Hawking radiation

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https://en.wikipedia.org/wiki/Hawking_radiation

Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's event horizon because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical argument for its existence in 1974. Hawking radiation is a purely kinematic effect that is generic to Lorentzian geometries containing event horizons or local apparent horizons.

Hawking radiation reduces the mass and rotational energy of black holes and is therefore also theorized to cause black hole evaporation. Because of this, black holes that do not gain mass through other means are expected to shrink and ultimately vanish. For all except the smallest black holes, this would happen extremely slowly. The radiation temperature is inversely proportional to the black hole's mass, so micro black holes are predicted to be larger emitters of radiation than larger black holes and should dissipate faster.

Overview

Black holes are astrophysical objects of interest primarily because of their compact size and immense gravitational attraction. They were first predicted by Einstein's 1915 theory of general relativity, before astrophysical evidence began to mount half a century later.

A black hole can form when enough matter or energy is compressed into a volume small enough that the escape velocity is greater than the speed of light. Nothing can travel that fast, so nothing within a certain distance, proportional to the mass of the black hole, can escape beyond that distance. The region beyond which not even light can escape is the event horizon; an observer outside it cannot observe, become aware of, or be affected by events within the event horizon. The essence of a black hole is its event horizon, a theoretical demarcation between events and their causal relationships.

Picture of space infalling into a Schwarzschild black hole at the Newtonian escape speed. Outside/inside the horizon (red), the infalling speed is less/greater than the speed of light. At the event horizon, the infalling speed equals the speed of light. Credit: Andrew Hamilton, JILA

Alternatively, using a set of infalling coordinates in general relativity, one can conceptualize the event horizon as the region beyond which space is infalling faster than the speed of light. (Although nothing can travel through space faster than light, space itself can infall at any speed.) Once matter is inside the event horizon, all of the matter inside falls inexorably into a gravitational singularity, a place of infinite curvature and zero size, leaving behind a warped spacetime devoid of any matter. A classical black hole is pure empty spacetime, and the simplest (nonrotating and uncharged) is characterized just by its mass and event horizon.

Our current understandings of quantum physics can be used to investigate what may happen in the region around the event horizon. In 1974, British physicist Stephen Hawking used quantum field theory in curved spacetime to show that in theory, the antimatter and matter fields were, instead of cancelling each other out normally, disrupted by the black hole, causing antimatter and matter particles to "blip" into existence as a result of the imbalanced matter fields, and drawing energy from the disruptor itself: the black holes (to escape), effectively draining energy from the black hole. In addition, not all of the particles were close to the event horizon, and the ones that were could not escape. In effect this energy acted as if the black hole itself was slowly evaporating (although it actually came from outside it).

An important difference between the black hole radiation as computed by Hawking and thermal radiation emitted from a black body is that the latter is statistical in nature, and only its average satisfies what is known as Planck's law of black-body radiation, while the former fits the data better. Thus, thermal radiation contains information about the body that emitted it, while Hawking radiation seems to contain no such information, and depends only on the mass, angular momentum, and charge of the black hole (the no-hair theorem). This leads to the black hole information paradox.

However, according to the conjectured gauge-gravity duality (also known as the AdS/CFT correspondence), black holes in certain cases (and perhaps in general) are equivalent to solutions of quantum field theory at a non-zero temperature. This means that no information loss is expected in black holes (since the theory permits no such loss) and the radiation emitted by a black hole is probably the usual thermal radiation. If this is correct, then Hawking's original calculation should be corrected, though it is not known how (see below).

A black hole of one solar mass (M_☉) has a temperature of only 60 nanokelvins (60 billionths of a kelvin); in fact, such a black hole would absorb far more cosmic microwave background radiation than it emits. A black hole of 4.5×10²² kg (about the mass of the Moon, or about 133 μm across) would be in equilibrium at 2.7 K, absorbing as much radiation as it emits.

Formulation

Hawking's discovery followed a visit to Moscow in 1973, where Soviet scientists Yakov Zel'dovich and Alexei Starobinsky convinced him that rotating black holes ought to create and emit particles. Even still, Russian physicist Vladimir Gribov believed that even a non-rotating black hole should emit radiation. Hawking would find this to be true once he did the calculation himself. In 1972, Jacob Bekenstein conjectured that the black holes should have an entropy, where by the same year, he proposed no-hair theorems. Bekenstein's discovery and results are commended by Stephen Hawking, leading him to think about radiation due to this formalism.

According to the physicist Dmitri Diakonov, there was an argument between Zeldovich and Vladimir Gribov at the Zeldovich Moscow 1972–1973 seminar. Zeldovich believed that only a rotating black hole could emit radiation, while Gribov believed that even a non-rotating black hole emits radiation due to the laws of quantum mechanics. This account is confirmed by Gribov's obituary in the Physics-Uspekhi by Vitaly Ginzburg and others.

Emission process

Hawking radiation is dependent on the Unruh effect and the equivalence principle applied to black-hole horizons. Close to the event horizon of a black hole, a local observer must accelerate to keep from falling in. An accelerating observer sees a thermal bath of particles that pop out of the local acceleration horizon, turn around, and free-fall back in. The condition of local thermal equilibrium implies that the consistent extension of this local thermal bath has a finite temperature at infinity, which implies that some of these particles emitted by the horizon are not reabsorbed and become outgoing Hawking radiation.

