Compactness and Size Constraints

Mass and size answer different questions. Dynamics may establish a large enclosed mass, while variability, resolved motion, interferometry, lensing, occultation, or surface emission constrain the region associated with it. Combining them yields compactness, but every size observable refers to a source model: an emitting photosphere, a brightness distribution, a causal region, or an orbital radius need not be a material surface. The observational examples here are current through 2026-08-18.

Required background. Probability and inference for gravity supplies likelihoods, correlated nuisance parameters, upper limits, and selection effects.

Helpful background. Black-Hole Geometry supplies horizon and photon-region scales; Relativistic Matter and Compact Objects supplies material compactness and stability limits; Gravitational-Wave Astronomy supplies compactness information inferred from waveforms under binary-source models.

One mass scale, several kinds of size

In the site units, the dimensionless compactness of a body with physical mass $M$ and circumferential radius $R$ is

$$\mathcal C=\frac{GM}{R}.$$

Restoring $c$ gives $\mathcal C=GM/(Rc^2)$. The gravitational length and angular scale are

$$r_g=\frac{GM}{c^2}, \qquad \theta_g=\frac{GM}{c^2D},$$

where $D$ is the angular-diameter distance for a cosmological source and is effectively the ordinary geometric distance for the nearby examples here. These scales convert an independently inferred mass into a prediction for strong-field angular structure. They do not say that the source has physical radius $r_g$. For orientation, a Schwarzschild horizon has circumferential radius $2r_g$, whereas its distant-observer critical curve has angular radius $3\sqrt3\,\theta_g$. Neither radius specifies where an accretion flow emits most brightly.

Different observations constrain different effective sizes:

Method	Quantity constrained	Assumption that turns it into a size
Variability	A timescale or cross-band lag	Causal connection, Doppler factor, geometry, and redshift
Orbital motion	Enclosed mass over an orbital radius	A gravitational orbit and controlled distributed mass
Eclipses or occultations	Projected emitting region	Orbital geometry and a model of the occulting body
Interferometry	Fourier components of sky brightness	Calibration, scattering, variability, and a brightness model
Surface-emission limit	Escaping luminosity or flux upper limit	Accretion power, thermalization, redshift, absorption, and steady state

The compactness argument is strongest when methods with different nuisance physics agree. It is weakest when several reported “sizes” are transformations of the same prior or source model.

Causality and variability bounds

For an observed variability time $\Delta t_{\rm obs}$, a common source-frame estimate is

$$R\lesssim\frac{\delta c\,\Delta t_{\rm obs}}{1+z},$$

where $\delta$ is a Doppler factor and $z$ a cosmological redshift. This is a conditional causal bound. A rotating beam can vary faster than the transverse emitting region; a compact fluctuation can modulate a larger photosphere; absorption can change without rearranging the whole source; gravitational lensing can create delayed subimages. Conversely, photon diffusion or viscous response can make variability slower than the light-crossing time.

The safe inference is therefore about a coherently varying component under a declared geometry. The data record must state the band, cadence, time standard, gaps, flare or state selection, variability statistic, red-noise model, and trials. A shortest observed bin is not automatically a physical timescale.

Resolved orbital motion supplies a complementary upper bound on the volume containing the mass. If a central point-mass model explains orbits down to $r_{min}$ while an extended component is constrained, then the characteristic density is at least of order

$$\bar\rho\gtrsim\frac{3M}{4\pi r_{min}^3}.$$

The argument against a dark cluster then uses its collision, evaporation, or luminosity timescale. It does not follow from density alone without specifying the candidate constituents and lifetime.

What horizon-scale interferometry measures

A very-long-baseline interferometer samples the complex visibility

$$V(u,v)=\int I(\alpha,\beta) e^{-2\pi i(u\alpha+v\beta)}\,d\alpha\,d\beta,$$

not an image pixel and not a surface radius. Here $I(\alpha,\beta)$ is sky brightness and $(u,v)$ is a projected baseline in wavelengths. Station gains, atmospheric phase, bandwidth and time averaging, scattering, sparse Fourier coverage, and intrinsic variability all sit between the sky and the released data.

Early 1.3-mm VLBI resolved compact structure in Sagittarius A* on gravitational scales, but the inferred intrinsic size required a scattering and brightness model (Doeleman et al. 2008, main result and Methods). The Event Horizon Telescope later found visibility features and reconstructed ring-like emission around M87*. Its comparison with geometric and GRMHD libraries links an emission-ring scale to a mass-distance scale; it explicitly separates measured visibilities, reconstruction, calibration, and physical interpretation (EHT Collaboration 2019, §§ 3–6).

