Science Friday: Einstein's Relativity 1

Albert Einstein, Relativity: The Special and the General Theory, trans. by R. W. Lawson (New York: Bonanza, 1961).

Part 1: The Special Theory of Relativity
I. Physical Meaning of Geometrical Propositions
Geometry reduces to assumptions, "axioms."  Propositions are built out of these axioms.  We prove things based on whether they proceed logically from the axioms.

At the same time, the basic ideas of geometry undoubtedly derived from nature originally. "Geometrical ideas correspond to more or less exact objects in nature, and these last are undoubtedly the exclusive cause of the genesis of those ideas" (2).

In fact, we can turn geometry into physics if we add the following proposition: "Two points on a practically rigid body always correspond to the same distance..., independently of any changes in position" (3).  So Einstein has now connected abstract geometry to physical objects.  We have connected what was an idea (geometry) to the real world (physics).

As the book continues, Einstein will go on to show that the "truth" of this proposition is actually limited.
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Note: The idea that the ideas of math originate in nature seems sound.  Pythagoras believed that numbers were the greatest reality, and that the world played them out in some way. Plato believed that the physical world was a copy of an ideal original. They had it backwards.  Aristotle probably came closer: numbers and mathematical "realities" are simply abstractions of the "real world."
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II. The System of Coordinates
We can identify the location of something with reference to the body it is on or we can take a rigid measuring rod from a body to it.  So we can locate a place on the earth by measuring off how many of the units on this rod it takes to get to it from some reference point or we can use a measuring pole to get to a point in a cloud above that point on the earth.

So we imagine locating any point by a number of units of some rigid measuring body to get to it from some point of reference. We don't always have to use a physical pole, since we can use other means to measure.  The Cartesian system of reference imagines three planes (x, y, z) from which we can construct perpendiculars to any point in space.

So in Euclidean geometry, "Every description of events in space involves the use of a rigid body to which such events have to be referred" (8).

III. Space and Time in Classical Mechanics
"The purpose of mechanics is to describe how bodies change their position in space with 'time'" (9). But the concepts of "position" and "space" are somewhat ambiguous. If I drop a rock straight down from a train, it looks like it falls in a straight line to me, but it looks likes it falls in the shape of a parabola to someone sitting on the ground.

First, let's do away with the notion of space ("of which, we must honestly acknowledge, we cannot form the slightest conception," 9) and replace it with "motion relative to a practically rigid body of reference." And by "rigid body of reference," we are thinking of a "system of coordinates" such as was defined in the previous chapter.

There is thus "no such thing as an independently existing trajectory... but only a trajectory relative to a particular body of reference" (10).

A complete description of the motion of a body includes how its position relative that frame of reference changes in relation to time.  The person dropping the rock off the train has a clock and the observer on the ground both have identical clocks measuring "ticks" on the clock as the rock drops.

IV. The Galilean System of Coordinates
The fundamental law of mechanics in physics is the law of inertia set down by Galileo and Newton.  A body at rest tends to stay at rest, and body in motion tends to stay in motion.  This law, however, only relates to a particular frame of reference, an "inertial frame of reference."  [A body at rest on the earth stays at rest on the earth, but it is constantly accelerating in relation to the sun because the earth is spinning.]

"A system of co-ordinates of which the state of motion is such that the law of inertia holds relative to it is called 'a Galilean system of co-ordinates'" (11).

V. The Principle of Relativity (in the restricted sense)
A "uniform translation" is when something is moving at a constant velocity and direction in relation to some frame of reference.  It is not rotating, for example.

"If K is a Galilean co-ordinate system, then every other co-ordinate system K' is a Galilean one, when, in relation to K, it is in a condition of uniform motion of translation" (13).  Accordingly, the mechanical laws of Galileo and Newton will hold good in K' just like they do in K.  In other words, the same physical laws work in K and K'.  This is the principle of relativity (in its restricted sense).

Developments in the study of electrodynamics in the late 1800s had called into question the principle of relativity. [Einstein's work would demonstrate that it could still hold.] But there were strong reasons to think it might hold. For example, "it supplies us with the actual motions of the heavenly bodies with a delicacy of detail little short of wonderful" (13). Why would it work in mechanics but not in electrodynamics?

Another complication if the principle of relativity didn't hold would be that we would have to have some basic frame of reference where the laws of mechanics hold most simply, but the rules would change somewhat in other frames of reference moving in relation to it.  So the laws of motion would apply straightforwardly in coordinate system K but they would be complicated by the motion of K' in relation to K.

Take the earth, for example, rotating around the sun.  Because it is moving in a circle, we would expect its movement to alter its velocity in relation to some absolute frame of reference throughout the course of the year.  Accordingly, we would expect the laws of motion to change in some way throughout the year as we moved in relation to the "absolute state of rest."

"The most careful observations have never revealed such anisotropic [different properties when something is moving in a different direction] properties in terrestrial physical space" (15). For Einstein, this was a very powerful argument for the principle of relativity, that the laws of motion apply the same way in every inertial frame of reference.
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Note: It would be very interesting to trace the history of rhetoric against relativism.  I've never heard anyone in Christian circles speak against relativity, but I can imagine some preachers in the early 20th century doing so.  Although the notion of relativism in ethics has been around forever, I have wondered if rhetoric against relativism in any way was affected or triggered by Einstein's theory of relativity at the turn of the twentieth century.