# Uniformity of black holes

 AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy Gravity, black holes and physics

4.5. The "black hole has no hair" theorem

In previous chapters, we have shown how the gravitational collapse of sufficiently massive stars creates black holes. But what specific "shape" do black holes have? The collapse of a spherical non-rotating star creates a spherical (spherically symmetrical) Schwarzschild black hole, the collapse of a rotating star creates an axially symmetric Kerr or Kerr-Newman black hole. However, these are all idealized cases. The question arises, what black hole will be created by the collapse of a real star? At first glance, it might seem that the collapse of a "deformed" star, whach may have a "bump" on the surface (protuberances, trace of planetary impact or gas accretion), could result in a "deformed" black hole with a "bump" on its horizon. As we will see below, according to the laws of the general theory of relativity, this cannot happen: either no black hole is formed at all, or an exactly symmetrical black hole is created.
Simply put, a black hole shows no trace of whether it formed from a star of round or deformed irregular shape.

The only thing the black hole manifests on the outside - the only thing left for the outside world from a collapsed object - is the outer field. It turns out that compared to the object before the collapse, the black hole has a very simple structure :

 Theorem 4.1 ("black hole has no hair")
 After the "completion" of the gravitational collapse (ie after the formation of the horizon and the disappearance of all gravitational and electromagnetic waves), the external electromagnetic and gravitational field of a stationary black hole in vacuum is completely determined by only three independent parameters : total mass M, electric charge Q and intrinsic rotational angular momentum J, regardless of what and how the black hole was formed from.

Metaphorically, this is expressed by the sentence "a black hole has no hair", ie it has no other specific independent characteristics other than mass, charge and rotational angular momentum *). We substantiate the statement "black hole has no hair" first by physical arguments, then we mention some geometric-topological theorems of uniqueness, on which the general mathematical proof is based, and finally we show the far-reaching physical consequences of this theorem.
*) The metaphorical wording "black hole has no hair" comes from J.A.Wheeler. By "hair" is meant here some specific symptoms that emerge from the horizon and show the individual characteristics of the matter from which the black hole originated. More precisely, we should say "a black hole has almost no hair", only weight, angular momentum and electric charge.
The "hairless" theorem he was just a hypothesis for several years, for which as evidenced by a number of physical arguments and findings from Zeldovich, Novikov, Ginzburg, Price, Israel, and Carter, but general proof was lacking. The definitive completion of the proof of Theorem 4.1 was performed in the works of B. Carter [44], W. Israel [142], S. Hawking [121] and D. Robinson [219].

Preservation of interaction with matter absorbed by a black hole
Mass
, charge and angular momentum are the only conserving quantities (which do not change with time during collapse), which are associated as sources with long-range force fields (mass and angular momentum with gravitational field, electric charge with electromagnetic field) on which they leave a unambiguous "reflection" or "trace". With the help of these unambiguous reflections (preserving integral flows over closed areas), these fields then continue to hold - maintain the values of these three quantities also after the horizon has been formed, even if their own sources disappeared below the horizon or were even perhaps already destroyed in singularity (in terms of their proper time). The creation of the horizon is therefore a decisive moment here: whatever then happens to the resources below the horizon, this cannot be reflected in any way in the external fields, which therefore retains its value corresponding to the resources just before crossing the horizon. This also justifies the use of Gauss's theorem for integral flows of external fields through closed surfaces, because we do not have to worry about possible unacceptable topological behavior (eg multiple connection or singularity) of spacetime inside the horizon. Therefore we can eg integral

