I'm searching for mathematical foundation of this model.
@Peter Enders: What is bad with Minkowski's 1908 talk?
Minkowski's 1908 talk was indeed very important. An excellent, detailed overview of Minkowski's views (including his 1908 talk) is given in
http://lt-jds.jinr.ru/record/52173/files/Minkowski%20Spacetime%20A%20Hundred%20Years%20Later.pdf
This talk includes what Minkowski wrote in German as well as the English translation (there are many helpful drawings showing how Minkowski viewed hypebolic space and Raum und Zeit (Space and Time), e.g., Fig. 2 on page xxvii, page 27 in the pdf file). Minkowski's portrait plus signature are given in the attached image (p. 13).
A particularly important part of this book is chapter 7, starting on page 200, on quantum space-times. Minkowski fused space and time into a 4D continuum. Of particular interest in this chapter is the discussion about big bang, black holes and information loss. A discussion about the quantum nature of the big bang is given in Section 7.2, starting on page 203. See the key questions in Section 7.3.
In his September 21, 1908 Hermann Minkowski included in his talk at the 80th Assembly of German Natural Scientists and Physicians the following observations:
A point of space at a point of time, that is, a system of values, x, y, x, t, I will call a world-point. The multiplicity of all thinkable x, y, x, t systems of values we will christen the world... Not to leave a yawning void anywhere, we will imagine that everywhere and everywhen there is something perceptible. To avoid saying "matter" or "electricity" I will use for this something the word "substance". We fix our attention on the substantial point which is at the world-point x, y, x, t, and imagine that we are able to recognize this substantial point at any other time. Let the variations dx, dy, dz of the space co-ordinates of this substantial point correspond to a time element dt. Then we obtain, as an image, so to speak, of the everlasting career of the substantial point, a curve in the world, a world-line, the points of which can be referred unequivocally to the parameter t from - oo to + oo. The whole universe is seen to resolve itself into similar world-lines, and I would fain anticipate myself by saying that in my opinion physical laws might find their most perfect expression as reciprocal relations between these world-lines. [1, p. 76]
This is a good question.
A good introduction to Minkowski space is given in
https://www.researchgate.net/publication/1913141_The_Rich_Structure_of_Minkowski_Space
More to the point, see
https://www.researchgate.net/publication/228398035_On_the_differential_geometry_of_the_curves_in_Minkowski_space-time_II
http://arxiv.org/pdf/gr-qc/9506088.pdf
Article The Rich Structure of Minkowski Space
Article On the differential geometry of the curves in Minkowski space-time II
There are many good books and I I would strongly recommend the book by Hermann Weyl called Space, Time, Matter.
Hope this helps!
@Peter Enders: What is bad with Minkowski's 1908 talk?
Minkowski's 1908 talk was indeed very important. An excellent, detailed overview of Minkowski's views (including his 1908 talk) is given in
http://lt-jds.jinr.ru/record/52173/files/Minkowski%20Spacetime%20A%20Hundred%20Years%20Later.pdf
This talk includes what Minkowski wrote in German as well as the English translation (there are many helpful drawings showing how Minkowski viewed hypebolic space and Raum und Zeit (Space and Time), e.g., Fig. 2 on page xxvii, page 27 in the pdf file). Minkowski's portrait plus signature are given in the attached image (p. 13).
A particularly important part of this book is chapter 7, starting on page 200, on quantum space-times. Minkowski fused space and time into a 4D continuum. Of particular interest in this chapter is the discussion about big bang, black holes and information loss. A discussion about the quantum nature of the big bang is given in Section 7.2, starting on page 203. See the key questions in Section 7.3.
In his September 21, 1908 Hermann Minkowski included in his talk at the 80th Assembly of German Natural Scientists and Physicians the following observations:
A point of space at a point of time, that is, a system of values, x, y, x, t, I will call a world-point. The multiplicity of all thinkable x, y, x, t systems of values we will christen the world... Not to leave a yawning void anywhere, we will imagine that everywhere and everywhen there is something perceptible. To avoid saying "matter" or "electricity" I will use for this something the word "substance". We fix our attention on the substantial point which is at the world-point x, y, x, t, and imagine that we are able to recognize this substantial point at any other time. Let the variations dx, dy, dz of the space co-ordinates of this substantial point correspond to a time element dt. Then we obtain, as an image, so to speak, of the everlasting career of the substantial point, a curve in the world, a world-line, the points of which can be referred unequivocally to the parameter t from - oo to + oo. The whole universe is seen to resolve itself into similar world-lines, and I would fain anticipate myself by saying that in my opinion physical laws might find their most perfect expression as reciprocal relations between these world-lines. [1, p. 76]
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