A Schwarzschild black hole has a metric

${\displaystyle (\mathrm{d}s{)}^2=-\left(1-\frac{2M}{r}\right)(\mathrm{d}t{)}^2+\frac{1}{\left(1-\frac{2M}{r}\right)}(\mathrm{d}r{)}^2+r^2(\mathrm{d}\mathrm{\Omega }{)}^2.}$

The black hole is the background spacetime for a quantum field theory.

The field theory is defined by a local path integral, so if the boundary conditions at the horizon are determined, the state of the field outside will be specified. To find the appropriate boundary conditions, consider a stationary observer just outside the horizon at position

${\displaystyle r=2M+\frac{{\rho }^2}{8M}.}$

The local metric to lowest order is

${\displaystyle (\mathrm{d}s{)}^2=-{\left(\frac{\rho }{4M}\right)}^2(\mathrm{d}t{)}^2+(\mathrm{d}\rho {)}^2+(\mathrm{d}{X}_{\perp }{)}^2=-{\rho }^2(\mathrm{d}\tau {)}^2+(\mathrm{d}\rho {)}^2+(\mathrm{d}{X}_{\perp }{)}^2,}$

which is Rindler in terms of τ = t/4M. The metric describes a frame that is accelerating to keep from falling into the black hole. The local acceleration, α = 1/ρ, diverges as ρ → 0.

The horizon is not a special boundary, and objects can fall in. So the local observer should feel accelerated in ordinary Minkowski space by the principle of equivalence. The near-horizon observer must see the field excited at a local temperature

${\displaystyle T=\frac{\alpha }{2\pi }=\frac{1}{2\pi \rho }=\frac{1}{4\pi \sqrt{2Mr\left(1-\frac{2M}{r}\right)}},}$

which is the Unruh effect.

The gravitational redshift is given by the square root of the time component of the metric. So for the field theory state to consistently extend, there must be a thermal background everywhere with the local temperature redshift-matched to the near horizon temperature:

${\displaystyle T(r^{\prime })=\frac{1}{4\pi \sqrt{2Mr\left(1-\frac{2M}{r}\right)}}\sqrt{\frac{1-\frac{2M}{r}}{1-\frac{2M}{r^{\prime }}}}=\frac{1}{4\pi \sqrt{2Mr\left(1-\frac{2M}{r^{\prime }}\right)}}.}$

The inverse temperature redshifted to r′ at infinity is

${\displaystyle T(\mathrm{\infty })=\frac{1}{4\pi \sqrt{2Mr}},}$

and r is the near-horizon position, near 2M, so this is really

${\displaystyle T(\mathrm{\infty })=\frac{1}{8\pi M}.}$

Thus a field theory defined on a black-hole background is in a thermal state whose temperature at infinity is

${\displaystyle T_{\text{H}}=\frac{1}{8\pi M}.}$

From the black-hole temperature, it is straightforward to calculate the black-hole entropy S. The change in entropy when a quantity of heat dQ is added is

${\displaystyle \mathrm{d}S=\frac{\mathrm{d}Q}{T}=8\pi M\mathrm{d}Q.}$

The heat energy that enters serves to increase the total mass, so

${\displaystyle \mathrm{d}S=8\pi M\mathrm{d}M=\mathrm{d}(4\pi M^2).}$

The radius of a black hole is twice its mass in Planck units, so the entropy of a black hole is proportional to its surface area:

${\displaystyle S=\pi R^2=\frac{A}{4}.}$

Assuming that a small black hole has zero entropy, the integration constant is zero. Forming a black hole is the most efficient way to compress mass into a region, and this entropy is also a bound on the information content of any sphere in space time. The form of the result strongly suggests that the physical description of a gravitating theory can be somehow encoded onto a bounding surface.

Black hole evaporation

When particles escape, the black hole loses a small amount of its energy and therefore some of its mass (mass and energy are related by Einstein's equation E = mc²). Consequently, an evaporating black hole will have a finite lifespan. By dimensional analysis, the life span of a black hole can be shown to scale as the cube of its initial mass, and Hawking estimated that any black hole formed in the early universe with a mass of less than approximately 10¹⁵ g would have evaporated completely by the present day.

In 1976, Don Page refined this estimate by calculating the power produced, and the time to evaporation, for a non-rotating, non-charged Schwarzschild black hole of mass M. The time for the event horizon or entropy of a black hole to halve is known as the Page time. The calculations are complicated by the fact that a black hole, being of finite size, is not a perfect black body; the absorption cross section goes down in a complicated, spin-dependent manner as frequency decreases, especially when the wavelength becomes comparable to the size of the event horizon. Page concluded that primordial black holes could only survive to the present day if their initial mass were roughly 4×10¹¹ kg or larger. Writing in 1976, Page using the understanding of neutrinos at the time erroneously worked on the assumption that neutrinos have no mass and that only two neutrino flavors exist, and therefore his results of black hole lifetimes do not match the modern results which take into account 3 flavors of neutrinos with nonzero masses. A 2008 calculation using the particle content of the Standard Model and the WMAP figure for the age of the universe yielded a mass bound of (5.00±0.04)×10¹¹ kg.

