Inside astronomically realistic black holes

TL;DR

This paper explores the internal structure of realistic black holes, revealing that singularities are surfaces with diverging radiation energy density and that accretion leads to oscillatory collapse to a strong singular surface.

Contribution

It provides a detailed analysis of the internal geometry of realistic black holes, including the effects of accretion and the nature of singular surfaces.

Findings

Singularity in Schwarzschild black holes is a surface with diverging radiation density.

Inside Kerr black holes, spacetime ends at the inner horizon due to mass inflation.

Accreting black holes undergo oscillatory collapse to a strong, spacelike singular surface.

Abstract

The singularity of a spherical (Schwarzschild) black hole is a surface, not a point. A freely-falling, non-rotating observer sees Hawking radiation with energy density diverging with radius as $\rho \propto r^{-6}$ near the Schwarzschild singular surface. Spacetime inside a rotating (Kerr) black hole terminates at the inner horizon because of the Poisson-Israel mass inflation instability. If the black hole is accreting, as all realistic black holes do, then generically inflation gives way to Belinski-Khalatnikov-Lifshitz oscillatory collapse to a strong, spacelike singular surface.

Equations

$$⟨T_{\mu }^{\mu }⟩=\frac{q_{\mathrm{eff}}}{60{\pi }^2}\frac{M^2}{r^6},$$

$$T=\frac{\kappa }{2\pi }.$$

$$\kappa\rightarrow{1\over 3|\tau|}\propto r^{-3/2}\quad\mbox{as $r\rightarrow 0$}\ .$$

$$\frac{q_{\mathrm{eff}}}{2880{\pi }^2{M}^4}.$$

$$\rho ={\rho }_{\mathrm{s}}{e}^{vt-\xi }.$$

$$\text{mbox}in=u-v=0.001,\text{mbox}out=u+v=0.003.$$

$$d{s}^2=-d{t}^2+a_1^{2}d{x}_1^2+a_2^{2}d{x}_2^2+a_3^{2}d{x}_3^2,$$

$$a_i=|t{|}^{q_i},$$

$$q_1+q_2+q_3=1,q_1^2+q_2^2+q_3^2=1.$$

$$\text{mbox}in=u-v=0.01,\text{mbox}out=u+v=0.03.$$

$$q_1=1,q_2=q_3=0\text{mbox}inflation.$$

$$q_1=-\frac{1}{3},q_2=q_3=\frac{2}{3}\text{mbox}collapse.$$

Full text

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1 Introduction

What really happens inside black holes? The question is of perennial fascination to students and the public. Yet the literature on the subject is surprisingly sparse. That is to say, there is a large literature on singularities and quantum gravity, but little on the more prosaic topic of what happens on the way to a singularity. In fact it’s a wilder ride than you might imagine.

What would you respond to a student or member of the public who asks what happens inside black holes? Here is a sample of possible answers:

Nothing can escape a black hole, so what happens inside is unknowable. 2. 2.

“All that makes up a black hole is crushed together at a single miniscule point at the black hole’s center.” 3. 3.

You are spaghettified. 4. 4.

You are burned up by a firewall at the horizon. 5. 5.

You are burned up by Hawking radiation as you approach the singularity. 6. 6.

There is a weak null singularity on the Cauchy horizon. 7. 7.

There is BKL oscillatory collapse to a spacelike singularity. 8. 8.

There is a naked singularity.

Here are my opinionated answers to the above questions: (1) False—see §8. (2) False—see §2. (3) True. Interesting fact: you are spaghettified a tenth of a second before you hit the singularity (tidal force and time both depend in the same way $GM/r^{3}$ on mass $M$ and radius $r$). (4) False—see §3. (5) True—see §4. (6) False, at least for astronomically realistic black holes—see §5. (7) True—see §7. (8) I suspect that classical naked singularities do not survive quantum back reaction.

