A journey inside a black hole: Does the singularity not exist?

Theoretical exploration of the interior of black holes has always been an important direction of physics research. One of the difficult questions to answer is whether singularities really exist. Although Penrose and Hawking obtained the singularity theorem based on classical gravity and confirmed the singularity of black holes, there is more than one definition of singularities. In order to explore this core issue of black hole theory, Roy Kerr, a pioneer in black hole research, recently published an article stating that singularities do not exist. Why did he come to this conclusion?

Written by An Yusen (School of Physics, Nanjing University of Aeronautics and Astronautics)

Looking up at the universe, the magnificent starry sky is fascinating, but beauty often also hides dangers. The most dangerous and mysterious area is the black hole.

Einstein's general theory of relativity, proposed in 1915, revolutionized people's concept of space and time. The introduction of general theory of relativity indicates that space and time are not a static stage, but will be distorted by the audience on the stage - matter in space and time. This effect can be described by solving the equations of motion (i.e., Einstein's equations). Generally speaking, Einstein's equations are extremely difficult to solve, but physicists can always find various situations with good symmetry and use them as ideal models for research. In 1916, Karl Schwarzschild obtained the famous solution named after him in the trenches of World War I, called the Schwarzschild metric:

The Schwarzschild metric can be used as a spacetime solution outside a star or as a spherically symmetric black hole solution. By observing this metric, it is obvious that there are two special positions, r=2M and r=0, at which the metric (1) diverges. The understanding of these two divergences has puzzled physicists for many years. People eventually discovered that these two divergent positions actually correspond to the two most important features of a black hole: the event horizon and the singularity. Understanding them is the most important part of black hole physics research.

Before introducing these two characteristics of black holes, let's review the formation of the concept of black holes. In the 1930s, J. Robert Oppenheimer, Subrahmanyan Chandrasekhar and others thought deeply about the problem of gravitational collapse after the burning of stars. They calculated that when the mass of the star is large enough, there is theoretically no sufficient repulsive force to prevent gravitational collapse. Therefore, they boldly predicted that gravitational collapse will not stop, and eventually the star will become extremely dense, which will cause the surrounding space-time to be highly distorted, forming a black hole. This view was questioned by many at the time, and more people believed that this prediction might be due to the oversimplification of the theoretical model. In reality, there will be some mechanism that people have not yet understood to stop the collapse, so black holes are not real physical reality.

To this day, the physics at the end of gravitational collapse has not been fully understood, and the singularity problem that gravitational collapse ultimately results in still puzzles physicists. However, strong evidence for the existence of black holes has been obtained in current gravitational wave observations. In addition, photos of black holes were taken in the past few years. These advances all reveal that such a mysterious celestial body really exists in our universe.

Figure 1 The first photo of a black hole | Source: Event Horizon Telescope Collaboration

Black hole horizon

Let's first introduce one of the two important features of a black hole - the event horizon. Taking the Schwarzschild black hole as an example, for the event horizon, which is the position r=2M in the Schwarzschild solution, people have found that the divergence in the metric solution appears to be due to the problem of our choice of coordinates. If we do not choose the coordinates t, r, θ, φ, the divergence will not appear in other special coordinate systems.

The physical characteristic of the event horizon is the exchange of space-time coordinates due to the high distortion of space-time. It can be easily seen that when passing through the event horizon, that is, from r>2M to r<2M, the original external time coordinate t will become a space coordinate inside the event horizon, and the original radial space direction r will become a time direction. This feature is the essential feature of a black hole. Even if the outside of a black hole is a static space-time that does not change with time, the inside of a black hole will no longer have static properties due to the exchange of space-time coordinates.

Another physical feature of the event horizon is that all light rays inside the event horizon, whether emitted inward or outward, will eventually converge. Therefore, no matter what enters a black hole, it can escape, not even light. It is like a glutton, greedily devouring everything around it, which is also the origin of the name of the black hole.

The above discussion is based on the Schwarzschild solution to understand the physics of the black hole horizon and the interior of the black hole. Shortly after the Schwarzschild black hole appeared, people constructed a charged spherically symmetric black hole, namely an RN black hole, by assuming that there was an electromagnetic field outside the black hole. For a long time, solving Einstein's field equations required the help of spherical symmetry. An important progress occurred in 1963, when Roy Kerr discovered the Kerr black hole named after him [1]. This black hole metric has fewer symmetries (only axial symmetry), so it can describe a black hole with rotation.

RN black holes and Kerr black holes are fundamentally different from Schwarzschild black holes inside their event horizons. The following mainly introduces some of their physical characteristics based on Kerr black holes.

