Hawking radiation

Hawking radiation is a form of black-body radiation emitted from beyond a black hole's event horizon, a phenomenon attributed to quantum effects, as theorized by Stephen Hawking in 1974. Earlier theoretical frameworks did not anticipate this radiation, operating under the premise that electromagnetic radiation, once within the event horizon, could not egress. The predicted intensity of Hawking radiation is exceedingly low, falling many orders of magnitude below the detection capabilities of contemporary astronomical instruments.

Hawking radiation is theorized to diminish the mass and rotational energy of black holes, thereby leading to their eventual evaporation. Consequently, black holes that do not accrete additional mass are projected to contract and ultimately dissipate. This process occurs at an exceedingly slow rate for all but the most diminutive black holes. The temperature of this radiation, designated as Hawking temperature, exhibits an inverse proportionality to the black hole's mass. This implies that micro black holes are anticipated to emit radiation at a higher rate than their more massive counterparts, leading to a more rapid dissipation relative to their mass. Therefore, should primordial black holes exist, as posited by theoretical models, their mass loss would accelerate as they shrink, culminating in a terminal burst of high-energy radiation. To date, no such radiation bursts have been empirically observed.

Background

The theoretical foundation for contemporary understanding of black holes was established by Albert Einstein's 1915 theory of general relativity. Astrophysical evidence supporting the existence of these objects, now known as black holes, accumulated approximately fifty years thereafter. Their profound gravitational pull and extreme compactness render them subjects of significant contemporary scientific inquiry. Pioneering investigations into black holes were conducted by researchers including Karl Schwarzschild and John Wheeler, who initially conceptualized these entities as possessing zero entropy.

Black holes originate when a sufficient quantity of matter or energy is compressed into a volume so minuscule that the resultant escape velocity exceeds the speed of light. Given that no entity can surpass this velocity, no object situated within a specific radius—a distance directly proportional to the black hole's mass—can egress beyond that boundary. This critical boundary, from which not even light can escape, is termed the event horizon. Consequently, an external observer is fundamentally unable to perceive, comprehend, or be influenced by any occurrences transpiring within the event horizon.

From an alternative perspective, employing infalling coordinates within the framework of general relativity, the event horizon can be conceptualized as the boundary where spacetime itself is collapsing inward at a velocity exceeding the speed of light. (While no object can traverse through space faster than light, the fabric of spacetime itself is not constrained by this limit in its inward flow.) Upon crossing the event horizon, all matter inexorably descends into a gravitational singularity—a point of infinite spacetime curvature and infinitesimal volume. This process leaves behind a distorted spacetime region devoid of matter. Thus, a classical black hole is essentially pure, empty spacetime, with its simplest form (non-rotating and uncharged) defined solely by its mass and the extent of its event horizon.

Discovery

In 1971, Soviet physicists Yakov Zeldovich and Alexei Starobinsky posited that rotating black holes should generate and emit particles, drawing an analogy with electromagnetic radiation from spinning metallic spheres. Subsequently, in 1972, Jacob Bekenstein formulated a theory suggesting that black holes possess an entropy directly proportional to their surface area. Initially, Stephen Hawking opposed Bekenstein's hypothesis, considering black holes to be simplistic entities devoid of entropy. However, following a meeting with Zeldovich in Moscow in 1973, Hawking synthesized these two concepts, integrating principles from both quantum field theory and general relativity.

In his 1974 paper, Stephen Hawking theoretically demonstrated that black holes emit particles, behaving akin to a blackbody. The escape of these particles effectively depletes energy from the black hole. Hawking's prediction posits that black holes radiate a modest amount of thermal energy at a temperature defined by the equation: ${\displaystyle {\frac {\hbar c^{3}}{8\pi GMk_{B}}}}$. Here, ${\displaystyle {\hbar }}$ represents the reduced Planck constant, ${\displaystyle c}$ denotes the speed of light, ${\displaystyle G}$ is the gravitational constant, ${\displaystyle M}$ signifies the black hole's mass, and ${\displaystyle k_{B}}$ is the Boltzmann constant. Through the application of quantum field theory to black holes, Hawking concluded that these celestial objects should continuously emit thermal blackbody radiation. This phenomenon is now recognized as Hawking radiation. This theoretical framework found corroboration in earlier research by Jacob Bekenstein, who postulated that black holes possess a finite entropy directly proportional to their surface area, thereby implying the existence of a temperature. Given Bekenstein's foundational contribution to black hole entropy, this radiation is also referred to as Bekenstein–Hawking radiation. Since Hawking's initial publication, numerous researchers have independently validated this result through diverse mathematical methodologies.

