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The space-time in the macrocosm
- THE RESTRICTED COSMOLOGICAL PRINCIPLE
- THE GENERAL COSMOLOGICAL PRINCIPLE
- THE LAW OF HUBBLE FOR AN EXPANDING UNIVERSE
- THE LAW OF NEWTON FOR GRAVITATION
- THE INVARIANCE OF THE SPACE TIME INTERVAL
- THE INTERVAL FOR A CURVED SPACE-TIME
- THE GEOMETRY OF THE CURVED SPACE-TIME
- THE ENERGY CONTENTS AND THE CURVATURE OF SPACE-TIME
- TABLE OF CONTENTS
a. The restricted cosmological principle.
Following the cosmological principle, this means that the space is homogeneous and isotropic, we can infer a lot of properties of the space-time. Homogeneity means, that the universe possesses everywhere the same properties and isotropy means, that the universe has in all directions the same structure. The universe has the same look everywhere, on which place and in which direction we look out like wise. A consequence of this is that pressure and density of matter and radiation is everywhere in the universe the same at all places and in all directions at a certain time.
A dependence of the time is however permitted, such as is required for an evolving universe from a big bang. This restricted cosmological principle is in conformity with the current observations, provided that we do the measurements, like for the density, for the average mass in large volumes, large in comparison with the distances between galaxies. We can abandon then the local clustering of masses in stars, stellar systems (galaxies) and clusters of stellar systems.
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b. The general cosmological principle.
Also there are cosmologists, such as the supporters of the 'steady state', who require a general cosmological principle. These extend the principle to the time also. The universe will have the same properties also for all times, instead of at a certain time, at all places and in all directions.
The big bang and an evolving universe is contradiction with this general principle. Also the background radiation of the universe must be given another meaning, than as a remainder of the big bang. The general law of conservation of the mass and energy, does not survive the eternal creation of here and there from time to time an atom in the 'steady state'. The definition of simultaneity, the terms 'before' and 'after', including those of present, past and future in the general theory of relativity become an extremely difficult, if not insoluble problem. Also for the geometry of a curved space-time another solution will have to be found.
Because of called difficulties at the maintenance of the 'steady state' the supporters of the general cosmological principle are small in number.
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c. The law of Hubble for the expanding universe.
In our expanding universe we see the galaxies removing from us in all directions. A galaxy, that is located two times farther away, is removing with two times larger velocity. The law of Hubble applies generally: expansion velocity and distance are straight proportional to each other.
Moreover the constant of proportionality in the law of Hubble is for all directions the same. A dependence of the constant of proportionality of the time however has been permitted. This law applies to every observer in the universe. For this reason each point can be considered as a center of the universe. Each other law for velocity and distance is in contradiction with the restricted cosmological principle mentioned above. We can also state that the law of Hubble is a property of the space-time.
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d. The law of Newton for gravitation.
When we apply the restricted cosmological principle and maintain we the law of conservation of matter, then also the gravitation law of Newton is to interpret as a property of the space-time. The only general law of force, which satisfies, is the straight proportionality of distance with force. The proportionality constant can only depend on the time. This law of force is simple to identify with the law of Newton by choosing the correct proportionality constant.
In formula we find for the gravitation F expressed in distance r and constant c: F=-c.r, with c=(4/3)pi.µ.G. Here is G the universal gravitation constant of Newton and pi=3.14. When we consider, that density µ=M/V (mass divided by volume) and that V=(4/3)pi.r3, then can this also be written in the better known form F=-GM/r².
The universal law of the gravitation is a general geometrical property of the space-time.
The law of Newton follows also directly from the general field equations of Einstein for the curved space-time, under the binding application of the restricted cosmological principle and the pure conservation of the mass only. When there are conversions of energy in mass, such as which took place at the origin of space and time, the big bang, then we can consider these field equations as the generalized law of Newton in four dimensions. Differently said: the found law by Newton applies only local and nowadays, seen on cosmically very large scale in distance and time.
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e. The invariance of the space-time interval.
The geometry of the space-time stretches out over four dimensions, in which the distance between galaxies is a three dimensional concept and the time plays the role of the fourth dimension. Instead of the spatial distance in three dimensions, we prefer to use the interval between two events in the space-time. We can consider the interval as the four dimensional 'distance' in the space-time.
The invariance of the interval between two events, this means, that each hypothetical observer, where and when also located, will measure the same interval between the two events, is a property of the space-time. From the invariance of the interval follows the Lorentz transformations for distance and time, that are the length shortening and time delay at high velocity.
Also this invariance of the interval is a direct consequence of the restricted cosmological principle and the finitude of the speed of light. This invariance of the interval applies as well as the law of Newton only nowadays and local in cosmic respect, as far as not corrected with a factor for the curvature of the space-time.
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f. The interval for a curved space-time.
In the ordinary three dimensional Euclidean space, with coordinates x, y and z, is the surface of a sphere the example for a curved two dimensional space. The mathematical formula for a sphere with radius r and center in the origin is x²+y²+z²=r². The curvature k of the sphere has been defined as k=1/r² and has in each point on the spherical surface the same value. If the curvature is positive, the radius is real and we talk of a normal spherical area. If the curvature is negative, then is the radius imaginary and we speak of a pseudo spherical area. If curvature becomes zero, then the radius approaches to infinitely large. An area on the sphere or pseudo sphere approaches in this last case to a flat area. Only for the a flat area with (length and width) coordinates x1 and x2 the theorem of Pythagoras applies to the distance dL between two points dL²=dx1²+dx2². In the two other cases the last expression must be corrected for the curvature of the curved area.