The metric of a Kerr black hole is:

Inside the event horizon, there is also a Cauchy horizon. The existence of the inner horizon has a great impact on the interior of the black hole. At this time, the event horizon is no longer a place where objects can enter but not exit, and at the same time, the space-time in the region within the inner horizon is still stable and does not evolve with time.

Black hole singularity

Let's introduce another feature of black holes: the singularity. For Schwarzschild black holes, at the position of r=0, the divergence of the metric at this time cannot be eliminated by any coordinate transformation. This can also be seen by calculating some scalars composed of space-time curvature, and it is found that these scalars are all divergent at this point. Because scalars do not depend on the choice of coordinate system, this divergence is a physical divergence, and this position of r=0 is called the singularity of the black hole.

For Kerr black holes, people further discovered that the structure of their singularities is very different from that of Schwarzschild black holes. First, because of the existence of the inner horizon, the black hole singularity will transform from a space-like singularity to a time-like singularity. At the same time, the structure of the singularity is also different, which is not easy to see from the spherical coordinates of equation (2), but if we convert it to Cartesian coordinates, it will become very clear. The coordinate transformation relationship is as follows:

If a=0, this coordinate relationship is the same as the familiar transformation from rectangular coordinates to spherical coordinates, but because of the existence of rotation, the result will change a little. x, y, and z have the following relationship:

Because of the existence of rotation, Kerr spacetime is also somewhat different from spherically symmetric spacetime for equal r surfaces. The equal r surfaces in the original spherically symmetric spacetime are spherical surfaces, but here they will become ellipsoidal surfaces. For the singularity position of r=0, the conditions are met at this time, x^2+y^2=a^2, z=0; when there is no rotation, the singularity position corresponds to x=y=z=0; but after rotation, x and y at the singularity may not be 0, and the x and y planes form a circle with a radius of a. Because of this feature, the position where the Kerr black hole singularity appears is also called a singular ring.

However, the theoretical fact that singularities appear in these analytical black hole solutions does not completely prove the existence of singularities. Because in the process of obtaining these black holes, special symmetries are more or less selected. A natural question is whether the singularities are just illusions caused by symmetry? Since the solution of Einstein's equations depends on symmetry, this question is actually not easy to answer.

In the 1960s and 1970s, Soviet physicists Vladimir Belinski, Isaak Khalatnikov and Evgeny Lifshitz tried to discuss the problem of singularity formation without adding symmetry[2] . In the end, this problem was solved with the proof of the singularity theorem by Roger Penrose and Stephen Hawking [3]. The singularity theorem is very clever. Penrose and Hawking used the techniques of global differential geometry such as geodesic sinks to prove that black holes must have singularities under quite general conditions without solving Einstein's equations. It is worth mentioning that the efforts of Soviet scientists were not in vain. The BKL hypothesis (a type of singularity model named after the surnames of the three people) they discovered simplified the gravitational dynamics behavior near the black hole-like singularity, making it possible to solve the equations of motion near the singularity. Although their work did not rule out the existence of singularities, it can help people more intuitively understand the law of change of the space-time metric near the singularity.

Singularity does not exist?

Readers may wonder, according to our introduction to singularities, singularities are the locations where scalars of various space-time curvature structures diverge. Since Penrose did not actually solve Einstein's equations, how did he know that curvature scalars diverge? In fact, there are some subtle differences between the singularities that Penrose talked about and the singularities we introduced.

There are actually two definitions of singularities: one is the definition mentioned above through the divergence of some scalars that do not depend on the choice of coordinates; the other is to characterize it through the finiteness of geodesic affine parameters in non-extendible spacetime (abbreviated as FALLs). The second definition of a singularity comes from the following intuition: for a normal spacetime, a particle should always be in spacetime; if a curve in spacetime suddenly disappears in this spacetime under a finite parameter (and the background spacetime cannot be analytically extended), then it must be because a singularity appears in the spacetime itself, causing the curve to end at this singularity. Although this definition of a singularity is abstract, it is general and does not depend on a specific metric solution. This feature makes it very useful in mathematical proofs. The first way to define a singularity is intuitive, but it depends on a specific black hole solution. The two definitions are very different, and it is difficult for people to intuitively establish the relationship between the two singularity descriptions. In response to this problem, Kerr, a pioneer in black hole research and who is about to turn 90, published another article to discuss [4] and proposed that black holes may not have singularities.

Kerr pointed out in his article that for a Kerr black hole, there is at least one FALL that does not end at a singularity. He found a simple counterexample: the trajectory of light moving along the symmetry axis of a Kerr black hole satisfies the following equation:

This allows us to solve for two rays,

and

At

Because the radial velocity dr/dt of this light ray is 0 at the inner and outer horizons, the geodesic can only be located in the area between the inner and outer horizons. At this time, this light-like geodesic has finite affine parameters

(The r coordinate can be used as the affine parameter of this curve). Therefore, Kerr pointed out in the article that he constructed a geodesic with finite affine parameters, but it obviously did not intersect any place with curvature singularity. From this, Kerr pointed out that the singularity theorem proved by Penrose may be incomplete. Although Penrose's proof pointed out that FALLs will definitely appear under very general conditions, it does not mean that singularities will definitely appear.

