Black Holes and Quantum Gravity

Although black holes were first imagined in the late eighteenth century, it was not until Karl Schwarzchild devised a solution to Einstein’s field equations in 1915 that they were accurately described. Despite Schwarzchild’s pioneering work, black holes were still widely thought to be purely theoretical, and so devoid of physical meaning. This view persisted until recent decades, an accumulation of observational evidence removing any lingering doubts about their existence. Beyond their obvious interest as astrophysical phenomena, black holes may, in time, come to be considered a laboratory for new physics. It is conceivable that black holes could be used to study quantum gravity; and a complete and consistent theory of quantum gravity remains the most elusive goal in theoretical physics.

The Schwarzschild Metric

In his solution to the Einstein field equations, Schwarzschild described the gravitational field for a static and spherically symmetric mass. Using natural units, the metric can be written as

1. \begin{equation} ds^2=e^{2\phi}dt^2-e^{2\Lambda}dr^2-r^2d\Omega^2, \end{equation}

where the parameters of the exponential functions can be determined using Einstein’s equations. In (1), the component (00) leads to

2. \begin{equation} e^{2\Lambda}=\left(1-\frac{2M}{r}\right)^{-1}, \end{equation}

where M is the mass of the black hole, while the component (11) leads to

3. \begin{equation} \phi=\frac{1}{2}ln\left(1-\frac{2M}{r}\right). \end{equation}

Schwarzschild’s metric can then be written as

4. \begin{equation} ds^2=\left(1-\frac{2M}{r}\right)dt^2-\left(1-\frac{2M}{r}\right)^{-1}dr^2-r^2d\Omega^2. \end{equation}

In fundamental terms, a black hole is an object with an effective radius r < 2M.¹ In comparison to the Schwarzschild metric, the Kerr–Newman metric is more complex, describing a solution for a spinning, charged mass. But for the purposes of investigating quantum gravity, the Schwarzschild metric is of greater interest.

The Schwarzschild metric tends toward the Minkowski metric for r → $\infty$ and M → 0 and so recovers the metric characteristic of special relativity. The coefficient $\left(1-\frac{2M}{r}\right)$ shows that the circumference of a massive spherical body differs from its radius multiplied by 2$\pi$. The coefficient $\left(1-\frac{2M}{r}\right)^{-1}$ may be attributed to gravitational redshift.

The energy conserved by a free-falling light-emitting particle can be written as

5. \begin{equation} E=m\left(1-\frac{2M}{r}\right)\frac{dt}{d\tau}, \label{ener} \end{equation}

where $\tau$ is the particle’s proper time. Consider an object initially at rest and that remains at rest. Its energy is E = m. From (5),

6. \begin{equation} \left(1-\frac{2M}{r}\right)\frac{dt}{d\tau}=1. \end{equation}

Whence

7. \begin{equation} \left(1-\frac{2M}{r}\right)^2=\left(1-\frac{2M}{r}\right)-\left(1-\frac{2M}{r}\right)^{-1}\left(\frac{dr}{dt}\right)^2. \end{equation}

And whence again

8. \begin{equation} \frac{dr}{dt}=-\left(1-\frac{2M}{r}\right)\left(\frac{2M}{r}\right)^{1/2}. \end{equation}

Shell coordinates are next:

9. \begin{equation} dt_{shell}=\left(1-\frac{2M}{r}\right)^{1/2}dt, \end{equation}

and

10. \begin{equation} dr_{shell}=\left(1-\frac{2M}{r}\right)^{-1/2}dt. \end{equation}

Equation (8) can be written as:

11. \begin{equation} \frac{dr_{shell}}{dt_{shell}}=-\left(\frac{2M}{r}\right)^{-1/2}. \end{equation}

The event horizon is the threshold at which the escape velocity from a black hole exceeds the speed of light. At the event horizon, the velocity in (8) tends toward 0, but in (12), toward –1. The particle is falling. A local observer sees an object in free fall entering a black hole at the speed of light. A distant observer sees the object frozen on the event horizon. Both points of view are correct and consistent within their respective frames of reference.

