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 is \(\Phi = -M/r\) with no \(G\), the tick rate is \(\sqrt{1 + 2\Phi}\) with no \(c^2\), and positions, masses, and times become pure numbers. Nothing physical is lost — only bookkeeping.

What M = 0.8 means

In these units a mass \(M\) has Schwarzschild radius \(r_s = 2M\). The demos place clocks within a handful of Schwarzschild radii of the hidden mass — deliberately deep in the relativistic regime, so that tick-rate differences are visible against 0.5% observation noise. The cell below computes the actual numbers for the 1D demo’s geometry:

Code

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

mass = 0.8
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 = 0.8: r_s = 1.6
  clock 0: r =  7.57  (r/r_s = 4.73)  rate = 0.8880
  clock 1: r =  2.69  (r/r_s = 1.68)  rate = 0.6370
  clock 2: r =  2.69  (r/r_s = 1.68)  rate = 0.6370

Read those r/r_s values: this is strong-field territory. A real observer at these separations would have larger worries than clock calibration.

Scaling to reality

At Earth’s surface, \(2\Phi/c^2 \approx 1.4 \times 10^{-9}\) — the dip the demos paint in percent is, for real planetary fields, parts per billion. 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. It does not change the math — the forward model, the degeneracies, and the filter are the same at \(10^{-1}\) and \(10^{-18}\); only the noise budget moves.

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).