Figure 2 The counterexample mentioned by Kerr is the green geodesic, but the coordinate system he used only covers the gray area. This area is not the complete space-time, but can be extended to the larger Kruskal space-time (that is, including the white part).

As for the specific physical image inside the black hole, he proposed that since the inner horizon is still a steady-state spacetime, there may be some kind of star (such as a neutron star) inside the inner horizon to replace the strange ring that appears in the Kerr black hole. The metric of this star is connected with the Kerr black hole outside to form a real physical image. In other words, the strange ring is just an ideal approximation of a real non-singular star.

There are still some questions worth exploring about Kerr's conclusion. First, in the counterexamples he found, the coordinate system used did not cover the entire spacetime, so the area covered by this part of the coordinate system does not represent an unextendable spacetime. In addition, the geodesic found by Kerr is not a complete geodesic. At least when studying the singularity theorem in the sense of mathematical solutions, exploring whether a singularity exists requires ensuring that the background spacetime is unextendable, so whether the counterexample he found constitutes a counterexample that violates the Penrose singularity theorem remains to be discussed.

Secondly, even if the space-time structure given by the analytical extension of the mathematical solution may not be the physical appearance of real space-time, the physical image of real space-time may also deviate from Kerr's idea to a certain extent. In Kerr's proposal, whether it is to find a counterexample that violates the singularity theorem or the physical image inside the black hole, it must rely on the existence of the inner horizon of the Kerr black hole. However, many studies have shown [5] that the inner horizon of a black hole is unstable. This instability comes from the dynamic instability of the inner horizon. At this time, tiny fluctuations will be amplified near the inner horizon, making the original inner horizon become the location of a new curvature singularity (i.e., the mass inflation effect).

At the same time, due to the existence of various material fields in space-time, when these material fields make the black hole hairy [Note 1] (such as scalar hair), the inner horizon of the black hole will also disappear. Therefore, the actual physical image may not have an inner horizon (that is, the area between the inner horizon and the singularity), which will exclude the possibility of a certain stellar metric existing stably within the inner horizon. Moreover, the type of FALLs that Kerr found that do not touch the singularity will also fail, because the geodesic in his example intersects at the inner horizon, but at this time the position of the inner horizon will become the location of the new singularity, so the geodesic in the counterexample will actually intersect with the singularity.

Inside a black hole: blackbody radiation for the 21st century?

It is worth noting that Kerr's discussion is only a solution to the singularity problem. Even if his proposal fails, it does not mean that the singularity will exist. The singularity theorem proved by Penrose and Hawking based on classical gravity is more of a sign of the failure of classical gravity. In this sense, it can be said that the black hole singularity bears more historical significance similar to the ultraviolet divergence of blackbody radiation. Quantum physics is a new paradigm. Any physical theory needs to be incorporated into the paradigm of quantum physics at a small scale. At present, only the gravitational theory represented by general relativity is still struggling. People believe that gravity needs to be quantized, and general relativity and quantum mechanics must also be integrated. In this process, giving a new quantum mechanical version of understanding to the black hole singularity where various physical quantities diverge, so as to solve this divergence is the only way for the birth of quantum gravity.

The image of the interior of a black hole predicted by classical general relativity must be incomplete. There must be more magical things hidden in this most secret corner of the universe, the inside of a black hole.

Notes

[1] Although black holes have the no-hair theorem, this is only within the Einstein-Maxwell framework. In a more general framework, there are many constructions that can go beyond the limitations of this loose theorem.

References

[1] RP Kerr, “Gravitational field of a spinning mass as an example of an algebraically special metric”, Phys. Rev. Lett. 11, p. 237 (1963).

[2] VA Belinskii, IM Khalatnikov and EM Lifshitz, “Oscillatory approach to a singular point in the relativistic cosmology,” Adv. Phys. 19, 525 (1970).

[3] R. Penrose, “Gravitational collapse and space-time singularities”, Phys. Rev. Lett. 14, p. 57 (1965).

[4] RP Kerr, Do Black Holes have Singularities? arXiv:2312.00841.

[5] E.Poisson, W.Israel, "Internal structure of black holes". Phys.Rev.D 41 (1990) 1796-1809.

This article is supported by the Science Popularization China Starry Sky Project

Produced by: China Association for Science and Technology Department of Science Popularization

Producer: China Science and Technology Press Co., Ltd., Beijing Zhongke Xinghe Culture Media Co., Ltd.

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