Surveying an Inertial Frame
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In the essay on Inertial Frames of Reference I described the use of ghostly rulers and clocks to mark out a four-dimensional grid representing an inertial frame. Now I want to consider in more detail actually doing such a thing: I want to unabstract the abstraction in order to clarify the abstraction. I want to consider whether we could, in fact, survey space and time to construct a bona fide inertial frame of reference. In setting up such a survey we will take yet another look at the basic nature of space and time and their relationship.
When I first encountered the ideas of Relativity, I felt much as I imagined ancient mariners felt when they first ventured out onto the open ocean and went out of sight of land. All familiar points of reference went away, leaving me with new and unfamiliar points of reference and a strong sense of disorientation. I had about me only empty space with nothing but the ghost of a measurement system, one not anchored to any familiar landmark, against which I could orient myself. I began to understand that our sense of space and time, which we derive from our day to day experiences, has more the nature of an illusion than of solid fact. That understanding led me to understand further that our knowledge of Relativity advances through the progressive stripping away of illusion, however persistent or insistent. We can include the process of abstraction in that stripping, the process of looking through what we perceive in order to conceive the Platonic Forms behind them.
We can see that process exemplified when we look at how our ancestors transformed their conception of the world from that of objects participating in events into that of measured objects participating in timed events. That transformation proceeded through the invention of rulers and other surveying tools, which allowed them to abstract space from the world, and the invention of clocks, which allowed them to abstract time from natural events. Such abstraction resemble Platonic Forms, the barest essence of the thing abstracted. Consider how the abstractions came about.
The invention of agriculture led to the abstraction of space. When farming began people had to divide the landscape into the wild and the domestic and then they had to further subdivide the domestic. The needs to allocate land to farmers and to mark out fields for plowing led to the devising of land measures and the means of determining them on the ground. Later, when the Greeks abstracted those measures and means into the discipline they called measuring the Earth (Geo Metria in Greek), they understood that the lines that they drew to depict geometric figures only crudely approximated the Platonic Forms that they represented. So too do we understand that our imaginary experiments (or verbal descriptions) only crudely approximate Ultimate Truth.
And just as the invention of agriculture led to land surveying and thence to Euclid's geometry, so too did the invention of the mechanical clock lead to modern physics. We have, by inventing the clock (in the 14th Century), separated the counting of time from observations of Nature and made our conception of time independent of natural events. In the next obvious step, we abstract time from the clock itself, conceiving it as a continuous, featureless elapse that underlies the dance of all events. By abstracting time from events we enable the mapping of physics, in much the way that the latitude/longitude grid enables the mapping of land as distinct from the mere description of land by indeterminate metes and bounds.
Before the invention of the mechanical clock people could not separate time from events. Thus only local time existed in human conception; a universal flow of time (Heraclitus' river of time) did not truly exist for people. As people developed ways to synchronize clocks (make them tell the same time) they extended local fluxes of time into a single uniform current flowing at the same speed for everyone. Thus we went from coordinating events with other events to coordinating events with counted time.
We can also reverse that process and paint imagery onto our abstractions if we need to examine them through our imaginary experiments. I now want to use that reverse process to create a useable inertial frame.
I have tacitly assumed into the premises underlying the Map of Physics the axioms and common notions of Euclidean geometry; that is, I have assumed that space in itself has a nature that follows the rules of Euclid's abstraction of land surveying. We have space described as the Platonic Form of distance in three dimensions, deformed only by observable phenomena in the affected region (Mach's principle). But in incorporating Euclid's axioms into the Map of Physics I have also tacitly assumed that space and time suffer no inherent distortions. Can we prove and verify those assumptions and thereby transform them into facts?
(Here I want to acknowledge that I have tacitly assumed the existence of space with three mutually perpendicular dimensions and of a one-dimensional time. For now we will have to accept that assumption as a postulate with the knowledge that we will eventually have to deduce the subject of that assumption from the bare fact of existence before we can say that we have completed the Map of Physics. So far I have not had any success in my efforts to make that deduction, but someday, I feel certain, someone will make it, prove it, and verify it.)
If we want to survey an inertial frame, we must clear out a region of space, remove all bodies that might exert forces, and then carry out the survey. Mach's Principle tells us that it takes a difference to make a difference - any deviation from a normal state of affairs must originate in some observable phenomenon. By removing all bodies from this region, taking them far away, we ensure that nothing exists in the volume in which we want to construct our inertial grid to absorb the equal and oppositely directed reactions that Newton's third law requires of all changes in momentum. Now we know that conservation of linear momentum ensures that all thrown bodies follow straight lines in all directions through this region; therefore, we have no distortions of space (which would curve the lines).
Here we have a precursor to Nöther's theorem. At about the time that Albert Einstein was conceiving the theory of Relativity, Amalie Emmy Nöther, a German mathematician, discovered that Reality relates conservation laws to symmetries of time and space. Among the relations that she discovered we find one that relates the homogeneity of space to conservation of linear momentum
Light gives us a good example of a thrown object that carries linear momentum; therefore, in empty space it must follow straight lines. Indeed, it gives us the perfect thrown object since Einstein's second postulate makes its motion an absolute standard and makes it a touchstone against which we can measure other phenomena. Thus, if we measure the length of an object from a distance, using standard surveying techniques, we must calculate the same length measured by an observer next to the object. And we can use beams of light to mark grid lines.
Speaking of touchstones reminds us that in proper science the truest touchstone against which we test our theories is Reality itself, through observation or experiment. All of our theoretical gassing off comes to nothing if our hypotheses do not match the phenomena we purport them to describe.
