The reciprocal theorem describes the relationship between two flows in the same geometry where inertial effects are insignificant compared to viscous effects. Consider a fluid filled region formula_13 bounded by surface formula_14 with a unit normal formula_15. Suppose we have solutions to Stokes equations in the domain formula_13 possessing the form of the velocity fields formula_4 and formula_18. The velocity fields harbor corresponding stress fields formula_19 and formula_20 respectively. Then the following equality holds:


The reciprocal theorem allows us to obtain information about a certain flow by using information from another flow. This is preferable to solving Stokes equations, which is difficult due to not having a known boundary condition. This particularly useful if one wants to understand flow from a complicated problem by studying the flow of a simpler problem in the same geometry.


One can use the reciprocal theorem to relate the swimming velocity, formula_22, of a swimmer subject to a force formula_23 to its swimming gait formula_24:


Now that we have established that the relationship between the instantaneous swimming velocity in the direction of the force acting on the body and its swimming gate follow the general form


where formula_27 and formula_28 denote the positions of points


on the surface of the swimmer, we can establish that locomotion is independent of rate. Consider a swimmer that deforms in a periodic fashion through a sequence of motions between the times formula_29 and formula_30. The net displacement of the swimmer is


Now consider the swimmer deforming in the same manner but at a different rate. We describe this with the mapping


This result means that the net distance traveled by the swimmer does not depend on the rate


at which it is being deformed, but only on the geometrical sequence of shape. This is the first key result.


If a swimmer is moving in a periodic fashion that is time invariant, we know that the average displacement during one period must be zero. To illustrate the proof, let us consider a swimmer deforming during one period that starts and ends at times formula_29 and formula_30. That means its shape at the start and end are the same, i.e. formula_37. Next, we consider motion obtained by time-reversal


symmetry of the first motion that occurs during the period starting and ending at times formula_38 and formula_39. using a similar mapping as in the previous section, we define formula_40 and formula_41 and define the shape in the reverse motion to be the same as the shape in the forward motion, formula_42. Now we find the relationship between the net displacements in these two cases:


This is the second key result. Combining with our first key result from the previous section, we see that formula_44. We see that a swimmer that reverses its motion by reversing its sequence of shape changes leads to the opposite distance traveled. In addition, since the swimmer exhibits reciprocal body deformation, the sequence of motion is the same between formula_38 and formula_39 and formula_29 and formula_30. Thus, the distance traveled should


be the same independently of the direction of time, meaning that reciprocal motion cannot be used for net motion in low Reynolds number environments.


The scallop theorem holds if we assume that a swimmer undergoes reciprocal motion in an infinite quiescent Newtonian fluid in the absence of inertia and external body forces. However, there are instances where the assumptions for the scallop theorem are violated. In one case, successful swimmers in viscous environments must display non-reciprocal body kinematics. In another case, if a swimmer is in a non-Newtonian fluid, locomotion can be achieved as well.


In his original paper, Purcell proposed a simple example of non-reciprocal body deformation, now commonly known as the Purcell swimmer. This simple swimmer possess two degrees of freedom for motion: a two-hinged body composed of three rigid links rotating out-of-phase with each other. However, any body with more than one degree of freedom of motion can achieve locomotion as well.


In general, microscopic organisms like bacteria have evolved different mechanisms to perform non-reciprocal motion:


The assumption of a Newtonian fluid is essential since Stokes equations will not remain linear and time-independent in an environment that possesses complex mechanical and rheological properties. It is also common knowledge that many living microorganisms live in complex non-Newtonian fluids, which are common in biologically relevant environments. For instance, crawling cells often


Non-Newtonian fluids have several properties that can be manipulated to produce small scale locomotion.


First, one such exploitable property is normal


stress differences. These differences will arise from the stretching of the fluid by the flow of


The Clausius theorem (1855) states that for a thermodynamic system (e.g. heat engine or heat pump) exchanging heat with external reservoirs and undergoing a thermodynamic cycle,


where formula_2 is the infinitesimal amount of heat absorbed by the system from the reservoir and formula_3 is the temperature of the external reservoir (surroundings) at a particular instant in time. The closed integral is carried out along a thermodynamic process path from the initial/final state to the same initial/final state. In principle, the closed integral can start and end at an arbitrary point along the path.


If there are multiple reservoirs with different temperatures formula_4, then Clausius inequality reads:


In the special case of a reversible process, the equality holds. The reversible case is used to introduce the state function known as entropy. This is because in a cyclic process the variation of a state function is zero. In other words, the Clausius statement states that it is impossible to construct a device whose sole effect is the transfer of heat from a cool reservoir to a hot reservoir. Equivalently, heat spontaneously flows from a hot body to a cooler one, not the other way around.


for an infinitesimal change in entropy "S" applies not only to cyclic processes, but to any process that occurs in a closed system.


