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Paradox Is Illuminating the Black Hole

How reconciling opposites is driving the science of black holes forward.

An Introduction to the Black Hole InstituteFittingly, the Black Hole Initiative (BHI) was founded 100 years after Karl Schwarzschild…By Gabriel Lynch

An Introduction to the Black Hole Institute

Fittingly, the Black Hole Initiative (BHI) was founded 100 years after Karl Schwarzschild solved Einstein’s equations for general relativity—a solution that described a black hole decades before the first astronomical evidence that they exist. As exotic structures of spacetime, black holes continue to fascinate astronomers, physicists, mathematicians, philosophers, and the general public, following on a century of research into their mysterious nature.

The mission of the BHI is interdisciplinary and, to that end, we sponsor many events that create the environment to support interaction between researchers of different disciplines. Philosophers speak with mathematicians, physicists, and astronomers, theorists speak with observers and a series of scheduled events create the venue for people to regularly come together.

As an example, for a problem we care about, consider the singularities at the centers of black holes, which mark the breakdown of Einstein’s theory of gravity. What would a singularity look like in the quantum mechanical context? Most likely, it would appear as an extreme concentration of a huge mass (more than a few solar masses for astrophysical black holes) within a tiny volume. The size of the reservoir that drains all matter that fell into an astrophysical black hole is unknown and constitutes one of the unsolved problems on which BHI scholars work.

We are delighted to present a collection of essays which were carefully selected by our senior faculty out of many applications to the first essay competition of the BHI. The winning essays will be published here on Nautilus over the next five weeks, beginning with the fifth-place finisher and working up to the first-place finisher. We hope that you will enjoy them as much as we did.

—Abraham (Avi) Loeb
Frank B. Baird, Jr. Professor of Science, Harvard University
Chair, Harvard Astronomy Department
Founding Director, Black Hole Initiative (BHI)


The way up and the way down are one and the same,” says the Sage.

Paradox has a way of plaguing thought. It illuminates the inconsistencies and inadequacies of the concepts with which we make sense out of our experience. It casts into doubt our framework of understanding as much as the understanding itself. Whatever tower of thought may be built in an attempt to see past its contradictions teeters and, with slight perturbation, collapses. However, it can also become a tool of epiphany, if only we wield it skillfully enough. The philosopher Heraclitus understood this, although his affinity for paradox earned him the epithet “the Obscure” from his peers. Today, physicists must also embrace paradox in order to reach a deeper understanding of the nature of black holes.

For Heraclitus, the difficulties of thought were inherent to the way in which we processed the world. Our external experience of nature is not processed directly, but rather is filtered through a web of concepts that develop and over time become linked with each other. If we are not careful, this mess of concepts and categories we use to understand nature becomes confused with nature itself. We develop models and theories that act as lenses to tell us what the world looks like. Sometimes, however, there is tension between our different models, our different concepts, that is a result of the world not fitting nicely into the web we have created. So it is with paradox. Paradox frustrates us because its appearance—how we understand what is paradoxical—depends on which lens we are using. In thinking, we introduce distinctions that are immaterial, and paradox cuts through them.

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“In the circle the beginning and the end are common,” says Heraclitus.

The situation is no less true in physics, and nowhere more evident within physics than in the study of black holes. Black holes occupy a privileged spot in modern physics, as it is at their core, the black hole’s singularity, that we expect two of our most successful physical models (those clouded, dark lenses) to meet. General relativity describes a framework in which “Spacetime tells matter how to move; matter tells spacetime how to curve,” as famously and succinctly explained by the physicist John Wheeler. The equations of the theory are geometric in nature, borrowing from fields of math that study the various shapes of abstract mathematical objects. It explains the phenomenon of gravity in terms of shape and curvature, and treats spacetime as an object in and of itself that bends and expands in response to the masses moving about within it. Space and time are no longer a stage in which action takes place, but rather players on that stage in their own right.

Quantum theory, the second of these frameworks, is altogether a different beast. It describes the behavior of particles and light at the smallest scales, often in confusing terms. But much of this confusion arises from our own tangled web of thought, developed and trained to understand the classical world of everyday experience. In any quantum theory, an object such as a particle has a given quantum state that evolves in time in a prescribed way through a sequence of successive states. But the properties of these states are strange. Entanglement, perhaps the most fundamentally quantum of all properties, gives rise to strange correlations between quantum objects here and quantum objects there, regardless of how far apart “here” and “there” are (it is a property so bizarre that Einstein deemed it “spooky action at a distance”). Spacetime is, if examined closely enough, teeming with virtual particles popping in and out of existence that render the vacuum into a quantum foam in flux. And yet, despite the quantum strangeness of the world, the theory adequately predicts the behavior of particles to an astounding degree.

