# Causality and chaos in nature

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

Chapter 3
GEOMETRY AND TOPOLOGY OF SPATIAL TIME
3.1. Geometric-topological properties of spacetime
3.2. Minkowski planar spacetime and asymptotic structure
3.3. Cauchy's problem, causality and horizons
3.4. Schwarzschild geometry
3.5. Reissner-Nordström geometry
3.6. Kerr and Kerr-Newman geometry
3.7. Spatio-temporal singularities
3.8. Hawking's and Penrose's theorems on singularities
3.9. Naked singularities and the principle of "cosmic censorship"

3.3. Cauchy's problem, causality and horizons

The causal relationships between events in space and time are the basis of our knowledge of objective reality. We know from the special theory of relativity, that the motion of matter and the transfer of energy and information can take place only within the space-time light cone (see §1.6). Because everything that applies in the special theory of relativity is true even in general relativity, but locally - the flow of matter, energy and information will always take place inside the local space-time light cone, as we postulated in §3.1. Local causality is thus given by the requirement of local validity of the special theory of relativity (STR). Questions of global causality are no longer so trivial and must be answered by the analysis of global geometric and topological properties of a given epecific spacetime, which is the solution of Einstein's equations.

The shape and inclination of the light cones is given by a metric tensor at a given point in space-time. Gravity is thus the force that determines the causal structure of the universe *) - it defines which events of space-time can be causally related and which cannot. As we will be see below, with a sufficiently strong gravitational field (curvature of space) occur event horizon and region of space-time appear, which are causally separated from other parts of space-time.
*) Which events are really causally related to each other depends on the specific situation. The properties of space and time give us only certain limitations - they determine which events in principle
can be related, or which cannot be causally related.
Local causality in spacetime M can also be expressed as follows according to Fig.3.8. Let a point pÎM, such that every world line of the time or light type passing through this point intersects the hyperplate S (spatial type). As UÌ S we denote the set of all points on S , in which it can be intersected by world lines of temporal or light character passing through the point p. In other words, U is the set of those points on S, from which the point p can be reached by moving along temporal or isotropic worldlines. The causality requirement then means that the values of all fields at point p will be anambiguously determined by the values of the fields and their derivatives (up to a certain finite order) just in the set U . Knowledge of fields (and their derivatives) only in part U is generally not enough to unambiguously determine the fields in point p. Such a formulation of the causality principle is already closely related to so called Cauchy's problem, see below. Fig.3.8. The field values f at a point p are uniquely determined by the initial values of the fields fo (and their derivatives `f) in a points x of subset U spatial hypersurface S .

Cauchy's problem in mathematical analysis generally represents a task - procedure for finding a solution for a differential equation under given initial conditions (eg at a given point x the default values of the function fo are entered, as is simplified indicated in Fig.3.8 on the right). These procedures were developed in 1842 by the French mathematician A.L.Cauchy. In our context of particle trajectories and field evolution in space-time, we will focus on how the initial conditions on certain sections (hyperpaces) in space-time can affect field evolution and metrics in other areas of space-time, according to Einstein's GTR differential equations.

Geometry and causality in spacetime
But before we formulate our own Cauchy problem, we will analyze the geometric aspects of causal relationships in spacetime. We will assume that spacetime is time-oriented, which means

 Definition 3.1 (time-oriented spacetime)
 Spacetime M is called time-oriented if, at each point in spacetime, all vectors of the time or isotropic type can be divided into two groups: vectors heading to the future and vectors heading to the past .

In practice, this distinction can be made with the help of irreversible physical processes such as the growth of entropy of isolated thermodynamics systems, radiation of energy by electromagnetic or gravitational waves, expansion of universe (which is, however, a very problematic indicator of the direction of the flow of time) - see §5.6, passage "Arrow time").
In time-oriented space-time, we can already establish a number of causal relationships between individual events and entire areas of space-time :

 Definition 3.2 (causal relationships in spacetime)
 Any worldline that has no spatial character at any point (ie is of the time or isotropic type everywhere) is called a causal world line (curve). We say that an event B chronologically follows for events A , where there is directed to the future world lines (i.e. whose tangent vector heading to the future) of the time type going from point A to point B . We will say that the event B causally followed for the event A , if there is future directed causal world line going from point A to point B . Causal past J-(A), resp. the causal future J+(A) of point A is called the set of all events that causively precede, resp. causally following events A. Let S is an area (a subset of) space-time M. Causal past J-(S), resp. the causal future J+(S) of the set S is the set of all events in M that causively precede, resp. causally follow least one worldpoint of S. We will denote the boundary of J-(S) by ¶J-(S) and the boundary of J+(S) by ¶J+(S). Analogous definitions as for causal relations (3rd, 4th, 5th) can also be introduced for chronological relations. Chronological past and future of point A is denoted by I-(A) and I+(A), similarly the I-(S) and I+(S) for a set S .

It should be noted that the statement "event B causally follows event A" does not mean that B must necessarily be a direct consequence of the events A, but only that event B could be principally influenced by the event A .

Closed World Lines and Time Travel
In §3.1, we sa
w that different kinds of "identification" in spatial dimensions can lead to different kinds of global topology of space, without affecting the local geometric structure, the local course of physical phenomena. If we performed a similar "identification" in time (time scale), closed time curves would appear in spacetime - closed world lines of the time type; however, they seem inadmissible for physical reasons because they are in conflict with the principle of causality. If we took two points (t1, x, y, z) and (t2, x, y, z) on the time axis such that t1<t2 and these identified (t1, x, y, z) º (t2, x, y, z), then an intervention in some physical phenomenon at (x, y, z) at time t2 would cause at this point changes at time t1, ie in the past.
Fictitious possibility of creating closed time loops we will show using the concept of the so-called wormhole (described in more detail in the conclusion of §4.4, passage "Black holes: bridges to other universes? Time machines?") .
Thus, the existence of closed time-type worldlines probably leads to logical paradoxes of the kind that along the closed time worldline, we could go back in time and kill our grandmother before she could conceive your mother (and she then you ...); or kill one's own parents before you were born (the paradox can be formulated through various family relationships). Your very existence, necessary to carry out such a murder, would then become inconsistent with the result of that crime. Or the astronaut in the rocket could, after such a closed time world line, return to space and time to its launch, damage the rocket and prevent itself from this original launch :

 The Astronaut Paradox - A failed launch and a space flight The astronaut launches into space in a rocket and flies into the mouth of a "wormhole", which acts as a "time machine". After a closed time worldline, he could then return in space and time to his launch, damage the rocket and prevent himself from this original launch .

We have changed the past - first what we wanted to do happened and then we decide to do it - or not. Haw, then, would it be possible to reconcile two controversial alternatives in the future: our existence, when we could not be born ?; or the flight of an astronaut in a rocket when he prevented himself from taking off ? Therefore, such a traveler could not even go back in time and carry out the interventions mentioned. A traveler through time into the past could prevent himself from traveling to it... How would it be possible for an event to happen without happening? A situation where a future reaction cancels a previous action has no logical solution. Such logical paradoxes should be avoided!
These strange, even "pathological" consequences of time travel naturally lead to an attempt to find mechanisms that would prevent space-time events from "doing such nonsense". S.Hawking put forward a hypothesis called the "principle of protection of chronology" - the protection of chronological order, which would forbid causal loops (cf. Penrose's "principle of cosmic censorship" forbidding nude singularities in §3.9 "Naked singularities and the principle of "cosmic censorship""). Some mechanisms of this kind will be discussed here and also at the end of §4.4, the passage "Black holes: bridges to other universes? Time machines?".
It would seem that such logical paradoxes and disputes with the principle of causality in time travel arise only if "freedom of will" - that the relevant subject may, at its discretion, in principle take any intervention in ongoing events. In the event that freedom will not exist (and in classical physics is really no such thing), may not cause a conflict with the principle of causality: a time traveler past unchanged, because it has always been an immanent part of it (it can fill the past, not change it). The universe can be imagined as a "finished" and unique 4-dimensional manifold into which the individual worldlines are already "incorporated". From this point of view, all events on closed world lines could already be "synchronized" so that they would interact with each other indisputably in a closed cycle - they would be self-consistent. However, if we take into account irreversibility evolution of the universe (existence of dissipative processes, 2nd law of thermodynamics), the existence of closed time curves is physically unlikely, because the situation at a later time t2 cannot be identical with the situation at an earlier time t1. Thus, closed curves leading to a "time machine" could perhaps function at most in the region of elementary particles.
The picture below - "The billiard ball paradox", however, shows an example of contradictory behavior, where there is no subjective decision-making and "freedom of will", it is a purely mechanical experiment..!..

