Topological shape of space-time

Chapter 3
GEOMETRY AND TOPOLOGY OF SPACE-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. Space-time singularities
3.8. Hawking's and Penrose's theorems on singularities
3.9. Naked singularities and the principle of "cosmic censorship"

3.1. Geometric-topological properties of spacetime

The gravitational field is a manifestation of the geometric properties of spacetime - this is the position of the general theory of relativity that we arrived at in the previous chapter. It is therefore useful to study the properties of spacetime in terms of geometry and topology. This will provide important knowledge of the general validity on the structure of spacetime and thus of the course of physical processes under the universal influence of gravity. Knowledge of the geometric structure of spacetime is not only interesting in itself, but is essential in the physics of black holes (see Chapter 4) and in cosmology (Chapter 5).
Note: Topological approaches and methods for the study of the properties of spacetime in the general theory of relativity were introduced in the 1960s by Roger Penrose.

T o p o l o g y
Before we proceed to our own study of geometric and topological properties of spacetime, we will roughly outline what is meant by topology and what is its relationship to geometry. A detailed explanation of the topology from a mathematical point of view is in a number of monographs, eg [151], [155], [60], we will outline only basic ideas here. Geometry (Greek. geos = earthly, terrestrial, metria = measurement - originally ie "surveying", land survey) formed as science of measurement (comparison) bodies - their length, shape, angle, area, volume, distance and the like *). The "scene" in which such measurements are made is space and we declare some common geometric properties of measured bodies to be geometric properties of this space. Space is a concept that expresses the mutual positional relationships of individual objects and their parts - it was created by abstraction from real material objects.
*) During the development of geometry, it gradually outgrew its original meaning and merged with all those parts of mathematics in which continuity plays a role. Geometry brings its great advantage to these general mathematical structures, which is its clarity. On two-dimensional analogies, sections, nesting diagrams, which contain almost all important features of multidimensional spatial shapes, many constructions can be clearly illustrated, which we cannot directly imagine in their general version - eg various transformations and representations can be interpreted as corresponding deformations (bending, stretching, gluing) of two-dimensional surfaces.
  The properties of space can be divided into quantitative - metric (related to the measurement of distances, angles, areas) - and qualitative - topological (Greek topos = place, logos = collect, study, calculate ). Topology, sometimes also called "qualitative geometry", is very roughly what is left of geometry when we take away from it everything that has some size (and in this sense a specific shape) *). It deals qualitatively with how points, sets and objects are internally and mutually interconnected (linked), or how they are adjacent to each other. Many geometric problems do not depend on the exact shape and size of objects, but only on the internal or external relationships that these objects have with each other.
*) Conversely, geometry is a topology provided with the concepts of distance and angle - the introduction of metrics .
  Topology studies such properties of geometric shapes, that do not change during continuous transformations ("deformations") - ie different expansions, compressions, rotations or bends *), provided that there are no tears or joints of the different parts; "near" points are transformed again into "near" points. It does not matter whether the object is small or large, round or square, because deformation can change these properties. From the point of view of topology, it is important whether the given object is holistic and continuous, whether it contains openings, "passages, tunnels", it is one-dimensional, planar or spatial, or multidimensional. In other words, the topology systematize our intuitive ideas and experience about the "possible" and "impossible" in space, in what ways it is possible or not to "get" to certain places.
*) We can imagine that the given shape is made of plasticine and we can smoothly and continuously reshape it into another shape without having to tear, puncture or join any parts (see below). During deformations, we must maintain the "neighborhood" of individual points, their surroundings.
  In terms of topology, a circle, ellipse, square or triangle are "the same" (they are one-dimensional objects that divide an area into two parts - inner and outer), they are homeomorphic to each other *) - using a topological mappig, you can deform a circle into an ellipse, a square or a triangle, and vice versa. All the more so are topologically equivalent circles with different radii, ellipses with different eccentricities, or squares with different side lengths. Similarly, sphere, ellipsoid, cube and pyramid. Such mutually homeomorphic formations are only different metric variants of the same topological set of points. Topology therefore studying the most fundamental global properties of space (and geometric shapes in it) as the integrity, continuity, the number of dimensions, limitations and limitlessness and etc. In this sense, the topology is deeper and more general than what is commonly considered to be geometry. Below we will see examples of spaces that have the same geometric (metric) properties, but completely different topological properties.
*) From the Greek. homeos = same; morphe = shape. Homeomorphic topological sets and shapes are the same from the topology point of view, they have the same "shape" and properties. Homeomorphism, also called isomorphism of topological spaces, will be defined below.
Topological similarities, modeling and transformation
Topology is not interested in specific "shapes", curvatures, sizes, but only in the internal interconnection of individual parts or points of the investigated formation. Let's take it closer to a a situation from everyday life.
  We will make a small afternoon siesta with coffee or tea and small vanilla wreaths. We hold a small coffee cup with  ear, we drink coffee and bite profiteroles. When we look at the objects of our sitting, the coffee cup has at first glance a completely different shape, size, curvature than the wreath. However, if the cup was made of perfectly malleable plasticine, it could be smoothly and continuously reshaped into a wreath shape, without tearing or piercing the plasticine (first we would press the walls of the cup down to the bottom and then press the resulting flat disk from the side until it merged with the ear; in the end, only the toroidal ear, shape equivalent to a wreath, remains). Conversely, from the plasticine wreath we could again smoothly and continuously model a cup with an ear. If we had a cup without a ear (as is the case with Japanese green tea), it would have a simple block topology - it would be equivalent in shape to a cube of nougat chocolate, or we could model it into a chocolate bar or round candy (we would proceed in the same way as in the previous case: we would press the walls of the cup to the bottom and then transform the resulting disk into a block or ball). However, we never continuously transform a cup without an ear into a cup with an ear, just as we do not remodel a bar of chocolate or a nougat ball into a wreath - a block or a ball cannot be transformed into a toroid homeomorphically !

Demonstration of how continuous deformations (homeomorphic mapping) can transform objects (point sets) into various topologically equivalent shapes.
Above: The coffee cup with a handle is topologically equivalent to a toroid - a vanilla wreath.
Bottom: A mug without a handle is topologically equivalent to a block or ball - a bar of chocolate or a round nougat candy.

Although it many seem strange at first glance, it is this small lug - ear that carries the overall toroidal topology of the cup! Similarly, in §3.5 "Reissner-Nordström geometry" and §3.6 "Kerr and Kerr-Newman geometry", we will see that in some specially curved spacetimes, relatively small structures inside the horizon carry a complex global spacetime topologies, even of topology the entire universe or more interconnected universes..!..

