First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way

As demonstrated using the suite of GRMHD simulations discussed in section VI, the EHT images of Sgr A∗ are consistent with the expected appearance of a Kerr black hole. Moreover, our images are consistent with an angular gravitational radius θg that matches the expectations from nearly Keplerian orbits of stars on scales of (10³−10⁵)RS.

To quantify this consistency, we compute θg using EHT measurements alone [40, for details, see]. Following the procedure developed in, we estimate θg by calibrating the observed emission ring diameter d to the known angular gravitational scale θg for a suite of synthetic datasets produced from the GRMHD library presented in. Specifically, we generate 100 datasets from GRMHD simulations with varying intrinsic θg and that span the explored range in accretion states, black hole spins, viewing inclinations, position angles, and phenomenological electron heating parameters Rhigh. Each synthetic dataset matches the baseline coverage and sensitivity of the EHT observations. We then estimate the ring diameter using multiple geometric modeling and imaging methods, deriving a separate calibration factor α≡d/θg for each method and for every dataset to attain a method-dependent scaling relationship and associated uncertainty.

With this approach, we estimate θg=4.8−0.7μas for Sgr A∗, where the uncertainties correspond to a 68% credible interval. This estimate is consistent with, but much less constraining than, measurements of θg using resolved stellar orbits (see Table 1). This procedure assumes the validity of the Kerr metric and relies on our suite of GRMHD simulations to provide a reasonable proxy for the space of viable emission models. The fractional uncertainty in our estimate of θg for Sgr A∗ is broader than our estimate for M87∗ (θg=3.8±0.4μas), primarily because of the increased calibration uncertainty associated with the unknown inclination of Sgr A∗ and because of the increased diameter measurement uncertainty associated with intrinsic variability of Sgr A∗.

We also used EHT images to constrain potential deviations from the Kerr metric and to test the nature of the compact object in Sgr A∗ [42, for details, see]. For instance, the brightness depression has a contrast fc<∼0.3, which provides support for the existence of an event horizon in Sgr A∗. If the compact object instead had an absorptive boundary that radiated the thermalized energy of infalling material M˙c², it could still produce a depression in the EHT images but would generate IR emission that greatly exceeds the measured spectrum of Sgr A∗. Alternatively, a partially reflecting surface would reduce the depth of the depression; the EHT images directly constrain the albedo of such a reflecting surface to be <∼ 0.3. The brightness depression also rules out several specific black hole alternatives, including some models of naked singularities [68, e.g.,] and some models of boson stars [92, e.g.,], although other horizonless black-hole mimickers produce apparent shadows that are consistent with the observed depression [114, e.g.,].

The measured ring diameter also provides constraints on the spacetime metric. To obtain these constraints, we generate a series of synthetic images to relate a measured emission ring diameter to that of the underlying black hole shadow. These images included a broad range of GRMHD simulations from, as well as images for which the GRMHD and underlying metric assumptions are relaxed. Specifically, we generated sets of images of analytic models for accretion onto 1) a Kerr black hole, 2) a black hole with parametric deviations from Kerr given by the Johannsen-Psaltis metric, and 3) a non-Kerr black hole defined by the Kerr-Sen metric. For each analytic model, we allow the emission prescriptions to vary within physically plausible limits [95, 121, for details, see]. We find a similar relationship between the diameter of the emission ring and that of the black hole shadow in all these cases, indicating that this relationship is insensitive to the details of the underlying spacetime.

Selecting a subset of these models, we then generate 145 synthetic datasets. We apply imaging and modeling methods to compute the ring diameter for each dataset to evaluate method-dependent measurement uncertainties and biases. We then use this calibration together with the EHT measurements of the ring diameter to determine the angular diameter of the black hole shadow for Sgr A∗: dsh=48.7±7.0μas. This result is tighter than the range for θg derived above because of differences in the calibration datasets and procedures used in and. Specifically, uses a calibration suite that consists entirely of dynamic GRMHD models, and it derives a scale factor between θg and the angular diameter of the emission ring. In contrast, uses a calibration suite containing both static Kerr and non-Kerr images and dynamic GRMHD models, and it derives a scale factor between dsh and the angular diameter of the emission ring. also excludes datasets for which the range of reconstructed diameters is more than 2−3 times the range that is measured using the Sgr A∗ data. Thus, comparison of these results provides a measure of the impact of our assumptions and procedures on the inference of the black hole properties.

By comparing this shadow diameter with stellar dynamical measurements of the mass of Sgr A∗, we also determine a deviation parameter δ, which quantifies the fractional difference between the inferred shadow diameter and its expected value for a non-spinning (Schwarzschild) black hole [42, for details, see]. We find δ=−0.08−0.09 when using VLTI measurements of θg and δ=−0.04−0.10 when using Keck measurements of θg (see Table 1). For comparison, a spinning (Kerr) black hole has −0.08≤δ≤0.

Under the assumption that this range of calibration factors applies generically to all non-Kerr spacetimes that have black hole shadows, we can translate measurements of δ into constraints on parameters of these spacetimes. In particular, our measurements exclude specific non-Kerr solutions for Sgr A∗, such as the traversible Morris-Thorne wormhole and the naked singularities in the Reissner-Nordström metric, which predict shadows that are significantly smaller than those of Kerr black holes.

Relative to M87∗, the primary strengths of testing GR with EHT observations of Sgr A∗ are the tight prior constraints on its mass-to-distance ratio θg and its shorter gravitational timescale that allows EHT observations to span many dynamical times. Together, these results show SMBHs consistent with predictions of the Kerr metric over a spread of three orders of magnitude in their mass (see Figure 6). The image of Sgr A∗ probes a similar gravitational potential to M87∗ but spacetime curvature ξ∝M⁻² that is six orders of magnitude larger. When combined with constraints from the measurement of gravitational waves from coalescing black-hole binaries with LIGO/Virgo, these results show a striking validation of the predictions of GR over a vast range of scales, from stellar mass black holes to supermassive black holes that are billions of times larger.