 The paradox of billiard balls - conflicting trajectories of bodies From a certain starting position, by tapping the cue, a billiard ball is sent as a suitable speed toward the right mouth of the wormhole, functioning as a "time machine" - along trajectory A. This sphere flies through a wormhole, goes back in time and flies out of the left mouth, before it has flown into trajectory A into the right mouth. It can then crash "into itself" (into its "younger version") , divert runway A into an alternative trajectory B , outside the wormhole. However, this prevents it from flying into the right mouth - and thus hitting itself.   When moving, starting with exactly the same initial conditions (from the same position and at the same speed), thus creating two conflicting trajectories A and B, along which the ball would move simultaneously. With proper orientation, the ball can fly through the wormhole several times - there are an infinite number of trajectories, differing in the number of passes through the wormhole...

Logical lessons learned from paradoxes :
No macroscopic object - the observer - can retroactively influence what has already happened. Apparently, there can be no "pilgrims over time" who could jeopardize the course of history. We can observe the past as witnesses, not as active participants. When we look into a distant universe, we actually see a distant past that has been brought to us by light during its journey of hundreds of thousands, millions, or even billions of years.
However, we cannot turn the future behind the past !
Many universes?

Everett's and Wheeler's quantum-mechanical hypothesis of "many worlds"  (discussed in §5.7 "
The Anthropic Principle and the Existence of Multiple Universes") could also provide a sci-fi way to circumvent disputes and logical paradoxes in time travel; according to which the universe contains not only a unique history of the world, but many histories parallel. Whenever there is an interaction (or, from the observer's point of view, a "decision" or a random event), there is a "branching of history" into different universes. If the time-traveler eventually flies back to the past and changes his history there (for example, kills his mother before his birth), there will be a "turn" to another history in the universe, which will coexist with the original universe - the traveler will it will actually move to another universe where it will be part of a changed history. There will be no unacceptable influence on the future of the present - our, default, original - universe.

Physical time travel?
The idea of time travel excites human imagination and evokes age-old dreams of eternal youth and overcoming death, correcting or averting past mistakes or predicting the future consequences of our decisions - an insight into what fate is preparing for us, or his prevention. Within Newtonian physics, due to its concept of absolute time, "time travel" was completely impossible *). It appeared only in the then beginning science fiction literature, especially in the famous sci-fi novel by H.G.Wells: "Time Machine" from 1895.
*) Note: Of course they have nothing to do with time traveltime shifts of several hours when traveling by plane, for example between Europe and America. Here, this is only related to the rapid flight between the agreed time zones, which are the result of our measurement of time of day by rotating the Earth around the axis. We do not move in time, but only in the indication of time by agreed numbers - daily hours, which are shifted according to longitude.
It was only within the special theory of relativity, which allows influencing the speed of time flow by motion, or the general theory of relativity showing the influence of gravity on the course of time, that time travel began to be considered in a sense a physically real possibility. If a body approaches the speed of light or resides in a strong gravitational field (perhaps near a black hole), time will slow down for it from the point of view of others, so it actually travels to the future. But neither of these ways makes it possible to go back to the starting time - the path to the future is one-way.
The relativistic time dilation in STR makes it relatively easy to travel to the future in principle. An observer moving at a high speed close to the speed of light passes time slower than a reference "rest" observer, so that even a large time interval in the rest frame can span a substantially shorter interval of its own time, ie "travel to the future" resting reference system; while it is observer still moving inside his space-light cone (see Minkowski diagram on fig.1.6). However, to travel into the past, the oserver's word-line would have to bend and turn back downwards and create a loop, which at some points would have to be tilted at an angle greater than 45° with respect to the vertical - it would be necessary to exceed the speed of light, which is not possible within STR
(Minkowski spacetime with the usual Euclidean topology). The kinematic effects of the theory of relativity offer - at least formally - two possibilities of time travel to the past : 1. In the planar spacetime STR by moving the superlight speed (pictured left); 2. In the curved spacetime of the general theory of relativity by moving at a locally sublight speed within a sequence of suitably inclined light cones in an area with a strongly curved geometry of spacetime (Fig. Right). Another hypothetical possibility is the complex topological structure of spacetime - "abbreviations, tunnels, wormholes" ( discussed below) .

The general theory of relativity, as the physics of gravity and curved spacetime, basically opens up the possibility of traveling to the future and the past. Traveling to the future is again relatively easy in principle: it is enough for the observer to stay long enough in a place with strong gravity (high gravitational potential), where the time according to relation (2.36) passes more slowly, to return to the starting point at a time, when a significantly longer time interval has elapsed in the meantime. The §4.3 describes an interesting phenomenon in the propagation of light in the field of a massive compact object - the effect of a gravitational lens. Such light propagation along curved paths leads not only to interesting optical effects of multiple images, but also to different time shifts along different beam paths. An observer moving along a suitable shorter path at a speed close to c could in principle "overtake" light photons moving along another (longer) path around a gravitational body; in extreme cases of motions close to c around a supermassive body (or even around a rotating Kerr black hole), these time differences could in principle be used to travel in time...
Gravity affects both the passage of time and spatial scales and proportions. In such globally curved spacetime, there may be some "shortcuts over spacetime" - hypothetical "wormholes", which allow the observer in a sense to" overtake "the light beam and "travel" back to the past. At the same time locally everything runs according to STR, the speed of light is not exceeded anywhere. It's similar, that a sailor sailing here on Earth across the ocean, still facing forward, may eventually find that he has returned to the place from which he sailed. When moving in curved spacetime, the observer can in principle find out over time that he has not only reached the starting point, but that he has "visited" an event from his past, albeit locally from his gaze all the time his time flowed towards the future..?..

