MATHEMATICAL APPENDIX II 
The spaces of the time
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- THE EUCLIDEAN SPACE AND UNIVERSAL TIME
- THE MINKOVSKY SPACE AND RELATIVISTIC TIME
- THE RIEMAN SPACE AND EIGEN-TIME
- THE AFFIEN SPACE
- TABLE OF CONTENTS
- Back to WELCOME TO THE WEB SITE
a. Euclidean space and universal time.
In euclidean space remains the spatial distance invariant. When
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are the position vectors of two points, then the relation applies:
whereby the spatial distance r remains invariant.
Herein the spatial distance between two points remains invariant under translations and
rotations of moving coordinate systems in space.
The interval of time between two events shall be also invariant under transformation of
coordinate systems. In this case we have a real universal time. That means that we can use
the same time in all coordinate systems.
In stead of universal time we also speak of absolute time, to underline the difference with relativistic time.
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b. Minkovsky space and relativistic time.
In this case the space-time distance s between two points remains invariant under transformation of coordinate systems. When
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are the space-time coordinates of two world points (space-time points) then applies
whereby the space-time interval (world-distance) remains invariant. From the invariance of
s under coordinate transformations follow the Lorentz transformations.
Instead of Minkovsky space we speak also of flat space or pseudo-Euclidean space. The laboratory times for observers in several coordinate systems no longer runs right, but are related to each other in a simple mathematical expression. We speak here of relativistic time.
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c. Riemann space and eigen-time.
In this case only the infinitesimal space-time interval remains invariant. This means that, only if the both world points lie very close to each other, the invariance of the interval applies. Moreover the space needs to be now no longer flat. The world distance ds satisfies to:
whereby
are the infinitesimal coordinate differences between both points. We
sum over i and k (all possible combinations from 0 to 3).
We call 
the metric tensor, which determinates
the local geometry (in the neighborhood of both world points).
Written out we get then for the interval s:
The flat space is a special case of the Riemann space, whereby the metric tensor has the
most simple form, such that,
I.e. all 
except 
and 
everywhere in space.
The quadratic expression for the interval we call also the geometric equation. The deviation of the geometry of the Riemann space from those of the flat space reflects himself in the geometric tensor and is related to the curvature of the concerning Riemann space.
In a general Riemann space the are no more
constants, but functions of the coordinates. Here the condition apply, that in this
general space, determinated by the geometrical equation, via a continue coordinate
transformation, the geometrical equation can be carried back to the equation for the flat
space and the other way round. The conditions, which have been linked to this, have been
established for the first time by Riemann, a half century before Einstein formulated the
general theory of relativity. When the geometric equation satisfies to these conditions of
Rieman space, then she represents in general covariant form a gravitation field of a
special type.
In a Riemann space it is frequently significantly to use a clock, which moves with the coordinate system. This clock indicates the eigen-time for the concerning system.
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In general we talk of a geometric space, when there exists a geometric equation, wherein a generalized distance (like the space-time interval) is defined. When this generalized distance is invariant under coordinate transformation, then the geometry of the space is determined by the form of the geometric tensor.
We demand for a Riemann space moreover the requirement, that the scalar (or dot) product of two vectors remains constant under parallel vector displacement. At this special displacement remains also the size of a vector constant. Under parallel vector displacement we understand the shift of a vector parallelly to himself along a geodetic line, whereby the vector makes always the same angle with these geodetic line.
In an affien space expires the aforesaid requirement for a parallel vector displacement. The size of a vector in this space will then no longer be invariant like in Riemann space. However, here a more general law applies to this parallel vector displacement.
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SEE ALSO: MATHEMATICAL APPENDIX PART 1
<< 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
<< 9 TERMINOLOGY FOR COSMOLOGY
<= 10 MATHEMATICAL APPENDIX PART 1 | PART 2
>> 11 SUMMARY OF THE CHAPTERS
>> 12 REFERENCES AND LITERATURE
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