Experiment: Black-Hole Observations
Tests Phenomenon 42.5, Equation (42.8) and Equation (50.4). Assuming Equation (50.2).
The stellar orbits around Sgr A*, the Event Horizon Telescope images, and the X-ray binary mass functions: this chapter assembles the observational case that the Schwarzschild and Kerr geometries of Schwarzschild Geometry and Black Holes are realized in Nature. The claim under test is specific—a mass confined within its gravitational radius, with no material surface—and it has been established four independent ways: dynamically, by the mass function of the X-ray binary Cygnus X-1 [Webster:1972] [Bolton:1972]; astrometrically, by the stellar orbits around Sgr A* [Ghez:2008] [Gillessen:2009]; by direct imaging at event-horizon scale in M87* [Akiyama:2019] and Sgr A* [Akiyama:2022]; and by the gravitational-wave mergers of Experiment: Gravitational Waves [Abbott:2016].
The four channels are treated as one experiment because they test one prediction from four directions, at masses spanning six orders of magnitude and field strengths spanning the gap between a stellar orbit a thousand gravitational radii out and light circling the photon sphere itself. No single channel proves the case alone; the concordance is the measurement, and Section 50.5 states with some care what it does and does not establish.
Historical context and the prediction under test
From invisible companions to relativistic collapse
The idea is older than the theory it now tests. In 1784 John Michell, reasoning with Newtonian corpuscles of light, computed that a body of the Sun's density and five hundred times its diameter would have an escape speed exceeding the speed of light, so that “all light emitted from such a body would be made to return towards it”—and he added the observation on which this entire chapter rests: such invisible bodies could nevertheless be detected through the motion of luminous ones revolving about them [Michell:1784]. Every dynamical mass in Section 50.4 is Michell's method carried out.
General relativity replaced the escape-speed argument with a sharper structure. Within weeks of the field equations, Schwarzschild produced the exact exterior solution for a spherical mass [Schwarzschild:1916], with its characteristic radius \(r_{\mathrm{s}}=2GM/c^{2}\); for decades the surface at \(r_{\mathrm{s}}\) was read as a coordinate accident or dismissed as physically unreachable. Two 1939 calculations made it unavoidable. Oppenheimer and Volkoff showed that cold neutron matter has a maximum mass, of order a solar mass in their equation of state, above which no static configuration exists at all [Oppenheimer:1939a]; and Oppenheimer and Snyder integrated the collapse of a pressureless ball past that point, finding that the matter crosses \(r_{\mathrm{s}}\) in finite proper time while a distant observer sees the surface freeze and redshift away [Oppenheimer:1939b]. Kerr's rotating generalization [Kerr:1963] and Penrose's theorem that collapse past a trapped surface ends in a singularity regardless of symmetry [Penrose:1965] completed the theoretical picture, which Schwarzschild Geometry and Black Holes develops in full: if masses above the cold-matter limit exist and lose their support, general relativity permits them no fate but a black hole.
Observation entered by another door. A 1962 sounding-rocket flight instrumented to look for X-ray fluorescence from the Moon instead recorded a bright source in Scorpius and a diffuse background—the birth of X-ray astronomy [Giacconi:1962]; compact accreting objects, it turned out, announce themselves most loudly at kiloelectronvolt energies. In 1963 Schmidt recognized the emission lines of the “radio star” 3C 273 as hydrogen lines redshifted by \(z=0.158\) [Schmidt:1963], which put the object at cosmological distance and its luminosity orders of magnitude beyond what any assembly of stars in so small a volume could supply. Lynden-Bell drew the inference that quasars are accreting supermassive collapsed objects, that dead quasars must therefore litter the present universe, and that the nuclei of ordinary galaxies—our own included—are where they hide [LyndenBell:1969]. The compact nonthermal radio source now called Sgr A* was found at the centre of the Milky Way in 1974 [Balick:1974]. The four observational programmes reported below are the systematic interrogation of these two populations, stellar and supermassive.
What general relativity predicts
The falsifiable signatures, each derived in Schwarzschild Geometry and Black Holes, are these.
First, a purely Newtonian instrument: for a spectroscopic binary the period and radial-velocity amplitude of the visible star alone bound the mass of its unseen companion from below, Equation (50.2). The bound involves no model of the companion, so wherever it exceeds the maximum mass of a cold compact star (Compact Stars and Relativistic Astrophysics), the companion can only be a black hole. This inequality is stated and proved with the Cygnus X-1 data in Section 50.4.1.
Second, a horizon at \(r_{\mathrm{s}}=2GM/c^{2}\) and hence no material surface: no thermonuclear flashes from accumulated fuel, no quiescent re-radiation of the accretion energy deposited on a surface. Accreted energy not radiated on the way down is lost. The comparison that turns this into a measurement is made in Section 50.5.
Third, the strong-field orbital mechanics of the Schwarzschild geometry: an innermost stable circular orbit truncating the accretion disc and fixing its radiative efficiency, gravitational redshift and periastron advance for test bodies—the same advance formula, Equation (42.8), that Mercury tests at solar field strengths, here applied four orders of magnitude deeper.
Fourth, a photon sphere. Light passing a black hole closer than the critical impact parameter is captured, so a black hole seen against its own emission shows a dark disc—a shadow—of angular diameter \(3\sqrt{3}\,r_{\mathrm{s}}/D\simeq5.2\,r_{\mathrm{s}}/D\) at distance \(D\), nearly independent of spin and viewing angle. Falcke, Melia and Agol computed the image for Sgr A* and showed that at millimetre wavelengths, where the accretion flow turns transparent and the atmosphere still permits interferometry, the shadow of the Galactic centre—about thirty microarcseconds for the mass then favoured—is resolvable from the ground [Falcke:2000]. Inverted, the shadow size weighs the hole: Equation (50.4).
