Experimental Signatures of Temporal Geometry

Why experimental signatures matter

Paper I showed that treating the lapse \(N=e^{\Phi}\) as primary recovers all of GR while clarifying its structure. But new perspectives need experimental grounding. The flux law \(\partial_t\Phi = -(G/c^4)L/r\) isn't a math trick, it's a sharp prediction about how spacetime responds to energy flow. Every clock becomes a \(\Phi\)-detector, and quantum systems provide unprecedented sensitivity.

The core observable: time's response to flux

When energy flows outward (luminosity \(L > 0\)), exterior clocks slow down. A direct, measurable redshift drift. The signature is unmistakable: it's locked to the flux direction and scales as \(1/r\). No flux, no drift.

Lever A: Flux-driven redshift drift

Any energy flux that crosses your clock's radius affects its rate. For a source with luminosity \(L(t)\) at distance \(r\):

where \(y = \ln(\nu_\infty/\nu_r)\) is the logarithmic frequency ratio. The detection pipeline:

1. Measure the flux \(L(t)\) at the clock location
2. Look for correlated drift in clock frequency
3. Verify the signature: sign-locked to flux direction, \(1/r\) scaling
4. Extract the coupling strength via matched filtering

Lever B: Quantum phase accumulation

Every quantum system accumulates phase according to its worldline:

Temporal fluctuations \(\delta\Phi\) add phase noise. In an interferometer with interrogation time \(T\):

where \(S_\Phi(0)\) is the zero-frequency spectral density of temporal fluctuations. Higher energy systems (\(\omega^2\) scaling) are more sensitive. Every massive particle is a \(\Phi\)-clock.

Case studies: from lab to cosmos

• Lab calibration (1 MJ pulse, 10 m): \(\partial_t\Phi \sim 10^{-32}\) s\(^{-1}\)—validates pipeline but far below detection
• Sgr A* flare (\(10^{29}\) W, 8 kpc): \(\partial_t\Phi \sim 10^{-29}\) s\(^{-1}\)—undetectable with current technology
• Galactic supernova neutrino burst (\(3 \times 10^{45}\) W, 10 kpc): \(\partial_t\Phi \sim 10^{-20}\) s\(^{-1}\)—approaching optical clock sensitivity!

The galactic supernova campaign

A core-collapse supernova in our galaxy releases \(\sim 10^{53}\) ergs in neutrinos over ~10 seconds. At 10 kpc:

This is within an order of magnitude of current optical clock stability (\(10^{-19}\) at 1000 s). The detection strategy:

1. Trigger on SNEWS 2.0 neutrino alerts
2. Record clock data ±6 hours around burst
3. Cross-correlate network of clocks at different distances
4. Look for the smoking gun: sign-locked drift scaling as \(1/r\)

Implementation protocols

For flux measurements (Lever A):

1. Align timing: account for light-travel delay between source and clock
2. Whiten data by the clock's noise spectrum \(S_y(f)\)
3. Apply matched filter with template \(s(t) = \int L(t')dt'\)
4. Validate with null tests: shuttered sources, time slides

For quantum experiments (Lever B):

1. Choose pulse sequence (Ramsey, Mach-Zehnder)
2. Measure visibility \(V(T)\) vs interrogation time
3. Extract bounds on \(S_\Phi(\Omega)\) from visibility loss
4. Separate gravity-gradient noise from temporal fluctuations

No new degrees of freedom

Important: \(\delta\Phi\) is not a propagating scalar field. In the ADM formalism, the lapse and shift are Lagrange multipliers enforcing constraints. After linearization, only the two transverse-traceless graviton modes propagate. The \(\delta\Phi\) correlator is the constraint-projected \(h_{00}\) component. Same physics, clearer bookkeeping.

What makes this approach unique

• Sharp predictions: The flux law gives exact formulas, not order-of-magnitude estimates
• Universal coupling: Every clock responds to \(\Phi\); heavier clocks are better detectors
• Clean signatures: Sign-locked, \(1/r\) scaling distinguishes signal from noise
• Near-term feasibility: Galactic supernovae are rare but within reach of current technology
• Quantum advantage: Interferometers gain \(\omega^2\) enhancement over classical clocks

Experimental summary

1. Classical channel: Energy flux causes measurable redshift drift in atomic clocks
2. Quantum channel: Temporal fluctuations reduce interferometer visibility
3. Key formula: \(\partial_t\Phi = -(G/c^4)L/r\) ties flux to time curvature
4. Best target: Galactic supernova neutrino burst (~once per century)
5. Current limits: \(S_\Phi(0) \leq 10^{-33}\) s from atomic interferometry

Does this violate known physics?

No. These are standard GR predictions written in lapse-first variables. The experimental signatures follow directly from Einstein's equations. We're just organizing the calculation differently. What's new is recognizing that modern quantum sensors have the precision to detect these effects for astrophysical sources.