of intensity of electric field E over a closed surface S, surrounding a black hole at a sufficient distance, to be considered as its electric charge Q, even if this real charge (its carriers) disappeared irreversibly below the horizon, was absorbed by singularity, or it emerged in another universe or another place in the same universe (all this, however, makes no sense to an outside observer, because it would take place in the absolute future). It is similar with the gravitational field and its source, mass M.
Thus, if we have two distant bodies or particles A and B , which are in mutual gravitational or electromagnetic interaction, after the absorption of one of them (for example A) this interaction does not disappear with a black hole. After radiating of dynamic components of fields arising during collapse or accretion, the long-range interaction with partner B remains. However, it will no longer be an interaction with the original individual body or particle A, but particle B will interact with the total field of the black hole of which the absorbed object A has become a part. And this field is determined by the total mass M, the rotational angular momentum J and the electric charge Q. It can be said that the particle A , even with its field, seems to "dissolve" in the total field of the black hole.
This mechanism also provides an answer to a frequently asked paradoxical question: "How can the gravitational force get out from under the horizon of a black hole and act on outer bodies? - when the horizon does not let anything out! ".
However, the gravitational field does not have to "penetrate" from the inside across the horizon. This field is already in that given external place and was there before the star collapsed or the body fell into a black hole. And when collapse or absorption body this field does not expire, but maintains that value it had just before the act of collapsing or absorption.
Note : The above response is in terms of the classical field theory . In quantum theory of fields (§1.1 "Atoms and nuclei" passage "Quantum field theory" in the monograph "Nuclear physics and ionizing radiation") the interaction is modeled by means of intermediate particles, which are constantly exchanging interacting bodies (particles). These should not be"let go" by the horizon of the black hole . The answer can be seen from two aspects :
1. It is only a model that is useful for dynamic processes between microparticles (discussed in the passage "
Virtual particles or real?" mentioned link); for a static force-field action between macroscopic bodies, it is probably irrelevant .
2. Intermedia particles are virtual, so for them to "ban" the passage of the horizon in principle does not apply. As will be shown in §4.7 "
Quantum radiation and thermodynamics of black holes", the laws of quantum physics allow near the horizon of generation (Hawking) radiation by the effects of the quantum tunneling phenomenon of virtual pairs of particles.

The way in which a black hole "
loses hairs" during a collapse can best be understood by analyzing the gravitational collapse, which differs from the symmetric case by only a small deviation (perturbation). The external field can then be decomposed into components related to multipole moments. During the collapse, the individual multipole moments will change over time, which will be accompanied by the emission of waves. It is the radiation of waves that is the mechanism that changes the value of some physical quantities during the collapse. Possibly asymmetries of electric charge cause the emission of electromagnetic waves during the collapse. More important, however, is the emission of gravitational waves (§2.7 "Gravitational waves", part "Sources of gravitational waves in space"), which carry away any initial asymmetries in the distribution of mass in a collapsed object.
According to the theory of radiation [166] can emit only the multipole moments, the "multi-polarity" is greater or equal to the spin of the respective field *), while the component with the smaller multipolarity is preserved and do not contribute to the radiation.
*) The spin of field here is sufficient to consider classically as a measure of symmetry in the plane wave of the field: the field has spin s, if its plane wave is invariant to rotation by an angle of 2p/s around the direction of propagation.
If the multipole moments, that are "allowed" to radiate completely, the resulting limit field will be fully characterized only by the remaining conserving multipole moments, whose multipolarity is less than the spin of field (and which cannot therefore radiate). For an electromagnetic field with a spin of s=1, all multipoles of the charge distribution are radiated in the form of electromagnetic waves, starting with the dipole moment, and only the monopole part - the total electric charge - is preserved. For a gravitational field with spin s=2, all mass distribution multipoles, starting with a quadrupole moment, are emitted in the form of gravitation waves; only the monopole part is preserved - the total mass, and the dipole part related to the own rotational angular momentum in the center of gravity system. For a (model) scalar field with spin s=0, no part would be preserved, everything will radiate.
A detailed analysis of the weakly perturbed gravitational collapse shows that before the formation of the horizon, all the higher multipole moments allowed are actually radiated. Price [212], showed that for contributions of such higher multipole fields there are no solutions with physically permissible asymptotic properties (either infinite on the horizon or divergent for infinite distances).
"Quantum Hair" ?

Quantum physics results in other conserving quantities associated with quantum fields. After forming a black hole, they could persist in the form of some "quantum hair". However, these are quantum fields short range; "quantum hair" could affect the properties of evaporating quantum black micro-hoes (such microholes are discussed in §4.7 "Quantum radiation and thermodynamics of black holes"). However, for macroscopic black holes, possible quantum micro-hairs have no consequences, because quantum physics is not applied at macroscopic scales. Extending the analysis to strong interactions [245] and weak interactions [116] shows that a macroscopic black hole cannot show the strengths of weak or strong interactions from leptons and baryons absorbed in the hole [10], [244].