Fifth, gravitational radiation: an inspiralling binary chirps and the merged remnant rings down at the quasinormal frequencies of a Kerr hole, per Gravitational-Wave Theory. That channel has its own chapter, Experiment: Gravitational Waves; here it contributes masses.
Apparatus
X-ray observatories
The stellar-mass channel is X-ray astronomy, because gas spiralling onto an object a few tens of kilometres across is heated to \(10^{7}\)–\(10^{8}\) kelvin and radiates at kiloelectronvolt energies, to which the atmosphere is opaque. The first generation of instruments were gas proportional counters flown above it: windowed tubes in which an incoming X-ray photoionizes the fill gas and the charge avalanche measures its energy, with sensitivity in the few- to tens-of-\(\mathrm{keV}\) band, no true imaging, and fields of view defined by mechanical collimators [Giacconi:1962]. Rockets gave minutes of exposure; the scanning and pointed satellites that followed gave years, all-sky monitors to catch transient outbursts, and—decisive for Cygnus X-1—timing at millisecond resolution, which revealed the rapid aperiodic flickering that marks an emitting region light-milliseconds across [Remillard:2006]. Later observatories added CCD and grating spectrometers resolving the iron K\(\alpha\) fluorescence line near \(6.4\,\mathrm{keV}\), whose gravitationally redshifted, relativistically broadened profile probes the innermost disc [Tanaka:1995]. The optical half of the apparatus is conventional stellar spectroscopy of the companion star, from which come the period and velocity amplitude that feed the mass function; the modern addition is very-long-baseline radio astrometry of the compact object itself, precise enough to measure its trigonometric parallax [MillerJones:2021].
Adaptive optics and interferometry on the Galactic Centre
The Galactic centre is invisible at optical wavelengths behind about thirty magnitudes of dust extinction, but opens up in the near-infrared at wavelengths near \(2.2\,\mu\mathrm{m}\). There the diffraction limit of a \(10\,\mathrm{m}\) telescope is \(\lambda/D\approx2.2\times 10^{-7}\,\mathrm{rad}\), about \(45\) milliarcseconds — fine enough to separate the stars of the central cluster, if the atmosphere can be defeated. Speckle imaging (freezing the turbulence in millisecond exposures and reconstructing afterwards) and then adaptive optics (measuring the wavefront on a reference star or laser beacon and correcting it with a deformable mirror at kilohertz rates) brought the Keck \(10\,\mathrm{m}\) and VLT \(8.2\,\mathrm{m}\) telescopes to that limit, and two decades of such imaging, tied to a radio reference frame and combined with spectroscopic radial velocities, produced the stellar orbits of Section 50.4.2 [Ghez:2008] [Gillessen:2009] [Genzel:2010].
The precision instrument is GRAVITY, the beam combiner that phase-references all four VLT unit telescopes into a single interferometer with baselines up to \(130\,\mathrm{m}\) [Abuter:2017]. The fringe spacing at \(2.2\,\mu\mathrm{m}\) is \(\lambda/B\approx1.7\times 10^{-8}\,\mathrm{rad}\), some \(3.5\) milliarcseconds, and because the quantity measured is a fringe phase against a nearby reference source, positions within that fringe are read to a small fraction of it: astrometry at the level of a few tens of microarcseconds, roughly a hundredfold gain on single-telescope imaging. This is the instrument that turned the S2 orbit from a mass measurement into a test of general relativity [Abuter:2018] [Abuter:2020].
The Event Horizon Telescope
Resolving an event horizon takes another three orders of magnitude. The two largest black-hole shadows on the sky—Sgr A* and M87*— subtend about fifty microarcseconds, so the required resolution is \(\sim2\times 10^{-10}\,\mathrm{rad}\), which at an observing wavelength of \(1.3\,\mathrm{mm}\) demands an aperture the size of the Earth: \(\lambda/D\approx1.2\times 10^{-10}\,\mathrm{rad}\), about \(25\) microarcseconds, for baselines near \(1.1\times 10^{7}\,\mathrm{m}\). The Event Horizon Telescope synthesizes exactly that, by very-long-baseline interferometry: in April 2017, eight radio telescopes at six sites from Hawaii to the South Pole observed simultaneously at \(1.3\,\mathrm{mm}\), each recording the raw voltage stream against a hydrogen-maser clock, with the phased ALMA array contributing the sensitivity of a \(73\,\mathrm{m}\) equivalent dish. Petabytes of recorded signal were later cross-correlated at dedicated correlator facilities, fringe-fitted, and calibrated through the chain described in the instrument papers accompanying [Akiyama:2019]. The wavelength is chosen twice over: it is short enough that the synchrotron emission of the accretion flow becomes optically thin, so the line of sight reaches the horizon scale, and long enough that the atmosphere and the antennas still support coherent interferometry [Falcke:2000].
Gravitational-wave interferometers, the fourth apparatus, are described in Experiment: Gravitational Waves [Aasi:2015].
Procedure
Four measurement protocols, one per channel.
Radial velocities. The optical companion's spectrum is observed over many orbits; the Doppler shift of its lines yields the period \(P\) and velocity semi-amplitude \(K\), and with them the mass function \(f=PK^{3}\!/2\pi G\) of Equation (50.1)—a strict lower bound on the unseen mass [Webster:1972] [Bolton:1972]. Converting the bound into a mass requires the inclination and the companion mass, which enter through ellipsoidal light-curve modelling, the spectral classification of the companion, and the distance; the modern Cygnus X-1 solution fixes the distance by VLBI parallax and fits all elements simultaneously [Orosz:2011] [MillerJones:2021]. For the transient systems the same spectroscopy is performed in quiescence, when the disc is faint and the companion's lines are clean [Remillard:2006].