A topologically complex, multiple connected universe ?
The local geometry of spacetime is determined by the distribution of matter in space - matter~energy curves the spacetime, in which bodies and particles then move along geotedics, representing the straightest possible trajectories. The curvature of space-time is described by Einstein's equations, whose application to the universe under appropriate simplifying assumptions leads to Fridman's equations (5.23) describing a universe whose space can have positive, negative or zero curvature, see §5.3.
However, this local geometry generally says nothing about the global shape, ie the overall topology space. In standard relativistic cosmology, a simply continuous universe (with a sphere topology) is considered, on which Einstein's, DeSitter's or Fridman's cosmological models work. Theoretically, however, the universe could have a more complex, multi-connected topology, with different topological tunnels or identifications of different parts; such a universe might even look like a kind of "emmental".
The complex multiple connected topological structure of the space of the finite universe would have interesting implications for what the observer sees in such a universe: in principle, he could see multiple images of galaxies, stars, and even himself, as in a mirror maze. And in the time at different stages of development. It could not be ruled out, that when we observe a distant galaxy, it could be our own galaxy, billions of years ago! However, it would be very difficult, if not hopeless, to astronomically recognize that the two galaxies observed are actually one and the same galaxy, represented by the passage of light through the complex topological structure of the universe. We would see them from different angles and, most importantly, due to the spatial scales of many billions of light-years, in completely different stages of development, changed beyond recognition.
A detailed way to obtain at least partial indications for certain topological structures of the universe could be a detailed measurement of the properties of microwave relic radiation.- its homogeneity, fluctuations (depending on the angular distance and wavelength), polarization. Already at the time of the separation of radiation from matter, there were nuclei of future structures in the universe, so these photons passed through places with different gravitational potentials, which led to small changes in their energy and wavelength - a slight cooling or heating. These fluctuations should be visible even now, as slightly warmer and colder "spots" in the otherwise isotropic distribution of relic radiation - they represent a kind of "paleontological imprint" of the structures of the early universe. The temperature difference is very small, of the order of 10-5 degrees, but a detailed "temperature map" of the early universe has already been measured using sensitive satellite apparatus - §5.4, passage "Microwave relic radiation - messenger of early space messages").
All these theoretical speculations have no justification in astronomical observations, so in interpreting of relativistic cosmology in Chapter 5, we will stick to the simplest and, from the current point of view, the natural assumption of a simply continuous topological structure of the universe.
Perhaps the only exception will be discussions about the possibility of multiple universes; however, even here it will not be a matter of introducing some a priori complex topology, but of hypothetical topological properties "induced" by turbulent quantum-gravitational processes at the beginning of the universe.

GTR in certain special solutions of Einstein's gravitational equations allows causal loops from a mathematical point of view. This is the case, for example, in Gödel's solution describing the rotating universe , or in spacetime around a long massive cylinder rotating at high speed (greater than c/2). The rotation of spacetime in such solutions carries with it light, and thus causal relationships between objects, in such a way that it allows the material object to move through a closed time worldline by suitable circulation without exceeding the speed of light around the object. However, such solutions are only certain mathematical curiosities that are not realized anywhere in nature. The same probably applies to the ring singularities in Kerr's geometry (§3.6 "Kerr and Kerr-Newman geometry") inside the rotating black holes, or to geometric-topological constructions involving so-called "wormholes" in spacetime - see §4.4.
Geometric-topological possibilities of "travel" in space and time in connection with the properties of space-time of black holes will be discussed in §4.4., Passage "Black holes: bridges to other universes? Time machines?", systematically in work (syllabus) "Journeys through time : fantasy or physical reality?". Some related considerations about the direction of time flow are further outlined in §5.6, passage "Arrow of time". For the topology "time travel" are important so-called Cauchy's horizonts (discussed in more detail below) , which, among other things, define and separate the space-time areas, in which time travel into future and past is and is not possible.

Cauchy region and horizon. Event horizon.
In the following, we will assume that in real space-time, closed worldlines of temporal or isotropic nature do not occur, that is, as it is sometimes said, a reasonable chronological condition is met. Then, in the spacetime for each hyperarea-hypersurface S of the spatial type, there is a certain maximum region of spacetime, in which it is possible to unambiguously and completely predict physical phenomena on the basis of knowledge of the initial conditions on S (Fig.3.9) : Fig.3.9. Based on knowledge of the initial conditions on the spatial hypersurface S , the future can be unambiguously predict in the Cauchy region D+(S), if the world-lines of time or isotropic type passing through any point p Î D+(S) in the past previously intersected the hypersurface S .
 Definition 3.3 (Cauchy region *):
 Cauchy's region of the future D+(S), resp. past D-(S), of hypersurface S of the spatial type, are called the set of all such points p Î M , for which each worldline of time or isotropic type passing through the point p intersects S in the past (resp. in the future).
 Definition 3.4 (Cauchy hypersurface *):
 Hypersurface S , which intersects each non-extendable worldline of the time or isotropic type, ie. for which D+(S) È D-(S) = M, is called the global Cauchy hypersurface .

Thus, if there is a global Cauchy hypersurface in space-time M, then based on the required set of initial conditions on this hypersuface it is possible to unambiguously determine the physical situation in the whole M, ie predict values of fields and positions and motions of all particles at any point in time in the future or past. Such is the situation, such as, in the flat Minkowski spacetime STR, where, for example, each hyperspace t=const. is Cauchy's hypersurface. In §3.5 "Reissner-Nordström geometry" and 3.6 "Kerr and Kerr-Newman geometry", however, we show that this "deterministic ideal of classical physics" is not fulfilled in some more complex cases, global Cauchy hypersurfaces do not exist there.
*) A task which, based on a set of initial conditions on a hyperplane S with the help of field equations it extends the solution further into the future (or into the past) is called Cauchy's problem (according to the French mathematician A.L.Cauchy, who dealt with the mathematical side of these solutions in the 19th century); hence the names "Cauchy region", "Cauchy hypersurface" and "Cauchy horizon". Fig.3.10. Examples of situations where there are no global Cauchy hypers
urfaces in spacetime M (and therefore Cauchy horizons H+C are present ).
a ) A certain point Q is "cut out" from the manifold M; then we can imagine a worldline C ' passing through the point p, which, when viewed into the past, ends at the place where Q was, and thus does not continue to the hypersurface t = const.
b ) Manifold with "angled" conformal boundary M (eg Kerr or Reissner-Nordström geometry has a similar structure). In addition to the worldlines C intersecting S, they can go to point p (uncontrollably from S) also world lines C ' from border areas M .

The fact, that some hypersurface S is Cauchy hypersurface, is a property not only the hypersurface S, but also of the whole surrounding spacetime M. Examples of situations, where there are no global Cauchy hyperssurfaces in space-time are shown in Fig.3.10. In Fig.3.10a there is an ordinary Minkowski spacetime, from which only a single point Q is "cut out". If it wheren't for that, each hypersurface S = (x, y, z, t | t = const.) would be a global Cauchy hypersurface. The state, for example, at point q is really unambigously determined by the initial conditions on S . However, if we take any point p inside the cone with the vertex at the cut-out point Q, then most of the worldlines passing through the point p will be intersect hypersurface S, but there also exis world lines which, when the extension of the point p in time to the past, encountered and hit the deleted point Q and thus can not be extended up to S. If we inverse this in time, we can say that from the cut-off point Q, additional "disturbing" influences (information) can come uncontrollably (by world lines not continuing into the past) to point p, which violates the prediction made from the hypersurface S for point p on the basis of knowledge of the complete set of initial conditions on S. There is no Cauchy hypersurface in such spacetime. A conical hypersurface diverging from the removed point Q separates the region of spacetime, in which evolution can be predicted based on the data on S from the region, where this is not possible; such an area is called the Cauchy horizon :

 Definition 3.5 (Cauchy horizon)
 Cauchy's horizon of the future H+C(S), resp. of the past H-C(S) of the hypersurface S is called the boundary of the Cauchy region D+(S) in the future, resp. boundaries of the area D-(S) in the past, ie H+C(S) = { p | pÎD+(S), I+(P) Ç D+(S) = 0 } , H-C(S) = { p | pÎD-(S), I-(P) Ç D-(S) = 0 } ·

It is obvious that the hypersurface S, which does not have a Cauchy horizon H+C(S) or H-C(S) is a global Cauchy hypersurface. Each bounded hypersurface has a Cauchy horizon (see Fig.3.9), so Cauchy horizons of this origin can be considered "trivial" - they do not tell us anything about the causal structure of a given spacetime. Only non-trivial Cauchy horizons are important, eg those that are for them for every (even unbounded) spatial hypersurface lying in a certain part of spacetime. Such non-trivial Cauchy horizons will be shown in §3.5 "Reissner-Nordström geometry" and 3.6 "Kerr and Kerr-Newman geometry". Fig.3.11. The horizon of the particles of the observer O (moving along the worldline C ) in the event p separates the particles which the observer O can see from the world point p from those particles which are hitherto unobservable from there.

