The no-hair theorem characterizes the fundamental nature of black holes in general relativity. This theorem can be tested observationally by measuring the mass and spin of a black hole as well as its quadrupole moment, which may deviate from the expected Kerr value. Sgr A*, the supermassive black hole at the center of the Milky Way, is a prime candidate for such tests thanks to its large angular size, high brightness, and rich population of nearby stars. In this paper, I discuss a new theoretical framework for a test of the no-hair theorem that is ideal for imaging observations of Sgr A* with very long baseline interferometry (VLBI). The approach is formulated in terms of a Kerr-like spacetime that depends on a free parameter and is regular everywhere outside of the event horizon. Together with the results from astrometric and timing observations, VLBI imaging of Sgr A* may lead to a secure test of the no-hair theorem.

According to the no-hair theorem, black holes are uniquely characterized by their masses and spins and are described by the Kerr metric [

Tests of the no-hair theorem have been suggested using observations in either the gravitational-wave [

Sgr A*, the supermassive black hole at the center of the Milky Way, is a prime target for testing strong-field gravity and the no-hair theorem with electromagnetic observations (see [

In this paper, I review this framework as well as the prospects for an observational test of the no-hair theorem with Sgr A*.

Spacetimes of rotating stellar objects in general relativity have been studied for several decades. Due to the nonlinearity of Einstein field equations, the construction of such metrics is plagued with sometimes incredible technical challenges. Following the discovery of the Schwarzschild [

Many exact solutions of the Einstein field equations are now known [

To date, there exist seven different approaches that model parametric deviations from the Kerr metric. Ryan [

Due to the no-hair theorem, the Kerr metric is the only asymptotically flat SAV in general relativity with an event horizon but no closed timelike loops [

For this reason, the emission from accretion flows around black holes is most interesting for strong-field tests of the no-hair theorem with observations across the electromagnetic spectrum ranging from X-ray observations of quasiperiodic variability, fluorescent iron lines, or continuum disk spectra [

These strong-field tests of the no-hair theorem require a very careful modeling of the inner region of the spacetime of black holes. Due to the pathologies of previously known parametric deviations, it has been necessary to impose an artificial cutoff at some radius outside of the event horizon that encloses all of the above pathologies and, thereby, shields them from the observer. Therefore, the application of parametric frameworks to such tests of the no-hair theorem in the electromagnetic spectrum has, so far, been limited to only slowly to moderately spinning black holes, for which the circular photon orbit and ISCO are still located outside of the cutoff radius [

Recently [

In [

(a) Values of the parameter

In Figure

Radius of (a) the ISCO and of (b) the circular photon orbit as a function of the spin

Contours of constant radius of the ISCO for values of the spin

In [

The location of the circular photon orbit determines the size of the shadow of Sgr A* (see [

In an optically thin accretion flow such as the one around Sgr A* at sub-mm wavelengths (e.g., [

Images of rings of light of (a) a Kerr and (b) a quasi-Kerr black hole at an inclination

The diameter of the ring of light as observed by a distant observer depends predominantly on the mass of the black hole and is nearly constant for all values of the spin and disk inclination as well as for small values of the deviation parameter. For nonzero values of the spin of the black hole, the ring is displaced off center in the image plane. In all cases, the ring of a Kerr black hole remains nearly circular except for very large values of the spin

(a) The ring diameter versus spin for inclinations

In addition to a strong-field test of the no-hair theorem with VLBI imaging of Sgr A*, there exist two other promising possibilities for performing such a test in the weak-field regime. The presence of a nonzero spin and quadrupole moment independently leads to a precession of the orbit of stars around Sgr A* at two different frequencies, which can be studied with parameterized post-Newtonian dynamics [

Evolution of the orbital angular momenta of stars around Sgr A* due to frame-dragging (dashed lines) and stellar perturbations (dotted lines) as measured by the angle

Yet another weak-field test can be performed by the observation of a radio pulsar on an orbit around Sgr A*. If present, timing observations may resolve characteristic spin-orbit residuals that are induced by the quadrupole moment and infer its magnitude (see Figure

Typical timing residuals for a radio pulsar in an orbit around a black hole with a mass of

The fundamental properties of the black hole in the center of our galaxy can be probed with three different techniques. The combination of the results of all three approaches will lead to a secure test of the no-hair theorem with Sgr A*.