Black hole starship

From Wikipedia, the free encyclopedia

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

In astronautics, a black hole starship is the suggestion that an artificial black hole could be used for spacecraft propulsion in a starship capable of interstellar travel. The concept was first discussed in science fiction, notably in the book Imperial Earth by Arthur C. Clarke, and in the work of Charles Sheffield, in which energy extracted from a Kerr–Newman black hole is described as powering the rocket engines in the story "Killing Vector" (1978).

A proposal to create an artificial black hole and using a parabolic reflector to reflect its Hawking radiation was discussed in 2009 by Louis Crane and Shawn Westmoreland. Their conclusion was that it was on the edge of possibility, but that quantum gravity effects that are presently unknown will either make it easier, or make it impossible. Investigation of the quantum gravity effects show it is not possible create a black hole from light alone.

Conceptual starship

One concept for a black hole starship creates a Kugelblitz, a small synthetic black hole from intense converging gamma rays, then intercepts the Hawking radiation from the black hole only in the direction of flight with a tungsten Dyson hemisphere. The spaceship pushes the black hole from behind, using particle beams.

Criteria

For a black hole to be used in space travel it must meet five criteria:

1. has a long enough lifespan to be useful,
2. is powerful enough to accelerate itself up to a reasonable fraction of the speed of light in a reasonable amount of time,
3. is small enough that we can access the energy to make it,
4. is large enough that we can focus the energy to make it,
5. has mass comparable to a starship.

These criteria imply a black hole weighing 606,000 metric tons (6.06 × 10⁸ kg) with a Schwarzschild radius of 0.9 attometers (0.9 × 10^–18 m, or 9 × 10^–19 m), a power output of 160 petawatts (160 × 10¹⁵ W, or 1.6 × 10¹⁷ W), and a 3.5-year lifespan. With such a power output, the black hole could accelerate to 10% the speed of light in 20 days, assuming 100% conversion of energy into kinetic energy. Assuming only 10% conversion into kinetic energy, it would take 10 times more.

Getting the black hole to act as a power source and engine also requires a way to convert the Hawking radiation into energy and thrust. One potential method involves placing the hole at the focal point of a parabolic reflector attached to the ship, creating forward thrust, if such a reflector can be built. A slightly easier, but less efficient method would involve simply absorbing all the gamma radiation heading towards the fore of the ship to push it onwards, and let the rest shoot out the back. This would, however, generate an enormous amount of heat as radiation is absorbed by the dish.

Advantages

Although beyond current technological capabilities, a black hole starship offers some advantages compared to other possible methods. For example, in nuclear fusion or fission, only a small proportion of the mass is converted into energy, so enormous quantities of fuel are needed. Another example, antimatter, is hugely energy-inefficient, and antimatter is difficult to contain. A black hole on the other hand is self-containing and very efficient in accepting mass which it radiates as energy.

Black holes from light

It has been shown that self-interaction effects of light lead to quantum dissipation effects, preventing the formation of a black hole from light alone. However these results have been challenged based on their assumption of adiabatic influx of radiation.

Engineering criticism

It is not clear that a starship powered by Hawking radiation can be made feasible within the laws of known physics. In the standard black hole thermodynamic model, the average energy of emitted quanta increases as size decreases, and extremely small black holes emit the majority of their energy in particles other than photons. In the Journal of the British Interplanetary Society, Jeffrey S. Lee of Icarus Interstellar states a typical quantum of radiation from a one-attometer black hole would be too energetic to be reflected. Lee further argues absorption (for example, by pair production from emitted gamma rays) may also be infeasible: A titanium "Dyson cap", optimized at 1 cm thickness and a radius around 33 km (to avoid melting), would absorb almost half the incident energy, but the maximum spaceship velocity over the black hole lifetime would be less than 0.0001c (about 30 km/s), according to Lee's calculations.

Govind Menon of Troy University suggests exploring the use of a rotating (Kerr–Newmann) black hole instead: "With non-rotating black holes, this is a very difficult thing...we typically look for energy almost exclusively from rotating black holes. Schwarzschild black holes do not radiate in an astrophysical, gamma ray burst point of view. It is not clear if Hawking radiation alone can power starships."