Astrometric orbits. Individual stars near Sgr A* are followed for decades in position (imaging, then interferometry) and in radial velocity (infrared spectroscopy). A simultaneous fit of all measured stars solves for each star's six orbital elements plus the central mass \(M\) and the Sun–centre distance \(R_{0}\), which the data determine jointly because the angular orbit scales as \(M^{1/3}\!/R_{0}\) while the velocities are absolute [Ghez:2008] [Gillessen:2009]. The relativistic tests are differential: the Keplerian orbit is fitted and the residual at pericentre—in radial velocity for the redshift, in the orbit orientation for the precession—is compared with the prediction, each effect published only after being measured on both sides of pericentre [Abuter:2018] [Abuter:2020] [Do:2019].
Interferometric imaging. The EHT data reduce to complex visibilities on every baseline. Calibration leans on closure quantities—combinations of phases around telescope triangles and of amplitudes around quadrangles from which station-dependent errors cancel exactly. Images were reconstructed by four teams working blind of one another with two independent algorithm families, and only then compared; in parallel, geometric crescent models and libraries of general-relativistic magnetohydrodynamic simulations were fitted directly to the visibilities. The ring diameter is the quantity on which all methods converge, and the procedure was validated end-to-end on synthetic data with known answers [Akiyama:2019] [Akiyama:2022].
Matched filtering. Gravitational-wave strain data are correlated against relativistic waveform banks and the significance established on time-shifted background, per Experiment: Gravitational Waves [Abbott:2016]. This chapter takes from that channel only the masses and the ringdown.
Observations and data
Cygnus X-1 and the X-ray binaries
Cygnus X-1 was among the brightest X-ray sources in the sky and the most erratic, flickering on timescales down to milliseconds. When a radio counterpart pinned its position in 1971, the field contained the ninth-magnitude B0 supergiant HDE 226868. In early 1972 Webster and Murdin [Webster:1972] and, independently, Bolton [Bolton:1972] showed the supergiant to be a single-lined spectroscopic binary with a period of \(5.6\) days, and both drew the same conclusion: for any mass consistent with the supergiant's spectral type, the velocity amplitude demands an unseen companion of several solar masses at least—too heavy for the white dwarfs and neutron stars then known, and dark at every wavelength searched. The phrase used was cautious (“a heavy companion”), but this was the first dynamical identification of a stellar black-hole candidate.
The half-century since has only sharpened the numbers. Table 50.1 collects the modern solution: the distance from VLBI parallax, and with it a black-hole mass of \(21.2\pm2.2\,M_{\odot}\) [MillerJones:2021]. The mass function computed from the period and the modern velocity amplitude is small — the orbit is seen nearly face-on — which is exactly why the full solution with inclination and companion mass is needed here, and why the transient systems below, where the mass function alone settles the question, carry independent weight.
| quantity | as published | SI |
|---|---|---|
| orbital period $P$ | \(5.600\) days | \(4.838\times 10^{5}\,\mathrm{s}\) |
| velocity semi-amplitude $K$ | $75.6\pm0.7$ \(\mathrm{km}/\mathrm{s}\) | \(7.56(7)\times 10^{4}\,\mathrm{m}/\mathrm{s}\) |
| mass function $f$ (computed) | $0.251\,M_{\odot}$ | \(5.0\times 10^{29}\,\mathrm{kg}\) |
| orbital inclination $i$ | about \(27.5^\circ\) | — |
| distance $d$ | $2.22^{+0.18}_{-0.17}\,\mathrm{kpc}$ | \(6.9(6)\times 10^{19}\,\mathrm{m}\) |
| black-hole mass $M_{\mathrm{X}}$ | $21.2\pm2.2\,M_{\odot}$ | \(4.22(44)\times 10^{31}\,\mathrm{kg}\) |
Cygnus X-1 is persistent, fed by the supergiant's wind. The larger population is transient: systems with low-mass companions that outburst for months and then sit quiescent for years, leaving the companion's spectrum clean. In the best of them the inequality proved below needs no help at all: the discovery measurement for V404 Cygni gave a mass function above \(6\,M_{\odot}\) [Casares:1992], refined to \(6.08\pm0.06\,M_{\odot}\) in the modern tabulation [Remillard:2006]—the absolute floor on the compact object's mass, from two spectroscopic observables, already twice the causal neutron-star bound of Compact Stars and Relativistic Astrophysics. Some twenty systems now have dynamical masses, running from about \(5\,M_{\odot}\) to about \(20\,M_{\odot}\), and their X-ray spectra and variability are read through the accretion-disc theory of Shakura and Sunyaev [Shakura:1973][Remillard:2006]. The gravitationally redshifted iron K\(\alpha\) line, first resolved in the active galaxy MCG–6–30–15, extends the same disc diagnostics to within a few gravitational radii of the horizon [Tanaka:1995].
In a number of X-ray binaries the spectroscopic orbit of the optical star requires an unseen companion heavier than any neutron star can be. The first such case was Cygnus X-1, identified in 1972 with the supergiant HDE 226868 [Webster:1972] [Bolton:1972]; its modern dynamical mass is \(21.2\pm2.2\,M_{\odot}\) [MillerJones:2021]. In the transient black-hole binaries the argument is sharper, because for several of them the mass function alone — a quantity built from the orbital period and the radial-velocity amplitude, and independent of both the inclination and the companion's mass — already exceeds the maximum neutron-star mass [Casares:1992][Remillard:2006]. Dynamically measured masses in this population run from about \(5\,M_{\odot}\) to about \(20\,M_{\odot}\). Rests on Theorem 27.30 and Phenomenon 27.35.