The Clausius theorem is a mathematical explanation of the second law of thermodynamics. It was developed by Rudolf Clausius who intended to explain the relationship between the heat flow in a system and the entropy of the system and its surroundings. Clausius developed this in his efforts to explain entropy and define it quantitatively. In more direct terms, the theorem gives us a way to determine if a cyclical process is reversible or irreversible. The Clausius theorem provides a quantitative formula for understanding the second law.


Clausius was one of the first to work on the idea of entropy and is even responsible for giving it that name. What is now known as the Clausius theorem was first published in 1862 in Clausius' sixth memoir, "On the Application of the Theorem of the Equivalence of Transformations to Interior Work". Clausius sought to show a proportional relationship between entropy and the energy flow by heating (δ"Q") into a system. In a system, this heat energy can be transformed into work, and work can be transformed into heat through a cyclical process. Clausius writes that "The algebraic sum of all the transformations occurring in a cyclical process can only be less than zero, or, as an extreme case, equal to nothing." In other words, the equation


with 𝛿"Q" being energy flow into the system due to heating and "T" being absolute temperature of the body when that energy is absorbed, is found to be true for any process that is cyclical and reversible. Clausius then took this a step further and determined that the following relation must be found true for any cyclical process that is possible, reversible or not. This relation is the "Clausius inequality".


Now that this is known, there must be a relation developed between the Clausius inequality and entropy. The amount of entropy "S" added to the system during the cycle is defined as


If the amount of energy added by heating can be measured during the process, and the temperature can be measured during the process, the Clausius inequality can be used to determine whether the process is reversible or irreversible by carrying out the integration in the Clausius inequality.


The temperature that enters in the denominator of the integrand in the Clausius inequality is actually the temperature of the external reservoir with which the system exchanges heat. At each instant of the process, the system is in contact with an external reservoir.


Because of the Second Law of Thermodynamics, in each infinitesimal heat exchange process between the system and the reservoir, the net change in entropy of the "universe", so to say, is formula_12.


When the system takes in heat by an infinitesimal amount formula_13(formula_14), for the net change in entropy formula_15 in this step to be positive, the temperature of the "hot" reservoir formula_16 needs to be slightly greater than the temperature of the system at that instant. If the temperature of the system is given by formula_17 at that instant, then formula_18, and formula_19 forces us to have:


This means the magnitude of the entropy "loss" from the reservoir, formula_21 is less than the magnitude of the entropy gain formula_22(formula_14) by the system:


Similarly, when the system at temperature formula_24 expels heat in magnitude formula_25 (formula_26) into a colder reservoir (at temperature formula_27) in an infinitesimal step, then again, for the Second Law of Thermodynamics to hold, one would have, in an exactly similar manner:


Here, the amount of heat 'absorbed' by the system is given by formula_29(formula_30), signifying that heat is transferring from the system to the reservoir, with formula_31. The magnitude of the entropy gained by the reservoir, formula_32 is greater than the magnitude of the entropy loss of the system formula_33


Since the total change in entropy for the system is 0 in a cyclic process, if one adds all the infinitesimal steps of heat intake and heat expulsion from the reservoir, signified by the previous two equations, with the temperature of the reservoir at each instant given by formula_34, one gets,


In summary, (the inequality in the third statement below, being obviously guaranteed by the second law of thermodynamics, which is the basis of our calculation),


For a reversible cyclic process, there is no generation of entropy in each of the infinitesimal heat transfer processes, so the following equality holds,


Thus, the Clausius inequality is a consequence of applying the second law of thermodynamics at each infinitesimal stage of heat transfer, and is thus in a sense a weaker condition than the Second Law itself.


Adiabatic quantum computation (AQC) is a form of quantum computing which relies on the adiabatic theorem to do calculations and is closely related to quantum annealing.


First, a (potentially complicated) Hamiltonian is found whose ground state describes the solution to the problem of interest. Next, a system with a simple Hamiltonian is prepared and initialized to the ground state. Finally, the simple Hamiltonian is adiabatically evolved to the desired complicated Hamiltonian. By the adiabatic theorem, the system remains in the ground state, so at the end the state of the system describes the solution to the problem. Adiabatic quantum computing has been shown to be polynomially equivalent to conventional quantum computing in the circuit model.


The time complexity for an adiabatic algorithm is the time taken to complete the adiabatic evolution which is dependent on the gap in the energy eigenvalues (spectral gap) of the Hamiltonian. Specifically, if the system is to be kept in the ground state, the energy gap between the ground state and the first excited state of formula_1 provides an upper bound on the rate at which the Hamiltonian can be evolved at time When the spectral gap is small, the Hamiltonian has to be evolved slowly. The runtime for the entire algorithm can be bounded by:


where formula_3 is the minimum spectral gap for


AQC is a possible method to get around the problem of energy relaxation. Since the quantum system is in the ground state, interference with the outside world cannot make it move to a lower state. If the energy of the outside world (that is, the "temperature of the bath") is kept lower than the energy gap between the ground state and the next higher energy state, the system has a proportionally lower probability of going to a higher energy state. Thus the system can stay in a single system eigenstate as long as needed.


