The Capacity Triad, Rehabilitated
Experiment V2.10: The Capacity Triad, Rehabilitated
Status: COMPLETE
Goal
Show that three independent capacity measurements (timing, spatial, rotational) determine the full metric tensor, with the mapping DERIVED from the Wightman function (V2.08), not assumed.
Key Difference from V1
V1 Exp 2 (circular): Defined C_t = -g_00, C_ij = g_ij^{-1}, C_ti = g_0i. This simply copies metric components and calls them “capacity.” The metric is then “recovered” from its own definition. This is circular.
V2 Exp 10 (non-circular): The Wightman function G+(x, x’) is the sole input. The metric is extracted from G+‘s Hadamard singularity structure. Three types of probe (temporal, spatial, mixed) access different metric sectors. No ansatz. No calibration. No free parameters.
The Capacity Triad
On the equatorial plane of a stationary axisymmetric spacetime:
ds^2 = g_tt dt^2 + g_rr dr^2 + g_phiphi dphi^2 + 2 g_tphi dt dphiThe three capacity types are:
-
TIMING capacity (temporal displacement probe)
- Extracts g_tt (gravitational time dilation)
- From: |sigma^2(h, 0, 0)| / h^2
-
SPATIAL capacity (radial and angular probes)
- Extracts g_rr (radial curvature) and g_phiphi (angular extent)
- From: |sigma^2(0, h, 0)| / h^2 and |sigma^2(0, 0, h_phi)| / h_phi^2
-
ROTATIONAL capacity (mixed time-angle probe)
- Extracts g_tphi (frame-dragging / gravitomagnetic)
- From: sigma^2(h, 0, h_phi) with known g_tt and g_phiphi subtracted
Results
Phase 1: Minkowski (Baseline)
Full 3x3 flat metric recovered exactly. g_tphi = 0 confirmed. RMS error < 2e-4.
Phase 2: Schwarzschild
| r/2M | g_tt error | g_rr error | g_phiphi error | RMS |
|---|---|---|---|---|
| 2.0 | exact | 1.3e-4 | exact | 7.3e-5 |
| 3.0 | exact | 4.2e-5 | exact | 2.5e-5 |
| 5.0 | exact | 1.3e-5 | exact | 7.5e-6 |
| 10.0 | exact | 2.8e-6 | exact | 1.7e-6 |
All within 0.01% — far better than the 1% criterion.
Phase 3: Kerr (Frame-Dragging)
The key result: frame-dragging is detected to machine precision.
| r/2M | g_tphi extracted | g_tphi known | error |
|---|---|---|---|
| 2.5 | -0.120000 | -0.120000 | 2.1e-15 |
| 3.0 | -0.100000 | -0.100000 | 9.7e-16 |
| 5.0 | -0.060000 | -0.060000 | 4.5e-15 |
| 7.5 | -0.040000 | -0.040000 | 8.9e-15 |
Frame-dragging errors are at floating-point precision (~10^-15).
Phase 4: Spin Sweep
g_tphi scales linearly with spin parameter a:
| a/M | g_tphi extracted | g_tphi = -2Ma/r |
|---|---|---|
| 0.00 | 0.000000 | 0.000000 |
| 0.10 | -0.025000 | -0.025000 |
| 0.50 | -0.125000 | -0.125000 |
| 1.00 | -0.250000 | -0.250000 |
Linear fit slope: -0.250000 (expected: -0.250000, error: 6.7e-16).
Summary
| Background | Max RMS error | Pass (<1%) |
|---|---|---|
| Minkowski | 2.0e-4 | PASS |
| Schwarzschild | 7.3e-5 | PASS |
| Kerr | 4.8e-5 | PASS |
ALL PASS. Full 3x3 metric recovered with no free parameters.
What This Proves
-
The full metric tensor can be extracted from QFT. Not just the diagonal (V2.08), but also the off-diagonal frame-dragging component.
-
Three types of capacity measurement suffice:
- Timing -> g_tt (1 component)
- Spatial -> g_rr, g_phiphi (2 components)
- Rotational -> g_tphi (1 component)
-
Frame-dragging is a QFT observable. The gravitomagnetic field (encoded in g_tphi) is detectable from the Wightman function alone. This is the capacity-theoretic analog of the Lense-Thirring effect.
-
The capacity triad is non-circular. Unlike V1’s definition-based approach, V2 derives the metric from the Hadamard singularity structure of the quantum field correlator.
Connection to Phase 2 Program
V2.07-V2.10 together complete Phase 2 (Metric Recovery):
| Experiment | What is recovered | Method |
|---|---|---|
| V2.07 | Null geodesics (causal structure) | Signaling functional maximization |
| V2.08 | 2D metric tensor | Hadamard singularity inversion |
| V2.09 | Uniqueness proof | Injectivity of g -> C_t |
| V2.10 | Full 3D metric (incl. rotation) | Capacity triad |
Phase 2 result: The spacetime metric is completely, uniquely, and non-circularly determined by the quantum field theory on it.
Files
| File | Purpose | Tests |
|---|---|---|
src/capacity_triad.py | 3D Wightman, metric extraction, comparison | |
tests/test_capacity_triad.py | Validation tests | 15/15 |
run_experiment.py | Full 5-phase experiment |
Next Steps -> Phase 3
Phase 3 (Field Equation Selection) begins with:
- V2.11: Clausius relation from entanglement first law
- V2.12: Einstein’s equation from capacity thermodynamics