The Edge of Space and the End of Time

The Black Hole Event Horizon

Douglas C. George

Eureka, CA

January 30, 2021

Overview

General relativity revealed that matter and radiation warp space and time. Shortly after Einstein published his theory in 1915, physicist and astronomer Karl Schwarzschild solved the theory’s field equations for the simplest possible case: a spherical, uncharged, non-rotating, massive object. His solution---the Schwarzschild metric---described how both time and distance in the surrounding space are altered by the object’s presence. Famously, the metric’s mathematics also revealed the most mysterious object ever, the black hole.

The defining feature of a black hole is its event horizon at the Schwarzschild radius. It marks where the escape velocity of objects must equal the speed of light. Otherwise, the event horizon is considered to be just a normal place in space indistinguishable from any other. In particular, and noteworthy for the discussion at hand, space is thought to extend uninterrupted across the event horizon.

That’s how it’s been for a hundred years.

Recently, however, the discovery of a paradox in quantum physics and direct observations in gravitational-wave astronomy has challenged this view. Now, indications are that spacetime may not be continuous and the black hole event horizon may not be a normal place in space. It may be the place where space and time come to an end.

The idea was first published by a group of quantum theorists in southern California in the year 2012.

The AMPS conjecture

The group---collectively referred to as AMPS[i]---at the Kavli Institute for Theoretical Physics at UC Santa Barbara, published a controversial paper (soon dubbed “Firewall Theory”) proposing that the event horizon of a black hole may actually be an unimaginably hot boundary where space and time come to a complete end. They described the event horizon as a “spherical singularity” in the spacetime manifold. Such a firewall was, according to AMPS, the “least-worse” scenario needed to resolve a paradox the group had discovered.

The LIGO Data

Four years later, in February of 2016, LIGO[ii]  researchers announced the first direct detection of gravitational waves, or ripples, in spacetime that were generated when two black holes merged[iii].

Soon thereafter, a team of physicists led by Vítor Cardoso at the Superior Technical Institute in Lisbon proposed that if there were any strange deviations from general relativity---such as the firewall proposed by AMPS ---or any other kind of structure at the event horizon---these black-hole mergers would release a predictable series of echoes after the initial gravitational-wave burst. Analysis of the data showed that, indeed, repeated gravity-wave echoes were detected and at precisely the predicted intervals. It was the first direct, physical evidence indicating a structure of some sort residing at the event horizon of a black hole.

The Kavli group’s paper and the LIGO data were not the only indications that black holes might be radically different than generally imagined. When taken collectively a number of seemingly unrelated observations over the years also point to a fundamentally different interpretation of these enigmatic objects.

Lynden-Bell and Katz

Years before AMPS, in 1985, Cambridge astrophysicist Professor Donald Lynden-Bell (first to determine that supermassive black holes reside at the centers of galaxies) and Professor Emeritus Joseph Katz at the Racah Institute of Physics published a paper titled Gravitational Field Energy Density for Spheres and Black Holes[iv]. They derived how energy---and therefore mass---is distributed in the gravitational field surrounding such an object. They concluded that the total coordinate-independent energy distributed in the gravitational field surrounding a black hole is mc^2. In other words, the total mass associated with a black hole is located outside the event horizon. They state explicitly that:

 "... all the energy remains outside the hole." 

The authors did not elaborate further but their calculations implied something quite remarkable. If empty space has its own intrinsic energy, the interior region of a black hole must be devoid of spacetime itself---as was later proposed by AMPS.

Topology Change in General Relativity

The question arises, then, as to whether or not the spacetime manifold can actually have such a hole or break in its topology.

On one side, arguments are given maintaining that the manifold’s continuity is sacrosanct; it cannot be broken. In the terminology of the experts, the spacetime manifold cannot undergo topology change. Nonetheless, a paper published in 2006 entitled Topology Change in General Relativity[v] by Gary T. Horowitz, Department of Physics at UCSB, appears to turn the question on its head. To quote Dr. Horowitz: "The question is not whether topology change can occur but rather how do we stop topology from changing? Why doesn’t the space around us suddenly split into disconnected pieces?"

Prior to Dr. Horowitz’s paper, the Russian mathematician Grigori Perelman, in proving the Poincaré conjecture, found it necessary to determine whether or not singularities could exist in topological manifolds. He found that, at least in the kind he was studying, certain singularities could exist and, notably, they were restricted to being oblong cavities---hollow spheres stretched out along a line. The simplest form of such a singularity is of course just a spherical hole.

Leaving aside the question of whether topology change is possible what does the Schwarzschild metric itself have to say about black holes and event horizons?

The Schwarzschild Metric:

Of the many solutions of Einstein’s field equations, Schwarzschild’s metric remained the most famous not only because it was the simplest and most useful case but because it first revealed the existence of black holes.

