Rotating black holes - energy extraction, portals in space and time?

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

4.4. Rotating and electrically charged Kerr-Newman black holes

In the previous chapters, it was gradually analyzed how by the the complete gravitational collapse of a spherically symmetric star (spherical non-rotating and electrically uncharged) creates a Schwarzschild black hole. However, the conditions of exact spherical symmetry are almost never actually met, with most stars rotating in particular. The question arises whether a black hole can also be caused by the collapse of a rotating star, or whether the accelerating rotation (centrifugal force) is able to prevent collapse? In Newtonian physics, even the initially slow rotation of the star would eventually prevail and stop the collapse. However, in the general theory of relativity, it is shown that even with the collapse of a rotating star, a black hole can eventually form, albeit in a more complex way [222], [215], [227].

Collapse of a rotating star
A rotating star, which is always "flattened" due to centrifugal force, compresses into a disk-shaped formation (Fig.4.14a), a rapidly rotating "pancake", during collapse (due to constantly accelerating rotation due to maintaining angular momentum), whose diameter may exceed its thickness several times. If the angular momentum is not too great (compared to the mass of the star), the collapse can continue also in the equatorial plane and a black hole is formed without further complications - a rotating black hole. A more complex situation occurs when the angular momentum is large, eg greater than the critical value J = M2 (see §3.6). In such a case, the constantly accelerating rotation can tear the disc-shaped "pancake" into several parts *) - see Fig.4.14c. Some parts that receive a high angular momentum will fly away (and accordingly reduces the remaining angular momentum of the system), but most will be gravitationally bound and orbit around a common center of gravity. During their circulation, these fragments will emit gravitational waves (the system has a large quadrupole moment rapidly changing with time) carrying away the rotational energy and thus they will brake and converge (Fig.4.14c, d), until they gradually merge (Fig.4.14 e) and creates the resulting rotating black hole with angular momentum J <M2. First, large fragments are slowed down and merged, because they (according to formula (2.82) ) emit the most intense gravitational radiation.
*) Theoretically, there is also the possibility of forming a rotating structure having the shape of a toroid, in which the collapse would take place first along a smaller radius and then (if the "longitudinal" density of mass ~ energy is sufficient) or collapse along a larger radius. In reality, however, it can be expected that in such a toroid, immediately upon its formation, there will be certain inhomogeneities and deformations, which will then grow with time - the toroidal formation will disintegrate into several fragments and further evolution will then proceed as described above in the text (and the lower part of Fig.4.14).

Fig.4.14. Gravitational collapse of a rotating star.
a) When a rotating star collapses, it compresses into a sharply rotating disk-like formation. If the rotation is not fast enough to stop the equatorial plane collapse, Kerr's black hole will form.
b) With a large angular momentum due to the constantly accelerating rotation, instabilities may be transported and the disk-shaped structure may be torn into several parts.
c , d) The individual fragments then revolve around a common center of gravity, brake by the emission of gravitational waves, gradually converge and connect.
e) Eventually, this fragments are absorbed and a rotating black hole is formed with a correspondingly reduced rotational angular momentum.

Kerr-Newman geometry
However, a black hole created by the collapse of a rotating star will no longer be spherically symmetric due to rotation, but can only be axially symmetric. The so-called Kerr-Newman space-time geometry is considered to be a general solution for an axially symmetric rotating and electrically charged black hole (for the reason see the following §4.5), which we described from a geometric point of view in §3.6 "Kerr and Kerr-Newman geometry". Kerr-Newman geometry is a generalization of Schwarzschild geometry roughly in the sense that Schwarzschild geometry is spherical, while Kerr-Newman geometry is generally elliptical. Some aspects of Kerr-Newman geometry have been discussed in the aforementioned §3.6; here we will continue this analysis in order to study the properties of rotating and electrically charged black holes, whose spacetime is described by this geometry.
The element of the space-time interval of Kerr-Newman geometry (3.41) in Boyer-Lindquist coordinates was given in §3.6; we will write it again here :

 (4.25)

where M is the total mass , Q is the electric charge , J is the rotational angular momentum of the axially symmetric source of this geometry; a = J / M is the specific rotational angular momentum per unit mass of the black hole. The event horizon r = rg+ has a radius

 (4.26)

The radius of the horizon is thus smaller than that of an uncharged non-rotating black hole of the same mass (for this case of the Schwarzschild black hole, relation (4.26) passes into relation (3.14), ie rg = 2M). In conventional spherical coordinates a horizon of a rotating black hole flattened ellipsoid shape (like the rotation centrifugal force deforms the Earth into a slightly flattened shape with a diameter of about 20km difference between the equator and poles; see also notice on Fig.4.16) and thus significantly, the faster the black hole rotates - for the higher angular momentum J. The question arises as to whether a rotation too fast could tear the horizon by "centrifugal force" (similar to what would happen with an excessively fast spin, such as a grinding wheel or a circular saw). In §4.6 "Laws of black hole dynamics" it will be shown that a black hole cannot be rotated above a certain maximum "allowed" or "insurmountable" speed of the so-called extreme Kerr black hole, when M2 = J2/M2. If we try to throw a body into a black hole rotating at almost extreme speed, which would spin it even more, centrifugal forces will prevent this body from penetrating close enough to the horizon, fly out and not be absorbed; see also the commentary on inequality below (4.36).
For the physics of black holes are especially interesting the areas of space r ³ rg+ above the outer horizon - event horizon *). Everything that takes place below the horizon r = rg+ is causally separated from the rest of the universe and has no effect. This applies even to the inner horizon r = rg - = M - Ö(M2 -a2 -Q2) as well as the annular singularity around r = 0, described in §3.6 "Kerr and Kerr-Newman geometry".
*) In §3.5 and 3.6, the analytical extension inside r = r
g + was discussed . However, this internal solution below the horizon does not seem to have physical significance. On the one hand, this area is filled with the interior of the collapsing body. In addition, the analogy of Birkhoff's theorem does not apply to collapse with rotation (§3.4, theorem 3.3); Kerr's metric does not describe the outer spacetime during the collapse, but the asymptotic shape of the metric after the end of all dynamic processes. Other reasons why the complex internal geometric-topological structure of spacetime inside a rotating black hole is not realized are discussed below in the section "Black holes - bridges to other universes? Time machines?", Fig.4.19.

Influence of black hole rotation. Ergosphere.
To clarify the nature of the physical phenomena that can take place around the Kerr-Newman black hole (and we will see that very interesting effects can be expected here), we will first look at the motion properties of the test particles. The properties of the orbits of the test particles in the field of the Kerr-Newman geometry of a general black hole are considerably more complex than in the case of the Schwarzschild geometry of a non-rotating uncharged black hole. However, for qualitative knowledge of some basic aspects of motion it is not necessary to solve complicated Carter equations (3.44), we will suffice with metric coefficients in the space-time element (4.25).

