Electromagnetic Mass and Spin Inference

Electromagnetic observations do not measure black-hole mass or spin as isolated numbers. They constrain a gravitational potential, emitting area, broadened transfer function, variability frequency, time delay, or polarization pattern; a physical model then maps that observable to $M$ or $a_*$. This chapter develops the map and its failure modes. Mass inference must distinguish a central point mass from tracer dynamics and distributed matter. Spin inference must establish a disk edge, mode, or geometry before identifying it with a Kerr scale. Cross-method agreement is meaningful only after shared distance, inclination, calibration, and source assumptions are counted once.

Helpful background. Black-Hole Geometry and Mechanics supplies Kerr orbital scales and photon transfer; Relativistic Matter and Compact Objects supplies disk, atmosphere, and plasma physics; Black-Hole Astrophysics supplies source states and observing context; and Probability and Inference for Gravity supplies likelihoods, selection effects, nuisance parameters, and posterior prediction.

Enter this chapter

The six pages form an inference sequence rather than a list of techniques:

1. Stellar, Gas, and Maser Dynamical Masses compares three tracer equations and their anisotropy, pressure, warp, distance, and resolution systematics.
2. Continuum-Fitting Spin Inference converts a thermal emitting area to an inner radius under a selected thin-disk state.
3. Reflection-Based Spin Inference asks whether relativistic broadening survives alternative continuum, absorption, density, ionization, emissivity, and truncation models.
4. Timing and Precession Spin Probes separates a measured QPO peak from its orbital, epicyclic, precession, or fluid-mode identification.
5. Reverberation and Polarization Constraints requires one disk–corona geometry to predict complex cross-spectra and Stokes observables.
6. Cross-Method Consistency and Systematics combines native likelihoods with shared nuisance parameters rather than multiplying published posteriors.

The scope is electromagnetic source inference. Kerr geometry supplies the relations used here, while detailed detector pipelines and production transfer codes require separate computational treatments. No method is presented as assumption free, and a compact dynamical mass by itself is not treated as proof of an event horizon.

Mass is a potential parameter

In the spherical approximation, the common circular-speed scale is

$$v_c^2(r)=\frac{G}{r} \left[M_\bullet+M_{\rm ext}(<r)\right].$$

For a flattened potential, use $v_c^2=R\,\partial\Phi/\partial R$ in the equatorial plane instead; mass inside a sphere alone does not determine the force. Stars sample a phase-space distribution whose projection depends on anisotropy, mass-to-light ratio, and deprojection. A gas disk adds pressure and noncircular flow. A maser disk can approach Keplerian motion but introduces warp, spot association, self-gravity, and a mass–distance covariance. The detector sees spectra, images, proper motions, accelerations, or spot catalogs, not $v_c(r)$ without convolution.

The methods therefore share $M_\bullet$, distance, center, and systemic velocity while retaining tracer-specific nuisance parameters. Resolving $r_{\rm infl}=GM_\bullet/\sigma_\star^2$ helps separate a point mass from extended matter, but it neither guarantees a correct tracer model nor proves a horizon. Kormendy and Ho compare the stellar and gas measurement architecture in Kormendy and Ho 2013, §§ 2.1–2.3.

Spin begins with an intermediate observable

The continuum method constrains an apparent area. Its leading scaling,

$$R_{\rm in}\propto \frac{D f_{\rm col}^2}{\sqrt{\cos i}}\, \frac{\sqrt{F_{\rm bol}}}{T_{\rm col}^2},$$

shows immediately why distance, inclination, and atmosphere hardening matter. Reflection instead constrains a relativistically broadened rest-frame spectrum whose inner radius is degenerate with illumination, ionization, density, abundance, absorption, and continuum. Both become spin methods only after supporting

$$R_{\rm in}=r_{\rm ISCO}(a_*)\frac{GM}{c^2}.$$

The state restriction is physical: a thin disk with a stable edge is not the same model as a truncated, thick, wind-dominated, or strongly Comptonized flow. Radius should be inferred and challenged before the nonlinear ISCO map is imposed. Reynolds lays out the two methods and their distinct systematic boundaries in Reynolds 2021, §§ 3.2–3.3.

