A Reader's Guide to Quantum Theory Meets Gravity
What each chapter is about, what it establishes, and why you should care
The book I co-edited, Quantum Theory Meets Gravity — From Curved Spacetime to Black Hole Information, was published in January 2026 and is now available open-access from IntechOpen and in hardcover from Amazon. It’s co-edited with Oliver K. Baker (Yale) and Djordje Minic (Virginia Tech).
I wrote a detailed post about my own chapter — the one on gravity’s effect on the double-slit experiment — when the chapter first appeared online. Here, I want to step back and give a guide to the whole book: what each chapter does, what it establishes, and why a curious non-physicist might care.
The book lives at the frontier where quantum mechanics and general relativity meet. These are our two best theories of the physical world, each confirmed to extraordinary precision in its own domain. But combining them fully — producing a complete theory of quantum gravity — remains one of the great unsolved problems in science. The chapters in this volume don’t claim to solve that problem. Instead, they map the territory where the two theories overlap, working out what can be calculated rigorously right now, and showing that the results are surprisingly rich.
A recurring theme, one I didn’t fully appreciate until the book came together, is that entropy and entanglement keep appearing across seemingly unrelated contexts.
A word about these two concepts, since they’ll come up repeatedly.
Entropy, in its most general sense, measures how much you don’t know about a system. If you know the exact state of every particle — their positions, velocities, everything — the entropy is zero. If all you know is the temperature and pressure (but the particles could be arranged in any of a vast number of ways consistent with those measurements), the entropy is large. The second law of thermodynamics says that entropy tends to increase: systems evolve from ordered to disordered states. A hot cup of coffee cools to room temperature; it never spontaneously heats back up. Entropy is the reason why. In the context of black holes, entropy takes on a geometric meaning — it turns out to be proportional to the area of the event horizon, not the volume. This connection between information, thermodynamics, and geometry is one of the deepest clues we have about quantum gravity.
Entanglement is a distinctly quantum phenomenon. Two particles can be “entangled,” which means their properties are correlated in a way that has no classical explanation. Measure the spin of one entangled electron and you instantly know the spin of its partner, no matter how far apart they are — not because any signal traveled between them, but because they share a single quantum state. Einstein famously called this “spooky action at a distance.” The crucial point for this book is that entanglement carries information. Two entangled systems have zero entropy individually (they appear random) but zero entropy jointly (the combined state is perfectly known). This interplay between local randomness and global order is central to the black hole information paradox.
These two concepts — entropy and entanglement — keep appearing in Hawking radiation, in quantum interference, in supersymmetric systems (supersymmetry is a proposed deep symmetry of nature that pairs every matter particle with a force-carrying partner and vice versa — more on this in Chapter 4), in black hole information theory. They seem to be telling us something deep about whatever the final theory of quantum gravity will look like.
Here’s what each chapter contributes.
Chapter 1: Quantum Field Theory in Curved Spacetime — Gravity’s Impact on the Double-Slit Experiment
Samir Varma and Arie Kapulkin
This is my chapter (with my co-author Arie Kapulkin), and I wrote about it at length here. What follows is a condensed version; the full post gives the complete picture.
The double-slit experiment is the canonical demonstration of quantum mechanics — fire particles at a barrier with two slits, and an interference pattern appears on the far side, as if each particle went through both slits simultaneously. Every textbook uses it to illustrate quantum weirdness. But every textbook version quietly assumes flat spacetime: no gravity, no curvature, no warping of time. We don’t live in flat spacetime. We live on a planet whose mass curves space and warps time.
What happens to quantum interference when you actually account for that curvature?
To answer this properly, you need quantum field theory — the framework that says particles aren’t little billiard balls but localized disturbances in fields that permeate all of spacetime. (I explain what fields are and why this matters in detail in the longer post.) Once you think of “particles” as field disturbances, interference isn’t mysterious at all — it’s what fields naturally do. The question becomes: how do the rules of field propagation change when spacetime is curved?
