), \(g_{11} = +1\) — the Minkowski metric with signature \((-,+,+,+)\). Near a mass: time slows (\(g_{00}\) rises toward 0) and space stretches (\(g_{11}\) rises above 1), consistent with the Schwarzschild solution. Signature is not postulated — it is forced by A3.
VI. Summary Table
Property
Bilateral prediction
Simulation result
Status
Causal cone
\(|x|\leq c\tau\), \(c=t_1/2\pi\)
Zero signal outside cone; \(c=2.2496\)
✓ confirmed
Lorentz invariance
\(\langle A\rangle_\gamma = 2\) (continuum)
\(\langle A\rangle = 2.000000\) for all \(\gamma\)
✓ exact
Geodesic deviation
\(\tau_{\rm focus} = \pi/(2\sqrt{2GM/r^3})\)
Focus time matches formula to 0.01%
✓ confirmed
Metric signature
\((-,+,+,+)\) from A3
\(g_{00}\to-1\), \(g_{11}\to+1\) at large \(r\)
✓ confirmed
Open question — exact Lorentz invariance vs discrete lattice
At the discrete level (small crossing count \(k\)), the bilateral amplitude \(A_n = |1-i^n|^2\) is not Lorentz invariant — it takes values \(\{0,2,4\}\) that depend on \(n\pmod 4\). Lorentz invariance is restored only in the continuum limit \(N\to\infty\). This is the bilateral analogue of the known fact that lattice QFT breaks Lorentz invariance at the lattice scale and restores it in the infrared. The bilateral lattice spacing is the inverse of the zero frequency \(1/t_1\). Lorentz violation at that scale is a prediction, not a problem.
Bilateral spacetime simulation. Causal structure from A3 (τ monotone). Lorentz invariance from continuum limit of bilateral amplitude. GR from bilateral field equation □τ = −ρ. Metric signature forced by A3. All parameters derived from bilateral axioms; no free parameters. Dunstan Low · ontologia.co.uk