Chapter 1: Classical Black Holes and the “No-Hair Theorem”

Behind the countless stars shining in the night sky lurk black holes, dark celestial bodies from which not even light can escape. Albert Einstein’s General Theory of Relativity, published in 1915, brought about the most beautiful paradigm shift in the history of physics by describing gravity as the “curvature of spacetime”. Shortly thereafter, in 1916, Karl Schwarzschild discovered an exact solution (the Schwarzschild solution) showing how a spherically symmetric mass distribution distorts the surrounding spacetime. This was the mathematical dawn of the concept of a black hole.

Within the framework of classical general relativity, a black hole is a surprisingly “simple” entity. The process by which a star ends its life and collapses under its own gravity is extremely complex, and the original star possesses a vast amount of information (degrees of freedom) such as its chemical composition, magnetic field structure, and shape asymmetries. However, once an Event Horizon is formed, all this complex information is either hidden inside the horizon or radiated away to the outside as gravitational waves.

This remarkable property was expressed by John Archibald Wheeler in his famous phrase “Black holes have no hair,” also known as the “No-Hair Theorem.” According to this theorem, a black hole that has settled into a steady state is completely described by just three physical quantities:

1. Mass (Mass, $M$)
2. Electric Charge (Electric Charge, $Q$)
3. Angular Momentum (Angular Momentum, $J$)

The Schwarzschild solution represents the simplest black hole with only mass, the Kerr solution is a rotating black hole with mass and angular momentum, the Reissner-Nordström solution is a black hole with mass and charge, and the Kerr-Newman solution is the most general solution possessing all three. No matter how complex the matter (be it a lump of iron, antimatter, or a mountain of encyclopedias) that formed the black hole, only these three parameters are observable from the outside.

However, this extreme simplicity caused a severe contradiction with another great pillar of physics, “Thermodynamics,” particularly the “Second Law of Thermodynamics.” The Second Law of Thermodynamics dictates that “the entropy of an isolated system never decreases.” Entropy is a measure of the number of microscopic states a system has, that is, the “randomness of information.”

If a black hole is a “hairless” entity completely specified by just the three parameters above, it means its number of microscopic states is exactly one (entropy is zero). This is a massive problem. Suppose we throw hot gas full of entropy or matter with complex structures into a black hole. After the black hole swallows it, the entropy of that matter completely disappears from the external universe. If the black hole’s own entropy is zero, the total entropy of the universe would have decreased, causing the collapse of the Second Law of Thermodynamics.

“Black holes are not the universe’s trash cans.” To save physics from this crisis, it was necessary to step into an unprecedented realm: the unification of gravitational theory, thermodynamics, and quantum mechanics. Does the absolute wall of the event horizon truly reduce everything to nothingness? Or does the black hole possess a “hidden entropy” that we do not yet know about? In the next chapter, let’s look at the groundbreaking ideas of a young physicist who tackled this mystery.

Chapter 2: The Bekenstein-Hawking Formula

In response to the severe paradox that black holes violate the Second Law of Thermodynamics, Jacob Bekenstein, a graduate student at Princeton University in 1972, proposed a bold and intuitive hypothesis. He focused on the “black hole area theorem” proved by Stephen Hawking (the area of a black hole’s event horizon never decreases).

Entropy never decreases either. The area of a black hole never decreases either. Is this similarity just a coincidence? Bekenstein argued that “the area of the black hole’s event horizon is the black hole’s entropy itself.” When matter falls into a black hole, the black hole’s mass increases, and consequently, the area of the event horizon expands. The entropy possessed by the matter is not lost but preserved in the form of the increased area of the black hole, and the generalized second law of thermodynamics (the sum of the entropy of the universe and the entropy of the black hole does not decrease) holds true.

This groundbreaking idea was later supported by Hawking’s quantum mechanical analysis and crystallized as the “Bekenstein-Hawking Entropy Formula”.