Therefore, according to Fig.4.15a, let us have a test particle (uncharged) in a place with specified coordinates r,J , which moves only in the j direction; such a motion is called stationary because the space-time geometry around the particle does not change. We will be interested in the angular velocity with respect to the asymptotically rest frame of reference W s dj /dt = (dj/dt)/(dt/dt) = uj/ut , where uj and ut are the respective components of the four-velocity (we investigate the stationary motion, ie only in the direction j, so uJ = ur = 0). This angular velocity cannot be arbitrary, only values of W are permissible for which the 4-velocity u lies inside the light cone of the future :

u 2   =  g tt u t u t + 2 g t j u t u j + g jj u j u j   <0   ,

which, expressed by W, gives: gtt (ut)2 + 2 gtj W (ut)2 + gjj W2 (ut)2 < 0, i.e.

 g tt + 2 W g t j + W 2 g jj   < 0   . (4.27)

Thus, the geometry of spacetime "dictates" to a body located at a location with given r and J a permissible angular velocity in the range

 (4.28)

where W-max is the maximum angular velocity of circulation against the direction of rotation and W+max is the maximum velocity of circulation in the direction of rotation of the black hole. For large distances r from a black hole (or at its small angular momentum J®0) is W-max » -1/r, W+max » 1/r, so the particle velocity u = W .r Î <-1, +1> can be at most equal to the speed of light, both in the direction of rotation and against it, as is the case in the plane space-time STR (movement allowed inside the light cone). As you approach to the black hole, the value W-max increases, so the body must move more slowly against the direction of rotation of the black hole, than if the rotation of the black hole were not (Fig.4.15a).

 Fig.4.15. Tearing down the motion of bodies near a rotating black hole. a) The effect of dragging of local inertial systems (tilting of light cones in the direction of rotation of a black hole) leads to a gradual limitation of the maximum possible speed u- movement of bodies against the direction of rotation of a black hole when approaching a black hole. Far from the black hole (for r®¥) is u+ = c, u- = -c, in finite distances outside the ergosphere it is u+ > 0, u- <0, | u- | <| u+ |. On the static limit, however, u- = 0, in the ergosphere even u- > 0 Þ the body must necessarily move in the direction of rotation of the black hole. b) Pparticles free-falling on a rotating black hole first move in the radial direction, but as they approach the horizon, the movement turns in the direction of rotation around the black hole; on the horizon the particle "falls" in the tangential direction and remains rotated along with the horizon. c) A particle thrown at a black hole in the opposite direction to the rotation of the black hole, when approached the horizon, is forced by the entrainment effect to change the direction of orbit and land on the black hole as co-rotating - just like the particle in Fig. b). In Fig. b), c) the particle trajectories for the case of non-rotating (Schwarzschild) black holes ( WH = 0) are plotted in dashed lines for comparison .

Rotation black hole with the outer geometry of space exhibits dragging (entrainment) local inertial system, who free body forces perform a rotational movement about a black hole, and the more so the nearer they are to the rotating black hole *). This is similar to a ball rotating in a viscous liquid entraining the liquid near its surface. The "space-time entrainment" effect of rotating gravity (by angular momentum) allows a certain amount of energy to be obtained from a black hole, which is related to its rotation (see "Extraction of rotational energy" below).
*) The effect of entrainment of inertial systems in the gravitational field of a rotating body (not just black holes) leads to a kind of "spin-orbital interaction" between the rotation ("spin") of the central gravitational body and the angular momentum of orbiting bodies: motion of bodies, eg their energy (see relation (4.39)), depends on the mutual orientation of the angular momentum of the black hole and the orbital moment of the body.
In the gravitational field of rotating bodies, there is also a "spin-spin interaction" with bodies that have their own rotation. For example, a gyroscope in the gravitational field of a body rotating at an angular velocity
W will perform a precessional motion with respect to an asymptotically inertial system - its axis of rotation having an angle j will rotate at an angular velocity w º dj/dt » W .rg+/r. This phenomenon is called the Lense-Thirring effect according to the authors who first studied it [248], see §2.5, passage "Rotating gravity".
It can be said that Kerr-Newman geometry in a sense rotates together with a black hole *) - spacetime light cones are tilted in the direction of rotation of the black hole. An observer located at a site with coordinates r, J is at rest with respect to local geometry, and the directions +j and -j are equivalent for it, only when it rotates (orbits) around a black hole at an angular velocity

 (4.29)

such an observer is called locally non-rotating.
*) For common rotating bodies (macroscopic objects, planets, ordinary stars), the Lense-Thirring gravitational-rotational effects are negligible, mostly at the limits of measurability (it is discussed in §2.5, section "
Rotating gravity", passage "Possibilities of verifying the effect of rotation"). However, an intense gravitational vortex similar to a massive tornado is created in the vicinity of a rapidly rotating black hole ! The rotation of the black hole carries the surrounding space with it and forces it to rotate in a "vortex" depending on the speed of rotation of the black hole and the distance from it. At a greater distance from the rotating black hole, the space rotates very slowly, the observer does not even feel it (similar to far from a tornado, air flows only slowly). As you approach the horizon, the space rotates faster and "dragging" the observer in the direction of rotation (similar to increasing wind speed near a tornado). At the horizon, space rotates at the same speed as the horizon itself, each observer is relentlessly drawn into the vortex of rotation.
The effect of entraining the motion of bodies near a rotating black hole has a significant effect on the motion of particles and bodies falling into a black hole. In Fig.4.15b,c the trajectories of two test bodies falling into a rotating black hole (from the point of view of the coordinate system of an external static observer) are schematically drawn. The particle in Fig.4.15b moves in a free fall in the radial direction towards the black hole. If only the black hole did not rotate (static Schwarzschild black hole), the particle would move in a straight line to the horizon (initially faster and faster, close to the horizon it would stop its fall from the point of view of an external observer and would remain permanently "frozen" at a given place on the horizon - discussed in § 4.2, passage "Two different views of gravitational collapse - external and internal"). However, the rotation of the black hole causes the particle's motion to turns in the direction of rotation around the black hole as it approaches the horizon. The particle "hits" the horizon in an almost tangential direction and remains rotated with the horizon (from the point of view of an external observer, the particle again "freezes" on the rotating horizon forever; from the point of view of the particle itself, it reaches the horizon in a short time of its own time, flies through it and heads towards the center of a black hole...). The second particle in Fig.4.15c is thrown at the black hole in the opposite direction to the rotation of the black hole. As you approach the horizon, the effect of entrainment gradually forces you to reverse the direction of orbit and lands on the black hole as a co-rotating - it ends up in the same way as the particles in case b).

Because real bodies can only move inside the light cone, this tilting of the space-time light cones limits the maximum possible speed of movement of the body against the direction of rotation of the black hole, from the original value equal to the speed of light to a value the lower the closer it is to the black hole (as shown above). On the surface given by the equation

 r  =  r S  =  M +  Ö (M 2 - a 2 cos 2 J - Q 2 ) (4.30)

the maximum possible speed of the stationary observer against the direction of rotation of the black hole is already zero. This surface (in the shape of a rotating ellipsoid, Fig.4.16), on which no object can move against the direction of rotation of the black hole (gt j = 0), is called the static limit; outside this limit there may be static (immobile with respect to infinity) bodies, inside not. The area spread between the static boundary and the horizon is called the ergosphere. Inside the ergosphere, the entrainment is already so strong that no body can remain still here; it is inevitably drawn into rotation and its rotation around the black hole is not prevented by any force - the light cones are completely turned in the direction of rotation of the black hole. The ergosphere (which only rotating black holes have) is the largest in the "equatorial" region and narrows towards the poles, where the static limit r = rS touches the horizon r = rg+ (Fig.4.16).

