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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
role, 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 of 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
= terrestrial, terrestrial, metria
= measurement - originally ie "surveying") 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 interconnected,
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 integral
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 view, 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 coherence, continuity, the number of dimensions,
limitations and limitlessness and under. In this sense, the
topology is deeper and more
general than what is
commonly considered to begeometry. 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 a closer look at a situation from everyday life.
^{ }We will make a small afternoon siesta with
coffee or tea and 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 tab; toroidal shape, equivalent to a wreath) . Conversely, from the plasticine wreath we could again
smoothly and continuously model a cup with an ear. If we had a
cup without a tab (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 mug handle 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 see that in some specially curved spacetimes, relatively small structures within the horizon carry complex global spacetime topologies, even 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 a
specified plurality of X include - denoted x Î X , or it does not: 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 , pointsgeometric 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 = inclusion -
integration into some whole) . The inclusion symbols " Ì, Í " are often indistinguishable 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 Y (which
is a set containing together all elements of X
and all elements of Y ) and the intersection
of X Ç Y (which 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 (X
Ç Y = 0 ) are denoted as mutually disjunctive .
^{ }Imaging operations
( binary sessions ) are used to compare sets .^{ }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 Î Y to each point x Î X.
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 . Display Id 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 ( per 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) view . 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 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, determining 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, the remainder remains 1).
l Rational
numbers are those which arise as a proportion
of two integers (lat. ratio = ratio
) - can be written as a fraction A / B two integers a
, b ¹^{ }0. 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 Ö 2 is the solution of the algebraic equation x ^{2} -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 intellectual
comprehension .
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
numberc , 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
(electrical circuits, waves, quantum physics) occur. In our
interpretation of the theory of relativity and gravity, we will
not use complex numbers
, with a few exceptions ...
Power 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
c _{0} (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 or 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} - power of the continuum . The set of all has
the same powerirrational numbers with the inclusion of transcendental
numbers - 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 power c _{1} , 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 sets can be used for modeling the
laws of our world, in general abstract sets of introducing mathematical structures - 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 support 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, 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 support
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 t
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 . Around U (x) the point x Î X we mean the open set U Î U , which contains the
point x . For each point x Î U, 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 is the set
of all of the elements of X , each of which contains at
least one near a given set point X and at least one point
among a plurality of X . This limit represents ¶ X
. A closed set is one that contains its
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 x Î X and for each neighborhood V ÎV at the point j
(x) Î Y there exists a neighborhood U
such that j ( U) Ì V.
It is therefore a display 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).
^{ }Display limit j : X ®^{ }Y between the topological spaces at the
point x _{o} Î X
is defined as the point y _{o} Î Y 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 x Î U ( x _{o}
) Þ y Î U (y _{o} ). Write lim _{x }_{® }_{x }_{o} 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 }_{® }_{x }_{o} f (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 xx _{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 }_{® }_{x }_{o} f (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 ter
inverse mapping j ^{-1 is} 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 opological 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 x
Î X and j
(x) Î Y 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 R ^{1} with a natural topology given by a set of subsets A Ì R ^{1} , which together with each of their points always contain a certain interval around it: for each point x Î A there are numbers a , b such that a <x <ba interval (a, b) Î A. The generalization is the n-dimensional Euclidean space R ^{n of} all n-tic real numbers (x ^{1} , x ^{2} , ..., x ^{n} ) at - ¥ <x ^{i} <+ ¥ 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 representations , which
are the above-mentioned simple mutually unambiguous
representations for which unambiguous invasive representations
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 singularitiesmetrics naturally
arise even under very general assumptions, which are probably
fulfilled in astrophysical practice. Besides removable coordinate
singularities ( "pseudosingularities") occur and
irreparable real physical singularity . Therefore, we
will deal with singularities in many places in our book - the
earliest 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 singularities " 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 ".
