Laws of black hole dynamics

Chapter 4
B L A C K    H O L E S
4.1. The role of gravity in the formation and evolution of stars
4.2. The final stages of stellar evolution. Gravitational collapse
4.3. Schwarzschild static black holes
4.4. Rotating and electrically charged Kerr-Newman black holes
4.5. The "black hole has no hair" theorem
4.6. Laws of black hole dynamics
4.7. Quantum radiation and thermodynamics of black holes
4.8. Astrophysical significance of black holes
4.9. Total gravitational collapse - the biggest catastrophe in nature

4.6. Laws of black holes dynamics

Let us now notice some general laws in physical phenomena with the participation of black holes, ie in the interaction of black holes with the surrounding matter and with each other. Based on the analysis of specific physical phenomena using physical laws of gravity, mechanics, electricity and other fields, as well as geometric-topological methods of studying the structure of space, the general laws of a global nature applicable to all processes with the participation of black holes, were derived. These general laws govern the behavior of some important global quanties describing a black hole and the surrounding matter. Due to a very close analogy with the laws of thermodynamics (as will be shown below), these laws are usually called "laws of black hole dynamics", or the "laws of black hole mechanics ", or "thermodynamics".
  In the following §4.7 of quantum processes around black holes to show that this striking similarity laws dynamics of black holes and the laws of thermodynamics is not accidental, but has profound physical meaning.

The physical significance of the second and third members in relation (4.50) is the work delivered (or taken) to a black hole to change its rotational angular momentum J and electric charge Q. The meaning of the first member is different and will only arise in connection with the second law of black hole dynamics a from analogy with thermodynamics.

Thus, the first law of black hole dynamics is simply the law of conservation of the most important physical quantities compatible with the "black hole has no hair" theorem (black holes do not have other conservation laws than the law of conservation of energy-momentum, angular momentum and electric charge *). The quantities mass-energy, momentum, angular momentum and charge are measured using its gravitational and electromagnetic effects in the asymptotically flat spacetime surrouding the studied system (using the Gauss theorem for integral field flows or by analysing the trajectories of test particles).
*) Note that, for example, the law of conservation of baryon charge (numbers) does not apply to processes with black holes!

The first law of black hole mechanics can be advantageously used in calculations of the motion and interactions of black holes, similarly as it is advantageous to use the laws of conservation of energy and momentum in the calculation of collisions of bodies in mechanics. It is however necessary to include energy, momentum and angular momentum carried away by gravitational waves. For example, in the collision and fusion of two black holes with masses M₁ and M₂ , the resulting black hole will have the mass M = M₁ + M₂ - M_wave , where M_wave is the energy-mass carried away by gravitational waves during the process of approaching and merging the two holes; analogously for momentum and angular momentum.

The second law of black hole mechanics, which plays a fundamental role in black hole physics, was arrived at by various methods on the one hand Hawking [120], [121], on the other hand Carter [43], Christotolou [53], Bardeen [9] and Ruffini [54]. Christodolou and Ruffini studied how the parameters of a black hole change when interacting with the surrounding matter and field, for example when small material objects are thrown into it. The absorption of such a small body changes the mass M, the angular momentum J and the charge Q of the black hole by dM, dJ and dQ. Based on the analysis of the trajectories of these bodies in the field of the Kerr-Newman black hole, Christodolu and Ruffini concluded that the changes dM, dJ, dQ cannot be arbitrary with respect to each other, but only those, that satisfy the inequalities resulting from the relation (4.36), are feasible :

(4.51)

Bodies that have too much angular momentum per unit mass (relative to the center of the black hole) or too much electric charge, the black hole simply does not "receive" - it flies out without crossing the horizon, they are not captured. If we try to throw a body into the maximally rotating extreme black hole, which would spin this black hole even more, centrifugal forces will prevent this body from approaching the horizon and falling into the black hole.

Inequality (4.51) can also be written in the form

where M_ired = (1/2).Ö(r_g⁺² + a²) is the irreducible mass of a black hole, which was introduced by relation (4.48) in §4.4. Since, according to Equation (4.48), the area of the horizon is proportional to the square of the irreducible mass (A = 16p M_ired ²), dA ³ 0 holds. The inequality (4.52), according to which the irreducible mass of a black hole cannot decrease when any body is absorbed, is a special case of the 2nd law of black hole dynamics; the origin of the name "irreducible mass" can also be seen from here.