We work this out using an approach called the path integral, one of the most powerful ideas in theoretical physics. In classical mechanics, a particle travels along a single trajectory — the one that minimizes the action (a quantity related to energy). In quantum mechanics, the particle doesn’t pick a single path. Instead, it takes every possible path simultaneously, and each path contributes a complex number (an “amplitude”) to the final result. The probability of the particle arriving at a given point is determined by adding up the amplitudes from all paths — and because these amplitudes are complex numbers with phases, they can interfere constructively or destructively, producing the characteristic patterns of quantum mechanics.
The path-integral approach, developed by Richard Feynman, is especially natural for our problem because it makes the effect of gravity transparent: curved spacetime changes the lengths and durations along each path, which changes the phase each path accumulates, which changes how the paths interfere.
Leo Stodolsky, in the late 1970s, developed a formalism that applies this path-integral logic specifically to interference experiments in gravitational fields. We adapt and extend his methods, treating scalar fields (spin 0), photons (spin 1), and electrons (spin 1/2) within a single unified framework. The central result is a compact formula for the gravitational phase shift:
Every symbol in this formula has a physical meaning. The Riemann curvature tensor (the R term) encodes the tidal gravitational field, which means how gravity varies from point to point. The ℓ terms represent the slit separation. The m is the particle’s mass and c is the speed of light. The ℏ (pronounced “h-bar”) is Planck’s constant divided by 2π, one of the fundamental constants of nature; its value is approximately 1.055 × 10⁻³⁴ joule-seconds, a number so tiny it explains why quantum effects are invisible in everyday life. The λ is the de Broglie wavelength — every particle in quantum mechanics has an associated wavelength, inversely proportional to its momentum. Slower particles have longer wavelengths, faster particles have shorter ones. This was proposed in 1924 by Louis de Broglie, a French prince who put the idea forward in his doctoral thesis, which was so radical that the examining committee sent it to Einstein for a second opinion before awarding the degree — de Broglie won the Nobel Prize for it five years later. Finally, the α term is a small spin-dependent correction. At leading order, all particle types — scalars, photons, electrons — couple to curvature in exactly the same way. The spin corrections are suppressed by a factor of about 10⁻¹¹ for electrons. This universality makes physical sense: at leading order, all particles follow paths dictated by the geometry of spacetime itself.
In rotating spacetimes — like the spacetime around a spinning black hole, or even Earth (which drags spacetime around as it rotates) — frame-dragging introduces a distinctive phase shift with odd parity. This means if you swap the two slits, the frame-dragging contribution changes sign. Static gravitational effects don’t do this — they don’t care which slit is which. This odd-parity signature provides, at least in principle, a smoking gun for detecting spacetime rotation through quantum interference.
The effects are small. For cold rubidium atoms in a 10-meter orbital interferometer at 400 km altitude, the tidal phase shift is about 2.7 × 10⁻²¹ radians (a radian is a unit of angle — 2π radians is 360 degrees — so this is an almost incomprehensibly tiny fraction of a full cycle of the interference pattern) — far below what any current instrument can detect. A futuristic kilometer-scale space interferometer might reach 10⁻⁷ radians. But the predictions are exact, derived from well-established physics, and they set the benchmarks that any future experiment would need to reach — and that any future theory of quantum gravity would need to reproduce.
Why you should care: This chapter shows that the most famous experiment in quantum mechanics has something new to teach us when we stop pretending gravity doesn’t exist. The gravitational effects we calculate are tiny, but they’re there — precise predictions sitting at the intersection of quantum mechanics and general relativity, waiting for technology to catch up. It’s also an accessible entry point into quantum field theory in curved spacetime, which is the framework that predicted Hawking radiation and the Unruh effect — two of the most important results in theoretical physics.