Sets and representations
The central abstract concept that forms the basis of mathematics is a set - a set of objects that are precisely determined either by their enumeration ("list") or by a characteristic property. For each object x can be unambiguously determined whether x belong to the given set of X - denoted x Î X, or does not belong to it: x Ï X. These objects belonging to a set are called elements of the set. Elements of sets can be basically anything in everyday life (apples, pears, trees, people, ...), in mathematics it is usually numbers, points of geometric shapes, functions and transformations, solution of equations. A set containing no element is called an empty 0. A part of the set A is called its subset B - it is such a set, all elements of which are also elements of the set A ; denoted BÍA. Each set is also its subset. A subset B that is not equal to the initial set A is called its own subset, denoted by B Ì A. The relations "Í, Ì" (they are analogous to " £, < ") between a set and a subset is called an inclusion (lat. inclusio = embracement - incorporation into some whole). The inclusion symbols " Ì, Í " are often do not distinguish in set applications (where their own subsets are usually used). In our text, we will generally denote any subset by the symbol "Ì ".
  Sets are usually viewed intuitively, but in fundamental mathematics the properties of sets are formalized using axiomatic set theory.
  Between two sets X and Y , the basic operations of unification XCY are defined (which is a set containing together all elements of X and all elements of Y ) and the intersection of XÇY (it is a set of elements belonging to both sets X and Y together) (other sometimes introduced operations, such as the difference of two sets or their symmetric difference and complement of one set in another, we will not use here). Sets with empty intersection (XÇY = 0) are denoted as mutually disjunctive.
  To compare sets with each other the imaging operations (binary sessions) are used.  The mapping j : X ® Y of the set X to the set Y means that we uniquely assign a certain point j(x) º y ÎY to each point xÎX. The element x is called a pattern and the element y its image. The display identifier j is also called a function, the set of patterns X is called a domain, and the set of images Y is called a domain of function values. The display identifier j (function) can be defined or written using a table, formula, calculation algorithm, graph, nomogram. The "opposite" or "reverse" display j^-1 : Y ® X is called the inverse display (but it cannot always be created). Depending on the uniqueness of the display, three types are distinguished: A display is surjective ( on set) when each image has at least one pattern. Injective, or simple representation, assigns two different images to each of two different patterns; therefore, an inverse view can be created for the injective (simple) imaging. If the representation is both surjective and injective, it is a mutually unique representation also called bijective - each pattern has exactly one image and each image has exactly one pattern.
Numbers and number sets
The basic abstract objects used for quantitative modeling of natural reality in everyday life, science and mathematics are numbers. From a mathematical point of view, we recognize several types of numbers :
l Natural numbers are positive integers 1,2,3,4, ..... They are the most common numbers that we encounter in everyday life in ordinary "numbering", determining the order, stating the quantity something, "merchant counting".
l Integers -3, -2, -1, 0, +1,2,3, .... are the addition of natural numbers by negative numbers (and also by zero, which is usually not classified as natural numbers), which we use to model reductions, missing amounts, "debt", values less than zero, and so on. Integers (and of course natural numbers) are divided into even (without a remainder divisible by 2) and odd (in which, after dividing 2, residue 1 remains).
l Rational numbers are those which arise as a proportion of two integers (lat. ratio = share) - can be written as a fraction a/b of two integers a, b¹0. The result can be either an integer (eg 6/2) or a non-integer value expressed either simply (eg 1/4 = 0.25) or a periodic decimal number (eg 5/3 = 1.666...) .
l Irrational numbers are generally those that cannot be expressed as a fraction of two integers. They are sometimes further divided into two subgroups :
- Algebraic irrational numbers are those that are the solution (root) of a polynomial with rational coefficients. A typical example is the square root, eg Ö2 is the solution of the algebraic equation x² -2 = 0.
- Transcendental numbers are those that are not by solving no algebraic equation with rational coefficients. They can only be expressed by infinite development. A typical example is Ludolf's number p or Euler's number e (base of natural logarithms). The name comes from lat. transcendent = transcending reasonal understanding .
l Real numbers are the unification of all rational and irrational numbers, they are all numbers that can be written by finite or infinite mathematical development. Thus, real numbers include all of the above natural, integer, rational, and irrational numbers, including transcentents. They can be used to quantify basically all events "really" taking place in nature.
l Imaginary and complex numbers. Complex numbers formally generalize real numbers by introducing roots from negative numbers (which cannot be defined in the field of real numbers, do not exist). The basic idea here is the introduction of an imaginary unit i , for the square of which the relation i ² = -1 applies. In other words, the imaginary unit is the square root of -1: i = Ö-1 . Complex numbers are then a kind of "combination" of real and imaginary numbers, they are written in the form c = a + bi, where a and b are real numbers. The number a is called the real part of a complex number c , the number b of its imaginary part. In the field of complex numbers, each algebraic equation has a corresponding number of solutions, corresponding to the degree of the polynomial. Although complex numbers do not have a direct physical meaning, they are a very useful tool for modeling a number of processes where periodic trigonometric functions occur (electrical circuits, waves, quantum physics). In our interpretation of the theory of relativity and gravity, we will not use complex numbers, with a few exceptions ...
Magnitude - mightiness - cardinality - of sets
The basic properties of sets are their "size", number of elements, range - how "mass" is. The size of a set is characterized by a term called cardinality. Sets containing only a limited (finite) number of elements are called finite sets - you can specify the number of elements and express it with a natural number. The cardinality of finite sets is equal to the number of elements. Infinite sets can be divided into two categories according to "size" :
¨ A countable set is one that can be unambiguously displayed on a subset of natural numbers - the elements of a set can be "calculated" by numbering them with natural numbers ; therefore, every finite set is automatically countable. The power of infinite countable sets is denoted by the symbol "alef-0" or c₀ (the letter c belongs mirror-inverted). The basic example of countability is the set of natural and rational numbers (it can be proved that they have the same cardinality). Even the set of algebraic irrational numbers is countable.
¨ An innumerable set is one that cannot be unambiguously displayed on any subset of natural numbers - its elements cannot be "calculated" by numbering nor by an infinite number of natural numbers. A basic example of innumerability is the set of real numbers ; its power is denoted by the symbol "alef-1" or c ₁ - magnitude of the continuum. The set of all irrational numbers, with the inclusion of transcendental numbers, has the same magnitude - it is these numbers that are responsible for innumerability. In mathematics, much more "infinite" or "innumerable" sets are introduced. A more detailed discussion of the cardinality of sets and an analysis of the nature of infinity in mathematics and physics is discussed below in the section "Infinity in Spacetime".
  For our purposes of modeling the geometric and topological structure of spacetime in relativistic physics, we suffice with sets of cardinality c₁, corresponding to the set of real numbers. Some properties of complex so - called fractal sets and shapes (sometimes referred to as "mathematical monsters") are briefly discussed in §3.3, section"Determinism-chance-chaos?").
Mathematical structures on the sets
For order for sets to be used for modeling the laws of our world, mathematical structures are introduced on general abstract sets - additional information on the properties and relationships between elements. They can be algebraic operations such as addition and multiplication (thus arise primarily groups - §B.6 "Unification of fundamental interactions. Supergravity. Superstrings", part "Symmetry in physics - Groups of transformations, calibration groups", further circuits, bodies, vector spaces ... ), ordering relations and logical operations (equations and inequalities, ordered sets, Boolean algebras, ...) , introduction of metrics for determining "distances" (metric spaces), topology - see below. The proper set on which such a structure is introduced is sometimes called a carrier set.
   Isomorphism (Greek isos = same, identical ; morph = shape) is a mapping between two sets with the same structure, which is mutually unique (bijective) and retains all the properties introduced by the mathematical structure on the set. Thus, each element of the first set corresponds to exactly one element of the structure of the second, whereas this assignment maintaining structural relations to the other elements. If such a representation exists - sets and structures are isomorphic - both sets have identical properties in terms of structure. In the case of an isomorphism, it is stated to which structure it relates, eg metric or group isomorphism. In this chapter, we will deal with a topological isomorphism called homeomorphism.