However, the general mathematical proof of the second law of black hole dynamics belongs to S.Hawking [120], [121]. According to Penrose's theorem 3.1, the horizon is generated by isotropic (zero) geodesics, which do not intersect anywhere and have no endpoints in the future. Due to the energy condition R_ik Vⁱ V^k ³ 0, the following relation holds for the convergence of C geodetics (and thus also zero geodetics generating the horizon): dC/dl ³ C² ³ 0 (see §2.6). If C were positive at some point (event) P , dC/dl would remain constantly positive and at least as large as at point P, so that after the final interval of the affine parameter l, a focal point would be reached, at which point the isotropic geodetic generators would start to intersect (§2.6). This would create an event through which more than one zero generator passes, which contradicts Penrose's Theorem 3.1. The convergence of isotropic horizon generators must therefore be less than or equal to zero C ³ 0 (their divergence must be non-negative). It follows that the two-dimensional surface the cross sections through which isotropic generators pass cannot decrease over time. Because at the same time, according to Theorem 3.1, no zero geodesics generating the horizon can disappear (they can only be created by new ones - by black hole absorption), this leads to the conclusion that the total horizon area of a black hole can never decrease with time. A necessary precondition, however, is the validity of the principle of cosmic censorship, which guarantees that the horizon cannot meet a (naked) singularity that would absorb and prematurely terminate (otherwise infinite in the future) zero geodesics generating the horizon.

For a general Kerr-Newman black hole, the area of horizon A is equal to

in the special case of the Schwarzschild black hole, A = 16p M². Based on expression (4.49), the total mass M of a black hole can be divided into three parts :

where M_ired is the above mentioned irreducible mass, M_elmag = Q²/4M_ired is the electromagnetic contribution and M_rot = J/2M_ired is the rotational contribution to the total mass.

In connection with the 2nd law of black hole dynamics, we can divide physically possible processes with black holes into two groups (similar to what happens in thermodynamics) :

• a) Reversible changes , in which the mass, charge and angular momentum change in such a way that the size of the surface A of the black hole remains the same. Such processes can in principle be reversed and the black hole can return to its original state.
• b) Irreversible changes , in which, together with the change in weight, charge and angular momentum, the size of the surface of a black hole increases (cannot decrease!). Such processes cannot be reversed and the black hole can never return to its original state.

The actual processes with black holes are irreversible; the reversible process is only a certain idealization, which can only be approached. The situation here is exactly the same as in thermodynamics.

Fig.4.21. Schematic representation of the processes of absorption of material bodies by a black hole and the fusion of two black holes. In all similar processes, the total area of the horizon increases .

The second law of black hole dynamics confirms (in addition to specific calculations of various types of processes) the extraordinary stability of black holes - a black hole cannot "split" or otherwise disintegrate, it is completely indestructible; it can only be increase in size by absorbing of next matter (Fig.4.21). However, we do not consider quantum processes here yet - see §4.7.

An interesting difference in the growth of the resulting surfaces of black holes and material bodies
The law of increasing the resulting surfaces of black hole horizons (according to 2nd law) differs from the behavior of ordinary material objects. If we take, for example, two identical plasticine small balls, join them together and knead a new larger ball from them, its surface will be about 20% smaller than the sum of the surfaces of both original balls (the volume of the ball is V = (4/3) p.r³, surface of the sphere S = 4pr², --> S = 2^2/3.V^2/3).
However, for black hole horizons, due to the enormous curvature of space, after merging, the resulting horizon will be larger than the sum of the two original holes.

As follows again from the relation (4.51), with the angular momentum J the mass M always increases accordingly - the extreme black hole is unreachable, it can only be approached. If it were possible to achieve J = M² with a finite number of steps, then we would achieve J> M² with the only other such step - Kerr's naked singularity would arise in conflict with the principle of cosmic censorship. Thus, the 3rd Law of Black Hole Dynamics shows that cosmic censorship cannot be violated by black holes.

Among he laws of black hole dynamics also sometimes include the following statements :

This "zero" law of black hole mechanics was derived by Bardeen, Carter and Hawking [9]; see also equation (4.47), where k does not depend on the coordinates j and J. The situation here can be compared with conventional gravitational bodies such as planets, where surface gravity k can differ at different points on the surface due to the inhomogeneous distributions of mass density inside the body (§1.2, passage "Surface gravity").

The laws of black hole dynamics (especially the second law in conjunction with the conservation laws that form the content of Law 1) place fundamental restrictions on the magnitude of changes in physical quantities during the interaction of black holes with the surrounding matter and with each other.