2 The Singularity is Not a Point

It is widely repeated in popular and indeed expert writing that the singularity at the centre of a (spherical) black hole is a point. The quote in item (2) of the Introduction is from Brian Greene’s “The Fabric of the Cosmos”, page 337.[1] It’s not true.

Of course, a mathematical relativist would say, correctly, that the central point is not part of the spacetime, and should be excised. Rigorously, in general relativity one should make statements about behaviour asymptotically close to, but not at, the singularity. But that is not the point I am making. The singularity is a surface, not a point. Let me prove it to you.

Figure 1 shows two infallers, blue and purple, who free-fall radially to the singularity of a Schwarzschild black hole. The defining feature of black holes is that space falls faster than light inside their horizons[2] (the rule is that nothing can move through space faster than light; spacetime itself can do whatever general relativity prescribes). So inside the horizon, light must move inward, regardless of the direction in which it may be pointed. There are lightrays that have the maximum possible sideways motion (technically, these lightrays have infinite angular momentum per unit energy). The cardioid-shaped111The region is in fact a mathematical cardioid. grey region in Figure 1 is bounded by maximally-sideways-moving lightrays that just hit the blue infaller as the latter hits the singularity. Anything in the grey region is invisible to the blue infaller: a signal emitted in the grey region would have to move through space faster than light to reach the infaller at (just outside) the singularity.

Therefore the last that the blue infaller sees of the purple infaller is at the boundary of the grey region, already well away from the singularity. Far from encountering each other at the singularity, the blue and purple infallers lose causal contact with each other already some distance from the singularity. Since the two infallers fall to causally disconnected places, the singularity cannot be a point: it must be a surface (as will be elucidated in a moment).

You might object that the spatial separation between the two infallers goes to zero at the singularity, as is evident from the Schwarzschild metric. The counter-intuitive thing here is that, contrary to familiar experience, being spatially close does not imply being causally close. The two red lines originating from the starred point in Figure 1 show the shortest causal path joining the two infallers as they approach the singularity. A pair of lightrays emitted from the starred point would just reach the infallers as they hit the singularity.

If the Schwarzschild singularity is a surface, what is the nature of that surface? Classically, the boundary of a 4-dimensional spacetime is 3-dimensional. Since it is the time dimension that gives up the ghost at the Schwarzschild singularity, classically the boundary is spacelike. However, quantum mechanically, if unitarity holds, then the singularity at different times (as seen by observers who fall in at different times) must be unitarily related to each other. Ordinarily, unitary evolution is what happens along a time dimension. Therefore an effective time dimension must emerge, and the singular surface should be thought of as a 2-dimensional surface that evolves unitarily in time. The notion that the singularity is a 2-dimensional surface accords with what Hawking taught us, that a black hole is a thermodynamic object with number of states proportional to the 2-dimensional area of its horizon.

3 The Horizon is Not a Barrier

An enduring feature of (mis)understanding of black holes is ongoing confusion over the nature of horizons. The classical confusion was cleared up by David Finkelstein in 1958,[3] who showed that light falls freely through the horizon, and does not get stuck at the horizon as a simplistic interpretation of the Schwarzschild metric had suggested.

In modern times the confusion has reappeared in a quantum context. Hawking radiation is emitted from the horizon, and the entropy of a black hole equals a quarter of its horizon area. What happens to an observer who falls to the horizon? Do they see Hawking radiation there? Do they encounter the states of the black hole there?

The clearest example of quantum confusion over the horizon is the “firewall paradox” posited by Almheiri, Marolf, Polchinski, and Sully in 2012.[4] More on this at the end of this section.

Consider the following thought experiment. You free-fall feet-first into a black hole. What do you see as you look down at your feet when your feet pass through the horizon? Obviously, since you are in a free-fall frame, you continue to see your feet below you. The light emitted by your feet upwards at the horizon remains at the horizon, and your eyes catch up with the light from your feet as your eyes fall through the horizon, but you do not catch up with your feet themselves. Continuing to free-fall inside the horizon, you continue to see your feet below you. Inside the horizon, where space is falling faster than light, you see an image of your feet below you as they used to be above your head.