Derivation. Derives Phenomenon 50.1. The inequality on which every dynamical black-hole mass rests is worth deriving in full. Let the compact object of mass \(M_{\mathrm{X}}\) and the optical star of mass \(M_{\mathrm{opt}}\) move on a circular orbit of separation \(a\) and period \(P\), inclined at \(i\) to the plane of the sky. The optical star circles the centre of mass at radius \(a_{\mathrm{opt}}=aM_{\mathrm{X}}/\left(M_{\mathrm{X}} +M_{\mathrm{opt}}\right)\), and spectroscopy measures the semi-amplitude of its line-of-sight velocity,
Out of the two observables \(P\) and \(K\) define the mass function
Substituting for \(K\) and then eliminating \(a\) with Kepler's third law \(4\pi^{2}a^{3}=G\left(M_{\mathrm{X}}+M_{\mathrm{opt}}\right)P^{2}\),
Since \(\sin i\leq1\) and \(M_{\mathrm{opt}}>0\), the right-hand side cannot exceed \(M_{\mathrm{X}}\), so
The mass function is therefore a strict lower bound on the mass of the unseen body, extracted from two numbers a spectrograph delivers directly, with no knowledge of the inclination, the distance or the companion star. Wherever \(f\) by itself exceeds the causal neutron-star bound of Compact Stars and Relativistic Astrophysics, the identification of the companion as a black hole follows with nothing further assumed — and this is the only one of the four measurement channels of this chapter that requires no relativistic input at all. For Cygnus X-1, \(P=4.838\times 10^{5}\,\mathrm{s}\) and \(K=7.56\times 10^{4}\,\mathrm{m}/\mathrm{s}\) give
which is why that system's mass rests on the full orbital solution of Table 50.1 rather than on the bound; for V404 Cygni the bound itself is \(6.08\,M_{\odot}\) [Casares:1992] [Remillard:2006] and the conclusion is immediate.
∎Sgr A*: stellar orbits
The decisive Galactic-centre observation was the closing of a single stellar orbit. The star S2, tracked from 1992 and followed through its pericentre passage in 2002, was shown to move on a Keplerian ellipse of period near fifteen years with the radio source Sgr A* at its focus—the first time the enclosed mass could be assigned to a volume as small as a single stellar orbit [Schoedel:2002]. Two decades of astrometry from Keck and the VLT then fixed the mass near \(4\times10^{6}\,M_{\odot}\): the Keck solution gave \((4.1\)–\(4.5)\times10^{6}\,M_{\odot}\) depending on the treatment of the distance [Ghez:2008], the VLT solution \((4.31\pm0.06_{\mathrm{stat}}\pm0.36_{R_{0}}) \times10^{6}\,M_{\odot}\) [Gillessen:2009], and the two programmes' agreement, star by star, is reviewed in [Genzel:2010].
GRAVITY made the orbit relativistic. At the May 2018 pericentre passage—the star moving at \(7650\) kilometres per second, \(2.6\%\) of \(c\)—the measured radial velocity exceeded the Keplerian prediction by about \(200\,\mathrm{km}/\mathrm{s}\), the combined gravitational redshift and transverse-Doppler shift; expressed as a parameter interpolating between Newtonian gravity (\(0\)) and general relativity (\(1\)), the result was \(0.90\pm0.17\) statistical, with a comparable systematic [Abuter:2018]. The Keck group, with independent instruments, astrometry and analysis, found \(0.88\pm0.17\) [Do:2019]. Continued interferometric monitoring through and beyond pericentre then resolved the Schwarzschild precession of the orbit, \(12\) minutes of arc per revolution in the plane of the orbit and prograde, at \(1.10\pm0.19\) of the prediction of Equation (42.8) [Abuter:2020]. The same fit delivers the current best mass and distance, collected in Table 50.2.
| quantity | as published | SI |
|---|---|---|
| orbital period $P$ | \(16.05\) yr | \(5.07\times 10^{8}\,\mathrm{s}\) |
| eccentricity $e$ | \(0.885\) | — |
| angular semi-major axis | \(125\) mas | \(6.06\times 10^{-7}\,\mathrm{rad}\) |
| pericentre distance $r_{\mathrm{p}}$ | $\approx120$ au | \(1.8\times 10^{13}\,\mathrm{m}\) |
| pericentre speed $v_{\mathrm{p}}$ | $\approx0.026\,c$ | \(7.65\times 10^{6}\,\mathrm{m}/\mathrm{s}\) |
| enclosed mass $M$ | $4.26\times10^{6}\,M_{\odot}$ | \(8.47\times 10^{36}\,\mathrm{kg}\) |
| distance $R_{0}$ | \(8.25\,\mathrm{kpc}\) | \(2.545\times 10^{20}\,\mathrm{m}\) |
| redshift parameter (VLT; Keck) | $0.90\pm0.17$; $0.88\pm0.17$ | — |
| precession parameter | $1.10\pm0.19$ | — |
Individual stars in the central arcsecond of the Galaxy have been followed through complete orbits about a common focus coincident with the compact radio source Sgr A*. The star S2, first traced through a pericentre passage in 2002 [Schoedel:2002], closes its orbit in \(16.05\,\mathrm{yr}\) [Abuter:2018], reaching a pericentre distance of about \(1.8\times 10^{13}\,\mathrm{m}\). Two decades of adaptive-optics astrometry from Keck and the VLT [Ghez:2008] [Gillessen:2009][Genzel:2010], later sharpened by interferometry [Abuter:2017], give an enclosed mass of about \(4\times10^{6}\,M_{\odot}\) within that pericentre. The orbits are Keplerian about a point mass to the precision of the data: no extended distribution of the required mass is consistent with them. Rests on Phenomenon 27.35 and Theorem 27.30.
