Einstein Master Equation Calculator

What this page does: lets you operate a GR–Quantum–Photon–CST self-consistency test. Enter tolerances & physical inputs, click Run, and read the residuals, stability, and validation badges. Numerics are placeholders (toy damping) so you can test the workflow. Replace the 3 hook functions (grStep, qmStep, cstFeedback) with your solvers to verify a physical system.
How to use
  1. Keep the example values or enter your own: steps, Δt, tolerances (εGR, εQM, εΔC), initial residuals, PD gains.
  2. Set photon–gravity inputs: base flux F₀, variability δI, redshift z, magnification M; optionally paste event times (e.g., solar flares) to drive flux via a kernel intensity.
  3. Click Run Simulation. If all residuals drop below tolerances for the hold window, you’ll see green badges (for this toy model).
  4. To make it a real verifier, swap in your GR/QM/CST kernels at the hooks—no UI changes needed.
Example that “solves” with toy kernels: N=500, Δt=0.02, ε’s=1e-6, RGR(0)=0.08, RQM(0)=0.06, ΔC(0)=0.10, kp=1.6, kd=0.3, F₀=1361 W/m², δI=0.03, z=0.0001, M=1.00.
Einstein “Master Equation” (coupled system)
Einstein field equations
\[ G_{\mu\nu} \;=\; 8\pi G\,T_{\mu\nu} \] Linearized wave form (Lorenz gauge, $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$):
\[ \square\,\bar h_{\mu\nu} \;=\; -16\pi G\,S_{\mu\nu}, \qquad S_{\mu\nu}=\langle \hat T_{\mu\nu}\rangle + T^{\text{class}}_{\mu\nu} + T^{\text{EM}}_{\mu\nu} + F^{\text{CST}}_{\mu\nu} \] EM (photons) via flux (effective):
\[ F_{\mathrm{eff}}(t)=\frac{F_0\,[1+\delta I + \delta I_{\text{events}}(t)]\,M}{(1+z)^4}, \qquad T^{\text{EM}}_{\mu\nu}\sim F_{\mathrm{eff}}/c \]
Quantum (Lindblad master equation)
\[ \dot{\hat\rho} = -\frac{i}{\hbar}\big[\hat H(g,A_\mu),\hat\rho\big] + \sum_j \kappa_j \mathcal D[L_j]\hat\rho + \mathcal L_{\text{CST}} \]
CST feedback (PD law)
\[ F^{\text{CST}}_{\mu\nu} = -k_p\,\Delta h_{\mu\nu}-k_d\,\Delta \dot h_{\mu\nu}, \quad \Delta h_{\mu\nu}=h_{\mu\nu}-h^{\text{target}}_{\mu\nu}(t_{\text{CST}}) \]
Point-process intensity (events → flux)
\[ \hat\lambda(t)=\sum_i K_h(t-t_i), \quad \delta I_{\text{events}}(t)=c_\lambda\,\hat\lambda(t) \]
Fix Lorenz gauge & monitor constraints Renormalize vacuum energy Avoid hidden power injection
Inputs

Simulation & Physics Parameters

This converts λ̂(t) into δIevents(t).
Idle — awaiting run
Run Log

Residuals vs. Time

Self-consistency: — Stability: — Physicality: —
Gauge: — ∇·T: — CFL: —
Fields

hμν Snapshots & Spectra (illustrative)

Snapshot: pre-control

Snapshot: post-control

Spectrum: pre

Spectrum: post

Energy

Input Power vs. Curvature Change (Efficiency)

Efficiency: — Total power: —
Validation

Observable Checks

TestTargetMeasuredStatus
Solar-system orbits (fractional err/orbit)< 1e-8——
Cosmology (FRW homogeneous limit)match——
Lab EM cavity (Δf vs small h)linear——
What it does & How to enter data

Einstein Master Equation — Purpose

The calculator couples four pieces into one loop: (1) GR curvature via the Einstein equations, (2) Quantum state evolution via a Lindblad master equation, (3) Photon forcing via flux (EM stress–energy), (4) CST feedback control that damps metric error. When all three residuals fall below tolerance and stay there (hold window), the system is self-consistent for those inputs.

How to fill each label

  • Steps (N): number of integration steps (e.g., 500).
  • Δt: time step (dimensionless here; obey CFL in real solvers), e.g., 0.02.
  • εGR, εQM, εΔC: tolerances for the GR residual, QM residual, and control error (e.g., 1e-6).
  • Consecutive steps for “solved”: how many steps in a row must be under tolerance (e.g., 50).
  • RGR(0), RQM(0): initial residual magnitudes at the start.
  • ΔC(0): initial metric error norm (toy scalar here).
  • h-spectrum imbalance: 0–1 slider that shapes the pre/post spectra visuals.
  • PD gains kp, kd: proportional/derivative gains for CST control (start low).
  • Photon base flux F₀ [W/m²]: e.g., 1361 at 1 AU (TSI).
  • Variability δI (fraction): slow baseline modulation (0.00–0.05 typical).
  • Lensing M: magnification factor (≈1 near Earth; >1 if lensed).
  • Redshift z: gravitational/cosmological redshift (near-Earth ≪1).
  • Noise τ, σ: Ornstein–Uhlenbeck correlation time (s) and amplitude (fraction) for jitter.
  • Event times (s): comma/space list of flare or burst times to spike flux.
  • Kernel & h: choose Gaussian / Epanechnikov / Exponential; set bandwidth/decay in seconds.
  • cλ: scales the event intensity λ̂(t) into fractional flux δIevents.

After you click Run Simulation, check badges and plots. To turn this into a proof for a real system, replace the three hook functions with your integrators.

Examples

Quick Fills (try these, then Run)

1) Sun–Earth (quiet Sun)

Stable baseline, minimal events; should damp smoothly.

2) Sun–Earth (flare events)

Adds bursty event times and larger variability.

3) Lab EM cavity (minimal flux)

Toy lab case: tiny flux & tighter tolerances.

After filling, press Run Simulation. Tweak gains or tolerances if residuals don’t converge with the toy kernels.

Example that passes all tests

Click the button to auto-fill parameters that (with these toy kernels) pass:
Solar-system orbits < 1e-8, FRW limit “match”, and Lab cavity “linear”. Then press Run Simulation.