The moral of this story is that when you fall through the horizon, you do not catch up with the source of any light that may have been emitted upwards at the horizon. For example, when you watch a black hole from outside the horizon, what you are looking at is not actually its horizon, but rather the dimming, redshifting surface of the star or whatever else collapsed to, or fell into, the black hole long ago. When you free-fall through the horizon, you do not catch up with the star or other stuff that collapsed long ago. It’s already gone. Its image remains ahead of you, still dimming and redshifting away.

Ray-tracing confirms these arguments. Figure 2 shows eight frames from a visualization[5] of the view seen by an observer who free-falls through the horizon of a Schwarzschild black hole. The red grid is nominally the past horizon. A real black hole formed from stellar collapse does not have a past horizon. Rather, the past horizon is replaced by the dimming, redshifting surface of the star that collapsed long ago. Gavin Polhemus and I[5] started calling this surface the “illusory horizon” to distinguish it from the true horizon that you actually fall through.

An observer who free-falls through the true horizon does not encounter a concentration of classical radiation emitted there by the star that collapsed long ago, and no more so do they encounter Hawking radiation there. The dimming, redshifting surface of the star still appears ahead of the observer, and so also does the Hawking radiation. The illusory horizon has all the properties that a source of Hawking radiation should have: it is the boundary that divides what an observer can see from what they cannot see; and from the perspective of the observer, objects infalling through the illusory horizon continue to redshift exponentially, albeit at an exponential rate that varies with time.

Where do the arguments of the firewall paradox go wrong? The fundamental problem is a confusion between the true and illusory horizons. Any argument involving regions outside and inside “the” horizon, for example quantum entanglement between outside and inside, must take into account that the notion of outside and inside is observer-dependent. It is the illusory horizon, not the true horizon, that divides what an observer can see from what they cannot. The “near-horizon zone” that so vexes firewallers is observer-dependent, just as it is in cosmology.

Figure 2 illustrates that a free-faller has the impression of reaching the illusory horizon as they hit the singularity. At the singularity, the illusory horizon has the appearance of a flat, 2-dimensional surface. This is where an infaller finally encounters the quantum states of the black hole. This is where the firewall resides.

4 Hawking Radiation inside a Schwarzschild Black Hole

Hawking radiation is widely considered to be a profound clue to quantum gravity. There are thousands of papers on Hawking radiation as seen by observers outside the horizon of a black hole, but practically nothing from the perspective of an observer who falls inside the horizon. This seems mysterious: if Hawking radiation is such a profound clue, would one not learn more by exploring it near the singularity?

Part of the difficulty is that the problem of Hawking radiation inside a black hole is mathematically challenging. Following Hawking’s original 1974 paper,[7] there was a spate of work on quantum field theory in curved spacetime.[8] There are two separate but related questions: (1) What does an observer see? (2) What is the quantum modification to the energy-momentum tensor? The first question is complicated by the fact that an accelerating observer sees quantum radiation—Unruh[9] radiation—that a free-faller does not. As regards the second question, a rigorous approach to calculating the quantum energy-momentum in a given curved spacetime involves summing over an infinite series of divergent terms. The divergent part of each term must be isolated and subtracted before the series can be summed. Unfortunately the problem proved intractable, even in the simple case of the Schwarzschild geometry.

