Logical Foundations of Science and Computing, Fall 2007

Chapter 1-12 Summary of A. Einstein's Special Relativity Paper

What follows is a brief summary of what I perceive are the key ideas of the first 14 chapters of A. Einstein's paper The Special Theory of Relativity found in the book Relativity: The Special and General Theory by Albert Einstein (translated by Robert W. Lawson). The paper is organized with the following architecture.

Chapter 1 - Physical Meaning of Geometric Propositions

Dr. Einstein shows us how to relate a mathematical model of space with "real" space as we perceive it. The mathematical model he discusses is a geometric model derived from Euclid's ancient Greek work on geometry.

Chapter 2 - The System of Co-ordinates

This chapter shows how to locate objects in space and measure distances between objects in a Euclidean model of space using a Cartesian coordinate system. Every object in space can be identified by three coordinates. All coordinate systems must be attached to a body of reference.

Chapter 3 - Space and Time in Classical Mechanics

In order to deal with objects in motion one needs to add time to a model of space. Chapter 3 introduces the model of space and time used by classical mechanics and describes what motion looks like. Dr. Einstein shows that bodies in motion do not have an independently existing trajectory, rather, only a trajectory relative to a body of reference. He implicitly suggests that we must think of every object in space and time as identified by 4 coordinates. The last coordinate is the moment in time of observing an object in space.

Chapter 4 - The Galilean System of Co-ordinates

Dr. Einstein introduces a requirement on any reference body and system of coordinates. In order to be a permissible reference body and system of coordinates all objects being observed must obey the law of inertia: A body removed sufficiently far from other bodies continues in a state of rest or of uniform motion in a straight line Such a permissible body of reference and coordinate system is referred to as a Galilean system of coordinates. (Sometimes I refer to a Galilean system of coordinates as a frame of reference.)

Chapter 5 - The Principle of Relativity (in the Restricted Sense)

This somewhat challenging chapter discusses what it means to observe the same phenomena from different frames of reference that are in uniform motion relative to each other. Dr. Einstein informs us that the principle of relativity is the widely held belief that the laws of mechanics remain equivalent independent of your frame of reference. Paraphrasing Dr. Einstein's words in paragraph 3 of Chapter 5: for two frames of reference K and K' in uniform motion relative to each other, natural phenomena run their course according to the same general laws. The restricted sense has to do with reference frames in uniform motion as opposed to more involved forms of relative motion that exhibit acceleration.

Chapter 6 - The Theorem of the Addition of Velocities in Classical Mechanics

Chapter 6 summarizes the common-sense understanding of the addition of velocities of moving objects assumed by classical mechanics. This is a direct setup for the conflict described in the next chapter.

Chapter 7 - The Apparent Incompatibility of the Law of Propagation of Light with the Principle of Relativity

Here the essential conflict of classical mechanics and the observed constancy of the speed of light is established. The conflict as described by Dr. Einstein reflects the foundational quandary physics was in at the turn of the century. According to the theorem of the addition of velocities in classical mechanics, the speed of light should be different in different frames of reference moving uniformly relative to each other. Yet the constancy of the speed of light was a law of physics that should remain the same in all frames of reference, according to the principle of relativity. Dr. Einstein claims that the special theory of relativity side-steps this conflict. How? Read on.

Chapter 8 - On the Idea of Time in Physics

Chapter 8 begins the process of re-conceptualizing time and space in order to achieve the new relativistic mechanics. What fundamental assumption was the physics community unwittingly clinging to at the time? Dr. Einstein will boldly propose that our concept of absolute time and space must go. But first he must carefully define time according to the method by which it is measured. All defintions in physics are operational definitions, meaning that all fundamental properties upon which physics is based are defined according to the method by which they are measured.

Chapter 9 - The Relativity of Simultaneity

Chapter 9 sets out to demonstrate that the concept or idea of Simultaneous Events can only be meaningful when expressed relative to a body of reference. The chapter presents a thought experiment that examines two lightening strikes as seen from an embankment or from a train moving with respect to the embankment. Einstein shows that if the two lightening strikes are perceived as simultaneous from one of the reference points then they will not be simultaneous from the other. A very important point of this chapter is that Dr. Einstein's conclusion that time itself cannot be perceived as the same from the two different reference points - there can be no absolute significance to any statement of time.

Chapter 10 - On the Relativity of the Concept of Distance.

Not only are time measurements relative to a reference body, but distance measurements are relative to a reference body as well. There can be no absolute significance to any statement of distance. In general Dr. Einstein is suggesting that we must be prepared to accept that time and distance are strictly local phenomena that change depending on motion. We must accept that time and length change for objects in motion relative to the observer. Identically constructed clocks can run at different rates. Biological aging (a kind of clock) can run at different rates.

Chapter 11 - The Lorentz Transformation

The Lorentz Transformation gives the method of translating the 4 dimensional spacetime coordinates of an event relative to one reference body into different 4 dimensional spacetime coordinates relative to another reference body. The transformation must be such that the laws of physics are the same in both reference frames -- even if the actual space and time coordinates are different.

Chapter 12 - The Behaviour of Measuring-rods and Clocks in Motion

This chapter derives the time dilation factor and the distance contraction factor for measurements on bodies in motion relative to an observer at rest.