Notation & Glossary
A consolidated lookup for every symbol and term used across the site. Symbols defined on their source pages are still listed here — this page exists so you never have to hunt for where a definition first appeared.
Physics notation
| Symbol | Meaning | Where introduced |
|---|---|---|
| \(\Phi\) | Newtonian gravitational potential (negative near a mass); \(-M/r\) for a point mass | Clocks as Gravimeters |
| \(G\), \(c\) | Newton’s constant and the speed of light; the simulation sets \(G = c = 1\) | Units and Scales |
| \(\tau\), \(t\) | Proper time vs coordinate time; a clock’s tick rate is \(d\tau/dt\) | Clocks as Gravimeters |
| \(r\) | Bare \(r\) is always a distance on this site | Clocks as Gravimeters |
| \(r_c\) | Tick rate of clock \(c\) in the filter equations — same letter as distance, distinguished by the clock index | The Particle Filter |
| \(r_{ij}\) | Distance from clock \(i\) to mass \(j\) | Into the Plane |
| \(r_s\) | Schwarzschild radius, \(r_s = 2M\) in simulation units | Units and Scales |
| \(R\) | Range from the clock lattice to an exterior mass; the differential signal falls as \(1/R^2\), the curvature term as \(1/R^3\) | Gravitational Echolocation |
| \(x\), \(y\) | Mass position coordinates (subscripted \(x_1, x_2\) with several masses) | The Search in One Dimension (\(x\)); Into the Plane (\(y\)) |
| \(M\), \(M_j\) | Mass (of mass \(j\)) | Clocks as Gravimeters |
| \(\mu\), \(\sigma_{\text{density}}\), \(A\) | Center, width, and peak amplitude of a continuous Gaussian mass profile | Beyond Point Masses |
Index conventions — context-dependent: \(i\) indexes clocks/evaluation points in the potential sum (\(r_{ij}\)) but particles in the filter equations (\(w_i\), \(\theta_i\)); \(j\) indexes masses; \(c\) indexes clocks (\(r_c\)).
Inference notation
| Symbol | Meaning | Where introduced |
|---|---|---|
| \(\sigma_{\text{obs}}\) | Observation noise standard deviation (0.005 in the demos); written plain \(\sigma\) on the inference pages | One Clock Is Not Enough |
| \(N\) | Number of particles in the filter’s cloud | The Particle Filter |
| \(w_i\) | Weight of particle \(i\) (weights sum to one) | The Particle Filter |
| \(\theta_i\) | Particle \(i\)’s full parameter hypothesis, e.g. \((x, M)\) | The Particle Filter |
| \(\beta\) | Tempering exponent for the newest observation, advanced adaptively from 0 to 1 | The Particle Filter |
| \(\pi_{t,\beta}\) | Intermediate target after observations before \(t\) and a fraction \(\beta\) of observation \(t\) | The Particle Filter |
| \(K\) | Number of masses a model assumes | How Many Masses? |
| ESS | Effective sample size, \(1/\sum_i w_i^2\) | The Particle Filter |
| evidence, log-evidence | SMC Monte Carlo estimate of a model’s marginal likelihood, accumulated before resampling at each tempering stage | How Many Masses? |
NoteTwo overloaded symbols
\(\sigma\) means observation noise on the inference pages but Gaussian profile width in Beyond Point Masses (which flags the reuse itself); this page writes \(\sigma_{\text{obs}}\) vs \(\sigma_{\text{density}}\) to keep them apart. \(\tau\) means proper time in \(d\tau/dt\), but in the echolocation coda it is a clock’s averaging time: there \(\sigma_y(\tau) = 10^{-16}/\sqrt{\tau}\) is the clock’s fractional-frequency instability after averaging for time \(\tau\).
Terms
- Weak field — the regime where gravity is a small correction (\(|2\Phi/c^2| \ll 1\)) and the tick rate \(\sqrt{1 + 2\Phi/c^2}\) is used as a pedagogical surrogate; this project enforces the conservative policy \(|2\Phi| \le 0.1\) in simulation units; see Clocks as Gravimeters.
- Time dilation — clocks deeper in a gravitational well tick slower relative to distant ones; the measurable effect this whole site is built on (Clocks as Gravimeters).
- Chronometric leveling — surveying height differences by comparing clock rates, as in the Tokyo Skytree experiment (Clocks as Gravimeters).
- Forward model — the physics direction: from a mass configuration to the clock rates it produces (Clocks as Gravimeters).
- Inverse problem — the detection direction: from noisy clock rates back to the masses that caused them (concept: One Clock Is Not Enough).
- Mass–distance degeneracy — a small mass nearby and a heavy mass far away can produce identical readings; reduced or broken by sufficiently informative array geometry (One Clock Is Not Enough).
- Particle filter — the site’s inference engine: adaptive tempered resample-move SMC over a cloud of hypotheses (The Particle Filter).
- Prior — what the filter assumes before any data: a uniform rectangle over the configured position and mass ranges, conditioned on ordering and weak-field validity (The Particle Filter).
- Likelihood — how well a hypothesis predicts an observed set of clock rates; Gaussian in the observation noise (The Particle Filter).
- Posterior — the belief after data. The particle cloud approximates it; the cloud’s spread is the filter’s claimed uncertainty (The Search in One Dimension).
- Evidence — the marginal likelihood a model assigned to the data it actually saw, estimated consistently by SMC but with finite-particle sampling error; what model comparison ranks (The Particle Filter).
- Resampling — redrawing the cloud in proportion to weight at an adaptive tempering stage; systematic, stratified, or residual (The Particle Filter).
- Rejuvenation — symmetric random-walk Metropolis-Hastings moves after resampling; accept/reject preserves the current tempered target (The Particle Filter).
- Effective sample size — \(1/\sum_i w_i^2\); how many particles are meaningfully alive (The Particle Filter).
- Model comparison — running one filter per candidate mass count \(K\) and comparing evidence (How Many Masses?).
- Label switching — with several masses the exchangeable posterior is symmetric under relabeling; the library represents its normalized form strictly ordered by the first spatial coordinate (Two Hidden Masses).
- Observation noise — the Gaussian noise \(\sigma_{\text{obs}}\) added to every clock reading; the floor that makes many observations necessary (One Clock Is Not Enough).