Identifying Astrophysical Black Holes

Astronomical black holes are identified by converging constraints, not by one iconic signature. Dynamics establishes a dark mass; size measurements restrict the volume associated with it; accretion supplies repeatable spectral and timing behavior; and surface-sensitive tests ask whether infalling energy returns. Alternative hypotheses must then reproduce the same evidence while also forming and remaining stable. This chapter builds that inference ladder and marks exactly where each rung stops.

Helpful background. Black-Hole Geometry supplies horizon and photon-region definitions; Relativistic Matter and Compact Objects supplies material compact objects and stability; Probability and inference for gravity supplies likelihoods, priors, upper limits, and selection functions; Gravitational-Wave Astronomy supplies detector-level mass and merger-response inference.

Route from motion to a black-hole interpretation

The six pages are ordered by how much model structure is needed.

1. Dynamical Mass Measurements compares binary, individual-star, stellar-ensemble, gas, and maser tracers. The immediate result is an enclosed-mass posterior with tracer-specific nuisance parameters.
2. Mass Functions in Compact Binaries derives the spectroscopic lower bound and shows how mass ratio and inclination turn it into a conditional compact-object mass.
3. Compactness and Size Constraints combines variability, orbital volume, interferometric brightness, and surface-emission information without calling any one of them a material radius.
4. Accretion Signatures and State Classification defines source states in measured spectral and timing coordinates before attaching a disk–corona or jet geometry.
5. Evidence for Horizons and the Absence of Surfaces tests specified surface models through energy balance, bursts, imaging, and merger response while retaining their evasion conditions.
6. Alternative Compact-Object Hypotheses requires formation, admissibility, stability, interaction, and detector predictions from each competing model.

The order prevents three shortcuts. A large mass does not prove compactness; compactness does not uniquely imply a horizon; and a phenomenological accretion state does not determine the central boundary condition.

The common inference chain

For a single-lined binary, spectroscopy supplies

$$f(M_1)=\frac{P K_2^3}{2\pi G}(1-e^2)^{3/2} =\frac{M_1\sin^3i}{(1+q)^2}\le M_1.$$

Here $M_1$ is the unseen mass, $q=M_2/M_1$ is the donor-to-unseen mass ratio, $P$ is the period, $K_2$ is the donor’s radial-velocity semiamplitude, $e$ is orbital eccentricity, and $i$ is inclination, with $i=90^\circ$ edge-on. The inequality is durable within the orbital model. Turning it into $M_1$ requires a joint light-curve, line, atmosphere, and inclination model. Casares and Jonker 2014, § 2 and eq. (1) is the anchor for this distinction; its displayed expression specializes to circular orbits.

Size information introduces a second forward map. In ordinary units,

$$\theta_g=\frac{GM}{c^2D}, \qquad R_{\rm var}\lesssim\frac{\delta c\,\Delta t_{\rm obs}}{1+z}.$$

Here $D$ is angular-diameter distance, $\delta$ is the emitting region’s Doppler factor, and $z$ is cosmological redshift. The first is a mass-distance conversion scale; the second bounds a coherently varying component under a specified causal geometry. An interferometer measures visibilities of a brightness distribution, while a variability analysis measures a time-series statistic. Shared distance or mass information should enter once, and source-specific nuisance parameters should not be forced to agree.

Surface tests add energy and boundary conditions. For

$$R=\frac{2GM}{c^2}(1+\epsilon), \qquad 1+z_s=\sqrt{\frac{1+\epsilon}{\epsilon}},$$

the model is a static spherical surface of circumferential radius $R$ in a Schwarzschild exterior. The parameter $\epsilon\gt0$ measures its fractional separation from $2GM/c^2$, and $z_s$ is its gravitational redshift to infinity. Small $\epsilon$ makes escape redshifted and delayed. A steady surface must nevertheless balance deposited energy unless it stores energy or uses another channel. The observation constrains a specified absorptivity, emissivity, heat capacity, age, and accretion model—not the word “surface.” Cardoso and Pani 2019, §§ 5–6 places thermal, imaging, tidal, and gravitational response in one model-dependent comparison.