Chapter 2: Dynamics and Quantum-Gravitational Interactions in Plasma-Surrounded Black Hole Spacetimes
Orchidea Maria Lecian
Real black holes aren’t sitting in empty space. They’re surrounded by swirling hot matter — accretion disks, plasma, jets of material blasting outward at nearly the speed of light. When the Event Horizon Telescope produced the first image of a black hole in 2019, what it captured wasn’t the black hole itself (which, by definition, emits no light) but the glowing ring of superheated plasma around it. To understand what those images actually mean — to extract the physics from the photons that reach our telescopes — you need to understand how light propagates through this extreme environment, where both intense gravity and hot dense matter simultaneously affect the signal.
This is what Lecian’s chapter tackles.
She works with a family of spacetimes called Kottler-Schwarzschild-Kiselev (KSK) spacetimes. To understand what these are, start with the simplest black hole solution: the Schwarzschild spacetime, which describes a non-rotating, uncharged black hole sitting in otherwise empty space. Karl Schwarzschild found this solution to Einstein’s equations in 1916, barely a month after Einstein published general relativity. It’s elegant but idealized — real black holes exist in a universe with a cosmological constant (the mysterious energy driving the universe’s accelerating expansion) and are surrounded by matter.
The KSK spacetimes generalize the Schwarzschild solution by incorporating both a cosmological constant and a surrounding matter field characterized by a “linear term” in the metric. This gives a more realistic starting point for astrophysical calculations.
Lecian then asks: what happens to a photon — treated quantum mechanically — as it propagates through this spacetime in the presence of plasma?
This requires working out the quantum gravitational Hamiltonian for the photon, which means writing down the equation that governs how the photon’s quantum state evolves in time, accounting for both the curvature of spacetime and the interaction with the plasma medium. She does this for three cases: vacuum (no plasma), cold unmagnetized plasma, and hot unmagnetized plasma. The plasma gives the photon an effective mass — just as photons traveling through glass slow down and behave as if they had mass, photons in a plasma acquire an effective mass that depends on the plasma’s density and the spacetime geometry.
Several concrete results emerge. The photon sphere — the radius at which light can orbit a black hole — exists only when the cosmological constant takes specific positive values. The gravitational lensing (the bending of light by gravity) is computed in the strong-field regime, which means close to the black hole where gravity is extreme and the gentle-bending approximations used for, say, galaxy clusters don’t apply. The chapter also analyzes accretion disk configurations, comparing the effective gravitational potentials that govern matter orbits with standard models used in astrophysics, and examines how adiabatic perturbations — slow, gradual disturbances — propagate through the hot plasma.
She also compares the KSK framework with alternative theories of gravity. This is important because general relativity, for all its success, might not be the final word on gravity — and different theories make different predictions for how black hole environments behave.
One alternative is Weyl conformal gravity, a higher-dimensional theory that modifies Einstein’s equations by demanding an additional symmetry: the equations must be unchanged if you locally stretch or shrink all distances by the same factor (a “conformal” transformation). This extra constraint changes how gravity behaves at very large distances and has been proposed as an alternative explanation for the rotation curves of galaxies — the same observations usually attributed to dark matter.
Another is massive gravity, which modifies general relativity by giving the graviton (the hypothetical particle that mediates gravity) a tiny but nonzero mass. In Einstein’s theory, the graviton is massless, which is why gravity has infinite range. If the graviton had even a tiny mass, gravity’s behavior would change at very large distances — potentially explaining the accelerating expansion of the universe without needing a cosmological constant.
Lecian maps the geometrical parameters of the KSK spacetimes onto these alternative frameworks, showing which features of the black hole environment are universal (present in any theory of gravity) and which are specific to general relativity. This kind of comparison is essential for turning observations into tests of fundamental physics: when we image a black hole’s shadow or measure the deflection of light near it, we need to know which theories are consistent with the data and which are ruled out.