 Fig.4.16. Schematic drawing of the horizon, static limits and region of the ergosphere of a rotating Kerr-Newman black hole with a rotational angular momentum J in the direction of the Z axis. This is a "side" view - from the equator side; the "top" view (from the pole side) is in Fig.4.17. Note: The r, J coordinates marked in the figure for better orientation are not common spherical coordinates! Near a rotating black hole, the coordinates r, J differ from the spherical coordinates because they are "elliptical" (spheroidal) Boyer-Lindquist coordinates (§3.6). In common spherical coordinates, the horizon of a rotating black hole has a flattened ellipsoidal shape.

In the ergosphere, there are orbits on which the particles have a negative total energy with respect to infinity - the binding energy exceeds the proper mass of the particle *). When passing over the static limit r = rS, the time component of the metric tensor goo changes sign and becomes negative. Therefore, the energy E = m gik uk = m(goou° + goaua) a particle of mass m moving at the velocity ua may become negative for some orbits of the test particle. Such a particle with negative energy cannot leave the ergosphere and, when absorbed by a black hole, brings negative energy below the horizon with respect to infinity - it reduces the weight of the black hole. Negative energy geodesics are completely enclosed within the ergosphere, so that no free-moving particles from the outer region can enter the negative energy orbit; to achieve a negative energy orbit, the body inside the ergosphere must be given additional (non-gravitational) acceleration - see the Penrose process below.
*) From the analysis of the relation (4.34 ") for energy of particles in the Kerr-Newman field implies that for the E <0 it is necessary an orbit with opposite-direction motion (L
j <0), whose parameters r,.r,Lj meet a certain (generally relatively complicated) inequality [8]; it follows from this inequality that the boundary of the region containing negative energy orbits is precisely the static limit given by equation (4.30).

Rotational energy extraction: Penrose process ; Superradiation; Blandford-Znajek process
Gaining a certain amount of energy from a black hole related to its rotation is generally made possible by the effect of "entrainment of space-time" (local inertial systems) by the angular momentum of a rotating gravitational field. The ergosphere has the interesting property (and hence its name - Greek ergos = work ) that it is possible to obtain from the black hole the part of energy (~ matter) that is related to rotation. Penrose [205] investigated the following effect (Fig.4.17): if body A enters the ergosphere and disintegrates here into two parts B and C so that one part, such as B , enters the opposite-direction orbit with negative energy and is absorbed by a black hole, the second part C can recoil and fly out of ergosphere with greater energy than the original body, while reducing the rotational angular momentum of the black hole. This phenomenon, which makes it possible to extract rotational energy and angular momentum from a black hole, is now called the Penrose process.

Fig.4.17. Penrose's process of obtaining energy from the ergosphere of a rotating black hole. Body A flies into the ergosphere and there, at the appropriate time, splits into two bodies B and C so that body B enters an orbit with negative energy (with respect to infinity) and is absorbed by a black hole. The second part C then flips from ergosphere with more energy than the original body A .

If the body A had (with respect to infinity) the total energy EA and the absorbed body B the energy EB <0 (got into orbit with negative energy), the change in the mass of the black hole when absorption will be DM = EB <0. Total energy balance then it will look like this:

 E C   =  E A - D M  > E A   ; (4.31)

therefore, the total energy obtained in this process is

 D E =  E C - E A   =   - D M  > 0  . (4.31 ')

The balance of kinetic energy is also interesting; for this purpose, we divide the total energy of the body into the kinetic energy Ekin and the rest mass: EA = EAkin + mA.c2, EC = ECkin + mCc2. The obtained kinetic energy DEkin = ECkin - EAkin = EC - mCc2 - EA +mAc2 then after substituting from (4.31) and mA = mB + mC it comes out

 D E kin   =   m B . c 2 - D M  ; (4.32)

in other words, all the rest mass of the absorbed body B and, in addition, part of the mass of the black hole were converted into the kinetic energy of the flying body C.
A black hole, if it has a maximum (extreme) rotational speed, can theoretically store up to 42% (of mc2) of its mass in the form of rotational energy. This energy is contained in a rotating gravitational field outside the horizon, so it can in principle be drawn.
Regarding the practical (astrophysical) significance of the Penrose process, various mechanisms and arrangements have been proposed for obtaining energy from rotating black holes [204], [205] *) .
*) Penrose even designed a curious sci-fi project "black hole power plant" using energy obtained by throwing waste into a rotating black hole. Waste containers are thrown into a black hole, the lid opens inside the ergosphere and the waste is "poured" into a path with negative energy. The containers thus gain considerable kinetic energy through the Penrose process, they fly out of the ergosphere quickly and can impact the turbine blades connected to the electric generator.

However, a more detailed analysis of the motion of bodies in the Penrose process by Bardeen, Press and Teukolský [8] showed, that the decay of a body within the ergosphere would have to take place with a very high mutual speed of both fragments - at least c/2. Therefore, for macroscopic bodies, the Penrose process is unlikely to be astrophysically significant; we do not know the physical processes that could give such great mutual velocities to macroscopic material objects. For charged particles, however, such a mechanism may be a strong magnetic field around a black hole - the so-called Blandford-Znajek process outlined below.
Another possibility would be to inject unstable particles (eg mesons m, p or K, see "Elementary particles and their properties") into the ergosphere, which would decay into muons, electrons and neutrinos inside the ergosphere. These secondary particles would fly from each other at relativistic velocities (> c/2), so that they could be accelerated by the Penrose process due to the rotation of the black hole. However, the "practical use" of the kinetic energy of these accelerated particles flying out of the ergosphere would be very difficult (cf. "Antiparticles - antiatoms - antimatter - antiworlds").

The wave (radiation) similar to Penrose's process is the effect of so-called superradiation [287], [8], when a wave incident on a rotating black hole is - if it has a suitable wavelength - amplified by a black hole at the expense of the rotational energy of a black hole. Let us have a (monochromatic) wave of a classical field with frequency w, angular momentum (axial quantum number) l and possibly an electric charge q, which hits (from infinity) on a black hole. In a stationary axially symmetric Kerr-Newman field, such a wave will be described by the function (solving the wave equation of the respective field) y(t,r,J,j) = F(r,J) e -i w t e i l j ; for scalar and electromagnetic fields it is possible to separate variables, ie F(r,J) can be written in the form of the product of two functions: F(r,J) = R(r). S(J) [43], [246]. Because the Kerr-Newman black hole is stationary and axisymmetrically, the quanties w and l will be integrals of motion. Part of the wave will be absorbed and after interacting with the black hole, the "scattered" wave, which will have the same frequency w, will continue to move, but generally a different amplitude. Changes in the mass, charge, angular momentum, and area of a black hole during this interaction are bound by the 1st law of Black Hole Mechanics (see §4.6, equation (4.50)) :

d M  =  ( k / 8 p ) d A + W d J + F d Q  .

The ratio of energy flow, angular momentum and charge (generally admit that the wave can also carry an electric charge q) in the incident and scattered wave is equal to w : l : q, so thanks to the conservation of energy and angular momentum, changes in the relevant parameters will be black in the same ratio holes dM : dJ : dQ = w : l : q . The first law of black hole dynamics then gives

d M ( 1 - W l / w - F q / w )  =  ( k / 8 p ) d A   .