“It is in changing that things find repose,” says Heraclitus.

Black holes occupy a place at the boundary of each theory and uncomfortably in between both. Here, our theories fall apart. The first indication something is amiss comes from the equations of general relativity. The equations describing the curvature of spacetime commit the cardinal sin of mathematics—division by zero—spitting out nonsensical infinities in the process. It is as though the fabric of spacetime is actually punctured from the immense stress it is placed under in the presence of the singularity, something that our equations are ill-equipped to describe. In other classical theories with similar breakdowns, quantum theory has come to the rescue. Much of the work in physics during the first half of the 20th century was done in “quantizing” classical theories: removing these nonsensical descriptions by describing objects in quantum terms. But when the same process is attempted with general relativity, attempting to write down a theory of quantum gravity, conceptual issues arise long before the mathematical ones. How can spacetime be described as having a quantum state evolving through time, when it is time?

A theory that looks like a quantum theory through one lens may, through another, look like a theory of gravity.

In such murky landscapes between theories, paradox is the way forward. The genius of Heraclitus is his ability to utilize paradox as an instrument to describe truths that language is inadequate to describe. In physics, the black hole information paradox is the guide out of the swamp of black hole ambiguity. This paradox captures the fundamental conflict between quantum theory and general relativity. It states that if black holes behave how general relativity demands, then the objects falling into a black hole cannot behave how quantum theory demands. Conversely, if the quantum objects falling into a black hole behave as expected, then the geometric description of black holes must be faulty. Whichever theory we privilege, the other must be cast aside.

The paradox arises from two facts. One is that quantum systems evolve in a “unitary” manner, which is to say that given the quantum state of a system at one time, the quantum state at other times can be uniquely determined. Information about the state is said to be preserved. The second is that black holes evaporate, and according to general relativity they take with them the information of what fell in. To an observer in spacetime after the black hole has evaporated the information available about the initial quantum system is incomplete. They can no longer determine the initial state, and it seems as though quantum information has been destroyed.

In grappling with this paradox, researchers have begun to find their footing. Searching for a resolution, physicists have noticed connections between quantum theory and the geometry intrinsic to gravity that offer to alleviate the paradox by offering a duality between quantum mechanics and geometry. A theory that looks like a quantum theory through one lens may, through another, look like a theory of gravity. This is the holographic principle (which does not, as has been suggested, propose that the universe is actually a hologram of the variety found in science fiction). A dictionary between the two viewpoints has started to develop, and researchers have been able to understand some aspects of one theory in terms of the other. One endpoint of this line of inquiry in a conjecture known as ER=EPR (standing for Einstein-Rosen bridges = Einstein-Podolsky-Rosen pairs, pairs of entangled objects). This idea, proposed by Juan Maldacena and Leonard Susskind, conjectures that the behavior of two entangled quantum systems can dually be described by two separated systems connected by a micro-wormhole. What in one lens appears as quantum weirdness is in the other strange geometry.

A true Heraclitean embrace of paradox would go further and recognize our theoretical distinctions between types of theories as immaterial. That we describe some objects as “quantum” and some as “geometric” is a consequence of that vexing web of concepts that processes our experience. To Heraclitus, the sorting of properties that results in some being labeled quantum and some geometric is artificial. Our reliance on these distinctions is what dooms us to see nature “through a glass, darkly.” But, armed with the contradictions of paradox and constantly pushing and probing the limits of our concepts, it is possible to attain a moment of clarity. Heraclitus would say that, even at the singularity, there is no conflict between gravity and quantum mechanics because gravity and quantum mechanics are one and the same. Entanglement is geometry.

The way up is the way down.

To be sure, the fullness of this moment of clarity is as of yet far off. The ER=EPR conjecture remains as conjecture, and likely will continue to do so for the immediate future. And other paths of black hole research present resolutions to the paradox that avoid the unity suggested by ER=EPR. What is clear is that physicists must embrace, often uncomfortably, the role of paradox in understanding nature. Such comfort with the contradictory is rarely easy, and the eventual clarity is by no means guaranteed. Perhaps it is well that it should be so.

“The hidden harmony is better than the obvious,” says Heraclitus.


Gabriel Lynch is a recent graduate of the University of Chicago, where he studied physics and conducted research on black holes and quantum information. He lives near Chicago.

This essay placed third in the Black Hole Institute’s essay contest.


Additional Reading

Maldacena, J. & Susskind, L. Cool horizons for entangled black holes. Fortschritte der Physik 61 (2013). Retrieved from DOI:10.1002/prop.201300020.

Wald, R.M. General Relativity University of Chicago Press (2009).

Wheelwright, P.E. Heraclitus Oxford University Press (1959).

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