"A ship in port is safe, but that's not what ships are built for."
Grace Hopper. (The computer term "bug" is also due to Grace Hopper)
Introduction to The Moving Target Laws
When I talk about the "path of light," I am really talking about the Moving Target Laws. It is critical for the reader to have an exact understanding of what the Moving Target Laws are in order to understand any of my experiments or the Lunar Laser Ranging experiments. This chapter will provide that understanding.
Consider an archer who is a perfect shot and always hits the exact center of the bulls-eye. Now consider the archer and his target as a three-dimensional coordinate system. The two-dimensional target is on the X-Z plane. The path of the arrow is the Y axis. The X axis, Y axis and Z axis all meet at a point that is also the center of the bulls-eye of the target. Suppose further that the archer is 100 feet from the target.
See graphics on next page.
Now suppose the archer aims at the target and that I am holding the target. Suppose that after the arrow leaves the bow, but before the arrow gets to the target, two things happen: 1) I move the target one-foot straight up and 2) the archer throws the bow into a trashcan. The arrow will obviously land one foot below the (center of the) bulls-eye. This example, as with all examples, assumes the archer is a perfect shot, there is no arch on the arrow, the experiment is done in a vacuum, etc.
The key to understanding the Moving Target Laws (MTLs) is to understand the slice of time between:
1) The instant after the arrow leaves the bow and
2) The instant before the arrow hits the target.
This slice of time is the time the arrow is "in the air." "In the air" means the arrow is no longer attached to the bow, and has not yet hit the target. While the arrow is "in the air," what happens to the archer and the bow is irrelevant because the arrow is no longer attached to the bow. As I implied above, while the arrow is in the air, the archer can throw the bow into the trashcan and this act will have absolutely no affect on where the arrow hits the target. While the arrow is "in the air" the bow and arrow are totally independent of each other.
But the same cannot be said about the target. While the arrow is in the air, it is headed towards the target. Thus any motion of the target, while the arrow is in the air, has a direct affect on where the arrow hits the target! This is why they are called the "moving target" laws.
The MTLs basically study the motion of the target during the slice of time the arrow is in the air. In the example just given, after the arrow leaves the bow, but before it gets to the target, the target is moved one-foot straight up. Thus the moving of the target has a direct affect on where the arrow hits the target.
More Examples of the MTL
If I were to move the target exactly one foot down while the arrow is in the air, the arrow would land one foot above the (center of the) bulls-eye. If I move the target exactly one foot to my right (while the arrow is in the air), which is to the archer's left, the arrow would miss the target by exactly one foot to my left. I am behind the target so the arrow would hit to the left of the bulls-eye from my perspective. But from the archer's perspective, the arrow would land to the archer's right (i.e. the archer would see the arrow land one foot to the right of the bulls-eye). And so on.
Now let us do thousands of these exercises. Suppose that after the arrow leaves the bow, that I am allowed to move the target exactly one foot on the two-dimensional X-Z plane, and that I can move the target in any of the 360 degrees of the plane. If we were to do this exercise thousands of times there would be up to 360 holes that would form a perfect circle (i.e. the holes would all be on the outside edge or perimeter of the circle), with the center of the bulls-eye being the center of the circle. No holes would be inside or outside of the perimeter of holes.
Now let us construct an imaginary three-dimensional sphere with the center of the sphere being the center of the bulls-eye. The imaginary sphere has a radius of exactly one-foot. Now let us change the rules. I can move the target exactly one foot while the arrow is in the air, but I can move it in any direction in three-dimensions, as long as the center of the target ends up being on the surface of the imaginary three-dimensional sphere when the arrow arrives.
In other words, the center of the bulls-eye will always be on the surface of the imaginary sphere by the time the arrow arrives. Since the sphere is in three dimensions, but the target is only in two dimensions, I can move the target such that the arrow can hit any point within a one-foot radius of the center of the bulls-eye.
As an example, if I were to move the target along the Y-Axis, directly towards the archer or directly away from the archer, ignoring the arch of the arrow, the arrow would hit the center of the bulls-eye even though I moved the target one-foot. If I were to move the target slightly off of the Y-Axis, the arrow would just miss the bulls-eye.
Now suppose this three-dimensional experiment were done thousands of times and I moved the target randomly. In this case an imaginary circle of one-foot radius on the target, with the bulls-eye at the center, would have many arrow holes in it or on its perimeter. In other words, instead of just being on the perimeter of the circle, the holes would also occupy the inside of the circle.
Now suppose we studied one specific hole in the target. Let us suppose that I moved the target one-foot at such an angle on the sphere that the arrow missed the center of the target by exactly 5 inches. Studying the angle that the hole is from the bulls-eye, the angle that I moved the target could easily be determined. However, since the target itself has only two-dimensions, it could not be determined whether I moved the target generally towards the archer or generally away from the archer (on the Y-Axis). Thus the solution to exactly how I moved the target in three-dimensions could only be reduced to two possibilities.
If an observer were standing on the side of the target, she could note whether the target were moved forward or backwards because she would see the Y-Z axis. Thus she could determine the Y-Axis movement and narrow the possibilities down to one.
More Complex Archer Examples
Now let us suppose that I am standing on a flatbed car on a moving train. Again, I am holding the target. We will assume the archer is standing on a train station platform. The flatbed car is moving with the train (along with the target), but the archer is not moving. Let the distance between the archer and the target be 100 feet.
Let us assume that the train is moving at such a speed that in the time it takes the arrow to travel to the target the train moves exactly 1 foot (the train is in constant motion). Now suppose the archer lets go of the arrow at the exact instant that the bulls-eye passes him. Now suppose that I do not move the target, but hold it still. The arrow will again miss the target by 1 foot. If the train is moving left to right (per the archer), the arrow will land to the left of the bulls-eye by 1 foot.
If additionally I had moved the target one-foot in the direction the train is traveling, the arrow would miss the bulls-eye by 2 feet. It would miss the bulls-eye by 1 foot because of my moving the target and another foot because of the moving train.
However, if I moved the target in the opposite direction the train is traveling, the arrow would hit the center of the bulls-eye. The 1-foot miss caused by the motion of the train would be offset by the 1-foot miss caused by my moving the target.
In these cases the platform that the target is standing on (i.e. the flatbed car) is in motion, thus the motion of the platform has a direct affect on the motion of the target while the arrow is in the air. The MTLs apply both to my movement of the target, and to the platform's movement of the target. Anything that moves the target while the arrow is in the air is significant to where the arrow hits the target.
In the experiments that are discussed in future chapters, the platform will be the earth, the archer will be a laser, and the arrow will be a laser beam.
One of the hardest things for people to grasp about the moving target laws is that once the arrow or laser beam is "in the air," the motion of the archer or the motion of the laser is irrelevant. People constantly wonder why the motion of the archer or laser is irrelevant. Many examples will be given to make this distinction clear.
The MTLs will be mentioned many times in this book. They are not only the basis of aberration of starlight, but are also directly involved with all of my experiments and any experiment that involves the path of light.