Derivation. Derives Phenomenon 50.2. The mass follows from Kepler's third law applied to the fitted orbit, which is what makes it independent of any model of the source. S2's angular semi-major axis is about \(125\) milliarcseconds; at the measured distance to the Galactic centre, \(R_{0}\approx8.2\,\mathrm{kpc}=2.53\times 10^{20}\,\mathrm{m}\), and with one milliarcsecond equal to \(4.848\times10^{-9}\) in radian measure, this is
With \(P=16.05\,\mathrm{yr}=5.07\times 10^{8}\,\mathrm{s}\),
that is about \(4.1\times10^{6}\,M_{\odot}\); the mass of S2 itself, of order \(10\,M_{\odot}\), is negligible against this and has been dropped. It is worth recording what this does and does not establish. The Schwarzschild radius of such a mass is \(2GM/c^{2}\approx1.2\times 10^{10}\,\mathrm{m}\), so S2's pericentre lies some \(1500\) gravitational radii out. The stellar orbits therefore demonstrate a very large mass in a very small volume — a mean density that no cluster of ordinary stars or stellar remnants could sustain against evaporation and collision for the age of the Galaxy — but they do not by themselves reach the horizon. That is what the imaging of Section 50.4.3 adds.
∎At S2's 2018 pericentre passage the measured radial velocity departed from the Keplerian prediction by an amount corresponding to about \(200\,\mathrm{km}/\mathrm{s}\), in the sense and of the magnitude predicted by the combined gravitational and transverse-Doppler shift [Abuter:2018], a result confirmed by the independent Keck programme [Do:2019]. Continued monitoring then detected the Schwarzschild precession of the orbit itself, an in-plane advance of the pericentre of about twelve minutes of arc per orbit, consistent with general relativity [Abuter:2020]. Both effects were predicted before they were measured, and neither has a Newtonian counterpart. Rests on Phenomena 38.13, 42.5 and 42.6.
Derivation. Derives Phenomenon 50.3. The redshift excess is first order in the small quantities and can be checked directly. A clock at radius \(r\) from a mass \(M\) moving with speed \(v\) is shifted, to first order in \(GM/\left(rc^{2}\right)\) and \(v^{2}/c^{2}\), by
the first term the gravitational redshift of The Equivalence Principle and Classical Tests and the second the transverse Doppler shift of Lorentz Transformations. Both are positive and both peak at pericentre, where the star is deepest in the potential and moving fastest. Taking the mass of Phenomenon 50.2, for which \(GM/c^{2}\approx6.1\times 10^{9}\,\mathrm{m}\), with S2's pericentre radius \(r_{\mathrm{p}}\approx1.8\times 10^{13}\,\mathrm{m}\) and pericentre speed \(v_{\mathrm{p}}\approx7650\,\mathrm{km}/\mathrm{s}\),
so \(z\approx6.7\times10^{-4}\) and \(cz\approx200\,\mathrm{km}/\mathrm{s}\), which is the measured excess.
That the two contributions come out nearly equal is not a coincidence. For a Keplerian orbit the vis-viva relation gives \(v_{\mathrm{p}}^{2}=\left(1+e\right)GM/r_{\mathrm{p}}\) at pericentre, so the transverse-Doppler term is exactly \(\left(1+e\right)/2\) times the gravitational one — a factor \(0.94\) for S2's eccentricity \(e\approx0.88\). The two therefore cannot be separated by this measurement; what is tested is their sum, and disentangling them would require an orbit of very different eccentricity.
The precession is the classical formula of Equation (42.8) evaluated on this orbit. With \(GM/c^{2}\approx6.1\times 10^{9}\,\mathrm{m}\), \(a\approx1.53\times 10^{14}\,\mathrm{m}\) and \(1-e^{2}\approx0.217\),
in radian measure per revolution — about twelve minutes of arc, which is what is measured [Abuter:2020]. Mercury, for comparison, accumulates \(5.0\times 10^{-7}\) radian per revolution (Experiment: The Classical Tests of General Relativity): the same formula, verified at a relativistic parameter \(GM/\left(c^{2}a\left(1-e^{2}\right)\right)\) some four orders of magnitude larger.
∎Event-horizon-scale imaging
In April 2019 the Event Horizon Telescope published the first image of a black hole: the nucleus of M87 resolved into a bright, asymmetric ring of \(42\pm3\) microarcseconds diameter around a deep central brightness depression [Akiyama:2019]. The ring persisted across four observing nights and across every independent imaging pipeline; its south side is brighter, as Doppler beaming requires for emission orbiting with the small approaching-side inclination inferred from the large-scale jet; and the inferred mass, \(\left(6.5\pm0.7\right)\times10^{9}\,M_{\odot}\), is the photon-capture cross-section of Equation (50.4) read off the sky. The polarized image that followed mapped an ordered, partly radial magnetic-field pattern in the ring, of the kind required to launch the jet and consistent with a dynamically important (“magnetically arrested”) field at the horizon [Akiyama:2021].