Here, $S_{BH}$ is the black hole’s entropy, and $A$ is the surface area of the event horizon. The proportionality constant is a gathering of all the fundamental constants of physics: the Boltzmann constant $k_B$, the speed of light $c$, Newton’s universal gravitational constant $G$, and the Dirac constant (Planck’s constant divided by $2\pi$) $\hbar$. This eloquently speaks to the fact that this formula is the pinnacle of “quantum gravity theory”, where thermodynamics ($k_B$), relativity ($c, G$), and quantum mechanics ($\hbar$) intersect.

The implication of this formula is extremely profound. Entropy is equivalent to information content. Normally, the entropy of matter increases in proportion to its volume (3-dimensional space) (e.g., the number of molecules making up a gas). However, the formula says the entropy of a black hole is proportional not to volume but to “area (2 dimensions).” If we rewrite the formula using the Planck length $l_p = \sqrt{G\hbar/c^3}$, it looks like this:

This means that if we tile the event horizon with tiny squares each with a side of one Planck length (about $1.6 \times 10^{-35}$ meters), one bit of information (entropy) is stored per four tiles (4 Planck areas). All the 3-dimensional information of the matter that fell into the black hole is recorded like a “hologram” on the 2-dimensional event horizon. This astonishing fact becomes the most crucial foreshadowing that leads to the later “Holographic Principle.”

However, the existence of entropy inevitably means, from the fundamental relations of thermodynamics, that a black hole possesses a “temperature”. By substituting the black hole’s mass energy $E = Mc^2$ and Bekenstein’s entropy into the first law of thermodynamics $dE = T dS$, it is derived that the black hole’s temperature $T$ is proportional to its “surface gravity $\kappa$”.

Initially, many physicists, including Hawking himself, were skeptical of Bekenstein’s hypothesis, thinking, “Since a black hole only sucks everything in and emits nothing, its temperature must be absolute zero.” An object with a temperature must emit thermal radiation (black-body radiation). Could it be possible that a black hole emits light?

To prove Bekenstein wrong, Hawking immersed himself in calculations applying quantum field theory in curved spacetime to black holes. However, what he arrived at was one of the most beautiful discoveries in the history of physics, fundamentally overturning his own expectations. Black holes were indeed shining due to quantum mechanical effects.

Chapter 3: The Discovery of Hawking Radiation

In 1974, Stephen Hawking published a shocking paper titled “Black holes aren’t black.” He theoretically derived the phenomenon where particles are emitted from the event horizon—from which classically nothing should escape—due to quantum mechanical tunneling effects. This is known as “Hawking Radiation.”

To understand this phenomenon, a quantum mechanical shift in perspective regarding the “vacuum” is necessary. According to quantum field theory, the vacuum is not “empty space” but a sea of “quantum fluctuations” where pairs of particles and antiparticles (virtual particles) are constantly created and annihilated. Normally, these virtual particle pairs vanish in an instant, so they cannot be observed directly.

However, if this pair creation occurs extremely close to a black hole, just outside the event horizon, a dramatic situation arises. The intense tidal force (gravity gradient) near the event horizon pulls the generated particle pair apart. Probabilistically, one particle may cross the event horizon and fall into the black hole, while the other particle escapes into the external universe.

To an external observer, it looks as if a particle has popped out of the black hole. This is Hawking radiation. So, what happens to the law of conservation of energy? The escaped particle becomes a real particle with positive energy. To conserve energy, the particle that fell into the black hole must possess “negative energy.” By absorbing negative energy, the black hole loses mass (i.e., energy). In other words, a black hole gradually evaporates by emitting radiation.

Hawking rigorously calculated the spectrum of this radiation using an advanced mathematical technique combining spacetime geometry and quantum field theory (the Bogoliubov transformation). He defined a vacuum state at past null infinity ($\mathscr{I}^-$) in flat spacetime and analyzed what kind of particle state it would be observed as at future null infinity ($\mathscr{I}^+$) after propagating through the spacetime of the black hole formed by stellar collapse.