Why you should care: We’re living in a golden age of black hole observation. The Event Horizon Telescope is producing increasingly sharp images of black hole environments. Gravitational wave detectors are recording the collisions of black holes. To interpret this data — to figure out what it tells us about gravity, about the nature of spacetime, about the matter spiraling into these objects — we need theoretical calculations like these, ones that account for both the extreme curvature near the black hole and the realistic astrophysical soup surrounding it. Lecian’s chapter provides part of the theoretical toolkit needed to turn observations of photons into statements about fundamental physics.
Chapter 3: Gravity in Extradimensional Space
Eugênio Maciel
Here’s a question that sounds like science fiction but is taken very seriously in theoretical physics: what if the universe has more than three dimensions of space?
The idea has a distinguished pedigree. In 1921, Theodor Kaluza did something remarkable. He took Einstein’s equations of general relativity — the equations that describe how matter and energy curve spacetime — and wrote them down in five dimensions instead of four (three of space plus one of time becoming four of space plus one of time). When he worked out what the five-dimensional Einstein equations implied for a four-dimensional observer, he found that they automatically produced not just gravity but also Maxwell’s equations of electromagnetism. Gravity and electromagnetism, two seemingly unrelated forces, emerged from a single geometric framework — you just needed one extra dimension.
Five years later, Oscar Klein added a crucial physical idea: maybe the extra dimension isn’t infinite. Maybe it’s curled up into a tiny circle, so small that we can’t see it. Imagine an ant walking along a garden hose. From far away, the hose looks like a one-dimensional line. But the ant knows the hose has a second dimension — it can walk around the circumference. If that circumference is small enough, a distant observer would never notice it. Klein proposed that the fifth dimension might be curled up at a scale near the Planck length — about 10⁻³³ centimeters — far too small for any experiment to resolve.
This “Kaluza-Klein” theory was beautiful but gathered dust for decades. Interest revived dramatically in 1998 with two new proposals that used extra dimensions to attack a different problem: the hierarchy problem.
The hierarchy problem is one of the most puzzling facts about our universe. Gravity is absurdly weak compared to the other forces. The electromagnetic force between two electrons is about 10⁴² times stronger than the gravitational force between them. That’s a one followed by forty-two zeros. Why? The Standard Model of particle physics provides no explanation. The relevant energy scales — the electroweak scale (about 10⁻¹⁸ cm) and the Planck scale (about 10⁻³³ cm) — differ by a factor of 10¹⁵, and nobody can explain why.
The ADD model (Arkani-Hamed, Dimapoulos, and Dvali) proposed a radical answer: gravity isn’t inherently weaker than the other forces. It just looks weaker because it spreads into extra dimensions that the other forces can’t access. Imagine rain falling on a large field versus being funneled through a narrow pipe. The total amount of water is the same, but the concentration is very different. In the ADD picture, all the forces except gravity are confined to our three-dimensional “brane” (a membrane in the higher-dimensional space), while gravity leaks into the full higher-dimensional “bulk.” This dilution makes gravity appear weak.
Maciel works through the mathematics of this idea. A key result is that in a space with D total dimensions, Newton’s gravitational law changes. Instead of the familiar inverse-square law (where the force falls off as 1/r²), gravity in higher dimensions falls off faster — as 1/r^(D−2) — at distances smaller than the size of the extra dimensions. At larger distances, the standard inverse-square behavior is recovered. This means extra dimensions could be detected by looking for deviations from Newton’s law at short distances. For two extra dimensions, the ADD model predicts deviations at scales around 0.01 mm — a distance that laboratory experiments are currently probing.
The Randall-Sundrum models take a different approach. Instead of flat extra dimensions, Randall and Sundrum showed that a single extra dimension with a strongly warped geometry — one where the metric (the mathematical object that defines distances and durations in spacetime; it tells you, given two nearby points, how far apart they are) varies exponentially along the extra dimension — can also solve the hierarchy problem. The RS1 model has two branes sitting at opposite ends of a compact extra dimension. The exponential warping of spacetime between the branes means that energy scales that are naturally of order the Planck scale on one brane get exponentially suppressed on the other brane — producing the enormous hierarchy between gravity and the other forces without requiring the extra dimension to be astronomically large.