One honest calibration must be stated. What the interferometer measures is the diameter of the emission ring, which simulations place close to, but systematically outside, the critical curve bounding the shadow — by roughly ten per cent, depending on where the emission peaks. The published mass therefore carries a theory-side calibration from the simulation library, and its error bar does [Akiyama:2019]. The Schwarzschild arithmetic below, which ignores this distinction, overshoots the full analysis by about the same ten per cent, exactly as it should.
In May 2022 the same technique, hardened against the source's minute-scale variability (the dynamical time at the horizon of a \(4\times10^{6}\,M_{\odot}\) hole is minutes, not days), yielded the image of Sgr A*: a ring of \(51.8\pm2.3\) microarcseconds [Akiyama:2022]. Here the prediction was available in advance with no freedom at all: the stellar-orbit mass and distance of Table 50.2 give a Schwarzschild shadow of \(53\) microarcseconds (computed here; spin and inclination lower it by a few per cent). The measured ring agrees. Table 50.3 collects both sources.
| quantity | as published | SI |
|---|---|---|
| M87* ring diameter | $42\pm3\,\mu\mathrm{as}$ | \(2.04(15)\times 10^{-10}\,\mathrm{rad}\) |
| M87* mass | $\left(6.5\pm0.7\right)\times10^{9}\,M_{\odot}$ | \(1.29(14)\times 10^{40}\,\mathrm{kg}\) |
| M87* distance | \(16.8\,\mathrm{Mpc}\) | \(5.18\times 10^{23}\,\mathrm{m}\) |
| Sgr A* ring diameter | $51.8\pm2.3\,\mu\mathrm{as}$ | \(2.51(11)\times 10^{-10}\,\mathrm{rad}\) |
| Sgr A* predicted shadow (computed) | $53\,\mu\mathrm{as}$ | \(2.57\times 10^{-10}\,\mathrm{rad}\) |
Very-long-baseline interferometry at \(1.3\,\mathrm{mm}\) resolves the nucleus of M87 into an asymmetric ring of emission surrounding a central brightness depression. The ring diameter is \(42\pm3\) microarcseconds, and the depression is deeper than any optically thin foreground could produce [Akiyama:2019]. The same technique applied to Sgr A* returns a ring of the diameter expected for the mass that the stellar orbits of Phenomenon 50.2 determine independently [Akiyama:2022]: one object weighed two ways, by Newtonian orbital dynamics and by the size of its photon capture cross-section, with concordant results. Rests on Phenomenon 45.52 and Theorem 45.1.
Derivation. Derives Phenomenon 50.4. The ring size measures a mass. From Schwarzschild Geometry and Black Holes, the photon capture cross-section of a black hole has apparent diameter \(2b_{\mathrm{c}}=3\sqrt{3}\,r_{\mathrm{s}}\simeq5.196\, r_{\mathrm{s}}\), and this is only weakly dependent on spin and viewing angle, so a measured angular diameter \(\theta_{\mathrm{sh}}\) at distance \(D\) gives
For M87, \(\theta_{\mathrm{sh}}=42\) microarcseconds, which is \(2.04\times10^{-10}\) in radian measure, and \(D=16.8\,\mathrm{Mpc}=5.18\times 10^{23}\,\mathrm{m}\), so \(\theta_{\mathrm{sh}}D\approx1.06\times 10^{14}\,\mathrm{m}\) and \(r_{\mathrm{s}}\approx2.03\times 10^{13}\,\mathrm{m}\). Then
about \(7\times10^{9}\,M_{\odot}\), in agreement with the \(\left(6.5\pm0.7\right)\times10^{9}\,M_{\odot}\) obtained from the full image-domain analysis [Akiyama:2019] — the few-per-cent excess of this crude inversion being the emission-ring calibration discussed above. The arithmetic makes plain what is being measured and what is being assumed: an angular size and a distance, converted into a mass by the single dimensionless number \(3\sqrt{3}\), which general relativity supplies with no freedom at all. A different theory of the strong field would predict a different number, and the concordance with the orbital mass of Sgr A* is the test: for \(M=8.47\times 10^{36}\,\mathrm{kg}\) at \(R_{0}=2.545\times 10^{20}\,\mathrm{m}\),
\(53\) microarcseconds, against a measured ring of \(51.8\pm2.3\) [Akiyama:2022].
∎Black-hole masses from gravitational waves
The fourth channel is treated in full in Experiment: Gravitational Waves; the discovery event and its parameter table are Section 47.3. What the channel adds to this chapter is mass, and a different kind of certainty. The dynamical and imaging channels infer a compact mass from the motion of matter and light outside it; a gravitational-wave chirp is emitted by the masses themselves, and its phase evolution measures them with no astrophysical intermediary. GW150914's component masses of about \(36\,M_{\odot}\) and \(29\,M_{\odot}\) [Abbott:2016] sit well above every X-ray binary mass, and the catalogue that followed extends the population to component masses of about \(50\,M_{\odot}\) in the first two observing runs [Abbott:2019] and beyond in later ones—black holes in a mass range that electromagnetic astronomy had never weighed. The remnant's ringdown, consistent in frequency and damping with the least-damped quasinormal mode of a Kerr black hole of the predicted mass and spin [Abbott:2016], is the one direct probe of the dynamical strong-field geometry in this chapter: the other channels test the stationary field.
The first detected gravitational-wave transient, GW150914, is the inspiral, merger and ringdown of two compact objects of about \(36\,M_{\odot}\) and \(29\,M_{\odot}\), leaving a remnant of about \(62\,M_{\odot}\) — roughly \(3\,M_{\odot}c^{2}\) radiated away as gravitational waves within a fraction of a second [Abbott:2016]. Neither body can be anything but a black hole: the signal sweeps up to frequencies at which the two are closer together than objects of those masses could be without merging. The catalogue that followed defines a population of black-hole masses and spins reaching well above the range accessible to the X-ray binaries of Phenomenon 50.1 [Abbott:2019]. Rests on Phenomenon 27.35 and Theorem 45.1.
