THE EVENT HORIZON OF SAGITTARIUS A*

The Schwarzschild metric presents the first example of a compact horizon: an imaginary surface delineating a compact region from which the rest of the universe is causally disconnected. Subsequently, a substantial theoretical effort was made to determine if such "black hole" solutions could practically come into existence (Oppenheimer & Snyder 1939; Penrose 1965; Wheeler 1966; Heger et al. 2003). Despite Einstein's misgivings, black holes are now believed to be the inevitable consequence of the demise of massive stars. In recent years, the existence of compact horizons has taken on a renewed significance due to efforts to construct a grand unified theory. Specifically, horizons play prominently in the well known "information paradox" (Hawking 2005; Mathur 2009). However, now we are in a position to constrain the existence of horizons observationally.

For astrophysical purposes, the critical features of horizons are (1) their compactness, allowing substantial amounts of gravitational binding energy to be liberated during accretion, and (2) their ability to hide the ultimate fate of accreting matter. Unlike a black hole, into which the kinetic and thermal energy gained during infall can disappear (adding only to the mass), objects with surfaces (e.g., stars) radiate the residual energy not emitted during the accretion process. This fact has been used in a number of efforts to test for the presence of horizons in a variety of black hole systems (Narayan et al. 1997; Narayan & Heyl 2002; McClintock et al. 2004; Broderick & Narayan 2006, 2007; Narayan & McClintock 2008). For many black hole candidates the tell-tale sign of surface emission, seen in accreting neutron stars, is absent, implying the lack of an analogous surface in these objects.

Sagittarius A* (Sgr A*), the radio point source associated with the dark mass located at the center of the Milky Way, is the best studied black hole candidate to date. Near-infrared (NIR) observations of massive stars in its vicinity have provided direct mass and distance measurements, M = (4.5 ± 0.4) × 10⁶ M_☉ and D = 8.4 ± 0.4 kpc, respectively, and confined it to within 40 AU (Ghez et al. 2008; Gillessen et al. 2009). With a luminosity of 10³⁶ erg s⁻¹, it is substantially underluminous relative to its limiting Eddington luminosity, 6 × 10⁴⁴ erg s⁻¹. The emission is strongly nonthermal, distributed from the radio to γ-rays, and believed to be powered by the release of gravitational binding energy by accreting gas.

Due to its mass and proximity, Sgr A* has the largest angular size of any known black hole and offers the best prospects for direct imaging of its silhouette (Falcke et al. 2000; Broderick & Loeb 2005, 2006a, 2006b). Unfortunately, millimeter imaging alone will be unable to verify or exclude the presence of a horizon (Broderick & Narayan 2006). This is because even a relatively bright central object (e.g., one radiating thermally with a luminosity comparable to $\dot{M} c^2$) will appear in silhouette against the much hotter surrounding accretion flow. However, here we show that recent millimeter-Very Long Baseline Interferometric (VLBI) observations, which have resolved sub-horizon scale structure for the first time (Doeleman et al. 2008), provide conclusive evidence for the presence of a horizon when coupled with the existing NIR and mid-infrared (MIR) flux limits (Ghez et al. 2005; Hornstein et al. 2007; Schödel et al. 2007). We do this by explicitly excluding emission from a putative compact infrared photosphere, lying inside of the millimeter-emitting region. Specifically, we show that for such surface emission to remain undetected would require unphysically large radiative efficiencies, greater than 99.6%, as compared to the typical efficiencies in active galactic nuclei (AGNs) of 10% (Kato et al. 2008) and the meager efficiency implicated in Sgr A*, 0.01%–1% (Narayan & McClintock 2008). Most importantly, we do this using only observable quantities, ensuring that these limits are largely independent of the gravitational theory employed.³

Section 2 describes the underlying physical assumptions and Section 3 presents the recent observational constraints, within the context of Sgr A*. Conclusions are collected in Section 4. Appendices A and B discuss photon propagation times, the notion of gravitational binding energy, and the apparent size of objects within the context of general geometric theories of gravity.

Our argument depends critically upon three underlying, physically well motivated assumptions: (1) Sgr A* is accretion powered; (2) has reached steady state; and (3) compact surfaces are approximate blackbodies. We wish to emphasize that relaxing any of these would require fundamental alterations to presently well understood physics (such as microscopic physics at pedestrian, and laboratory accessible, densities, temperatures, and magnetic field strengths) or the invention of exotic new states of matter. Nevertheless, we discuss each in detail, describing how these are justified in the context of Sgr A*.