Hawking radiation originates from quantum vacuum fluctuations. Specifically, a quantum fluctuation within the electromagnetic field can generate a pair of photons: one situated outside the black hole's event horizon and its counterpart located inside. The event horizon facilitates the escape of one photon in each direction.

Emission Process

The phenomenon of Hawking radiation is predicated upon the Unruh effect and the application of the equivalence principle to black hole event horizons. In the immediate vicinity of a black hole's event horizon, a localized observer must undergo acceleration to avoid being drawn into the singularity. Such an accelerating observer perceives a thermal ensemble of particles that spontaneously emerge from the local acceleration horizon, reverse direction, and subsequently free-fall back into the black hole. The principle of local thermal equilibrium suggests that the coherent extrapolation of this localized thermal bath maintains a finite temperature at infinite distances. Consequently, a subset of these particles emitted by the horizon avoids reabsorption and manifests as outgoing Hawking radiation.

A Schwarzschild black hole is characterized by a metric

( d s ) §1819§ = −<!-- − --> ( §3132§ − §3940§ M r ) ( d t ) §6667§ + §7475§ ( §8081§ − §8889§ M r ) ( d r ) §117118§ + r §127128§ ( d Ω ) §146147§ . {\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} \Omega )^{2}.}

The spacetime of a black hole serves as the foundational background for a quantum field theory.

The field theory is characterized by a local path integral, implying that the determination of boundary conditions at the horizon dictates the external field's state. To ascertain the suitable boundary conditions, one must consider a stationary observer positioned infinitesimally beyond the horizon at:

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

Expressed to the lowest order, the local metric is:

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

This metric is expressed in Rindler coordinates, with τ defined as ⁠t/§1112§M⁠. It characterizes a reference frame that accelerates to prevent its infall into the black hole. The local acceleration, α = ⁠§2324§/ρ⁠, diverges as the radial coordinate ρ approaches zero (ρ → 0).

The event horizon does not constitute a unique physical boundary, allowing objects to traverse it. Consequently, a local observer would perceive acceleration within ordinary Minkowski spacetime, consistent with the principle of equivalence. An observer situated near the horizon would detect the field exhibiting excitation at a specific local temperature.

T = α<!-- α --> §1617§ π<!-- π --> = §2829§ §3132§ π<!-- π --> ρ<!-- ρ --> = §4647§ §4950§ π §5657§ M r ( §6667§ − §7475§ M r ) , {\displaystyle T={\frac {\alpha }{2\pi }}={\frac {1}{2\pi \rho }}={\frac {1}{4\pi {\sqrt {2Mr\left(1-{\frac {2M}{r}}\right)}}}},}

This phenomenon is recognized as the Unruh effect.

The gravitational redshift is defined by the square root of the metric's temporal component. Consequently, for the field theory state to achieve consistent extension, a pervasive thermal background is essential, with its local temperature precisely redshift-matched to the temperature near the horizon:

T ( r ′ ) = §2223§ §2526§ π §3233§ M r ( §4243§ − §5051§ M r ) §7273§ − §8081§ M r §9192§ − §99100§ M r ′ = §120121§ §123124§ π §130131§ M r ( §140141§ − §148149§ M r ′ ) . {\displaystyle T(r')={\frac {1}{4\pi {\sqrt {2Mr\left(1-{\frac {2M}{r}}\right)}}}}{\sqrt {\frac {1-{\frac {2M}{r}}}{1-{\frac {2M}{r'}}}}}={\frac {1}{4\pi {\sqrt {2Mr\left(1-{\frac {2M}{r'}}\right)}}}}.}

The inverse temperature, when redshifted to a radial coordinate of r′ at an infinite distance, is expressed as:

T ( ∞<!-- ∞ --> ) = §1920§ §2223§ π §2930§ M r , {\displaystyle T(\infty )={\frac {1}{4\pi {\sqrt {2Mr}}}},}

Given that r represents the position near the event horizon, specifically in the vicinity of 2M, the expression can be refined to:

T ( ∞<!-- ∞ --> ) = §1920§ §2223§ π M . {\displaystyle T(\infty )={\frac {1}{8\pi M}}.}

Consequently, a field theory established within a black-hole spacetime background exists in a thermal state, with its temperature at infinite distance being:

T H = §1819§ §2122§ π M . {\displaystyle T_{\text{H}}={\frac {1}{8\pi M}}.}

If a black hole possessed a mass equivalent to that of Earth, its temperature would register at approximately 10⁻¹² K. The characteristic wavelength of a Hawking radiation quantum approximates the black hole's event horizon dimension.

