Fibrewise Born–Infeld Capacity Lapse and Gravitational Time Dilation

Current Zenodo release: version 1.5 (2026-10-26). Official title: 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 normalised capacity factor of TempProj, fixed by the Born–Infeld sphere and the filling law; it is not a Lorentzian structure, the Lorentzian interval being that of the supplied co-metric. The residual capacity per ordering step, $|F^\tau(x,n)| = R(x)/\gamma$, follows from the local capacity budget; it is a capacity increment, distinct from the constant ordering step $d\tau/dn = \Delta\tau$, and its reading as gravitational time dilation, the proper-time rate of an observer at $x$, requires [H-cone], which no source derives. The postulated metric identification $R(x)^2 = g_{\tau\tau}(x)/g_{\tau\tau}^\infty$ connects it to a supplied Lorentzian metric, whose flat temporal reference is postulated.

Scope statement. The fibrewise extension is structural and works on a supplied Lorentzian metric: no paper of the programme constructs that metric (Q5b gives a rank-three co-metric, Q11 leaves $A_\tau$ free, and the Gravity paper works on supplied Riemannian metrics), and the weak-field corollary restates the general-relativistic lapse of a supplied metric rather than deriving it. 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:

$|F^\tau(x,n)| = R(x)\cdot\dfrac{1}{\gamma}$

This is the residual capacity per ordering step; it is read as a proper-time rate $d\tau_u/d\hat\tau|_x$ under [H-cone] and the postulated occupancy–lapse identification.

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. Under [H-cone] and that postulate, special-relativistic time dilation is the homogeneous capacity trade-off, and 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 for a supplied Lorentzian metric, which no source in the programme constructs. The open problem LC-O2-O1 -- deriving $\epsilon(x) = 1 - R(x)^2 = 1 - g_{\tau\tau}(x)/g_{\tau\tau}^\infty$ within the capacity-flow language without a supplied metric -- remains open. The companion paper LCIIo1 establishes a no-go for the scalar free-fraction reduction, so the equivalence of the capacity and metric definitions of $R(x)$ stays a declared postulate (Remark 3.2).

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 $R(x)^2 = g_{\tau\tau}(x)/g_{\tau\tau}^\infty$ (postulated). Residual per ordering step: $|F^\tau(x,n)| = R(x)/\gamma$, read as gravitational time dilation under [H-cone] and the occupancy–lapse postulate. For a supplied weak-field metric: $R(x) \approx 1 + \Phi_N(x)/c^2$.

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 normalised capacity factor of TempProj, not a Lorentzian structure.
Gravitational time dilation: the residual capacity per ordering step $R(x)/\gamma$, read as gravitational time dilation under [H-cone] and the occupancy–lapse postulate.
Metric identification: $R(x)^2 = g_{\tau\tau}(x)/g_{\tau\tau}^\infty$, a declared postulate, connects fibrewise capacity to a supplied spacetime metric; $R(x)^2 = -2g_{\tau\tau}(x)$ for the normalisation $A_\tau^\infty = 2$, a choice of units.

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 normalised capacity factor).
Gravitational time dilation — the residual $|F^\tau(x,n)| = R(x)/\gamma$ from the local budget, read as a proper-time rate under [H-cone] and the occupancy–lapse postulate.
Metric identification and weak-field limit — the lapse ratio of a supplied metric and its Newtonian weak-field form, taken from general relativity.

Dependencies

TempProj: homogeneous temporal residual map (flat-space baseline).
LorentzCapacity: algebraic BI target and the O7 saturation framework.
Q5b: BFS-to-Carnot–Carathéodory convergence; a temporal splitting with signature $(-,+,+)$ on the non-degenerate block under [H-L], [H-lift] and [H-hyp].
Q11: the homogeneous form $\mathrm{diag}(-A_\tau^\infty, 2\lambda, 2\lambda, 2\lambda)$ of Corollary 6.1, for a supplied extension; $A_\tau$ is free, so the flat temporal reference $A_\tau^\infty$ is postulated here.
Gravity paper (H): not a dependency. It studies an Einstein term on supplied Riemannian metrics and supplies no Lorentzian $g_{\tau\tau}(x)$; LCII imports no result from it.
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.20316528

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 the gravitational time dilation of a supplied Lorentzian metric. Under [H-cone], the clean factorisation $R(x) \times \sqrt{f(2-f)}$ shows 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 restate relativistic time dilation in the projective capacity language, given [H-cone], a supplied metric and the postulated identification of the capacity radius with its lapse.