The precise nature of the event horizon represents another historically important question. The Schwarzschild metric appears to diverge at R = 2M, the factor dr² tending toward $\infty$ at the event horizon. Consider coordinates defined in free fall by using a Lorentz transformation over the shell coordinates:

12. \begin{equation} dt_{fall}=-\gamma V_{rel}dr_{shell}-\gamma dt_{shell}, \end{equation}

where V_rel is the relative velocity between the two frames of reference. It follows that

13. \begin{equation} dt=\frac{dt_{fall}}{\gamma \left(1-\frac{2M}{r}\right)^{1/2}}+\frac{V_{rel}dr}{\left(1-\frac{2M}{r}\right)}. \end{equation}

Replacing $\gamma$ with $(1-\frac{2M}{r})^{-1/2}$ yields

14. \begin{equation} ds^2= \left(1-\frac{2M}{r}\right)dt^2_{fall}-2 \left(\frac{2M}{r}\right)^{1/2} dt_{fall}dr-dr^2. \end{equation}

Although this expression uses a mixed coordinate system, it does show that the singularity at the event horizon is an artifact and so devoid of physical meaning. On the other hand, the central singularity is real, so much so that the Kretschmann invariant diverges from it—something it could not do if the central singularity were purely a mathematical construction.

The behavior of light in radial motion inside a black hole can be probed by starting from (14) and writing ds² = 0:

15. \begin{equation} \frac{dr^2}{dt^2_{fall}}+2\left(\frac{2M}{r}\right)^{1/2}\frac{dr}{dt_{fall}}-\left(1-\frac{2M}{r}\right)=0. \end{equation}

This equation has two solutions,

16. \begin{equation} \frac{dr}{dt_{fall}}=-\left(\frac{2M}{r}\right)^{1/2}\pm1, \end{equation}

corresponding to movements toward the center and the exterior. Inside the black hole, where r < 2M, both solutions are negative. An emitted photon moves inward. Since nothing can move locally faster than light, it is impossible to escape from a black hole.

Consider a particle initially at rest within a black hole. Introducing the proper time, $\tau$, in (8), where dr/d$\tau$ = –(2M/r)^1/2, its time inside the black hole is

17. \begin{equation} \tau=-\int_{2M}^0\left(\frac{r}{2M}\right)^{1/2}dr=\frac{4}{3}M. \end{equation}

The particle reaches a singularity in a finite time. Since the event horizon is mathematically but not physically real,² particles within a super massive black hole may enjoy a certain grim existence if tidal effects remain weak at the surface. That singularity is represented as a horizontal line in the Penrose diagram of a Schwarzschild black hole, but it could equally be considered the end of time.

Black Hole Thermodynamics

In recent decades, much energy has been devoted to investigating the degree to which the properties of black holes conform to the laws of thermodynamics. In general relativity, a black hole is described according to three fundamental parameters: mass, angular momentum, and electrical charge.³  This suggests that they may be amenable to thermodynamic analysis, if only because thermodynamics studies systems whose properties can also be described by a small number of parameters. A black hole can expand but not contract. It is a one-way job. In 1973, Jacob Bekenstein suggested that the entropy of a black hole might be proportional to the area of its event horizon,⁴

18. \begin{equation} S=\frac{A}{4}, \end{equation}

where the proportionality constant is determined by consistency considerations.⁵ From this conjecture, four laws followed, each with its analogue in old-fashioned thermodynamics:

Zeroth Law: The surface gravity of a black hole is constant at its event horizon. The analogue: The temperature of a body is homogeneous at equilibrium.

First Law: Black hole perturbations are given by

19. \begin{equation} SdE=\frac{\kappa}{8\pi}dA+\Omega dJ+\Phi dQ, \end{equation}

where E is the energy, $\kappa$, the surface gravity, $\Omega$, angular velocity, J, angular momentum, $\Phi$, electrical potential, and Q, charge. The analogue: Energy conservation in thermodynamics.⁶

Second Law: Given the weak energy condition $T_{\mu\nu}X^{\mu}X^{\nu}\le0$ for an inhomogeneous vector field, the surface area of a black hole increases in the obvious way:

20. \begin{equation} \frac{dA}{dt}\ge 0. \end{equation}

The analogue: The entropy of a closed system can only increase.