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) M^{n} is such a topological space,
each point of which has a neighborhood homeomorphic with R ^{n}
(with a certain neighborhood in R ^{n} ). The homeomorphic mapping j of an open (sub) set A Ì M ^{n} to R ^{n} assigns to each point x Î A an n-tuple of numbers j (x) = (x ^{1} , x ^{2} , ..., x ^{n} ) Î R ^{n} , which call coordinates of
the point x .
We say on a set A the coordinate
system
(coordinate system) x ^{i} is introduced . By selecting another homeomorphic mapping j' from A Ì M ^{n} to R ^{n} , other different coordinate values (x' ^{1} , x' ^{2} , ..., x' ^{n} ) will be assigned to individual
points x Î A Î R ^{n} - we will go to another
coordinate system in a subset 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-Besikoviè
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 M ^{n} to R ^{n }in this way for many topological spaces (eg the mapping S ^{2} to R ^{2} introducing the spherical coordinates J , j on the spherical surface S ^{2} ceases to be mutually unique on the poles). Thus, in general we manifold M ^{n} appear in R ^{n} piecewise - generate local coordinate "map" (A _{a} , j _{a} ) separate "domain" (coordinate vicinity) A_{a} Ì M . Set of maps of individual domains A _{a} Ì M , covering M (i.e. _{a}CA_{a} =M ) form the "tlas" manifold M . Only manifolds topologically equivalent to R ^{n} 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 M ^{n} , the images f _{a} (p) and f _{b} (p) of the point p from the intersection of the two domains A _{a} and A _{b are} bound by continuous transformations, including derivatives of the r-th order. |
The manifold M ^{n} is
called a differentiable class C ^{r} if it is given an atlas of maps
(A _{a} , j _{a} ) of individual domains A _{and} Ì M ^{n}
represented by mutually unique representations j _{a} on open sets in R ^{n}
satisfying the conditions :
a ) A _{a} forms the cover M , i.e. _{a}CA_{a} = M ;
b ) If two
domains have A _{a} and A _{b} non-empty intersection, then the
points p Î A _{and}
Ç A _{b }of this overlapping part will be
assigned by the representation j _{a} to the n-tuple of coordinates x ^{i }_{a} (p) Î R ^{n}
and by the representation j _{b }at the same time to the n-tuple of
coordinates x ^{i }_{b} (p) Î
R ^{n}
such that the transformations x ^{i }_{b} (p) = x ^{i} [x ^{k }_{a} (p)] are in R ^{n} 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 ® x ^{i} (x)) and (A ', j ': x ®
x ' ^{i}
(x)) such that A' = A = A Ç A 'but j ' ¹ j , then the transition from the coordinate
system x ^{i} 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), 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 havecountable base , i.e. that existed in ala such a countable set of open sets whose uniting 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. Coherence of sets.
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 E ^{n} . In order for this similarity to be true, it is necessary to construct the above-mentioned conditions of separability and paracompactivity .
Curves and
surfaces
The curve
(line) l (t) on the manifold M means the representation of a certain
section R ^{1} ® M , ie the set of points in M , which are representations of the curve
points x ^{i} = x ^{i} (t) in R ^{n} parameterized by the variable t Î R ^{1} . The basic topological
characteristic of each set (geometric shape) is the connection . 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 ( about br.3.2b). Connected set is called
simply continuous , if for any two points A and B are any connectors between mutually
equivalent topogicky ( 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 "cuts" 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 M ^{n} is the p
- dimensional area C ^{p} (p £^{ }n), which is a representation of
the corresponding p-dimensional subspace in R ^{n}
. 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 ^{and} £ 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) -rozmìrná boundaries (p-1) -rozmìrné
boundaries p-dimensional cube is zero: ¶¶ K
= O . This follows also from the structure
boundaries floes c lo using the sum of squares forming the
boundaries of the walls of the cube, each side of the square is
counted twice with opposite orientation and therefore cancels.
^{ }The general area S can be decomposed into a number of cubes
(appropriate dimensions) K _{i} : S = _{i} S a ^{i} 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 = _{i} S and ^{i} ¶ 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
¶ ¶ S = |
(3.1) |
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 ^{p }_{1} and C ^{p
}_{2} 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 ^{p} i } consists of all closed p-dimensional
surfaces C ^{p} which are homologous to each
other.