The mathematics showed that, under certain circumstances, a dying star would collapse under its own weight, imploding into an unimaginably dense object later characterized by physicist Kip Thorne as made “wholly and solely” of warpages in spacetime. It appeared possible that, in its final death-collapse, a star that is massive enough would crush its own matter completely out of existence, transforming itself into an object built entirely of deformations of space and time. All of its original matter, its electrons, protons, quarks and such would be crushed down to an infinitely small point. The original matter that made up the star would be essentially gone.

But, remarkably, the mass of the star would remain.

The same mass that was previously carried by the particles is now, as Lynden-Bell and Katz discovered, living outside the event horizon as the gravitational field of the black hole.

The creation of a black hole appears, in essence then, to be a transformation of normal matter into a gravitational field arrayed around what is probably a void in the spacetime manifold. To see this more clearly, we need to examine the Schwarzschild metric and its dramatic mathematical property called metric stretching.

Metric Stretching

The spacetime interval, ds in the first panel of the illustration below, is the basic measure of length and time in the Schwarzschild metric. The distance part of the interval (shown in yellow) displays the remarkable property known as metric stretching. Its leading fraction is plotted as the white curve in the center panel. It clearly demonstrates how the two spacetime intervals shown are different sizes according to how close they are to the black hole’s event horizon.

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The closer to the horizon, the more stretching there is. Significantly, as the event horizon is approached from the outside, this radial stretching phenomenon increases exponentially to infinity and, given that any object that expands must also thin out proportionally, the energy density of the associated spacetime would be expected to quickly drop away to nothing (as shown in the third panel).

Perpendicular to the radial directions, space is compressed a bit but in the radial direction the manifold is stretched according to its distance from the event horizon. (See addendum for further explanation).

If one assumes the spacetime manifold to have a physical structure, it seems reasonable to assume that the structure, whatever its nature, would likewise be stretched to infinity in the radial directions---and more importantly, it would be thinned out to nothing at and below the event horizon as shown in the third panel.

In the overview then, we appear to have a spherical black hole that has spacetime all around it but has no spacetime inside. Metric stretching, in other words, seems to be describing a spherical cutout in the manifold, or, as AMPS later put it, a spherical singularity.

If this is so, a black hole could only be accurately described as a cutout, or cavitation bubble, in the manifold, or literally as, the edge of space and the end of time.

The above infinity at the event horizon in the Schwarzschild metric is normally considered to be only an artifact of the coordinate system itself, a problem which can be readily eliminated by choice of a more suitable frame of reference. But, if space ends at the event horizon, this usual reasoning would no longer hold. There would be no choice related to the interior of the hole because the manifold itself has ended at the event horizon. Any coordinate system describing the manifold would necessarily end at the event horizon along with the manifold.

The Implications

The implications are significant.

The mathematician G.D. Birkhoff's theorem[vi] states that the general solution to Einstein’s field equations for any spherically symmetrical object must be the Schwarzschild metric. According to Birkhoff, then, a spherical cutout or void in the manifold would appear to be the same as a normal massive object. And importantly, black holes can be any size at all, including subatomic. Indeed, Stanford physicist Leonard Susskind and others have noted that elementary particles would be indistinguishable from black holes of equal size.

If the spacetime manifold has a structure, all forms of matter might ultimately be seen as built of distortions in that structure caused by the various kinds of black hole cavitation bubbles (charged or uncharged, spinning or not spinning, right or left-handed, etc.). All matter, in other words, may be, at the deepest possible level, nothing more than the warped space caused by voids in the spacetime manifold.

Stephen Wolfram, in his 2002 publication A New Kind of Science, appears inclined toward the same conclusion:

"... it seems much more plausible that both space and its contents should somehow be made of the same stuff---so that space becomes the only thing in the universe."

 

Addendum

The Dust Particle Argument

As a thought experiment, consider a group of eight theoretical dust particles arranged to form the corners of a cube. These imaginary particles have no mass or charge (this is a thought experiment) and they are situated far, far away from any massive object. Since they, themselves, have no mass, charge or motion relative to each other, they would be expected to remain in a fixed position relative to each other.

Now insert a large massive body into the nearby space and measure the positions of the particles again. One would find that the particles are no longer positioned at the corners of a cube. Their positions relative to each other will have changed. They now measure further apart along an axis radial to the massive object and squeezed a bit in the orthogonal directions.

The particles have moved relative to each other only because the underlying space they are embedded in has been changed. The spacetime manifold itself has been stretched out of shape by the massive object.

If the manifold has intrinsic energy content, i.e., if spacetime has an actual structure, the energy density of the manifold must thin out radially as the stretching proceeds. Since metric stretching in the radial direction goes to infinity at the event horizon, the spacetime manifold, itself, must disappear in the radial direction at the event horizon.

About the Author

For a number of years, I was employed as a physicist doing laboratory research in atmospheric physics. With respect to black holes, I’m a self-taught amateur. It’s a hobby.