Universality results in the adiabatic model are tied to quantum complexity and QMA-hard problems. The k-local Hamiltonian is QMA-complete for k ≥ 2. QMA-hardness results are known for physically realistic lattice models of qubits such as


where formula_5represent the Pauli matrices Such models are used for universal adiabatic quantum computation. The Hamiltonians for the QMA-complete problem can also be restricted to act on a two dimensional grid of qubits or a line of quantum particles with 12 states per particle. If such models were found to be physically realisable, they too could be used to form the building blocks of a universal adiabatic quantum computer.


In practice, there are problems during a computation. As the Hamiltonian is gradually changed, the interesting parts (quantum behaviour as opposed to classical) occur when multiple qubits are close to a tipping point. It is exactly at this point when the ground state (one set of qubit orientations) gets very close to a first energy state (a different arrangement of orientations). Adding a slight amount of energy (from the external bath, or as a result of slowly changing the Hamiltonian) could take the system out of the ground state, and ruin the calculation. Trying to perform the calculation more quickly increases the external energy; scaling the number of qubits makes the energy gap at the tipping points smaller.


Adiabatic quantum computation solves satisfiability problems and other combinatorial search problems. Specifically, these kind of problems seek a state that satisfies


This expression contains the satisfiability of M clauses, for which clause formula_7 has the value True or False, and can involve n bits. Each bit is a variable formula_8 such that formula_7 is a Boolean value function of formula_10. QAA solves this kind of problem using quantum adiabatic evolution. It starts with an Initial Hamiltonian formula_11:


where formula_13 shows the Hamiltonian corresponding to the clause formula_7. Usually, the choice of formula_13 won't depend on different clauses, so only the total number of times each bit is involved in all clauses matters. Next, it goes through an adiabatic evolution, ending in the Problem Hamiltonian formula_16:


where formula_18 is the satisfying Hamiltonian of clause C.


For a simple path of adiabatic evolution with run time T, consider:


which is the adiabatic evolution Hamiltonian of our algorithm.


According to the adiabatic theorem, we start from the ground state of Hamiltonian formula_11 at the beginning, proceed through an adiabatic process, and end in the ground state of problem Hamiltonian formula_16.


We then measure the z-component of each of the n spins in the final state. This will produce a string formula_25 which is highly likely to be the result of our satisfiability problem. The run time T must be sufficiently long to assure correctness of the result. According to adiabatic theorem, T is about formula_26, where


is the minimum energy gap between ground state and first excited state.


Adiabatic quantum computing is equivalent in power to standard gate-based quantum computing that implements arbitrary unitary operations. However, the mapping challenge on gate-based quantum devices differs substantially from quantum annealers as logical variables are mapped only to single qubits and not to chains.


The D-Wave One is a device made by Canadian company D-Wave Systems, which claims that it uses quantum annealing to solve optimization problems. On 25 May 2011, Lockheed-Martin purchased a D-Wave One for about US$10 million. In May 2013, Google purchased a 512 qubit D-Wave Two.


The question of whether the D-Wave processors offer a speedup over a classical processor is still unanswered. Tests performed by researchers at Quantum Artificial Intelligence Lab (NASA), USC, ETH Zurich, and Google show that as of 2015, there is no evidence of a quantum advantage.


The positive energy theorem (also known as the positive mass theorem) refers to a collection of foundational results in general relativity and differential geometry. Its standard form, broadly speaking, asserts that the gravitational energy of an isolated system is nonnegative, and can only be zero when the system has no gravitating objects. Although these statements are often thought of as being primarily physical in nature, they can be formalized as mathematical theorems which can be proven using techniques of differential geometry, partial differential equations, and geometric measure theory.


Richard Schoen and Shing-Tung Yau, in 1979 and 1981, were the first to give proofs of the positive mass theorem. Edward Witten, in 1982, gave the outlines of an alternative proof, which were later filled in rigorously by mathematicians. Witten and Yau were awarded the Fields medal in mathematics in part for their work on this topic.


An imprecise formulation of the Schoen-Yau / Witten positive energy theorem states the following:


The meaning of these terms is discussed below. There are alternative and non-equivalent formulations for different notions of energy-momentum and for different classes of initial data sets. Not all of these formulations have been rigorously proven, and it is currently an open problem whether the above formulation holds for initial data sets of arbitrary dimension.


The original proof of the theorem for ADM mass was provided by Richard Schoen and Shing-Tung Yau in 1979 using variational methods and minimal surfaces. Edward Witten gave another proof in 1981 based on the use of spinors, inspired by positive energy theorems in the context of supergravity. An extension of the theorem for the Bondi mass was given by Ludvigsen and James Vickers, Gary Horowitz and Malcolm Perry, and Schoen and Yau.


