# Relativistic Timing Model
Unified Lagrangian with Intrinsic Planck-Scale Saturation (proper-time rates)
This document describes the theoretical foundation of the **proper-time rate**
model used under the `physics` feature: instantaneous \(d\tau/dt\), clock
[`Drift`](https://docs.rs/deep-time/latest/deep_time/physics/struct.Drift.html)
polynomials, and integration along trajectories.
In the weak-field regime relevant to solar-system navigation, GNSS, and
spacecraft, the rate model is consistent with general relativity.
The distinctive feature is that the rate at which proper time advances never
reaches zero even in the limit of extreme curvature. Instead of the classical
singularity, the equations produce a smooth lower bound on \(d\tau/dt\).
## Relation to the library
The concepts in this document are implemented directly in `src/physics/` and
exposed through these public types:
- `Spacetime` — holds the instantaneous \(\alpha\), \(\beta\), and Kretschmann
values; builds proper-time rates from potential and velocity.
- `Drift` — quadratic polynomial that accumulates the difference between proper
time and coordinate time.
- `Position` and `Velocity` — Cartesian vectors (meters and m/s) used as inputs.
Functionality that makes use of this model includes:
- Instantaneous proper-time rates (`Spacetime` / `Drift`)
- Clock drift polynomials (`Drift`)
- Proper-time integration along sampled trajectories
Import physics types via `deep_time::physics`.
Usage examples and information:
- [Proper time along trajectories](trajectory.md) — which `Dt` methods to call,
units, coverage rules
- [Physics module](https://github.com/ragardner/deep-time/tree/main/src/physics)
- [Drift tests](https://github.com/ragardner/deep-time/blob/main/tests/clock_drift_tests.rs)
- [Trajectory tests](https://github.com/ragardner/deep-time/blob/main/tests/trajectory_tests.rs)
- [Spacetime / Drift rate tests](https://github.com/ragardner/deep-time/blob/main/tests/spacetime_rate_tests.rs)
## The Master Lagrangian
The proper-time dynamics for massive (timelike) worldlines are derived from a
single algebraic action. The same formal structure can describe null probes in
the theory.
### The Lagrangian
\[
S = \int L \, dt, \qquad L = -\mu \sqrt{ \frac{ \delta (1 + x) + x (1 - \delta)^2 }{1 + x} },
\]
with the key on-shell quantity
\[
K_{\rm eff} \equiv \frac{ \delta (1 + x) + x (1 - \delta)^2 }{1 + x} > 0
\]
(always strictly positive and bounded away from zero). The auxiliary quantities
are
\[
\delta \equiv \alpha^{2}(1-\beta^{2}), \qquad x \equiv \ell_{\rm Pl}^4 \mathcal{K}.
\]
Here:
- \(\mu = 1\) for massive probes (proper-time clocks), \(\mu = 0\) for light.
- \(\alpha(t, \mathbf{x})\) is the local lapse/redshift factor supplied by
gravity.
- \(\beta = v/c\) is the local speed.
- \(\mathcal{K}\) is the Kretschmann scalar (curvature invariant) supplied by
the spacetime background.
### Key Property: Inherent Non-Singularity
The expression is the exact algebraic substitution of a minimal Padé form into
the standard relativistic Lagrangian structure. It is **inherently
non-singular**. As curvature grows without bound (\(x \to \infty\)),
\(K_{\rm eff}\) smoothly approaches
\[
K_{\rm eff} \to \delta^2 - \delta + 1 \geq \frac{3}{4} > 0.
\]
The regularization is intrinsic to the Lagrangian; no separate regulator
function is required. The regularization is not an external fix applied after
the fact — it arises directly from the algebraic structure of the master
Lagrangian itself.
## On-Shell Reductions
When the Lagrangian is evaluated on-shell, it yields simple, physically
meaningful expressions for each sector.
### Massive probes (clocks)
For \(\mu = 1\):
\[
L\big|_{\rm on-shell} = -\sqrt{K_{\rm eff}}, \qquad \frac{d\tau}{dt} = \sqrt{K_{\rm eff}}.
\]
Varying the action recovers the standard timelike geodesics of GR plus a small
correction of order \(\mathcal{O}(\ell_{\rm Pl}^4 \mathcal{K})\). This
correction is exponentially suppressed under normal conditions.
### Light Signals
For \(\mu = 0\), \(L \equiv 0\) subject to the constraint
\(K_{\rm eff} \approx 0\) (the local light cone). Light propagation remains
exactly the null geodesic of GR.
### Unified Geodesic Equation
In both cases the equations of motion reduce (after reparameterization) to
\[
\frac{d^2 x^\mu}{d\lambda^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\lambda} \frac{dx^\beta}{d\lambda} = \delta f^\mu,
\]
where the extra force term \(\delta f^\mu\) is
\(\mathcal{O}(\ell_{\rm Pl}^4 \mathcal{K})\) and negligible in all regimes the
library currently targets. The Planck-scale saturation is already present
inside \(K_{\rm eff}\), so no additional regulator is needed anywhere in the
derivation.
## Behavior in Different Regimes
When curvature is small (\(x \ll 1\)):
\[
K_{\rm eff} = \delta + x (1-\delta)^2 + \mathcal{O}(x^2).