Let us have a Kerr black hole of mass M₁ and angular momentum J₁ = a₁ .M₁. Let this black hole interact with matter, bodies and radiation during a certain time interval, so that the resulting black hole will have the parameters M₂ and J₂ = a₂ .J₂. According to the 2nd law of Black Hole Dynamics, the horizon must increase, so there must be an inequality between the two states of the black hole:

where M and J are determined with respect to the asymptotic planar region, where the laws of conservation of total energy and angular momentum apply. In the case of a₁ > 0 (ie when the original black hole rotated), the inequality (4.55) can be satisfied even if M₂ is less than M₁.

Thus, in principle, it is possible to "extract" a certain amount of energy-mass from the rotating black hole. One way to do this is to take advantage of the entrainment effect of inertial systems by rotating the black hole: a suitable set of bodies is placed around the black hole and the entrainment forces perform the work by means of an appropriare arrangement. An example of such a mechanism was shown in Fig.4.20a where although the rigid frame radiated gravitational waves, but with a suitable arrangement could also perform mechanical work. The second method of extraction of rotational energy from a rotating black hole is the Penrose process described in §4.4, Fig.4.17 (or its superradiation analogy), in which the body flies into the ergosphere, there it is divided into two parts, one of which is absorbed and the other part flies out of the ergosphere with more energy than the original body. By similar processes, in principle, all rotational energy (--> a₂ = 0) can be taken away from a black hole ; the limit of the energy obtained is given by the inequality (resulting from (4.55) for a₂ = 0)

(4.56)

Let us now have two Kerr black holes with the parameters M₁,a₁ and M₂,a₂ , which are initially so far apart that their mutual interaction can be neglected. The areas of their horizons will be A₁= 8pM₁[M₁ + Ö(M₁² - a₁²)], A₂= 8pM₂[M₂ + Ö(M₂² - a₂²)]. If these two black holes move towards each other so that they "collide" and merge, during this process a certain amount of energy is radiated in the form of gravitational waves and finally the resulting black hole is stabilized with the parameters M₃ and a₃ and the area horizon A₃= 8pM₃[M₃ + Ö(M₃² - a₃²)]. According to the 2nd law of Black Hole Mechanics, A₃ ³ A₁ + A₂ , ie.

M ₃ [ M ₃ + Ö (M ₃ ² - a ₃ ² ) ]   ³   M ₁ [ M ₁ + Ö (M ₁ ² - a ₁ ² ) ]  +  M ₂ [ M ₂ + Ö (M ₂ ² - a ₂ ² ) ]  .        

At the same time, the energy carried away by gravitational waves is (according to the 1st Act - the law of conservation) equal to E_wave = M₁ + M₂ - M₃ ; we could define the "efficiency" of mass conversion to gravitational radiation h = (M₁ + M₂ -M₃) / (M₁ + M₂). This inequality (ie the 2nd law of black hole dynamics) places a basic upper limit on the magnitude of this radiated energy. For example, in a head-on collision of two non-rotating black holes with masses M₁ = M₂ = M/2, when the resulting black hole will also be non-rotating (a₁ = a₂= a₃ = 0), we obtain M₃ ³ M Ö2, so that the total radiated energy E_wave < M (1 - Ö2) @ 0,293.M and efectivity of conversion of matter into energy gravitational wave is h <1 - Ö2 @ 0.293.

The laws of black hole dynamics, as we have formulated them here, are at first glance similar to the laws of thermodynamics. In order to clearly show and concretize this similarity, we compare the relevant laws of black hole mechanics and the laws of thermodynamics together in a clear table :

The similarity between the laws of mechanics of black holes and the laws of thermodynamics is very tight, and the surface gravity k being analogous to the temperature T and the total area of the horizon A being similar to the entropy S. It can already be seen here that the analogy between the entropy S in thermodynamics and the area of the horizon A in the mechanics of black holes is not only a purely formal similarity of some laws by which these quantities are governed. These quantities also have a similar role: just as in thermodynamics the law of entropy growth forbids the complete conversion of the thermal energy of a heated body into useful work, the law of growth of the black hole horizon area makes it impossible to convert all the mass of a black hole into useful work. In the following paragraphs we will see, that due to quantum processes in the field of black holes, the analogy between the laws of black holes mechanics and the laws of thermodynamics has a deep physical meaning, and black hole can actually be attributed to the thermodynamic temperature and entropy.

4.5. The "black hole has no hair" theorem		4.7. Quantum radiation and thermodynamics of black holes

Vojtech Ullmann