In fiction

This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Black hole starship" – news · newspapers · books · scholar · JSTOR (July 2026) (Learn how and when to remove this message)

• Arthur C. Clarke, Imperial Earth (1976)
• Charles Sheffield, Killing Vector (1978)
• Peter Watts, The Freeze Frame Revolution (2016)
• In John Varley's 1977 novel The Ophiuchi Hotline, a black hole engine is used to move a moon of Jupiter to Alpha Centauri.
• In the 2014 Hannu Rajaniemi science fiction novel The Causal Angel, Jean le Flambeur's ship Leblanc has a black hole that emits Hawking radiation which is used for propulsion.
• In the Star Trek universe, the Romulan D'deridex-class warbird uses an artificial quantum singularity as a power source for its warp propulsion drive.
• In the 1997 Paul W. S. Anderson science fiction horror film Event Horizon, the eponymous starship uses an artificial black hole drive to achieve faster-than-light travel.
• In the MMO Eve Online, starships designed by the Triglavian faction utilize naked singularities contained on the external hull as their vessel's primary power source.
• In the Foundation TV series, jump ships appear to use black holes to power their jumpdrives, enabling faster-than-light (FTL) travel over interstellar distances. This differs from the book series on which the show is based, where FTL travel is facilitated through hyperspace travel.
• In the universe of Doctor Who, the TARDIS is powered by a black hole which enables it to travel through time and space.
• In Sequoia Nagamatsu's novel How High We Go in the Dark, an interstellar starship is used to take passengers in cryo-sleep 582 light years from Earth at 10% the speed of light. The starship uses an engine powered by Hawking radiation.

Molecular vibration

From Wikipedia, the free encyclopedia

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

A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical vibrational frequencies range from less than 10¹³ Hz to approximately 10¹⁴ Hz, corresponding to wavenumbers of approximately 300 to 3000 cm⁻¹ and wavelengths of approximately 30 to 3 μm.

Vibrations of polyatomic molecules are described in terms of normal modes, which are independent of each other, but each normal mode involves simultaneous vibrations of parts of the molecule. In general, a non-linear molecule with N atoms has 3N − 6 normal modes of vibration, but a linear molecule has 3N − 5 modes, because rotation about the molecular axis cannot be observed. A diatomic molecule has one normal mode of vibration, since it can only stretch or compress the single bond.

A molecular vibration is excited when the molecule absorbs energy, ΔE, corresponding to the vibration's frequency, ν, according to the relation ΔE = hν, where h is the Planck constant. A fundamental vibration is evoked when one such quantum of energy is absorbed by the molecule in its ground state. When multiple quanta are absorbed, the first and possibly higher overtones are excited.

To a first approximation, the motion in a normal vibration can be described as a kind of simple harmonic motion. In this approximation, the vibrational energy is a quadratic function (parabola) with respect to the atomic displacements and the first overtone has twice the frequency of the fundamental. In reality, vibrations are anharmonic and the first overtone has a frequency that is slightly lower than twice that of the fundamental. Excitation of the higher overtones involves progressively less and less additional energy and eventually leads to dissociation of the molecule, because the potential energy of the molecule is more like a Morse potential or more accurately, a Morse/Long-range potential.

The vibrational states of a molecule can be probed in a variety of ways. The most direct way is through infrared spectroscopy, as vibrational transitions typically require an amount of energy that corresponds to the infrared region of the spectrum. Raman spectroscopy, which typically uses visible light, can also be used to measure vibration frequencies directly. The two techniques are complementary and comparison between the two can provide useful structural information such as in the case of the rule of mutual exclusion for centrosymmetric molecules.

Vibrational excitation can occur in conjunction with electronic excitation in the ultraviolet-visible region. The combined excitation is known as a vibronic transition, giving vibrational fine structure to electronic transitions, particularly for molecules in the gas state.

Simultaneous excitation of a vibration and rotations gives rise to vibration–rotation spectra.

Number of vibrational modes

Main article: Degrees of freedom (physics and chemistry)

For a molecule with N atoms, the positions of all N nuclei depend on a total of 3N coordinates, so that the molecule has 3N degrees of freedom including translation, rotation and vibration. Translation corresponds to movement of the center of mass whose position can be described by 3 cartesian coordinates.

