Searching for quantum black hole structure with the Event Horizon Telescope

The impressive images from the Event Horizon Telescope sharpen the conflict between our observations of gravitational phenomena and the principles of quantum mechanics. Two related scenarios for reconciling quantum mechanics with the existence of black hole-like objects, with"minimal"departure from general relativity and local quantum field theory, have been explored; one of these could produce signatures visible to EHT observations. A specific target is temporal variability of images, with a characteristic time scale determined by the classical black hole radius. The absence of evidence for such variability in the initial observational span of seven days is not expected to strongly constrain such variability. Theoretical and observational next steps towards investigating such scenarios are outlined.

The impressive success [1] of the Event Horizon Telescope (EHT) in imaging of the apparent black hole in M87 has opened a second observational window to the regime of strong gravity, following the detection of gravitational waves from black hole mergers [2].
It has been widely believed that, due to the smallness of the curvature near the horizon of a large black hole (BH), this is a regime governed by classical general relativity (GR).
However, it has been increasingly appreciated that the inconsistency of BH evolution with the principles of quantum mechanics, given the Hawking effect [3], strongly motivates the need to modify the classical description of a BH at horizon scales [4][5][6][7][8][9][10]. This suggests searching for signatures of such modification in EHT or gravitational wave observations [11][12][13]. As this note will elaborate on, a particularly interesting target is temporal variability of EHT images [14].
Without such modification of BH behavior on horizon scales, BH evolution appears to contradict quantum mechanics. Suppose we think of a BH as a subsystem of a larger quantum system including its environment. Either through interaction with surrounding matter and radiation, or just through the Hawking process itself, the BH subsystem will absorb information from its environment. 1 However, the Hawking process also indicates that BHs ultimately evaporate away. Locality of the geometric description of a BH states that a BH cannot release information while evaporating. And, if an informationcontaining subsystem ceases to exist without releasing its information, that violates the basic quantum-mechanical principle of unitary evolution.
To preserve unitarity, BHs must release their information; simple arguments [15,16] indicate that this should begin midway through a BH's decay, while the BH is comparable to its original size. Such information release requires some modification to the description of a BH that is based local quantum field theory (LQFT) evolution on a classical BH geometry; this modification needs to occur on scales comparable to the size of the event horizon, for an arbitrarily large black hole.
In searching for the physics that makes BHs consistent with quantum mechanics, a reasonable approach is to seek a minimal departure from the conventional GR/LQFT description of BHs, possibly corrected for small backreaction effects. In fact, such a "correspondence principle" is a third postulate for BH evolution, beyond the postulates of consistency with quantum mechanics and of approximate validity of a subsystem description [17].
While there are various other conjectured scenarios for how BH evolution can be consistent with quantum mechanics [18,7,9], the correspondence postulate strongly suggests an effective parameterization of BH evolution in terms of interactions between the BH quantum state, and the quantum degrees of freedom of the BH atmosphere. Such an effective description has been developed in [8,[19][20][21]17].
Beyond the three preceding postulates, a fourth [17] is that such interactions between a black hole and its environment couple universally to all degrees of freedom. This is plausible for an effect that is intrinsically gravitational, and is also strongly motivated both by Gedanken experiments involving black hole mining [22,23] and by the desire to maintain as many features as possible of the beautiful story of BH thermodynamics.
Here dV is the volume element on the time slices and g tt is the metric component in slice-time t, T µν (x) is the stress tensor for field excitations, 3 and H µν (x) parameterize the interactions with the BH. 4 Specifically, the H µν are operators that act on the BH quantum state, and these have spatial dependence that localizes them near the BH. The simplest way to satisfy the correspondence principle is if these are localized within a distance of O(R) from the event horizon of the classical geometry, of radius R, although more complicated scalings are possible [12]. One immediately sees that the interactions (1) behave like BH state dependent metric perturbations. And this raises the prospect of observational signatures in EHT images, due to the effect of such interactions on light propagation near the horizon.
A key question for prospects of observing such a "quantum halo" is the size of the H µν . In units,h = c = 1, H µν is dimensionless, as is appropriate for a metric perturbation.
Since H µν is an operator on BH states, one characteristic of its size is its expectation value H µν in a typical BH state. Another key question is the time dependence of H µν . In 2 For further details, see [17].