If black holes evaporate under Hawking radiation, a solar mass black hole will evaporate over 10⁶⁴ years which is vastly longer than the age of the universe. A supermassive black hole with a mass of 10¹¹ (100 billion) M_☉ will evaporate in around 2×10¹⁰⁰ years. Some monster black holes in the universe are predicted to continue to grow up to perhaps 10¹⁴ M_☉ during the collapse of superclusters of galaxies. Even these would evaporate over a timescale of up to 2 × 10¹⁰⁶ years.

The power emitted by a black hole in the form of Hawking radiation can be estimated for the simplest case of a nonrotating, non-charged Schwarzschild black hole of mass M. Combining the formulas for the Schwarzschild radius of the black hole, the Stefan–Boltzmann law of blackbody radiation, the above formula for the temperature of the radiation, and the formula for the surface area of a sphere (the black hole's event horizon), several equations can be derived.

The Hawking radiation temperature is:

${\displaystyle T_{\mathrm{H}}=\frac{\hslash c^3}{8\pi GM{k}_{\mathrm{B}}}}$

The Bekenstein–Hawking luminosity of a black hole, under the assumption of pure photon emission (i.e. that no other particles are emitted) and under the assumption that the horizon is the radiating surface is:

${\displaystyle P=\frac{\hslash c^6}{15360\pi G^2{M}^2}}$

where P is the luminosity, i.e., the radiated power, ħ is the reduced Planck constant, c is the speed of light, G is the gravitational constant and M is the mass of the black hole. It is worth mentioning that the above formula has not yet been derived in the framework of semiclassical gravity.

The time that the black hole takes to dissipate is:

${\displaystyle t_{\mathrm{e}\mathrm{v}}=\frac{5120\pi G^2{M}^3}{\hslash c^4}=\frac{480c^{2}V}{\hslash G}\approx 2.1\times {10}^{67}\text{years}{\left(\frac{M}{M_{\odot }}\right)}^3,}$

where M and V are the mass and (Schwarzschild) volume of the black hole. A black hole of one solar mass (M_☉ = 2.0×10³⁰ kg) takes more than 10⁶⁷ years to evaporate—much longer than the current age of the universe at 1.4×10¹⁰ years. But for a black hole of 10¹¹ kg, the evaporation time is 2.6×10⁹ years. This is why some astronomers are searching for signs of exploding primordial black holes.

However, since the universe contains the cosmic microwave background radiation, in order for the black hole to dissipate, the black hole must have a temperature greater than that of the present-day blackbody radiation of the universe of 2.7 K. A study suggests that M must be less than 0.8% of the mass of the Earth – approximately the mass of the Moon.

Black hole evaporation has several significant consequences:

• Black hole evaporation produces a more consistent view of black hole thermodynamics by showing how black holes interact thermally with the rest of the universe.
• Unlike most objects, a black hole's temperature increases as it radiates away mass. The rate of temperature increase is exponential, with the most likely endpoint being the dissolution of the black hole in a violent burst of gamma rays. A complete description of this dissolution requires a model of quantum gravity, however, as it occurs when the black hole's mass approaches 1 Planck mass, when its radius will also approach two Planck lengths.
• The simplest models of black hole evaporation lead to the black hole information paradox. The information content of a black hole appears to be lost when it dissipates, as under these models the Hawking radiation is random (it has no relation to the original information). A number of solutions to this problem have been proposed, including suggestions that Hawking radiation is perturbed to contain the missing information, that the Hawking evaporation leaves some form of remnant particle containing the missing information, and that information is allowed to be lost under these conditions.

Problems and extensions

Trans-Planckian problem

The trans-Planckian problem is the issue that Hawking's original calculation includes quantum particles where the wavelength becomes shorter than the Planck length near the black hole's horizon. This is due to the peculiar behavior there, where time stops as measured from far away. A particle emitted from a black hole with a finite frequency, if traced back to the horizon, must have had an infinite frequency, and therefore a trans-Planckian wavelength.

The Unruh effect and the Hawking effect both talk about field modes in the superficially stationary spacetime that change frequency relative to other coordinates that are regular across the horizon. This is necessarily so, since to stay outside a horizon requires acceleration that constantly Doppler shifts the modes.

An outgoing photon of Hawking radiation, if the mode is traced back in time, has a frequency that diverges from that which it has at great distance, as it gets closer to the horizon, which requires the wavelength of the photon to "scrunch up" infinitely at the horizon of the black hole. In a maximally extended external Schwarzschild solution, that photon's frequency stays regular only if the mode is extended back into the past region where no observer can go. That region seems to be unobservable and is physically suspect, so Hawking used a black hole solution without a past region that forms at a finite time in the past. In that case, the source of all the outgoing photons can be identified: a microscopic point right at the moment that the black hole first formed.

The quantum fluctuations at that tiny point, in Hawking's original calculation, contain all the outgoing radiation. The modes that eventually contain the outgoing radiation at long times are redshifted by such a huge amount by their long sojourn next to the event horizon that they start off as modes with a wavelength much shorter than the Planck length. Since the laws of physics at such short distances are unknown, some find Hawking's original calculation unconvincing.