Let us have an observer O , which moves in space-time M along the world-line C (of a temporal nature). Let us imagine that the spacetime M is filled with a system of test particles moving along time worldlines. An observer O located at worldpoint (event) p can see some of these particles (those that intersect the light cone past point p ); however, there may be particles in M whose light lines do not intersect this light cone and therefore the observer O cannot see them from the point p (some of them will see later) - Fig.3.11. We say that for the observer O in the event p exists the particles horizont, separating region of space-time, in which the particles are observable geodesics from p, from a region of spacetime in which the moving particles cannot be observed from p. The particle horizon arises, for example, when the infinity of the past J- is of a spatial character, as in Fig.3.11. Fig.3.12. The horizon of the particle P moving along the worldline C separates the regions of space-time from which the particle P can in principle be observed during its movement along the worldline C, from the areas from which it can never be seen.

Particles moving along the world line C can be observed from certain areas of space-time, but in general there can be areas in M from which it cannot be observed in principle (Fig.3.12). The boundary separating these two regions of spacetime is called the horizon of particle, moving along the world line C.

As can be seen from Fig. 3.13, in general spacetime for an observer O moving along the world line C, there may be events that he can never see - we say that these events are hidden behind the horizon of events for the observer O. Event horizon (the future) for observer O, traveling along the world-line C, we call the area separating those areas of space-time whose worldpoints (events) can be seen by the observer O as he moves along the world C, from the areas that the observer O can never see from C. From Fig.3.13 we see that the event horizon for a geodetic observer arises, for example, when J+ is of a spatial character. In Minkowski spacetime, no geodetically moving observer has an event horizon, because his light cone gradually runs through spacetime. However, an observer moving, for example, uniformly accelerated has a horizon of future events (the so-called Rindler's horizon). Fig.3.13. The event horizon for an observer O moving along the worldline C separates those events that are observable to O during his movement along the C worldline, from the events that he can never observe from the worldline C.

The event horizon for observer O depends on its specific movement and therefore does not provide completely unambiguous and objective information about the causal structure of a given spacetime. We can make more specific judgments about the causal structure of spacetime, when we find in it a horizon of events that will be for a certain wider class of observers, for example, for all observers moving in a certain area of spacetime. As we will see below (§3.5-3.6 and Chapter 4), the most important are the event horizons, which are them for every observer located in a certain external asymptotically planar region, ie for all observers distant at infinity. It is horizons of this kind that we will keep in mind when we talk about the horizons of events in the next :

 Definition 3.6 (event horizon)
 Event horizon (future) is the boundary of the space-time region, from which worldlines can be led to the isotropic infinity of the future J+, which at each of their points lie inside or on the mantle of the light cone of the future (i.e. causal curves) : horizon = ¶ J-(J+) - is the boundary of the past of the area J+ . Anologously, the horizon of past events is ¶ J+(J-) .

The event horizon thus separates the regions of spacetime from which particles can reach infinity J+ from the regions from which no particle can escape to infinity (Fig.3.14). Fig.3.14. The event horizon is the boundary dividing those space-time regions, in which events can be observable from infinity J+, and regions from J+ in principle unobservable.

The reasons why event horizons arise will be explained below on specific solutions of Einstein's equations; this is most often because in certain areas gravity is so strong, that it does not let any body or light out. Here we will notice only some geometric-topological properties of the event horizon.
Horizon generators are called those isotropic geodesics, that lie in the horizon (at least for a certain finite interval of the affine parameter). It can be imagined that the escape velocity on the horizon is equal to the speed of light and therefore the photons are constantly "floating" on the horizon. R.Penrose  proved an important theorem on the structure of the horizon of future events :

 Theorem 3.1 (horizon generators)
 The event horizon is generated by isotropic geodesics, that do not have endpoints in the future. The generating geodetic, that enters the horizon already remains there and cannot intersect with any other generator. Every event on the horizon, which is not the world point of the generator input, goes through exactly one generating zero geodetic.

The properties of spacetime horizons play a key role in the physics of black holes , as we will see several times in Chapter 4.

Cauchy's problem and the evolution of spacetime
So much in brief on general issues of causality in spacetime
(further details can be found especially in the monograph ). The specific realization of these causal links, is expressed in the so-called Cauchy task, which consists roughly of the following:
Consider the Cauchy hypersurface
spatial type S on which known (measure) the initial value of the field; if the field is described by second-order differential equations, the initial conditions for them must include the distribution of the field potentials and their first time derivatives. These Cauchy initial values satisfy certain binding conditions resulting from the field equations (to be consistent with the field equations). Then we can use field equations to extend this initial solution to the immediate future (or past), ie to the infinitely close hypersurface S'. By repeatedly continuing this procedure, it is then possible to extend the solution further and further into the future (and past) and thus determine the field values in the entire Cauchy space-time region D+(S); if S is a global Cauchy hypersurface, the field in the whole spacetime M can be determined.
Cauchy's problem expresses the deterministic character of the whole state of physics: the evolution of each physical system (field) is unambiguously determined by the equations of motion (field equations) only when the appropriate initial conditions are given. This can be seen in the simplest case of Newton's classical equations mechanics that completely determine the trajectory of a particle only by entering the appropriate initial conditions, such as the position of the particle and its velocity at a certain point in time (eg t = 0). Electrodynamics has a similar character , where it is necessary to :
a ) In space-time choose a spatial-type hypersurface ;
b
) On this initial hypersurfaced, enter the intensities of the electric field E and the magnetic field B so that they are consistent with the Maxwell equations div B = 0, div E = 4 pr, which play the role of binding conditions for the initial values ;
c ) Then, using the second pair of Maxwell's equations rot E = - B/t, rot B = j + E/t, the whole evolution of the electromagnetic field in the future (and in the past) can be determined.
The actual initial conditions are obtained by measurement; they are the results of observations and cannot be obtained or derived from equations of motion (these equations impose only certain limitations on them). There is so far no theory (nor does anyone know if such a theory can exist at all..?..), which together with the equations of motion would determine the initial values *).
*) A new interesting approach to the problem of initial conditions is now emerging in quantum cosmology in connection with the concept of inflationary expansion of the very early universe. According to this concept, the structure and evolution of the universe is not determined by the initial conditions during the Big Bang, but is the product of only the very fundamental laws of physics - see §5.5, passage "A complete cosmological theory ?".

When applying the Cauchy problem to Einstein's gravitational equations, it is useful to divide this system of equations into two groups. The first group consists of four equations

 Ri° - 1/2 di° R = 8p Ti° , (3.8)

which contain time derivatives of the metric tensor only of the 1st order and do not contain its second time derivatives. These equations are the binding conditions for the initial values . The second group consists of six equations

 Rab - 1/2 dab R = 8p Tab , (3.9)

which contain the second time derivatives of the metric tensor and thus describe the evolution of the field. Cauchy's problem here is that :
a
) We have (enter, measure) the values of the metric and its first derivatives on suitable initial hypersurface of spatial character, which must satisfy the binding conditions (3.8) ;
b
) By integrating equations (3.9) we can then extend the initial solution further, ie obtain the values of the metric tensor on other spatial hypersurfaces .