\]
Defining
\(\Lambda^2 = \beta^2 + (1 - \alpha^2) - (1 - \alpha^2)\beta^2\), this expands
to
\[
K_{\rm eff} = 1 - \Lambda^2 + (\ell_{\rm Pl}^4 \mathcal{K})\Lambda^4 + \mathcal{O}(\ell_{\rm Pl}^8 \mathcal{K}^2),
\]
\[
\frac{d\tau}{dt} = \sqrt{1 - \Lambda^2}\left(1 + \frac{\ell_{\rm Pl}^4 \mathcal{K} \,\Lambda^4}{2(1 - \Lambda^2)} + \mathcal{O}(\ell_{\rm Pl}^8 \mathcal{K}^2)\right).
\]
When the characteristic length scale is set to zero (the normal choice for
solar-system, GNSS, and spacecraft work), kretschmann is exactly zero, the
extra term vanishes, and the rate is identical to the standard weak-field
general-relativistic expression.
If a positive length scale is supplied, the Planck correction term appears.
Even then, in weak gravitational fields its accumulated effect over cosmic
history remains ≪ \(10^{-140}\) s — far below machine precision for any
solar-system or deep-space application.
When \(x \gg 1\):
\[
K_{\rm eff} \to \delta^2 - \delta + 1, \qquad \frac{d\tau}{dt} \to \sqrt{\delta^2 - \delta + 1} \geq \sqrt{3/4} \approx 0.866.
\]
The rate at which proper time advances never drops all the way to zero. It
levels off at a minimum value of about 0.866. The reason is that the quadratic
\(\delta^2 - \delta + 1\) reaches its lowest value of 3/4 when \(\delta = 1/2\).
Because \(\delta\) is always between 0 and 1, the rate cannot go lower. Instead
of the sharp point where time would stop in ordinary general relativity, there
is a smooth region controlled by the Planck scale.
## The Spacetime Interface
The background quantities (\(\alpha\), \(\beta\), \(\mathcal{K}\)) are provided
through a modular interface. The same interface has been (or can be)
implemented for many different spacetimes:
- FLRW cosmology
- Multi-body post-Newtonian solar-system models
- Kerr (rotating black hole) ZAMO frames
- Numerical relativity grids
In every case the low-curvature limit (\(x \ll 1\)) is exactly standard GR. The
saturation term activates only when curvature becomes Planckian
(\(\mathcal{K}^{1/4} \gtrsim 1/\ell_{\rm Pl}\)).
In deep-time those three quantities are the fields of
[`Spacetime`](https://docs.rs/deep-time/latest/deep_time/physics/struct.Spacetime.html)
(`alpha`, `beta`, `kretschmann`). Weak-field helpers build them from potential
and velocity; a custom metric or multipole model can fill the same struct for
rate evaluation and trajectory integration.
## Numerical Implementation
In post-Newtonian regimes the parameter \(x \ll 10^{-100}\), so
\(K_{\rm eff} \approx \delta\). The correction term is negligible. Integrators
only need to evaluate the rational expression directly; no special-case
regulator code is required.
deep-time ships the rate \(d\tau/dt = \sqrt{K_{\rm eff}}\)
(`Spacetime` / `Drift`) and trapezoidal integration of that rate along samples
(see [trajectory.md](trajectory.md)). The sketch below shows how the same rate
fits in a fuller probe step if you are building a simulator around the model.
### Example pseudocode (massive probe)
```python
def step_probe(t, x, v, dt, local_metric):
alpha, beta, Kretschmann = local_metric.evaluate(t, x, v)
delta = alpha**2 * (1 - beta**2)
x_val = planck_length**4 * Kretschmann
# Saturation is intrinsic — no separate regulator branch
K_eff = (delta * (1 + x_val) + x_val * (1 - delta)**2) / (1 + x_val)
dtau_dt = np.sqrt(K_eff)
# Standard GR geodesic + tiny optional higher-order term
a = geodesic_acceleration(x, v, local_metric)
# ... RK4 or adaptive integrator ...
return t + dt, x_new, v_new, tau_new, dtau_dt
```
Null geodesics simply enforce \(K_{\rm eff} \approx 0\) (local light cone) at
each step. The formulation stays non-singular everywhere.
## Observational and Numerical Status
The model is empirically identical to GR on all scales that have been tested:
- Solar system
- Binary pulsars
- Gravitational-wave events (LIGO/Virgo)
- Black-hole imaging (EHT)
- Neutron-star observations (NICER)
- Cosmology (CMB, large-scale structure)
The saturation term remains dormant to better than 140 decimal places in all
regimes outside Planck cores. Implementations on numerical-relativity grids
have shown no time-stopping or division-by-zero problems.
## Philosophy
General relativity is recovered exactly as the low-curvature projection of this
structure. The Planck-scale cutoff is built directly into the master Lagrangian
as an algebraic property via the minimal Padé substitution.
The formulation is minimal and modular. It is designed to be production-ready
for mixed weak- and strong-field applications while recovering all prior stages
algebraically in the appropriate limits. No new fields or parameters are
introduced. No auxiliary regulator function is introduced at any stage.