No hair theorem
Encyclopedia
The no-hair theorem postulates that all black hole
solutions of the Einstein-Maxwell equations of gravitation
and electromagnetism
in general relativity
can be completely characterized by only three externally observable classical parameters: mass
, electric charge
, and angular momentum
. All other information (for which "hair" is a metaphor) about the matter
which formed a black hole or is falling into it, "disappears" behind the black-hole event horizon
and is therefore permanently inaccessible to external observers.
whereas the second is made out of antimatter
; nevertheless, they will be completely indistinguishable to an observer outside the event horizon. None of the special particle physics
pseudo-charges (baryon
ic, lepton
ic, etc.) are conserved in the black hole.
(point of view of the observer). The theorem therefore says nothing about a black hole's position or velocity.
More generally, every unstable black hole decays rapidly to a stable black hole; and (modulo quantum fluctuations) stable black holes can be completely described at any moment in time by these eleven numbers:
These numbers represent the conserved attributes of an object which can be determined from a distance by examining its gravitational and electromagnetic fields. All other variations in the black hole will either escape to infinity or be swallowed up by the black hole.
, obeying the Einstein field equation of general relativity
with zero cosmological constant
, in the presence of electromagnetic fields, or optionally other fields such as scalar field
s and massive vector field
s (Proca
fields, spinor fields
, etc.).
is positive (which recent observations are tending to support).
Magnetic charge
, if detected as predicted by some theories, would form the fourth parameter possessed by a classical black hole.
Yang-Mills fields, non-abelian Proca fields, some non-minimally coupled
scalar fields, or skyrmion
s; or in some theories of gravity other than Einstein’s general relativity. However, these exceptions are often unstable solutions and/or do not lead to conserved quantum numbers so that "The 'spirit' of the no-hair conjecture, however, seems to be maintained". It has been proposed that "hairy" black holes may be considered to be bound states of hairless black holes and soliton
s.
, each of which can be chosen as similar as desired to any of the others (hence the loss of information).
do away with this picture. Rather than having a potentially infinite information capacity, it is suggested that the entropy
of a quantum black hole should be a strictly finite A/4, where A is the area of the black hole in Planck units.
Along with a finite (non-infinite) entropy, quantum black holes acquire a finite (non-zero) temperature, and with it the emission of Hawking radiation
with a black body spectrum characteristic of that temperature. At a statistical level, this can be understood as a consequence of detailed balance
following from the presumed micro-reversibility (unitarity) of the interaction between the quantum states of the radiation field and the quantum states of the black hole. This implies that if black holes can absorb radiation, they should therefore also emit radiation, with a black body spectrum characteristic of the temperature of the relevant part of the system.
of an astronomical-sized black hole these corrections are tiny. The classical infinite information density is actually quite a good approximation to the finite but large black hole entropy, the black hole temperature is very nearly zero, and there are very few Hawking particles to disrupt the classical trajectories.
states for sensibly talking about the system. Further in than this, the core of the system needs to be treated in its own, specifically quantum, terms.
can still manage to look like the black hole of classical general relativity, not just at the event horizon but also for a substantial way inside it, despite actually possessing only finite entropy.
A quantum black hole only has finite entropy and therefore presumably exists in one of a limited effective number of corresponding states. With reference to a careful description of the available states, this granularity may be revealed. However, trying to enforce a purely classical description represents a projection into a much bigger space, made possible presumably by probabilities supplied by environmental decoherence. Any structure implicit in the finite entropy against a quantum description could then be totally washed out by the huge injection of uncertainty this projection represents. This may explain why even though Hawking radiation has non-zero entropy, calculations so far have been unable to relate this to any fluctuations from perfect isotropy.
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
solutions of the Einstein-Maxwell equations of gravitation
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
and electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
can be completely characterized by only three externally observable classical parameters: mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
, electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
, and angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
. All other information (for which "hair" is a metaphor) about the matter
Matter
Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...
which formed a black hole or is falling into it, "disappears" behind the black-hole event horizon
Event horizon
In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...
and is therefore permanently inaccessible to external observers.
