Timing Tests and Orbital Frequencies

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Accreting compact objects vary on orbital timescales, and relativistic orbital, radial-epicyclic, and vertical-epicyclic frequencies contain spacetime information. A power-spectrum peak is not itself one of those frequencies. Precession, diskoseismic modes, turbulence, radiative transfer, and instabilities can create or shift timing features. The useful test asks which frequency scaling and cross-spectral behavior one model predicts across mass, state, radius, and energy. The evidence boundary is 2026-08-19.

Helpful background. Weak-Field Gravity supplies the far-field frequency limits; Probability and Statistical Inference supplies stochastic-process likelihoods; Black-Hole Geometry and Mechanics supplies Kerr orbital structure; and Black-Hole Astrophysics supplies disks, coronae, magnetic fields, and source states.

Characteristic frequencies are model inputs

For a neutral test particle on a circular, prograde, equatorial Kerr orbit, define the Boyer–Lindquist radius $\bar r=r/(GM)$ and spin $0\leq a_*=J/(GM^2)\leq1$ . The frequencies below are angular frequencies relative to time measured by a distant stationary observer in the asymptotically flat geometry. The radial and vertical frequencies describe small excursions about that circular orbit. Then

GM\Omega_\phi=\frac{1}{\bar r^{3/2}+a_*},

and

\frac{\Omega_r^2}{\Omega_\phi^2} =1-\frac{6}{\bar r} +\frac{8a_*}{\bar r^{3/2}} -\frac{3a_*^2}{\bar r^2}, \qquad \frac{\Omega_\theta^2}{\Omega_\phi^2} =1-\frac{4a_*}{\bar r^{3/2}} +\frac{3a_*^2}{\bar r^2}.

Use these as real oscillation frequencies only on the stable circular-orbit branch, outside the innermost stable circular orbit (ISCO). At the ISCO, $\Omega_r^2=0$ ; inside it, a negative square describes instability rather than an observable periodic radial motion. For Schwarzschild ( $a_*=0$ ), $\Omega_\theta=\Omega_\phi$ and $\Omega_r=\Omega_\phi\sqrt{1-6/\bar r}$ , providing immediate spin and stability checks.

Periastron and nodal-precession frequencies are $\Omega_\phi-\Omega_r$ and $\Omega_\phi-\Omega_\theta$ . Measured cycles per second require $\nu=\Omega/(2\pi)$ and, for a cosmological source, an additional division by $1+z$ . For example, a Schwarzschild black hole of $10M_\odot$ has $\nu_\phi\simeq220\,\mathrm{Hz}$ at $\bar r=6$ , using $GM_\odot/c^3\simeq4.9255\,\mu\mathrm{s}$ ; its radial epicyclic frequency there is zero, not $220\,\mathrm{Hz}$ . The $1/M$ scaling at fixed $\bar r$ and $a_*$ is a powerful check. Yet a thick, pressured, magnetized disk is not a test particle: mode trapping, sound speed, radial drift, stress, and emissivity change the connection between these frequencies and observed photons. Bambi reviews the competing orbital, resonance, and diskoseismic identifications in Bambi 2017, §III.D.

The most discriminating data are not only centroid frequencies. Given energy-band light curves $x(t)$ and $y(t)$ , define the averaged cross spectrum

C_{xy}(f)=\langle X^*(f)Y(f)\rangle, \qquad \tau_{y\leftarrow x}(f)=\frac{\arg C_{xy}(f)}{2\pi f},

with the synthesis convention $x(t)=\int X(f)e^{-2\pi i f t}df$ . A positive $\tau_{y\leftarrow x}$ means that $y$ is delayed relative to $x$ : setting $y(t)=x(t-\tau_0)$ gives $Y=e^{2\pi if\tau_0}X$ and $\arg C_{xy}=2\pi f\tau_0$ . This checks the sign without borrowing a potentially opposite Fourier convention. The argument is measured modulo $2\pi$ , so the principal-value lag wraps once the true phase leaves $(-\pi,\pi]$ ; broad transfer functions need not have a single frequency-independent delay. Phase lags, coherence, quality factor, harmonic structure, energy dependence, and their evolution through a source state constrain whether a feature comes from global precession, reverberation, an oscillation mode, or an instability. Uttley and collaborators develop the estimators in Uttley et al. 2014, §§2–5, with the signs here fixed by the stated synthesis convention.

