Strong-Field Observables and Interpretation Boundaries

A strong-field observable is a theory-level map from a specified compact-object solution and perturbation problem to an invariant or operationally defined number. It is not yet a measurement and not yet a bound on a coupling. This page constructs that first map for charges, circular orbits, tides, oscillations, and absorption, then explains which waveform, image, population, calibration, and likelihood models are needed to compare it with data.

Helpful background. Einstein Equations supplies the background solution; Gravitational Actions and Boundary Terms fixes charges and variational response; Formulations of General Relativity controls gauge and asymptotic definitions; Black-Hole Geometry and Mechanics supplies horizon and geodesic invariants; Black-Hole Perturbation Theory supplies response and radiative spectra; and Relativistic Matter and Compact Objects supplies surfaces, equations of state, and matter-frame observables.

Fix the solution, frame, and asymptotic data

For the compact formulas on this page use geometrized units $G=c=1$: $M=GM_{\rm phys}$ is a length and $D$ has dimensions of length when $\varphi$ is dimensionless. For example the Schwarzschild ISCO frequency in ordinary units is $\Omega_{\rm ISCO}=c^3/(6^{3/2}GM_{\rm phys})$.

Before extracting a number, state the action and couplings, solution branch, horizon or material-surface boundary condition, asymptotic sources, and approximation order. If matter follows a physical metric $\widetilde g_{ab}=A^2(\varphi)g_{ab}$, clocks and freely falling emitters probe $\widetilde g_{ab}$; geodesics of an auxiliary Einstein-frame metric are not directly observable. An asymptotically flat spherical solution may have

$$g_{tt}=-1+\frac{2M}{r}+O(r^{-2}), \qquad \varphi=\varphi_\infty+\frac{D}{r}+O(r^{-2}).$$

Here $M$ and the scalar charge $D$ are outputs only after the asymptotic radial and time normalization have been fixed. A sensitivity such as $\partial\ln M/\partial\varphi_\infty$ additionally states what is held fixed. For stationary rotating solutions, define mass and current multipoles in one asymptotically Cartesian, mass-centered convention; coordinate-series coefficients in an arbitrary gauge are not multipole moments.

Circular timelike and null diagnostics

Consider the physical, areal-radius metric

$$d\widetilde s^2=-A(r)dt^2+\frac{dr^2}{B(r)}+r^2d\Omega^2.$$

For equatorial circular timelike geodesics, eliminating the conserved energy and angular momentum gives

$$\Omega^2=\frac{A'}{2r}, \qquad E^2=\frac{2A^2}{2A-rA'}, \qquad L^2=\frac{r^3A'}{2A-rA'}.$$

Here $E=-u_t$ and $L=u_\varphi$ are energy and angular momentum per unit rest mass, and $\Omega=d\varphi/dt$ uses asymptotically normalized time. The formulas require a static exterior with $A\gt0$, $B\gt0$, $A'\gt0$, and $2A-rA'\gt0$ for a timelike circular orbit. A smooth marginally stable orbit satisfies $dL^2/dr=0$; identifying it as the innermost stable orbit also requires following the connected stable family. A circular null orbit instead satisfies

$$rA'-2A=0, \qquad b_{\mathrm{ph}}^2=\frac{r_{\mathrm{ph}}^2}{A(r_{\mathrm{ph}})}.$$

Sotiriou and Zhou apply precisely these degeneracy and photon-orbit conditions to scalar–Gauss–Bonnet black holes in Sotiriou–Zhou 2014, § III.F, Eqs. (67)–(71), pp. 9–10 (PDF). The Schwarzschild check $A=1-2M/r$ gives $r_{\mathrm{ISCO}}=6M$, $M\Omega_{\mathrm{ISCO}}=6^{-3/2}$, $r_{\mathrm{ph}}=3M$, and $b_{\mathrm{ph}}=3\sqrt3M$.

These are geodesic outputs, not image diameters or spectral-line measurements. A shadow or critical curve also depends on observer position and inclination; a brightness image further requires an emission, absorption, scattering, and instrumental model. In theories with nonminimal photon couplings, even the physical-metric null geodesic may cease to be the correct ray equation.

Tidal response, oscillations, and absorption

For a static quadrupolar tide, choose a large-radius basis that separates the applied field from the body’s response and define, in one sign convention,

$$\delta Q_{ij}=-\lambda_2\mathcal E_{ij}, \qquad \Lambda_2=\frac{\lambda_2}{M^5}.$$

Integrate the coupled perturbations from a regular center, horizon, or material surface and match them to that basis. Extra scalar or vector tides enlarge the response matrix; setting their applied amplitudes to zero is a boundary choice, not an automatic property. Cardoso and collaborators give relativistic polar/axial definitions, multipole extraction, and the normalization conversion between mass- and radius-scaled Love numbers in Cardoso et al. 2017, § II, Eqs. (2)–(4), pp. 3–5 (PDF). Their Appendix B, pp. 18–20, shows the asymptotic source/response extraction. Logarithms, field redefinitions, or overlapping long-range fields can make an individual coefficient scheme-dependent; the matching prescription must accompany the number.