A nonlinear molecule can rotate about any of three mutually perpendicular axes and therefore has 3 rotational degrees of freedom. For a linear molecule, rotation about the molecular axis does not involve movement of any atomic nucleus, so there are only 2 rotational degrees of freedom which can vary the atomic coordinates.

An equivalent argument is that the rotation of a linear molecule changes the direction of the molecular axis in space, which can be described by 2 coordinates corresponding to latitude and longitude. For a nonlinear molecule, the direction of one axis is described by these two coordinates, and the orientation of the molecule about this axis provides a third rotational coordinate.

The number of vibrational modes is therefore 3N minus the number of translational and rotational degrees of freedom, or 3N − 5 for linear and 3N − 6 for nonlinear molecules.

Vibrational coordinates

The coordinate of a normal vibration is a combination of changes in the positions of atoms in the molecule. When the vibration is excited the coordinate changes sinusoidally with a frequency ν, the frequency of the vibration.

Internal coordinates

Internal coordinates are of the following types, illustrated with reference to the planar molecule ethylene,

• Stretching: a change in the length of a bond, such as C–H or C–C
• Bending: a change in the angle between two bonds, such as the HCH angle in a methylene group
• Rocking: a change in angle between a group of atoms, such as a methylene group and the rest of the molecule
• Wagging: a change in angle between the plane of a group of atoms, such as a methylene group and a plane through the rest of the molecule
• Twisting: a change in the angle between the planes of two groups of atoms, such as a change in the angle between the two methylene groups
• Out-of-plane: a change in the angle between any one of the C–H bonds and the plane defined by the remaining atoms of the ethylene molecule. Another example is in BF₃ when the boron atom moves in and out of the plane of the three fluorine atoms.

In a rocking, wagging or twisting coordinate the bond lengths within the groups involved do not change. The angles do. Rocking is distinguished from wagging by the fact that the atoms in the group stay in the same plane.

In ethylene there are 12 internal coordinates: 4 C–H stretching, 1 C–C stretching, 2 H–C–H bending, 2 CH₂ rocking, 2 CH₂ wagging, 1 twisting. Note that the H–C–C angles cannot be used as internal coordinates as well as the H–C–H angle because the angles at each carbon atom cannot all increase at the same time.

Note that these coordinates do not correspond to normal modes (see § Normal coordinates). In other words, they do not correspond to particular frequencies or vibrational transitions.

Vibrations of a methylene group (−CH₂−) in a molecule for illustration

Within the CH₂ group, commonly found in organic compounds, the two low mass hydrogens can vibrate in six different ways which can be grouped as 3 pairs of modes: 1. symmetric and asymmetric stretching, 2. scissoring and rocking, 3. wagging and twisting. These are shown here:

Symmetrical stretching	Asymmetrical stretching	Scissoring (Bending)
Rocking	Wagging	Twisting

(These figures do not represent the "recoil" of the C atoms, which, though necessarily present to balance the overall movements of the molecule, are much smaller than the movements of the lighter H atoms).

Symmetry-adapted coordinates

See also: Molecular symmetry

Symmetry–adapted coordinates may be created by applying a projection operator to a set of internal coordinates. The projection operator is constructed with the aid of the character table of the molecular point group. For example, the four (un-normalized) C–H stretching coordinates of the molecule ethene are given by $${\displaystyle \begin{array}{rl}{Q}_{s1}& =q_1+q_2+q_3+q_4\\ Q_{s2}& =q_1+q_2-q_3-q_4\\ Q_{s3}& =q_1-q_2+q_3-q_4\\ Q_{s4}& =q_1-q_2-q_3+q_4\end{array}}$$ where ${\displaystyle q_1-q_4}$ are the internal coordinates for stretching of each of the four C–H bonds.

Illustrations of symmetry–adapted coordinates for most small molecules can be found in Nakamoto.

