Fibrewise Born–Infeld Capacity Lapse and Gravitational Time Dilation

Overview

The paper TempProj constructed the temporal residual map $\mathcal{P}_\tau^{\mathrm{Q5b}}$ in the homogeneous (flat-space) regime where the Born–Infeld capacity radius is globally $R = 1$. The present paper extends this construction to non-homogeneous regimes, where a localised spectral occupancy $\epsilon(x)$ is present in the fibre over position $x$.

The key structural result is that a localised spectral occupancy reduces the local BI capacity radius to $R(x)^2 = 1 - \epsilon(x)$. The fibrewise temporal residual map factorises cleanly as $R(x)\sqrt{f(x,n)(2-f(x,n))}$, where $R(x)$ carries all local physics and the second factor is the universal Lorentzian structure from TempProj. Gravitational time dilation $d\tau/dn|_x = R(x)/\gamma$ emerges directly from the local capacity budget, and the metric identification $R(x)^2 = -2g_{\tau\tau}(x)$ connects the framework to the Gravity paper.

Scope statement. The fibrewise extension is structural: it imports the metric from the Gravity paper and does not independently derive Einstein's equations. The homogeneous case $R(x) \equiv 1$ reduces exactly to TempProj as a corollary.

Conceptual interpretation: unified capacity arbitration

The Born–Infeld budget is a local capacity arbitration principle. Both types of relativistic time dilation are manifestations of the same trade-off, separated in the product:

$\dfrac{d\tau}{dn}\bigg|_x = R(x)\cdot\dfrac{1}{\gamma}$

The factor $1/\gamma$ measures the capacity diverted into spatial mobility (kinematic time dilation).
The factor $R(x)$ measures the capacity already depleted by the local spectral configuration (gravitational time dilation).

In this reading, curvature is the effective metric expression of local projective-capacity depletion. Special-relativistic time dilation is the homogeneous capacity trade-off; gravitational time dilation is its fibrewise generalisation.

Epistemic boundary. The present result does not derive Einstein's equations. It gives the fibrewise capacity interpretation of the metric lapse once the effective metric is supplied by the infrared Einstein response. The open problem LC-O2-O1 -- deriving $\epsilon(x) = 1 - R(x)^2 = 1 + 2g_{\tau\tau}(x)$ within the capacity-flow language without importing the Einstein equations -- is closed by the companion paper LCIIo1.

Main result

The fibrewise temporal residual map is:

$\mathcal{P}_{\tau,x}^{\mathrm{Q5b}}\,\Sigma^{\mathrm{tot}}(x,n) = R(x)\,\sqrt{f(x,n)\bigl(2-f(x,n)\bigr)}$

with $R(x)^2 = -2g_{\tau\tau}(x)$. Gravitational time dilation: $d\tau/dn|_x = R(x)/\gamma$. In the weak-field regime: $R(x) \approx 1 + \Phi_N(x)/c^2$.

Key contributions

Fibrewise extension: the homogeneous case $R(x) \equiv 1$ reduces to TempProj (corollary).
Clean factorisation: $R(x)$ carries all local physics; $\sqrt{f(2-f)}$ is the universal Lorentzian structure.
Gravitational time dilation: identified as position-dependence of the temporal residual capacity.
Metric identification: $R(x)^2 = -2g_{\tau\tau}(x)$ connects fibrewise capacity to the spacetime metric.

Structure

The homogeneous baseline — recap of TempProj and the flat-space formula.
Fibrewise spectral occupancy — definition of $\epsilon(x)$ and reduction of the local BI radius.
Construction of the fibrewise map — derivation of $\mathcal{P}_{\tau,x}^{\mathrm{Q5b}}$ from the local capacity budget.
Factorisation theorem — proof that the map factorises as $R(x) \times$ (universal Lorentzian factor).
Gravitational time dilation — emergence of $d\tau/dn|_x = R(x)/\gamma$ from the local budget.
Metric identification and weak-field limit — connection to the Gravity paper and Newtonian potential.

Dependencies

TempProj: homogeneous temporal residual map (flat-space baseline).
LorentzCapacity: algebraic BI target and the O7 saturation framework.
Q5b: BFS-to-Carnot–Carathéodory convergence; Lorentzian temporal splitting.
Q11: co-metric closure fixing the Lorentzian signature $g^{\mu\nu} = 2\eta^{\mu\nu}$.
Gravity paper (H): metric $g_{\tau\tau}(x)$ imported for the identification $R(x)^2 = -2g_{\tau\tau}(x)$.
Born–Infeld paper: mobility budget $c_{\mathrm{BI}}$ and the capacity-flow sphere.
Foundation: fibre structure of the substrate and the spectral occupancy concept.

References

Jérôme Beau. Fibrewise Born–Infeld Capacity Lapse and Gravitational Time Dilation. Preprint. 10.5281/zenodo.20317235

Relation to the Cosmochrony program

LCII is the third paper of the Lorentz Capacity sub-programme. It extends the flat-space result of TempProj to curved-space regimes, connecting the spectral capacity framework to gravitational time dilation from the Gravity paper. The clean factorisation $R(x) \times \sqrt{f(2-f)}$ reveals that gravitational and kinematic time dilation arise from independent mechanisms within the same capacity budget: $R(x)$ from local spectral occupancy, and $\sqrt{f(2-f)}$ from Lorentzian filling. Together, the three papers LorentzCapacity, TempProj, and LCII form a complete derivation of relativistic time dilation from the projective capacity framework.