Timing replaces an emitting area with a frequency identification. Kerr predicts linked orbital, radial-epicyclic, and vertical-epicyclic frequencies, but a single QPO centroid can trade radius against mass and spin. A stronger model predicts a simultaneous frequency set, harmonics, quality factors, energy-dependent phase, state evolution, and mass scaling. The rare GRO J1655−40 triplet illustrates both the promise and the scarcity of this test Ingram and Motta 2019, § 6.4.

Geometry must agree in observable space

Reverberation measures a complex impulse response, not simply a height divided by $c$. Direct-continuum dilution, propagation lags, phase wrapping, coronal extent, disk thickness, and reference-band choice modify the lag. Polarimetry measures an event azimuth distribution that becomes $(I,Q,U)$ only after modulation, angle, response, leakage, and background calibration. Disk scattering, Comptonization, returning radiation, and plasma propagation can produce competing polarization patterns.

These observables are valuable because they challenge the geometry used by spectral and timing methods. The strongest workflow fits a common inner edge, inclination, and corona to count spectra, cross-spectra, and Stokes spectra, then predicts a held-out observable. The Fourier construction and its dilution boundary are reviewed in Uttley et al. 2014, §§ 2.1–2.4.

For several methods $k$, the organizing joint model is

$$p(z,\{\eta_k\}\mid\{d_k\}) \propto p(z)\prod_k p(d_k\mid z,\eta_k)p(\eta_k\mid z),$$

where $z$ contains each shared mass, distance, inclination, spin, inner-edge, and calibration quantity once. A published posterior already contains its prior; multiplying it by another posterior can duplicate the same dynamical information. Cross-method consistency should instead be assessed through native likelihoods, explicit covariance, and leave-one-method-out predictions. The five frontier pages use the evidence cutoff 2026-08-18 because calibration, source constraints, and model verdicts can change.

Review the chapter

Transfer diagnostic. A continuum analysis has fixed flux, color correction, temperature, and inclination, so its physical radius scales as $R_{\rm in}\propto D$. New astrometry raises the distance by $5\%$, while a revised dynamical model raises the mass by $10\%$. Determine the change in the dimensionless radius $r_{\rm in}=c^2R_{\rm in}/(GM)$, the direction of the prograde-spin shift if $r_{\rm in}$ is identified with the ISCO, and the change in any Kerr timing frequency evaluated at fixed dimensionless radius. Explain why a reflection posterior using the same dynamical model is not independent.

Solution

The physical continuum radius grows by $1.05$, but the gravitational radius grows by $1.10$. Hence

$$\frac{r_{\rm in,new}}{r_{\rm in,old}} =\frac{1.05}{1.10}\simeq0.955,$$

a $4.5\%$ decrease. On the prograde Kerr branch, a smaller ISCO corresponds to a larger $a_*$, so the conditional continuum spin shifts upward. At fixed dimensionless radius and spin, orbital and epicyclic frequencies scale as $c^3/(GM)$; the revised prediction is therefore smaller by $1/1.10\simeq0.909$, or about $9.1\%$.

A reflection analysis may have little direct distance dependence, but if it uses the same mass–inclination likelihood, alignment assumption, or ISCO mapping, those inputs are shared. Multiplying its posterior by the revised continuum posterior as independent would count the dynamical information twice. The native spectral likelihoods should instead be joined to one copy of the new dynamical likelihood.

References

• Ingram, Adam, and Sara Motta. 2019. “A Review of Quasi-Periodic Oscillations from Black Hole X-Ray Binaries: Observation and Theory.” New Astronomy Reviews 85:101524. DOI.
• Kormendy, John, and Luis C. Ho. 2013. “Coevolution (Or Not) of Supermassive Black Holes and Host Galaxies.” Annual Review of Astronomy and Astrophysics 51:511–653. DOI.
• Reynolds, Christopher S. 2021. “Observational Constraints on Black Hole Spin.” Annual Review of Astronomy and Astrophysics 59:117–154. DOI.
• Uttley, Phil, et al. 2014. “X-Ray Reverberation around Accreting Black Holes.” Astronomy and Astrophysics Review 22:72. DOI.