Topological spaces and their representation
A part of mathematics called topology, which is based on the refinement of the intuitive concepts of "continuity", "proximity", "limit", deals with a kind of "topography" of point sets. It studies the qualitative concept of "proximity" of individual points by specifying what is meant by the surroundings of each point of the set. The initial step of a set topology is to cover a given carrier set with a suitable set of subsets.
We say that a topology is given on the (carrier) set X , if the system U of subsets U Ì X it is determined and holds that :
a ) The intersection of a finite number of sets from U also belongs to U (U ₁ Ç U ₂ Î U) ;
b ) Unification any system of sets of U also belongs to U .
  The set X (which is also an element U) together with a given topology is called a topological space (X, U). The system U is called the topology on the set (X, U). The sets U Î U are called open sets. Surroundings U(x) of the point xÎX we mean the open set UÎU, which contains the point x. For each point xÎU, some of its surroundings also belong to this open set. Thus, with each of its points, the open set also contains points that are "close enough" to it.
  Boundary of a set X represent the set of all such elements of X, each neigborghood of which contains at least one point of a given set X and at least one point outside the set X. This boundary is denoted ¶X. A closed set is one that contains its own boundary.  
  Implementing a topology allows you to specify other important properties of a set imaging. The mapping of the topological space (X, U) to the space (Y, V) is called a continuous map if for every point xÎX and for each neighborhood VÎV at the point j(x) Î Y there exists a neighborhood U such that j(U) Ì V. It is therefore a imaging that displays sufficiently close points again close to each other - it preserves the surroundings of the points.
It is a topological generalization of a continuous function in mathematical analysis, the graph of which does not have sharp jumps and can be represented as a continuous curve. We say that the function f(x) is continuous at the point x = x_o, if for each positive number e there exists a positive number d such that for all values of x in the interval x-x_o < d < x+x_o the functional values satisfy the inequality f(x) -f(x_o) < e < f(x) + f(x_o). The close neighborhoods of the independent variable x are displayed in the close neighborhoods of the function values f(x).
   Limit of imaging j : X®Y between the topological spaces at the point x_oÎX is defined as the point y_oÎY such that for each neighborhood U(y_o) of the point y_o there exists a neighborhood U(x_o) of the point x_o, for which the implications xÎU(x_o) Þ yÎU(y_o) applies. It is writen lim_{x ® xo}j(x) = y_o .
This topological definition is a generalization of the limits of a function, used in mathematical analysis to investigate the behavior of functions around a certain point. The limit lim_x®xof(x) = y_o expresses the fact that if the value of the independent variable x approaches the value x_o, the value of the function f(x) approaches indefinitely close to the value y_o - the limit of the function at the point x_o. This is defined by the behavior of the function in the infinitesimal vicinity of the investigated point x_o: The function f(x) has a limit y_o at the point x_o, if there is a positive number d for each positive number e so that for all values of x from the neighborhood x-x_o < d < x+x_o the functional values satisfy the inequality f(x)-y_o < e < f(x)+y_o. A function can have a well-defined limit even at a point where the actual function value is not defined (eg the function [e^x -1]/x has a limit equal to 1 at the point x=0). For continuous functions, the limit is equal to the functional value at the given point: lim_x®xof(x) = f(x_o); the opposite is also true. For discontinuous functions, limits are introduced from the left and right; if these limits are the same, we denote this value as the limit of the function at the given point. If they differ, there is no limit at this point.
The concept of limit is the initial basis of differential and integral calculus, which examines changes in functional values depending on the infinitesimal changes of the independent variable. Using the function change limit, the derivative and the inverse integration process are introduced.
  Mutually unique (bijective) continuous mapping of j space (X, U) on (Y, V), for which the inverse mapping j^-1 is also continuous, is called homeomorphism  (it is obvious that j^-1 is then also a homeomorphic mapping of the space Y on X). Homeomorphic mapping is thus an topological isomorphism - such a mutually unique mapping of sets X and Y, in which the near points of one set are converted to the near points of the other set (open subsets in X and Y forming the vicinity of points xÎX and j(x)ÎY they are in a mutually unambiguous relationship) - the surroundings of the points are preserved. The sets X and Y, between which such a homeomorphism exists, are called homeomorphic and are considered topologically equivalent. Homeomorphism is the expression of those "continuous deformations" (compression or expansion) mentioned above. Topological concepts and topological properties are those concepts and properties that remain in homeomorphism *).
*) For example, an electrical circuit is a topological term, because for its operation it is not essential the geometric arrangement of individual components, but their mutual electrical interconnection. If we change the spatial arrangement of components without interrupting their electrical connection, the circuit will work the same (this is not quite true for high-frequency technology, where the phenomena of capacitance, electromagnetic induction or wave radiation may apply differently for different component distributions).

The most illustrative example of a topological space is a set of real numbers R¹ with a natural topology given by a set of subsets AÌR¹, which together with each of their points always contain a certain interval around it: for each point xÎA there are numbers a, b, such that a < x < b and interval (a, b)ÎA. The generalization is the n-dimensional Euclidean space Rⁿ of all n-tuples real numbers (x¹,x²,...,xⁿ) at -A < xⁱ < +A with the usual topology. And it is the well-known properties of Euclidean space, "learned" from the behavior of macroscopic bodies, that allow (by means of a suitable mapping) to introduce additional structures on an otherwise amorphous topological space and thus make it a suitable tool for modeling physical processes.

Regular and singular behavior
In everyday life, surrounding objects have the usual and expected physical, geometric and topological properties. They are (relatively) smooth and continuous, have finite dimensions and finite values of their physical quantities. We call such behavior regular. And it can be mathematically modeled using regular displays - transformations, which are the above-mentioned simple mutually unambiguous representations for which unambiguous inverse transformations are available.
  However, in mathematical modeling in theoretical physics, we also encounter situations where the relevant equations diverge and formally give infinite or indeterminate values of physical quantities. A simple example is the idealization of a point electric charge, where according to Coulomb's law there is an infinitely large electric field intensity in a place with zero distance (r = 0). Such anomalous behavior of a physical quantity is called singular (it is the opposite of regular behavior). And the place or point of anomalous behavior of a physical quantity is called singularity (Latin singularis = unique, exceptional, isolated ) .
  In classical physics, singularities are more or less formal, they arise from the idealization of a given model and in realistic cases they do not occur. In the general theory of relativity, however, it turns out that gravitational or spacetime singularities of metrics naturally arise even under very general assumptions, which are probably fulfilled in astrophysical practice. In addition to removable coordinate singularities ("pseudo-singularities"), there also occur the irreparable real physical singularities. Therefore, we will deal with singularities in many places in our book - first the mathematical aspects from that §3.1 (in its final passages) through §3.4 "Schwarzschild geometry" 3.5. , 3.6 "Kerr and Kerr-Newman geometry", up to §3.7 "Spatio-temporal ingularities" and 3.8 "Hawking and Penrose theorems about singularities". From the astrophysical point of view then in chapter 4 "Black holes", "The final stages of stellar evolution. Gravitational collapse. Formation of a black hole." and §4.4 "Rotating and electrically charged Kerr-Newman black holes", in cosmology in §5.3. Friedman's dynamic models of the universe , §5.4. Standard cosmological model. Big Bang. Forming the structure of the universe. , §5.5. Microphysics and cosmology. Inflationary universe. .

M a n i f o l d s
In order to be able to quantify the topological space numerically, it is useful to introduce numerical coordinates in it - to assign numerical values of real numbers to the points of the topological space. This creates the so-called manifold.
The name "manifold" here means diversity, variety, ductility, adaptibility. So far, it is completely general, no relationships are established in it (such as connections or metrics). It can model objects of various shapes and structures.
  Manifold of dimension n (n-dimensional manifold) Mⁿ is such a topological space, each point of which has a neighborhood homeomorphic with Rⁿ (with a certain neighborhood in Rⁿ). The homeomorphic mapping j of an open (sub) set AÌMⁿ to Rⁿ assigns to each point xÎA an n-tuple of numbers j(x) = (x¹,x²,...,xⁿ) Î Rⁿ, which call coordinates of the point x. We say that on a set A the coordinate system (system of coordinates) x ⁱ is introduced. By selecting another homeomorphic mapping j' from AÌMⁿ to Rⁿ, other different coordinate values (x'¹,x'²,...,x'ⁿ) Î Rⁿ will be assigned to individual points xÎ A - we will go to another coordinate system in a set of A. We performed a coordinate transformation.

Topological and Hausdorff dimension
The above-mentioned dimensionality - the number of dimensions - of a set or object is the usual topological dimension. It is an integer n indicating the number of parameters (coordinates) by which the position of individual points of this unit is unambiguously defined. In addition to the topological dimension, an alternative metric variant of the dimension is introduced, the so-called Hausdorff-Besikovic dimension, which for geometrically smooth sets and shapes is equal to the respective topological dimension, but for so-called fractal shapesit can be higher and usually non-integer. We postponed the analysis of this kind of dimension (for formal-technical reasons of interpretation of physical phenomena in relation to the causal structure of spacetime) until the end of §3.3, passage "Determinism in principle, chance and chaos in practice?", where we use it to discuss the behavior of chaotic systems, described by so-called strange attractors in phase space.

However, it is not possible to display the whole Mⁿ to Rⁿ in this way for many topological spaces (eg the mapping S² to R² introducing the spherical coordinates J, j on the spherical surface S² ceases to be mutually unique on the poles). Thus, in general we can display the manifold Mⁿ in Rⁿ in parts - to create local coordinate "maps" (A_a, j_a) of individual "domains" (coordinate surroudings) A_aÌ M. Set of maps of individual domains A_aÌ M, covering M (i.e. _aCA_a=M) form the "atlas" of manifold M. Only manifolds topologically equivalent to Rⁿ can be completely covered by a single map (M, j). With the introduction of the coordinate system, the points of the manifold M lose their "anonymity" and the manifold can be investigated using well-known and well-developed mathematical operations with real numbers.

Fig.3.1. In the differentiable manifold Mn, the images fa(p) and fb(p) of the point p from the intersection of the two domains Aa and Ab are bound by continuous transformations, including derivatives of the r-th order.