Normal coordinates

The normal coordinates, denoted as Q, refer to the positions of atoms away from their equilibrium positions, with respect to a normal mode of vibration. Each normal mode is assigned a single normal coordinate, and so the normal coordinate refers to the "progress" along that normal mode at any given time. Formally, normal modes are determined by solving a secular determinant, and then the normal coordinates (over the normal modes) can be expressed as a summation over the cartesian coordinates (over the atom positions). The normal modes diagonalize the matrix governing the molecular vibrations, so that each normal mode is an independent molecular vibration. If the molecule possesses symmetries, the normal modes "transform as" an irreducible representation under its point group. The normal modes are determined by applying group theory, and projecting the irreducible representation onto the cartesian coordinates. For example, when this treatment is applied to CO₂, it is found that the C=O stretches are not independent, but rather there is an O=C=O symmetric stretch and an O=C=O asymmetric stretch:

• symmetric stretching: the sum of the two C–O stretching coordinates; the two C–O bond lengths change by the same amount and the carbon atom is stationary. Q = q₁ + q₂
• asymmetric stretching: the difference of the two C–O stretching coordinates; one C–O bond length increases while the other decreases. Q = q₁ − q₂

When two or more normal coordinates belong to the same irreducible representation of the molecular point group (colloquially, have the same symmetry) there is "mixing" and the coefficients of the combination cannot be determined a priori. For example, in the linear molecule hydrogen cyanide, HCN, The two stretching vibrations are

• principally C–H stretching with a little C–N stretching; Q₁ = q₁ + a q₂ (a << 1)
• principally C–N stretching with a little C–H stretching; Q₂ = b q₁ + q₂ (b << 1)

The coefficients a and b are found by performing a full normal coordinate analysis by means of the Wilson GF method.

Newtonian mechanics

The HCl molecule as an anharmonic oscillator vibrating at energy level E₃. D₀ is dissociation energy here, r₀ bond length, U potential energy. Energy is expressed in wavenumbers. The hydrogen chloride molecule is attached to the coordinate system to show bond length changes on the curve.

Perhaps surprisingly, molecular vibrations can be treated using Newtonian mechanics to calculate the correct vibration frequencies. The basic assumption is that each vibration can be treated as though it corresponds to a spring. In the harmonic approximation the spring obeys Hooke's law: the force required to extend the spring is proportional to the extension. The proportionality constant is known as a force constant, k. The anharmonic oscillator is considered elsewhere. $${\displaystyle \mathrm{F}=-kQ}$$ By Newton's second law of motion this force is also equal to a reduced mass, μ, times acceleration. $${\displaystyle \mathrm{F}=\mu \frac{d^{2}Q}{d{t}^2}}$$ Since this is one and the same force the ordinary differential equation follows. $${\displaystyle \mu \frac{d^{2}Q}{d{t}^2}+kQ=0}$$ The solution to this equation of simple harmonic motion is $${\displaystyle Q(t)=A\cos (2\pi \nu t);\nu =\frac{1}{2\pi }\sqrt{\frac{k}{\mu }}.}$$ A is the maximum amplitude of the vibration coordinate Q. It remains to define the reduced mass, μ. In general, the reduced mass of a diatomic molecule, AB, is expressed in terms of the atomic masses, m_A and m_B, as $${\displaystyle \frac{1}{\mu }=\frac{1}{m_A}+\frac{1}{m_B}.}$$ The use of the reduced mass ensures that the centre of mass of the molecule is not affected by the vibration. In the harmonic approximation the potential energy of the molecule is a quadratic function of the normal coordinate. It follows that the force-constant is equal to the second derivative of the potential energy. $${\displaystyle k=\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{Q}^2}}$$

When two or more normal vibrations have the same symmetry a full normal coordinate analysis must be performed (see GF method). The vibration frequencies, ν_i, are obtained from the eigenvalues, λ_i, of the matrix product GF. G is a matrix of numbers derived from the masses of the atoms and the geometry of the molecule. F is a matrix derived from force-constant values. Details concerning the determination of the eigenvalues can be found in.

Quantum mechanics

In the harmonic approximation the potential energy is a quadratic function of the normal coordinates. Solving the Schrödinger wave equation, the energy states for each normal coordinate are given by $${\displaystyle E_n=h\left(n+\frac{1}{2}\right)\nu =h\left(n+\frac{1}{2}\right)\frac{1}{2\pi }\sqrt{\frac{k}{m}},}$$ where n is a quantum number that can take values of 0, 1, 2, ... In molecular spectroscopy where several types of molecular energy are studied and several quantum numbers are used, this vibrational quantum number is often designated as v.