Third Law: It is impossible for the surface gravity of a black hole to be precisely zero. The analogue: It is impossible to reach absolute zero in a finite number of operations.

The Hawking Effect

In 1975, Stephen Hawking established that, contrary to conventional vision, black holes can evaporate and emit radiation.⁷ This effect can be understood in a number of different ways. The most basic view considers the tidal effect on vacuum fluctuations at the boundary of a black hole, the razor’s edge. Particle creation is pair-wise. One of the particles drops back into the black hole, the other is ejected beyond the event horizon.

The Unruh effect demonstrates that this might well be so.⁸ An observer subjected to constant acceleration perceives a thermal bath of particles at the temperature T = a/(2$\pi$). This phenomenon is established using Bogoliubov transformations. This approach involves starting from the Schwarzschild metric and considering a stationary observer in the expression

21. \begin{equation} r=2M+\frac{\rho^2}{8M}. \end{equation}

The associated Rindler metric, in which the Rindler coordinates represent a hyperbolic acceleration reference frame, can then be written in lowest order as $\tau$ = t/(4M). Einstein’s equivalence principle establishes that, given the Unruh effect, the observer would perceive an excited field at the temperature

22. \begin{equation} T_{loc}=\frac{a}{2\pi}=\frac{1}{2\pi\rho}=\frac{1}{4\pi\sqrt{2M(r-2M)}}. \end{equation}

The temperature at distance R is obtained by simply applying the gravitational shift factor g₀₀(r) / g₀₀(R) to (22):

23. \begin{equation} T(R)=\frac{1}{4\pi\sqrt{2Mr(1-\frac{2M}{R})}}. \end{equation}

The value at infinity is therefore

24. \begin{equation} T(\infty)=\frac{1}{4\pi\sqrt{2Mr}}, \end{equation}

which leads, with r = 2M, to

25. \begin{equation} T_H=\frac{1}{8\pi M}. \end{equation}

In complete units, the temperature is written as $T_H=\hbar c^3/(8\pi G k_B M)$. This is one of the few simple formulas in physics to include all of the fundamental constants. The Hawking effect invokes gravitation, quantum physics, statistical physics, and relativity, all at the same time.

In reality, the Hawking effect is obviously much more complex and the spectrum is not entirely thermal. The number of spin particles s emitted per unit of time t and energy Q is

26. \begin{equation} \frac{d^2N}{dQdt}=\frac{\Gamma_s}{2\pi\left(e^{\frac{Q}{\kappa/(4\pi^2)}}-(-1)^{2s}\right)}, \end{equation}

with the gray-body factor

27. \begin{equation} \Gamma_s=4\pi \sigma_s(Q,M,\mu)(Q^2-\mu^2), \end{equation}

where $\mu$ is the mass of the emitted particle and $\sigma$, the effective absorption cross section. The absorption cross section is not trivial: it contains information about the structure of space-time.⁹ It also serves to express the probability of backscattering from the emitted particle in the gravitational potential.

The Hawking effect is explosive. Unlike a piece of metal, for example, the more a black hole radiates, the hotter it becomes. Although this process is negligible for massive astrophysical black holes with an extremely low Hawking temperature, it becomes important for low mass black holes. The existence of such black holes remains unproven, but they may have arisen from particular conditions in the primordial universe.¹⁰ The Hawking effect has nevertheless been observed in similar acoustic black hole systems.¹¹

Although the evaporation of black holes is well understood, it is nonetheless linked to a central paradox in theoretical physics. The notion that information is lost to the outside universe when it enters a black hole is not inherently problematic. The paradox arises from the fact that evaporation transforms the black hole into a thermal, or quasi-thermal, spectrum. As a result, the information loss that occurs conflicts with quantum field theory.¹² Although many solutions have been proposed, ranging from stable relics to subtle correlations between emitted particles, a consensus is yet to emerge.