When the past vacuum state $|0\rangle_{in}$ is expanded using future particle creation and annihilation operators, it is described using Bogoliubov coefficients $\alpha, \beta$. The core of Hawking’s calculation lay in showing that the phase of light traveling through the surface of a collapsing star to reach future infinity undergoes a logarithmic delay (redshift) due to the formation of the event horizon. This geometric redshift is the decisive factor that makes the Bogoliubov coefficient $\beta$ (the coefficient governing particle creation) non-zero and produces a thermal spectrum.

Astonishingly, the resulting radiation perfectly matched the Planck spectrum of black-body radiation. Its temperature (Hawking temperature $T_H$) is given by the following equation:

What is noteworthy here is that the Hawking temperature is “inversely proportional” to the mass $M$ of the black hole. This means that as a black hole loses mass and shrinks, its temperature rises, and it radiates more intensely (it has a negative specific heat).

In the case of a solar-mass black hole, the Hawking temperature is an extremely low temperature of about $6 \times 10^{-8}$ Kelvin, much lower than the current cosmic microwave background radiation (about 2.7 Kelvin), making it practically impossible to observe the radiation. However, as the mass decreases, the temperature rises rapidly. The lifetime until the black hole completely evaporates (evaporation time) is proportional to the cube of the mass, $t_{evap} \sim \frac{G^2 M^3}{\hbar c^4}$. It would take a mind-boggling $10^{67}$ years for a solar-mass black hole to evaporate, but a microscopic black hole would eventually release a massive amount of energy and disappear explosively.

The discovery of Hawking radiation completed black hole thermodynamics and became a monumental achievement in theoretical physics. But at the same time, it opened a Pandora’s box to the deepest and most profound paradox that would shake the foundations of physics. That is the “Black Hole Information Paradox.”

Chapter 4: The Outbreak of the Information Paradox

Hawking radiation predicted that black holes would ultimately evaporate and vanish. From this arises one of the greatest mysteries in modern physics, the “Information Paradox.”

One of the most fundamental and sacred principles of quantum mechanics is “Unitarity.” This principle dictates that as a system’s state evolves over time, the sum of probabilities is always conserved at 1, and information is deterministically preserved from the past to the future and from the future to the past. In other words, if you know the complete information (pure state) of a system at a certain point in time, you can perfectly predict and restore its state at any past or future point by solving the Schrödinger equation (or its extended equations), guaranteeing the reversibility of physical laws.

Let’s apply this principle to the process of a black hole’s formation and evaporation. Suppose matter in a pure state (for example, a group of particles precisely prepared in a specific quantum state, or an encyclopedia) undergoes gravitational collapse and forms a black hole. At this point, all information resides inside the black hole.

Then, Hawking radiation begins. According to Hawking’s calculations, the radiation originating at the event horizon is “perfect thermal radiation.” The spectrum of thermal radiation is determined solely by temperature (the black hole’s mass, charge, and angular momentum); there is no quantum correlation (entanglement) among the radiated particles, and it contains absolutely no “information” about what the original matter that fell into the black hole was. If you burn an encyclopedia into ash and smoke, in principle, you could reconstruct the original text if you perfectly measured the motion of all the ash and smoke, but Hawking argued that Hawking radiation is “truly random.”

And finally, the black hole completely evaporates and disappears. All that is left is thermal radiation devoid of information (a mixed state). The universe, which was supposed to be in a pure state, has transitioned into a mixed state lacking information. The vast amount of information possessed by the original matter has completely vanished from the universe along with the black hole.

This signifies the collapse of the unitarity of quantum mechanics. Faced with this paradox where quantum mechanics and general relativity collide, physicists divided into camps and engaged in fierce debates. This is what is famously known as the “Black Hole War.”

The standard-bearer of one camp was Stephen Hawking himself. He argued, “In black holes, the rules of quantum mechanics break down, and information is truly lost.” Together with Kip Thorne, he made a bet with John Preskill taking the stance that “information is lost.”

Fiercely opposing this were particle physicists like Leonard Susskind and Gerard ’t Hooft. “If information is lost, even the law of conservation of energy would break down, and the very foundation of modern physics would collapse.” They were convinced that when a black hole evaporates, some unknown mechanism must encrypt the information into the Hawking radiation and carry it out.