Because according to the 2nd law of black hole dynamics (see §4.6; we assume that the considered wave satisfies the energy condition (2.60) of non-negative local energy density for each observer) is dA> 0, the inequality will apply

d M ( w - l W - q F )   ³ 0   .

If it is

w <  l W + q F

*) Based on the effect of superradiation, some thought projects [210] have also been proposed to allow highly developed civilizations to draw large amounts of energy from rotating black holes, as well as the "black hole bomb" mechanism: when a black hole is surrounded by a spherical mirror, the electromagnetic waves will be reflected many times to the black hole and amplified by superradiation ("positive feedback" will be created), so that their intensity (energy) will increase avalanche until the explosion.
This is how the process of superradiation for "boson" classical arrays with integer spin takes place. In the case of "fermion" classical fields, it turns out that superradiation does not occur. This is caused by the fact that the energy-momentum tensor of the classical field does not meet the half-integer spin energy conditions (2.60), and therefore can not be applied mechanics 2.law black holes, and from a quantum viewpoint, Pauli principle allows the presence of only one particle at a time for a given frequency of scattering wave, so that the scattered wave cannot be stronger than the incident wave. Quantum aspects of superradiation and the related effect of quantum evaporation of black holes will be discussed in §4.7, section "Mechanism of quantum radiation".

Electromagnetic extraction of rotational energy - Blandford-Znajek mechanism
Another interesting modification of the Penrose process
, with the participation of a strong magnetic field, was discussed by R.Blandford and R.Znajek in an extensive work [20]. However, the charged particles swirling in the plasma of the accretion disk around the black hole, represent an effective electric current generating a magnetic field.
It turns out that an external very strong magnetic field around a rotating black hole by its force acting can the fast-moving charged particles to introduce into negative energy orbits in the ergosphere of a rotating Kerr black hole, which could lead to the extraction of rotational energy
(and angular momentum) from black hole by the Penrose process. The extracted energy in the ergosphere would be transmitted to an electromagnetic field, which could then accelerate other charged particles. Such the Blandford-Znajek mechanism could work inside accretion disks around rotating black holes, where it could contribute to the energy of relativistic jets from quasars and active galactic nuclei, see §4.8 "Astrophysical significance of black holes", part "Thick accretion disks. Quasars".
When a plasma of charged particles orbits a rotating black hole, it creates a strong poloidal magnetic field by rotating toroidal currents flowing in the equatorial plane. The entrainment of space, and thus of magnetic field lines by the rotation of a black hole
(magnetic power lines twist into a spiral), then induces an intense electric field, acting on the charged plasma particles in the direction along the axis of rotation of the black hole. It accelerates these particles at relativistic speeds - a powerful electric generator is formated in the form of a stream of charged particles. Some of them enter orbits with negative energy in the ergosphere and fall into a black hole, while the extracted energy strengthens the electromagnetic field. Other charged particles are then electromagnetically accelerated by the extracted rotational energy and transfer this energy to the plasma in the jets by magnetohydrodynamic effects. Such a "gravito-magnetic dynamo", driven by the rotation of a black hole, could supply a considerable amount of energy to the jets from the accretion disk.
Thus, the above mechanisms show that rotating black holes are "alive" unlike "dead" Schwarzschild black holes, from which no energy can be obtained (if we do not take into account accretion discs and quantum phenomena). This "liveliness" of rotating black holes can have considerable astrophysical significance (in §4.8, section "Accretion disks around black holes" - "Thick accretion disks. Quasars" will be discussed) .

The motion of particles in the field of a rotating black hole
The motion of test particles in the Kerr-Newman field of a general black hole is given by equations (3.44). From equations (3.44a) and (3.44b) we get for a particle with rest mass mo, electric charge q, energy (with respect to infinity) E and axial component of the angular momentum Lj (with respect to the axis of rotation of a black hole) the relation

 (4.33)

For the coefficients at individual powers of energy E (which are functions of the place and parameters of the test particle and the black hole), introduces the designation [43] :

 (4.33a) (4.33b) (4.33c)

The above equation then has the form

 (r2 + a2 cos2 J) [ (dr/dl)2 + (dJ/dl)2 ]  =  a E2 - 2 b E + g  . (4.34)

If we express energy from here

 (4.34 ')

is seen (given that a > 0 everywhere outside of the horizon) that satisfies the energy inequality

 E    ³ . (4.35)

On the horizon r = rg+ is a = (rg+ 2 + a2)2, b = (Lj a + q Q rg+)(rg+ 2 + a2), g = (Lj a + q Q rg+)2, so the relation (4.35) is simplified here to :

 (4.36)

only if this condition is met can the particle reach the horizon and be absorbed. Inequality (4.36), which from below limits the amount of energy brought below the horizon when absorbing a body with charge q and angular momentum Lj , plays an important role in the dynamics of black holes - see §4.6 "Laws of black hole dynamics", where the relation (4.51) follows directly from it.

The analysis of the motion of test particles in Kerr-Newman geometry according to Carter's equations (3.44) is generally quite complicated [228], [48] , [237]. We achieve a certain simplification if we investigate only the motion of particles in the "equatorial" plane of a rotating black hole. Equatorial orbits in the Kerr-Newman black hole field are at the same time the most interesting, the most characteristic and the most important for practice. Due to the effect of dragging of local inertial systems, for example, the accretion disk around the rotating black hole (§4.8) is rotated into the equatorial plane and the actual accretion takes place mainly from the equatorial orbits. At great distances from the black hole, the effect of its rotation is small again and the trajectory of the particles here does not depend too much on in which plane and in which direction with respect to the axis of rotation of the black hole the particle moves (the motion of the particle at great distances is similar to that in the Schwarzschild field of a non-rotating black hole).
The equation of the radial component of motion in the equatorial plane of the Kerr-Newman geometry is obtained from equation (4.34) by substituting J = p/2 and dJ/dl = 0 :

 r 4 ( dr / d t ) 2   =   a E 2 - 2 b E + g   . (4.37)

a, b, g are again denoted by quantities according to (4.33a,b,c), where, however, sin2 J = 1 and cos2 J = 0. This equation can also be rewritten in the form

 (4.37 ')

The only factor that can change the sign here is factor II; factors I and III are positive (if energy E is positive). Quantity

 V (r) = (4.38)

here, therefore, it plays the role of an effective potential for the radial component of the motion. A particle with energy E can only get to places where V(r) £ E in accordance with the general relation (4.35). The places where V(r) = E are the turning points where the radial component of the motion changes its direction. Circular orbits are (together with the condition V(r) = E) given by the condition dV(r)/dr = 0. Thus, circular orbits are at the extremes of the potential V(r), while the minima V(r) correspond to stable circular orbits. The conditions V(r) = E and dV(r)/dr = 0 lead to the relations (for movement in the equatorial plane, Lj = L)

 (4.39)

wher` E º E / mo and` L º L / mo are the energy and the angular momentum per unit mass of the test particles [8], [81]. The angular and orbital velocities in orbit (from the point of view of a distant observer) are equal