And here we come to a good place to note that so far Reality conforms to our theoretical expectations. If light flew at different speeds in different directions, Michaelson and Morley would have detected the difference in their rotating interferometer, which they had designed specifically to detect differences in the speed of light due to Earth's motion through the ćther, the presumed medium of electromagnetic propagation. And if the speed of light varied with location, it would cause observable optical distortions due to refraction, the phenomenon that enables glass, water, and other transparent substances to deform the paths that light rays follow. Einstein arcs, the thin blue curves that astronomers find in some pictures of distant galaxy clusters, would seem to provide evidence of just such distortions, but astronomers only find them associated with galaxy clusters and thus attribute them to the action of the clusters' gravitational fields and not to any inherent warping of space.
If time were to elapse at different rates at different places in the same inertial frame, we would detect other anomalous effects. We would see the most obvious effects in our efforts to synchronize pairs of clocks. If we synchronized clocks at different locations, then we would see some time later that the clocks tell different times. But we know that any repeating event gives us the basis for a clock, so we know that light, the vibration of electromagnetic fields propagating through space, must obey the same rules that clocks obey. That fact means that electromagnetic waves propagating through regions of variable time would increase or decrease their rate of vibration and, thereby, gain or lose energy gratuitously (that is, absent any interaction with another body or system of bodies to provide or absorb the difference in the energy). Such a phenomenon would violate the law requiring the conservation of energy, so to the extent that we have confidence in the full validity of the conservation law to that extent we have confidence that time does not elapse at different rates in different places. Here again we see a precursor to Nöther's theorem. In this case we relate the homogeneity of time to the law of conservation of energy.
Thus we validate our assumption of the proposition that space has such a nature that light flies through it in straight lines. Now we know that we can establish a truly Euclidean grid as the foundation of our depiction of an inertial frame of reference. To set up our coordinate grid we must accomplish two tasks:
1) we must establish the three-dimensional grid of straight lines that meet at right angles, and
2) we must synchronize the clocks at the intersections of those straight lines.
In the imaginary experiments of Relativity we make our inertial frames countable with rulers and clocks, in essence painting lines and tick-tocks onto the Platonic Form of Euclidean space and time. And as the Greeks used the crude lines drawn in the dirt or on parchment as a means of deducing the theorems of plane geometry, so we can use our crude coordinate grids as a means to deduce the laws of physics. And we use our equally crude tools to survey those grids, though we strive to ensure that those tools give us as much precision and accuracy as we need for the purpose of rendering distances and durations countable.
We can ensure that our rulers will put the same numbers of standard distances to the same measured distances by bringing them into coincidence and ensuring that the marks that we have made upon them line up adjacent to each other. We cannot do that with clocks, though, because clocks are dynamic devices whose operations may change when they move, thereby taking away our guarantee that they tell the same time. (Later we will deduce the theorem of time dilation, which tells us that clocks definitely tick at different rates when they move, thereby justifying this assumption.) We will have to apply some clever tricks to ensure that the clocks on our grid do indeed tell the same time: we will discover that we can either synchronize two clocks at one point and then move them in a way that preserves their synchronization or we can preset the clocks at their assigned locations and start them with a signal that preserves the synchronization.
With the assurance that we can draw straight lines in space and determine distances along them in a consistent way, we begin to lay out our coordinate grid. Our surveyors project three thin beams of light in three directions in such a way that at one point those beams cross each other at right angles. They then project more beams parallel to those original three to fill space with a kind of ćtherial jungle gym. At the point where the original beams cross each other they establish a massless spacemark that identifies the point as the origin of our coordinate grid, the point that has coordinates (0, 0, 0). That act makes the original beams the coordinate axes of our grid. Our surveyors then use their rulers to measure distances along the beams, establishing more massless spacemarks to show the coordinates of the points that they mark, like mileposts set along a railroad track. We can then calculate the distance between any two points in our grid by subtracting the coordinates of one from the coordinates of the other and then applying Pythagoras' theorem.
Now comes the hard part, the one that requires of us some cleverness - the synchronization of the clocks that our surveyors have put at each intersection of the beams. We have two possible methods available to us.
Suppose that we put two clocks together and start them at the same time. Then we move the clocks apart at the same speed for the same interval of time. We would end up with two clocks that tell the same time, as we can verify by looking at the difference between the times that we actually see the clocks displaying: that difference should equal the distance between the clocks divided by the speed of light, because it represents the time it takes the light carrying the image of one clock to reach the other clock.
Alternatively we can have two clocks motionless at points A and B and preset them to show the same time. Euclidean bisection of the line AB gives us the midpoint of that line and we put a mirrored wedge on that point. A pulse of light fired at the wedge splits and flies equal distances to the clocks and starts them simultaneously; thus we have synchronized the clocks. Given the second postulate of Relativity, we know that equal distances correspond to equal times for the motion of light. That fact gives us our guarantee that the clocks tell the same time.
We can test that synchronization with the aid of observers standing by each of the clocks with cameras. Each of the observers sends a pulse of light to the other. In the dark the pulses illuminate the clocks and the times that they display at the instant of illumination. From an examination of the relevant photographs the observers take note of the times that the clocks display upon the emission and the reception of the pulses. Each observer then subtracts the time their clock displayed when it emitted its pulse from the time the other clock displays when it receives that same pulse of light. If both observers calculate the same number, then they can say that their clocks are properly synchronized.
Now we extend the system to span the inertial frame: in two mutually perpendicular directions (the y- and z-directions by definition) oriented perpendicular to the line AB (the x-direction by definition) we move pairs of preset clocks in opposite directions a distance d away from one of the already running clocks. We preset the new clocks to a time t+d/c and at the time t send light pulses in opposite directions from the old clock to start the new clocks. Then using the new clocks as references, we establish yet more pairs of clocks until we have a clock running at every point where three beams of light intersect.
Thus we have our inertial
frame of reference surveyed and ready for use.
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