In the Euclidean space R ^{n} , all p-dimensional
(p <= n) closed surfaces can be compressed to a point, so that
they are all logical zeros and belong to the zero homology class
{C ^{p }_{0} } = {0}.
The number of independent homology classes {C ^{p }_{1} }, {C ^{p }_{2} },.,., {C ^{p
}_{Bp}
} of areas of dimension p is called the p-
th Betti number of manifold M (class {C ^{p }_{o} } = {0} is not included here areas
homologous to zero). The quantity c = _{p = 0} S ^{n}
(-1) ^{p} B p is called the Euler characteristic of this manifold. The so-called topological genus 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 M ^{2} between the genus g and the Euler characteristic c, the relation c = 2
- 2 g belongs .
^{ }Because addition 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 R ^{2}
(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 B _{0} =
1, B _{1} =
1, B _{2} =
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" 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 its
(local) metric properties are not enough to completely determine
the character of a space, but its (global) topological properties
must also be taken into account. In addition to the Euclidean
space R ^{n} , 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 (x ^{1} ) ^{2} + (x ^{2}
) ^{2} + (x ^{3} ) ^{2} = 1. Analogously, the n-dimensional sphere S ^{n} (as a subspace in R ^{n + 1} ) is the geometric place of points in R ^{n
+ 1} satisfying
the condition _{i = 1} S ^{n + 1} (x ^{i} ) ^{2} = 1. The sphere S ^{n}
isfinite (compact) simply continuous manifold. For a two-dimensional spherical
surface S ^{2} , the Betti numbers B _{0}
= 1, B _{1} = 1, B _{2} = 1 and the Euler characteristic
c ( S ^{2} ) = 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. The toroid T ^{2} , which is generated coincident d m (x + a,
y + b) º (x, y) points in R ^{2}
, 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 T ^{n}
is the space that results from the identification of (x ^{i}
+ and ^{i} ) º (x ^{i} ), i = 1,2, ..., n, points in R ^{n}
. The two-dimensional toroid T ^{2} has Betti numbers equal to B _{o}
= 1 (corresponds to the class of all points - all points are
homologous to each other), B _{1} = 2 (there are two independent
classes {C ^{1 }_{1} } and {C ^{1
}_{2} }
of closed curves passing around the smaller s
it toroid
circumference and greater), B _{2}= 1 (corresponds to the toroid
itself); Euler's characteristic c (T ^{2}
) = 0.
From the n-dimensional manifold M ^{n} and m-dimensional manifold M ^{m} we can construct the (n + m) -dimensional manifold M ^{n} ´ M ^{m }by the "Cartesian product" , whose points are pairs (x, y ), where x is any point z M ^{n} ay any point of M ^{m} . E.g. Euclidean space R ^{3} is the product of R ^{2} ´ R ^{1} , R ^{n} can be written as R ^{n} = R ^{1} ´ R ^{1} ´ ... ´ R ^{1} ( Cartesian product of n-coefficients). The cylindrical surface C ^{2} can be considered as the product of a circle and the Euclidean line, ie C ^{2} = S ^{1} ´ R ^{1} . As far as the toroid is concerned, it is especially clear that the one-dimensional toroid T ^{1} and the one-dimensional sphere S ^{1} (circle) are homeomorphic to each other, i.e. T ^{1} = S ^{1} . Therefore , from an topological point of view , the n-dimensional toroid T ^{n} is a Cartesian product of n circles: T ^{n} = S ^{1} ´ S ^{1} ´ ... ´ S ^{1} .