2.1. Accretion Power and the Luminosity of Sgr A*

2.2. Steady State

2.3. Compact Objects and Blackbodies

Our argument will hinge upon our ability to specify the spectral signatures of any surface emission from black hole alternatives. Subject to the condition outlined in Section 2.2, it is natural to expect any such emission to be thermal. This is obvious if the object accrues an optically thick atmosphere of baryonic material. However, for any compact surface which lies within the photon orbit this is generally an excellent assumption.

By definition, at the photon orbit whether or not a photon impacts the surface is determined by the radial component of the momentum. Thus, if photons are emitted isotropically by a surface located at the photon orbit, half will impact the surface and half will escape to infinity. As the surface shrinks inside the photon orbit the fraction of escaping photons also decreases, falling to zero when the surface coincides with the horizon.⁴ Thus, the higher the redshift of the surface, the more effectively different regions on the surface are coupled to one another via radiation and the closer the surface approaches thermodynamic equilibrium. Consequently, high-redshift surfaces present a perverse realization of the canonical pin-hole cavity, becoming ideal blackbodies as z goes to ∞ (Broderick & Narayan 2006). For a Schwarzschild spacetime this is shown in Figure 1; however, this behavior is generic to spherically symmetric spacetimes. Thus if the system has sufficient time to have reached steady state, it must be a blackbody.

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The primary astrophysical importance of a horizon is that the gravitational binding energy liberated by material as it accretes can be advected into the black hole without any further observational consequence. This is very different from accretion onto other compact objects, e.g., neutron stars, in which this liberated energy ultimately must be emitted by the stellar surface. Importantly, this argument is not dependent upon the particulars of the compact object. Any object powered by accretion, whose surface is visible from the external universe, should show evidence of surface radiation. We will use this fact to rule out the possibility that accreted material in Sgr A* settles in a region visible to outside observers, and in doing so make the argument that a horizon must exist. That is, we imagine that material comes to rest at some surface where it radiates its remaining kinetic energy and observationally constrain the associated surface luminosity.

The gravitational binding energy released by a particle falling onto the putative surface depends upon the details of the gravitational theory. For those theories which admit notions of energy conservation (in the test particle limit), including all stationary spacetimes, we may write this in terms of the specific binding energy at the putative surface, Δ_g. That is, the liberated energy due to a particle of mass m as measured at infinity is E_∞ = Δ_gmc². For a continuous accretion flow, this provides a total power, as measured at infinity, of

$$\begin{equation} L_\infty = \Delta \epsilon _g \dot{M} c^2\,, \end{equation} \tag{ 1 }$$

where $\dot{M}$ is the mass accretion rate at the surface.

The particular form of Δ_g also depends upon the nature of the accretion flow. For arbitrary stationary, spherically symmetric spacetimes this is computed in Appendix A for two typical cases: zero-angular momentum accretion flows in which matter does not orbit the black hole (e.g., Bondi accretion) and Keplerian accretion disk (e.g., appropriate for thin disks). In the latter case Δ_g is generally smaller since a fraction of the liberated binding energy is necessarily converted into the kinetic energy associated with the orbital motion. Within the context of GR black holes⁵ Δ_g is straightforward to compute explicitly for these cases, and is shown in Figure 2 for both nonrotating and rapidly rotating black holes. However, we shall see that our ultimate constraints are independent of the form of Δ_g.

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Only a fraction of L_∞ is converted into radiation. The observed electromagnetic luminosity at infinity may be parameterized in terms of a radiative efficiency, η_r:

$$\begin{equation} L_{\rm obs} = \eta _r L_{\infty }\,. \end{equation} \tag{ 2 }$$

Gravitational binding energy may also be converted into the kinetic energy of relativistic outflows, which we similarly parameterize in terms of outflow efficiency, η_k:

$$\begin{equation} L_{\rm out} = \eta _k L_{\infty }\,. \end{equation} \tag{ 3 }$$

It is important to note that η_r and η_k are primarily a function of the accretion flow, dependent upon the microphysics, and thus relatively insensitive to the character of strong gravity.