Gary Gibbons, Stephen Hawking, Horowitz and Perry proved extensions of the theorem to asymptotically anti-de Sitter spacetimes and to Einstein–Maxwell theory. The mass of an asymptotically anti-de Sitter spacetime is non-negative and only equal to zero for anti-de Sitter spacetime. In Einstein–Maxwell theory, for a spacetime with electric charge formula_1 and magnetic charge formula_2, the mass of the spacetime satisfies (in Gaussian units)


with equality for the Majumdar–Papapetrou extremal black hole solutions.


An initial data set consists of a Riemannian manifold and a symmetric 2-tensor field on . One says that an initial data set :


Note that a time-symmetric initial data set satisfies the dominant energy condition if and only if the scalar curvature of is nonnegative. One says that a Lorentzian manifold is a development of an initial data set if there is a (necessarily spacelike) hypersurface embedding of into , together with a continuous unit normal vector field, such that the induced metric is and the second fundamental form with respect to the given unit normal is .


This definition is motivated from Lorentzian geometry. Given a Lorentzian manifold of dimension and a spacelike immersion from a connected -dimensional manifold into which has a trivial normal bundle, one may consider the induced Riemannian metric as well as the second fundamental form of with respect to either of the two choices of continuous unit normal vector field along . The triple is an initial data set. According to the Gauss-Codazzi equations, one has


where denotes the Einstein tensor of and denotes the continuous unit normal vector field along used to define . So the dominant energy condition as given above is, in this Lorentzian context, identical to the assertion that , when viewed as a vector field along , is timelike or null and is oriented in the same direction as .


The ends of asymptotically flat initial data sets.


In the literature there are several different notions of "asymptotically flat" which are not mutually equivalent. Usually it is defined in terms of weighted Hölder spaces or weighted Sobolev spaces.


However, there are some features which are common to virtually all approaches. One considers an initial data set which may or may not have a boundary; let denote its dimension. One requires that there is a compact subset of such that each connected component of the complement is diffeomorphic to the complement of a closed ball in Euclidean space . Such connected components are called the ends of .


Let be a time-symmetric initial data set satisfying the dominant energy condition. Suppose that is an oriented three-dimensional smooth Riemannian manifold-with-boundary, and that each boundary component has positive mean curvature. Suppose that it has one end, and it is "asymptotically Schwarzschild" in the following sense:


Schoen and Yau's theorem asserts that must be nonnegative. If, in addition, the functions formula_6 formula_7 and formula_8 are bounded for any formula_9 then must be positive unless the boundary of is empty and is isometric to with its standard Riemannian metric.


Note that the conditions on are asserting that , together with some of its derivatives, are small when is large. Since is measuring the defect between in the coordinates and the standard representation of the slice of the Schwarzschild metric, these conditions are a quantification of the term "asymptotically Schwarzschild". This can be interpreted in a purely mathematical sense as a strong form of "asymptotically flat", where the coefficient of the part of the expansion of the metric is declared to be a constant multiple of the Euclidean metric, as opposed to a general symmetric 2-tensor.


Note also that Schoen and Yau's theorem, as stated above, is actually (despite appearances) a strong form of the "multiple ends" case. If is a complete Riemannian manifold with multiple ends, then the above result applies to any single end, provided that there is a positive mean curvature sphere in every other end. This is guaranteed, for instance, if each end is asymptotically flat in the above sense; one can choose a large coordinate sphere as a boundary, and remove the corresponding remainder of each end until one has a Riemannian manifold-with-boundary with a single end.


Let be an initial data set satisfying the dominant energy condition. Suppose that is an oriented three-dimensional smooth complete Riemannian manifold (without boundary); suppose that it has finitely many ends, each of which is asymptotically flat in the following sense.


Suppose that formula_10 is an open precompact subset such that formula_11 has finitely many connected components formula_12 and for each formula_13 there is a diffeomorphism formula_14 such that the symmetric 2-tensor formula_15 satisfies the following conditions:


The conclusion is that the ADM energy of each formula_12 defined as


For each formula_30 consider this as a vector formula_31 in Minkowski space. Witten's conclusion is that for each formula_32 it is necessarily a future-pointing non-spacelike vector. If this vector is zero for any formula_33 then formula_34 formula_35 is diffeomorphic to formula_36 and the maximal globally hyperbolic development of the initial data set formula_37 has zero curvature.


According to the above statements, Witten's conclusion is stronger than Schoen and Yau's. However, a third paper by Schoen and Yau shows that their 1981 result implies Witten's, retaining only the extra assumption that formula_20 and formula_21 are bounded for any formula_40 It also must be noted that Schoen and Yau's 1981 result relies on their


As of April 2017, Schoen and Yau have released a preprint which proves the general higher-dimensional case in the special case formula_42 without any restriction on dimension or topology. However, it has not yet (as of May 2020) appeared in an academic journal.