Quantum Gravity

The search for a quantum theory of gravitation is considered one of the most important and difficult problems in theoretical physics. The conceptual and technical obstacles are formidable, encompassing, as they do, problems of non-renormalizability and the question of fundamental invariance. It is generally thought that the effects of quantum geometry are only felt close to the singularity at the center of a black hole. By contrast, the outer and measurable zone of a black hole, are faithful to the predictions of general relativity. A number of solutions have been proposed to overcome this somewhat disappointing and limited conclusion.

Loop quantum gravity (LQG) is an attempt to formulate a non-perturbative and invariant quantification of general relativity.¹³ LQG has been developed in both a canonical form, using the Ashtekar connection and the flow of triad densities,¹⁴ and a covariant form, using spin networks. The description of black holes under LQG is based on the concept of an isolated event horizon,¹⁵ a quasi-local notion liberated from a global description of space-time.¹⁶ In essence, the event horizon acts as a surface intercepting the spin network. Each intersection has corresponding quantum numbers (j, m),¹⁷ where j is a half-integer associated with the area and m is the associated projection corresponding to the curvature. These values verify

28. \begin{equation} A-\Delta\leq 8\pi \gamma \sum_{p}{\sqrt{j_p(j_p+1)}}\leq A+\Delta, \end{equation}

where $\gamma$ is the Immirzi parameter,¹⁸ while $\Delta$ is a smoothing scale that refers to the different intersections.

Physicists using a Monte Carlo simulation have demonstrated that Hawking evaporation retains a footprint of the discrete structure of this area¹⁹—an imprint emanating from the effects of quantum geometry; yet if the density of black hole states grows exponentially, it is still not possible to discern emission lines for high mass black holes. For masses relatively close to the Planck mass, the discrete character of the area, given by

29. \begin{equation} A_j=8 \pi \gamma \sum_{p=1}^N\sqrt{j_p(j_p+1)}, \end{equation}

where N is the total number of intersections, is revealed in the evaporation spectrum. A Kolmogorov–Smirnov test shows that the observation of 4 × 10⁵ events would be required to discriminate LQG-type behavior from what might be considered purely Hawking behavior at 3$\sigma$ for 20% experimental resolution. In this sense, the non-perturbative effects of quantum gravity can alter the semi-classical effects associated with evaporation.

It is conceivable that the consequences of discretization would be visible for much more massive black holes. Suppose that transitions between the quantum states of a black hole during evaporation are not associated with a complete state reconfiguration, but only a change of state for a single wafer of the elementary area.²⁰ In this case, only the quantum states associated with one of the terms in (29) need be considered, rather than the quantum state associated with its sum. A simple calculation shows that the relative spacing between the rays no longer depends on the mass of the black hole, suggesting that the measurement of quantum gravitational effects might be achievable when at some arbitrary distance from the Planck mass.

The reasoning is straightforward, if intuitive. When a high mass black hole evaporates, it should emit a very low quantum of energy since the temperature is given by T = 1/(8$\pi$M) for large values of M. Its area should, therefore, decrease by a tiny and possibly sub-Planckian value. Not so. Classically, A = 16$\pi$M² and thus dA = 32$\pi$MdM. Given that dM ~ T, dA ~ 4, it follows that a variation in area is not dependent on mass. At the peak of the spectrum, the independence of the variation in area from mass is the underlying reason why a local perspective can induce measurable quantum gravity effects for massive black holes. The diffuse background of gamma rays created by the decay of the neutral pions emanating from the quarks and gluons emitted by the black hole is on the same order of magnitude as this signal and does not mask it.²¹

Even if the area spectrum were continuous, the mere existence of a minimum value for the area, probably close to the Planck area, would induce a truncation of the emitted flux by prohibiting excessively low energy values. At a given fixed temperature T, the spectrum would be truncated below

30. \begin{equation} E_{min}=\frac{T}{4}A_{min}\sim\frac{T}{4}. \end{equation}

This is another potential avenue for observation.