However, Hawking’s calculations seemed rock-solid. Once information crosses the event horizon, it cannot escape outside unless it travels faster than light. The inside and outside of the event horizon are causally disconnected, and the general theory of relativity (locality) forbids information from escaping. If one tries to protect the conservation of information (quantum mechanics), causality (relativity) is violated; if one tries to protect causality, information is lost. Resolving this dilemma required overturning the very concept of spacetime from its foundations.

Susskind and his colleagues, armed with quantum entanglement and String Theory, took on this impregnable paradox. In the next chapter, let’s delve into the dramatic theoretical leap to protect information, “Black Hole Complementarity,” and the new spark it ignited, the “Firewall Problem.”

Chapter 5: Black Hole Complementarity and the Firewall Problem

Convinced that information is conserved, Susskind and others proposed the groundbreaking concept of “Black Hole Complementarity” in 1993, drawing inspiration from Niels Bohr’s principle of complementarity, to resolve the contradiction of the event horizon.

The root cause of the paradox was that if “information falling inside a black hole” and “information escaping outside as Hawking radiation” occur simultaneously, the information would be “copied” (violating the no-cloning theorem of quantum mechanics).

Black hole complementarity argues as follows: From the perspective of an observer remaining outside (Alice), the matter falling onto the superheated membrane near the event horizon (the stretched horizon) is incinerated, and its information is slowly brought back outside, riding on Hawking radiation. On the other hand, for an observer in free fall into the black hole (Bob), the event horizon is nothing but an ordinary transit point in space, and he falls harmlessly inside while carrying the information.

At first glance, it seems the information has split in two, but Susskind argued that since “it is in principle impossible for Alice and Bob to communicate and compare their results,” no contradiction arises. Two different, yet complementary descriptions of the same physical phenomenon exist, and there is no absolute perspective (God’s perspective) that describes the whole. This meant that in the extreme realm where gravity and quantum mechanics intersect, the common sense of the “locality” of spacetime breaks down.

It seemed that black hole complementarity had resolved the paradox. However, in 2012, a paper published by four physicists at the University of California, Santa Barbara (Almheiri, Marolf, Polchinski, Sully: commonly known as AMPS) sent shockwaves through the physics community once again. It was the introduction of the “AMPS Firewall Paradox.”

Assuming that Hawking radiation carries information out (unitarity is preserved), AMPS focused on the complex distributional relationship of quantum entanglement that occurs after the black hole has passed half of its lifetime (the Page time).

In order to carry information outside, a newly radiated Hawking particle (B) must be strongly entangled with the set of all previously emitted Hawking particles (A). On the other hand, to pass through the event horizon unharmed (preserving the equivalence principle), there must also be a strong entanglement between the particle pair generated from vacuum fluctuations: the outward-escaping particle (B) and the inward-falling particle (C).

However, quantum mechanics has a strict rule called the “monogamy of entanglement.” It is impossible for one quantum system (B) to be maximally entangled with two independent systems (A and C) simultaneously.

To resolve this dilemma, something must be sacrificed. AMPS concluded that if one wishes to maintain black hole complementarity, there is no choice but to sever the entanglement between B and C. But severing the entanglement between B and C means that an extremely high-energy wall of particles, a “Firewall,” would appear at the event horizon.

If a firewall exists, Bob, attempting to pass through the event horizon, would not pass unharmed but would instantly burn to a crisp upon colliding with this high-energy wall. This signifies the complete breakdown of the “Equivalence Principle” (an observer in free fall feels no gravity, meaning nothing special happens at the event horizon), which is the cornerstone of Einstein’s general theory of relativity.

If you try to protect information, the event horizon becomes a wall of fire (collapse of the equivalence principle). If you try to protect the equivalence principle, either information is lost, or the foundations of quantum mechanics collapse. Physicists faced a paradox deeper and sharper than ever before. The key to saving this desperate crisis lay hidden in the dimension-crossing “Holographic Principle” and the latest theories of quantum gravity.