The trans-Planckian problem is nowadays mostly considered a mathematical artifact of horizon calculations. The same effect occurs for regular matter falling onto a white hole solution. Matter that falls on the white hole accumulates on it, but has no future region into which it can go. Tracing the future of this matter, it is compressed onto the final singular endpoint of the white hole evolution, into a trans-Planckian region. The reason for these types of divergences is that modes that end at the horizon from the point of view of outside coordinates are singular in frequency there. The only way to determine what happens classically is to extend in some other coordinates that cross the horizon.

There exist alternative physical pictures that give the Hawking radiation in which the trans-Planckian problem is addressed. The key point is that similar trans-Planckian problems occur when the modes occupied with Unruh radiation are traced back in time. In the Unruh effect, the magnitude of the temperature can be calculated from ordinary Minkowski field theory, and is not controversial.

Large extra dimensions

The formulas from the previous section are applicable only if the laws of gravity are approximately valid all the way down to the Planck scale. In particular, for black holes with masses below the Planck mass (~10⁻⁸ kg), they result in impossible lifetimes below the Planck time (~10⁻⁴³ s). This is normally seen as an indication that the Planck mass is the lower limit on the mass of a black hole.

In a model with large extra dimensions (10 or 11), the values of Planck constants can be radically different, and the formulas for Hawking radiation have to be modified as well. In particular, the lifetime of a micro black hole with a radius below the scale of the extra dimensions is given by equation 9 in Cheung (2002) and equations 25 and 26 in Carr (2005).

${\displaystyle \tau \sim \frac{1}{M_{\ast }}{\left(\frac{M_{\text{BH}}}{M_{\ast }}\right)}^{\frac{n+3}{n+1}},}$

where M_∗ is the low-energy scale, which could be as low as a few TeV, and n is the number of large extra dimensions. This formula is now consistent with black holes as light as a few TeV, with lifetimes on the order of the "new Planck time" ~10⁻²⁶ s.

In loop quantum gravity

A detailed study of the quantum geometry of a black hole event horizon has been made using loop quantum gravity. Loop-quantization does not reproduce the result for black hole entropy originally discovered by Bekenstein and Hawking, unless the value of a free parameter is set to cancel out various constants such that the Bekenstein–Hawking entropy formula is reproduced. However, quantum gravitational corrections to the entropy and radiation of black holes have been computed based on the theory.

Based on the fluctuations of the horizon area, a quantum black hole exhibits deviations from the Hawking radiation spectrum that would be observable were X-rays from Hawking radiation of evaporating primordial black holes to be observed. The quantum effects are centered at a set of discrete and unblended frequencies highly pronounced on top of the Hawking spectrum.

Experimental observation

Astronomical search

In June 2008, NASA launched the Fermi space telescope, which is searching for the terminal gamma-ray flashes expected from evaporating primordial black holes. As of Jan 1st, 2023, none have been detected.

Heavy-ion collider physics

If speculative large extra dimension theories are correct, then CERN's Large Hadron Collider may be able to create micro black holes and observe their evaporation. No such micro black hole has been observed at CERN.

Experimental

Under experimentally achievable conditions for gravitational systems, this effect is too small to be observed directly. It was predicted that Hawking radiation could be studied by analogy using sonic black holes, in which sound perturbations are analogous to light in a gravitational black hole and the flow of an approximately perfect fluid is analogous to gravity (see Analog models of gravity). Observations of Hawking radiation were reported, in sonic black holes employing Bose–Einstein condensates.

In September 2010 an experimental set-up created a laboratory "white hole event horizon" that the experimenters claimed was shown to radiate an optical analog to Hawking radiation. However, the results remain unverified and debatable, and its status as a genuine confirmation remains in doubt.

Black hole information paradox

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Black_hole_information_paradox 

 

The first image (silhouette or shadow) of a black hole, taken of the supermassive black hole in M87 with the Event Horizon Telescope, released in April 2019.

The black hole information paradox is a puzzle that appears when the predictions of quantum mechanics and general relativity are combined. The theory of general relativity predicts the existence of black holes that are regions of spacetime from which nothing — not even light — can escape. In the 1970s, Stephen Hawking applied the rules of quantum mechanics to such systems and found that an isolated black hole would emit a form of radiation called Hawking radiation. Hawking also argued that the detailed form of the radiation would be independent of the initial state of the black hole and would depend only on its mass, electric charge and angular momentum. The information paradox appears when one considers a process in which a black hole is formed through a physical process and then evaporates away entirely through Hawking radiation. Hawking's calculation suggests that the final state of radiation would retain information only about the total mass, electric charge and angular momentum of the initial state. Since many different states can have the same mass, charge and angular momentum this suggests that many initial physical states could evolve into the same final state. Therefore, information about the details of the initial state would be permanently lost. However, this violates a core precept of both classical and quantum physics—that, in principle, the state of a system at one point in time should determine its value at any other time. Specifically, in quantum mechanics the state of the system is encoded by its wave function. The evolution of the wave function is determined by a unitary operator, and unitarity implies that the wave function at any instant of time can be used to determine the wave function either in the past or the future.