It seems likely that the gray-body coefficients, which encode the possible backscattering of particles in the gravitational potential of a black hole, are affected by quantum gravitational effects.²² The result would be a distortion of the potentially measurable Hawking spectrum.

Bouncing Black Holes

Might black holes rebound from the effects of quantum gravity? Some physicists have suggested as much.²³ Something similar happens in loop quantum cosmology when the Big Bang is replaced by a Big Bounce. The current expansion phase would then have been preceded by a phase in which the universe contracted. A number of general arguments lend support to the supposition that non-perturbative quantum geometry effects could cause a transition between a black hole and white hole type of state. From the nodal point, the expected rebound time is proportional to M², with a proportionality constant on the order of 5 × 10^–2 set for internal consistency. The Hawking evaporation time is on the order of M³. In this model, the black holes bounce before evaporating; the Hawking effect acts as a dissipative correction. Primordial black holes formed just after the Big Bang with a mass of 10²⁶g or less should have already rebounded.

The phenomenology associated with these ideas is rich and complex. A more delicate point concerns the energy of the signal emitted by bouncing black holes. Two hypotheses are forthcoming, the first for a low energy, and the second for a high energy signal. On the first hypothesis, only scale counts, and so the wavelength of the emitted radiation must be on the order of the black hole’s size. On the second, the energy of the radiation emitted by the white hole is equal to the energy of the radiation that collapsed to form the black hole in the first place. Since the mass of a primordial black hole is almost in bijective correspondence with its formation time, it is possible to determine the wavelength of the emitted radiation for each mass.

Fast radio bursts—the result of a still unidentified astrophysical process—might be phenomena of this type.²⁴ These mysterious bursts of radio waves could be due to the low-energy component of bouncing black holes when the stochastic nature of their lifespans is taken into account. Although alternative explanations have also been proposed, this model has the advantage of being testable in principle. Indeed, redshift dependence is extremely specific. If fast radio bursts are explained by an astrophysical or particle physics phenomenon—such as the annihilation of dark matter particles—their characteristic frequency must be redshifted in a way that matches the host galaxy. In the case of bouncing black holes, the black holes located further away rebound earlier and are lighter as a result. They emit a higher energy signal, which compensates for the rebound.

In the future, it would be interesting to study the effects of a gravitational, rather than a cosmological redshift. In this type of redshift, radiation would be emitted in the vicinity of the black hole as an intense gravitational field, its energy red-shifted by a factor of $\left(1-\frac{2M}{r_e}\right)$, where r_e is the source of emission. More precise modeling is required for a quantitative estimate, but the characteristic frequencies could be substantially reddened. In this context, it is pertinent to consider varying the coefficients of proportionality between the rebound time and the square of the mass, while also taking care to ensure that the coefficient remains lower than the Hawking time. In doing so, it is little short of remarkable that the excess of gamma rays observed by the Fermi satellite can also be explained by bouncing black holes.²⁵ A potential link via the high energy component to the events detected by the Auger collaboration, an ongoing effort to study ultra-high energy cosmic rays, is also conceivable.

Quasi-Normal Modes

Gravitational waves were first measured by the LIGO interferometry project in September 2015.²⁶ This discovery not only represented the first step in a new area of astrophysics, but also provided the basis for new high-precision tests of general relativity. The black hole coalescence events observed by LIGO exhibit three distinct phases: orbit, fusion, and relaxation. During the final phase, the resulting coalesced black hole emits so-called quasi-normal modes (QNMs), which correspond to its de-excitation by gravitational wave emissions. The radial part of the perturbed metric is written as