Example
Suppose two black holes have the same masses, electrical charges, and angular momenta, but the first black hole is made out of ordinary matterMatter
Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...
whereas the second is made out of antimatter
Antimatter
In particle physics, antimatter is the extension of the concept of the antiparticle to matter, where antimatter is composed of antiparticles in the same way that normal matter is composed of particles...
; nevertheless, they will be completely indistinguishable to an observer outside the event horizon. None of the special particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
pseudo-charges (baryon
Baryon
A baryon is a composite particle made up of three quarks . Baryons and mesons belong to the hadron family, which are the quark-based particles...
ic, lepton
Lepton
A lepton is an elementary particle and a fundamental constituent of matter. The best known of all leptons is the electron which governs nearly all of chemistry as it is found in atoms and is directly tied to all chemical properties. Two main classes of leptons exist: charged leptons , and neutral...
ic, etc.) are conserved in the black hole.
Independent from the reference frame
Like most ideas based on the general theory of relativity, the "no hair" theorem is concerned only with properties which are independent of the frame of referenceFrame of reference
A frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
(point of view of the observer). The theorem therefore says nothing about a black hole's position or velocity.
More generally, every unstable black hole decays rapidly to a stable black hole; and (modulo quantum fluctuations) stable black holes can be completely described at any moment in time by these eleven numbers:
- mass-energy M,
- linear momentum P (three components),
- angular momentumAngular momentumIn physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
J (three components), - position X (three components),
- electric chargeElectric chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
Q.
These numbers represent the conserved attributes of an object which can be determined from a distance by examining its gravitational and electromagnetic fields. All other variations in the black hole will either escape to infinity or be swallowed up by the black hole.
Four-dimensional space-time
The no-hair theorem was originally formulated for black holes within the context of a four-dimensional spacetimeSpacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
, obeying the Einstein field equation of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
with zero cosmological constant
Cosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
, in the presence of electromagnetic fields, or optionally other fields such as scalar field
Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
s and massive vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
s (Proca
Proca action
In physics, in the area of field theory, the Proca action describes a massive spin-1 field of mass m in Minkowski spacetime. The field involved is a real vector field A...
fields, spinor fields
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
, etc.).
Extensions
It has since been extended to include the case where the cosmological constantCosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
is positive (which recent observations are tending to support).
Magnetic charge
Magnetic monopole
A magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole . In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring...
, if detected as predicted by some theories, would form the fourth parameter possessed by a classical black hole.
Counterexamples
Counterexamples in which the theorem fails are known in spacetime dimensions higher than four; in the presence of non-abelianNon-abelian
In theoretical physics, a non-abelian gauge transformation means a gauge transformation taking values in some group G, the elements of which do not obey the commutative law when they are multiplied. The original choice of G in the physics of electromagnetism was U, which is commutative.For a...
Yang-Mills fields, non-abelian Proca fields, some non-minimally coupled
Minimal coupling
In physics, minimal coupling refers to a coupling between fields which involves only the charge distribution and not higher multipole moments of the charge distribution...
scalar fields, or skyrmion
Skyrmion
In theoretical physics, a skyrmion is a mathematical model used to model baryons . It was conceived by Tony Skyrme.-Overview:...
s; or in some theories of gravity other than Einstein’s general relativity. However, these exceptions are often unstable solutions and/or do not lead to conserved quantum numbers so that "The 'spirit' of the no-hair conjecture, however, seems to be maintained". It has been proposed that "hairy" black holes may be considered to be bound states of hairless black holes and soliton
Soliton
In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
s.
Black holes in quantum gravity
The no-hair theorem is formulated in the classical spacetime of Einstein's general relativity, assumed to be infinitely divisible with no limiting short-range structure or short-range correlations. In such a model, each possible macroscopically-defined classical black hole corresponds to an infinite density of microstatesMicrostate (statistical mechanics)
In statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations...
, each of which can be chosen as similar as desired to any of the others (hence the loss of information).