However, Cauchy's problem for the gravitational field (ie for GTR) is somewhat different from the corresponding problem for other physical fields. If we have two metrics g1 and g2 between which there is a diffeomorphism converting to each other, these metrics are physically equivalent. Thus, there are whole classes of equivalent metrics, so the solution of gravitational equations can be found with only precision to diffeomorphism. In order to eliminate this ambiguity, it is necessary to prescribe certain conditions, similarly to the introduction in electrodynamics of Lorentz conditions for the elimination of arbitrariness in calibration transformations. The equations of most physical fields are linear; the nonlinear equation appears only when several fields interact with each other. However, due to its universality, the gravitational field exhibits self-gravity (interacts "with itself"), and Einstein's equations of the gravitational field are nonlinear with and about ourselves, even without the presence of other fields. The gravitational field also determines the metric and thus the structure of spacetime, in which we solve Cauchy's problem. Therefore, we generally do not know in advance what the Cauchy region of evolution of the initial hypesurface will be - we do not know the space-time region, in which the solution is to be determined (evolution can give us "surprises", perhaps in the form of horizon or singularity).
It can be proved that if the initial values satisfy the coupling equations and if the postulate of local causality is satisfied for a possible non-gravitational field (see §3.1), Cauchy's problem for Einstein's equations (and non-gravitational mass equations of motion) has an unambiguous solution (with diffeomorphism accuracy). Moreover, this solution depends in a sense continuously on the initial conditions, as claimed in the sentence, the simplified wording of which is as follows :

 Theorem 3.2 (continuity and stability of the solution of the Cauchy problem)
 Let g represents in U solution Cauchy problem for the initial condition w to spatial hypersurface S . Then for changed initial conditions w + Dw such that their change Dw will be small in the region J-(U) Ç S, we get in the region U a new solution g ', which will be close to the original solution g .

This theorem justifies the use of perturbation analysis of the solution of gravitational equations mentioned in §2.5, where from a known solution under certain initial conditions (eg solution for the case of exact symmetry) we try to obtain information about a new solution under slightly changed initial conditions (eg slight symmetry violation). These methods are of considerable importance especially in the analysis of the gravitational collapse, in which the known course of collapse in the case of a spherically symmetric infer the course of real collapse without exact symmetry, see Chapter 4.
However, a truly consistent use of Cauchy's problem to determine the evolution of physical systems in nature is not possible, Cauchy's problem is only a theoretical model and a guide on how to find such a solution in principle. The reason is that it is not possible to determine the complete set of initial values of physical quantities on the Cauchy hypersurface of spatial type sufficiently densely and precisely, partly due to the finite speed of propagation of interactions would have been necessary to get into the future at all (even very distant) points, which is also practically impossible. When solving the Cauchy problem, the initial condition for the Cauchy hypersurface are only awarded on the basis of certain theoretical model (e.g. everywhere vacuum or homogeneous distribution of a certain density, etc.) and we follow the evolution of this model.
In addition, there may be (at least theoretically, see §3.5-3.9) situations where the considered spacetime does not have global Cauchy hyperfields, and even the best knowledge of the complete set of initial conditions is not sufficient to determine its evolution. Furthermore, the Cauchy problem is the epitome of classical deterministic spirit of the classical physics; quantum processes preserve deterministic relations only at the level of wave functions, while globally the exact determinism of classical physics is already disturbed (see the following passage "Determinism, radomnesss, chaos" and also §4.7).

Determinism in principle, randomness and chaos in practice ?
Cauchy problem embodies Laplace mechanistic conception of the universe as a "clockwork" :
"A perfect reason, knowing at given moment all the forces governing nature and the relative positions of objects in it, and that would be powerful enough to analyze this data, could be summarized into a single system the motion of the largest cosmic bodies and the lightest atoms: for such a mind there would be nothing uncertain and the future as well as the past would be present before his sight" (Pierre Simon Laplace, 1812) - a mechanistic conception of the world.
Is our universe deterministic, as Laplace claimed, or is ruled by chance, how often does this seem to us in everyday life? The first weakness of Laplace's idea is that we can never measure the initial state of a system absolutely accurately, so even future deductions from it cannot be completely accurate. However, it was assumed that if we perform the initial measurement with an accuracy of, for example, 12 decimal places, then all subsequent predictions will also have an accuracy of 12 decimal places - the initial error does not disappear, but it does not increase either. Unfortunately, it turned out that the error or deviation actually increases - at each step of the evolution of the system, the prediction error increases by a certain percentage, so after a few steps or tens of steps, we can no longer predict practically anything ...
This amplification of errors is the second weakness, negating the perfect Lapace determinism. Sensitivity to initial conditions makes the behavior of the system irregular and unpredictable - chaotic (Greek cháos = empty, gaping abyss, formless state; it means a state without order and laws). The individual elementary stages of the system's behavior are governed by deterministic laws, but result in such irregularities that they appear to be completely random. Such (apparent) chaos is thus a complex and (seemingly) irregular behavior, which in fact has a simple deterministic basis. Because this chaos is generated by systems themselves governed by deterministic laws, it is often referred to as "deterministic chaos". This chaos is probably the main reason that our nature is so varied, diverse, variable. And perhaps also the "driving force" of mental activity and our human "freedom will"..?..
Everywhere in the universe, matter behaves according to the same laws of physics and chemistry. However, the specific behavior, the course of events and their outcome, depend on current conditions - these have also evolved according to these exact laws, but often through a complex combination of circumstances, that already have the character of chance.
The role of chance in the origin and evolution of life is discussed in the work "Anthropic principle and/or cosmic God", passage "Origin and evolution of life".
Quantum physics also shows that at the microscales of space and time, nature is truly and principally controlled by chance, for example, whether or not a particular elementary particle or radioactive nucleus decays at a given moment is a purely coincidental matter; between the nucleus which is to disintegrate immediately and the one which does not (or only after a long time) can not be found absolutely no difference. Only file a large number of particles or radioactive nuclei appear accurate statistical regularity. After all, a fundamental question arises as to where these statistical regularities come from; are they traces of some hidden determinism on a more fundamental level? Quantum physics resolutely denies this ...
It turns out that the deterministic idea works only in the simplest idealized cases - determinism can be considered only "in principle", not in practice. In real systems of many bodies, which follow not only the laws of classical but also quantum mechanics, we observe that the slightest uncertainty in determining the state of a system at a given initial time usually leads to a complete loss of the ability to accurately determine its state even after a relatively short period of time. Small differences in the initial conditions can cause large changes in the resulting phenomena. In other words, almost identical states of presence can evolve towards very different futures - any prediction becomes impossible, the phenomenon effectively becomes random; we are talking about the chaotic behavior of the system *).
*) On this instability and indeterminism the behavior of complex systems is often encountered by meteorologists in their efforts to forecast the weather for a longer period of time (which usually fails). They refer to it metaphorically as the "butterfly wing effect" - that the mere flutter of a butterfly wing, causing a slight swirl of air in Europe, for example, can "cause" a storm or cyclone on the other side of the Earth - in America or Australia.
However, it should be borne in mind that the "butterfly wing effect" is just an ad-absurdum of increased mystification! In fact, atmospheric events are constantly influenced by local fluctuations in the surrounding natural conditions, which are many orders of magnitude greater than that distant butterfly effect ...

Lyapunov instability
Calculations and computer simulations show that in such unstable systems a small change
do of the initial conditions causes, that the initially close trajectories to diverge exponentially with time t : d = do .e -l .t . After a long enough time, the system eventually becomes chaotic. The degree of linear stability or instability - "chaoticism" of such a system can be characterized by the so-called Lyapunov time TL = 1/l , for which the system deviates 2.7 times (by this factor is increases each initial deviation); parameter l= 1/TL is sometimes called Lyapunov exponent.
Interestingly, even seemingly stable systems such as the solar system are probably chaotic.
The orbits of the planets are mutually affected by gravitational perturbations - the symmetry of Kepler's orbits is slowly broken. For the inner planets of the solar system (excluding Mercury), the Lyapunov time is estimated at TL » 5.106 years. The high value of this time explains the extraordinary accuracy of astronomical predictions of planetary motions over time horizons of hundreds and thousands of years. However, at intervals of hundreds of millions to billions of years, the chaotic nature of the planet's orbits would already be decisive; some of the planets could even leave the bound system of the solar system.