The RS2 model goes further: a single brane in an infinite extra dimension. Despite the extra dimension being infinite, gravity on the brane still looks four-dimensional at large distances because the warped geometry effectively traps the graviton near the brane.
Maciel works through all of these models in detail — the Kaluza-Klein construction, the ADD scenario, both Randall-Sundrum models — deriving the graviton mass spectra (the tower of massive particles that a four-dimensional observer would see as a consequence of the extra dimension) and showing how each model addresses the hierarchy problem through different geometric mechanisms.
Why you should care: Extra dimensions aren’t just theoretical curiosities. They’re a central ingredient in string theory, which requires 10 or 11 spacetime dimensions to be mathematically consistent. They offer concrete, calculable mechanisms for unifying gravity with the other forces and for explaining why gravity is so much weaker. And they make testable predictions: deviations from Newton’s inverse-square law at short distances, and new particles (Kaluza-Klein excitations) that could appear at high-energy colliders. Some of these predictions are actively being tested. If extra dimensions exist, they would fundamentally change our understanding of what spacetime is.
Chapter 4: Thermodynamics of a Supersymmetric Oscillator
Jishad Kumar
This chapter might seem like the odd one out — a detailed analysis of the thermodynamic properties of a supersymmetric harmonic oscillator. It’s more technical and less cosmic in scope than the chapters on black holes and extra dimensions. But it illuminates something deep about the relationship between two classes of particles that make up everything in the universe, and it connects directly to one of the most embarrassing unsolved problems in all of physics.
Let me build up the context.
Bosons and fermions. Every particle in the universe falls into one of two categories. Bosons are the force carriers: photons (electromagnetism), gluons (strong force), W and Z bosons (weak force), and the Higgs boson. They obey Bose-Einstein statistics, which means any number of them can occupy the same quantum state. This is why lasers work — you can pile up photons in identical states. Fermions are the matter particles: electrons, quarks, neutrinos. They obey Fermi-Dirac statistics and the Pauli exclusion principle, which forbids any two of them from occupying the same quantum state. This is why atoms have structure, why the periodic table works, why matter is solid instead of collapsing. That last point deserves emphasis: atoms are overwhelmingly empty space. If an atom were the size of a football stadium, the nucleus would be a grain of sand at the center and the electrons would be specks of dust in the upper stands. Yet when you press your hand against a table, it feels solid. The reason is the Pauli exclusion principle. The electrons in the atoms of your hand and the electrons in the atoms of the table cannot be forced into the same quantum states — they resist being squeezed together with an effective pressure (called “degeneracy pressure”) that has nothing to do with electromagnetic repulsion. It’s a purely quantum mechanical effect, and it’s what keeps matter from collapsing into nothing. The same effect holds up white dwarf stars against the crush of gravity.
These two classes of particles have very different thermodynamic behavior. A bosonic oscillator — think of a vibrating mode of the electromagnetic field — has an infinite ladder of equally spaced energy levels given by:
Here ω (omega) is the oscillator’s natural frequency — how many times per second it vibrates — and n can be any non-negative integer: 0, 1, 2, 3, and so on. A fermionic oscillator, by contrast, has only two states: n = 0 or n = 1 (because of the exclusion principle). Their thermodynamic properties — how they store energy, how their entropy changes with temperature — are correspondingly very different.
The vacuum energy problem. Here’s the embarrassment. Every quantum field, even in its ground state (the “vacuum”), has a residual energy — the zero-point energy. For a bosonic oscillator, the ground state energy is +ℏω/2 (half of one quantum of energy). When you add up the zero-point energies of all the quantum fields in the universe across all their modes, you get an absurdly large number for the vacuum energy density. General relativity says this vacuum energy should gravitate — it should curve spacetime. The predicted vacuum energy is roughly 10¹²⁰ times larger than what we actually observe from the measured cosmological constant. This is the “cosmological constant problem,” and it’s been called the worst prediction in the history of physics.