It is now generally believed that information is preserved in black-hole evaporation. This means that the predictions of quantum mechanics are correct whereas Hawking's original argument that relied on general relativity must be corrected. However, views differ as to how, precisely, Hawking's calculation should be corrected. In recent years, several extensions of the original paradox have been explored. Taken together these puzzles about black hole evaporation have implications for how gravity and quantum mechanics must be combined, leading to the information paradox remaining an active field of research within quantum gravity.

Relevant principles

In quantum mechanics, the evolution of the state is governed by the Schrödinger equation. The Schrödinger equation obeys two principles that are relevant for the paradox. These are the principles of quantum determinism, which means that given a present wave function, its future changes are uniquely determined by the evolution operator and also the principle of reversibility, which refers to the fact that the evolution operator has an inverse, meaning that the past wave functions are similarly unique. The combination of the two means that information must always be preserved. In this context "information" is used to refer to all the details of the state and the statement that information must be preserved means that details corresponding to an earlier time can always be reconstructed at a later time.

Mathematically, the Schrödinger equation implies that the wavefunction at a time t₁ can be related to the wavefunction at a time t₂ by means of a unitary operator.

$${\displaystyle |\mathrm{\Psi }(t_1)⟩=U(t_1,t_2)|\mathrm{\Psi }(t_2)⟩.}$$

Since the unitary operator is bijective, the wavefunction at t₂ can be obtained from the wavefunction at t₁ and vice versa.

The reversibility of time-evolution described above applies only at the microscopic level since the wavefunction provides a complete description of the state. It should not be conflated with thermodynamic irreversibility. A process may appear to be irreversible if one keeps track only of coarse-grained features of the system and does not keep track of its microscopic details, as is usually done in thermodynamics. However, at the microscopic level the principles of quantum mechanics imply that every process is completely reversible.

Starting in the mid-1970s, Stephen Hawking and Jacob Bekenstein put forward theoretical arguments that suggested that black-hole evaporation loses information, and is therefore inconsistent with unitarity. Crucially, these arguments were meant to apply at the microscopic level and suggested that black-hole evaporation is not only thermodynamically irreversible but microscopically irreversible. This contradicts the principle of unitarity described above and leads to the information paradox. Since the paradox suggested that quantum mechanics would be violated by black-hole formation and evaporation, Hawking framed the paradox in terms of the "breakdown of predictability in gravitational collapse".

The arguments for microscopic irreversibility were backed by Hawking's calculation of the spectrum of radiation emitted by isolated black holes. This calculation utilized the framework of general relativity and quantum field theory. The calculation of Hawking radiation is performed at the black-hole horizon; for a large enough black hole the curvature at the horizon is small and therefore both these theories should be valid. Hawking relied on the no-hair theorem to arrive at the conclusion that radiation emitted by black holes would depend only on a few macroscopic parameters such as the black hole's mass, charge and spin and not on the details of the initial state that led to the formation of the black hole. Moreover, the argument for information loss relied on the causal structure of the black-hole spacetime, which suggests that information in the interior should not affect any observation in the exterior including observations performed on the radiation emitted by the black hole. If so, the region of spacetime outside the black hole would lose information about the state of the interior after black-hole evaporation leading to the loss of information.

Today, some physicists believe that the holographic principle (specifically the AdS/CFT duality) demonstrates that Hawking's conclusion was incorrect, and that information is in fact preserved.

Black hole evaporation

Main article: Hawking radiation

 

The Penrose diagram of a black hole which forms, and then completely evaporates away. Time shown on vertical axis from bottom to top; space shown on horizontal axis from left (radius zero) to right (growing radius).

In 1973–1975, Stephen Hawking showed that black holes should slowly radiate away energy and Hawking later argued that this leads to a contradiction with unitarity. Hawking used the classical no-hair theorem to argue that the form of this radiation – called Hawking radiation – would be completely independent of the initial state of the star or matter that collapsed to form the black hole. Hawking argued that the process of radiation would continue until the black hole had evaporated completely. At the end of this process, all the initial energy in the black hole would have been transferred to the radiation. But, according to Hawking's argument, the radiation would retain no information about the initial state and therefore information about the initial state would be lost.

More specifically, Hawking argued that the pattern of radiation emitted from the black hole would be random, with a probability distribution controlled only by the initial temperature, charge and angular momentum of the black hole and not by the initial state of the collapse. The state produced by such a probabilistic process is called a mixed state in quantum mechanics. Therefore, Hawking argued that if the star or material that collapsed to form the black hole started in a specific pure quantum state, the process of evaporation would transform the pure state into a mixed state. This is inconsistent with the unitarity of quantum-mechanical evolution discussed above.

The loss of information can be quantified in terms of the change in the fine-grained von Neumann entropy of the state. A pure state is assigned a von Neumann entropy of 0 whereas a mixed state has a finite entropy. The unitary evolution of a state according to Schrödinger's equation preserves the entropy. Therefore Hawking's argument suggests that the process of black-hole evaporation cannot be described within the framework of unitary evolution. Although this paradox is often phrased in terms of quantum mechanics, the evolution from a pure state to a mixed state is also inconsistent with Liouville's theorem in classical physics.