The difference in energy when n (or v) changes by 1 is therefore equal to ${\displaystyle h\nu }$, the product of the Planck constant and the vibration frequency derived using classical mechanics. For a transition from level n to level n+1 due to absorption of a photon, the frequency of the photon is equal to the classical vibration frequency ${\displaystyle \nu }$ (in the harmonic oscillator approximation).

See quantum harmonic oscillator for graphs of the first 5 wave functions, which allow certain selection rules to be formulated. For example, for a harmonic oscillator transitions are allowed only when the quantum number n changes by one, $${\displaystyle \mathrm{\Delta }n=\pm 1}$$but this does not apply to an anharmonic oscillator; the observation of overtones is only possible because vibrations are anharmonic. Another consequence of anharmonicity is that transitions such as between states n = 2 and n = 1 have slightly less energy than transitions between the ground state and first excited state. Such a transition gives rise to a hot band. To describe vibrational levels of an anharmonic oscillator, Dunham expansion is used.

When it comes to polyatomic molecules, it is common to solve the Schrödinger Equation using Watson's nuclear motion Hamiltonian. Similar as for diatomics, this can be done within the harmonic approximation as stated above. For the anharmonic calculation of vibrational spectra of polyatomic molecules, more sophisticated approaches are used. Prominent examples in computational chemistry are 2nd order vibrational perturbation theory (VPT2) or vibrational configuration interaction theory (VCI).

Intensities

In an infrared spectrum the intensity of an absorption band is proportional to the derivative of the molecular dipole moment with respect to the normal coordinate. Likewise, the intensity of Raman bands depends on the derivative of polarizability with respect to the normal coordinate. There is also a dependence on the fourth-power of the wavelength of the laser used.

Hawking radiation

From Wikipedia, the free encyclopedia

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

Hawking radiation is radiation released outside a black hole's event horizon due to quantum effects according to a model developed by Stephen Hawking in 1974. This black-body radiation was not predicted by previous models which assumed that once electromagnetic radiation is inside the event horizon, it cannot escape. Hawking radiation is predicted to be extremely faint and is many orders of magnitude below the current best telescopes' detecting ability.

Hawking radiation would reduce the mass and rotational energy of black holes and consequently 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 happens extremely slowly. The radiation temperature, called Hawking 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 per their mass. Consequently, if small black holes exist, as permitted by the hypothesis of primordial black holes, they will lose mass more rapidly as they shrink, leading to a final cataclysm of high energy radiation alone. Such radiation bursts have not yet been detected.

Background

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

Modern black holes were first predicted using Albert Einstein's general theory of relativity. Evidence of the astrophysical objects termed black holes began to mount half a century later, and these objects are of current interest primarily because of their compact size and immense gravitational attraction. Early research into black holes was done by individuals such as Karl Schwarzschild and John Wheeler, who modeled black holes as having zero entropy.

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. Because nothing can travel that fast, 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.

Alternatively, using a set of infalling coordinates in general relativity, one can conceptualize the event horizon as the region beyond which space is falling faster than the speed of light. (Although nothing can travel through space faster than light, space itself can fall at any speed.) Once matter is inside the event horizon, all of the matter inside falls inevitably 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 (non-rotating and uncharged) is characterized just by its mass and event horizon.

Discovery

Diagram by Jacob Bekenstein of the entropy on a black hole's event horizon.

In 1971, Soviet physicsists Yakov Zeldovich and Alexei Starobinsky proposed that rotating black holes ought to create and emit particles, reasoning by analogy with electromagnetic spinning metal spheres. The process is now called Zeldovich amplification. In 1972, Jacob Bekenstein developed a theory and reported that the black holes should have an entropy proportional to their surface area. Initially Stephen Hawking argued against Bekenstein's theory, viewing black holes as a simple object with no entropy. After meeting Zeldovich in Moscow in 1973, Hawking put these two ideas together using his mixture of quantum field theory and general relativity.