Supersymmetry and cancellation. Supersymmetry (SUSY) is a proposed symmetry that pairs every boson with a fermion partner and vice versa. In a supersymmetric system, something remarkable happens: the positive zero-point energy of the bosonic modes is exactly canceled by the negative zero-point energy of the fermionic modes. The ground state energy is exactly zero.
This isn’t a hand-waving argument. Kumar demonstrates it explicitly with a supersymmetric oscillator — a system that combines a bosonic oscillator and a fermionic oscillator in a specific way. The energy spectrum of the SUSY oscillator is:
Note the crucial difference from the ordinary bosonic oscillator: there’s no +1/2. The ground state energy (n = 0) is exactly zero. What happened? The positive vacuum energy of the boson (+ℏω/2) has been precisely canceled by the negative vacuum energy of the fermion (−ℏω/2). The two contributions are equal and opposite, and their sum is zero.
Above the ground state, a beautiful structure emerges. Every energy level (except the ground state) has a two-fold degeneracy: for each value of the bosonic excitation number, there are two states with the same energy — one that is purely bosonic and one that includes a fermionic excitation. This pairing of bosonic and fermionic states at every energy level is a hallmark of supersymmetry.
Kumar then works out the full thermodynamics: the partition function, the free energy, the entropy, and the heat capacity, as functions of temperature. At low temperatures, the heat capacity of the SUSY oscillator behaves differently from either a bosonic or fermionic oscillator alone. It exhibits a distinctive peak at intermediate temperatures, a feature that reflects the underlying supersymmetric structure.
The chapter then turns to a real physical system: an electron in a magnetic field, the “Landau problem.” When an electron moves in a uniform magnetic field, its motion perpendicular to the field is quantized into discrete Landau levels — equally spaced energy levels, just like a harmonic oscillator. If you ignore the electron’s spin, this is a purely bosonic oscillator. But when you include the spin — and use the specific value of the electron’s g-factor (g = 2, as predicted by Dirac’s equation) — the ground state energy drops to exactly zero. The spin-up ground state with zero orbital excitation has precisely zero energy. The system reveals a hidden supersymmetric structure: you can construct a “supercharge” operator whose square equals the Hamiltonian, and which commutes with the Hamiltonian (meaning it doesn’t change the energy when it acts). This is the algebraic signature of supersymmetry.
Finally, Kumar examines what happens when supersymmetry is broken — specifically, by adding a confining potential (as occurs in quantum dots, where electrons are trapped in a small region). In this “Fock-Darwin” model, the ground state energy is no longer zero, and the boson-fermion degeneracy is lifted. The chapter traces how the thermodynamic properties change when SUSY breaking occurs, connecting the abstract algebra to observable physical consequences.
Why you should care: The vacuum energy problem is one of the biggest unsolved puzzles in physics. It sits at the intersection of quantum field theory and general relativity — exactly the intersection this book is about. Supersymmetry offers one of the very few known mechanisms for taming vacuum energy, and understanding exactly how this cancellation works in controlled examples is essential groundwork. Kumar’s chapter shows that the cancellation isn’t just an abstract theoretical trick — it shows up in a real, physical system (the Landau problem), and it can be broken in understandable ways that have measurable thermodynamic consequences. This kind of detailed, worked-out toy model is how physicists build the intuition needed to attack the bigger problem.
Chapter 5: Black Hole as an Information Mirror
Xuanhua Wang
If you throw a book into a fireplace, the information in the book isn’t really destroyed — in principle, if you tracked every photon of light, every particle of ash, every molecule of gas released by the fire with perfect precision, you could reconstruct the text. This isn’t practically possible, but physics guarantees it: information is conserved. The laws of quantum mechanics are “unitary,” which means they preserve information. You can always run the movie backward.