A geometric critical curve is fixed by spacetime and observer. The observed bright ring also depends on emissivity, absorption, Doppler beaming, lensing order, optical depth, and finite resolution. A central brightness depression can be influenced by capture, lack of background illumination, foreground absorption, or source morphology. Thus a horizon-scale ring is strong evidence for an ultracompact gravitational field under the forward model, not direct imaging of a material surface or event horizon.

Combining constraints without multiplying assumptions

Let $d_j$ denote data from method $j$ and let $\eta_j$ denote its method-specific nuisance parameters. A coherent combination shares only genuinely common variables:

$$p(M,R,\{\eta_j\}\mid\{d_j\},S) \propto p(M,R,\{\eta_j\}\mid S) \prod_j p(d_j\mid M,R,\eta_j,S_j).$$

Distance may be shared by dynamics and angular imaging. Inclination may connect an orbit and disk model but only if alignment is justified. A mass prior used to calibrate a predicted ring diameter cannot then be counted again as independent image evidence for that mass. Upper limits require the detection process and selection threshold, not an invented Gaussian measurement centered at zero.

The data and calibration boundary must be explicit. For an EHT example, bind the analysis to the EHT 2017 M87 calibrated data release and its DOI in the EHT data-products repository, retain station, scan, baseline, frequency, and flag selections, and state gain and scattering treatment. For timing, name the event or light-curve product, clock correction, response, background, state window, and red-noise likelihood. Statistical uncertainty is variation conditional on these choices; systematic and model dependence comes from calibration, distance, emissivity, opacity, scattering, Doppler factor, variability, and priors.

A practical reproduction path is two-stage. First recover a published visibility-domain angular scale from the released data with the authors’ selections. Then compute an independent light-crossing or orbital bound from a separately released time series or orbit. Replace the brightness model by an extended or optically thick alternative and verify which conclusion survives. Reconstructing the same image with the same prior is not an independent size measurement.

Compactness is not a horizon theorem

An exterior close to Kerr can be shared by a black hole and by some horizonless models outside their surface or throat. Extremely compact material models face formation, stability, causality, light-ring, accretion, and dissipation constraints, but those constraints must be evaluated model by model. Cardoso and Pani 2019, §§ 2–6 reviews the distinction between compactness evidence and evidence for a particular boundary condition.

The appropriate conclusion has three levels: data constrain an emitting or orbital scale; a source model maps that scale to compactness; additional formation, stability, surface, and response tests discriminate a horizon from alternatives. Evidence for horizons and the absence of surfaces develops the last step.

Check the gravitational angular scale

Take Sagittarius A* to have $M=4.0\times10^6M_\odot$ and distance $D=8.2\ {\rm kpc}$. Using $GM_\odot/c^2=1.4766\ {\rm km}$, compute $r_g$ and $\theta_g$ in microarcseconds. Explain why this is not the predicted full diameter of a brightness ring.

Solution

The gravitational length is

$$r_g=(4.0\times10^6)(1.4766\ {\rm km}) =5.91\times10^6\ {\rm km} =5.91\times10^9\ {\rm m}.$$

With $D=8.2\times10^3\ {\rm pc}=2.53\times10^{20}\ {\rm m}$,

$$\theta_g=\frac{r_g}{D} =2.34\times10^{-11}\ {\rm rad} \simeq4.8\ \mu{\rm as}.$$

This is one gravitational angular radius. The critical-curve diameter has a metric- and inclination-dependent multiple of $\theta_g$; the peak brightness ring additionally depends on emission, absorption, scattering, and resolution. Treating $4.8\ \mu{\rm as}$ as a measured material radius would confuse a conversion scale with both geometry and source structure.

References

• Cardoso, Vitor, and Paolo Pani. “Testing the Nature of Dark Compact Objects: A Status Report.” Living Reviews in Relativity 22 (2019): 4. DOI.
• Doeleman, Sheperd S., et al. “Event-Horizon-Scale Structure in the Supermassive Black Hole Candidate at the Galactic Centre.” Nature 455 (2008): 78–80. DOI.
• Event Horizon Telescope Collaboration. “First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole.” Astrophysical Journal Letters 875 (2019): L6. arXiv:1906.11243.