Until recently, it was assumed in classical physics that simple systems behave simply, and that therefore every complex behavior must have "complex causes." However, the analysis of the behavior of the systems in recent years has shown some surprising findings. That the behavior of even simple systems can be very complex, seemingly chaotic. And again, some complex systems can behave surprisingly simply - for example, due to self-regulatory or synergistic mechanisms. Within this new theory of chaos, the chaos and order are like connected vessels: under certain circumstances, order changes into chaos, under other conditions chaos turns into ordered structures. And even (seemingly) chaotic behavior leaves traces albeit of complex and seemingly disordered, but the theory of chaos often finds a surprising order in them - a kind of "organized chaos". These traces chaotic behavior usually have a complex geometrical structure, for a description of which is no longer suitable classical Euclidean geometry, demonstrates, however, that it can be well modeled with a new type of so-called fractal geometry (see below).
To describe the behavior of dynamic systems is often used so-called phase space (introduced by H.Poincaré) - an imaginary mathematical space whose points represent all possible states of a dynamic system. The behavior of the system is then expressed by a certain phase trajectory - a curve in this phase space.The dynamics of the system can be represented by geometric shapes in the phase space. If we start a dynamic system from some initial state (point in phase space) and observe its long-term behavior, it often happens that its trajectory in phase space will converge to some defined geometric shape, sometimes even with different choices of initial conditions. Such an end formation in phase space, to which the dynamic behavior of the system is directed, is called the attractor (lat. Atractio = to attract) - this structure seems to "attract" the trajectories of the system in phase space, so that the system either rests in it, or "orbits" around it.
A system that settles in a constant quiescent state has only one point as an attractor. In a system that stabilizes by repeating its state periodically, the attractor will be a closed loop around which the system revolves. Systems with quasi-periodic motion have more complex attractors with multiple and split loops. Chaotically behaving systems describe in phase space very complex trajectories around special formations called strange attractors, which have a fractal structure :

Fractals - infinitely segmented formations
Objects in nature, although mostly irregular and complex, are usually modeled using simple geometric shapes such as line, triangle, square and rectangle, circle or ellipse, plane, cube, sphere and other shapes described by Euclidean geometry. Formulas for calculating their length, area or volume are also well known for these basic structures. Similarly, in other more complex shapes, which can be composed as a combination of finite or infinite (but countable) number of basic geometric shapes.
We can assign a certain integer to all such formations - the number of dimensions or the (topological) dimension of a given shape, determined by the number of numbers, coordinates or parameters determining the position of points in these shapes: a line or curve (albeit heavily bent) has dimension 1, the surface has dimension 2, cubes, spheres and all spatial shapes have dimension 3 (because the position of each point in them it is uniquely determined by 3 numbers - coordinates).
However, it turns out that very articulated, "rough" and irregular shapes in nature (as well as the behavior of chaotic systems) better than classical Euclidean geometry models the so-called fractal geometry (lat. Fractus = broken, break ), which allows to describe even infinitely segmented shapes. Euclidean geometry is a kind of abstraction of real shapes, while fractal geometry reflects the real complexity and articulation of shapes: fractal shapes are no longer a simple combination of ideal geometric shapes, but are of infinite complexity - the more detailed we examine them, the more complex details appear. Fractal geometry tries to capture all the holes, bumps, distortions, entanglements that occur in natural formations. Despite this bizarre complexity of fractal geometry, it reveals certain regularities of so-called self-similarity, where each part of the object is similar to the whole (see below) - in nature there are often branching fractal structures. It turns out that fractal geometry is a suitable mathematical tool for describing the structures and dynamics of natural processes.
For readers who do not come into contact with sets more often, it may be useful to read in §3.1 "Geometric-topological properties of spacetime", the introductory part "Topology", the passage "Sets and representations" and also the section "Infinity in space and time", passage "Infinity in mathematics".

From the history of fractals
The roots of these concepts go back in a way to the founder of set theory G.Cantor, who in 1883 built a peculiar purely speculative set - the so-called Cantor's discontinuum. Cantor's discontinuum arises from a line segment of unit length by first removing the middle third, then in the remaining two third segments always the middle third again, etc., to infinity. Then a set of isolated points remains, Cantor's discontinuum or Cantor's dust. The sum of the lengths of all omitted intervals is exactly equal to 1. Cantor's dust is, at first glance, a negligible group of points. However, Cantor proved that there are exactly as many of these "dust grains" as there were points on the original line segment (!) - they can be unambiguously assigned to each other. The part is, in a sense, as numerous as the whole - compare with the discussion of the concept of infinity in mathematics in §3.1 "Geometry and topology of spacetime", the passage "
Infinity in space and time". If we stack all the omitted third sections from the previous structure on top of each other in steps (the same height as the width), a so-called devil's staircase will be created. This "staircase", despite its complex fractal structure of infinitely many degrees, has a finite length of 2.
Other features of unusual internal structure, such as the Koch flake or the Sierpinic rug, will be discussed below in connection with fractal geometry and the Hausdorff dimension. The paradoxical properties of these artificially constructed objects or structures with the mathematicians of the time, "bred" by classical algebra and mathematical analysis, seemed so bizarre and contrary to intuition and common sense, that they called them a kind of perverted "mathematical monsters".
However, the main founder of modern fractal geometry is Benoit Mandelbrot, who revealed new and unexpected structural properties of geometrically complex shapes and sets - the anomalous dimension and periodicity of structures
in different scales. In the 1960s, Mandelbrot analyzed noise and errors in electronic signal transmission. He observed that alternating time intervals of correct and erroneous transmission occur on different time scales - a kind of "self-similarity". Coincidentally, he linked this knowledge to empirical data on the measurement of the length of sea coasts (specifically the coast of the island of Corsica), collected by L.Richardson. He found that the determination of the length of such a coast depends significantly on the scale, ie on the "length of the rod" with which we perform the measurement. On a larger scale, on the map, we do not see all the real irregularities, bends, protrusions and other intricacies of the coast, which will "bridge" the larger scale - we will measure the length shorter. In smaller and smaller scales of a more detailed view, we have to copy smaller and smaller fragments when measuring with a short rod, so with a finer scale, the detected length of the coast will become larger and larger - theoretically to infinity (the so-called Richardson effect). For the length of the coast L measured by a rod of length e , Richardson determined the empirical dependence L(e) = K. e 1 - DR , where the constant K is a certain "normal" length of a particular coast and the constant DR (called Richardson's constant ) characterizes the fragmentation from about 1.05-1.3; the average value of the Richardson constant is taken to be 1.26. Mandelbrod analyzed Richardson's empirical formula by of a given coast. For different coasts, the value of DR ranged Mandelbrod analyzed Richardson's empirical formula by introducing another parameter of the number of intersections of the measuring rod N(e), so that L(e) = e .N(e). He thus adjusted the equation to the form K = L(e). e DR - 1 = e .N(e). e DR - 1 = N(e). e DR , from which it follows that K can be considered as a Hausdorff measure and DR as a Hausdorff dimension of a set of points describing the coast. Furthermore, Mandelbrod analyzed and supplemented the so-called Julius sets. By elaborating and generalizing these findings, Mandelbrod came to the concept of fractal .

Properties of fractals
Fractal formations (fractals) have two basic remarkable properties (which can also serve as definitions of fractals) :

• 1. Self-similarity
Fractals are ivariant to certain transformations consisting in change the scale - scaling. Fractal geometry studies shapes in which the same or similar shape is repeated on an ever smaller and smaller scale. It can be said that a self-similar formation looks the same, even if we look at in at any scale or magnification.
In the simplest case, a structure is similar, in the sense that it can be divided into several parts, where each of these parts is a reduced copy of the whole. No matter how small a piece of shape we look at, this cutout contains everything that is in the great part of the image.