In his 1974 paper Hawking showed that in theory, black holes radiate particles as if it were a blackbody. Particles escaping effectively drain energy from the black hole. In Hawking's prediction, black holes emit small amounts of thermal radiation at temperature $${\displaystyle T=\frac{\hslash c^3}{8\pi GM{k}_{\mathrm{B}}},}$$ where ${\displaystyle \hslash }$ is the reduced Planck constant, ${\displaystyle c}$ is the speed of light, ${\displaystyle G}$ is the gravitational constant, ${\displaystyle M}$ is the mass of the black hole and ${\displaystyle k_{\mathrm{B}}}$ is the Boltzmann constant. By applying quantum field theory to black holes, Hawking determined that a black hole should continuously emit thermal blackbody radiation. This effect has become known as Hawking radiation. This theory was supported by previous work by Jacob Bekenstein, who theorized that black holes should have a finite entropy proportional to their surface area, and therefore should also have a temperature. Due to Bekenstein's contribution to black hole entropy, it is also known as Bekenstein–Hawking radiation. Since Hawking's 1974 publication, he and others have mathematically verified the result through different approaches, such as by path integrals.

Hawking radiation derives from vacuum fluctuations. A vacuum fluctuation in the electromagnetic field can result in a photon outside of the black hole horizon paired with one on the inside. The horizon allows one to escape in each direction.

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

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 $${\displaystyle \begin{array}{rl}(ds{)}^2& =-{\left(\frac{\rho }{4M}\right)}^2(dt{)}^2+(d\rho {)}^2+(d{X}_{\perp }{)}^2\\ & =-{\rho }^2(d\tau {)}^2+(d\rho {)}^2+(d{X}_{\perp }{)}^2,\end{array}}$$ which is the Rindler coordinates in terms of ⁠${\displaystyle \tau =\frac{t}{4M}}$⁠. The metric describes a frame that is accelerating to keep from falling into the black hole. The local acceleration, ⁠${\displaystyle \alpha =\frac{1}{\rho }}$⁠, diverges as ⁠${\displaystyle \rho \to \mathrm{\infty }}$⁠.

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: $${\displaystyle \begin{array}{rl}T(r^{\prime })& =\frac{1}{4\pi \sqrt{2Mr\left(1-\frac{2M}{r}\right)}}\cdot \frac{\sqrt{1-\frac{2M}{r}}}{\sqrt{1-\frac{2M}{r^{\prime }}}}\\ & =\frac{1}{4\pi \sqrt{2Mr\left(1-\frac{2M}{r^{\prime }}\right)}}.\end{array}}$$

The inverse temperature redshifted to ${\displaystyle r^{\prime }}$ at infinity is $${\displaystyle T(\mathrm{\infty })=\frac{1}{4\pi \sqrt{2Mr}},}$$ and r is the near-horizon position, near ⁠${\displaystyle 2M}$⁠, 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 $${\displaystyle T_{\text{H}}=\frac{1}{8\pi M}.}$$

For a black hole mass equal to the Earth, this temperature would be 10⁻¹² K. The wavelength of a typical quantum of Hawking radiation is similar to the size of the black hole.

From the black-hole temperature, it is straightforward to calculate the black-hole entropy ⁠${\displaystyle S}$⁠. The change in entropy when a quantity of heat ${\displaystyle dQ}$ is added is $${\displaystyle dS=\frac{dQ}{T}=8\pi MdQ.}$$

The heat energy that enters serves to increase the total mass, so $${\displaystyle dS=8\pi MdM=d(4\pi M^2).}$$

So the entropy of a black hole is proportional to its surface area: $${\displaystyle S=4\pi M^2=\frac{A}{4},}$$ where, since the radius of the black hole is twice its mass, we have the area ${\displaystyle A}$ being given by $${\displaystyle A=4\pi R^2=16\pi M^2.}$$

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 (by the holographic principle) onto a bounding surface.

Greybody factors

Greybody factors are functions of frequency and angular momentum that characterize the deviation of the emission-spectrum of a black hole from a pure black-body spectrum. As a result of quantum effects, an isolated black hole emits radiation that, at the black-hole horizon, matches the radiation from a perfect black body. The rate at which a black hole emits particles with energy between ${\displaystyle \hslash \omega }$ and ${\displaystyle \hslash (\omega +d\omega )}$ and with angular momentum quantum numbers ${\displaystyle \ell ,m}$ is given by $${\displaystyle d\mathrm{\Gamma }=\frac{{\sigma }_{\omega ,\ell ,m}}{\exp \left(\frac{\hslash \omega }{k_{\mathrm{B}}T}\right)\pm 1}d\omega }$$ where ${\displaystyle k_{\mathrm{B}}}$ is the Boltzmann constant and ${\displaystyle T}$ is the Hawking temperature of the black hole. The constant in the denominator is 1 for Bosons and −1 for Fermions. The factors ${\displaystyle {\sigma }_{\omega ,\ell ,m}}$ are called the greybody factors of the black hole. For a charged black hole, these factors may also depend on the charge of the emitted particles.