The manifold Mⁿ is called a differentiable of class C^r, if it is given an atlas of maps (A_a, j_a) of individual domains A_aÌ Mⁿ represented by mutually unique displays j_a on open sets in Rⁿ satisfying the conditions :
a ) A_a forms the cover M, i.e. _aCA_a = M ;
b ) If two domains A_a and A_b have non-empty intersection, then the points pÎA_aÇ A_b of this overlapping part will be assigned by the representation j_a to the n-tuple of coordinates xⁱ_a(p) Î Rⁿ and by the imaging j_b at the same time to the n-tuple of coordinates xⁱ _b(p) Î Rⁿ such that the transformations xⁱ_b(p) = xⁱ[x^k_a(p)] are in Rⁿ continuous functions with continuous derivatives up to the r-th order (Fig.3.1) .
  If we apply the property b) to two domains (A, j : x®xⁱ(x)) and (A', j' : x®x'ⁱ( x)) such that A' = A = AÇA' but j' ¹ j, then the transition from the coordinate system xⁱ to another coordinate system x'ⁱ will be given by the regular and continuous transformation x'ⁱ(x)= x'ⁱ[x^k(x)] r-times derivable. In differential geometry, we mostly deal with local geometric properties within a single local map, while global geometry studies the geometric-topological structure of the whole manifold.

In order for the manifold to have the usual local properties (and be usable for the classical description of physical processes), still two additional requirements are placed on it : Hausdorff and paracompactivity. A space is called Hausdorff if there are different surroundings for every two different points. The paracompactivity requirement means that for each coverage of manifold M by a system of open subsets, there is a refinement in which each point of the manifold has an environment intersecting only a finite number of subsets of that refined coverage (ie, this refinement is locally finite) [155]. When Hausdorff is met, the paracompactivity is equivalent to the requirement that M have countable base, i.e. that be such a countable set of open sets whose union is any open set of M (spaces whose topology has a countable base are called separable ). Paracompactivity allows the introduction of a connection on M (see below).

Fig.3.2. Contiguity of sets (manifolds).
a ) Continuous set. b ) An incoherent set, which is a union of two disjoint parts.
c ) Simply continuous set - all connections between two points are topologically equivalent, each closed curve is homologous to zero.
d ) Double continuous set - there are two classes of connections between points, some closed curves (eg C ) cannot be shrunk to a point.

In short, an n-dimensional manifold is a topological space that locally (in a sufficiently small vicinity of each of its points) "looks" like the Euclidean space Eⁿ. In order for this similarity to be plausible, it is necessary to establish the above-mentioned conditions of separability and paracompactivity.

Curves and surfaces
The curve (line) l(t) on the manifold M means the imaging of a certain section R¹ ® M, ie the set of points in M, which are imaging of the curve points xⁱ = xⁱ(t) in Rⁿ parameterized by the variable tÎ R¹. The basic topological characteristic of each set (geometric shape) is the connection, integrity. As a continuous denote we such a manifold that is not formed by the unification of several disjoint non-empty parts; then each of its two points can be connected by a line that is a whole part of this set (Fig.3.2a). Otherwise, it is an incoherent set (Fig.3.2b). A continuous set is called simply continuous, if for every two points A and B, all connections between them are topogically equivalent to each other (homologous); in other words, each closed curve can be continuously "pulled" to a point (each closed curve is homologous to zero) - Fig.3.2c. If there are several types of connections between some points that are not topologically equivalent to each other, it is multiple continuous set (Fig.3.2.d), where some closed lines cannot be "compressed" until they disappear at a point. Here, the "multiplicity" of a coherence is defined as s = c + 1, where c is the number of topologically independent closed lines that cannot be shrunk to a point (c is also equal to the number of "cutting" after which the set becomes simply continuous); the quantity s indicates how many topologically different paths it is possible to get from one place of the manifold to another place.
  The generalization of a one - dimensional curve in the manifold Mⁿ is the p - dimensional area (surface) C^p (p £  n), which is a imaging of the corresponding p-dimensional subspace in Rⁿ. Such an area C^p can be considered as the sum (unification) of elementary p-dimensional "parallelograms", resp. "cubes" K^p (which are generally "curvilinear") 0 £ x^a £ 1 (a = 1,2, ..., p). Orientation and addition are introduced here in a suitable way, which makes it possible to study the connections between different surfaces C and their boundaries ¶C, eg during integration [217]. The oriented p-dimensional cube K^p has a (p-1) -dimensional boundary ¶K formed by individual walls. This area is closed and therefore does not itself has no limit, so that the (p-2) -dimensional boundaries (p-1) -dimensional boundaries p-dimensional cube is equal zero: ¶¶ K = Ø. This follows also from the construction of the boundaries of the cube by the sum of the squares forming the boundaries of the individual walls of the cube, where each side of the square is counted twice with the opposite orientation and is therefore canceld.
  The general surface-area S can be decomposed into a number of cubes (appropriate dimensions) K_i : S = _iS aⁱ K_i ; then we define the boundary of the surface S as the sum of the boundaries of the "cubes" of which it is composed: ¶S = _iS aⁱ ¶K_i (in fact, most of these contributions from internal regions are canceled because they are counted twice with the opposite orientation, similar to the usual derivation of a Gaussian or Stokes theorem). If the boundary of a p-dimensional surface S is equal to zero (¶S = 0), it is a closed (compact) surface. Boundaries ¶S of each surface (not only closed) is a closed surface that no longer has its boundary, so it always applies

this is referred to as the topological principle "the boundary of boundary is equal to zero", which is of great importance for the conservation laws in general field theory [181], see also §2.5.
  If two closed surfaces C^p₁ and C^p₂ form the boundary of the (p + 1) -dimensional region in M, we say that they are homologous to each other (they can be converted into each other by continuous deformation); if the closed surface C^p itself forms the boundary (C^p = ¶A^p+1) of the region A Ì M, it is called homologous zero (it can be retracted to a single 0-dimensional point by continuous deformation). The homology class {C^pi} consists of all closed p-dimensional surfaces C^p which are homologous to each other.
In the Euclidean space Rⁿ, all p-dimensional (p <= n) closed surfaces can be compressed to a point, so that they are all homological zeros and belong to the zero homology class {C^p₀} = {0}.
The number of independent homology classes {C^p₁}, {C^p₂}, .,., {C^p_Bp} of areas of dimension p is called the p- th Betti number of manifold M (the class {C^p_o} = {0} areas of homologous zeros is not included here). The quantity c = _p=0Sⁿ(-1)^pBp is called the Euler characteristic of this manifold. The so-called topological genus of manifold is also used to describe the topological complexity (multiple connections) of manifolds, which is a number indicating the number of groups of closed curves that cannot be pulled to a point by a continuous transformation because they run around a topological tunnel or cut-out area. For the two-dimensional manifold M² between the genus g and the Euler characteristic c, the relation c = 2 - 2g belongs.
  Because summation is defined between the surfaces C^p, the set of these surfaces in the manifold M forms a group; a set of classes of mutually homologous p-dimensional closed surfaces then form a p-dimensional group of homologies of a given space. Relationships between sets and their boundaries can thus be studied by algebraic methods in the so-called algebraic topology [151], [106].
Note: H.Poincaré was at the birth of algebraic topology around 1900, who assigned elements of a certain group (called the fundamental group of the manifold ) to the curves on the 3-manifold .

The reason for the multiple connection of the area according to Fig.3.2d is obvious: the part of M is "cut out", so that the given area has, in addition to the outer boundary, also an inner boundary, through which no line may go. However, there are formations and entire spaces without boundaries that are multiple contiguous, as we will show in the following simple examples :
  We take a straight sheet of paper, which can be considered as part of the Euclidean plane R² (Fig.3.3a). This sheet is simply continuous and the axioms of Euclidean geometry apply here (therefore, for example, the sum of the angles in the drawn triangle will be equal to 180°). If we curl this sheet of paper and glue the opposite sides, ie we make the identification (x+a, y) º (x, y), we get a cylindrical surface. The Euclidean character of the geometry did not change locally - the distances between the individual points remained the same, the angles and areas did not change. However, due to its global topological properties, this cylindrical surface is a completely different two-dimensional space than the original Euclidean plane. Between any two points, there are two topologically distinct classes of connectors, a closed circle surrounding the cylinder does not download to the point, while others closed curve yes; the cylindrical surface is doubly continuous and finite in one direction (dimension). The Betti numbers here are B₀ = 1, B₁ = 1, B₂ = 1.