Chapter 6: The Holographic Principle, AdS/CFT Correspondence, and the Island Formula

The decisive paradigm shift that saved physics from the crisis of the firewall paradox was the “Holographic Principle” and its rigorous mathematization, the “AdS/CFT Correspondence (Anti-de Sitter/Conformal Field Theory Correspondence).”

The Holographic Principle was proposed by Gerard ’t Hooft in 1993 and incorporated into the framework of string theory by Leonard Susskind in 1995. As suggested by the Bekenstein-Hawking formula (entropy is proportional not to volume but to area) discussed in Chapter 2, this principle asserts that “all physical phenomena (including gravity) occurring in a certain 3-dimensional spatial region can be completely described by the laws of physics, excluding gravity, on the 2-dimensional boundary surface (boundary) surrounding that region.” Our 3-dimensional universe might just be a phantom “hologram” projected on a distant 2-dimensional screen.

In 1997, Juan Maldacena used superstring theory to provide the first concrete model in which this holographic principle rigorously holds. This is the “AdS/CFT correspondence.” Maldacena proved that a quantum gravity theory (superstring theory) in an Anti-de Sitter (AdS) space with a negative cosmological constant is mathematically completely equivalent (dual) to a conformal field theory (CFT) without gravity defined on the boundary at infinity of that space.

This was a miraculous discovery in physics. Two theories with entirely different dimensions and laws—bulk physics including gravity and boundary physics (CFT) not including gravity—were actually just two different translations of the same physical phenomena. Using this duality dictionary (holographic dictionary), it becomes possible to transform hopelessly difficult quantum gravity problems into easy-to-calculate quantum field theory problems on the boundary and solve them.

In the context of the information loss paradox, the AdS/CFT correspondence provided a crucial answer. Because the CFT on the boundary obeys standard quantum mechanics, unitarity is perfectly guaranteed (information is never lost). If the AdS/CFT correspondence is correct, the process of black hole formation and evaporation in the dual AdS space must also be unitary as a whole, and information must be safely preserved.

However, while it was proven that “information is conserved,” the dynamical mechanism of “exactly how information is transferred from inside the black hole to the external Hawking radiation” remained a mystery. The ultimate breakthrough answering this question was brought about by the discovery of the “Island Formula,” published in 2019.

The story began with the concept of the “Page Curve,” proposed by Don Page in 1993. Page predicted how the entanglement entropy of Hawking radiation should change over time if information is preserved (unitarity is maintained). In the initial stages of radiation, the emitted particles are entangled with particles inside the black hole, so the entanglement entropy of the total radiation monotonically increases over time. But when the black hole reaches half its lifetime (the Page time), the situation reverses dramatically. Newly emitted particles start to become entangled with previously emitted particles, and the entanglement entropy of the total radiation begins to decrease. When the black hole finally disappears, all radiation should return to a pure state, and the entropy should become zero again. This mountain-shaped curve is the Page curve.

On the other hand, in Hawking’s classical calculation (the stance that information is lost), the entropy continues to increase forever and never decreases (the Hawking curve).

In 2019, Geoff Penington, Almheiri, Mahajan, Maldacena, Zhao, and others recalculated the entropy of Hawking radiation using a holographic entanglement entropy calculation method derived from string theory (the quantum extremal surface formula, a generalization of the Ryu-Takayanagi formula). As a result, they discovered that after a certain point (the Page time), the geometric region used to calculate the entanglement entropy changes discontinuously.

The formula they derived showed that when calculating the entropy of Hawking radiation, one must add the contribution not only of the radiation outside the black hole but also of a specific region “inside” the event horizon (called an “Island”).

Amazingly, when proceeding with the calculations using this island formula, the island’s contribution becomes dominant after the Page time, and the results perfectly reproduced Don Page’s curve. In other words, what was missing from Hawking’s calculations were non-perturbative quantum effects of gravity (changes in spacetime topology called replica wormholes), and by incorporating these, it was proven that information is indeed encoded in Hawking radiation and escapes.