A black hole firewall is a hypothetical phenomenon where an observer falling into a black hole encounters high-energy quanta at (or near) the event horizon. The "firewall" phenomenon was proposed in 2012 by physicists Ahmed Almheiri, Donald Marolf, Joseph Polchinski, and James Sully as a possible solution to an apparent inconsistency in black hole complementarity. The proposal is sometimes referred to as the AMPS firewall, an acronym for the names of the authors of the 2012 paper. The use of a firewall to resolve this inconsistency remains controversial, with physicists divided as to the solution to the paradox.<ref name="nature/Fire in the hole">Astrophysics: Fire in the hole!</ref>


According to quantum field theory in curved spacetime, a single emission of Hawking radiation involves two mutually entangled particles. The outgoing particle escapes and is emitted as a quantum of Hawking radiation; the infalling particle is swallowed by the black hole. Assume a black hole formed a finite time in the past and will fully evaporate away in some finite time in the future. Then, it will only emit a finite amount of information encoded within its Hawking radiation. Assume that at time formula_1, more than half of the information had already been emitted.


According to widely accepted research by physicists like Don Page and Leonard Susskind, an outgoing particle emitted at time formula_1 must be entangled with all the Hawking radiation the black hole has previously emitted. This creates a paradox: a principle called "monogamy of entanglement" requires that, like any quantum system, the outgoing particle cannot be fully entangled with two independent systems at the same time; yet here the outgoing particle appears to be entangled with both the infalling particle and, independently, with past Hawking radiation.


In order to resolve the paradox, physicists may eventually be forced to give up one of three time-tested theories: Einstein's equivalence principle, unitarity, or existing quantum field theory.<ref name="sciam/Confound"> Originally published in Quanta, December 21, 2012.</ref>


Some scientists suggest that the entanglement must somehow get immediately broken between the infalling particle and the outgoing particle. Breaking this entanglement would release large amounts of energy, thus creating a searing "black hole firewall" at the black hole event horizon. This resolution requires a violation of Einstein's equivalence principle, which states that free-falling is indistinguishable from floating in empty space. This violation has been characterized as "outrageous"; theoretical physicist Raphael Bousso has complained that "a firewall simply can't appear in empty space, any more than a brick wall can suddenly appear in an empty field and smack you in the face."


Some scientists suggest that there is in fact no entanglement between the emitted particle and previous Hawking radiation. This resolution would require black hole information loss, a controversial violation of unitarity.


Others, such as Steve Giddings, suggest modifying quantum field theory so that entanglement would be gradually lost as the outgoing and infalling particles separate, resulting in a more gradual release of energy inside the black hole, and consequently no firewall.


Juan Maldacena and Leonard Susskind have suggested in ER=EPR that the outgoing and infalling particles are somehow connected by wormholes, and therefore are not independent systems; however, , this hypothesis is still a "work in progress".<ref name="nytimes/Mystery Wrapped"></ref>


The fuzzball picture resolves the dilemma by replacing the 'no-hair' vacuum with a stringy quantum state, thus explicitly coupling any outgoing Hawking radiation with the formation history of the black hole.


Stephen Hawking received widespread mainstream media coverage in January 2014 with an informal proposal to replace the event horizon of a black hole with an "apparent horizon" where infalling matter is suspended and then released; however, some scientists have expressed confusion about what precisely is being proposed and how the proposal would solve the paradox.


The firewall would exist at the black hole's event horizon, and would be invisible to observers outside the event horizon. Matter passing through the event horizon into the black hole would immediately be "burned to a crisp" by an arbitrarily hot "seething maelstrom of particles" at the firewall.


In a merger of two black holes, the characteristics of a firewall (if any) may leave a mark on the outgoing gravitational radiation as "echoes" when waves bounce in the vicinity of the fuzzy event horizon. The expected quantity of such echoes is theoretically unclear, as physicists don't currently have a good physical model of firewalls. In 2016, cosmologist Niayesh Afshordi and others argued there were tentative signs of some such echo in the data from the first black hole merger detected by LIGO; more recent work has argued there is no statistically significant evidence for such echoes in the data.


The no-hair theorem states 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 that formed a black hole or is falling into it "disappears" behind the black-hole event horizon and is therefore permanently inaccessible to external observers. Physicist John Archibald Wheeler expressed this idea with the phrase "black holes have no hair", which was the origin of the name. In a later interview, Wheeler said that Jacob Bekenstein coined this phrase.


The first version of the no-hair theorem for the simplified case of the uniqueness of the Schwarzschild metric was shown by Werner Israel in 1967. The result was quickly generalized to the cases of charged or spinning black holes. There is still no rigorous mathematical proof of a general no-hair theorem, and mathematicians refer to it as the no-hair conjecture. Even in the case of gravity alone (i.e., zero electric fields), the conjecture has only been partially resolved by results of Stephen Hawking, Brandon Carter, and David C. Robinson, under the additional hypothesis of non-degenerate event horizons and the technical, restrictive and difficult-to-justify assumption of real analyticity of the space-time continuum.