But in the 1970s, Stephen Hawking showed that black holes seem to violate this rule, and the resulting paradox has shaped theoretical physics for fifty years.
Hawking radiation. Hawking’s calculation combined quantum field theory with general relativity in the semiclassical regime (the same regime my chapter works in). He showed that the intense gravitational field near a black hole’s event horizon causes the quantum vacuum to radiate particles. The black hole glows, albeit incredibly faintly, with a temperature inversely proportional to its mass. A black hole with the mass of our sun would have a temperature of about 60 nanokelvins — far colder than the cosmic microwave background — but the radiation is real in principle.
Here’s the problem. The radiation Hawking calculated appears to be exactly thermal — it has a perfect blackbody spectrum, just like the glow from a hot piece of metal, carrying no information whatsoever about what fell into the black hole. Throw in an encyclopedia, a cat, or a Ferrari — the Hawking radiation looks exactly the same. And since the black hole is radiating energy, it’s slowly losing mass. Eventually, it evaporates completely. When it’s gone, all that remains is featureless thermal radiation. The information about everything that ever fell in seems to have vanished from the universe.
This violates unitarity. Quantum mechanics says information can’t be destroyed. General relativity says it can. Something has to give.
The Page curve. In 1993, Don Page made a crucial observation. If a black hole really is an ordinary quantum system (just one that happens to be very hot and very scrambled), then the entropy of its radiation should follow a specific pattern. Early on, as the black hole emits radiation, the entropy of the radiation should rise — each new particle adds to the disorder. But at a certain point (the “Page time,” roughly when the black hole has emitted half its initial entropy), the entropy should stop rising and start falling. This is because the radiation is becoming increasingly entangled with the earlier radiation in a specific way that a unitary process guarantees. By the time the black hole has fully evaporated, the entropy of the radiation returns to zero — all the information has been transferred out. This rise-then-fall graph is called the “Page curve.”
Hawking’s calculation, by contrast, predicts that the entropy just keeps rising, monotonically, forever — no turn-around, no information recovery. Reproducing the Page curve from a gravitational calculation became the central challenge.
The breakthrough: entanglement islands. Wang’s chapter surveys the remarkable recent progress. The key development, emerging around 2019, was the “island formula” for gravitational entropy. The idea is startling: when computing the entropy of Hawking radiation after the Page time, you must include contributions from a region insidethe black hole — the “island” — that, through the mathematics of quantum entanglement, turns out to be encoded in the radiation outside.
The formula says: to compute the entropy of the radiation, minimize the “generalized entropy,” which has two terms. The first is the area of a special surface (the “quantum extremal surface”) divided by 4G (where G is Newton’s gravitational constant) — this is essentially the Bekenstein-Hawking entropy, the same formula that gives a black hole its entropy. The second is the entropy of quantum fields in the region between this surface and the boundary where the radiation is collected. Before the Page time, the quantum extremal surface shrinks to zero and you recover Hawking’s monotonically increasing entropy. After the Page time, the quantum extremal surface jumps to just outside the event horizon, and the entropy is dominated by the area term, which decreases as the black hole shrinks.
The result: the Page curve is reproduced. The entropy rises, then falls, exactly as unitarity demands.
This might sound like a mathematical trick, but the derivation is grounded in the gravitational path integral — the same mathematical framework that underlies all of quantum gravity. The island formula was derived by considering “replica wormholes,” additional saddle-point contributions to the path integral that become important when computing entanglement entropy using the replica trick. These aren’t optional additions; they’re contributions that the mathematics demands.
The Hayden-Preskill protocol. Wang then turns from entropy (can information be recovered?) to a more operational question: how quickly can it be recovered?