Derivation. Derives Phenomenon 50.5. The compactness argument requires only Kepler's third law. Take the total mass \(M=70\,M_{\odot}=1.39\times 10^{32}\,\mathrm{kg}\) read off the inspiral, and the moment in the signal at which the gravitational-wave frequency reaches \(f_{\mathrm{GW}}=150\,\mathrm{Hz}\). A binary radiates principally at twice its orbital frequency (Gravitational-Wave Theory), so the orbital angular velocity there is \(\omega=\pi f_{\mathrm{GW}}=471\,/\mathrm{s}\), and two bodies orbiting with that angular velocity are separated by
Against this, the Schwarzschild radius of the total mass is
so the two bodies were still orbiting freely at a separation of only about \(1.7\,r_{\mathrm{s}}\). No neutron star can be involved: its radius, which Compact Stars and Relativistic Astrophysics places near \(12\,\mathrm{km}\), is small enough, but its mass cannot approach \(30\,M_{\odot}\). No main-sequence or degenerate star of any kind is remotely this compact at this mass. Only black holes remain. The Newtonian estimate is moreover conservative — the true relativistic separation at a given orbital frequency is smaller still — so the conclusion is not an artefact of applying Kepler's law outside its proper regime.
∎Interpretation
What the ensemble establishes is best stated as three nested claims, in decreasing order of certainty.
First, and beyond reasonable doubt: dark compact masses exist, from about \(5\,M_{\odot}\) in the X-ray binaries through \(21\,M_{\odot}\) in Cygnus X-1 and \(\sim60\,M_{\odot}\) in the merger remnants to \(4.3\times10^{6}\,M_{\odot}\) at the Galactic centre and \(6.5\times10^{9}\,M_{\odot}\) in M87. The stellar-mass cases rest on the Newtonian bound of Equation (50.2); the Galactic centre on Kepler's law applied to a closed stellar orbit. Each such mass exceeds every cold-equilibrium limit (Compact Stars and Relativistic Astrophysics), and for the S2 volume no non-collapsed alternative — a cluster of remnants, a ball of fermions — survives its own dynamics for the age of the Galaxy (Phenomenon 50.2).
Second, and directly measured: the gravitational field of these objects is the field general relativity assigns them, wherever it has been probed. The S2 orbit verifies the redshift-plus-Doppler prediction and the Schwarzschild precession at the ten-to-twenty per cent level, fifteen hundred gravitational radii out (Phenomenon 50.3). The photon-ring diameters of both imaged sources match Equation (50.4), whose only theoretical content is the strong-field number \(3\sqrt{3}\) — and for Sgr A* the mass on the right-hand side was measured first, by an independent Newtonian method, so the image was a prediction cashed (Phenomenon 50.4). The GW150914 ringdown is consistent with the quasinormal spectrum of the Kerr geometry (Phenomenon 50.5). Nothing yet observed requires any parameter beyond the mass and spin of the Kerr family.
Third, and here the honesty matters most: no material surface is seen, where every known class of surface would show itself. This is the closest observation comes to the horizon, and it is an absence, argued quantitatively below (Phenomenon 50.6).
Against these stand the things the data do not establish. No observation reaches the horizon itself: an event horizon is a globally defined null surface, and no electromagnetic or gravitational-wave measurement of finite duration can certify one — what is measured is light and matter behaving, down to a few gravitational radii, exactly as if one were there, with no sign of anything else. The interior, the singularity theorems' conclusion, and any Planck-scale modification of the horizon are untested by everything in this chapter. Exotic horizonless alternatives — objects contrived to be as compact as a black hole without its causal structure — are constrained by the ringdown, by the shadow depth and by the surface arguments, but the most contrived of them are not excluded [Akiyama:2019]; the honest assessment of that frontier, and of what would settle it, is continued in Quantum Gravity: The Honest Status. Interpretation throughout leans on the theory of Schwarzschild Geometry and Black Holes and Geometric Formulation of Gravity.
Accreting neutron stars betray their surfaces twice over: the fuel accumulating on them ignites periodically in thermonuclear flashes — the type I X-ray bursts — and in quiescence they re-radiate the accretion energy deposited on them. The black-hole candidates of Phenomenon 50.1 do neither. No burst has ever been recorded from one, and at comparable accretion rates their quiescent luminosities fall orders of magnitude below those of the neutron-star systems [Narayan:1997][Remillard:2006]. The energy that arrives is not returned. That is what a horizon does and what a surface of any composition cannot do, and it is the closest the electromagnetic observations come to a direct test of the defining property of a black hole. Rests on Theorem 19.32 and Proposition 45.18.
Derivation. Derives Phenomenon 50.6. The comparison is an energy budget, and the budget is Newtonian bookkeeping plus one relativistic fact. Let matter fall steadily at rate \(\dot{M}\) onto a compact object of mass \(M\) and radius \(R_{*}\). Each unit of mass, arriving from far away, has given up its gravitational binding energy \(GM/R_{*}\) by the time it settles on the surface, whatever route it took; in steady state that energy can only leave as radiation. An object with a surface therefore shines with
independently of how efficiently the infalling flow itself radiates. For a neutron star, with \(M\approx1.4\,M_{\odot}\) and \(R_{*}\approx12\,\mathrm{km}\) (Compact Stars and Relativistic Astrophysics), \(\eta_{*}\approx0.17\): a fixed seventeen per cent of the accreted rest energy must come back out. If instead the object has a horizon, nothing obliges the flow to radiate at all: the luminosity is \(\epsilon\,\dot{M}c^{2}\) with \(\epsilon\) whatever radiative efficiency the flow manages before crossing, and the balance is carried through the horizon and gone.