If a horizon is present, the remainder of L_∞ may be advected across it without further observational consequence. However, in the presence of a surface, if Sgr A* has reached steady state, this remainder must ultimately be radiated. Thus the surface luminosity, as measured at infinity, is

$$\begin{equation} L_{\rm surf} = L_\infty - L_{\rm obs}-L_{\rm out} = \frac{1-\eta _r-\eta _k}{\eta _r} L_{\rm obs}\,, \end{equation} \tag{ 4 }$$

where we have written this entirely in terms of the unknown efficiencies and the observed luminosity. (Specifically, note that neither Δ_g nor $\dot{M}$ appears in this expression.)

As we have argued in Section 2.3, for compact surfaces this radiation will be in the form of a blackbody spectrum. That is, in terms of the apparent radius (R_a) and temperature (T_∞) of the putative surface, as measured at infinity,

$$\begin{equation} L_{\rm surf} = 4\pi \sigma R_a^2 T_\infty ^4\,. \end{equation} \tag{ 5 }$$

Alternatively, this provides a means to estimate the surface temperature (and thus spectrum) given L_surf and R_a. Combined with Equation (4), we may write T_∞ in terms of L_obs, R_a, η_r, and η_k:

$$\begin{equation} T_\infty = \left(\frac{1-\eta _r-\eta _k}{\eta _r} \frac{L_{\rm obs}}{4\pi \sigma R_a^2} \right)^{1/4}\,. \end{equation} \tag{ 6 }$$

In terms of this surface temperature, the expected flux seen by distant observers is then

$$\begin{equation} F_\nu = \pi \left(\frac{R_a}{D}\right)^2 B_\nu \left(T_\infty \right)\,, \end{equation} \tag{ 7 }$$

where B_ν is the blackbody spectrum.

However, no thermal surface component is observed in Sgr A*'s spectrum. Thus, if such surface emission is present, it must be hidden under the emission from the accretion flow. As a consequence, each flux measurement constitutes an independent upper limit upon the surface flux, and thus surface temperature. Explicitly, given an observed flux F^obs_ν, T_∞ ⩽ T_max(ν, F^obs_ν, R_a/D) where

$$\begin{equation} T_{\rm max}\left(\nu,F^{\rm obs}_{\nu };\frac{R_a}{D}\right) = h\nu \bigg / k \ln \left(1 + \frac{2 \pi h \nu ^3 R_a^2}{c^2 F^{\rm obs}_{\nu } D^2} \right)\,. \end{equation} \tag{ 8 }$$

Note that this is generally a function of R_a, a consequence of the fact that larger surfaces are correspondingly cooler, and therefore easier to hide under the observed emission.

This maximum temperature then implies a limit upon L_surf directly via the blackbody condition:

$$\begin{equation} \frac{L_{\rm surf}}{L_{\rm obs}} \le \frac{L_{\rm surf, max}}{L_{\rm obs}} \equiv \frac{\sigma R_a^2}{D^2 F_{\rm obs}} T_{\rm max}^4\left(\nu,F^{\rm obs}_\nu ;\frac{R_a}{D}\right)\,, \end{equation} \tag{ 9 }$$

where F_obs = L_obs/4πD² is the integrated observed flux from Sgr A*. Alternatively, we may rewrite this in terms of a lower limit upon the efficiencies:

$$\begin{equation} \eta _r + \eta _k \ge \frac{1}{1 + L_{\rm surf, max}/L_{\rm acc}} \,. \end{equation} \tag{ 10 }$$

That is, if the efficiencies are sufficiently high the unradiated liberated energy in the accretion flow is sufficiently small that the surface could escape detection. Unlike L_surf/L_acc, we have natural scales against which to compare η_r + η_k. In prodigiously accreting systems, such as AGNs, X-ray binaries, and gamma-ray bursts, which are believed to be radiatively efficient, the radiative efficiency η_r + η_k is estimated to be ∼10% (Kato et al. 2008). In Sgr A*, on the other hand, η_r is thought to be quite small (0.01%–1% for typical accretion models) as a consequence of the weak coupling between accreting ions and electrons within the gas (Narayan & McClintock 2008), and there is presently no direct evidence for energetic outflows.

Within the context of GR black holes and standard accretion theory, there are fundamental limits upon how large η_r + η_k may be for compact surfaces. This arises from the fact that matter inside of the innermost stable circular orbit (ISCO) plunges rapidly onto the surface, and thus does not have sufficient time to radiate. In contrast, outside of the ISCO material may linger on stable orbits for long periods of time, and at least in principle can radiate efficiently. Thus, even if the "intrinsic" radiative efficiency outside of the ISCO reached unity, the accreting material would still accrue additional kinetic energy during the plunge from the ISCO to the putative surface. If this surface is located inside of the ISCO, we may therefore place a lower bound upon the additionally liberated energy, and thus an upper bound upon η_r + η_k. These are shown in Figure 3 for zero-angular momentum accretion flows⁶ and orbiting accretion disks surrounding both nonrotating and rapidly rotating black holes (see Appendix A for details). For orbiting disks the maximum value of η_r + η_k is generally less than 33%.

