Units and Scales

G = c = 1, and what that buys

The simulation sets \(G = c = 1\). One choice of units removes two constants from every equation: the potential simplifies to \(\Phi = -M/r\) (setting \(G=1\)), and the tick rate simplifies to \(\sqrt{1 + 2\Phi}\) (setting \(c=1\)), turning positions, masses, and times into dimensionless simulation numbers. This unit choice removes bookkeeping; it does not make the model itself more exact.

What M = 0.10 means

In these units a mass \(M\) has length scale \(r_s = 2M\). The demos enforce \(|2\Phi| \le 0.1\) so the pedagogical rate map stays within its declared weak-field domain. The cell below computes the 1D demo’s geometry:

Code

import numpy as np
from clocks import ClockArray, MassConfig, clock_rates
from clocks.physics import compute_distances

mass = 0.10
truth = MassConfig(positions=np.array([[2.5]]), masses=np.array([mass]))
clock_array = ClockArray(positions=np.array([[-5.0], [0.0], [5.0]]), track_offset=1.0)

distances = compute_distances(
    clock_array.positions, truth.positions, clock_array.track_offset
)[:, 0]
rates = clock_rates(truth, clock_array)
r_s = 2 * mass

print(f"Schwarzschild radius of M = {mass}: r_s = {r_s}")
for i, (r, rate) in enumerate(zip(distances, rates)):
    print(f"  clock {i}: r = {r:5.2f}  (r/r_s = {r / r_s:4.2f})  rate = {rate:.4f}")

Schwarzschild radius of M = 0.1: r_s = 0.2
  clock 0: r =  7.57  (r/r_s = 37.83)  rate = 0.9867
  clock 1: r =  2.69  (r/r_s = 13.46)  rate = 0.9621
  clock 2: r =  2.69  (r/r_s = 13.46)  rate = 0.9621

The project treats \(\sqrt{1+2\Phi}\), together with summed Newtonian potentials \(\Phi_i=-\sum_j M_j/r_{ij}\), as a pedagogical weak-field surrogate. It is not an exact strong-field solution and the potential sum is not a general superposition law of general relativity. Direct physics calls reject nonfinite, positive, singular, or overly deep potentials; inference assigns candidates outside the same domain zero prior support. The conservative operational policy \(|2\Phi| \le 0.1\) keeps the model in its declared regime rather than merely keeping the square root real.

Scaling to reality

At Earth’s surface, \(2\Phi/c^2 \approx 1.4 \times 10^{-9}\) — the dip the demos paint at a few percent is, for real planetary fields, parts per billion. Under the enforced policy, the largest possible rate decrease is \(1-\sqrt{0.9}\approx5.13\%\); the shipped ground truths are shallower. Detecting anomalies (a buried density contrast, not the whole planet) means resolving fractional rate differences around \(10^{-18}\) to \(10^{-17}\), which is exactly why the gravimeters page leads with optical lattice clocks: that is the instrument class where this stops being a toy.

The plain summary: the simulation compresses the dynamic range so the inference story is visible to the eye. The qualitative inverse-problem geometry carries over to much weaker signals, but this surrogate deliberately omits higher-order relativistic physics and real instrument systematics.

Convention

Every number on this site is in simulation units unless it is explicitly tied to a real experiment (GPS microseconds, Skytree meters, lattice-clock fractions).