We represent nowadays the cosmic space as a curved three dimensional space embedded in a fictitious four dimensional Euclidean space. We do this in analogy with the above defined curved two dimensional Euclidean space embedded in an ordinary three dimensional Euclidean space. We introduce a four dimensional Euclidean space with coordinates x1, x2, x3 and x4.
We let coincide the three of these coordinates with the well known three spatial coordinates. The fourth coordinate must be considered as a fictitious spatial coordinate, which moreover has nothing in common with the fourth coordinate of our space-time. A hyper sphere with radius a has in the this four-space the mathematical form x1²+x2²+x3²+x4²=a². The distance between two points on the hyper sphere is dL²=dx1²+dx2²+dx3²+dx4². We eliminate the fourth coordinate and find for the formula for the spatial distance, also called the geometrical equation:
dL²=dx1²+dx2²+dx3²+(x1.dx1+x2.dx2+x3.dx3)²/(a²-x1²-x2²-x3²).
We proceed to ordinary spherical coordinates r, X and Ø, defined by x1=r.cosØ.sinX, x2=r.sinØ.sinX and x3=r.cosX. We find then for the geometrical equation:
dL²=(dr²+r²dX²+r²sin²Xdز)+(r²dr²)/(a²-r²).
We can simplify this to:
dL²=(dr²)/(1-r²/a²)+r²dO² with dO²=(dX²+sin²Xdز).
The radius of curvature a we call also world radius and the curvature R of the space has been defined by R=1/a². For R positive, negative or zero we talk of a three dimensional space, which is spherical, pseudo-spherical or flat. If the radius becomes very large, the curvature approaches to zero. In this case approach the spherical and pseudo-spherical space to the flat space with the Pythagorean expression for the distance (dL²=dx1²+dx2²+dx3²). A growing radius in the four-space corresponds to an expanding three dimensional space.
The infinitesimal space-time interval ds gets the expression ds²=dL²-c²dt² with for dL the above mentioned general expression for a curved three dimensional space. In an expanding universe is the curvature of the spatial distance a function of the time and this will have its effect on the real structure of the four dimensional space-time.
More concerning the geometry above described worlds, can we find in the Mathematical appendix.
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g. The geometry of the curved space-time.
The geometry of the curved space-time obeys in the three different cases to different geometrical laws. The geometry of Riemann applies for a spherical space. To the pseudo spherical space the geometry of Lobatsjewski applies. Euclidean geometry only applies to a flat space. If our universe corresponds to a spherical space then this universe is finite large and unlimited in analogy with a spherical surface. A pseudo spherical universe is infinitely large and unlimited, just as as a flat universe.
Because the world radius is everywhere in space just as large, the universe has no edge. For this reason the universe has to be considered as unlimited. The denomination 'radius of the universe' instead of radius of curvature is misguiding and we will not use. The universe is not a sphere with an edge which expands. The radius of curvature of the curved hyper surface, what is spherical, flat or pseudo-spherical curved, stands in each point perpendicular to the our well known three spatial dimensions. It is the unlimited space itself, which expands in each point, as a result of which the galaxies escape from each other, with speeds, which become larger, as there is more space between those galaxies.
Also has our universe no favored center, from where the expansion takes place. Each point can be considered as a center of expansion.
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h. The energy content and curvature of the space-time.
In the general theory of relativity, in which the mathematical foundation lies for the properties of the space-time, one assumes the principle simplicity. This is, that our space-time must have the most simple structure. For this reason the curvature of the space-time must be invariant under transformations from one to another place in the universe, in conformity with the restricted cosmological principle, that our universe has everywhere and in all directions the same properties at a certain time. The radius of curvature can change, however, in the course of the time in conformity with an expanding or contracting universe.
In the case of the most general properties of the space-time, concerning the general form of the laws of nature, the principle of simplicity must even be enlarged to the time. We assume after all obviously, that the laws of nature are valid, everywhere and always. An experiment done in a laboratory on a certain place and time will produce the same result under the same circumstances on each other place and time. An expanding universe with a big bang corresponds to an evolving universe with a beginning and possibly an end. The our well known laws of nature apply no longer for the beginning and ending of the universe.
In the modern theories concerning the space-time we speak of curved spaces in positive or negative sense. Also a flat space with curvature zero is possible. This agrees to an always expanding universe (negative curvature), an universe, where the expansion eventually reached a standstill (zero curvature) and an universe, which first expands and then after a given moment starts to contract (positive curvature). This curvature R results from the energy contents T of the space, which can be positive, zero or negative. From the general field equations of Einstein the relation follows between these last called two quantities: R = - kT. (*)
In the curved time-spaces the term distance is practically without meaning, when that distance can be measured only by mean of rays of light. Because large distances change in an expanding universe in the course of the time and measurings of large distances take large time spans, it is a practically impossible task to determine at the current point of view of our science in which universe we live and which geometry is applicable to our universe. The large mystery of the real properties of space and time continues to exist.
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<< 1 TEXTS ABOUT OUR SPACE-TIME
<< 2 THE SPACE-TIME CONTINUUM
== 3 THE SPACE-TIME IN THE MACROCOSM
>> 4 THE SPACE-TIME IN THE MICROCOSM
>> 5 SPACE-TIME DIAGRAMS FOR PARTICLES AND PHOTONS
>> 6 TABLE FOR QUARKS AND HADRONS
>> 7 TABLE FOR QUARKS AND LEPTONS
>> 8 THE WINDOWS OF TIME FOR OUR SPACE
>> 9 TERMINOLOGY FOR COSMOLOGY
>> 10 MATHEMATICAL APPENDIX PART 1 | PART 2
>> 11 SUMMARY OF THE CHAPTERS
>> 12 REFERENCES AND LITERATURE
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