Suppose two black holes have the same masses, electrical charges, and angular momenta, but the first black hole was made by collapsing ordinary matter whereas the second was made out of antimatter; nevertheless, then the conjecture states they will be completely indistinguishable to an observer "outside the event horizon". None of the special particle physics pseudo-charges (i.e., the global charges baryonic number, leptonic number, etc., all of which would be different for the originating masses of matter that created the black holes) are conserved in the black hole, or if they are conserved somehow then their values would be unobservable from the outside.


Every isolated unstable black hole decays rapidly to a stable black hole; and (excepting quantum fluctuations) stable black holes can be completely described (in a Cartesian coordinate system) 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.


By changing the reference frame one can set the linear momentum and position to zero and orient the spin angular momentum along the positive "z" axis. This eliminates eight of the eleven numbers, leaving three which are independent of the reference frame: mass, angular momentum magnitude, and electric charge. Thus any black hole that has been isolated for a significant period of time can be described by the Kerr–Newman metric in an appropriately chosen reference frame.


The no-hair theorem was originally formulated for black holes within the context of a four-dimensional spacetime, 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 fields and massive vector fields (Proca fields, etc.).


It has since been extended to include the case where the cosmological constant 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.


Counterexamples in which the theorem fails are known in spacetime dimensions higher than four; in the presence of non-abelian Yang–Mills fields, non-abelian Proca fields, some non-minimally coupled scalar fields, or skyrmions; 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 solitons.


In 2004, the exact analytical solution of a (3+1)-dimensional spherically symmetric black hole with minimally coupled self-interacting scalar field was derived. This showed that, apart from mass, electrical charge and angular momentum, black holes can carry a finite scalar charge which might be a result of interaction with cosmological scalar fields such as the inflaton. The solution is stable and does not possess any unphysical properties; however, the existence of a scalar field with the desired properties is only speculative.


The LIGO results provide some experimental evidence consistent with the uniqueness of the no-hair theorem. This observation is consistent with Stephen Hawking's theoretical work on black holes in the 1970s.


A study by Stephen Hawking, Malcolm Perry and Andrew Strominger postulates that black holes might contain "soft hair", giving the black hole more degrees of freedom than previously thought. This hair permeates at a very low-energy state, which is why it didn't come up in previous calculations that postulated the no-hair theorem.


In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons (NGBs) are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries. They were discovered by Yoichiro Nambu in particle physics within the context of the BCS superconductivity mechanism, and subsequently elucidated by Jeffrey Goldstone, and systematically generalized in the context of quantum field theory. In condensed matter physics such bosons are quasiparticles and are known as Anderson-Bogoliubov modes.


These spinless bosons correspond to the spontaneously broken internal symmetry generators, and are characterized by the quantum numbers of these.


They transform nonlinearly (shift) under the action of these generators, and can thus be excited out of the asymmetric vacuum by these generators. Thus, they can be thought of as the excitations of the field in the broken symmetry directions in group space—and are massless if the spontaneously broken symmetry is not also broken explicitly.


If, instead, the symmetry is not exact, i.e. if it is explicitly broken as well as spontaneously broken, then the Nambu–Goldstone bosons are not massless, though they typically remain relatively light; they are then called pseudo-Goldstone bosons or pseudo-Nambu–Goldstone bosons (abbreviated PNGBs).


Goldstone's theorem examines a generic continuous symmetry which is spontaneously broken; i.e., its currents are conserved, but the ground state is not invariant under the action of the corresponding charges. Then, necessarily, new massless (or light, if the symmetry is not exact) scalar particles appear in the spectrum of possible excitations. There is one scalar particle—called a Nambu–Goldstone boson—for each generator of the symmetry that is broken, i.e., that does not preserve the ground state. The Nambu–Goldstone mode is a long-wavelength fluctuation of the corresponding order parameter.


By virtue of their special properties in coupling to the vacuum of the respective symmetry-broken theory, vanishing momentum ("soft") Goldstone bosons involved in field-theoretic amplitudes make such amplitudes vanish ("Adler zeros").


Consider a complex scalar field , with the constraint that formula_1, a constant. One way to impose a constraint of this sort is by including a potential interaction term in its Lagrangian density,


and taking the limit as . This is called the "Abelian nonlinear σ-model".


The constraint, and the action, below, are invariant under a "U"(1) phase transformation, . The field can be redefined to give a real scalar field (i.e., a spin-zero particle) without any constraint by


where is the Nambu–Goldstone boson (actually formula_4 is) and the "U"(1) symmetry transformation effects a shift on , namely


but does not preserve the ground state (i.e. the above infinitesimal transformation "does not annihilate it"—the hallmark of invariance), as evident in the charge of the current below.