From a mathematical point of view, a self-similar set A from the n-dimensional Euclidean space En is a set for which there are finally many so-called contracting representations f1 , f2 , ..., fn (these are such imaging En to En, which reduce the distance between two points lying in En), such that A arises as a unification of A = i=1Cn fi(A). Such self-like sets are created by repeating "myself" in a certain transformation, such as scaling, rotation or shifting. Self-similar sets are invariant to scaling - they look similar when zoomed in or out. It can be said that a self-like set arises "from itself" - it arises by repeating the same basic motif. E.g. Cantor's discontinuum consists of its repeating exact copies, reduced to 1/3.

We can simplified say that a fractal is a geometric shape (or set) that consists of a certain number of its suitably reduced "copies". Such a principle of repeating similar shapes in reduced or enlarged form we can often observe in nature, where many complicated and complex formations are created by repeating simple structures and rules. It is, for example, the growth of tree branches, coral reefs in the sea, snowflakes, weathered rocks, or branching of the vascular system in the body from large aortic vessels to the finest capillaries. Structures of this kind are very effective in packing a large surface area into a small volume.

• 2. Hausdorff dimension
A fractal is a set whose so-called Hausdorff-Besikoviè dimension (also called fractal dimension or similarity dimension - see below) differs from the topogic dimension - it is usually non-integer. The degree of difference between the fractal and topological dimensions indicates the "level of articulation-diversity" of a given formation.

Topological dimension
The usual dimension - the number of dimensions - of an object, also called the topological dimension (see §3.1 "
Geometry and topology of spacetime") is an integer D indicating the number of parameters which unambiguously define the position of individual points of this object. A line, line fragment, a circle, a parabola, a sine wave, and any other curve have the dimension D = 1 (it is one-dimensional), because the position of a point on it can be parameterized by a single number (coordinate). Each smooth surface - plane, triangle, circle, spherical or cylindrical surface, has the dimension D = 2, because the position of the point here must be defined using two coordinates. Bodies such as a cube, cylinder, a pyramid, a sphere, as well as the entire usual space around us, have the dimension D = 3, because the position of each point in them is uniquely determined by 3 coordinates. By analogy, we can formally construct formations with higher dimensions, even if we have no direct experience with them and cannot imagine them; in our treatise we often use 4-dimensional spacetime.

Fractal dimension
Hausdorff's dimension

However, we can look at the dimension from a non-topological point of view - from the metric point of view, which models the process of measuring a given geometric shape - determining its length, area, volume, measure in general. Let us first consider the measurement of a line fragment (dimension D = 1) of total length L, which we cover N with equally long intrervals ("scales") of length
e. The number of intervals N(e) that cover the whole line fragment depends on the length of the intervals according to the relation N(e) = L. (1/e) and the length of the line segment can be calculated based on the number of coverage L = N(e). e . If we measure the length of the curve in a similar way, we will refine the coverage, so that the length will be expressed by the relation L = lim0N(e). e . Similarly, in the case of dimension D = 2, the area of a square of side L can be covered by the number N(e) of squares of side e , and for the total area of the square we get S º L(2) = N(e). e2. For a general planar shape, its area L(2) can be expressed as L(2) = lim0N(e).e2. In general, for a D-dimensional structure, the relationship between its size (measure) L(D) and the number of coverage N(e) is a measure of length e : L(D) = lim0N(e).eD. By logarithmization, this makes it possible to express the dimension D of a shape of "volume" L(D) using the relation D = lim0[ln N(e)]/[ln L(D)+ ln(1/e)]. By normalizing to the unit volume, we get a relation for the dimension, which we can consider as an alternative independent metric definition of the dimension - so-called Hausdorff- Kolmogorov dimensions DH :
D H = lim 0 [ln N ( e )] / [ln (1 / e )] ,
where N(
e) is the number of coverage of the measured structure by scales (lines segments) of length e .
If the formation is regular, it is not necessary to calculate the limit for infinite refinement of the scale; it is enough to compare which resultant factor will change the determined length of the examined unit when the scale is refined by a given certain factor. The relation for determining the Hausdorff dimension is then simplified to:
D H = ln N / ln (1 / e ) ,
where N
is the change factor of the determined length, e is the length of one newly formed scale fraction when the original shape is divided by the scale change factor 1/e .
If we calculate the Hausdorff-Besikov dimension for any smooth one-dimensional shape (line or curve), we get D
H = 1, for a smooth area shape we get DH = 2, for ordinary three-dimensional geometric shapes DH = 3. For geometrically smooth shapes in general always holds DH = D - Hausdorff's dimension is equal to the topological dimension. However, for complex articulated formations - fractals - their Hausdorff dimension is higher than the topological dimension and is usually given by a non-integer value. The Hausdorff dimension can be considered as a certain generalization of the usual (topological) dimension, which better captures the behavior of complex articulated formations than the topological dimension. The fractal dimension quantifies the degree of complexity or articulation of an object by how fast its measured length, content, or volume increases depending on the size of the scale by which we measure (a generalized Richardson effect is used).
Similarity dimension

For self-similar sets, another alternative definition of dimension can be expressed - the similarity dimension. This dimension quantifies how many copies of itself, reduced by an appropriate factor, are contained in a given set. If the examined object contains N copies of themselves, reduced by factors 1/k
i (i = 1,2, ..., N) , the similarity dimension DS is given by the relation
i = 1S N (1 / k i ) D S = 1 .
For the most common case that the factors k
i are the same (k i = k), we obtain a similarity dimension by solving the equation i=1SN(1/k)DS = 1, which gives DS = ln N / ln k = DH. It can be proved that the similarity dimension is equal to the Hausdorff dimension. The concept of similarity dimension thus making it easy to determine the fractal dimension of symmetrical fractals of geometric origin. It is enough to find out how many "copies of itself" and to what extent the structure contains, and its fractal dimension will be given by the proportion of logarithms of these values - see below.
Grid dimension

For more complex fractal features whose structures do not show self-similarity, the fractal dimension can be determined empirically by grid counting - "box-counting". A grid (of the dimension given by the topological dimension of the studied formation) of cell size d is overlaid over the given formation and the number of cells containing some points of the observed formation is calculated. This gives the number N , which depends on the size d grid cells N = N(d) - the denser grid (smaller d), the larger the N. We gradually refine the grid (reduce d) and analyze the function N(d) by plotting it in the log/log graph [ln N(d)
« ln (1/d)]. We use the obtained points to intersect a line, the direction of which then indicates the fractal dimension of the investigated formation.
In the simplest case, when the test grid is refined by a factor of 2 (twice as thick), the number of cells counted between two consecutive coats is multiplied by 2
D, where D is the fractal dimension.
For self-similar objects, this method gives the same values as the similarity and Hausdorff dimensions; however, it is easy to algorithmize and works even for complicated fractal formations.