 (4.39 ')

which is a generalization of 3.Kepler's law for circular equatorial orbits in the Kerr metric of rotating black holes. Formulas (4.39) are given for simplicity only for the Kerr black hole, ie for Q = 0 (otherwise the rest mass mo could not be included in E and L and in addition would depend on the specific charge q/mo of the test particles). The upper signs applies to the movement in the direction of rotation of the black hole (L.a> 0), the lower signs for orbits against the direction of rotation (L.a <0).
From relations (4.39) it can be seen that circular orbits can exist only at such distances from the black hole for which the condition is met

 r 3/2 -   3 M r 1/2  ± 2 a .M 1/2   > 0   . (4.40)

The photon circular orbit (on which` E = ¥,` L = ± ¥) in the equatorial plane of the Kerr black hole r = rf , which is also the innermost (boundary) circular orbit, has a radius rf given by the equation r3/2 - 3 M r1/2 ± 2 a.M1/2 = 0, ie.

 r f   =  2m [1 + cos 2 / 3 arccos (± a / m)]   . (4.41)

For a = 0 we get rf = 3M in accordance with the relation (4.5) for the photon sphere of the Schwarzschild black hole, for the extreme black hole a = M is rf = M for the co-rotating orbits and rf = 4M for the opposite photon paths. As in the Schwarzschild field, not all circular orbits are bound here: for orbits close enough to the boundary photon orbit, the energy value (4.39) becomes greater than one (` E º E/mo > 1) - particles on such a circular orbit E is not tied to a black hole and, under the influence of the slightest upward perturbation, flies out of this orbit into the infinity.
Bound circular orbits exist only for r ³ rmv , where

r mv   = 2M + 2M (M ± a) 1/2  ± a

is the boundary radius of the bound orbit corresponding to E/mo = 1. This radius rmv is also the minimum "perihelion" of all parabolic orbits (ie orbits with E/mo = 1, eg bodies falling from infinity); every parabolic trajectory of a body that penetrates closer than rmv ends at a black hole. For a = 0 we get rmv = 4M (compare with Fig.4.6 on the left), for a = M it is for the co-rotating circular paths rmv = M, resp. rmv = 5.83 M for opposite orbits. In order for a circular orbit to be stable to radial perturbations (even all bounded circular orbits are not stable!), it must correspond to the minimum effective potential, ie the condition d2V(r)/dr2 L 0 must still be met, leading to inequality

 r 2 - 6 M r ± 8 a Ö (M r) - 3 a 2  ³  0   . (4.42)

Only for radii satisfying this inequality can there exist stable circular orbits, where the equation v (4.42) corresponds to the innermost (lowest, limit) stable circular orbit r = rms [8] :

 (4.42 ')

For a = 0 we again get the radius of the innermost stable circular orbit rms = 6M in the Schwarzschild geometry, for an extremely rotating black hole a = M the correspondingly co-rotating innermost stable circular orbit has a radius rms = M, while for the innermost stable opposite circular orbit it is rms = 9M.

At large distances r from the rotating black hole, the parameters (binding energy, specific angular momentum, velocity of circulation) of the co-rotating and counter-rotating orbits are almost the same. Approaching the black hole (with decreasing r ), however, increases the influence of the angular momentum of a black hole - the binding energy of the co-rotating orbits increases, while the binding energy of the opposite orbits decreased in comparison with Schwarzschild orbits. The most striking difference motion of test particles compared to the Schwarzschild geometry will be near the extreme Kerr black hole (a = M), where the trajectories of particles moving in the direction and against the direction of rotation of the black hole will also differ the most. This is markedly reflected in the parameters of boundary stable circular orbits. The innermost stable circular orbit around the Schwarzschild black hole had the following characteristics (see §4.3): radius r = 6M, specific angular momentum` L = 2Ö(3) M, energy E = mo .8/9 and binding energy of particle Ebind = mo - E = 5.72% mo.
For the extreme Kerr black hole, according to formulas (4.42) and (4.39), the
innermost stable co-rotating orbit will have a radius r = M, a specific angular momentum` L = 2M/Ö3, energy E = mo/Ö3 and the binding energy of the particle will be Ebind = 42.26% mo; in circulation against the direction of rotation of the black hole is the innermost stable circular orbit in the equatorial plane have a radius r = 9M, specific angular momentum of` L = 22M/Ö27 , energy E = 5M/Ö27 and the binding energy of the particle will only Ebind = 3.77% from mo.
If we compare this with the results obtained in the previous §4.3 (passage "Emission of gravitational waves when moving in the field of a black hole"), specifically with relation (4.21), the most remarkable is that a body orbiting in a circular orbit (co-rotating) in the equatorial plane extreme Kerr's black holes glow out in the form of gravitational waves more than 40% of its rest weight! This value of the binding energy at the marginal stable co-rotating orbit is also of great importance for assessing the potential efficiency of energy release in accretion disks around black holes - see §4.8, section "Accretion disks".

Light propagation in the field of a rotating black hole
Similar to the trajectories of test particles, the rotation of a black hole also affects the propagation of light in its field - it affects photon orbits. The angle of deflection of the photons by a spherically symmetric Schwarzschild black hole depends only on the impact parameter, not on the direction from which they come to the black hole. There is only one photon sphere around Schwarzschild's black hole, on which photons can move at all angles accordig on the direction from which they reached.
However, if the black hole rotates, the situation is more complicated - the geometry is not spherically symmetrical, but only axially symmetrical, so that for rays coming from different directions, spacetime appears differently curved. The angle of deflection of the rays (and the whole character of their trajectory) can significantly depend on the direction of movement of the photons with respect to the direction of rotation of the black hole. In general, rays directed against the direction of rotation are deflected more than photons going in the direction of rotation - Fig.4.18. These differences are most pronounced for photons passing in close proximity to a black hole. Indeed, from relation (4.41) it can be seen that there are two photon circular orbits in the equatorial plane around a rotating black hole: an opposite circular photon orbit with a larger diameter than Schwarzschild photon sphere, and a co-rotating photon orbit lying lower than for a non-rotating black hole. The total effective cross-section of the gravitational capture of photons by a rotating black hole does not differ much from the corresponding value (4.15) for the Schwarzschild black hole. However, the capture angle is asymmetric - photons going in the opposite direction of rotation are captured with a larger effective cross section than photons coming along the direction of rotation - Fig. 4.18b.

Fig.4.18. Light propagation in the gravitational field of a rotating black hole.
a) Light geodesics in the equatorial plane of the Kerr black hole. The angle of deflection Dj- of a photon moving against the direction of rotation is (at the same collision parameter) greater than the angle Dj+ of deflection of a photon along the direction of rotation of a black hole. The co-rotating photon orbit r = rf+ has a larger radius than the opposite photon orbit r = rf-. b) The effective cross-section of capturing photons (as well as other particles) by a rotating black hole is asymmetric - counter-moving photons are captured more efficiently than photons going in the direction of rotation. c , d) Rotating (Kerr) gravitational lens: c - asymmetrical beam path in equatorial plane; d - deflection of rays in the general plane.