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Therefore, each combined size and flux measurement of Sgr A* places a direct constraint upon the luminosity of a putative surface, or, equivalently, upon the radiative efficiency of the accretion flow. Thus, we now turn our attention to describing the current best constraints upon R_a/D and F^obs_ν.

3.1. Millimeter Size Constraints

3.2. Infrared Flux Limits

3.3. Constraints Upon Surface Existence

Each of the infrared flux limits listed in Table 2 places an upper limit upon L_surf/L_obs via Equation (9), given a value of R_a. Figure 4 shows these limits as a function of surface size, together with their 3σ upper bounds (denoted by the hatched regions). Generally, larger surfaces are cooler. This has two consequences. First, since the strongest constraints are placed by flux measurements near the peak in the thermal spectrum, the infrared band providing the most stringent limit is a function of R_a as well; near R_a/D ≃ 100 μas the infrared band dominating the constraint changes from M_S (4.7 μm)⁷ to N (3.8 μm). The constraint due to the combination of all of the infrared bands is the lower envelope of all the bands, and defines the excluded region, corresponding to small, luminous surfaces. Second, since cooler surfaces are more easily masked by the emission from the accretion flow, the limit becomes less stringent as R_a increases. Thus, the infrared constraint upon L_surf/L_obs must be supplemented with an independent limit upon the size of the putative surface.

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Upper limits upon R_a/D are found directly via the radio VLBI observations collected in Table 1. These are shown in Figure 5, with their 3σ upper bounds (again denoted by the hatched regions) together with the combined infrared limit. The recent 1.3 mm detection is the strongest, and excludes R_a/D > 27 μas at the 3σ level. When combined with the infrared constraints, only a small corner of the R_a–L_surf/L_obs is still permitted, corresponding to small, dim surfaces. Earlier measurements at 7 mm and 3 mm already required L_surf/L_obs ≲ 0.02. The new 1.3 mm detection improves these by nearly an order of magnitude, requiring L_surf/l_obs ≲ 0.004 at the 3σ level.

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A more physical interpretation is provided by the accretion efficiencies the latest observations now demand, shown on the right-hand vertical axis. In order for a surface to be present η_r + η_k ≳ 99.6% (at 3σ). That is, as matter falls onto Sgr A*, somehow 99.6% of the liberated gravitational binding energy must be radiated, powering either the observed luminosity or kinetic outflows. Otherwise the emission of the remainder upon settling onto the surface would have been detected. This needed efficiency is considerably larger than even the 10% efficiencies implicated in rapidly accreting systems, let alone the meager efficiencies (0.01%–1%) inferred in Sgr A*.

More striking is the comparison of this to the maximum efficiencies within GR spacetimes. A variety of black hole alternatives are grafted onto such spacetimes outside of some compact region, and thus in these the GR limits explicitly hold. For other black hole alternatives, including those associated with different gravity theories, these limits at least provide a scale for comparison. These fundamental constraints upon the efficiencies arise from the fact that accreting material cannot radiate efficiently inside of the ISCO. Inside of the ISCO accreting material plunges inward rapidly in comparison to the radiative timescales. Thus, even if the intrinsic radiative efficiency outside the ISCO is 100% (which is extremely unlikely), the fraction of the total binding energy released by accreting gas in the course of its inward flow is limited to that fraction that is emitted outside the ISCO. Therefore, within the context of GR, even with the most conservative of assumptions (zero-angular momentum accretion flow with 100% radiative efficiency down to the ISCO) we obtain an absolute upper limit on η_r + η_k for surfaces which lie within the photon orbit of 91%, while for a more reasonable accretion scenario involving orbiting gas the limit is 33% (see Appendix A for more details). The combined observational limits upon L_surf/L_acc and η_r + η_k are compared against those inferred in GR spacetimes for zero-angular momentum and Keplerian flows in Figure 6. In particular, there is no longer any allowed region given the recent 1.3 mm size constraint.