However, some exact results are calculable. The so-called trace anomaly $\langle T_{\mu}^{\mu}\rangle$, or conformal anomaly, which is the trace of the expectation value of the quantum energy-momentum, equals a certain combination of squares of Riemann curvature components. For a Schwarzschild black hole of mass $M$, the trace anomaly is[10] (in Planck units $c=G=\hbar=1$)

[TABLE]

where $q_{\rm eff}$ is an effective number of particle species. The trace anomaly (1) diverges as $r^{-6}$ near the singular surface. Further calculable constraints emerge from the requirement that the expectation value $\langle T_{\mu\nu}\rangle$ of the quantum energy-momentum be covariantly conserved, an approach pioneered by Christensen et al.[11]

There is a second reason why there has been so little progress on Hawking radiation inside the horizon, namely confusion between the illusory and true horizons. As remarked in §3 above, it is the illusory horizon that is the source of Hawking radiation, for both inside and outside observers. The illusory horizon is the surface where an observer see objects redshifting exponentially.

Although a full quantum calculation is still intractable, it is relatively straightforward to go to the classical high-frequency limit of geometric optics, and ray-trace from the illusory horizon to the observer. A quantity of central interest is the acceleration $\kappa$, the rate of redshifting at the illusory horizon. If the acceleration $\kappa$ were constant, then the standard Hawking calculation shows that the emission would be thermal with temperature $T$ equal to the acceleration divided by $2\pi$,

[TABLE]

Figure 4 shows the acceleration on the illusory horizon directly below as seen by an observer who free-falls radially into a Schwarzschild black hole.[12] Since a free-faller accelerates away from the sky above, they also see Hawking radiation from the sky above. Figure 4 shows the acceleration on the sky directly above. The Figure recovers the usual results well outside the black hole, that the acceleration on the illusory horizon is $1/(4M)$, while the acceleration on the sky is zero. As the free-faller falls through the true horizon, they see both accelerations increasing smoothly. Near the singularity, the acceleration on both illusory horizon below and sky above diverges, tending to one third the reciprocal of the proper time $|\tau|$ left until the observer hits the singularity,

[TABLE]

A more extensive calculation[12] shows that, for a non-rotating observer (who stares fixedly in constant directions), the perceived acceleration near the singularity tends to the same constant (3) in all directions, except for an ever-thinning band in the horizontal directions. This points to the remarkable conclusion that the Hawking radiation seen by a freely-falling, non-rotating observer becomes isotropic as they approach the singular surface.

The fact that the perceived acceleration (3) changes in time means that the radiation is non-thermal. Nevertheless, the acceleration (3), which is the only timescale in the problem, sets the natural frequency of Hawking radiation. Encouragingly, when the same calculation is carried out in 1+1 dimensions, the result coincides with the known exact result.[13]

Figure 5 shows the resulting quantum energy density in radiation and vacuum. A density of one in the Figure corresponds to (in Planck units)

[TABLE]

Radiation has a relativistic equation of state, so its energy-momentum tensor has zero trace. The vacuum energy then follows directly from the trace anomaly (1). It should be cautioned that the calculation is not exact (an exact calculation being intractable); nevertheless there does not seem to be much wiggle room to adjust the curves substantially.

The central conclusion of the calculation is that Hawking radiation diverges near the singular surface, with characteristic frequency going as $r^{-3/2}$, and the energy density going as $r^{-6}$, the fourth power of the frequency. The singular surface is hot.

5 Mass inflation at the inner horizon

Real astronomical black holes rotate. The most important difference between the Kerr geometry for a rotating black hole and the Schwarzschild geometry for a spherical black hole is that the Kerr geometry has an inner as well as an outer horizon. Figure 6 illustrates the geometry of a Kerr black hole with spin parameter $a=0.96$.

A major advance in understanding the interior structure of black holes was Poisson & Israel’s[15] 1990 discovery of the mass inflation instability at the inner horizon of a charged, spherical (Reissner-Nordström) black hole. The inflationary instability is the nonlinear realization of the infinite blueshift at the inner horizon first pointed out by Penrose.[16] Barrabès, Israel & Poisson[17] soon generalized Poisson & Israel’s argument to the case of a rotating black hole.