Thus,
there are strong physical arguments in favor of the "black hole has no hair" theorem. However, since a detailed analysis of the course of gravitational collapse is possible only in considerably simplified cases, this does not prove the validity of theorem 4.1 for a general black hole. General matematical proof of the theorem "black hole has no hair" is based on a global geometric-topological methods to study the structure of space-time, through which you can get some information about the global structure of spacetime on the basis of certain general assumptions, without necessity of detailed solutions gravitational equations. Using these methods, some important theorems on the uniqueness of solutions in empty stationary asymptotically prognostic spacetime were derived.
The simplest "prototype" of such theorems on unitarity can already be considered Schwarzschild-Birkhoff theorem 3.3 (derived in §3.4 "Schwarzschild geometry") , according to which vacuum asymptotically planar spacetime with spherical symmetry has Schwarzschild geometry and is thus described by only one parameter M regardless of how, for example, the mass is distributed radially (if spherical symmetry is maintained). The relevant more general sentences on unambiguity are the following :

 Theorem 4.2 (Israel [141]):
 Each empty (Tik = 0) static spacetime, in which there is a regular event horizon with a sphere topology, has a Schwarzschild geometry (and is therefore centrally symmetric and characterized only by the parameter of the total mass M ).
 Theorem 4.3 (Israel [142]):
 Every static spacetime, which contains only a static electromagnetic field and which has a regular event horizon with a sphere topology, has a Reissner-Nordström geometry (it is thus centrally symmetrical and characterized by a total mass M and an electric charge Q ).
 Theorem 4.4 (Hawking [120], [121]):
 The stationary black hole must be either axially symmetrical or static (with the event horizon having a sphere topology) .

If the black hole were not axially symmetrical, it would emit gravitational waves (time-varying quadrupole moment) during rotation and would therefore not be stationary; only an axially symmetrical black hole can rotate and be stationary at the same time.

To illustrate theorem 4.4, imagine the trought experiment according to Fig.4.20. We surround the rotating axially symmetrical black hole at a sufficiently large distance from the horizon with a mass with a fixed axially non-asymmetrical distribution - for example, we make a rigid material frame around it (for example from iron bars), Fig.4.20a. The gravitational action of the frame causes the metric to no longer be axially symmetric. However, this is not a violation of theorem 4.4, because if the black hole rotates, the situation according to Fig.4.20a will no longer be stationary. Due to the entrainment by the angular momentum, the frame begins to rotate in the same direction as the black hole. Since the frame is axially asymmetrical, it will emit gravitational waves during this rotation and thus lose the kinetic energy of the rotation. Finally, the rotation of both the black hole and the frame brakes by radiation; the black hole becomes non-rotating (static) in accordance with Hawking's theorem 4.4. A somewhat more complicated situation would be if the material frame were not free, but fixed (by some non-gravitational forces) so that it could not begin to rotate and emit gravitational waves - Fig.4.20b. Even in this case, theorem 4.4 can be defended. The gravitational action of the frame deforms the horizon. However, the horizon rotates and the deformation of the horizon as it moves will emit gravitational waves (inside the hole), which will slow down its rotation. Finally, the rotation of the black hole to brake again.

Fig.4.20. Imaginary experiment to physically illustrate the theorem 4.4.
a) A rigid material frame is freely placed around the rotating black hole. Due to the entrainment of the rotation of the black hole, the frame will spin, gravitational waves will be emitted and the whole system will be radiation braked. This arrangement is also one of the ways to obtain energy from a rotating black hole (see §4.4).
b) Rigid material frame fixed around a black hole by external forces.

 Theorem 4.5 (Carter [44]):
 For an empty (Tik = 0) stationary axially symmetric spacetime in which there is a regular event horizon with a sphere topology, there are two different groups of solutions, each of which depends on only two parameters; as these parameters we can take the total mass M and the rotational angular momentum J .

One such group of solutions is Kerr's geometry at a2 £ M2; as Robinson showed, no other distinct group of solutions already exists. Carter's theorem 4.5 was further generalized in the works of Robinson [219], Bose and Wang [24] to the case of the presence of an electromagnetic field (ie for an electrically charged black hole), whereby the Kerr-Newman geometry of spacetime is obtained as a general solution (analyzed in §3.6 "Kerr and Kerr-Newman geometry") .