Even more shocking is the fact that although the “island” is inside the event horizon, its information is completely contained within the Hawking radiation. This means that a part of the inside of the black hole is actually holographically projected outside through the Hawking radiation. As the ER=EPR hypothesis (the hypothesis that Einstein-Rosen bridges = wormholes, and Einstein-Podolsky-Rosen entanglement are fundamentally the same) suggests, quantum entanglement formed a bridge through spacetime (a wormhole), connecting the inside of the black hole with the external Hawking radiation.

Appendix A: Bogoliubov Transformation of Hawking Radiation and Rigorous Derivation of the Thermal Spectrum

To truly understand the profound nature of Hawking radiation, one cannot avoid the detailed derivation process of the mathematical structure of quantum field theory in curved spacetime, especially the “Bogoliubov Transformation.” In this appendix, we will trace the core part of the calculation developed in Stephen Hawking’s historic 1974 paper as rigorously and in as much detail as possible.

1. Differences in the Definition of “Vacuum” in Flat vs. Curved Spacetime

In quantum field theory, a scalar field $\phi(x)$ is mode-expanded using creation operators $\hat{a}^\dagger$ and annihilation operators $\hat{a}$. In flat Minkowski spacetime, the solution to the field equation (Klein-Gordon equation) $(\square + m^2)\phi = 0$ is expanded using plane waves $e^{ikx}$ as a basis. Using positive-frequency modes $f_k(x)$ and negative-frequency modes $f_k^*(x)$, the field is written as follows:

However, when spacetime is curved (a gravitational field is present), global Poincaré symmetry is lost, and so there is no universal time coordinate that uniquely defines the “positive-frequency modes.” Depending on the observer’s state of motion and position (the geometry of spacetime), which modes are considered “positive” becomes relative.

2. Mode Expansions at Past Null Infinity ($\mathscr{I}^-$) and Future Null Infinity ($\mathscr{I}^+$)

Hawking considered a dynamic spacetime where a star collapses under its own gravity to form a black hole. In this spacetime, there is a flat asymptotic region before the collapse begins, called “Past Null Infinity ($\mathscr{I}^-$),” and a flat asymptotic region after black hole formation, called “Future Null Infinity ($\mathscr{I}^+$).”

Let the mode functions at past null infinity $\mathscr{I}^-$ be $f_k$, the corresponding operators be $\hat{a}_k, \hat{a}_k^\dagger$, and the vacuum defined by a past observer be $|0_{in}\rangle$. Let the mode functions at future null infinity $\mathscr{I}^+$ be $p_j$, the corresponding operators be $\hat{b}_j, \hat{b}_j^\dagger$, and the vacuum defined by a future observer be $|0_{out}\rangle$.

Since the field $\hat{\phi}$ can be expanded using either basis, the following equality holds:

3. Bogoliubov Transformation and Particle Creation

Because the two different sets of mode functions $f_k$ and $p_j$ form complete sets, one can be expressed as a linear combination of the other.

This linear transformation is the Bogoliubov transformation, and the expansion coefficients $\alpha_{jk}$ and $\beta_{jk}$ are called Bogoliubov coefficients. Using this to express the future annihilation operator $\hat{b}_j$ in terms of past operators yields:

A decisive fact is derived here. Suppose the initial state was the past vacuum $|0_{in}\rangle$ (meaning initially there were no particles at all). Let us calculate the expected value of particles with wavenumber $j$ (the expected value of the number operator $\hat{N}_j = \hat{b}_j^\dagger \hat{b}_j$) observed by a future observer in this state.

If $\beta_{jk} \neq 0$, then even though the initial state was a vacuum, it will be observed by the future observer as if “particles exist”! The fluctuation of spacetime curvature (stellar collapse) made $\beta_{jk}$ non-zero, effectively “creating” particles from the vacuum.

4. Logarithmic Phase Delay (Redshift) by the Event Horizon

Hawking’s greatest achievement lies in explicitly calculating this $\beta_{jk}$ through geometric ray tracing in the black hole formation spacetime.

Consider the process where a wave packet originating from past null infinity $\mathscr{I}^-$ passes through the collapsing star and reaches future null infinity $\mathscr{I}^+$. Light rays (modes) that pass through the star just before the event horizon forms undergo extreme redshift as they climb out of the strong gravitational field.

