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Affine Geometry and Relativity

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Abstract

We present the basic concepts of space and time, the Galilean and pseudo-Euclidean geometry. We use an elementary geometric framework of affine spaces and groups of affine transformations to illustrate the natural relationship between classical mechanics and theory of relativity, which is quite often hidden, despite its fundamental importance. We have emphasized a passage from the group of Galilean motions to the group of Poincaré transformations of a plane. In particular, a 1-parametric family of natural deformations of the Poincaré group is described. We also visualized the underlying groups of Galilean, Euclidean, and pseudo-Euclidean rotations within the special linear group.

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Notes

  1. The book is written in the form of discussions. As an illustration, we present the following beautiful part, taken from [13], page 18: “When you have observed all these things carefully (though there is no doubt that when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stem than toward the prow even though the ship is moving quite rapidly, despite the fact that during the time that you are in the air the floor under you will be going in a direction opposite to your jump. In throwing something to your companion, you will need no more force to get it to him whether he is in the direction of the bow or the stem, with yourself situated opposite. The droplets will fall as before into the vessel beneath without dropping toward the stem, although while the drops are in the air the ship runs many spans.”

  2. In the 4-dimensional affine world \(\mathcal A^4\), two orthonormal bases are related by the elements of the group SO(3).

  3. While the legacy of Lorentz, Poincaré, and Einstein in the founding of special relativity is well known, we would like to mention here a contribution of Mileva Marić (1875–1948) as well [4].

  4. Let G be a subgroup of the general linear group GL(n). The connected component of \(A=(a_{ij})\in G\) is the subset of the matrixes \(B=(b_{ij})\in G\), such that there exist a matrix curve \(C(t)=(c_{ij}(t))\in G\) (\(t\in [0,1]\)), where \(c_{ij}(t)\) are continuous functions and \(c_{ij}(0)=a_{ij}\), \(c_{ij}(1)=b_{ij}\) (\(i,j=1,\dots ,n\)).

  5. In the literature, both linear maps, given by \(\mathrm L_c(u)\) and by \(\tilde{\mathrm L}_c(u)\), are called Lorentz transformations. Also, both affine transformations \(P_c(u,\textbf{b})\) and \({\tilde{P}}_c(u,\textbf{b})\) are called Poincaré transformations.

  6. The above consideration proves Problem 4. What is missing here is the prof that the vector \(\mathrm R(-\alpha )\textbf{w}\) belongs to the line \(\gamma =\{\sigma \mathrm A(u)\textbf{e}_2\,\vert u\in \mathbb R\}\). For a general Iwasawa decomposition of semisimple Lie groups, e.g., see [6].

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Acknowledgements

The author is very grateful to Vladimir Dragović for useful discussions. The research was supported by the Project No. 7744592 MEGIC “Integrability and Extremal Problems in Mechanics, Geometry and Combinatorics” of the Science Fund of Serbia.

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  1. Mathematical Institute SANU, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000, Belgrade, Serbia

    Božidar Jovanović

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  1. Božidar Jovanović
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Correspondence to Božidar Jovanović.

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Jovanović, B. Affine Geometry and Relativity. Found Phys 53, 60 (2023). https://doi.org/10.1007/s10701-023-00700-2

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