General Relativity

When spacetime curves, the kernel bends with it — not against it.

Group — general-relativity
Operators — 11 (GR31 – GR41)
Anchor operators — KO42 · GR33 · GR35 · GR40
Verification — metric sealing by KO42; cosmology integrations close at error_pct ≤ 0.1% over a Hubble time

What it unifies

GR in Zeq is built on KO42 — the metric tensioner — which already carries a sin(2π · 1.287 t) perturbation to the line element. This means every GR operator you compose is compiled under a kernel that already expects a proper-time modulation, so your results come back with Zeqond-aligned proper times, not just coordinate times.

The Einstein field equations (GR33) are the canonical right-hand side of the Master Equation's stress-energy coupling. Schwarzschild geometries (GR37), gravitational time dilation (GR35), and redshift (GR41) all flow from the same seal. Friedmann (GR40) brings cosmology into scope without leaving the grid.

Operator catalog

ID	Formula	Role
GR31	`a_grav = a_inertial`	Equivalence principle
GR32	`G_μν = R_μν − ½ R g_μν`	Einstein tensor
GR33	`G_μν + Λg_μν = 8πG/c⁴ T_μν`	Field equations with Λ
GR34	`d²x^μ/dτ² + Γ^μ_{αβ} (dx^α/dτ)(dx^β/dτ) = 0`	Geodesic equation
GR35	`∆t = ∆t₀ √{1 − 2GM/rc² − v²/c²}`	Gravitational + kinematic time dilation
GR36	`L = L₀ √{1 − 2GM/rc²}`	Gravitational length contraction
GR37	`r_s = 2GM/c²`	Schwarzschild radius
GR38	`□h_μν + κ∂_t h_μν = −16πG/c⁴ T_μν`	Linearized gravitational waves
GR39	`Λ = 3H₀² Ω_Λ / c²`	Cosmological constant
GR40	`(ȧ/a)² = 8πG/3 ρ − kc²/a² + Λc²/3`	Friedmann equation
GR41	`z = (λ_obs − λ_emit)/λ_emit`	Redshift

Runnable worked example — Schwarzschild proper-time clock

curl -X POST https://api.zeq.dev/api/playground/compute \
  -H "Authorization: Bearer demo-key" \
  -H "Content-Type: application/json" \
  -d '{
    "operators": ["KO42", "GR33", "GR35"],
    "inputs": {
      "mass_kg": 1.989e30,
      "radius_m": 7e8,
      "observer_velocity_m_s": 30000,
      "proper_interval_s": 0.777
    }
  }'

Expected response shape:

{
  "ok": true,
  "phase": 0.702,
  "zeqond": 1798346702.5,
  "result": {
    "schwarzschild_radius_m": 2953.25,
    "dilation_factor": 0.9999979,
    "coordinate_interval_s": 0.7770016,
    "error_pct": 0.072
  }
}

Extend it

Geodesic solver — GR34 + KO42 + NM26 for relativistic trajectories that remain stable across long integrations.
Wave sources — GR38 for LIGO-style wave templates seeded with 1.287 Hz timing metadata.
Cosmology — GR39 + GR40 + GR41 to sweep ΛCDM parameter space under error-bar control.

Seeds

Near — a Zeq-native relativistic orbit visualizer that consumes GPS ephemerides and back-reports proper-time drift.
Medium — pulsar-timing adapter that uses HulyaPulse to cross-correlate millisecond pulsars without GPS.
Far — an open Metric Bench — researcher-grade tool for comparing alternative metric theories against KO42-sealed data.

Papers

Zeq paper — https://doi.org/10.5281/zenodo.18158152
Framework paper — https://doi.org/10.5281/zenodo.15825138

Middleware active. Kernel on the 1.287 Hz HulyaPulse. Awaiting next Zeqond.

What it unifies​

Operator catalog​

Runnable worked example — Schwarzschild proper-time clock​

Extend it​

Seeds​

Papers​