Any black hole with an inner horizon is subject to the mass inflation instability. Regardless of their orbital parameters, light and matter falling into a black hole focus at the inner horizon into one of just two directions, the principal ingoing and outgoing null directions. Figure 7 illustrates that an ingoing observer at the inner horizon of a Reissner-Nordström black holes sees outgoing light and matter focused into a divergingly blueshifted bright point. An outgoing observer similarly sees ingoing light and matter focused along the opposite direction. Movies of journeys into a Reissner-Nordström black hole are at https://jila.colorado.edu/~ajsh/insidebh/rn.html. Unfortunately I have not yet implemented comparable movies of Kerr black holes.

Already in his seminal note,[16] Penrose suggested that the diverging concentration of energy density at the inner horizon would destabilize it, but in subsequent work he fell short of proving as much. Poisson & Israel’s great contribution was to calculate for the first time the back-reaction of the counter-streaming streams on the black hole. They showed that the counter-streaming would drive exponential growth of the interior mass.

Figure 8 shows a Penrose diagram that illustrates the physical reason behind the Poisson-Israel mass inflation instability. Between the outer and inner horizons of the Kerr (or Reissner-Nordström) geometry, the Killing time coordinate $t$ (the coordinate associated with time translation symmetry) is a spacelike coordinate. Infalling objects are ingoing or outgoing depending on whether they are moving backwards or forwards in time $t$. There is not one but two inner horizons, an ingoing inner horizon through which ingoing objects fall, and an outgoing inner horizon through which outgoing objects fall. While all infalling objects are necessarily ingoing at the outer horizon, objects on prograde222Prograde here means having specific angular momentum greater than that of the principal null geodesics. geodesics turn around and become outgoing at the inner horizon, while objects on retrograde geodesics remain ingoing.

In their original work, Poisson & Israel envisaged that ingoing and outgoing streams would be produced by a Price tail of gravitational radiation generated by the initial collapse of a black hole (for simplicity they used a spin-0 scalar field as a surrogate for spin-2 gravitational waves). Most of the literature on mass inflation since then has focused on this same scenario. The outcome is a “weak null singularity on the Cauchy horizon” (the Cauchy horizon, the boundary of (un)predictability, is what I have been referring to as the inner horizon).

The singularity is weak in the sense that, although the tidal force diverges to infinity, it does so in such a short proper time that the spacetime is scarcely distorted. The reason for this behaviour is that the exponential timescale over which inflation grows is proportional to the accretion rate: smaller accretion rates imply faster inflation.[18] The Price tail of radiation that is assumed to drive inflation decreases to zero as a power law with time, causing the growth rate of inflation, and with it the tidal force, to diverge before the spacetime has had time to respond to the tidal force.

All real astronomical black holes accrete, cosmic microwave background photons if nothing else. Accretion fuels both ingoing (retrograde) and outgoing (prograde) streams. It is straightforward to estimate that when a black hole forms from stellar collapse, the energy density in the Price tail of gravitational radiation, which decays with time as $t^{-12}$ in its slowest decaying mode (the quadrupole), falls below the density of the cosmic microwave background in at most several seconds. For black holes that continue to accrete, the outcome is collapse, not a weak null singularity.[18]

6 Conformally separable solutions for accreting, rotating black holes

One of the remarkable features of the Kerr geometry is that geodesics in it are Hamilton-Jacobi separable.[19] Indeed, the hypothesis that geodesics are Hamilton-Jacobi separable provides one way to derive the Kerr and related geometries: Hamilton-Jacobi separability imposes conditions on the line-element that permit the Einstein equations to be separated systematically.

It proves possible to impose the weaker condition of conformal separability, and still separate the Einstein equations.[20] Conformal separability requires only that null geodesics, not all geodesics, are Hamilton-Jacobi separable. Whereas standard separability imposes that the spacetime is stationary, conformal separability admits a time-dependent conformal factor $\rho$ in the line-element. The conformal factor $\rho$ in a conformally separable black hole is a product of separable (Kerr) $\rho_{\rm s}$, time-dependent $e^{vt}$, and inflationary $e^{-\xi}$ factors,

[TABLE]

The factor $e^{vt}$ means that the spacetime expands in a self-similar fashion. The inflationary exponent $\xi$, which is a function only of the self-similar radial coordinate $x$, begins to deviate from zero only close to the inner horizon.