Winding the light into the spiral by the black hole's rotation
In §2.5, the passage "
Rotating gravity", we showed that the space-time around a rotating body also rotates in a certain sense - the Lense-Thirring effect of entrainment of local inertial systems in the direction of rotation of the source. The space-time around the massive rotating body is deformed not only in the radial direction, but the resulting potential well also twists in the direction of the source's rotation. In the vicinity of rotating black holes, this effect is very pronounced and is also visible in light. The angular momentum of a rotating black hole should generally cause the electromagnetic wave to spiral in the direction of rotation. Astrophysically, this effect of "twisted" space-time could be most evident in the radiation emitted by the accretion disk in the equatorial plane :
-> This light would not travel to us from the black hole in a straight line, but initially along a curved path.
-> When an electromagnetic wave passes through rotating space-time near a black hole, its wavefront will be deformed. Phase of the wave - alternation of maximum and minimum - with the influence of entrainment by rotating gravity. they will move through the field and acquire a helix shape along the beam, they will form a spiral. It would be a kind of "rotating, twisting or spiraling light", the polarization of which would be different from the usual circular one. If it hits charged particles, it causes, among other things, their rotation around the direction of propagation of the beam, which can be considered as a manifestation of the angular momentum of this radiation. So it behaves like a wave with orbital angular momentum - cf. §1.5, passage "Intrinsic and orbital angular momentum of electromagnetic waves".

Parameters of the Kerr-Newman black hole
In addition to the expressions (4.26) and (4.30) for the horizon and the ergosphere, we present other important relationships between the quantities characterizing a general black hole. The horizon area of the Kerr-Newman black hole is

 A =  r=2M,nt=const.|gJJ gjj|1/2 dJ dj = 4p (rg+2 + a2)  = =  4p [ 2 M2 - Q2 + 2M Ö(M2 - Q2 - a2) ] (4.43)

in geometrodynamic units; in ordinary units, the area of the horizon is given by the formula

 A  =  4pG/c4 [ 2 G M2 - Q2 + 2 Ö(G2 M4 - J2 c2 - G M2 Q2 ) ]   . (4.43 ')

The angular velocity of a black hole horizon (which is the angular velocity with which a locally non-rotating observer orbits on the horizon with respect to infinity) is calculated from (4.29) by placing r = rg+ (which can be further adjusted by substituting from (4.26) and (4.43) for rg+ and area A ) :

 (4.44)

It can be seen that this angular velocity is the same at all points on the horizon (the black hole rotates like a "solid body"), so WH is defined as the angular velocity of the black hole.
For completeness, we will also state the
electrical potential of the horizon :

 (4.45)

the four-potential of the electromagnetic field Ai(r, J) around the Kerr-Newman black hole at the location with the coordinates r, J is (in the Lorentz calibration Ai ; i = 0) given by

 A t  =  Q r / (r2 + a2cos2 J) ,  A j = - a Q r sin2 J / (r2 + a2cos2 J)   ; (4.46)

(due to stationarity and axial symmetry, the potential does not depend on time t and on angle j and its other components are equal to zero).
The surface gravity k on the horizon (see §4.3), which is the gravitational force acting on the test particle on the horizon and having the same angular velocity WH as the horizon (ie on the locally non-rotating particle on the horizon) is equal to

 (4.47)

It can be seen at first glance, that for an extreme black hole (for which rg+ = rg- = M) the surface gravity is zero (!).
It is useful to define the so-called irreducible mass of a black hole Mired (the meaning of this quantity and its name will be shown in §4.6 in connection with the laws of dynamics of black holes), which is the mass that would have Schwarzschild black hole with the same surface of the horizon - 16p Mired = 4p(rg+ 2 + a2) :

 M ired  º  Ö (A / 16 p )   =  1 / 2 Ö (r g + 2 + a 2 )   . (4.48)

For a black hole of total mass M , the quantity M ired is limited by inequality

M / Ö 2    £   M ired  £   M   ,

where the upper limit corresponds to a Schwarzschild black hole and the lower limit to an extreme black hole. From (4.48), (4.43) and (4.44) the following relations can be deduced :

 (4.49)
 W H  =   a / (4 M 2 ired )   =   J / (4 M . 4 M 2 ired )   . (4.44 ')

The physical significance of the important relationship (4.49) will be discussed in §4.6 "Laws of black hole dynamics" in connection with the 2nd law of Black Hole Dynamics .

Classification of black holes
In general Kerr-Newman geometry, which in the next paragraph we will show is probably the most general space-time geometry of a general stationary black hole, we can distinguish some significant special cases (we can compare with the classification of Kerr-Newman geometry in §3.6) :

• a) Especially for Q = J = 0 we get the Schwarzschild geometry of a centrally symmetric non-rotating uncharged black hole.
• b) For Q = 0, 0 <J <M2 it is Kerr's geometry of axially symmetric rotating uncharged black holes.
• c) For J = 0, 0 <Q <M we obtain the Reissner-Nordström geometry of non-rotating centrally symmetric electrically charged black holes.
• d) At M2 = Q2 + J2/M2 it will be the Kerr-Newman geometry of the so-called extreme black hole - the fastest possible rotation (or the largest possible electric charge) of the black hole with respect to its total mass M; the horizon has a radius rg = M, surface gravity on the horizon k = 0.
• e) Finally, in the case of Q2 + J2/M2 > M2, Kerr-Newman geometry no longer describes a black hole having a horizon (expression (4.26) becomes imaginary), but the so-called Kerr naked singularity without horizon (properties of naked singularities and their existence we discussed in §3.9 "Naked singularities and the principle of "cosmic censorship" ").

Kerr-Newman geometry has a horizon (and thus describes a black hole) only if the condition M2 l Q2 + J2/M2 is met, ie if it "does not rotate too fast" or is not "too electrically charged" compared to its total mass M.
Rotation
(intrinsic angular momentum J) plays an important role for the geometry of spacetime around a black hole formed by gravitational collapse, because most stars are rotating and angular momentum of the rotation due to the law of conservation of angular momentum during the collapse would not change significantly (unless the case shown in Fig.4.14 occurs, where a considerable part of the angular momentum can be carried away by gravitational waves; however, only the "excess" angular momentum is carried away, the remaining rotational momentum is still considerable). However, it is different with the electric charge of black holes :
Electrically charged black holes? - no !
The electric charge Q is probably not important for black holes in space. In order for an electric charge to leave a noticeable mark on the space-time metric, it would have to have a huge value comparable to the total mass M (in geometrodynamic units). Because stars are usually electrically virtually neutral, the creation and maintenance of such a large charge is very unlikely; in addition, such large electrical repulsive forces would probably prevent collapse to dimensions of ~ 2M, because in areas r>2M these electrical repulsive forces would be completely dominant. Even if a highly electrically charged black hole formed, it would soon "discharge" in a real situation. If there is a material environment around the black hole (interstellar matter), opposite charges (particles, ions) will be attracted and captured from the surroundings by the charged black hole - by this selective absorption(accretion) of oppositely charged particles with the charge neutralizes the black hole quickly *). But even when a heavily charged black hole is in a vacuum, quantum electron-positron pair formation processes in a strong electric field will play a neutralizing role, repelling black-hole charges and attracting and absorbing opposite charges through the black hole. The electric charge of a stationary black hole is therefore practically zero . The reason why we dealt with the influence of charge and Reissner-Nordström geometry here and in §3.5 was rather theoretical - to show interesting (and at least physically possible in principle) properties of spacetime.
*) If a rotating black hole is immersed in an environment with a strong magnetic field, a certain equilibrium very small value of electric charge, determined by the speed of rotation and the intensity of the external magnetic field, may persist even in the steady state.
The electric charge is therefore irrelevant for black holes - black holes are electrically uncharged, Q = 0 .