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Recent infrared and millimeter-VLBI observations imply that if the matter accreting onto Sgr A* comes to rest in a region visible to distant observers, the luminosity associated with the surface emission from this region satisfies L_surf/L_acc ≲ 0.004. Equivalently, these observations require that 99.6% of the gravitational binding energy liberated during infall is radiated in some form prior to finally settling. These numbers are inconsistent by orders of magnitude with our present understanding of the radiative properties of Sgr A*'s accretion flow specifically and relativistic accretion flows generally. Therefore, it is all but certain that no such surface can be present, i.e., an event horizon must exist.

Our conclusions rest upon three rather conservative physical assumptions: (1) Sgr A* is gravitationally powered, (2) Sgr A* has reached an approximate steady state and (3) a putative surface would be sufficiently compact and/or optically thick to be well modeled by a thermal spectrum. Black holes evade our argument since they violate the second of these, failing ever to reach steady state and growing in mass as a consequence. Some black hole alternatives have been proposed which would also violate the steady-state condition intrinsically, though these depend upon exotic new physical models and it is likely that the baryonic atmospheres they will inevitably develop via accretion will indeed reach steady state separately. Alternatively, there do exist astronomical objections that presumably violate the first assumption, namely rotationally powered objects such as pulsars. However, since we have fundamentally constrained the luminosity of a putative surface, independent of the mechanism powering the emission, rotation powered objects must satisfy a similar constraint, though a full discussion is beyond the scope of this paper.

Critical to our analysis is the recent submillimeter constraint upon the size of the emitting region of Sgr A* since it (1) justifies the assertion that the putative photosphere is sufficiently compact to be treated as a blackbody, (2) limits the photosphere's temperature from below, and (3) within the context of GR provides a strong constraint upon possible values of the radiative efficiency that is easily excluded. Future millimeter-VLBI observations will be critical to understanding the morphology of Sgr A*'s emitting region, and thus validating the interpretation of current observations as the approaching side of an orbiting accretion flow, with the attendant implications for Sgr A*'s size. However, unless the spacetime around Sgr A* deviates substantially from that of a GR black hole, future observations of the intrinsic size of Sgr A* will be unable to further restrict the size of a putative surface due to the influence of gravitational lensing. The reason is that all objects which lie within the photon orbit generally have the same apparent radius. Thus, obtaining more sensitive near-infrared flux measurements will remain a crucial avenue for strengthening this kind of argument.

We were able to place these constraints completely in terms of observable quantities: fluxes and the apparent size of Sgr A*. Beyond using GR spacetimes to provide scales for the purpose of comparison, we did not need to make use of the particular structure of GR black hole spacetimes. As a result, our conclusions may be applied more generally to all gravitational theories that admit notions of energy conservation in the test-particle limit. Specifically, these include all geometric gravitational theories that admit stationary solutions, including all of the f(R) theories and black hole alternatives that exist within the context of GR. As a consequence, we cannot yet say that Sgr A* is described by a GR black hole despite being able to conclude that a horizon must exist.

As material falls into the gravitational potential well induced by Sgr A* it necessarily converts a portion of its gravitational binding energy into luminosity. In Section 3 we simply parameterized the amount of liberated energy in terms of some Δ_g, finding it unnecessary to specify this function further. In this appendix we derive a general expression for the magnitude of the binding energy available. We necessarily assume that gravity is described by a metric theory, and that this theory admits a stationary solution. For simplicity, we will not explicitly discuss nonspherical solutions, however this makes no formal difference. We take the metric signature to be −+ + + and choose units such that G = c = 1.

The state of an infalling particle, characterized by its 4-momentum, p^μ, can be used to determine the energy available to radiation as measured at infinity. Given an initial momentum for an infalling particle, p^μ_i, the momentum of the radiated photon, p^μ_γ, and the final particle momentum, p^μ_f, conservation of momentum gives

$$\begin{equation} p_\gamma ^\mu = p_i^\mu - p_f^\mu \,. \end{equation} \tag{ A1 }$$

The steady state of the spacetime implies that along geodesics (e.g., the null geodesic followed by the photon, or the free-fall of the gas particle) p_t is explicitly conserved, corresponding to a conserved energy. Thus, the photon's energy at infinity is simply Δ_g ≡ −p_γt = p_{f t} − p_{i t}. Since this quantity is observed at infinity, where the spacetime is flat, we are guaranteed that this quantity is not affected by the choice of coordinates deep in the gravitational well. In addition, this has the virtue of being precisely what is measured.