Fig.3.3. On the relationship between (geo)metric and topological properties.
a ) A sheet of paper is part of the Euclidean plane. By twisting and gluing it, we get a cylindrical surface with locally preserved Euclidean geometry, but a different global topology.
b ) If an additional 180° twist is performed during twisting, a Möbi sheet (strip) is formed.
c ) By twisting and gluing a section of the cylindrical surface, a toroid (annulus) is formed.

Or similarly by bending, twisting by 180° and gluing - ie by identifying (x+a, y) º (x, -y) - paper tapes with originally Euclidean geometry and topology, we get the known Möbi sheet (strip, Fig.3.3b), whose local geometry again does not differ from Euclidean, but topological properties have other. This is a one-sided surface (a known unsuccessful attempt to color the "front" and "back" sides in one stroke with the same color), on which orientation cannot be introduced, because after one "round" cycle, what was left appears on the right, direction "up "changes to" down "and vice versa.
  The given examples show that to completely determine the nature of space are not enough its (local) metric properties, but its (global) topological properties must also be taken into account. In addition to the Euclidean space Rⁿ, on which the concept of manifold is based, there are also more general manifolds with other topological properties. Here are some more cases.

One of the most important types of manifold is the spherical surface. A two-dimensional spherical surface (sphere) S ² of unit radius is, as is known, an area in R ³ , the points of which are given by the equation (x¹)² + (x²)² + (x³)² = 1. Analogously, the n-dimensional sphere Sⁿ (as a subspace in Rⁿ⁺¹) is the geometric place of points in Rⁿ⁺¹ satisfying the condition _i=1Sⁿ⁺¹(xⁱ)² = 1. The sphere Sⁿ is finite (compact) simply continuous manifold. For a two-dimensional spherical surface S² , the Betti numbers B₀ = 1, B₁ = 1, B₂ = 1 and the Euler characteristic c(S²) = 1.
  If we twist a two-dimensional cylindrical surface (made of elastic material) and glue the opposite bases, a toroid (anuloid, Fig.3.3c) is formed, which, unlike the original cylindrical surface, has its internal geometry curved. This toroid T², which is formed by identifying (x+a, y+b) º (x, y) points in R², is an example of triple-continuous surfaces: there are two classes of closed curves - circles along the "large" and "small" perimeter of the toroid - that cannot be shrunk to a point. In general, the n-dimensional toroid Tⁿ is the space that results from the identification of (xⁱ+aⁱ ) º (xⁱ), i = 1,2, ..., n, points in Rⁿ. The two-dimensional toroid T² has Betti numbers equal to B_o=1 (corresponds to the class of all points - all points are homologous to each other), B₁=2 (there are two independent classes {C¹₁} and {C¹₂} of closed curves passing around the smaller an larger circumference of the toroid), B₂=1 (corresponds to the toroid itself); Euler's characteristic c(T²) = 0.

From the n-dimensional manifold Mⁿ and m-dimensional manifold M^m we can construct the (n+m) -dimensional manifold Mⁿ x M^m by the "Cartesian product", whose points are pairs (x, y), where x is any point z Mⁿ and y any point of M^m. E.g. Euclidean space R³ is the product of R²´R¹, Rⁿ can be written as Rⁿ = R¹´R¹´...´R¹ (Cartesian product of n-coefficients). The cylindrical surface C² can be considered as the product of a circle and the Euclidean line, ie C² = S¹´R¹. As far as the toroid is concerned, it is especially clear that the one-dimensional toroid T¹ and the one-dimensional sphere S¹ (circle) are homeomorphic to each other, i.e. T¹ = S¹. Therefore , from an topological point of view, the n-dimensional toroid Tⁿ is a Cartesian product of n circles: Tⁿ = S¹´S¹´...´S¹.

The topological structure of the manifold Mⁿ ´ M^m is naturally given by the structure Mⁿ and M^m : for any points x Î Mⁿ and y Î M^m having coordinate neighborhoods A Ì Mⁿ and B Ì M^m is the point (x, y) Î Mⁿ ´ M^m contained in the coordinate neighborhood A ´ B Ì Mⁿ ´ M^m and has coordinates there (xⁱ , y^j), where xⁱ are the coordinates of point x in domain A y^j coordinate of the point y in the domain B .
  The function f (scalar field) on the manifold Mⁿ is a mapping from Mⁿ to R¹. We say that this function is differentiable of class C^r at the point p Î M, if it is defined in the open vicinity of the point p and and its expression f(x) = f(x¹,x²,...,xⁿ) using the coordinates xⁱÎRⁿ in some local coordinate system has a continuous derivation up to the r-th order according to xⁱ. It follows from this definition that in the differentiable manifold M of the class C^s, the coordinates xⁱ(x) is a differentiable function of class C^s.

Tensors in a manifold
Other geometric objects that are naturally related to the structure of a manifold are tensors and tensor (especially also vector ) fields. The r-order tensor at the point "p" of the n-dimensional manifold Mⁿ means the summary of n^r numbers
            T^il_jlⁱ²_j2^._.^._.^._.^ia_jb   ,   jl,j2,...,jb, il,i2,...,ia = l, 2, 3, ..., n
with a £ r contravariant (upper) and b = r- a covariant (lower) indices, which during the transformation of coordinates x'ⁱ(p) = x'ⁱ(x^j(p)), ie. dx'ⁱ =(¶x'ⁱ/¶x^j) dx^j, transform in contravariant indices as products of a- coordinate differentials and in covariant indices as products of b - inverse differentials at point p :

(3.2)

These transformation properties guarantee that the tensor equations are invariant (covariant) with respect to coordinate transformations. The rules for arithmetic operations between tensors are the same as in the Euclidean space Rⁿ.
  The possibility of introducing any tensor field on a manifold is generally conditioned by the topological properties of the manifold [151], [106]. E.g. each non-compact manifold admits the existence of a constant vector field. However, for the existence of a constant vector field on a compact manifold, it is a necessary and sufficient condition that the Euler characteristic c manifold equal to zero. For example, a cylinder or toroid allows a constant vector field, while a spherical surface does not ("we can not smoothly comb the hair on the tennis ball").
Connections and metrics in a manifold. Curvature of space.
In order to be able to compare vectors and tensors entered in different points of the manifold, a connection (from the Latin connectio = joining, connection, binding) is introduced, ie a rule (prescription) for parallel transfer of vectors and tensors between different points; the manifold thus becomes a space of affine connection (lat. affinis = adjacent, connected, related). And here he can come to the word the differential geometry - calculating covariant derivatives of tensor fields, quantification of curvature using the curvature tensor, determination of geodetic lines, etc., as outlined in §2.4 "Physical laws in curved space - time", part"Curvature of space. Curvature tensor".
  The concept of curvature ( flexion ) in differential geometry plays an important role. It generalizes, formalize and quantify our intuitive experience with the shapes of uneven objects - lines (curves), surfaces, solids. During the development of differential geometry was introduced several expressions of curvature, especially internal and the outer curvature :
The outer and inner curvature
Resolution of these two types of curvature (stemming from a different point of view) can be clearly illustrated by two-dimensional surfaces on which they live two-dimensional beings that, in principle, cannot leave their 2-dimensional world. The inner curvature of a surface is the curvature that our two-dimensional beings can observe without leaving their 2-dimensional world: to measure, for example, around the circumference of each point, the circumference of the circles L and their radii r. From the detected differences from the Euclidean relation l = 2p r they can determine the internal curvature, quantified eg by the so-called Gaussian curvature C_G=6.(1-L/2pr)/r². This internal curvature can have different values at different points on the surface.
  To imagine the external curvature, draw a few triangles and circles on a sheet of paper representing the Euclidean plane. If we bend and glue this sheet of paper into a cylindrical surface, distances or angles, this action will not change from the original Euclidean values, the resulting cylindrical surface still has zero intrinsic curvature C. However, on the drawn triangles and circles, we, as three-dimensional beings, will observe that the sum of the angles in the triangle is greater than 180^o and the circumference of the circle is less than 2p r-times the radius, measured over three-dimensional space. In this way we will observe the external curvature of the cylindrical surface. However, for our analysis of gravity as a curved spacetime, only internal curvature is essential .
  Finally, a metric is introduced into the manifold, ie a rule for determining the distances between individual points, thus creating a metric space. The distance between the point xⁱ and the adjacent infinitely close point xⁱ + dxⁱ is given by the coordinates given by the differential form ds² = g_ik dxⁱ dx^k (i, k = 1,2, ..., n), where g_ik is a metric tensor expressing the relationship between coordinates and actual distances. In order for connections to be compatible with metrics (connections and metrics are generally independent structures introduced into a manifold), the rules of tensor algebra must be observed during parallel transmission and the size of the transmitted vector must be preserved. This leads to the law of parallel transfer (2.8) and an unambiguous relationship (2.2b) between the coefficient of the connection and the components of the metric tensor [214], see §2.1 "Acceleration and gravity from the point of view of special relativity" and 2.4 "Physical laws in curved spacetime". A metric space with a connection (compatible with metrics) is called a Riemann space. Differential geometry provides precise analytical tools for quantifying the curvature of space - §2.4, part "Curvature tensor".