The topological
structure of the manifold M ^{n} ´ M ^{m} is naturally given by the
structure M ^{n}
and M ^{m}
: for any points x Î M ^{n} ay Î M ^{m}
having coordinate neighborhoods A Ì M ^{n}
and B Ì M ^{m} is the point (x, y) Î M ^{n} ´ M ^{m} contained in the coordinate
neighborhood A ´ B Ì M ^{n} ´ M ^{m} and has coordinates there (x ^{i}
, y^{j} ), where x ^{i} 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 M ^{n} is a mapping from M ^{n} to R ^{1} . 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 pa and its expression f (x) = f (x ^{1}
, x ^{2} , ..., x ^{n} ) using the coordinates x ^{i} Î R ^{n}^{ }in some local coordinate system it has
continuous derivatives up to the r-th order according to x ^{i}
. This definition implies that in the differentiable manifold M Class C^{s} is the coordinates x ^{i}
(x) 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 t-order tensor at the point "p" of the n-dimensional manifold M ^{n} means the sum of n ^{r}
numbers
T ^{i }^{l }_{j }_{l}^{ i }^{2
}_{j }_{2}^{ . }_{. }^{. }_{.
}^{. }_{. }^{i }^{a }_{j }_{b} , j l , j 2 , ..., j b , i l , i 2 , ... , i a = l, 2, 3, ..., n
with a £ r contravariant (upper) and b = r- a covariant (lower mi) 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 R ^{n} .
^{ }The possibility of introducing
any tensor field on a manifold is generally conditioned by the topological properties of the manifold [1 51], [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
("you can't comb your hair smoothly on a 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 = connection, 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 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 shapes rough objects - lines (curves), the
area of the body. 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 graphically 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 = 2 p r they can
determine the internal curvature , quantified eg by the
so-called Gaussian curvature C_{G}=6.(1-L/2pr)/r^{2}. 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 the fire and glue the sheet 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 2 p r-times the
radius, measured over three-dimensional space. We will observe
the external curvature of the cylindrical
surface. However, only internal curvature is essential
for our analysis of gravity as a curved spacetime .
^{ }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 x ^{i} and the adjacent infinitely
close point x ^{i} + dx ^{i} is given by the coordinates
given by the differential form ds ^{2} = g _{ik} dx ^{i} 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 and the
size of the transmitted vector must be observed in parallel
transmission. This leads to the law of parallel transfer (2.8)
and an unambiguous relationship (2.2b)
between the coefficient y 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 M ^{4} , which is Hausdorff and
paracompact with Riemann metric g . We will denote it ( M , g ). The relationship manifold M we want, because nothing can exist and
move beyond space and time, so if it were incoherent, it would be
any information for one part of M on the other disjoint parts M fundamentally deficient, t upná, so
that such extraneous would 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 spinor relations can be
replaced by equivalent (albeit more complicated) tensor
equations.
We will assume that the real physical field M will tyt on two fundamental features :
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
"fabric content" is realized in that in the space- M are satisfied Einstein equation R_{ik} - ^{1}/_{2 }g_{ik}R = 8pT_{ik} . As shown
in §2.5, the local law of conservation of energy and momentum T ^{ik }_{; k}^{ }= 0 as a consequence of
Einstein's equations of the excited gravitational field.
Formally each spacetime ( M , g ) be considered as solving equations of R_{ik} - ^{1}/_{2 }g_{ik}R = 8pT_{ik} in the sense that the component-based metric tensor g _{also} can calculate the magnitude (R_{ik}-^{ 1}/_{2 }g_{ik}R)/8p and we define this as the tensor T _{ik} . 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
(with the exception of singularities) interesting, 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 experimentally
verified 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 (the 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 Maxw '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 ds ^{2} = dx ^{2} + dy ^{2} and twist it into a cylindrical surface (ie we perform the identification (x + a, y) º (x, y), where a = 2 p r, ris 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 relation to the concepts or " infinity " and " absoluteness " of space and time
generally is 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. Doing what is "very large" (ie. better to
say "big enough") totally depends on the situation - as
great a cally infinitely large is considered a
number that is very large in comparison to all the other values
of the variables occurring in the analysis certain problem. 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 :^{ }
Beneath the infinity of
space and time, there are actually two different aspects :
a ) Global infinity "in terms of
width" - 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 infinite 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 - the most powerful - are now based on methods
mathematical tool for applications in physics, natural 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 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 power 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 power
is called "alef _{0} " or c _{0} .
