# Geometrical Representations of Relativistic Motion

6-3-1966

## Abstract

This investigation begins with the development of a transformation or mapping in velocity space of such a character that velocities are mapped into a quantity that adds linearly, while velocities add non-linearly according to the Einstein Law of Addition, and has no upper limit to its magnitude, i.e., when the speed of an object approaches that of light in a vacuum, this quantity becomes infinitely large. Kinematic relationships, in Newtonian form, for a particle undergoing acceleration, are developed in a frame of reference that remains attached to the particle using the new variable (W). It is found that relativistic energy and momentum, when divided by the quantity MC, where M is the rest mass of the particle under consideration, and C is the speed of light in vacuo, can be put in the form of the equations for a catenary with parameter, C. Further relationships are revealed when the involute of the catenary, a curve called the tractrix, is brought into the picture. W/C is used as the independent parameter in all equations. When the tractrix is revolved about the "W" axis,a surface called a pseudosphere is formed. On this surface, a constant gaussian curvature of -1/C2, vectorial relationships may be determined in terms of the properties of geodesic triangles. The pseudosphere is a surface on which hyperbolic geometry is applicable. Some of the major characteristics of hyperbolic geometry are discussed and an attempt is made to relate these to Special Relativity. The catenary, when revolved about the "W" axis forms a surface of minimum area, i.e., when the catenary connects any two points, the surface of revolution formed will have a minimum surface area. This property is then used to show that the equations for relativistic energy and momentum can be determined by specifying that the integral of energy with respect to momentum be minimized. When more than one particle is considered, interactions between them are described in terms of relocation of the particles involved on the catenary. A very simple static model of the universe, in velocity space is suggested. The second half of the study is devoted to an investigation of hyperbolic motion. Particular emphasis is placed on investigating the situation that a light signal starting out a distance C2 /g behind an object undergoing hyperbolic motion, i.e., constant rest acceleration, g, in one direction, will never catch that object. Furthermore, the world lines of the light signal and the object become indistinguishable when the rest acceleration of the object becomes infinitely large, independent of the inertial frames in which they are observed. Arguments are given to support the idea that the kinematical behavior of light cannot be distinguished from an infinitely accelerating (rest accel.) object in any frame of reference including non-inertial frames. During the course of this section of the study, various characteristics of hyperbolic motion are discussed. In particular, the idea is brought out that instants in time on a moving particle propagate through a stationary frame in the same direction as the particle at speed C2 /V, where V is the speed of the particle. It is hypothesized that states in time always travel at speeds greater than or equal to the speed of light and that its speed represents an inverse point, in velocity space, to the speed of the corresponding particle, with the circle or sphere of inversion having the "radius" C. The symmetry that is suggested between the motion of time and the motion of material particles is investigated with the aid of the Space-Time diagram. In the final section, a brief geometrical discussion concerning the finiteness of the speed of light is presented.

Physics

Masters

## Department Name

Physics & Astronomy

## First Committee Member (Chair)

Christopher Pratt Leavitt

Derek B. Swinson

## Third Committee Member

Howard Carnes Bryant

English

Thesis

COinS