We will consider two scenarios, corresponding to different choices of p^μ_f. The first of these is the most extreme: the particle was initially at rest at infinity (p_{i t} = −m), fell to some radius and came to rest, with⁸

$$\begin{equation} p^{\rm rest}_\mu = \left(\frac{m}{\sqrt{-g^{tt}}},0,0,0 \right)\,, \end{equation} \tag{ A2 }$$

at which point its accumulated binding energy was emitted. The available energy, per unit mass is then

$$\begin{equation} \Delta \epsilon _g = \frac{p_t - p^{\rm rest}_t}{m} = \frac{\sqrt{-g^{tt}}-1}{\sqrt{-g^{tt}}} = \frac{z}{z+1}\,, \end{equation} \tag{ A3 }$$

where the redshift is defined in the usual way: $z\equiv \sqrt{-g^{tt}}-1$.

Our second scenario involves the presence of an accretion disk. Previously, we assumed that the accreting material has no angular momentum, and therefore explicitly ignored orbital motion. Generally, the gas is expected to orbit the black hole, and the kinetic energy in the orbital motion will be supplied by the liberated gravitational binding energy, decreasing the reservoir of energy available to radiation. Hence we have placed a firm upper limit upon the luminosity of the accretion flow itself. However, we can compute the decrease explicitly for the simple case of stationary, spherically symmetric spacetimes. If

$$\begin{equation} ds^2 = g_{tt} {\rm d}t^2 + g_{rr} {\rm d}r^2 + r^2 {\rm d}\Omega ^2\,, \end{equation} \tag{ A4 }$$

for orbits in the equatorial plane, p^orbit_t and p^orbit_ϕ are conserved, p^θ_orbit vanishes due to symmetry and

$$\begin{equation} p_{\rm orbit}^r = m \frac{{\rm d}r}{{\rm d}\tau } = \sqrt{-\frac{1}{g_{rr}} \left(m^2 + g^{tt} {p^{\rm orbit}_t}^2 + \frac{{p^{\rm orbit}_\phi }^2}{r^2} \right)} = 0\,. \end{equation} \tag{ A5 }$$

If the orbit is to remain closed, we also require

$$\begin{equation} m \frac{{\rm d}^2 r}{{\rm d}\tau ^2} = -\frac{1}{2} \frac{\partial }{\partial r} \frac{1}{g_{rr}} \left(m^2 + g^{tt} {p^{\rm orbit}_t}^2 + \frac{{p^{\rm orbit}_\phi }^2}{r^2} \right) = 0\,. \end{equation} \tag{ A6 }$$

Solving these for p^orbit_t and p^orbit_ϕ gives

$$\begin{eqnarray} p^{\rm orbit}_t &=& \frac{1}{\sqrt{-g^{tt}}} \bigg / \sqrt{1 - \frac{1}{2} \frac{\partial \ln (-g^{tt})}{\partial \ln r }}\nonumber\\ &=& \frac{1}{z+1} \bigg / \sqrt{ 1 + \frac{\partial \ln (z+1)}{\partial \ln r} }\,, \end{eqnarray} \tag{ A7 }$$

which is generally larger than for the zero-angular momentum case (since z generally decreases with radius). Correspondingly, Δ_g in this case is generally lower.

While our discussion of the liberated gravitational binding energy is quite general, requiring only that gravity is described by a metric theory that admits a stationary solution, GR can provide some intuition regarding the magnitude of energy that can reasonably be released. Figure 2 shows the specific binding energy released by material as a function of radius. Typically, the ISCO is the final radius at which matter can efficiently radiate since inside this point material plunges into the black hole on a free fall timescale,⁹ rapid in comparison to the relevant radiative timescale for the rather pedestrian densities and magnetic fields observed in Sgr A*.¹⁰ As a consequence, even if 100% of the gravitational binding energy released outside of the ISCO is radiated or goes into outflows, the maximum fraction of the rest mass that can be emitted is shown by the blue line in Figure 2. If the region where the accreted material finally comes to rest lies within the ISCO, additional gravitational binding energy will be released, and must subsequently be radiated in the stopping region. Thus, the radiative efficiency of the accretion flow, defined by

$$\begin{equation} \eta _r + \eta _k \equiv \frac{L_{\rm acc}}{\Delta \epsilon _g(R) \dot{M}} + \frac{L_{\rm out}}{\Delta \epsilon _g(R) \dot{M}}, \end{equation} \tag{ A8 }$$