What each class of evidence establishes

Evidence	Robust conclusion under stated assumptions	What remains to be shown
Keplerian orbit or binary mass function	A dark enclosed mass or lower mass bound	Inclination, distributed mass, compactness, and boundary condition
Small orbital or emitting scale	High compactness for a declared size map	Whether the scale is a surface, critical curve, photosphere, or variable component
Spectral–timing state	Reproducible behavior in declared bands and time intervals	A unique disk, corona, jet, inner radius, or central object
Weak surface-like radiation	Exclusion of surfaces with the modeled deposition and thermal response	Long storage, unusual absorption, another channel, formation, and stability
Ring-like horizon-scale image	Ultracompact lensing and emission structure under transfer and calibration models	A model-independent event horizon or unique plasma geometry
Agreement with a Kerr black-hole source model	Consistency across tested data and alternatives	Exhaustion of all possible horizonless theories

The strongest black-hole case is overconstrained: mass, compactness, lifetime, accretion response, and multimessenger behavior agree while credible alternatives fail for independent reasons. It is not made stronger by calling correlated transformations of one dataset independent evidence.

Data, calibration, and model boundaries

Every observational result in the chapter is attached to a named data release or machine-readable table, a calibration boundary, selection rules, a forward likelihood, priors, and a systematics test. Radial velocities need wavelength and barycentric calibration; astrometry needs a reference frame and distortion model; X-ray states need response, background, dead time, energy bands, and timing intervals; interferometry needs gains, flags, scattering, and visibility-domain analysis.

Statistical uncertainty is conditional on that machinery. Model dependence includes donor irradiation, inclination, anisotropy, distributed mass, Doppler factor, emissivity, opacity, disk atmosphere, red noise, surface heat storage, and the prior volume of an alternative family. The chapter’s observational and frontier claims use the evidence cutoff 2026-08-18; a later calibration, catalog, source reclassification, or stable alternative can change an assessment without changing the underlying orbital or redshift equations.

A useful reproduction sequence is to recover the closest-to-data quantity first: a velocity curve, mass function, visibility scale, hardness–rms track, or flux upper limit. Only then add the source interpretation. An alternative model should be passed through the same response and selection rather than compared to a deconvolved summary produced under the reference model.

Connections to other topics

This chapter uses but does not redevelop horizon geometry or material equilibrium. The construction and stability of boson stars, gravastars, wormholes, and other compact alternatives belongs in Relativistic Matter and Compact Objects. The astrophysical question here is whether a complete member of one of those families can match the mass, compactness, environment, and response of a particular source population.

Detector-specific ringdown and residual likelihoods remain in Gravitational-Wave Astronomy, while operational deviations from general relativity continue in Tests of Gravity. Later chapters build the source physics that this chapter initially treats as nuisance structure: accretion flows, radiation, jets, orbital environments, imaging, and cross-method inference.

Compare what the observations establish

A binary has $f(M_1)=6.0M_\odot$, $q=0.10$, and $i=60^\circ$. Its fastest coherent X-ray variation is $1.0\ {\rm ms}$. Assume no cosmological redshift or beaming for this diagnostic. Compute $M_1$, the causal scale $R_{\rm var}$, and $GM_1/(R_{\rm var}c^2)$. Does this establish an event horizon?

Solution

The conditional mass is

$$M_1=6.0M_\odot\frac{1.10^2}{\sin^3 60^\circ} \simeq11.2M_\odot.$$

The light-crossing estimate gives

$$R_{\rm var}\lesssim c\Delta t=300\ {\rm km}.$$

Using $GM_\odot/c^2=1.4766\ {\rm km}$,

$$\frac{GM_1}{R_{\rm var}c^2} \gtrsim\frac{11.2(1.4766\ {\rm km})}{300\ {\rm km}} \simeq0.055.$$

The exercise combines a conditional dynamical mass with a conditional size of the varying component. It shows a compact source but does not establish that the whole emitter has radius $R_{\rm var}$, that this radius is a material boundary, or that an event horizon exists. A horizon argument additionally needs tighter spatial information and model-explicit surface, stability, and response tests.

References

• Cardoso, Vitor, and Paolo Pani. “Testing the Nature of Dark Compact Objects: A Status Report.” Living Reviews in Relativity 22 (2019): 4. DOI.
• Casares, Jorge, and Peter G. Jonker. “Mass Measurements of Stellar and Intermediate-Mass Black Holes.” Space Science Reviews 183 (2014): 223–252. DOI.