Whereas stationary black holes cannot accrete, conformal separability yields solutions for rotating black holes that grow by accretion. In contrast to stationary black holes, conformally separable black holes necessarily undergo inflation at their inner horizons.

The conformally separable black hole solutions are not exact, but hold asymptotically in the limit $v\rightarrow 0$ of small but non-zero accretion rates. Furthermore the solutions require a symmetric (monopole, essentially) accretion rate. However, what happens in the inflationary zone at any angular position on the inner horizon depends largely on the accretion rate of ingoing and outgoing streams on to that position. Inflation takes place over such a small proper time that different angular positions on the inner horizon are causally separated from each other, and become progressively more causally separated as inflation proceeds. Therefore the conformally separable solutions ought to be a reliable guide to what happens at the inner horizons of real black holes. Full numerical relativity would be needed to test this proposition.

It might seem remarkable that conformally separable solutions for accreting, rotating black holes exist. The reason is that, in the limit of small accretion rates, the geometry above the inner horizon is the usual Kerr geometry, and then the behaviour at the inner horizon is constrained by the fact that all matter incident on the inner horizon becomes highly focused along either of just two directions, the principal ingoing and outgoing null directions. The Einstein equations enforce conservation of energy-momentum, and the symmetry imposed by conformal separability then fixes the solution.

The solutions Figure 9 shows the radial ($x$) and angular ($y$) components $T_{xx}$, $T_{xy}$, $T_{yy}$ of the energy-momenta in a conformally separable, accreting, rotating black hole. The solutions depend on the accretion rates $u\mp v$ of ingoing and outgoing streams incident on the inner horizon. The solution shown in Figure 9 is for ingoing and outgoing accretion rates

[TABLE]

The energy-momenta in Figure 9 are shown as a function of the inflationary exponent $\xi$ in the conformal factor $\rho$, equation (5). The inflationary exponent is a function of the self-similar radial coordinate $x$, but $x$ itself scarcely budges away from its value at the inner horizon, whereas $\xi$ changes. The inflationary exponent $\xi$ is almost zero well above the inner horizon, remains small during inflation, and then grows large as the spacetime collapses.

Figure 9 illustrates that, for a black hole that continues to accrete, as all astronomically realistic black holes do, inflation gives way to collapse.[18]

The angular components of energy-momentum shown in Figure 9 are relative to the frame defined by the principal null directions. The fact that the angular components are non-zero means that the ingoing and outgoing streams in a rotating black hole are slightly misaligned with the principal directions.

7 Inflation gives way to BKL oscillatory collapse

Collapse causes rotational motions to grow. The conformally separable solution shown in Figure 9 fails once the spacetime has collapsed to the point that angular energy-momentum is comparable to the radial energy-momentum. What happens then?

During the 1970s, Belinskii, Khalatnikov & Lifshitz[21] (BKL) (see also Belinskii[22]) developed analytic arguments about the generic behaviour of spacetime during collapse to a singularity. They assumed that spacetime was already deep into collapse, and they argued that under those circumstances the gradient in the time direction would dominate gradients in spatial directions. Thus it would be reasonable to suppose that the spacetime was spatially homogeneous. The eigenvectors and eigenvalues of the $3\times 3$ spatial tetrad metric can be thought of as describing 3 orthogonal axes with 3 different lengths. As the spacetime evolves, the lengths and directions of the 3 orthogonal axes evolve.