must satisfy

$$\begin{equation} \eta _r+\eta _k \le \frac{\Delta \epsilon _g(r_{\rm ISCO})}{\Delta \epsilon _g(R)} \end{equation} \tag{ A9 }$$

where R is the radius of the stopping region. This limit shown as a function of R in Figure 3 for zero-angular momentum accretion flows and accretion disks in Schwarzschild and Kerr spacetimes. Generally, once R is constrained to lie within the photon orbit, the maximum η_r + η_k possible is 91%, corresponding to a zero-angular momentum accretion flow in a rapidly rotating (a = 0.998) Kerr spacetime. When an accretion disk is present this limit declines to 33%. For the Schwarzschild spacetime, η_r + η_k is bounded from above by 43% (zero-angular momentum accretion) and 14% (accretion disk). In all cases, these maximum values are substantially smaller than the limits set by recent submillimeter and infrared observations if Sgr A* did not have a horizon.

Observations of the size of a compact emitting region are necessarily impacted by strong gravitational lensing. In metric theories of gravity, objects associated with deep potential wells will appear larger to observers at infinity. The apparent size of the region is directly related to both the physical size and the redshift of the compact object. Thus, relating the actual size of a compact emitting region to the observed size requires some understanding of the spacetime structure around the object. This is, of course, one of the reasons we chose to cast the constraints upon the existence of a horizon, described in Section 3, in terms of R_a and not a physical object size. Nevertheless, for completeness, we discuss the procedure here.

It is typically very difficult to compute the relationship between the physical and observed size and shape of a compact emitting object. However, in the special case of a spherically symmetric spacetime, this is generally tractable, independent of the particular form of the metric. We assume

$$\begin{equation} {\rm d}s^2 = g_{tt} {\rm d}t^2 + g_{rr} {\rm d}r^2 + r^2 {\rm d}\Omega ^2\,, \end{equation} \tag{ B1 }$$

where g_tt and g_rr are functions of r alone. Then, in the equatorial plane, the null geodesics are defined by

$$\begin{eqnarray} \frac{{\rm d}t}{{\rm d}\lambda } &=& g^{tt} \,,\nonumber\\ \frac{{\rm d}r}{{\rm d}\lambda } &=& \sqrt{ \frac{1}{g_{rr}} \left(-g^{tt}-\frac{b^2}{r^2} \right)}\nonumber\\ &=& \sqrt{\frac{-g^{tt}}{g_{rr}}\left[ 1 - \frac{b^2}{(z+1)^2r^2} \right] } \,,\nonumber\\ \frac{{\rm d}\theta }{{\rm d}\lambda } &=& 0\nonumber \end{eqnarray}$$

and

$$\begin{equation} \hspace{-7.5pc}\frac{{\rm d}\phi }{{\rm d}\lambda } = \frac{b}{r^2}\,, \end{equation} \tag{ B2 }$$

where the equations for dt/dλ and dϕ/dλ are associated with the existence of a time-like and azimuthal Killing vectors, respectively, the equation for dθ/dλ is fixed by vertical symmetry and the equation for dr/dλ arises from the null ray condition. In these b is the impact parameter at infinity and λ is an arbitrary affine parameter. The minimum radius reached by a given null geodesic occurs at its inner turning point, at which $b = r \sqrt{-g^{tt}} = r (z + 1)$. Alternatively, this corresponds to the maximum b that a null geodesic can have and still impact a surface of radius r. Thus, the apparent radius, R_a, of an object with physical radius R is simply

$$\begin{equation} R_a = R \left(z + 1\right)\,. \end{equation} \tag{ B3 }$$

Some care must be taken, however, when R is smaller than the photon orbit. This is because rays which cross the photon orbit have no radial turning points, and therefore will in all cases be captured. This is clear from the definition of the photon orbit, r_γ:

$$\begin{equation} \left. \frac{{\rm d}}{{\rm d}r} \frac{1}{(z+1)^2 r^2} \right|_{r_\gamma }= 0, \end{equation} \tag{ B4 }$$

which corresponds to the position of the maximum of the "effective potential" in the radial equation. Thus, if

$$\begin{equation} b < r_\gamma [z(r_\gamma)+1 ] \Rightarrow \frac{b^2}{(z+1)^2 r^2} < 1 \quad \mbox{for all}\quad r. \end{equation} \tag{ B5 }$$