Black holes - bridges to other universes? Time machines?
Rotating or electrically charged black holes have a complex geometric and topological structure of spacetime
(as we have shown in §3.5 "Reissner-Nordström geometry" and §3.6 "Kerr and Kerr-Newman geometry"); there has been speculation that they could serve as bridges - tunnels or portals - to other universes, or tunnels between distant places or various times in the same universe. We will try to critically assess these possibilities from a physical point of view.
Wormholes

Hypothetical "tunnels" in space-time that connect the shorter route two remote sites in the same space, or two places in various otherwise separate universes, are called "worm holes" - according to the similarity with connecting paths that a worm biting through an apple.
A worm that is located on the surface of an apple and has to get from one side to the other can do this in two ways. If it considers its space to be two-dimensional (the surface of the apple), it must crawl over the entire circumference of the apple. However, if he perceives the apple as a three-dimensional object, he can choose the path inside it and bite through to the desired opposite place along a much shorter path (or also longer in the case of a crooked corridor ...). It does not travel two-dimensionally on the surface of the apple, but uses the three-dimensional shortcut in the way - "wormhole".
The wormhole in the apple allows only a slight shortening of the path between the two places on the surface-peel. However, with a suitable geometric-topological structure in space, a wormhole could represent an absolutely fundamental shortening of the path between very distant places in space (eg from many light-years to hundreds of meters or kilometers)! An observer who passes through a wormhole does not exceed the limiting speed of light c anywhere, but can still cover the distance between the starting point and the destination much faster than a light beam flying straight. Hypothetically, wormholes could thus represent peculiar shortcuts not only in space, but also in time - enabling "time travel" (cf. the passage "
Closed world lines and time travel" in §3.3 and "Time travel: fantasy or physical reality?").
According to this analogy with the worm, we intuitively imagine that the shortcut path between different places or universes takes place through another, higher dimension, through hyperspace. In mathematical topology (§3.1 "Geometric-topological properties of spacetime"), however, a space (manifold) can be multiply connected by its internal structure, so there is no need to use additional dimensions to "travel" between different parts.
Something that could be a forerunner of the worm hole model, was first described in 1935 by A.Einstein and N.Rosen as the so-called Einstein -Rosen bridge (see §3.4 "
Schwarzschild geometry", Fig.3.18) in Schwarzschild's solution of the centrally symmetric gravitational field.

 Fig.3.18. a) Schematic representation of the geometric structure of the section (spatial hyperplanes) v = t = 0, J = p/2 Schwarzschild spacetime in the form of nesting into auxiliary three-dimensional Euclidean space. This auxiliary three-dimensional space has no physical significance (it is only a means of representation); only the internal geometry of the nested surface, which shows the two asymptotically planar regions A and A' connected by the Einstein-Rosen bridge, is relevant. b) A topological tunnel between two places in the same universe. c) "Wormhole" between two places in the same universe.

The name "worm hole" was proposed in 1957 by J.A.Wheeler. The mouth or gullet of a wormhole would look like a two-dimensional sphere. Unlike the spherical horizon of a black hole, which allows only a "one-way path without return", mostly to perdition, however, the estuary of a wormhole is a bidirectional through surface (at least in principle). The wormhole has an entrance and an exit, we can pass them both inside the wormhole and out into the outer universe. If a hypothetical astronaut penetrates a wormhole, he could find himself in another place in space and at a different time, in the future or in the past (depending on the values of the metric tensor in both mouths).. Possibly, even in another causally separated universe (according to the global topology of spacetime)..?.. The wormhole has no direct connection with the black hole, but in many mathematical models of black holes there is also a wormhole inside.
From a geometrical-topological point of view, a wormhole can be defined as a compact region of spacetime, whose spatial boundary is topologically equivalent to a simple sphere S3, but whose interior is topologically connected to places outside this initial region. It is thus a compact region W in asymptotically flat spacetime, whose topology has the form W ~ R ´S, where R is the Riemann space and S is a 3-dimensional tube whose boundary has a topology of the shape ¶S ~ S2 and the hyper-surface S described in space-time has a spatial (space-like) character.
From a geometric point of view, we have two types of wormholes :

a)
Tunnels connecting different separate universes ; b) Connections between different distant places of the same universe (ours).
In terms of "throughput", we also recognize two types of wormholes :
- Traversible - stable, allowing movement across space (spacetime), wormholes connecting two different and distant places in our universe or different universes. To stabilize these wormholes need matter with a negative energy density (see the discussion below "Can wormholes really exists?") .
- Non - traversible - very short-lived, vanishing so fast that no real object can fly through them without being absorbed by singularity. Such are the characteristics of wormholes inside black holes (such as the Einstein-Rosen bridge), connecting various separate universes. Of course, all virtual worm micro-holes in quantum topological foam are non-traversible (again discussed below in the section "Can Worm Holes Really Exist ?") .
Ordinary tunnel
(such as railway) connects two places in space, a wormhole connects two places in space-time. If there were a wormhole, it would theoretically make it possible to bridge two very distant places in space in a much shorter path. An observer (astronaut) through a wormhole - even if it does not exceed the speed of light anywhere - could cover the distance between the two places much faster than a light beam flying in the usual straight path (through the space outside the wormhole). He would reach a distant place before the light. Due to the fact that the observer "overtakes" light with this shortcut, from the point of view of the theory of relativity he can "overcome time" - he can travel in time to the future and back to the past. The possibilities of fictitious or real time travel are discussed in §3.3, part "Closed worldlines and time travel ", then systematically in the syllabus "Time Travel: Fantasy or Physical Reality?".
Can wormholes really exists ?
Large "wormholes" could occur in space as remnants of turbulent processes with matter and space-time during the Big Bang, which cosmic inflation has expanded to macroscopic or even astronomical sizes. Microscopic wormholes measuring about 10
-33 cm everywhere and constantly created and destroyed as a result of quantum gravity fluctuations of spacetime metric and topology (see §B.4 "Quantum geometrodynamics ") - in quantum topological foam.
Wormholes inside black holes they have a "mayfly" life. In addition to gravitationally dynamic effects, quantum radiation effects also contribute to this. A random quantum of radiation falling on a black hole from the outside will accelerate to high energies due to gravity and will "bomb" even the esophagus of a wormhole, which will shrink and close quickly as a result. Any object that would attempt to pass through such a wormhole during its short life will disappear at the moment of the wormhole's rupture, along with it, in the resulting singularity. Due to this extremely short duration of topological tunnel is problematic even speak of the existence of wormhole ...
In order for a macroscopic wormhole could maintain stable and "through - traversible" must be present or mass field inducing antigravity effects (material with negative energy density), which would avoid the tendency of gravity to "necking" in singularities and the formation of a black hole - by Raychaudhuryho equation (2.59) deviation geodesics (
§2.6 " deviation and focusing of geodesics ") energy condition opposite to (2.60) causes expansion of geodesics after passage through the wormhole. The only hypothetical possibility of stable wormholes is to "reinforce" them with a specific type of substance that would have antigravity effects and thus effectively "push" the walls of the wormhole apart; in terms of energy-momentum tensor components, this substance must have a large stress in the radial direction, exceeding the energy density. Therefore, a mass with a negative energy density *) is needed to stabilize the wormholes (violating the weak energy condition (2.60)), which K.Thorne called "exotic matter".
*) The exotic substance must have a negative energy from the point of view of the observer or the light beam passing through the wormhole, not necessarily from the point of view of the observer who is at rest inside the wormhole. In the theory of relativity, the energy density can be negative in one frame of reference, positive in another.
Black holes are (at least at the level of current astrophysical knowledge) a legitimate physical consequence of stellar evolution
(§4.1. "The role of gravity in the formation and evolution of stars" and §4.2, section "Complete gravitational collapse. Black hole.") and their existence is evidenced by strong astronomical observational data. However, wormholes are still a mere hypothesis - we do not know of any natural way in which wormholes in the universe could form outside the interiors of black holes.
So far there are no observational indications for wormholes !

