Observables and Parameters

An observable is not merely a symbol in an equation. It is a quantity tied to an operation: an integral over a specified boundary, a reading made by a declared observer, a field in a fixed asymptotic frame, a calibrated detector channel, or an estimate obtained from data through a forward model. This directory makes those defining operations visible before collecting formulas.

Choose the defining operation

The six records are ordered from geometric and boundary quantities toward increasingly model-dependent inference.

Immediate question	Record	Boundary to check first
Which mass, momentum, or angular-momentum definition applies at this surface or infinity?	ADM, Bondi, Komar, and Quasilocal Charges	Phase space, integration surface, symmetry generator, reference, and flux or integrability condition
Which area, surface gravity, angular velocity, or potential belongs to this horizon?	Horizon Area, Surface Gravity, Angular Velocity, and Potentials	Horizon class and normalization of its null generator
What do the clock, photon, image angle, or arrival-time comparison actually measure?	Proper Time, Redshift, Deflection, and Time Delay	Observer, emitter, receiver, null path, screen, and approximation
How do strain, news, curvature, flux, modes, and a detector channel fit together?	Gravitational-Wave Strain, News, Flux, and Ψ₄	Dyad, tetrad, Bondi frame, Fourier sign, extraction procedure, and response
Is this a pole, an excited response, a tail, a Love quantity, or a detector fit?	Quasinormal Frequencies, Tails, and Response	Operator and domain, boundary conditions, analytic sheet, normalization, source, subtraction, and fit window
Which data and forward model support a quoted black-hole parameter?	Astrophysical Black-Hole Parameters	Calibrated data, likelihood, shared nuisance parameters, selection, and degeneracy

These are lookup records. The linked Library pages remain responsible for full derivations, proofs, horizon mechanics, waveform construction, and observational inference.

Five layers that must not be collapsed

Local geometric quantities are defined at an event, along a worldline, or on a finite surface. Proper time and horizon area are examples. Their values can be invariant while still depending physically on the chosen curve or surface. An invariant scalar has the same value in every coordinate chart; a covariant tensor may instead transform nontrivially while representing the same geometric object. Neither property removes dependence on a physically chosen observer, surface, tetrad, or asymptotic frame.

Asymptotic or boundary quantities require a phase space, falloff conditions, a conformal or Hamiltonian frame, and a normalized generator. ADM energy, Bondi mass, news, and asymptotic waveform modes belong here. They are not local densities in the spacetime interior.

Response-defined quantities belong to an operator, boundary-value problem, source, and projection. Quasinormal poles, Green-function residues, scattering coefficients, branch-cut tails, and tidal response can be related without being interchangeable. A pole position does not supply its excitation amplitude, and neither one is automatically a detector observable.

Detector-level observables include a clock comparison, a signed lens map, or a calibrated gravitational-wave channel. They depend on an apparatus worldline, orientation, transfer function, bandwidth, and calibration even when the final number is coordinate independent.

Model-inferred parameters are estimands in a likelihood. A black-hole mass, spin, accretion rate, or emission-ring scale is not copied directly from raw data: it is supported by a declared measurement model and remains conditional on its nuisance parameters, priors, and alternatives.

The word observable is therefore useful only after its layer and defining operation have been named.

A reliable lookup sequence

For any quantity, proceed in the same order:

1. identify the spacetime, observer, surface, asymptotic end, or detector;
2. state the symmetry, frame, gauge, orientation, and normalization;
3. write the operational definition before a convenient coordinate formula;
4. declare units and distinguish a physical mass from the length $GM$;
5. name the propagation, response, or forward model connecting theory to data;
6. test a limit, balance law, gauge change, independent method, or predictive observable that could expose a wrong interpretation.

This sequence prevents several common substitutions: a Komar integral for an ADM charge without stationarity, a horizon-coordinate radius for invariant area, a Doppler-plus-gravitational story for an invariant frequency ratio without a comparison frame, $\Psi_4$ for strain without integration data, or a reconstructed image diameter for a metric critical curve without an emission model.

Stable definitions, dated values

The directory contains durable definitions and convention maps. It does not freeze event parameters, catalog counts, experimental bounds, calibration status, population hyperparameters, or claims that one inference method is currently most accurate. Such statements require a dated analysis specifying the data release, model family, software and calibration versions, likelihood, selection function, uncertainty budget, and conditions that would change the conclusion.

When a lookup will support a proof, precision calculation, or empirical claim, follow its links to the full treatment. A compact formula is safe only within the domain and comparison procedure that give it meaning.