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 of mass–energy equivalence, ⁠${\displaystyle E=m{c}^2}$⁠). 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. The time that the black hole takes to dissipate is:$${\displaystyle t_{\mathrm{e}\mathrm{v}}=\frac{5120\pi G^2{M}^3}{\hslash c^4}=\frac{480c^{2}V}{\hslash G}=5120\pi t_{\text{P}}{\left(\frac{M}{m_{\text{P}}}\right)}^3\approx 2.140\times {10}^{67}\text{years}{\left(\frac{M}{M_{\odot }}\right)}^3,}$$ where ${\displaystyle M}$ and ${\displaystyle V}$ are the mass and volume of the (Schwarzschild) black hole, ${\displaystyle m_{\mathrm{P}}}$ the Planck mass and ${\displaystyle t_{\mathrm{P}}}$ Planck time. 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.

Black hole evaporation takes a long time relative to the current age of the universe, for black holes larger than a proton in diameter.

The Hawking radiation temperature is:

$${\displaystyle T_{\mathrm{H}}=\frac{\hslash c^3}{8\pi GM{k}_{\mathrm{B}}}\approx \frac{{10}^{23}}{M}.}$$Larger mass black holes have lower Hawking radiation temperatures. For the smallest predicted stellar black hole, about 3 solar masses, this temperature is 10⁻⁷ K. Since the universe contains the cosmic microwave background radiation at 2.7 K, no stellar black holes can evaporate: they are colder than outer space.

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 ${\displaystyle P}$ is the luminosity or radiated power, ${\displaystyle \hslash }$ is the reduced Planck constant, ${\displaystyle c}$ is the speed of light in a vacuum, ${\displaystyle G}$ is the gravitational constant, and ${\displaystyle M}$ is the mass of the black hole.

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, 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.

Evaporation of primordial black holes

The relationship between mass and temperature for Hawking radiation then implies the mass would need to be less than 0.8% of the mass of the Earth. This in turn means any black hole that could dissipate cannot be one created by stellar collapse. Only primordial black holes might be created with this little mass.

Hawking estimated that any black hole formed in the early universe with a mass of less than approximately 10¹² kg 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 ${\displaystyle M}$. 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 survive to the present day only 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.

Some pre-1998 calculations, using outdated assumptions about neutrinos, were as follows: 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 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. Post-1998 science modifies these results slightly; for example, the modern estimate of a solar-mass black hole lifetime is 10⁶⁷ years.

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, so Hawking used a different black hole solution without a past region, one 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$${\displaystyle \tau \sim \frac{1}{M_{\ast }}{\left(\frac{M_{\text{BH}}}{M_{\ast }}\right)}^{\frac{n+3}{n+1}},}$$

where ⁠${\displaystyle M_{\ast }}$⁠ is the low-energy scale, which could be as low as a few TeV, and ${\displaystyle n}$ 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.

Tunneling picture

An alternative description of the Hawking effect as a tunneling process has been developed by Parikh and Wilczek, and by Padmanabhan and Srinivasan. In this approach, the probability of tunneling through the horizon is compared with a Boltzmann distribution, leading to the same temperature as in Hawking's original derivation. Owing to its local formulation, the tunneling picture can be applied to various types of horizons, including mildly dynamical ones.

Empirical observation

Astronomy

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 January 2024, none have been detected.

Neutrino detector KM3NeT observed a 120 PeV event labeled KM3-230213A in 2023; one of the proposed explanations is evaporation of a primordial black hole. A broad range of measurements at KM3NeT and IceCube are consistent with primordial black hole evaporation assuming such primordial black holes account for a significant fraction of dark matter.

Particle 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.

Laboratory experiments

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 setup 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 debated, and the experiment's status as a genuine confirmation remains in doubt.