Examples of fractal formations
Koch's flake

One of the simplest and most interesting examples of how a complex fractal formation can be created from an originally Euclidean-shaped geometric construction is the so-called Koch's curve ; this curve was constructed in 1904 by the Swedish mathematician H. van Koch as a geometric model approximation of the perimeter of a snowflake - hence the name "Koch snowflake". We can construct the Koch snowflake by a series of successive triangular iterations as shown in the figure. We start from an equilateral triangle with unit side length (1st iteration); the perimeter therefore consists of three sections of unit length.
In the 2nd iteration, we connect to the middle third of each side another equilateral triangle with a third length of the side - a 6-pointed star is formed, the circumference of which consists of 12 sections of length 1/3. In the next iteration we add another smaller triangle to each middle third of each of the 12 sides. The length of the curve will be extended by 1/3 at each step (four parts of the line will form four of the same length.) In the infinite limit of the number of steps n®¥ we get a curve whose length 3.(4/3)n is infinite, but the content of the area bounded by this infinite line remains finite (less than the area of the circle circumscribed by the original triangle, since the Koch curve does not intersect the circumscribed circle anywhere; at each iteration of e®e/3, 4 (self-similar) parts are formed, ie N(e/3)®4.N(e). According to the above definition formula (in the simplified version), the Hausdorff dimension of the Koch curve comes out : DH = ln4/ln[1/(1/3)] = ln4/ln3 @ 1.261, ie higher than the topological dimension D = 1. Thus, such a curve "fills" a space (plane) somewhat more than a mere line or line segment with dimension D = DH = 1- it is a set or shape metric "denser" than would be expected from its topological dimension D = 1. Construction and structure of Koch fractal curve. Above: 1., 2., 3. and 4. iteration of the curve. Bottom: Enlarged view of the rugged structure of the curve.

Such "dense" fractal formations with a Hausdorff dimension higher than the topological one are created by adding infinitely many refining parts into geometrically smooth formations .
Cantor's discontinuum

By removing infinitely many infinitely decreasing parts, on the contrary, "thinner" sets are created, whose Hausdorff dimension is smaller than the topological dimension. The oldest example is the already mentioned Cantor's discontinuum with D = 1 and D
H = ln2 / ln3 @ 0.63 (Cantor's discontinuum is the unification of its two copies reduced by a coefficient of 1/3).
Sierpinski triangle
it is created by cutting an inner smaller triangle formed by the middle bars of the original triangle from the initial triangular area (D=2). We repeat this procedure in the three remaining triangles, etc. - an infinitely many infinitely small triangles are created. This formation consists of three of its copies reduced in half, so that D
H = ln3 / ln2 @ 1.58.
The Sierpinski carpet

is created by an analogous carving procedure from a square area, which we always divide into 9 identical squares (using perpendicular partitions in 1/3 and 2/3 of each side) and cut out the middle one. Repeat the same procedure with the remaining 8 squares, etc. The Sierpinski square with the default D = 2 consists of 8 of its copies reduced to 1/3, so D
H = ln8 / ln3 @ 1.89.
Menger's sponge
Generalizing to three-dimensional objects creates a Menger's sponge : Divide the cube into 27 identical smaller cubes and remove the middle 7 (whose edge is not part of the edge of the original large cube). We repeat this procedure with the remaining 20 smaller cubes, etc. In the end, an infinitely segmented three-dimensional grid with an infinite surface is created (each wall of the Menger sponge is also a Sierpinski carpet), but with a zero (infinitesimal small) volume. Interestingly, in addition to fractal properties, it also contains topological equivalents of all curves existing in space. Since this formation (with the initial topological dimension D = 3) consists of 20 copies of the original cube reduced to 1/3, the fractal dimension of the Menger sponge D
H = ln20 / ln3 @ 2.73. Some typical fractal sets and formations. a: Cantor's discontinuum (first 6 iterations). b: Sierpinsky triangle (first 7 iterations). c: Sierpinsky carpet (first 5 iterations). d: Menger's sponge (after about 5 iterations). e: Example of a polynomial Julius-Mandelbrot fractal (detail - cut-out from a complex plane).

Polynomial fractals
Other interesting fractal shapes arise as a geometric place of points in the Gaussian plane of complex numbers z = x + y.i ( i is an imaginary unit), for which iterative methods for solving some algebraic equations converge. The simplest example is a gradual iteration of the function of a complex parabola
zn+1 = zn2 + c, where a set of all complex numbers z0 is drawn into the complex plane, for which the sequence zn converges (zn is finite for n®¥). These so-called Julius- Mandelbrot sets  forms a great variety of often beautiful patterns, depending on the value of the constant c ; they can be continuous or discrete. They are now popularly rendered using computer graphics , including spectacular color modulations *). However, most fractals are "invented" by nature itself ..!..
*) The color representation of individual points of a bit-map is modulated, for example, by the number of iterations needed to reach a certain value | z |.
Fractals in nature

For complex fractal sets, the calculation of the Hausdorff dimension cannot usually be performed directly algebraically according to the above formula. It is necessary to proceed "experimentally" or empirically - to construct a graph [ln N(
e) « ln (1/e)] and determine the dimension by extrapolating the graph directive for 1/e®¥ ; or use lattice methods to determine the fractal dimension. In nature, we also encounter complex objects whose shape or behavior cannot be described by a single fractal dimension; such "multifractals" are characterized by two or more dimensions (manifested by different linear segments on a log / log graph).
From the fractal formations occurring in nature, values were empirically determined for the Hausdorff dimension: seashore D
H @ 1.26 (Richardson's constant); rock surface DH @ 2.3; human brain membrane surface DH @ 2.76.

Theory of relativity, quantum physics + chaos theory ® new conception of reality ?
As mentioned above, in nature we encounter a number of phenomena leading to the formation of fractal structures *). In the field of astrophysics and cosmology, it is possible that groups or clusters of galaxies and galaxies arising from germinal inhomogeneities spread by expansions of the universe to different sizes (§5.4) have a fractal structure. In §5.5 "Microphysics and Cosmology. Inflation Universe", the passage "Chaotic Inflation and Quantum Cosmology", we will see that the presumed set of spontaneously emerging universes from quantum fluctuations creates a "fractal tree" of new and new "universes"...
*) Even fractal geometry is just a model, to some extent idealized. The fragmentation of natural formations and the complexity of the behavior of dynamical systems is colossal, but not infinite. The fractal model loses its validity at the level of atomic dimensions, where the idea of self-similarity ceases to apply (atoms do not have a fractal structure) and the determination of the metric dimension loses the possibility of realization.
The behavior of nonequilibrium and nonlinear dynamical systems is dealt with by synergetics, which can be described with a bit of exaggeration as the doctrine of organized chaos - against the background of seemingly chaotic behavior, hidden patterns are sought, which are often very non-trivial and remarkable. The self-similarity of fractals corresponding to strange attractors of chaotic systems is a kind of hidden order that is a symptom of some kind of "deterministic chaos".
It turns out that the theory of chaos and infinite arcticulation of formations is probably a new fundamental idea of the surrounding world, complementing two already elaborated and proven fundamental conceptions of modern physics: the theory of relativity and quantum physics. All three fundamental theories, on the one hand, lead to the enrichment and deepening of our understanding of nature and the universe, but at the same time, unfortunately, they set our knowledge at the same time
universal limit restrictions, a kind of "gnoseological barriers", through which we fundamentally can not get :
¨ The theory of relativity
shows an impassable limit the spread of fields and information given speed of light in vacuum and then in combination with gravity as a theory of curved spacetime, the existence of the event horizons, which are repeatedly discussed in our book.

¨
Quantum physics,
with its relations of uncertainty and stochastic character, sets the limits of recognizability of the course of individual processes in the microworld
(the individuality of particles and processes is actually blurred).
¨
Chaos
and nonlinear dynamics fundamentally limit the possibilities of long-term predictability of the exact behavior of all systems, even seemingly simple processes
(the most powerful computer or the most accurate numerical methods will not help us in this, it is a fundamental limitation!) .
However, we will not deal with these last aspects of the behavior of physical systems; in our book we will deal with a deterministic analysis of gravity and the structure of spacetime - that is, deterministic, at least in principle, within the general theory of relativity.   3.2. Minkowski planar spacetime and asymptotic structure 3.4. Schwarzschild geometry

 Gravity, black holes and space-time physics : Gravity in physics General theory of relativity Geometry and topology Black holes Relativistic cosmology Unitary field theory Anthropic principle or cosmic God Nuclear physics and physics of ionizing radiation AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy

Vojtech Ullmann