3 This is also expected to contain perturbative gravitons. 4 In (1), H I is given in Schrödinger picture. The size of H µν is constrained by the requirement of unitary evolution. Arguments related to those of [15,16] indicate that the interaction (1) needs to transfer information from the BH at a rate dI dt i.e., at roughly one qubit per light crossing time. A bilinear hamiltonian of the form (1) will generically transfer information, but one needs to determine the rate at what it does so.
The simplest way to achieve the required rate (2) follows purely from dimensional analysis. All important dimensional quantities have scale governed by R, and so this rate arises from for x an O(R) distance from the horizon. Such an O(1) metric perturbation raises the prospect of important observational consequences.
One can inquire whether (3) is the minimum possible size. This leads to general question in information theory [17,24]: given two subsystems coupled by a Hamiltonian with a bilinear interaction like (1), how fast does information transfer? In the BH context, ref. [17] argued that the operators H µν could have a much smaller typical size than (3) and still achieve the rate (2) where, for example S bh might be given by the Bekenstein-Hawking entropy, Area/4G. A simple argument for the rate estimate follows from Fermi's Golden Rule, and yields the result that can achieve the rate (2).
Thus, there are two distinct possible scenarios, the strong/coherent scenario of (3), and the weak/incoherent scenario of (5).
Clearly for an astronomical-sized black hole, (5) leads to tiny metric perturbations, which might be naïvely expected to have small effects. However, consider scattering from a BH, mediated by the interaction (1). Fermi's Golden Rule can be again used to estimate the transition probability, which takes the form where the states α, β include the initial and final states of the scattered particle, and ρ is the density of final states. By the same logic as above, this can be O(1), given a typical The initially-reported EHT observations [1] spanned a seven day period; within instrumental resolution, there was no clear indication for temporal variability. In order to test the strong/coherent scenario, it is obviously important to go beyond these initial impressive results, with both further theoretical and observational work.
On the theoretical side, one problem is to more completely explore the parameter space of possible perturbations, and the dependence of observable signatures on these parameters.
Ref. [14] considered a simple parameterization of such coherent perturbations, in terms of their radial wavenumber k, their radial extension R G above the horizon, their angular profile determined by spherical harmonics with angular momenta l, m, their amplitude, and their frequency ω. The spectrum of perturbations assumed in [14] produced a rather dramatic time dependence of images. However, variation of these parameters can of course yield much less dramatic effects. It is also important to extend the models of [14] to the case of a black hole with angular momentum.
In particular, the fact that the initial EHT images produced a clearly identifiable black hole shadow should place an important bound on this parameter space, which could be inferred with further work to study dependence of the simulated images on the parameters.
A particularly promising signature is the temporal variability, which has prospects to be a key test The corresponding periods are for Sgr A * and M87, with respective masses M ≃ 4.3 × 10 6 M ⊙ and 6.5 × 10 9 M ⊙ .
For Sgr A * , such a period is short as compared to the multi-hour imaging scan time used by EHT, except in case of high spin, 5 and so EHT images will average over fluctuations on these time scales. The EHT images are not produced from a simple average, but rather based on a more complex algorithm involving the magnitudes of Fourier components and closure phases of baseline triangles. This presents a problem requiring more analysis, 5 Note that a spin parameter a = .98 multiplies the period by 3.0.
to determine the impact of the perturbations on these averaged images, once they are obtained, and their distinguishability from other sources of variability, such as turbulence in the accretion.
For M87, it appears easier to see such a time dependence directly [14]. However, the currently-reported observations span a time of seven days. While relaxing the demand for agreement with BH thermodynamics could yield shorter-time scale variations, investigating the scale (8) requires longer-span observations. If the spin is high, this period may be significantly extended. Thus, on the observational end, longer-span observations will be useful in searching for strong/coherent fluctuations.
It is also important to disentangle any observed variability from other possible origins; in addition to turbulence in the accretion flow, there can be scattering from the interstellar medium. This is another place where combined theoretical and observational efforts may be needed, both in order to better model such time-dependence, particularly from a conventional accretion origin, and to investigate ways to better distinguish these possible origins observationally. One important aspect of this [14] is the achromaticity of effective metric perturbations; they affect light of all frequencies uniformly, so that their effects should be observed at different EHT operating wavelengths.
Further such theoretical and observational investigations are important for determining EHT sensitivity to the strong/coherent scenario. Ultimately such efforts could reveal the existence of strong/coherent perturbations, or rule them out over a large range of parameter space. The latter would point towards the weak/incoherent scenario, or to an even more exotic reconciliation of the existence of black-hole like objects with quantum mechanics.