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Affine Geometry and Relativity

Bozidar Jovanovic

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.

Affine Geometry and Relativity

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.
Paper Structure (14 sections, 5 theorems, 84 equations, 10 figures)

This paper contains 14 sections, 5 theorems, 84 equations, 10 figures.

Table of Contents

  1. Introduction.
  2. Affine world and the principle of relativity.
  3. Galilean geometry.
  4. Galilean transformations.
  5. From the Galilean to the Poincaré transformations.
  6. 1-parametric deformation of the Poincaré group.
  7. Pseudo-Euclidean 2-dimensional world.
  8. 4-dimensional affine world.
  9. The full groups of isometries in 2-dimensional Galilean and pseudo-Euclidean worlds.
  10. Galilean world.
  11. Pseudo-Euclidean isometries.
  12. Visualisation of groups
  13. The Iwasawa decomposition of $SL(2)$.
  14. $SL(2)$ as a quadric in $\mathbb R^4$.

Key Result

Proposition 1

The Galilean transformation $G$ between the coordinates with respect to the inertial frames $\mathbf I'$ and $\mathbf I$ is of the form where $\mathbf b=(b_1,b_2)$ are the coordinates of $O'$ in the frame $\mathbf I$. The parameter $u$ has the following kinematical interpretation: the motions that are at rest in the reference frame $\mathbf I'$ have the velocity $\mathbf u=u\mathbf e_1$ in the re

Figures (10)

  • Figure 1: Galilean world
  • Figure 2: Galilean transformation of coordinates. The uniform motion $\gamma$ that is at rest in the reference frame $\mathbf I'$ has the velocity $\mathbf u=u\mathbf e_1$, $u=\Delta x/\Delta t$, in the reference frame $\mathbf I$. Note that $u$ equals to the tangent of the angle between $\gamma$ and $t$-axis in the reference frame $\mathbf I$. Also, the time-like velocity $\mathbf V$ of $\gamma$ is equal to $\mathbf e_2'$.
  • Figure 3: Galilean transformation of a plane
  • Figure 4: Four lines through an event $A$ in two coordinates systems related by an affine transformation of the form \ref{['opsta']}, \ref{['opste-A']}, in particular by a Poincaré transformation \ref{['poenkareove']}. $\gamma_1$ and $\gamma_3$ are uniform motions with the speed of light (we take $c=1$). $\gamma_2=\{x'=x_0\}$ is at rest in the reference frame $\mathbf I'$. $\gamma_4=\{t'=t_0'\}$ is the space of simultaneous events in the frame $\mathbf I'$. In both coordinates systems $\gamma_1$ and $\gamma_3$ are axis of symmetries for the union of intersecting lines $\gamma_2\cup\gamma_4$.
  • Figure 5: The region of future and past events of $A$ in the inertial reference frame $\mathbf I=[O,\mathbf e_1,\mathbf e_2]$ with the speed of light equals $c=1$ and $c=2$. Note that if we use the system of units with meters for $x$-axis and seconds for $t$-axis, and the value $c=3\cdot 10^8 m/s$, then the space of future events would not be distinguished from the upper half plane.
  • ...and 5 more figures

Theorems & Definitions (17)

  • Definition
  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 12
  • Definition
  • Proposition 16: Twin paradox
  • ...and 7 more