The comparison becomes decisive in quiescence, precisely because the accretion there is inefficient. At low accretion rates the inflow is a hot, tenuous, advection-dominated flow: the ion gas is too dilute to transfer its heat to the radiating electrons within an infall time, so it carries its thermal energy inward, and \(\epsilon\lesssim10^{-2}\) [Narayan:1997]. The predicted contrast between two otherwise similar transients, one with a horizon and one with a surface, is then
one to two orders of magnitude, before allowing for the gravitational redshift of the surface emission, which only deepens it. This is what is observed: at matched orbital periods — the available proxy for matched quiescent accretion rate — the black-hole transients sit roughly a factor of one hundred below the neutron-star transients in quiescent luminosity [Narayan:1997][Remillard:2006].
The burst argument is the same bookkeeping applied to matter rather than energy. On a surface, accreted hydrogen and helium accumulate, compress, and heat until the layer ignites; the burning is thermally unstable in a thin shell, so it runs away, and the neutron-star systems duly flash — thousands of type I bursts stand recorded. A horizon stores no fuel. No type I burst has ever been observed from any system whose mass function exceeds the neutron-star bound [Remillard:2006].
Two limits of the argument are stated as plainly as the argument. It is differential — it compares two source classes assuming the same kind of quiescent flow in both, so it inherits the advection-dominated model's assumptions, though not its details: any flow radiating a fraction \(\epsilon\ll\eta_{*}\) before arrival gives the same contrast. And it bounds what a surface would emit; it does not observe the horizon. It is evidence of absence obtained where presence was obligatory, which is the strongest statement an electromagnetic observation can make about a horizon.
∎Modern repetitions and precision
Every one of the four channels is a running programme, and each has a stated next decimal place.
At the Galactic centre, GRAVITY continues to track S2 away from pericentre, where the precession signal accumulates, and has added fainter stars on shorter orbits to the fit; each pericentre passage — S2's next falls in 2034 — multiplies the relativistic leverage [Abuter:2020]. The two-programme redundancy that checked the redshift [Abuter:2018] [Do:2019] remains the model: the Keck and VLT groups share neither instruments nor analysis chains. The targets beyond the first post-Newtonian order are the spin of Sgr A* through frame dragging of the innermost orbits, and any deviation of the enclosed mass from a point — both within reach of the existing astrometry extended by a decade (Schwarzschild Geometry and Black Holes supplies the predictions).
The Event Horizon Telescope has repeated and extended its campaigns: the polarized image of M87* [Akiyama:2021] turned the ring into a magnetometer, and the Sgr A* result [Akiyama:2022] showed that even a source varying on its minutes-long horizon-crossing time can be imaged. Precision here scales directly with the array: every added station fills the aperture, and a shorter observing wavelength tightens the beam as \(\lambda/D\) — the path to resolving the photon ring's substructure, time-resolved imaging of the flow, and a sharper shadow-diameter test than the current few-per-cent agreement of Table 50.3.
In the X-ray binaries the frontier is spin and population. The relativistically broadened iron line [Tanaka:1995] and the disc-continuum method, both calibrated on the innermost-stable-orbit physics of Schwarzschild Geometry and Black Holes, are applied across the twenty-odd dynamically confirmed systems [Remillard:2006], while VLBI parallaxes of the kind that re-weighed Cygnus X-1 [MillerJones:2021] remove the distance systematics that dominated a generation of mass estimates.
The gravitational-wave catalogue grows by design: from one event [Abbott:2016] to ten binary black holes in the first two observing runs [Abbott:2019] to, at the time of writing, of order a hundred (Experiment: Gravitational Waves) — a population experiment on black-hole masses and spins, and, as the loudest events accumulate, a spectroscopy of the ringdown: more than one quasinormal mode measured in a single remnant would test the Kerr no-hair property, not merely be consistent with it. The projected sensitivities of the next-generation interferometers are cross-referenced in Experiment: Gravitational Waves.
Primary references
The identification of Cygnus X-1: [Webster:1972] [Bolton:1972]; the modern orbital solution and mass: [Orosz:2011] [MillerJones:2021]; the transient population and its phenomenology: [Casares:1992][Remillard:2006]; the disc theory and the iron-line diagnostic: [Shakura:1973] [Tanaka:1995]. The birth of the field: [Giacconi:1962] for X-ray astronomy, [Schmidt:1963] [LyndenBell:1969] [Balick:1974] for the quasar argument and the Galactic-centre source, [Michell:1784] for the prehistory.
The Galactic-centre orbits: [Schoedel:2002] for the first orbit, [Ghez:2008] [Gillessen:2009] for the two long-term programmes, reviewed in [Genzel:2010]; the interferometric instrument: [Abuter:2017]; the relativistic tests: [Abuter:2018] [Do:2019] [Abuter:2020].
Event-horizon-scale imaging: [Akiyama:2019] (M87*), [Akiyama:2021] (polarization), [Akiyama:2022] (Sgr A*); the shadow prediction: [Falcke:2000].
Gravitational waves: [Abbott:2016][Abbott:2019], with the full experimental record in Experiment: Gravitational Waves [Aasi:2015].
The horizon-versus-surface argument: [Narayan:1997]. The theoretical prehistory: [Schwarzschild:1916] [Oppenheimer:1939a] [Oppenheimer:1939b] [Kerr:1963] [Penrose:1965].