Given the black hole's temperature, the calculation of its entropy, S, is direct. The incremental change in entropy, occurring upon the addition of a heat quantity dQ, is expressed as:

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

The absorbed thermal energy contributes to an augmentation of the black hole's total mass; consequently,

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

Thus, the entropy of a black hole is directly proportional to its surface area, expressed as:

${\displaystyle S=4\pi M^{2}={\frac {A}{4}},}$

Considering the black hole's radius is equivalent to twice its mass, the area A can be calculated using:

${\displaystyle A=4\pi R^{2}=16\pi M^{2}.}$

The premise that a small black hole exhibits zero entropy implies a null integration constant. The formation of a black hole represents the maximal efficiency for mass compression within a given spatial region, and this entropy also establishes an upper limit for the information capacity of any spherical region within spacetime. The structure of this outcome robustly indicates that the physical characteristics of a gravitational theory may be intrinsically encoded upon a delimiting surface.

Evaporation of Black Holes

As particles emanate, the black hole experiences a reduction in its energy, and consequently, a diminution of its mass (a relationship governed by Einstein's equation: E = mc§56§). Therefore, a black hole undergoing evaporation possesses a finite operational lifespan. Through dimensional analysis, it can be demonstrated that a black hole's lifespan is proportional to the cube of its initial mass. The duration required for a black hole to completely dissipate is given by:

${\displaystyle t_{\mathrm {ev} }={\frac {5120\pi G^{2}M^{3}}{\hbar c^{4}}}={\frac {480c^{2}V}{\hbar G}}=5120\pi \,t_{\text{P}}\left({\frac {M}{m_{\text{P}}}}\right)^{3}\approx 2.140\times 10^{67}\,{\text{years}}\ \left({\frac {M}{M_{\odot }}}\right)^{3},}$

In this context, M represents the black hole's mass, and V denotes its Schwarzschild volume. Additionally, m_P signifies the Planck mass, and t_P refers to the Planck time. Consequently, a black hole possessing one solar mass (M_☉ = 2.0ק303132§ kg) would require over §3637§67 years to fully evaporate, a duration substantially exceeding the universe's current estimated age of 1.4ק444546§ years.

The temperature associated with Hawking radiation is defined as follows:

$${\displaystyle T_{\mathrm {H} }={\frac {\hbar c^{3}}{8\pi GMk_{\mathrm {B} }}}\approx {\frac {10^{23}}{M}}.}$$ Black holes possessing greater mass exhibit proportionally lower Hawking radiation temperatures. For instance, the smallest hypothesized stellar black hole, estimated at approximately three solar masses, registers a temperature of ${\displaystyle 10^{-7}}$ Kelvin. Given that the cosmic microwave background radiation throughout the universe maintains a temperature of 2.7 Kelvin, stellar black holes are unable to evaporate, as their intrinsic temperature is lower than that of the surrounding cosmos.

The Bekenstein–Hawking luminosity of a black hole is derived under specific assumptions: primarily, that emission consists solely of photons (excluding other particle types), and secondarily, that the event horizon functions as the radiating surface.

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

In this equation, P represents the luminosity, which quantifies the radiated power; ħ denotes the reduced Planck constant; c signifies the speed of light; G refers to the gravitational constant; and M indicates the black hole's mass.

The phenomenon of black hole evaporation yields several notable implications:

• Black hole evaporation contributes to a more coherent understanding of black hole thermodynamics by elucidating the thermal interactions between black holes and the broader universe.
• In contrast to most celestial bodies, a black hole's temperature rises as it emits radiation and consequently loses mass. This temperature increase occurs at an exponential rate, culminating most probably in the black hole's complete dissolution via an intense gamma-ray burst. A comprehensive theoretical framework for this dissolution necessitates a quantum gravity model, particularly since this event transpires when the black hole's mass nears one Planck mass, at which point its radius concurrently approaches two Planck lengths.
• The most straightforward models of black hole evaporation precipitate the black hole information paradox. Within these models, the information initially contained within a black hole seemingly vanishes upon its dissipation, primarily because the emitted Hawking radiation is posited as random, bearing no correlation to the original informational content. Various resolutions to this paradox have been advanced, encompassing hypotheses that Hawking radiation is subtly altered to encode the lost information, that the evaporative process leaves behind a remnant particle preserving the information, or that information loss is permissible under these specific physical conditions.

Evaporation of Primordial Black Holes

The inverse relationship between mass and temperature in Hawking radiation suggests that a black hole capable of dissipation must possess a mass less than 0.8% of Earth's mass. Consequently, black holes formed through stellar collapse are precluded from dissipating via this mechanism, leaving only primordial black holes as potential candidates for such low-mass formation.