As a consequence, the apparent radius of objects for which R < r_γ is the same as that for objects with R = r_γ. Thus, generally,

$$\begin{equation} R_a = \left\lbrace \begin{array}{@{}ll@{}} r_\gamma [ z(r_\gamma) + 1 ] &\mbox{if }\quad R\le r_\gamma \\[9pt] R [ z(R) + 1 ] &\mbox{otherwise}. \end{array} \right. \end{equation} \tag{ B6 }$$

In the case of the Schwarzschild metric, this gives the well known result

$$\begin{equation} R_a = \left\lbrace \begin{array}{@{}ll@{}} 3 \sqrt{3} M & \mbox{if}\quad R\le 3M\\[6pt] R \displaystyle\sqrt{\frac{R}{R-2M}} & \mbox{otherwise}. \end{array} \right. \end{equation} \tag{ B7 }$$

For a rapidly rotating Kerr spacetime, the apparent radius in the equatorial plane may also be computed without undue difficulty (though in this case care must be taken into account for the non-diagonal components of the metric). Generally, this is given by

$$\begin{equation} R_a = \case{1}{2}(b_+ - b_-), \end{equation} \tag{ B8 }$$

where

$$\begin{eqnarray} b_\pm &=& \pm {\rm max}\left(R \frac{R\sqrt{R^2-2MR+a^2} \mp 2 a M}{R^2-2MR} \,,\right.\nonumber\\ &&\left. r_{\pm \gamma } \frac{r_{\pm \gamma }\sqrt{r_{\pm \gamma }^2-2Mr_{\pm \gamma }+a^2} \mp 2 a M}{r_{\pm \gamma }^2-2M r_{\pm \gamma }} \right) \end{eqnarray} \tag{ B9 }$$

in which r_±γ is the radius of the prograde/retrograde photon orbit. Since these differ for rotating black holes, we generally have three conditions. In the case of a maximally rotating black hole (a = 1), this expression is especially simple:

$$\begin{equation} R_a = \left\lbrace \begin{array}{@{}ll@{}} \displaystyle\frac{9}{2} M & \mbox{if}\quad R\le r_{+\gamma }\\[-2pt]\\[-2pt] \displaystyle\frac{R + 8M}{2} & \mbox{if}\quad r_{+\gamma } < R \le 4M\\[-2pt]\\[-2pt] R \displaystyle\frac{R-1}{R-2} & \mbox{otherwise}\,, \end{array} \right. \end{equation} \tag{ B10 }$$

where R is the object radius in Boyer–Lindquist coordinates. While r_+γ = M in these coordinates for a = 1, there remains a finite proper distance between the photon orbit and the horizon, the equality being an artifact of the coordinates themselves. For this reason, we distinguish between these formally, though it makes no difference (since R_a = 9/2 for all R between r_+γ and the horizon), as it must not given that R_a is a gauge invariant quantity as a consequence of its definition.

Most important for the present discussion is the fact that R_a for the Schwarzschild and equatorial Kerr spacetimes differs by only about 15%. Thus, despite the vastly different coordinate sizes, an object with R = 1 M embedded in a maximal Kerr spacetime has roughly the same apparent size as an object with R = 3 M embedded in a Schwarzschild spacetime. As a consequence, if the present limit upon the size of the submillimeter emitting region in Sgr A* of 37 μas (R_a = 3.5 M) is interpreted as a photosphere surrounding the stopping region, it constrains the size of a central emitting region to lie well within the photon orbit of both a Kerr and Schwarzschild black hole.

On the other hand, the anomalously small apparent radius might appear unphysical within the context of GR. However, this is easily rectified if the emission region is interpreted instead as the visible arc of an oncoming accretion disk (as a result of Doppler boosting and Doppler shifts, the receding side being considerably dimmer for the same reason; Broderick & Loeb 2006a). While the equatorial extent of the arc can be significantly smaller than the minimum apparent radius in this situation, the vertical extent is still roughly 2R_a (see, e.g., Broderick et al. 2009; Broderick & Narayan 2006). Hence, unless the projected baseline was extraordinarily fortuitously aligned, again we would expect large measured sizes for central emission regions larger than the photon orbit radius. It is possible to coincidentally fit the existing spectral, polarization, and millimeter-VLBI observations using orbiting accretion flow models (Broderick et al. 2009). However, this is due at least in part to the fact that the existing millimeter-VLBI size constraint is essentially restricted to the east–west direction. Future millimeter-VLBI observations will be critical to unambiguously determining the morphology of the emitting region (Fish et al. 2009).