In equations, Hawking showed that if one denotes the creation and annihilation operators at a frequency ${\displaystyle \omega }$ for a quantum field propagating in the black-hole background by ${\displaystyle a_{\omega }}$ and ${\displaystyle a_{\omega }^{†}}$ then the expectation value of the product of these operators in the state formed by the collapse of a black hole would satisfy

$${\displaystyle ⟨a_{\omega }{a}_{\omega }^{†}{⟩}_{\mathrm{h}\mathrm{a}\mathrm{w}\mathrm{k}}=\frac{1}{1-e^{-\beta \omega }}}$$

where ${\displaystyle \beta =1/(kT)}$ where k is the Boltzmann constant and T is the temperature of the black hole. There are two important aspects of the formula above. The first is that the form of the radiation depends only on a single parameter, the temperature, even though the initial state of the black hole cannot be characterized by one parameter. Second, the formula implies that the black hole radiates mass at a rate given by

$${\displaystyle \frac{dM}{dt}=-a{T}^4}$$

where a is constant related to fundamental constants, including the Stefan–Boltzmann constant and certain properties of the black hole spacetime called its greybody factors.

The temperature of the black hole is in turn dependent on the mass, charge and angular momentum of the black hole. For a Schwarzschild black hole the temperature is given by

$${\displaystyle T=\frac{\hslash c^3}{8\pi kGM}}$$

This means that if the black hole starts out with an initial mass ${\displaystyle M_0}$, it evaporates completely in a time proportional to ${\displaystyle M_0^3}$.

The important aspect of the formulas above is they suggest that the final gas of radiation formed through this process depends only on the black hole's temperature and is independent of other details of the initial state. This leads to the following paradox. Consider two distinct initial states that collapse to form a Schwarzschild black hole of the same mass. Even though the states were distinct to start with, since the mass (and, hence, the temperature) of the black holes is the same, they will emit the same Hawking radiation. Once the black holes evaporate completely, in both cases, one will be left with a featureless gas of radiation. This gas cannot be used to distinguish between the two initial states, and therefore information has been lost.

It is now widely believed that the reasoning leading to the paradox above is flawed. Several solutions have been put forward that are reviewed below.

Popular culture

The information paradox has received coverage in the popular media and has been described in popular-science books. Some of this coverage resulted from a widely publicized bet made in 1997 between John Preskill on the one hand with Hawking and Kip Thorne on the other that information was not lost in black holes. The scientific debate on the paradox was described in a popular book published in 2008 by Leonard Susskind called The Black Hole War. (The book carefully notes that the 'war' was purely a scientific one, and that at a personal level, the participants remained friends.) The book states that Hawking was eventually persuaded that black-hole evaporation was unitary by the holographic principle, which was first proposed by 't Hooft, further developed by Susskind and later given a precise string theory interpretation by the AdS/CFT correspondence. In 2004, Hawking also conceded the 1997 bet, paying Preskill with a baseball encyclopedia "from which information can be retrieved at will" although Thorne refused to concede.

Solutions

Since the 1997 proposal of the AdS/CFT correspondence, the predominant belief among physicists is that information is indeed preserved in black hole evaporation. However, there are broadly two main streams of thought about how this happens. Within, what might broadly be termed, the "string theory community", the dominant idea is that Hawking radiation is not precisely thermal but receives quantum correlations that encode information about the black hole's interior. This viewpoint has been the subject of extensive recent research and received further support in 2019 when researchers amended the computation of the entropy of the Hawking radiation in certain models, and showed that the radiation is in fact dual to the black hole interior at late times. Hawking himself was influenced by this view and in 2004, published a paper that assumed the AdS/CFT correspondence and argued that quantum perturbations of the event horizon could allow information to escape from a black hole, which would resolve the information paradox. In this perspective, it is the event horizon of the black hole that is important and not the black-hole singularity.

On the other hand, within, what might broadly be termed, the "loop quantum gravity community", the dominant belief is that to resolve the information paradox, it is important to understand how the black-hole singularity is resolved. These scenarios are broadly called remnant scenarios since information does not emerge gradually but remains in the black-hole interior only to emerge at the end of black-hole evaporation.

Other possibilities are also studied by researchers, which include a modification of the laws of quantum mechanics to allow for non-unitary time evolution.

Some of these solutions are described at greater length below.

Small-corrections resolution to the paradox

This idea suggests that Hawking's computation fails to keep track of small corrections that are eventually sufficient to preserve information about the initial state. This can be thought of as being analogous to what happens during the mundane process of "burning": the radiation produced appears to be thermal but its fine-grained features encode the precise details of the object that was burnt. This idea is consistent with reversibility, as required by quantum mechanics. It is the dominant idea in, what might broadly be termed, the string-theory approach to quantum gravity.

More precisely, this line of resolution suggests that Hawking's computation is corrected so that the two point correlator computed by Hawking and described above becomes

$${\displaystyle ⟨a_{\omega }{a}_{\omega }^{†}{⟩}_{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}=⟨a_{\omega }{a}_{\omega }^{†}{⟩}_{\mathrm{h}\mathrm{a}\mathrm{w}\mathrm{k}}(1+{ϵ}_2)}$$

and higher-point correlators are similarly corrected

$${\displaystyle ⟨a_{{\omega }_1}{a}_{{\omega }_1}^{†}{a}_{{\omega }_2}{a}_{{\omega }_2}^{†}\dots a_{{\omega }_n}{a}_{{\omega }_n}^{†}{⟩}_{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}=⟨a_{\omega }{a}_{\omega }^{†}{⟩}_{\mathrm{h}\mathrm{a}\mathrm{w}\mathrm{k}}(1+{ϵ}_n)}$$

The equations above utilize a concise notation and the correction factors ${\displaystyle {ϵ}_i}$ may depend on the temperature, the frequencies of the operators that enter the correlation function and other details of the black hole.

