AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy | Gravity, black holes and physics |
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 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 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 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 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 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 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_{o}ÎX is defined as the point y_{o}ÎY 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 }_{® }_{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 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}_{®}_{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 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 R^{1} with a natural topology given by a set of subsets AÌR^{1}, 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 R^{n} of all n-tuples real numbers (x^{1},x^{2},...,x^{n}) at -A < x^{i} < +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) 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 AÌM^{n} to R^{n} assigns to each point xÎA 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 that on a set A the coordinate
system
(system of coordinates) x^{ i} is introduced. By selecting another homeomorphic mapping j' from AÌM^{n} to R^{n}, other different coordinate values (x'^{1},x'^{2},...,x'^{n}) Î R^{n} 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 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 can display the manifold M^{n} in R^{n} in parts - to create local coordinate "maps" (A_{a}, j_{a}) of individual "domains" (coordinate surroudings) A_{a}Ì M. Set of maps of individual domains A_{a}Ì M, covering M (i.e. _{a}CA_{a}=M) form the "atlas" of 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 of class C^{r}, if it is given an atlas of maps
(A_{a}, j_{a}) of individual domains A_{a}Ì M^{n}
represented by mutually unique displays 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 A_{a} and A_{b} have non-empty intersection, then the
points pÎA_{a}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 imaging 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 = AÇA' 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), 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 E^{n}. 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 R^{1} ® M, ie the set of points in M, which are imaging of the curve points x^{i}
= x^{i}(t) in R^{n} parameterized by the variable tÎ R^{1}. 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 M^{n} is the p
- dimensional area (surface) C^{p} (p £^{ }n), which is a imaging 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^{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 = _{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 a^{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 homological 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 (the class {C^{p}_{o}} = {0} areas of homologous zeros is not included here). The
quantity c = _{p=0}S^{n}(-1)^{p}Bp 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 M^{2} 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 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" 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 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}
is finite (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. This toroid T^{2}, which is formed
by identifying (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}+a^{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 an larger circumference of the
toroid), 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} x M^{m }by the "Cartesian product", whose points are pairs (x, y), where x is any point z M^{n} and y 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} and y 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 p and and its expression f(x) = f(x^{1},x^{2},...,x^{n})
using the coordinates x^{i}ÎR^{n} in some
local coordinate system has a continuous derivation up to
the r-th order according to x^{i}. It
follows from this
definition that in the differentiable manifold M of the class C^{s}, the
coordinates x^{i}(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 M^{n} means the summary of n^{r}
numbers
T^{i}^{l}_{j}_{l}^{i}^{2}_{j}_{2}^{.}_{.}^{.}_{.}^{.}_{.}^{i}^{a}_{j}_{b} ,
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 R^{n}.
^{ }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)/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 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 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 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 M^{4}, 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_{ik}R = 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_{ik}R = 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_{ik}R)/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 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 = 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 :^{ }
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 c_{0} .
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}.x^{n} + a_{n-1}.x^{n-1} + ... + a_{2}.x^{2} +a_{1}.x +a_{0} = 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 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 cardinality of
infinite sets (i.e.^{ }c_{0} , c_{1} 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: 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"
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 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 - 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 S_{2} and S_{1}, but on the isotropic hyperplanes 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 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} 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_{®A}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) 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 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,
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.
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