Geometric calculations show that the phase of the wave reaching future null infinity at time $u$ suffers a logarithmic delay as it approaches the formation time of the event horizon. Specifically, near the event horizon, a logarithmic relationship (a consequence of the Raychaudhuri equation) holds between the time $v$ at past null infinity and the time $u$ at future null infinity:

Here, $v_0$ is the departure time of the light ray that forms the event horizon, and $\kappa$ is the surface gravity of the black hole. This logarithmic phase delay $e^{-\kappa u}$ crucially dictates the functional form of $\beta_{jk}$ through the Fourier transform.

5. Derivation of the Planck Spectrum and Hawking Temperature

The Bogoliubov coefficients $\alpha_{jk}$ and $\beta_{jk}$ are calculated using an inner product (the Klein-Gordon inner product). When the wave with the above logarithmic phase delay is Fourier-transformed and integrated (making full use of Gamma functions and complex integration), an astonishing analytic relationship is derived between the absolute values of $\alpha_{jk}$ and $\beta_{jk}$:

Here, $\omega$ is the energy (angular frequency) of the radiated particle. Furthermore, from the normalization condition of the Bogoliubov transformation (preservation of commutation relations), $\sum_k (|\alpha_{jk}|^2 - |\beta_{jk}|^2) = 1$ holds. Solving these two equations simultaneously for $|\beta_{jk}|^2$ yields:

This result is phenomenal. The left side represents the number (expected value) of particles created from the vacuum, but the right side has precisely the form of the thermal distribution (Planck distribution) of bosons radiated from a black body with temperature $T$!

The standard form of the Planck distribution is $\frac{1}{e^{\frac{\hbar \omega}{k_B T}} - 1}$. Comparing this, the black hole’s temperature (Hawking temperature $T_H$) is immediately derived:

In the case of a Schwarzschild black hole, the surface gravity is given by $\kappa = \frac{c^4}{4 G M}$, so substituting this gives:

Thus, the formula for the Hawking temperature has been beautifully derived. The complete picture of Hawking’s formidable mathematical insight—that the extreme geometric redshift (logarithmic phase delay) caused by a black hole’s event horizon produces a perfect thermal mixed state (Planck distribution) from a purely quantum mechanical initial state—is laid bare here. This astounding calculation, integrating relativity ($\kappa$), quantum mechanics ($\hbar$), and thermodynamics ($k_B$) into a single formula, can be called a pinnacle of human intellect.

Appendix B: The Ryu-Takayanagi Formula and the Depths of Holographic Entanglement Entropy

At the foundation of the “Island Formula,” which enabled the reproduction of the Page curve in solving the information loss paradox, lies the “Ryu-Takayanagi Formula” generated from superstring theory and the AdS/CFT correspondence. This formula is one of the most important equations in modern physics, directly linking the entanglement of quantum information with the geometry of spacetime (gravity). Here, we will explain its concepts and mathematical background in detail.

1. The Geometrization of Entanglement Entropy

In quantum field theory, the quantity measuring the degree of quantum entanglement between a certain spatial region $A$ and its external region $B$ is the “entanglement entropy (von Neumann entropy)” $S_A = -\text{Tr}(\rho_A \log \rho_A)$. Here, $\rho_A$ is the reduced density matrix of region $A$. However, calculating entanglement entropy directly in an interacting continuous quantum field theory is extremely difficult, and often faces the challenge of ultraviolet divergence (infinity).

In 2006, Shinsei Ryu and Tadashi Takayanagi proposed a groundbreaking conjecture (formula) stating that by using the framework of the AdS/CFT correspondence, the entanglement entropy in the CFT on the boundary can be calculated geometrically as the “area of a minimal surface” in the bulk (the internal AdS space).

Here:

• $S_A$ is the entanglement entropy of region $A$ in the CFT on the boundary.
• $\gamma_A$ is a $(d-1)$-dimensional minimal surface (like a soap film) spanned within the bulk space. The boundary of this surface must match the boundary $\partial A$ of region $A$ on the CFT ($\partial \gamma_A = \partial A$).
• $G_N^{(d+1)}$ is the $(d+1)$-dimensional Newton constant in the bulk space.