Such corrections were initially explored by Maldacena in a simple version of the paradox. They were then analyzed by Papadodimas and Raju who showed that corrections to low-point correlators (such as ${\displaystyle {ϵ}_2}$ above ) that were exponentially suppressed in the black-hole entropy were sufficient to preserve unitarity and significant corrections were required only for very high point correlators. The mechanism that allowed the right small corrections to form was initially postulated in terms of a loss of exact locality in quantum gravity so that the black-hole interior and the radiation were described by the same degrees of freedom. Recent developments suggest that such a mechanism can be realized precisely within semiclassical gravity and allows information to escape. See § Recent developments.

Fuzzball resolution to the paradox

Some researchers, most notably Samir Mathur, have argued that the small corrections required to preserve information cannot be obtained while preserving the semiclassical form of the black-hole interior and instead require a modification of the black-hole geometry to a fuzzball.

The defining characteristic of the fuzzball is that it has structure at the horizon scale. This should be contrasted with the conventional picture of the black-hole interior as a largely-featureless region of space. For a large enough black hole, tidal effects are very small at the black-hole horizon and remain small in the interior until one approaches the black-hole singularity. Therefore, in the conventional picture, an observer who crosses the horizon may not even realize that they have done so until they start approaching the singularity. In contrast, the fuzzball proposal suggests that the black hole horizon is not empty. Consequently, it is also not information free since the details of the structure at the surface of the horizon preserve information about the initial state of the black hole. This structure also affects the outgoing Hawking radiation and thereby allows information to escape from the fuzzball.

The fuzzball proposal is supported by the existence of a large number of gravitational solutions called microstate geometries.

The firewall proposal can be thought of as a variant of the fuzzball proposal except that it posits that the black-hole interior is replaced with a firewall rather than a fuzzball. Operationally, the difference between the fuzzball and the firewall proposals has to do with whether an observer crossing the horizon of the black hole encounters high-energy matter, suggested by the firewall proposal, or merely low-energy structure, suggested by the fuzzball proposal. The firewall proposal also originated with an exploration of Mathur's argument that small corrections are insufficient to resolve the information paradox.

The fuzzball and firewall proposals have been questioned for lacking an appropriate mechanism that can generate structure at the horizon scale.

Strong-quantum-effects resolution to the paradox

In the final stages of black-hole evaporation, quantum effects become important and cannot be ignored. The precise understanding of this phase of black-hole evaporation requires a complete theory of quantum gravity. Within, what might be termed, the loop-quantum-gravity approach to black holes, it is believed that understanding this phase of evaporation is crucial to resolving the information paradox.

This perspective holds that Hawking's computation is reliable until the final stages of black-hole evaporation when information suddenly escapes. An alternative possibility along the same lines is that black-hole evaporation might simply stop when the black hole becomes Planck-sized. Such scenarios are called "remnant scenarios".

An appealing aspect of this perspective is that a significant deviation from classical and semiclassical gravity is needed only in the regime in which the effects of quantum gravity are expected to dominate. On the other hand, this idea implies that just before the sudden escape of information, a very small black hole must be able to store an arbitrary amount of information and have a very large number of internal states. Therefore, researchers who follow this idea must take care to avoid the common criticism of remnant-type scenarios, which is that they might may violate the Bekenstein bound and lead to a violation of effective field theory due to the production of remnants as virtual particles in ordinary scattering events.

Soft-hair resolution to the paradox

In 2016, Hawking, Perry and Strominger noted that black holes must contain "soft hair". Particles that have no rest mass, like photons and gravitons, can exist with arbitrarily low-energy and are called soft particles. The soft-hair resolution posits that information about the initial state is stored in such soft particles. The existence of such soft hair is a peculiarity of four-dimensional asymptotically flat space and therefore this resolution to the paradox does not carry over to black holes in anti-de Sitter space or black holes in other dimensions.

Information is irretrievably lost

A minority view within the theoretical physics community is that information is genuinely lost when black holes form and evaporate. This conclusion follows if one assumes that the predictions of semiclassical gravity and the causal structure of the black-hole spacetime are exact.

However, this conclusion leads to the loss of unitarity. Banks, Susskind and Peskin argued that, in some cases, loss of unitarity also implies violation of energy–momentum conservation or locality, but this argument may possibly be evaded in systems with a large number of degrees of freedom. According to Roger Penrose, loss of unitarity in quantum systems is not a problem: quantum measurements are by themselves already non-unitary. Penrose claims that quantum systems will in fact no longer evolve unitarily as soon as gravitation comes into play, precisely as in black holes. The Conformal Cyclic Cosmology advocated by Penrose critically depends on the condition that information is in fact lost in black holes. This new cosmological model might in the future be tested experimentally by detailed analysis of the cosmic microwave background radiation (CMB): if true, the CMB should exhibit circular patterns with slightly lower or slightly higher temperatures. In November 2010, Penrose and V. G. Gurzadyan announced they had found evidence of such circular patterns, in data from the Wilkinson Microwave Anisotropy Probe (WMAP) corroborated by data from the BOOMERanG experiment. The significance of the findings was subsequently debated by others.