Spacetime as a manifold
After this fleeting excursion into the field of general geometric-topological structures, we can return to our own object of interest - gravity and spacetime. All known physical phenomena take place in space and time - within four-dimensional space-time. Experience teaches us that spacetime has the properties of a continuum in common macroscopic scales (unlimited divisibility of spatial scales and time intervals) and can be modeled as a four-dimensional differentiable manifold with Riemannian metrics. When studying the geometric properties of spacetime, we will start from this basic model :
A. Spacetime is a continuous four-dimensional differentiable manifold M⁴, which is Hausdorff and paracompact with Riemann metric g. We will denote it (M, g). We require the connection of the manifold M because nothing can occur and move outside of space and time, so if it were incoherent, any information for one part M about the other disjoint parts of M would in principle be inaccessible, so such unrelated parts would not effectively exist.
B. In spacetime M there may be various "physical" ("substance", non-gravitational) fields, such as electromagnetic fields, which will follow certain equations. These equations will have the character of relations between tensors (tensor fields) *) in M (general covariance principle) and their covariant derivatives according to spacetime coordinates with respect to the connection G induced by the metric g.
*) We do not consider spinor fields here. The introduction of spinor formalism is advantageous in some cases [97], but in general the spinor relations can be replaced by equivalent (albeit more complicated) tensor equations.
We will assume that the real physical field in M will have the following two basic properties :
1.  The equations describing the behavior of the fields must be such that the signal (energy) transmission takes place locally inside or on the mantle of the space-time light cone. Thus, the transfer of signal and energy between two points (events) of spacetime is possible only if these points can be connected by a light line that lies everywhere inside or on the mantle of a local light cone (tangent vector is at each point either time or light type). This property is an expression of local causality.
2.  For each physical field in M there exists a symmetric tensor T^ik - energy and momentum tensor, which depends on the potentials (intensities) of the fields and their covariant derivatives in the metric g. The energy-momentum tensor has the following properties :
a) T^ik = 0 in some subset M if the material fields are zero.
b) T^ik_{; k} = 0 applies - ie the local law of conservation of energy and momentum.
c) The relationship between the geometries of space-time and the "substance content" is realized so that in spacetime M are satisfied the Einstein equation R_ik - ¹/₂ g_ikR = 8pT_ik. As shown in §2.5, the local law of conservation of energy and momentum T^ik _{; k} = 0 can be considered as a consequence of Einstein's equations of the excited gravitational field.

Formally each spacetime (M, g) can be considered as a solution of Einstein's equations R_ik - ¹/₂ g_ikR = 8pT_ik in the sense, that based on the components of metric tensor g_ik we can calculate the quantity (R_ik- ¹/₂ g_ikR)/8p and we define this as a T_ik tensor. In the general case, however, the energy-momentum tensor thus defined may not have physically permissible properties. Only the spacetime of very specific geometric properties will describe the real one gravitation fields excited by the actual distribution of matter ~ energy.

Local and global properties of spacetime
Geometric and topological properties of spacetime are usually divided into local and global *). Within the classical general theory of relativity, local geometry and topology are not interesting (with the exception of singularities), because according to the principle of equivalence, spacetime is locally Euclidean everywhere. Currently, the adequacy of the concept of manifolds, i.e. the idea of the continuous space and time is experimentally verified by scattering experiments of elementary particles at high energies to scales of the order of about 10^-16 cm [229]. However, if we take into account quantum regularities (universal influence of uncertainty relations), they can be local geometric and topological properties of spacetime within very small (subnuclear ~ 10^-33 cm) regions strongly different from the usual Euclidean regions. We can illustrate this by looking at a perfectly polished surface of a mirror (whose local geometric and topological properties normally appear to us perfectly Euclidean) under a microscope, we see very significant local differences from ideal flatness and even smoothness and continuity - the microstructure has no Euclidean geometry and not even topology. We will postpone the questions of the local topological structure to "Appendix B" (§B.4 "Quantum Geodynamics") and otherwise we will consider the local geometry and topology ii of spacetime to be Euclidean.
*) However, this distinction may not always be completely unambiguous - for example, in the presence of a local naked singularity, there would be no global Cauchy hypersurface (§3.3).
  Before the creation of the general theory of relativity, even the questions of global geometry and the topology of spacetime did not seem interesting; the structure and evolution of the universe was mostly understood as the distribution and evolution of substances and fields in space, while the structure of space and time itself was taken for granted - Euclidean. Even within Newtonian physics or STR can formally consider more complex topological structure of the space, but there is not any physical reason for this, it would only purposeless structure (about pun). However, the general theory of relativity shows that spacetime is curved (and this curvature can be strong), so its global properties can differ significantly from those of Euclidean. From a two-dimensional analogy, we know that compared to planar surfaces, curved surfaces have a great manifold of shapes with different geometric and topological properties - these surfaces can be open, closed, variously "intertwined" (multiple continuous) and the like. In the curved spacetime of GTR can be expected situation where not only geometrical but also globally topological properties of space can be completely different than the usual Euclidean. This will indeed manifest itself in almost all the cases that we will investigate in §3.4-§3.6 as exact solutions of Einstein's gravitational equations.
  As we showed in Chapter 2, the basis of GTR is the local principle of equivalence, which is the connecting "bridge" between non-gravitational and gravitational physics. Einstein's equations of the gravitational field (like Maxwel's equations of the electromagnetic field) are local equations: they describe how the gravitational field (ie the metric tensor g_ik and its first and second derivatives) is excited at a certain point in space-time by the distribution of mass and energy (ie tensor of energy and momentum T_ik) at the same place (event) of spacetime. However, Einstein's equations do not give direct information about the global geometric and topological structure of spacetime. The global topological structure of spacetime has the nature of the "boundary conditions" that we actually have to enter on the basis of certain physical (or philosophical?) assumptions. The solution of Einstein's equations gives us only some indirect information about the global topology, eg that in certain cases it cannot be eukleidic. However, the specific global topology remains to some extent a matter of choice.
  We can illustrate the ambiguity of global topology in a simple case. Let us have an empty Minkowski planar spacetime with metrics  

ds ²   =   - dt ² + dx ² + dy ² + dz ²   ,      

in which Einstein's equations are identically satisfied. Space has Euclidean geometry and can of course have the usual Euclidean topology, where x, y, z Î (-¥ , +¥). However, if we make an identification

(x+a, y, z)  º  (x, y, z) ,  (x, y+b, z)  º  (x, y, z) ,  (x, y, z+c)  º  (x, y , z) ,      

the local geometry remains Euclidean and Einstein's equations will continue to be satisfied, but globally it is a topology of a three-dimensional toroid. Thus, there can be a planar yet closed (with a final total volume V = a.b.c) three-dimensional space!
This can be clearly shown by a two-dimensional analogy: when we take a sheet of paper with Euclidean geometry ds² = dx² + dy² and twist it into a cylindrical surface (ie we perform the identification (x+a, y) º (x, y), where a = 2p r, r is the radius of the cylinder), the local geometry will remain Euclidean, but the topology will be different - Fig.3.3a. It is not possible to twist the two-dimensional cylindrical surface and glue it to the toroid while maintaining the local Euclidean geometry, but by adding another dimension it is already possible (three-dimensional space becomes a hyperplate) and the above-mentioned option is obtained.