The set of all rational ones is also countable
numbers, as these 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 ( Ö 2, Ö 3, ...) 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 and _{n} .x ^{n} + and _{n-1} .x ^{n-1} + ... + a _{2} .x ^{2} + a _{1} .x + a _{0} = 0 with integer coefficients and _{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 power
than the set of natural numbers. The cardinality of the set of
real numbers R is denoted by c _{1} , ie alef _{1} , and c _{1} > c _{0} 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 powers of
infinite sets (i.e.^{ }c _{0} , c _{1} and possibly others)
are called cardinal numbers , or cardinals
for short . Cardinal numbers can count on cardinal
numbers in the same way as on "ordinary" numbers,
expressing the size of finite sets. A peculiar "arithmetic
of cardinal numbers" was created, the basic rules of which
are: c _{0} + c _{0} = c _{0} , c _{1} = 2 ^{c}^{ 0} (the fact that the
cardinality of the set R is expressed by the expression 2 ^{c}^{ 0} is related to the
fact that 2 ^{n} indicates the number of all subsets of
the set by nelements). From this follows the equality 2 ^{c}^{ 0} ´ 2 ^{c}^{ 0} = 2 ^{c}^{ 0} , or c _{1} ´ c _{1} = c _{1} , 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 c _{0} < k < c _{1}, 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 c _{1}^{ }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" powers) 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 power c _{1} , 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 - commonspatial 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 representations ,
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 vhen a locally linear (geodetic)
motion along the main circle, it returns to the starting point,
traversing only the final distance and meeting no 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 event, 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 S _{2} and S _{1} , but on the isotropic hyperpaces I _{2} and I _{1} . |
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) M _{1} on the hyperplate S _{1} , 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 M _{2} . 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 M_{1} on the hypersurface S _{1}
does not cause fundamental problems (if the system did not
radiate before!): Time component of the metric tensor g _{oo} » -1 + 2M _{1} / r at r ®¥ .
However, if we determine the total mass on the hyper-surface S _{2 in a} similar way , we again get the value M _{1}
regardless of how much energy was carried away by the radiation,
because the hyper-surface S _{2} always intersects all the
outgoing waves at appropriate distances. This problem does not occur, if we use the isotropic
(zero) type hypersurface
I _{1} and I _{2} , instead
of the spatial type S _{1}
and S _{2} hypersurface, as shown in Fig.3.4. Both
hyperfields I _{1} and I _{2} pass outside the cone of
radiated waves and the asymptotic behavior of the metric on I _{1}
defines the mass M _{1} and the asymptothic behavior of the metric on I _{2} gives the mass M _{2}
. And the difference M _{1} - M _{2} 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 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 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. Penrose's
conformal methods [201], [106], [203] are very useful for monitoring the global
properties of spacetime and the asymptotic behavior of physical
quantities (ie their behavior in infinitely distant regions of
spacetime) . 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} dx ^{i} dx ^{k} = W ^{2} .ds ^{2} . Conformal coefficient
W = W (x ^{i} ) 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 _{ik} A ^{i} A ^{k} / g _{ik} B ^{i} B ^{k} = g ^ _{ik} A ^{i} A ^{k} / g ^ _{ik} B ^{i} B ^{k} . Thus , the conformal
representation does not change the structure of light cones, ie
their shapes and inclinations locally:
/ | > 0 | / | > 0 | |||
g _{ik} X ^{i} X ^{k} | - | = 0 | Þ | g ^ _{ik} X ^ ^{i} X ^ ^{k} | - | = 0 |
\ | < 0 | \ | < 0 |
The Penrose method consists in using a suitable conformal mapping performing for infinity regions in M an infinitely large "compression" of all dimensions ( _{x}i _{®¥} lim W (x^{i} ) = 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 ) an isometric proper subset ( 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 ' . 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) Einstein's equations for a given
physical situation in some x ^{i} 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 , M was not "whole" and while
removing the singular behavior of metric components, we also
managed 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.
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