Along similar lines, Modak, Ortíz, Peña and Sudarsky, have argued that the paradox can be dissolved by invoking foundational issues of quantum theory often referred as the measurement problem of quantum mechanics. This work was built on an earlier proposal by Okon and Sudarsky on the benefits of objective collapse theory in a much broader context. The original motivation of these studies was the long-standing proposal of Roger Penrose wherein collapse of the wave-function is said to be inevitable in the presence of black holes (and even under the influence of gravitational field). Experimental verification of collapse theories is an ongoing effort.

Other proposed resolutions

Some other resolutions to the paradox have also been explored. These are listed briefly below.

• Information is stored in a large remnant

This idea suggests that Hawking radiation stops before the black hole reaches the Planck size. Since the black hole never evaporates, information about its initial state can remain inside the black hole and the paradox disappears. However, there is no accepted mechanism that would allow Hawking radiation to stop while the black hole remains macroscopic.

• Information is stored in a baby universe that separates from our own universe.

Some models of gravity, such as the Einstein–Cartan theory of gravity which extends general relativity to matter with intrinsic angular momentum (spin) predict the formation of such baby universes. No violation of known general principles of physics is needed. There are no physical constraints on the number of the universes, even though only one remains observable. However, it is difficult to test the Einstein–Cartan theory because its predictions are significantly different from general-relativistic ones only at extremely high densities.

• Information is encoded in the correlations between future and past

The final-state proposal suggests that boundary conditions must be imposed at the black-hole singularity which, from a causal perspective, is to the future of all events in the black-hole interior. This helps to reconcile black-hole evaporation with unitarity but it contradicts the intuitive idea of causality and locality of time-evolution.

• quantum-channel theory

In 2014, Chris Adami argued that analysis using quantum channel theory causes any apparent paradox to disappear; Adami rejects black hole complementarity, arguing instead that no space-like surface contains duplicated quantum information.

Recent developments

Significant progress was made in 2019, when starting with work by Penington and Almheiri, Engelhardt, Marolf and Maxfield, researchers were able to compute the von Neumann entropy of the radiation emitted by black holes in specific models of quantum gravity. These calculations showed that, in these models, the entropy of this radiation first rises and then falls back to zero. As explained above, one way to frame the information paradox is that Hawking's calculation appears to show that the von Neumann entropy of Hawking radiation increases throughout the lifetime of the black hole. However, if the black hole formed from a pure state with zero entropy, unitarity implies that the entropy of the Hawking radiation must decrease back to zero once the black hole evaporates completely. Therefore, the results above provide a resolution to the information paradox, at least in the specific models of gravity considered in these models.

These calculations compute the entropy by first analytically continuing the spacetime to a Euclidean spacetime and then using the replica trick. The path integral that computes the entropy receives contributions from novel Euclidean configurations called "replica wormholes". (These wormholes exist in a Wick rotated spacetime and should not be conflated with wormholes in the original spacetime.) The inclusion of these wormhole geometries in the computation prevents the entropy from increasing indefinitely.

These calculations also imply that for sufficiently old black holes, one can perform operations on the Hawking radiation that affect the black hole interior. This result has implications for the related firewall paradox, and provides evidence for the physical picture suggested by the ER=EPR proposal, black hole complementarity and the Papadodimas–Raju proposal.

It has been noted that the models used to perform the Page curve computations above have consistently involved theories where the graviton itself has a mass, unlike the real world where the graviton is massless. These models have also involved a "nongravitational bath", which can be thought of as an artificial interface where gravity ceases to act. It has also been argued that a key technique used in the Page-curve computations, called the "island proposal", would be inconsistent in standard theories of gravity with a Gauss law. This would suggest that the Page curve computations are inapplicable to realistic black holes and only work in special toy models of gravity. The validity or otherwise of these criticisms remains under investigation and there is no general agreement in the research community.

In 2020, Laddha, Prabhu, Raju and Shrivastava argued that, as a result of the effects of quantum gravity, information should always be available outside the black hole. This would imply that the von Neumann entropy of the region outside the black hole always remains zero, as opposed to the proposal above, where the von Neumann entropy first rises and then falls. Extending this, Raju argued that Hawking's error was to assume that the region outside the black hole would have no information about its interior.

Hawking formalized this assumption in terms of a "principle of ignorance". The principle of ignorance is correct in classical gravity, when quantum-mechanical effects are neglected, by virtue of the no-hair theorem. It is also correct when only quantum-mechanical effects are considered but gravitational effects are neglected. But Raju has argued that, when both quantum mechanical and gravitational effects are accounted for, the principle of ignorance should be replaced by a "principle of holography of information" which would imply just the opposite: all the information about the interior can be regained from the exterior through suitably precise measurements.

The two recent resolutions of the information paradox described above — via replica wormholes and the holography of information — share the common feature that observables in the black-hole interior also describe observables far from the black hole. This implies a loss of exact locality in quantum gravity. Although this loss of locality is very small, it persists over large distance scales. This feature has been challenged by some researchers.