Quasinormal frequencies $\omega_n$ are poles of the coupled retarded response with ingoing future-horizon behavior and outgoing, normalizable, or otherwise physically specified outer behavior. State the time convention, mode labels, physical unit used to scale $\omega$, and all coupled channels. A reflecting surface produces a different spectrum from a horizon even when the exterior metrics nearly agree. Likewise, an absorption probability is a flux ratio: normalize an incident channel, compute conserved flux into the future horizon or dissipative material boundary, and divide by incident flux. Negative asymptotic-time energy flux into the horizon is superradiant amplification. For a mode with $\omega\gt0$ and azimuthal number $m$, that flux is proportional to $\omega(\omega-m\Omega_H)$, whereas the flux of energy associated with the horizon generator is proportional to $(\omega-m\Omega_H)^2$ for a standard positive-energy bosonic field. Negative Killing energy measured at infinity supplies the amplified outgoing wave; the generator-energy flux is nonnegative.

Comparing theory predictions with data

theory output	extraction criterion	essential boundary control	independent theory check	comparison with data
$M,J,D$ and higher multipoles	invariant asymptotic charge or fixed multipole convention	frame, gauge, falloff, branch	GR and weak-coupling limits	waveform or orbital model
ISCO and photon critical orbit	degeneracy of physical-metric effective potential	physical matter/photon metric	Schwarzschild values above	source and image model
tidal response	asymptotic response-to-source ratio	horizon, surface, EOS, extra-field tides	zero-coupling or known-star benchmark	inspiral phase model
quasinormal spectrum	pole or homogeneous radiative boundary problem	ingoing/outgoing convention and all channels	residual plus GR spectrum	ringdown waveform model
absorption	conserved absorbed-to-incident flux ratio	dissipative boundary and mode normalization	flux balance	population and detector likelihood

The first four columns belong to theory prediction. The final column begins a different calculation. Turning a predicted correction into an empirical constraint requires source populations, nuisance parameters, waveform or radiative-transfer errors, detector calibration and selection effects, and a declared likelihood and prior. Berti, Yagi, and Yunes make this distinction concrete for inspiral generation effects, multipole degeneracies, and tidal parameters in Berti–Yagi–Yunes 2018, § 3, pp. 12–24 (PDF). Those inference tasks belong in Tests of Gravity; this volume supplies theory inputs and their domains, not observational exclusion claims.

Boundaries on interpretation

Every quoted deviation should survive three limits: the GR or zero-coupling limit, enlargement of the numerical domain, and variation of the matching surface inside a genuine overlap region. It should also remain below the effective cutoff: a high-frequency mode or near-horizon gradient can invalidate a truncated action even when the background correction appears small. Report a dimensionless quantity together with the mass, spin, branch, asymptotic scalar value, EOS where relevant, and coupling normalization used to dimensionalize it.

Degeneracy is not inconsistency. The same change in $M\Omega_{\mathrm{ISCO}}$ can be mimicked by spin or a matter-model change; a nonzero tidal phase can arise from a material surface, an extra field, or ordinary neutron-star structure. Theory calculations identify response patterns and correlated outputs. Only an explicit joint inference can decide how data distribute support among them.

Exercise: recover the Schwarzschild markers

Use $A(r)=1-2M/r$ to derive the photon-orbit radius and impact parameter, then find the ISCO from $dL^2/dr=0$.

Solution

The null condition gives

$$r\frac{2M}{r^2}-2\left(1-\frac{2M}{r}\right)=0,$$

so $r_{\mathrm{ph}}=3M$. Since $A(3M)=1/3$, $b_{\mathrm{ph}}=3\sqrt3M$. For timelike circular orbits,

$$L^2=\frac{Mr^2}{r-3M}.$$

Differentiating gives $dL^2/dr=Mr(r-6M)/(r-3M)^2$, hence the marginally stable orbit is $r_{\mathrm{ISCO}}=6M$. Finally $\Omega^2=A'/(2r)=M/r^3$, so $M\Omega_{\mathrm{ISCO}}=6^{-3/2}$. These checks test the metric and normalization; they do not validate an emission or likelihood model.

References

• Berti, Emanuele, Kent Yagi, and Nicolás Yunes, “Extreme Gravity Tests with Gravitational Waves from Compact Binary Coalescences: (I) Inspiral-Merger,” General Relativity and Gravitation 50 (2018) 46. DOI; Open PDF.
• Cardoso, Vitor, Edgardo Franzin, Andrea Maselli, Paolo Pani, and Guilherme Raposo, “Testing strong-field gravity with tidal Love numbers,” Physical Review D 95 (2017) 084014. DOI; Open PDF.
• Sotiriou, Thomas P., and Shuang-Yong Zhou, “Black Hole Hair in Generalized Scalar-Tensor Gravity: An Explicit Example,” Physical Review D 90 (2014) 124063. DOI; Open PDF.