By combining these (partially overlapping) theorems 4.2 to 4.5, we conclude that (if the gravitational field of matter outside the horizon can be neglected) all stationary black holes must be axially symmetric and the geometry of spacetime is generally expressed by Kerr-Newman's solution, which is described only three free parameters: mass M, rotating angular momentum J and the electric charge Q. And this is theorem 4.1 "a black hole has no hair".
Thus, it turns out that in addition to mass, charge and angular momentum, all other characteristics (which would not be deducible from these three parameters) - such as inhomogeneities and asymmetries of mass distribution and charge of the body before collapse, magnetic fields, currents, pressures, turbulence, etc. - completely "radiates" in the form of gravitational and electromagnetic waves, part of which expands into space, part is absorbed by a black hole. This radiation will carry away the "hair" of the black hole. Inhomogeneities, as well as any other phenomena occurring already below the horizon, do not affect the external field, because information about them (in the form of appropriate disturbances and waves) of the space-time geometry does not let to the external observer.

Uniformity of black holes
Black holes of different origins are
indistinguishable from each other if they have the same mass, charge and rotational angular momentum. It is not recognizable whether the black hole originated from hydrogen or iron, from ordinary matter or from antimatter, from protons, electrons, neutrons and in what proportion *). Thus, black holes are very uniform compared to the great variety of different species and types of stars. All black holes are similar, like eggs with eggs: almost all of them are Kerr's variously fast rotating (elliptical), sometimes a Schwarzschild (spherical) hole may appear between them (probably not electrically charged). Other qualitative differences between black holes are not, are only differences in their "size". With the formation of a black hole, are disappearing almost all the information about what this black hole came from; only the total weight, angular momentum and electric charge can be detected.
*) In principle, it is possible, for example, to achieve such a concentration of light that part of its electromagnetic energy succumbs to gravitational collapse and forms a black hole. Or similarly, part of the (non-local) gravitational energy may succumb to gravitational collapse with sufficient concentration of gravitational waves
(see §2.7 "Gravitational waves" and B.3 "Wheeler's geometrodynamics. Gravity and topology."). Such a "purely electromagnetic" (or "purely gravitational") black hole will then be indistinguishable from the black hole created by the collapse of an (electrically neutral) star with the appropriate mass and angular momentum. However, the conditions for such a massive concentration of gravitational or electromagnetic waves could probably arise only in the initial stages of the universe's evolution after a hypothetical "big bang", or perhaps during the gravitational collapse of entire galaxies...
Black holes are the only macroscopic objects that are very simple. After the completion of the gravitational collapse and the departure of the radiation that carries away their "hair", a very simple object is created in the final state, which can be precisely and completely describe with only three parameters and a few mathematical formulas - without any idealization or limitation of simplifying models! We do not encounter this anywhere else in the macro world *). It has an analogy only in the microworld of "elementary" particles, which can also be completely described by a few of their quantum numbers (§1.5 "Elementary particles and accelerators" in the book "Nuclear Physics and Physics of Ionizing Radiation") .
*) It is wonderful and paradoxical that such a mysterious and exotic object as a black hole is, from a physical-theoretical point of view, the best described and explored "body" in nature ..!..

Loss of determinism? - An information paradox ?
Deterministic laws of classical physics make it possible to
predict the state of a system at any time in the future and in the past, based on knowledge of the complete set of state conditions of a given system at a certain moment (on Cauchy's spatial type hypersurface ). However, if black holes are present in the universe, even on the basis of a perfect knowledge of its state at a certain point in time, it is not possible to determine the state that was in the past, because the properties of the matter from which the black hole formed (and also the matter that was at any time in the past absorbed by a black hole), almost nothing can be detected; the black hole not only engulfed this matter, but also engulfed all the information about it with the exception only of summary mass, charge and angular momentum. The presence of a black hole - its property of "not having hair" - thus, even within classical physics, in principle limits the possibility of reconstructing the past on the basis of the best knowledge of the present state; it is only possible to predict the future (and only in the outer asymptotically planar region of spacetime - see §3.5). This "information paradox" thus leads to a partial loss of determinism. But as we will see in §4.7 "Quantum radiation and thermodynamics of black holes", the effect of quantum radiation of black holes even takes us (at least in principle) this last deterministic ability..!.. From the point of view of quantum physics, the "paradox of information loss" is discussed in §4.7, passages "Quantum evaporation: return of matter and information from a black hole?" and "Paradox of information loss?".

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Vojtech Ullmann