Looking at the form of this formula, many readers will be unable to hide their surprise. This equation has exactly the same structure as the Bekenstein-Hawking entropy formula $S_{BH} = \frac{A}{4G\hbar}$ explained in Chapter 2! The thermodynamic entropy possessed by a black hole’s event horizon and the entanglement entropy possessed by a quantum field in a vacuum state were perfectly unified as a geometric entity called “area” through a holographic perspective.

2. Minimal Surfaces and “Replica Wormholes”

The Ryu-Takayanagi formula was initially a “conjecture” based on intuition and various circumstantial evidence, but it was later mathematically proven by Aaron Wall, Alexander Maloney, Juan Maldacena, and others through the gravitational version of the “Replica Trick” (replica wormholes) using path integral representations in quantum gravity.

The replica trick is a method to avoid calculating $\log \rho_A$ directly by instead calculating the trace of the $n$-th power of $\rho_A$, $\text{Tr}(\rho_A^n)$, and finally taking the limit as $n \to 1$.

Calculating $\text{Tr}(\rho_A^n)$ in CFT corresponds to opening up the original spacetime at region $A$, preparing $n$ copies (replicas), and calculating the partition function on the complicated multiple Riemann surface stitched together. According to the AdS/CFT correspondence, this partition function on the boundary should be equal to the partition function (saddle-point evaluation of the action) of the bulk gravity theory that satisfies the boundary conditions.

Maldacena and his colleagues showed that in the gravitational path integral within the bulk, when the boundary is an $n$-replica spacetime, a “wormhole (replica wormhole)” solution where the $n$ spacetimes are connected to each other within the bulk becomes dominant. And when taking the limit as $n \to 1$, they rigorously proved that the geometric backreaction of this wormhole appears exactly as the “minimal surface $\gamma_A$” of the Ryu-Takayanagi formula.

3. Quantum Extremal Surfaces (QES) and the Emergence of Islands

The Ryu-Takayanagi formula was a formula for when gravity in the bulk space is classical (quantum effects can be ignored). To deal with phenomena where quantum effects play an essential role, such as black hole evaporation (Hawking radiation), an extension incorporating quantum field theory effects within the bulk into this formula was necessary. This is the prescription of the “Quantum Extremal Surface (QES)” proposed by Engelhardt and Wall in 2014.

In the QES formula, the object to be minimized is replaced not just by “area” but by the Generalized Entropy $S_{gen}$, which is the sum of “area + quantum entanglement entropy within the bulk.”

Here, $X$ is a surface in the bulk, and $\Sigma_X$ is the region of the bulk space sandwiched between $X$ and region $A$ on the boundary. We extremize (make stationary) this generalized entropy and further find the surface $X$ that minimizes it (this is the QES), and the value of $S_{gen}$ at that time becomes the entropy $S_A$ of the boundary CFT.

When calculating the entropy of Hawking radiation, let the external region that collects the radiation be $A$. In the initial stages, the QES is a trivial surface (zero area), and the entropy monotonically increases (Hawking curve). However, past the Page time, a non-trivial QES suddenly appears “inside” the event horizon, and a dramatic phase transition occurs where the value of $S_{gen}$ there becomes smaller (dominant).

This very region inside the horizon enclosed by this new QES is the “Island”. When an island appears, in the calculation of the bulk entropy term $S_{bulk}$, because the particles within the island (particles that fell into the black hole) and the Hawking radiation (particles that escaped outside) are strongly entangled, the sum of the entropies cancels out significantly and turns to a decrease.

Through this, Don Page’s predicted Page curve was perfectly derived from gravity theory calculations. The resolution of the information loss paradox went beyond a mere conceptual discussion, reaching the point of being proven as a rigorous mathematical formula by considering the non-perturbative topological changes of quantum gravity known as “replica wormholes.” Spacetime emerges from quantum entanglement, and the profound information of a black hole is indeed delivered to the ends of the universe along the invisible threads of wormholes.