RELATIVE REALITY _______________ A thesis presented to the faculty of the College of Arts & Sciences of Ohio University _______________ In partial fulfillment of the requirements for the degree Master of Science ________________ Gary W. Steinberg June 2002 © 2002 Gary W. Steinberg All Rights Reserved This thesis entitled RELATIVE REALITY BY GARY W. STEINBERG has been approved for the Department of Physics and Astronomy and the College of Arts & Sciences by David Onley Emeritus Professor of Physics and Astronomy Leslie Flemming Dean, College of Arts & Sciences STEINBERG, GARY W. M.S. June 2002. Physics Relative Reality (41pp.) Director of Thesis: David Onley The consequences of Einstein’s Special Theory of Relativity are explored in an imaginary world where the speed of light is only 10 m/s. Emphasis is placed on phenomena experienced by a solitary observer: the aberration of light, the Doppler effect, the alteration of the perceived power of incoming light, and the perception of time. Modified ray-tracing software and other visualization tools are employed to create a video that brings this imaginary world to life. The process of creating the video is detailed, including an annotated copy of the final script. Some of the less explored aspects of relativistic travel—discovered in the process of generating the video—are discussed, such as the perception of going backwards when actually accelerating from rest along the forward direction. Approved: David Onley Emeritus Professor of Physics & Astronomy 5 Table of Contents ABSTRACT........................................................................................................4 LIST OF FIGURES .............................................................................................6 LIST OF TABLES...............................................................................................7 I. INTRODUCTION .......................................................................................8 II. EINSTEINIAN OPTICS ..............................................................................9 A. Aberration of Light.............................................................................................. 12 B. Doppler Effect..................................................................................................... 13 C. Power Distortion.................................................................................................. 14 D. Temporal Distortion............................................................................................ 14 III. VISUALIZATION..................................................................................16 A. Ray-Tracing......................................................................................................... 16 B. Color Shifting...................................................................................................... 19 C. Power Shifting..................................................................................................... 26 IV. THE VIDEO .........................................................................................26 A. Introduction ......................................................................................................... 27 B. Temporal Distortion............................................................................................ 28 C. Spatial Distortion................................................................................................. 32 D. Spectral Distortion............................................................................................... 38 E. Power Distortion.................................................................................................. 40 F. Combined Effect ................................................................................................. 41 G. Conclusion........................................................................................................... 41 6 List of Figures Fig. 1. The capturing of a sky on a unit sphere. .............................................................. 11 Fig. 2. The perspective camera. ...................................................................................... 18 Fig. 3. The CIE-XYZ color-matching functions.............................................................. 22 Fig. 4. Chromaticity Diagram for sRGB......................................................................... 24 7 List of Tables Table I. Chromaticity Coordinates of sRGB primaries and white point.......................... 25 8 I. INTRODUCTION George Gamow’s popular treatment of modern physics, Mr. Tompkins in Wonderland1, was first published in 1940 and has remained in continual print for the past sixty years. The adventures of its protagonist, C.G.H. Tompkins, have captivated generations of readers. Mr. Tompkins’ first adventure, and possibly his most famous, occurs after he falls asleep during a lecture on Albert Einstein’s Special Theory of Relativity and dreams that he has awakened in a world where the speed of light does not have its conventional value; instead, it has a drastically smaller value, one closer to a human scale. Due to this change Mr. Tompkins witnesses many marvelous sights from bicycles that contract in size to seemingly ageless travelers. Although, as we shall see, Gamow’s description of a world with a much slower speed of light has some flaws, it nevertheless conveys to a general audience the bizarre nature of relativity and is the inspiration for this thesis. This is an attempt to create a virtual reality that obeys Gamow’s principal conceit, which is to realize a world with a much slower speed of light and to examine the consequences. As this exercise is also motivated by a desire to popularize and elucidate as Gamow had done, the accompanying video presentation, intended for a broad, popular audience, has been created. Properly speaking, the video is the thesis: it is intended to stand alone and be understood without this paper. 1 G. Gamow, Mr. Tompkins in Paperback: Containing Mr. Tompkins in Wonderland and Mr. Tompkins Explores the Atom (Cambridge University Press, New York, 1965). 9 This paper, then, is meant as a supplement to the video. It seeks to explain what is being shown there and the science behind it in more specific terms than the medium of video allows. It is also meant to justify some of the choices made in the presentation. Information on the process of generating the video and other technical issues are also discussed. II. EINSTEINIAN OPTICS The description of the alternate reality with the slower speed of light in Mr. Tompkins contains a major flaw, although quite a common one. Gamow’s hero sees a mysteriously contracted bicyclist pass him on the street. Later, as he rides on his own bicycle in pursuit, he discovers that the bicyclist is now completely normal while the city blocks and other stationary objects are contracted. The problem is that Mr. Tompkins could not possibly see length contraction as simply as described. Although the bicyclist and the city are each, in turn, contracted relative to Mr. Tompkins, he cannot perceive it in a clear fashion. This is because simultaneously released light signals from different parts of the bike or street do not reach Mr. Tompkins’ eyes simultaneously: the light reaching Mr. Tompkins at any given instant conveys images of objects as they were at the time the light left them. His view either of moving objects while stationary or of the stationary landscape while he is moving is distorted. In other words, the Lorentz transformations, which describe length contraction and other aspects of relativity, cannot alone describe visual phenomena; a new Einsteinian optics2 must be developed. Dealing with Einsteinian optics is simplified considerably by the realization that the totality of light that reaches a certain point at a certain time is what the observer interacts with. To simplify discussion, we will refer to the entirety of light that reaches an observer at a given point and time as his ‘sky’. This sky can be captured on a unit sphere centered on the observer (Fig. 1) that would thus contain a snapshot of all the light the 10 observer would receive at a given instant.3 That the observer may be in motion through the point does not alter the information contained in his sky, it merely distorts how he perceives it or, alternately, how it is to be presented in the spherical snapshot. Consider an observer traveling through a static landscape in an arbitrary fashion. Calculating how the terrain would look from his perspective—creating a snapshot of his sky—would be a difficult task if tackled directly. The observer’s sky is different depending on his velocity. By developing methods to map one observer’s snapshot into another’s, we can avoid the computationally daunting task of directly constructing the sky for an observer moving in arbitrary fashion through a static landscape. Instead, we first generate a snapshot of the landscape as it would appear to an imaginary observer who is at the same point at the same time but at rest relative to the terrain; that is, we record the terrain’s contribution to this observer’s sky. For objects that are stationary relative to an observer the process of rendering their contribution to his sky is comparatively straightforward since we can call upon traditional computer graphic techniques in the absence of the distorting phenomena caused by relativistic motion. Once this is complete the relevant section of the sky can be mapped