Infinity in spacetime
In this context, the terms of "infinity" and "unlimitedness" of space and time are generally not enough to understand the intuitive sense of what they have in classical physics. No one has any experience with true "infinity", no human has ever seen it; the concept of infinity originated as an idealization of very large (with respect to ordinary), but finite distances, times or other quantities. At the same time, what is "very large" (ie. better to say "large enough") totally depends on the situation - a number that is very large in comparison with all other values of a given quantity, occurring in the analysis of a certain problem, is considered to be practically infinitely large. For example, a distance of 10^-8 cm is very small in terms of macroscopic physics, but it can be considered practically infinite in terms of the structure of elementary particles; or a distance of 100 light years is practically infinite for astrophysics of solar system, but at the same time very small in terms of structure of the universe as a whole.
  According to its basic logical nature, infinity is divided into two categories :  

• Potential infinity
  is an expression of a permanently possible continuation of a certain process "to infinity", eg the process of gradual approach to a value (limit), or the principal possibility of a constant increase of a certain quantity "above all limits".
• Actual, current infinity
  expresses the infinite size of a certain real (already "finished") whole, it is conceived as an object of other mathematical manipulations.

Beneath the infinity of space and time, there are actually two different aspects :
a ) Global infinity "in terms of width, extend" - the so-called extensive infinity ;
b ) Local infinity at each point "in terms of depth" in the sense of unlimited divisibility into ever smaller and smaller parts - "intense" infinity .
Intense infinity has been discussed above in connection with the properties of the continuum and the adequacy of the space-time manifold model for space-time.
  In the following, we will understand infinity at infinity in the metric sense.

Infinity in Mathematics
In reflecting on the word "infinity", a thoughtful person often engages in a kind of subconscious feeling of mystery, even sacred horror, from something unknown, hidden somewhere beyond the horizon of our comprehension. Infinity somehow does not belong in our rational world; it is considered either a vague outgrowth of an obscure fantasy of philosophical or theological directions, or a theoretical construction based on an incomprehensible mathematical apparatus. Yes, it was mathematics, which was based on the analysis of real macroscopic objects and later generalized to abstract exact science, that developed procedures for dealing with the concept of infinity as precisely as with the calculation of numbers in arithmetic.
Note: Mathematical symbol for infinity "¥" ("horizontal eight") is of ancient Greek origin and symbolizes a serpent, devouring itself as an endless process. It was introduced into mathematics in the 17th century by J.Wallis.
  The original concept of infinity was based on potential infinity. This infinity represents the possibility to continue "indefinitely" in the process of gradually approaching to a certain sought value - to the limit. Indications of these considerations have already appeared in ancient philosophy (Zeno's paradox) and mathematics (counting the content of a circle). Differential and integral calculus are now based on methods of approximations with infinitely approaching values - the most powerful mathematical tool for applications in physics, science and technology.
Actual infinity as a term expressing the size, abundance or "countability" of infinite sets of objects, it began to be studied or modeled in the mid-19th century in the formulation of set theory, founded by B.Bolzano and developed especially by G.Cantor. For two finite sets A and B, they are "equally large" - they have the same number of elements, when each element a Î A can be assigned exactly one element b Î B , and vice versa, each element b Î B can be assigned exactly one element a Î A ; it is a mutually unambiguous (simple) representation. This comparison can be generalized even to infinite sets: if exactly one element from the second set can be assigned to each element from one set so that all elements from the second set are assigned, we say that both sets have the same cardinality. The cardinality of a set is a generalization of the concept of size to infinite sets. The cardinality of the set is denoted by the symbol "alef", which is the first letter of the Hebrew alphabet, which can be written in the Greek transcription with a similar character "c" (the Hebrew character is its mirror image; it is not available in standard fonts).
  The most basic example of an infinite set is the set of all natural numbers N (positive integers 1,2,3,4, ......). More generally, every set whose elements can be arranged in some infinite sequence and assigned to each element a natural number corresponding to its order in that sequence (the sequence can be "numbered") has the same cardinality as the set of all natural numbers N - such sets are called countable, their the cardinality is called "alef ₀" or c₀ .
The set of all rational numbers is also countable, because they can be expressed as fractions, where both numerator and denominator are integers; such fractions can then be arranged in a sequence (eg alternately according to an increasing numerator and denominator), which can be numbered and thus natural numbers can be assigned to all rational numbers. Even when irrational algebraic numbers (Ö2, Ö3, ...) are included, the cardinality of the set remains the same as for natural numbers: the roots can be expressed as real solutions of polynomial equations a_n.xⁿ + a_n-1.x^n-1 + ... + a₂.x² +a₁.x +a₀ = 0 with integer coefficients a_i and we can again sort these polynomials into countable sequences according to coefficients and exponents.
However, the set of real numbers R also contains so-called transcendental numbers (the best known of which are Ludolf's number p and Euler's number e - the basis of natural logarithms), which are not solutions of any polynomials with integer coefficients, nor the result of any finite developments. Transcendental numbers can no longer be sorted into any sequence according to natural numbers. Indeed, in 1873, Cantor proved with his famous diagonal method that the set of all real numbers R is innumerable. That real numbers cannot be arranged in a (infinite) sequence, the members of which could be assigned natural numbers - the set of real numbers has a different, greater cardinality than the set of natural numbers. The cardinality of the set of real numbers R is denoted by c₁ ie alef₁, and c₁ > c₀ holds .
Note: Some properties of complex so-called fractal sets and shapes (sometimes referred to as "mathematical monsters") are briefly discussed in §3.3, passage "Determinism-chance-chaos?").
  "Numbers" denoting different cardinality of infinite sets (i.e. c₀ , c₁ and possibly others) are called cardinal numbers, or cardinals for short. With cardinal numbers the set theory he can calculate similarly as to "ordinary" numbers, expressing the size of finite sets. A peculiar "arithmetic of cardinal numbers" was created, the basic rules of which are: c₀ + c₀ = c₀, c₁ = 2^c0 (the fact that the cardinality of the set R is expressed by the expression 2^c0 is related to the fact that 2ⁿ indicates the number of all subsets of the set by nelements). From this follows the equality 2^c0 ´ 2^c0 = 2^c0, or c₁´ c₁ = c₁, which means that there is a simple mapping of a line to a plane, ie the set of points of the line has the same cardinality as the set of all points of the plane .
  The so-called continuum hypothesis was stated that there is no cardinal number k such that c₀ < k < c₁, or there is no set whose power is greater than the sets of natural numbers, but less than the power of the set of real numbers. The continuum hypothesis failed to prove; was later included axiomatically, similar to the so-called axiom of choice .........
  Modern set theory is built axiomatically and led to the construction of several different models of set theory. Axioms postulating the existence of large cardinals were also introduced, describing cardinalities of sets that are much larger than the cardinality of c₁ sets of real numbers. It is not yet known what the consequences of the bizarre theory of such "insanely large sets" (even with infinite and "unattainable" cardinalities) for future mathematics. And it is no longer clear at all whether it could have any relation to the real world, some "practical significance". For the study of the geometric and topological structure of spacetime in relativistic physics, however, we are quite sufficient (at least for the time being ..? ..) with sets of cardinality c₁, corresponding to modeling using a set of real numbers .

So much in brief about complicated mathematical structures of infinity from the point of view of set theory. In geometric modeling, the idea of infinity is more complex from another point of view. In two- and multidimensional metric spaces (such as 2-dimensional plane, 3-dimensional Euclidean or curved space, 4-dimensional spacetime) we have several different infinities - places with infinite distance in different directions. In Euclidean space, where all dimensions are equivalent, this infinity need not be distinguished; we will formally add one "point" to the space, which will have the properties of infinity. In the spacetime of the theory of relativity, where the temporal dimension differs in its metric properties from the spatial dimensions, we have several kinds of infinity - common spatial infinity, temporal infinity of past and future, and finally isotropic (zero or luminous) infinity of future and past. These types of infinity will be defined and analyzed in the following §3.2. For a clearer insight into the structure of infinity, it is possible to use such unambiguous representations of the whole space on itself, after the application of which "ordinary" points in finite (perhaps even unit) distances correspond to infinity. The most suitable for this purpose are the so-called conformal transformations, which we will often use in the following chapters.
  In Newtonian physics (and in STR) space and spacetime are Euclidean, so the terms "infinity" and "unlimity" need not be distinguished there. According to "common sense" here from any, however distant point of space we can "throw a stone" even further, repeat the same from the point thus reached, etc. There is no point beyond which there are no more distant places, it is possible to move indefinitely from each starting point. However, in GTR, which deals with the non-Euclidean geometry of space and spacetime, the concepts of infinity and unlimity can be substantially different. The simplest two-dimensional example of this is a spherical surface, on which a two-dimensional creature during a locally linear (geodetic) motion along the main circle, it returns to the starting point, traversing only the final distance and not hitting any boundaries. The total two-dimensional "volume" (area) is finite here - it is unlimited, but finite space. An analogous situation in a closed three-dimensional space (which may be relativistic cosmology our universe): it is unbounded, but in terms of volume finite space into which only a finite number of galaxies and stars "can fit" - see Chapter 5, §5.2 "Einstein's and deSitter's universe. Cosmological constant.".