In 2007, Hayden and Preskill considered a thought experiment: Alice throws a few qubits of information into a black hole that has already been radiating for a long time (past its Page time). Bob has been carefully collecting all the earlier Hawking radiation. How long does Bob have to wait before Alice’s information appears in the new radiation?
The answer, derived using tools from quantum information theory, is surprising: Bob only needs to collect slightly more new qubits than Alice threw in. The information comes back almost immediately (in “scrambling time” — roughly the time it takes for the black hole to thoroughly mix the new information into its internal degrees of freedom). The black hole doesn’t hoard information; it reflects it back like a mirror, just thoroughly scrambled.
Wang’s chapter reviews this result and then goes further, discussing explicit decoding protocols — the Yoshida-Kitaev protocol and the Horowitz-Maldacena model — that specify concrete quantum circuit operations Bob could perform to extract Alice’s information from the radiation. These aren’t just theoretical curiosities; they’ve been tested on small quantum computers, simulating the information-recovery process with a handful of qubits.
Why you should care: The black hole information paradox sits at the exact collision point of quantum mechanics and general relativity, and its resolution is likely to reveal something fundamental about the nature of spacetime. For decades, the paradox seemed intractable. The progress surveyed here — entanglement islands, the Page curve from gravitational path integrals, explicit decoding protocols — represents a genuine breakthrough. The emerging picture is that black holes aren’t information destroyers but information mirrors: what falls in comes back out, encoded in the subtle quantum correlations of the radiation. Understanding how this works may be the best clue we have about what a full theory of quantum gravity will look like.
The Common Thread
Reading these five chapters together, a pattern emerges that wouldn’t be visible from any single one alone.
Chapter 1 shows that spacetime curvature modifies quantum interference in precise, calculable ways — the phase shift depends on the Riemann curvature tensor, which is the mathematical object encoding how spacetime bends. Chapter 2 extends quantum field theory to the realistic environment around black holes, where plasma and extreme curvature interact. Chapter 5 shows that the entropy of Hawking radiation — a quintessentially quantum-gravitational quantity — can be computed using entanglement and quantum extremal surfaces, and that information is preserved through the entanglement structure of the radiation. Chapter 4 reveals that supersymmetry produces exact cancellations in vacuum energy through the pairing of bosonic and fermionic degrees of freedom, with the entropy and thermodynamic properties carrying signatures of the underlying symmetry. Chapter 3 reminds us that the geometry of spacetime may be richer than the three spatial dimensions we perceive, with extra dimensions offering new mechanisms for how gravity operates.
The connecting thread is entropy and entanglement. These concepts appear everywhere — in the phase shifts of interferometry, in the thermodynamics of supersymmetric systems, in the Page curve of an evaporating black hole, in the entanglement islands that encode a black hole’s interior in its radiation. They seem to be the language in which the quantum-gravity interface is written, even before we have a complete theory.
The book doesn’t deliver that complete theory. Nobody has it yet. But it does provide a clear map of what can be calculated now, what the open questions are, and where the next breakthroughs might come — from atom interferometry to black hole imaging to tabletop experiments probing Newton’s law at sub-millimeter scales. For anyone who wants to understand the frontier where quantum mechanics meets gravity — the most important unsolved problem in fundamental physics — this is a good place to start.
Quantum Theory Meets Gravity — From Curved Spacetime to Black Hole Information is available as open-access from IntechOpen and in hardcover from Amazon. Edited by Oliver K. Baker, Djordje Minic, and Samir Varma.
And my other book, The Science of Free Will is available on Amazon too. In that I talk about all kinds of cool stuff. Why does traffic suck so much? Why don’t we trade with ants? Why do dolphins save drowning humans? And much more.
Fascinating guide. In Leveille terms black-hole interiors and ringdown cannot reach exact closure — regulated tolerance band Δ > 0 forces a scar that keeps everything regular. Paper 58 gives the big picture. No paywall archive: exactlyinfinite.substack.com