Thus, the vacuum is degenerate and noninvariant under the action of the spontaneously broken symmetry.


The corresponding Lagrangian density is given by


Note that the constant term formula_8 in the Lagrangian density has no physical significance, and the other term in it is simply the kinetic term for a massless scalar.


The charge, "Q", resulting from this current shifts and the ground state to a new, degenerate, ground state. Thus, a vacuum with will shift to a "different vacuum" with . The current connects the original vacuum with the Nambu–Goldstone boson state, .


In general, in a theory with several scalar fields, , the Nambu–Goldstone mode is massless, and parameterises the curve of possible (degenerate) vacuum states. Its hallmark under the broken symmetry transformation is "nonvanishing vacuum expectation" , an order parameter, for vanishing , at some ground state |0〉 chosen at the minimum of the potential, . Symmetry dictates that all variations of the potential with respect to the fields in all symmetry directions vanish. The vacuum value of the first order variation in any direction vanishes as just seen; while the vacuum value of the second order variation must also vanish, as follows. Vanishing vacuum values of field symmetry transformation increments add no new information.


By contrast, however, "nonvanishing vacuum expectations of transformation increments", , specify the relevant (Goldstone) "null eigenvectors of the mass matrix",


The principle behind Goldstone's argument is that the ground state is not unique. Normally, by current conservation, the charge operator for any symmetry current is time-independent,


Acting with the charge operator on the vacuum either "annihilates the vacuum", if that is symmetric; else, if "not", as is the case in spontaneous symmetry breaking, it produces a zero-frequency state out of it, through its shift transformation feature illustrated above. Actually, here, the charge itself is ill-defined, cf. the Fabri–Picasso argument below.


But its better behaved commutators with fields, that is, the nonvanishing transformation shifts , are, nevertheless, "time-invariant",


thus generating a in its Fourier transform. (This ensures that, inserting a complete set of intermediate states in a nonvanishing current commutator can lead to vanishing time-evolution only when one or more of these states is massless.)


Thus, if the vacuum is not invariant under the symmetry, action of the charge operator produces a state which is different from the vacuum chosen, but which has zero frequency. This is a long-wavelength oscillation of a field which is nearly stationary: there are physical states with zero frequency, , so that the theory cannot have a mass gap.


This argument is further clarified by taking the limit carefully. If an approximate charge operator acting in a huge but finite region is applied to the vacuum,


a state with approximately vanishing time derivative is produced,


Assuming a nonvanishing mass gap , the frequency of any state like the above, which is orthogonal to the vacuum, is at least ,


Letting become large leads to a contradiction. Consequently 0 = 0. However this argument fails when the symmetry is gauged, because then the symmetry generator is only performing a gauge transformation. A gauge transformed state is the same exact state, so that acting with a symmetry generator does not get one out of the vacuum.


The argument requires both the vacuum and the charge to be translationally invariant, , .


Consider the correlation function of the charge with itself,


so the integrand in the right hand side does not depend on the position.


Thus, its value is proportional to the total space volume, formula_16 — unless the symmetry is unbroken, . Consequently, does not properly exist in the Hilbert space.


There is an arguable loophole in the theorem. If one reads the theorem carefully, it only states that there exist non-vacuum states with arbitrarily small energies. Take for example a chiral N = 1 super QCD model with a nonzero squark VEV which is conformal in the IR. The chiral symmetry is a global symmetry which is (partially) spontaneously broken. Some of the "Goldstone bosons" associated with this spontaneous symmetry breaking are charged under the unbroken gauge group and hence, these composite bosons have a continuous mass spectrum with arbitrarily small masses but yet there is no Goldstone boson with exactly zero mass. In other words, the Goldstone bosons are infraparticles.


A version of Goldstone's theorem also applies to nonrelativistic theories (and also relativistic theories with spontaneously broken spacetime symmetries, such as Lorentz symmetry or conformal symmetry, rotational, or translational invariance).


It essentially states that, for each spontaneously broken symmetry, there corresponds some quasiparticle with no energy gap—the nonrelativistic version of the mass gap. (Note that the energy here is really and not .) However, two "different" spontaneously broken generators may now give rise to the "same" Nambu–Goldstone boson. For example, in a superfluid, both the "U(1)" particle number symmetry and Galilean symmetry are spontaneously broken. However, the phonon is the Goldstone boson for both.


In general, the phonon is effectively the Nambu–Goldstone boson for spontaneously broken Galilean/Lorentz symmetry. However, in contrast to the case of internal symmetry breaking, when spacetime symmetries are broken, the order parameter "need not" be a scalar field, but may be a tensor field, and the corresponding independent massless modes may now be "fewer" than the number of spontaneously broken generators, because the


Goldstone modes may now be linearly dependent among themselves: e.g., the Goldstone modes for some generators might be expressed as gradients of Goldstone modes for other broken generators.