Finite entropy
Proposals towards a theory of quantum gravityQuantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
do away with this picture. Rather than having a potentially infinite information capacity, it is suggested that the entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
of a quantum black hole should be a strictly finite A/4, where A is the area of the black hole in Planck units.
Along with a finite (non-infinite) entropy, quantum black holes acquire a finite (non-zero) temperature, and with it the emission of Hawking radiation
Hawking radiation
Hawking radiation is a thermal radiation with a black body spectrum predicted to be emitted by black holes due to quantum effects. It is named after the physicist Stephen Hawking, who provided a theoretical argument for its existence in 1974, and sometimes also after the physicist Jacob Bekenstein...
with a black body spectrum characteristic of that temperature. At a statistical level, this can be understood as a consequence of detailed balance
Detailed balance
The principle of detailed balance is formulated for kinetic systems which are decomposed into elementary processes : At equilibrium, each elementary process should be equilibrated by its reverse process....
following from the presumed micro-reversibility (unitarity) of the interaction between the quantum states of the radiation field and the quantum states of the black hole. This implies that if black holes can absorb radiation, they should therefore also emit radiation, with a black body spectrum characteristic of the temperature of the relevant part of the system.
Near the event horizon
From a different perspective, if it is correct that the properties of a quantum black hole should correspond at a broad level more or less to a classical general-relativistic black hole, then it is believed that the appearance and effects of the Hawking radiation can be interpreted as quantum "corrections" to the classical picture, as Planck's constant is "tuned up" away from zero up to h. Outside the event horizonEvent horizon
In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...
of an astronomical-sized black hole these corrections are tiny. The classical infinite information density is actually quite a good approximation to the finite but large black hole entropy, the black hole temperature is very nearly zero, and there are very few Hawking particles to disrupt the classical trajectories.
Within the event horizon
Very little changes for a test particle as the event horizon is crossed; classical general relativity is still a very good approximation to the quantum gravity outcome. But the further the particle falls down the gravity well, the more the Hawking temperature increases, the more Hawking particles there are buffeting the test particle, and the greater become its deviations from a classical path as the increasingly limited density of quantum states starts to pinch. Ultimately, much further in, the density of the quantum "corrections" becomes so pronounced that the classical variables cease to be good quantum numbers to describe the system. This deep into the black hole it becomes the quantum gravitational forces, above all else, that dominate the environmental interactions which determine the appropriate decoheredQuantum decoherence
In quantum mechanics, quantum decoherence is the loss of coherence or ordering of the phase angles between the components of a system in a quantum superposition. A consequence of this dephasing leads to classical or probabilistically additive behavior...
states for sensibly talking about the system. Further in than this, the core of the system needs to be treated in its own, specifically quantum, terms.
A quantum black hole compared to a classical black hole
In this way, the quantum black holeMicro black hole
Micro black holes are tiny black holes, also called quantum mechanical black holes or mini black holes, for which quantum mechanical effects play an important role....
can still manage to look like the black hole of classical general relativity, not just at the event horizon but also for a substantial way inside it, despite actually possessing only finite entropy.
A quantum black hole only has finite entropy and therefore presumably exists in one of a limited effective number of corresponding states. With reference to a careful description of the available states, this granularity may be revealed. However, trying to enforce a purely classical description represents a projection into a much bigger space, made possible presumably by probabilities supplied by environmental decoherence. Any structure implicit in the finite entropy against a quantum description could then be totally washed out by the huge injection of uncertainty this projection represents. This may explain why even though Hawking radiation has non-zero entropy, calculations so far have been unable to relate this to any fluctuations from perfect isotropy.
External links
- Hawking, S. W.Stephen HawkingStephen William Hawking, CH, CBE, FRS, FRSA is an English theoretical physicist and cosmologist, whose scientific books and public appearances have made him an academic celebrity...
(2005). Information Loss in Black Holes, arxiv:hep-th/0507171. Stephen Hawking’s purported solution to the black hole unitarityUnitarity (physics)In quantum physics, unitarity is a restriction on the allowed evolution of quantum systems that insures the sum of probabilities of all possible outcomes of any event is always 1....
paradox, first reported in July 2004.