Topological tunnels inside black holes
About the complete analytical extension of the Kerr-Newman geometry shown in Figs.3.21 to 3.25 and the resulting geometric-topological consequences can be said essentially the same as the extension of the Schwarzschild geometry (see end of §4.3). Let us first notice Schwarzschild's geometry, for which a Kruskal diagram was constructed in §3.4 (Fig.3.17) showing the presence of two mirror-inverted universes - Fig.3.18a. However, in a gravitational collapse, Schwarzschild's solution describes the geometry of spacetime only above the surface of a collapsing star. Therefore, most part of the idealized (an extended) Schwarzschild solution on the Kruskal diagram is therefore not realized - is "cut off" by the inside of a collapsing star (Fig.4.19a).
In §3.5 "Reissner-Nordström geometry" and §3.6 "Kerr and Kerr-Newman geometry" we have shown, that an observer who, during his motion in Reissner-Nordström or Kerr-Newman spacetime, intersects the inner (Cauchy) horizon r = rg-, can get into another "universe". Accordingly, in the event of a gravitational collapse (of an electrically charged or rotating mass), there would be a possibility that after crossing the Cauchy horizon r = rg- the shrinking mass may avoid singularity and begin to expand again into another region of spacetime ("another universe") - Fig.4.19b.
The Einstein-Rosen bridge and analogous structures of Kerr-Newman geometry can be understood as a "bridges" connecting two different asymptotically planar universes (assuming the usual Euclidean global topology of each). With a suitable topology, such a bridge could connect two different places of the same (multiple conected) universe - Fig.3.18b, c. Such a "worm hole" (as defined above) could create a "topological shortcut" between two distant regions of spacetime. There is even speculation about the possibility that a certain movement of both ends of the wormhole could form a closed time curve. From here, it is only a step towards the sci-fi idea of how a technically advanced civilization, with technology that allows it to manipulate (through gravity) a wormhole, triggers a "time machine" ("Time Travel: Fantasy or Physical Reality?") ...
Such travel between different universes, or between different distant places in the same universe, or even time travel, may be an exciting topic for science fiction, but the reality is probably more prosaic. The exact Schwarzschild, Reissner-Nordström or Kerr solution containing "tunnels" between different universes is valid only under the conditions for which it was derived, ie for an otherwise completely empty asymptotically planar universe.
At the end of §3.5, it was noted that an observer moving so that it intersects the inner horizon r = rg-, will see during the final interval of his own time the whole further history of the "universe" he is leaving. Any body from this universe would be seen by an observer approaching rg- with a violet shift increasing to infinity. In connection with this, it has been shown [114], [192] that the Cauchy horizon r = rg- inside a black hole is unstable to electromagnetic and gravitational perturbations occurring outside the black hole (classical instability) *). Analysis of quantum processes of particle formation in strong fields inside a black hole further shows [192 ] (see also §4.7) that there is also a quantum instability of the inner horizon **), which manifests itself even in the case when the space is empty from the classical point of view.
*) Figuratively speaking, a particle that would fly through such a black hole into another universe would "demolish" behind it this theoretical tunnel into other universes.
**) In addition, the existence of closed time worldlines would allow particles to interfere with their own past through a time loop. Quantum considerations suggest that the disturbances resulting from such phenomena would intensify spontaneously and, with their high energy, would eventually destroy the topological tunnel.
..

Fig.4.19. The space-time structure of a real black hole.
a) In Schwarzschild's spacetime of a static black hole created by the collapse of a (non-rotating) star, a large part of the structure of the Kruskal's diagram is cut off by interior of collapsing star.
b) The collapse of a rotating star into a black hole could theoretically create a space-time structure allowing it to travel to another universe. The collapsing matter could then emerge in the second universe in the form of a "white hole".
c) However, due to the diverging intensity of the incoming radiation at the inner horizon and the quantum production of particles around the singularity, much of the theoretical structure of Kerrr-Newman geometry (including inner horizons and other universes on the Penrose diagram) cannot actually be realized. The collapse creates a singularity that is in the future and takes on a spatial character, which does not allow any travel to other universes.

These instabilities (to perturbations and quantum processes) lead in practice to the destruction and disappearance of the Cauchy horizon and the emergence of a singularity of the spatial type - Fig.4.19c. Therefore, it can be expected that in the real situation the structure of spacetime within the event horizon r = rg+ will be qualitatively similar to Schwarzschild's black hole and the complex topological construction according to Fig.3.25 will not be realized (matter would have to pass through the singular region before).
Again, we come to the conclusion that the "exotic" possibilities of traveling between different universes, or time travel, could work at most within elementary particles (or only a single elementary particle !?) - cf. with a discussion of closed time curves in §3.3., passage "Closed worldlines and time travel". The individual mechanisms are described in more detail in the work (syllabus) "Journeys through time: fantasy or physical reality?". At the end of §3.4, the note "Dual solutions in analytic extension - reality or fiction?", doubts are discussed about the very physical meaning of the full extensions of Schwarzschild and Kerr-Newman geometry of spacetime, on which all ideas about traveling between different universes or in time are ideologically based..?..
Some related considerations about the direction of time flow are further outlined in §5.6, section "
Time arrow".

Black holes - "hatcheries" of new universes ?
In addition to the idea that black holes are portals to other worlds (universes), an even bolder sci-fi hypothesis has emerged (recently revived and supplemented by L.Smolin): that new universes are being born in black holes . Our entire universe could have been born of a black hole from another universe; big-bang would really just be a widening of a black hole in another universe. The idea also emerged that these emerging universes also carried away the physical laws of the original ("mother") universe - as if there was a kind of "cosmic inheritance". This speculative "evolutionary cosmology of natural selection" would then be similar in some sense to Darwinian evolutionary biology, as it said that the most common types of universes would be the ones that make the most "copies of themselves"; universes with a large number black holes formed would have a certain "reproductive advantage" passing on to the next "generations" of universes...
From the point of view of the sober physical approach of general relativity and quantum theory, however, these speculative concepts appear to be unfounded :
1. Black holes do not produce new universes - from a classical (non-quantum) point of view they do not arise inside black holes at all (as discussed above ), from a quantum point of view, spontaneous fluctuations in space-time, which can potentially lead to the emergence of new universes, are taking place everywhere and constantly, not just inside black holes.
2. There is no mechanism by which any particular physical information can be transferred from one universe to another. Within classical GTR, this is forbidden by Hawking's and Penrose's theorems on singularities. In the approach of quantum cosmology, chaotic quantum field fluctuations and metrics in the "topological foam" of space-time effectively erase and randomize any macroscopic physical structure and information from another (previous) universe..!..

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Vojtech Ullmann