BKL showed that, in the most generic case, spacetime collapse could be described by a series of “Kasner epochs” punctuated by bounces. The Kasner line-element is[23]

[TABLE]

which is a spatially homogeneous solution of the vacuum Einstein equations provided that the scale factors $a_{i}$ depend as power laws with time $t$,

[TABLE]

with exponents $q_{i}$ related by

[TABLE]

The time coordinate $t$ is negative, and goes to zero at the singularity, $t\rightarrow-0$. The relations (9) between the Kasner exponents $q_{a}$ imply that two axes collapse while one expands, the mean scale factor ${\bar{a}}\equiv(a_{1}a_{2}a_{3})^{1/3}=|t|^{1/3}$ decreasing monotonically. During a BKL bounce, the exponents $q_{i}$ change abruptly, with one of the collapsing axes turning into expansion, and the expanding axis going into collapse. Generally, the bounce is accompanied by a change in the directions of the 3 orthogonal axes.

To find out what happens after the conformally separable solutions of §6 failed, I resorted to numerical relativity. The problem is challenging: the numerical method must remain stable while following robustly the physical divergence in curvature and energy-momentum. For this purpose I developed a new covariant Hamiltonian tetrad approach[24] that permits greater freedom in the choice of tetrad frame.

Because of the difficulty of the problem, I followed the lead of BKL in assuming that spatial gradients in horizontal directions are subdominant compared to radial gradients, and followed the integration into the black hole along lines of constant latitude.

Figure 10 illustrates the result of a numerical calculation at latitude $22.5^{\circ}$ into a rotating black hole of spin $a=0.96$. The initial conditions were taken to be those of the conformally separable solution, and the integration was started already deep into the inflationary epoch. Because smaller accretion rates result in larger curvatures and energy-momenta, it was advantageous to chose the accretion rates to be “large,” so that the numerical calculation would continue as long as possible before over/underflow stuttered the calculation to a halt. On the other hand, the accretion rates must be small enough that the conformally separable solution provides a satisfactory approximation to the initial conditions. The compromise initial conditions in Figure 10 are $u=0.02$ and $v=0.01$, implying ingoing and outgoing accretion rates of

[TABLE]

The evolution shown in Figure 10 is entirely characteristic of BKL collapse. The cosmic scale factors $a_{i}$ evolve as power laws with time $|t|={\bar{a}}^{3}$ through four Kasner epochs separated by BKL bounces. The calculation terminates when numerical over/underflow occurs.

Figure 11 shows the Kasner exponents $q_{i}$ measured from the evolution shown in Figure 10. During Kasner epochs, the exponents indeed satisfy the Kasner condition $\sum_{i}q_{i}^{2}=1$.

The first Kasner epoch is that of inflation, during which the Kasner exponents are

[TABLE]

The second Kasner epoch is that of conformally separable collapse, during which the Kasner exponents are

[TABLE]

Interestingly, the Schwarzschild geometry approximates a Kasner geometry near the Schwarzschild singularity, and the Kasner exponents (12) during the collapse epoch coincide with those of the Schwarzschild geometry, suggesting an elegant continuity between rotating and spherical black holes.

8 The Future of Research on Black Hole Interiors

What happens inside black holes is unobservable, and inaccessible to experiment, therefore it is not very useful to carry out theoretical research on the subject. Does that thought resonate with you? If so, welcome to what I believe is the principal reason for the paucity of research on the interiors of astronomically realistic black holes. My own view is that it is time to recognize and renounce the taboo.

The top priority areas for future research on black hole interiors are, firstly, Hawking radiation in the inflationary and collapse zones of rotating black holes, and secondly, what happens when two rocks collide in the inflationary zone of a rotating supermassive black hole at the centre of a galaxy. The first problem is interesting first to see how Hawking radiation modifies the classical evolution, and second because Hawking radiation gives clues about quantum gravity, and it makes sense to look for clues near singularities. The second problem is interesting because this is the one place in our Universe where Nature is carrying out collision experiments at Lorentz gamma factors of $10^{40}$ and higher, reaching Big Bang densities and beyond.