Stephen Hawking posited that any black hole originating in the early universe with a mass below approximately 10¹² kg would have undergone complete evaporation by the current epoch.

In 1976, Don Page advanced these estimations by computing the power output and evaporation duration for a non-rotating, uncharged Schwarzschild black hole of mass M. These computations are intricate because a black hole, possessing a finite dimension, does not behave as an ideal black body; its absorption cross-section diminishes in a complex, spin-dependent fashion with decreasing frequency, particularly when the wavelength approaches the event horizon's scale. Page's findings indicated that primordial black holes could persist until the present only if their initial mass was approximately 4×10¹¹ kg or greater. It is noteworthy that Page's 1976 work was based on the then-prevalent, but now superseded, assumption that neutrinos were massless and existed in only two flavors. Consequently, his calculated black hole lifetimes diverge from contemporary results, which incorporate three neutrino flavors with non-zero masses. A subsequent calculation in 2008, leveraging the particle content of the Standard Model and the WMAP data for the universe's age, established a mass boundary of (5.00±0.04)ק161718§ kg.

Prior to 1998, calculations based on outdated neutrino assumptions indicated that a solar-mass black hole, if evaporating via Hawking radiation, would require over 10⁶⁴ years to dissipate, a duration significantly exceeding the universe's current age. A supermassive black hole, with a mass of 10¹¹ (100 billion) M_☉, was estimated to evaporate in approximately §10^11§×§1213§100 years. Furthermore, certain colossal black holes within the cosmos are projected to expand to potentially 10§1617§ M_☉ during the gravitational collapse of galactic superclusters; even these would eventually evaporate over a period extending up to 2 × 10¹⁰⁶ years. Subsequent scientific advancements post-1998 have marginally revised these figures; for instance, the contemporary estimate for a solar-mass black hole's lifetime is 10⁶⁷ years.

Challenges and Further Research

The Trans-Planckian Problem

The trans-Planckian problem refers to a conceptual difficulty arising from Hawking's initial calculations, which involve quantum particles whose wavelengths become shorter than the Planck length in the vicinity of a black hole's event horizon. This phenomenon stems from the anomalous relativistic effects near the horizon, where time appears to halt from a distant observer's perspective. Consequently, a particle emitted from a black hole with a finite observed frequency, when theoretically traced back to the horizon, would necessitate an infinite frequency and, by extension, a trans-Planckian wavelength.

Both the Unruh effect and the Hawking effect describe the behavior of field modes within a seemingly stationary spacetime, where these modes exhibit a frequency shift relative to coordinate systems that remain regular across the event horizon. This inherent characteristic arises because maintaining a position outside the horizon necessitates continuous acceleration, which perpetually Doppler shifts these modes.

When an outgoing photon of Hawking radiation is traced backward in time, its frequency diverges significantly from its value at a great distance as it approaches the event horizon, implying an infinite "scrunching" of the photon's wavelength at the black hole's boundary. Within a maximally extended external Schwarzschild solution, the photon's frequency remains regular only if its mode is extrapolated into a past region inaccessible to any observer. To circumvent this, Hawking employed an alternative black hole solution that lacked a past region, instead positing its formation at a finite point in the past. Under this model, the origin of all outgoing photons can be precisely identified as a microscopic point coinciding with the black hole's initial formation.

Hawking's initial calculations posited that quantum fluctuations at a singular point encapsulate all emitted radiation. The modes responsible for this outgoing radiation, after an extended proximity to the event horizon, undergo significant redshifting, originating with wavelengths substantially shorter than the Planck length. Consequently, the lack of understanding regarding physical laws at such minuscule scales leads some to question the validity of Hawking's original derivation.

Currently, the trans-Planckian problem is largely regarded as a mathematical artifact inherent in horizon calculations. An analogous phenomenon manifests when ordinary matter accretes onto a white hole solution. This matter accumulates on the white hole without any accessible future region. Projecting the future trajectory of this matter reveals its compression onto the white hole's ultimate singular endpoint, entering a trans-Planckian domain. These divergences arise because modes terminating at the horizon, when viewed from external coordinates, exhibit singular frequencies. Classical determination of these events necessitates an extension into alternative coordinate systems that traverse the horizon.

Alternative theoretical frameworks exist that account for Hawking radiation while simultaneously resolving the trans-Planckian problem. A crucial observation is that comparable trans-Planckian issues emerge when tracing back in time the modes associated with Unruh radiation. Within the Unruh effect, the temperature's magnitude is derivable from conventional Minkowski field theory, a result that remains widely accepted.