Data response, selection, and stochastic likelihood

A versioned timing analysis identifies event-mode light curves, detector and calibration release, good-time intervals, energy response, background, dead-time correction, gaps, time binning, barycentric convention, and segment averaging. Source-state selection based on hardness, count rate, or variability can narrow the physical regime but also select the timing feature. Search ranges, number of sources, harmonics, and multiple time windows determine trials.

For an approximately stationary Gaussian process, a Whittle likelihood for periodogram $I_k$ and model spectrum $S_k(\vartheta,\eta)$ is schematically

-\ln\mathcal L=\sum_k \left[\ln S_k+\frac{I_k}{S_k}\right],

with the appropriate distribution and covariance for averaged powers and cross spectra. Poisson noise, dead time, red-noise leakage, window convolution, and nonstationarity modify this expression. The parameters $\vartheta$ describe the spacetime-frequency map; nuisances $\eta$ describe broad-band noise, feature widths, energy response, disk thickness, corona, magnetic geometry, and state evolution. Priors on the number, ordering, and identity of peaks can dominate a claimed metric constraint.

Reproduction requires the observation identifiers, calibrated event files, interval and energy selections, response, time and Fourier conventions, segment rules, noise subtraction, software version, frequency model, priors, sampler diagnostics, and machine-readable power and cross spectra. A peak found after looking across many sources and intervals requires the full search selection, not a local likelihood ratio.

Results and limitations through 2026-08-19

Evidence or position	Result supported	Limitation or scope
Stable or recurring power-spectrum features	Establish preferred variability timescales and state dependence	A centroid does not identify an orbital, epicyclic, or mode frequency
Simultaneous peaks and mass-scaled frequency relations	Can test a constrained frequency-identification model	Radius, spin, pressure, resonance choice, and source selection remain degenerate
Energy-dependent lags, coherence, harmonics, and held-out states	Discriminate some precession, reverberation, diskoseismic, and instability predictions	Radiative transfer, corona, magnetic geometry, and nonstationarity remain model-dependent

The conservative position treats timing as a test of complete variability models, not a metric inversion. A competing position identifies a small set of peaks directly with Kerr test-particle frequencies; it gains precision by assuming the most disputed step. Evidence against that shortcut includes frequency drift with luminosity, inconsistent spin across identification schemes, and cross-spectral behavior not predicted by the assigned orbit. A null result for one resonance ratio does not exclude other timing models or geometries.

The adversarial check injects frequencies and lags into realistic Poisson, dead-time, gap, and red-noise backgrounds, then recovers them across energy bands and source states. Orbital, precession, diskoseismic, and instability models should predict withheld segments and reproduce mass and radius scaling with calibrated trials. The spacetime interpretation remains unresolved if changing peak identity or disk thickness moves the inferred geometry.

What could change the conclusion. The interpretation could change after a superseding timing or response release, a reproducible simultaneous-feature detection, or a blinded comparison shows that one physical model predicts withheld cross spectra and source-state evolution substantially better than its competitors.

Exercise

At fixed $\bar r$ and $a_*$ , two systems have masses $M$ and $10M$ . Compare their Kerr orbital frequencies and state one reason observed quasi-periodic peaks might not follow this scaling.

Solution

Because $GM\Omega_\phi$ is dimensionless and fixed, the larger system’s orbital frequency is one tenth of the smaller system’s. The same inverse-mass scaling applies to the epicyclic frequencies at the same dimensionless radius and spin. Observed peaks can violate it if state-dependent pressure or magnetic stresses move the emitting radius, or if different sources select different modes. Therefore mass scaling tests a complete identification only after those source variables are controlled.

References

Bambi, C. (2017), “Testing Black Hole Candidates with Electromagnetic Radiation,” Reviews of Modern Physics 89, 025001. DOI.
Uttley, P., Cackett, E. M., Fabian, A. C., Kara, E., and Wilkins, D. R. (2014), “X-Ray Reverberation Around Accreting Black Holes,” Astronomy and Astrophysics Review 22, article 72. DOI.