Spontaneously broken global fermionic symmetries, which occur in some supersymmetric models, lead to Nambu–Goldstone fermions, or "goldstinos". These have spin ½, instead of 0, and carry all quantum numbers of the respective supersymmetry generators broken spontaneously.


Spontaneous supersymmetry breaking smashes up ("reduces") supermultiplet structures into the characteristic nonlinear realizations of broken supersymmetry, so that goldstinos are superpartners of "all" particles in the theory, of "any spin", and the only superpartners, at that. That is, to say, two non-goldstino particles


are connected to only goldstinos through supersymmetry transformations, and not to each other, even if they were so connected before the breaking of supersymmetry. As a result, the masses and spin multiplicities of such particles are then arbitrary.


The theorem was proved first by Gell-Mann and Low in 1951, making use of the Dyson series. In 1969 Klaus Hepp provided an alternative derivation for the case where the original Hamiltonian describes free particles and the interaction is norm bounded. In 1989 Nenciu and Rasche proved it using the adiabatic theorem. A proof that does not rely on the Dyson expansion was given in 2007 by Molinari.


Let formula_1 be an eigenstate of formula_2 with energy formula_3 and let the 'interacting' Hamiltonian be formula_4, where formula_5 is a coupling constant and formula_6 the interaction term. We define a Hamiltonian formula_7 which effectively interpolates between formula_8 and formula_2 in the limit formula_10 and formula_11. Let formula_12 denote the evolution operator in the interaction picture. The Gell-Mann and Low theorem asserts that if the limit as formula_13 of


exists, then formula_15 are eigenstates of formula_8.


Note that when applied to, say, the ground-state, the theorem does not guarantee that the evolved state will be a ground state. In other words, level crossing is not excluded.


As in the original paper, the theorem is typically proved making use of Dyson's expansion of the evolution operator. Its validity however extends beyond the scope of perturbation theory as has been demonstrated by Molinari. We follow Molinari's method here. Focus on formula_17 and let formula_18. From Schrödinger's equation for the time-evolution operator


and the boundary condition formula_20 we can formally write


Focus for the moment on the case formula_22. Through a change of variables formula_23 we can write


This result can be combined with the Schrödinger equation and its adjoint


The corresponding equation between formula_28 is the same. It can be obtained by pre-multiplying both sides with formula_29, post-multiplying with formula_30 and making use of


The other case we are interested in, namely formula_32 can be treated in an analogous fashion


and yields an additional minus sign in front of the commutator (we are not concerned here with the case where


formula_33 have mixed signs). In summary, we obtain


We proceed for the negative-times case. Abbreviating the various operators for clarity


Now using the definition of formula_36 we differentiate and eliminate derivatives formula_37 using the above expression, finding


i \hbar \epsilon g \partial_g | \Psi_\epsilon \rangle &=


5. A.L. Fetter and J.D. Walecka: "Quantum Theory of Many-Particle Systems", McGraw–Hill (1971)


In mathematics, Bogoliubov's edge-of-the-wedge theorem implies that holomorphic functions on two "wedges" with an "edge" in common are analytic continuations of each other provided they both give the same continuous function on the edge. It is used in quantum field theory to construct the analytic continuation of Wightman functions. The formulation and the first proof of the theorem were presented by Nikolay Bogoliubov at the International Conference on Theoretical Physics, Seattle, USA (September, 1956) and also published in the book "Problems in the Theory of Dispersion Relations". Further proofs and generalizations of the theorem were given by R. Jost and H. Lehmann (1957), F. Dyson (1958), H. Epstein (1960), and by other researchers.


In one dimension, a simple case of the edge-of-the-wedge theorem can be stated as follows.


In this example, the two wedges are the upper half-plane and the lower half plane, and their common edge is the real axis. This result can be proved from Morera's theorem. Indeed, a function is holomorphic provided its integral round any contour vanishes; a contour which crosses the real axis can be broken up into contours in the upper and lower half-planes and the integral round these vanishes by hypothesis.


The more general case is phrased in terms of distributions. This is technically simplest in the case where the common boundary is the unit circle formula_1 in the complex plane. In that case holomorphic functions "f", "g" in the regions formula_2 and formula_3 have Laurent expansions


absolutely convergent in the same regions and have distributional boundary values given by the formal Fourier series


Their distributional boundary values are equal if formula_6 for all "n". It is then elementary that the common Laurent series converges absolutely in the whole region formula_7.


In general given an open interval formula_8 on the real axis and holomorphic functions formula_9 defined in formula_10 and formula_11 satisfying


for some non-negative integer "N", the boundary values formula_13 of formula_14 can be defined as distributions on the real axis by the formulas


Existence can be proved by noting that, under the hypothesis, formula_16 is the formula_17-th complex derivative of a holomorphic function which extends to a continuous function on the boundary. If "f" is defined as formula_14 above and below the real axis and "F" is the distribution defined on the rectangle formula_19