Large Extra Dimensions

The equations presented previously are valid exclusively under the assumption that gravitational laws remain approximately consistent down to the Planck scale. Specifically, for black holes possessing masses below the Planck mass (approximately 10⁻⁸ kg), these formulas predict unfeasible lifetimes shorter than the Planck time (around 10⁻⁴³ s). This outcome is generally interpreted as evidence that the Planck mass constitutes the minimum mass threshold for a black hole.

Within theoretical models incorporating large extra dimensions (specifically, 10 or 11), the magnitudes of Planck constants can diverge significantly, necessitating corresponding modifications to the Hawking radiation formulas. Notably, the lifetime of a micro black hole, whose radius is smaller than the scale of these extra dimensions, is described by equation 9 in Cheung (2002) and by equations 25 and 26 in Carr (2005).

τ<!-- τ --> ∼<!-- ∼ --> §1415§ M ∗ ( M BH M ∗ ) n + §6263§ n + §7071§ , {\displaystyle \tau \sim {\frac {1}{M_{*}}}\left({\frac {M_{\text{BH}}}{M_{*}}}\right)^{\frac {n+3}{n+1}},}

In this context, M_∗ represents the low-energy scale, potentially reaching values as low as a few TeV, while n denotes the quantity of large extra dimensions. This revised formula now accommodates black holes with masses as minimal as a few TeV, predicting lifetimes on the order of the "new Planck time," approximately 10⁻²⁶ s.

Within Loop Quantum Gravity

Loop quantum gravity has facilitated a comprehensive investigation into the quantum geometry of a black hole's event horizon. However, loop quantization fails to inherently reproduce the black hole entropy result initially established by Bekenstein and Hawking, unless a specific free parameter is adjusted to nullify various constants, thereby recovering the Bekenstein–Hawking entropy formula. Nevertheless, the theory has enabled the calculation of quantum gravitational corrections to black hole entropy and radiation.

Quantum black holes, characterized by fluctuations in their horizon area, are predicted to exhibit deviations from the standard Hawking radiation spectrum. These deviations, which would manifest as observable X-rays from evaporating primordial black holes, are concentrated at distinct, unblended frequencies that are highly prominent within the Hawking spectrum.

Tunneling Perspective

An alternative conceptualization of the Hawking effect, framing it as a quantum tunneling process, has been advanced by Parikh and Wilczek, as well as by Padmanabhan and Srinivasan. This methodology involves correlating the probability of quantum tunneling across the event horizon with a Boltzmann distribution, thereby yielding a temperature consistent with Hawking's initial derivation. Due to its localized formulation, this tunneling framework is applicable to diverse types of horizons, including those exhibiting mild dynamic properties.

Empirical Observation

Astronomical Investigations

In June 2008, NASA launched the Fermi space telescope, which is dedicated to searching for the terminal gamma-ray flashes anticipated from evaporating primordial black holes. As of January 1, 2024, no such phenomena have been detected.

The KM3NeT neutrino detector recorded a 120 PeV event, designated KM3-230213A, in 2023; one hypothesis for its origin involves the evaporation of a primordial black hole. Furthermore, a comprehensive array of measurements conducted by both KM3NeT and IceCube aligns with the primordial black hole evaporation model, provided that such black holes constitute a substantial component of dark matter.

Heavy-Ion Collider Physics

Should the speculative theories concerning large extra dimensions prove accurate, CERN's Large Hadron Collider might possess the capability to generate micro black holes and subsequently observe their evaporative processes. However, no such micro black holes have been detected at CERN to date.

Analog Experiments

Under the experimentally attainable conditions for actual gravitational systems, the Hawking effect is of insufficient magnitude for direct observation. Consequently, it has been posited that Hawking radiation could be investigated through analog models, specifically utilizing sonic black holes. In these models, sound perturbations serve as analogs to light within a gravitational black hole, and the flow of an approximately perfect fluid simulates gravitational effects (see Analog models of gravity). Reports of Hawking radiation observations have emerged from sonic black holes created using Bose–Einstein condensates.

In September 2010, an experimental setup successfully generated a laboratory "white hole event horizon," which its creators asserted exhibited radiation analogous to optical Hawking radiation. Nevertheless, these findings remain unverified and contentious, thus casting doubt on their definitive confirmation status.

References

References

• An article by A. Barrau and J. Grain, titled "The case for mini black holes," elucidates potential methods for detecting Hawking radiation at particle colliders.
• The case for mini black holes A. Barrau & J. Grain explain how the Hawking radiation could be detected at colliders