31. \begin{equation} \Psi = A e^{-i\omega t} = A e^{-i(\omega_R + i\omega_I)t}, \end{equation}

where $\omega$_R characterizes the oscillations and $\omega$_I is the relaxation time $\tau = \frac{1}{\omega_I}$. QNMs form a discrete set.²⁷ They are composed of axial and polar perturbations that are described by the Regge–Wheeler and Zerilli equations. Modified gravitation models—which can break the Lorentz invariance, the equivalence principle, and even diffeomorphism invariance—generically lead to QNM modifications.²⁸ It is conceivable that approaches to quantum gravity might lead to this type of effect. In 2016, a model along these lines was developed by Hal Haggard and Carlo Rovelli.²⁹ The idea they proposed was straightforward. The scale of curvature is on the order  $l_R \sim {\cal R}^{-1/2}$, where the Kretschmann scalar is ${\cal R}^2:=\cal R_{\mu\nu\rho\lambda}\cal R^{\mu\nu\rho\lambda}$. From this formulation it seems that quantum effects are negligible for the massive black holes encountered in astrophysics. Yet this assertion does not take into account cumulative effects, which may turn out to be significant. On the basis of dimensional arguments, the so-called quanticity of space-time can be estimated and integrated over a proper time $\tau$ at $q=l_P \ {\cal R} \ \tau$. Relating proper time to Schwarzschild time by

32. \begin{equation} \tau=\sqrt{1-\frac{2M}{r}}\ t, \end{equation}

yields

33. \begin{equation} q(r) = \frac{M}{r^3} \ \left(1-\frac{2M}{r}\right)^{1/2} t. \end{equation}

The maximum of this function is reached for $r=2M\left(1+\frac{1}{6}\right)$, which is where the integrated quantum effects should be most important. It is remarkable that this is outside the event horizon.

For the moment, these arguments remain heuristic and the exact location of maximum quantum correction has not been rigorously established. For this reason, it is relevant to model the effect of the Schwarzschild metric distortion³⁰

34. \begin{equation} ds^2=-f(r)dt^2+f^{-1}(r)dr^2-r^2d\Omega^2, \end{equation}

with

35. \begin{equation} f(r)=\left(1-\frac{2M}{r}\right)\left(1+Ae^{-\frac{(r-\mu)^2}{2\sigma^2}}\right)^2. \end{equation}

It is then possible to calculate the QNMs in this approach and to examine the parameters for which deviations from general relativity become important. The effects are greatest for a modification at r = 3M. This is not entirely surprising because such a result corresponds to the maximum potential. A numerical analysis shows that the relative variation for the real part of the QNMs is given by 0.8 × A, for $\mu$ = (7/6)r_S, and that the relative variation of the imaginary part is given by 2.7 × A. Given that these coefficients of proportionality are known, it then becomes easy to estimate the magnitude of the quantum corrections that give rise to an observation. This is an enticing prospect. In a little more than a decade, the Einstein Telescope should allow for a measurement of the fundamental and first harmonics of QNMs with a precision of a few percent.³¹

Event Horizon Telescope

The information paradox demonstrates that there is some tension between the theories of relativity and quantum mechanics and the principle of locality. To resolve this tension, a new and radical development is needed. Several arguments have been made emphasizing that significant changes are expected on the event horizon, or beyond, even for supermassive black holes.³² In 2019, the Event Horizon Telescope captured an image of the black hole at the heart of the M87 galaxy using interferometric radio astronomy techniques.³³ It turns out that such images could also be used to probe quantum gravity effects.³⁴ The presence of metric fluctuations with time and length scales determined by the size of the black hole should result in a temporal variation in images. This is potentially measurable for the most massive black holes. The period is given by

36. \begin{equation} P\simeq 0.93 \left(\frac{M}{4.3\times 10^6 M_\odot}\right)~\mbox{hours}, \end{equation}

where M$_\odot$ is the mass of the Sun.

At this stage, the measurements being made are few in number and the complex averaging technique in use makes interpretation in terms of stability a challenge. Conventional astrophysical effects, it should be noted, can also generate temporal effects. Although there are still limitations, this approach holds much promise for future research and should also be considered in relation to gravitational waves.

New Physics

It has long been assumed that the quantum gravity effects associated with black holes are confined to their centers and are unobservable as a result. Despite such constraints, black holes still have much to offer researchers investigating quantum gravitation. The emergence of a true black hole astronomy based on the measurement of gravitational waves and radio interferometry has the potential to bring quantum gravity into the field of experimental or observational science. On this view, black holes should rightly be considered incomparable laboratories for the development of new physics.

Translated and adapted from the French by the editors.