Asymptotic properties of spacetime
Let us now notice some general aspects of the asymptotic structure of spacetime, ie its properties at infinity. If we observe a spatially bounded process, such as the evolution of a star or an entire galaxy, the curvature values in a sufficiently large vicinity of such a process will be many orders of magnitude greater than the average curvature of the "background" (global cosmological curvature of the universe). From the point of view of such a phenomenon, the cosmological curvature of the background (universe) can be neglected, considered to be zero, and the given process investigated against the background of asymptotically flat spacetime. In fact, in almost all physical situations, except for observing the universe as a whole (ie in cosmology), we can consider spacetime as asymptotically planar. This is of great importance, because only in asymptotically flat spacetime do some basic physical characteristics, such as energy, charge, momentum, have a well-defined global meaning (see also §2.8 "Specific properties of gravitational energy").

Fig.3.4. Spatio-temporal diagram of the evolution of the island's physical system, during which part of the total mass ~ energy radiated in the form of electromagnetic or gravitational waves. This radiated energy is given by the difference of the total mass not on the spatial hyperplanes S2 and S1, but on the isotropic hyperplanes I2 and I1 .

One of the typical situations occurring in the study of bounded physical processes is schematically shown in the space-time diagram according to Fig.3.4. Let an island physical system, having a total mass (~ energy) M₁ on the hyperplate S₁, radiate part of its energy in the form of gravitational or electromagnetic waves during a relatively short time interval (it may be a non-spherical gravitational collapse of a star), so then it will have a lower mass M₂. As shown in §2.8, under a total mass (energy) physical systems in general relativity means the gravitational mass measured at asymptotically planar region of the relavant spatial hyperplate. Determination of mass M₁ on the hypersurface S₁ does not cause fundamental problems (if the system did not radiate before!): time component of the metric tensor g_oo » -1 + 2M₁/r at r ®¥. However, if we determine the total mass on the hyper-surface S₂ in a similar way, we again get the value M₁ regardless of how much energy was carried away by the radiation, because the hyper-surface S₂ always intersects all the outgoing waves at appropriate distances. This problem does not occur, if we use the isotropic (zero) type hypersurface I₁ and I₂, instead of the spatial type S₁ and S₂ hypersurface, as shown in Fig.3.4. Both hyperfields I₁ and I₂ pass outside the cone of radiated waves and the asymptotic behavior of the metric on I₁ defines the mass M₁ and the asymptothic behavior of the metric on I₂ gives the mass M₂. And the difference M₁ - M₂ is the total energy of the radiated waves.
  In general, when observing physical processes in asymptotically planar spacetime, it is necessary to investigate the asymptotic behavior of the respective fields. E.g. the electric charge of a physical system is given by the asymptotic behavior of an electric potential or electric intensity vector ("how fast" they go at infinity to zero), the mass and angular momentum is determined by the asymptotic form of the metric. The example just shown in Fig.3.4 shows that it is probably not enough to perform the asymptotic analysis only in the "spatial" infinity r ®¥, but it is necessary to detect asymptotic shape of metrics and fields in the "isotropic" infinity. Specifically, we will deal with individual types of infinity in the following §3.2 "Minkowski planar spacetime and asymptotic structure".

Conformal asymptotic analysis
In an asymptotic analysis, it is annoying that it is necessary to monitor the behavior of physical quantities somewhere at infinity, not only in the usual type of "spatial" or "temporal" infinity. It is necessary to calculate the limits for infinite coordinate values and, in addition, it is difficult to imagine the structure of the respective asymptotic regions of spacetime. For monitoring the global properties of spacetime and the asymptotic behavior of physical quantities (ie their behavior in infinitely distant regions of spacetime), the Penrose's conformal methods [201], [106], [203] are very useful. By conformal mapping of spacetime (M , g) to spacetime (M^, g^) is called the mapping M ® M^ such that the metric is transformed according to the relation g^_ik = W. g_ik and the space-time element of the interval ds^ ² = W ² .g_ik dxⁱ dx^k = W ² .ds ². Conformal coefficient W = W(xⁱ) may of course be different at each point, but the dimensions in all directions (including time) at a given point are always multiplied by the same number. All scales at a given location are isotropic "stretched" or "contracted"; therefore, in a conformal representation, the surroundings of the points, angles and length ratios are preserved locally : g_ikAⁱA^k/g_ikBⁱB^k = g^_ikAⁱA^k/g^_ikBⁱB^k . Thus, the conformal representation does not change the structure of light cones, ie their shapes and inclinations locally :

The Penrose method consists in using a suitable conformal mapping performing for infinity regions in M an infinitely large "compression" of all dimensions (_xi_®Alim W(xⁱ) = 0) so that these regions of infinity can have finite coordinates in M^. With such a conformal mapping converts entire infinite spacetime M to a specific finite region M^ whose boundary ¶M^ are the conformal image of the region of the original unlimited infinity spacetime M (fig.3.5). The asymptotic properties of geometry and physical quantities can then be monitored by analyzing their behavior at the boundaries of the conformal image, where the coordinates have finite values. The condition here, however, is a conformal invariance of the equations of the respective physical quantities.

Fig.3.5. Using a suitable conformal mapping can whole infinite spacetime M converted to a finite area M^ so that their own points of M are displayed on the inside of the M^ and area and infinity in M on the border ¶ M^. The structure of the light cones in M^ is accordingly the same as in the original M.

A suitable function W for such a conformal representation is, for example, the arctangent function, which converts the interval (-¥ , +¥) into an interval (-p/2, +p/2). Usefulness of the Penrose method is reflected in several places in another interpretation, where the conformal spacetime diagram will be used to display global structure of various models of universe and study the properties of black holes.

Analytical extension of spacetime
Spacetime (M', g') is called analytical enlargement (extension) of space (M, g), if (M, g) is isometric to some proper subset of (M', g'). If such an extension exists, the spacetime M is extensible, i.e. it can be "increased" as spacetime; then we must consider as points of this spacetime also the points M'. Indeed, there is no reason why the structure of spacetime should be limited to the stage of spacetime M, when the same right could continue to the stage of spacetime M'. Only an inextensible spacetime can be considered as "complete"; extensible spacetime, on the other hand, raises the suspicion that it is only a "part" of real spacetime.
  If we are looking for a solution of Einstein's equations, we work in a certain coordinate system in which we find the appropriate solution, ie the space-time metric g. It often happens that the metric found in this way is not regular in all places (eg Schwarzschild's solution - §3.4). To conclude that the geometrical properties of spacetime are singular in these places would be premature (hasty), because the singular behavior of metric tensor components can only be caused by the unsuitability of the coordinate system used (see §3.4, Fig.3.15). In such cases, we first try to remove the singular behavior of the metric by switching to another coordinate system; if it succeeds in at least some places, the solution in this new coordinate system will be an analytical extension of the original solution, because it will cover a larger portion of the space-time.
  The procedure of analytical extension can thus be roughly as follows :
We have found a certain solution (M , g) of Einstein's equations for a given physical situation in some xⁱ coordinate system. We move to the new coordinate system x' ⁱ , eg in order to remove the pathological behavior of the metric coefficients g_ik due to the inappropriate original coordinate system - the metric g'_ik is created. The analytical extension (M', g') is obtained by using g' as the metric and the maximum manifold on M', on which g' has the required analytical properties (ie it has continuous derivatives of the second order). It may happen that the space-time M' thus obtained is "larger" than M, so that the original space-time M was not "whole" and while removing the singular behavior of metric components, we also manage to find an analytical extension. If the space-time M' thus found is no longer extensible, it is a complete (maximum) analytical extension of the respective solution (geometry). A specific illustration of these procedures will be shown in §3.4 and 3.5 on the Schwarzschild and Reissner-Nordström solutions.

2.9. Geometrodynamic system of units		3.2. Minkowski planar spacetime and asymptotic structure

Vojtech Ullmann