5/29/01

Hi Aurino,

50. You have given me a very difficult challenge. I have spent a lot of time in the past couple weeks trying to come to grips with relativity so that I could confirm or deny or even just understand what you wrote in your "Geometries, speed, and relativity" letter. I failed in all respects.

51. I have never formally studied relativity and have only read popular accounts. None has been satisfying to me. The closest was Brian Greene's "The Elegant Universe". At least I understood his description line by line, but I still didn't come away from the reading with an overall understanding or a comfortable feeling. I felt as if I had been paying careful attention to a magician's trick but still ended up being fooled.

52. I guess I am stuck back in the 19th century being baffled by the results of the Michelson-Morley experiment. No explanation for it has ever made sense to me. When I try to make sense of it, I get hung up at the very beginning trying to define 'speed'. I will not be satisfied with any definition of 'speed' that does not make intuitive sense to me or that is not mathematically rigorous.

53. You defined 'speed' as "the rate at which an object moves in relation to a point of reference." Brian Greene says that "Speed is a measure of how far an object can travel in a given duration of time."

54. The mysterious quantities are 'distance' and 'time'. Both of these quantities are things we need to be able to measure. But what is measurement? I think that's where all the complexity comes in.

55. Measurement comes in two parts. First is the definition of the measure itself, i.e. the definition of the units of measure. The second part is the act of measurement, i.e. the process or method of discovery of the number which answers the question, How much?

56. In my study of mathematics, one of the hardest topics for me was the Theory of Functions of a Real Variable. And one of the hardest sub-topics within that was Measure Theory. I ended up understanding the concepts and appreciating the problem, but I by no means mastered Lebesgue Integration or any of the rest of the technical details. The striking things I learned were, number one, how difficult it is to rigorously define units of measure at all, and secondly that the definitions are arbitrary and it is possible to define multiple, self-consistent systems of measure (called 'metrics') on the same set but which are completely different.

57. So, how do we go about defining units of length? Well, the old-fashioned way starting with Euclid, was to assume that space sort of stood still so that each point could be identified. Then since it stays still, we can super-impose a coordinate system over it so that we can assign a set of numbers to each point, which stay the same for the period of interest. Then we define a metric over this space which gives us the numeric distance between any two points. This is the familiar Euclidean metric which is the square root of the sum of the squares of the component distances between the two points in each of the coordinate directions.

58. That worked very well for a thousand years The earth is reasonably flat so the curvature of the earth didn't foul up surveying calculations much until precision improved and the tracts got large. Even then, it wasn't too bad because we simply needed to acknowledge that the surface we were measuring was really curved and that we simply needed to add another Euclidean dimension and do spherical geometry instead of plane geometry. The metric over the three dimensional space was still Euclidean. No problem.

59. Of course, the Flat Earth Society proposed an alternative metric, the hyperbolic metric, which placed all points of the Earth on a flat disk which did not include the circular boundary. This scheme matches what we see. Things on Earth don't appear to recede infinitely far in the distance but they only go out to the horizon. More importantly, things get smaller as they go out there so they never can quite reach the outer edge. They get too small and can't make any appreciable progress as they travel. This metric provides a consistent way of measuring distance in the world that works as well, so I have heard, as the Euclidean metric does.

60. So the first problem is that since we can define different metrics on space to define length, is there one that is 'true', or that aligns somehow with 'reality'? I don't think the answer is obvious.

61. I think the second problem is harder. That is the problem of the act of measurement. Let's say that we do have the 'correct' metric defined on space so that the units of length are 'correctly defined'. How would we go about measuring the distance between two points?

62. Well, the old-fashioned way, starting well before Euclid, was to place one end of a graduated rod at one of the points and align it so the rod lay alongside the other point. Then you could read off the length by the graduations on the rod. This also worked well for thousands of years, but with Einstein, we learn that the graduated rod doesn't stay the same length all the time.

63. The measuring rod shrinks in the direction of motion.

64. Now, to me, that statement doesn't even make sense. By what right can we claim that it shrinks? Certainly not by the old-fashioned method of measuring length: the rod is one rod long no matter what its orientation.

65. We could claim that the coordinate system we super-imposed over space has accurate 'marks' on it and when the rod is moving, it doesn't always line up exactly with these 'marks'. How could we check that out? Well, let's say that there were two stationary points in space separated by exactly one meter. If we lay a stationary meter stick between them, we see that we get the correct measurement. Now, if we slide a meter stick past the two points, how will we make the measurement? If we are watching one end to know when it is aligned with the first point, we would have to instantaneously be able to shift our attention to the other point to see if it lined up with the end of the stick. Since we can't do that in zero time, the stick will have moved in the meantime so we can't get an accurate measurement.

66. This, of course brings in the second of the two problem quantities, 'time'.

67. We could give up on rods, since they are said to shrink anyway, and use time to measure distance. We just need something that travels at a constant 'rate' and measure the time it takes to get from one point to the other. That is what Brian Greene did in his description. He described a simple clock which was two parallel facing mirrors with a single photon bouncing between them. It was presumed that the speed of the photon was constant and the distance between the mirrors was constant so the interval between bounces would be constant.

68. To me, these assumptions are unwarranted. How do we know the speed of light is constant, since the whole exercise of trying to define 'space' and 'time' and 'rate' is in an effort to explain the apparent constancy of light speed? I think that Einstein's approach was to try to make that constant light-speed the only assumption.

69. So, to sum up my difficulty, I don't see any way of defining length or time without assuming that something stays constant. And, I don't see any justification for assuming anything is constant. Any assumption would influence the inferences drawn from then on.

70. Now, let me comment on what you wrote. You proposed a thought experiment of observing a group of constantly moving cars on a straight road. The very definition of speed in that scenario seems to involve all of the problems I mentioned. First of all, the size of the cars changes from my perspective as they move. How can I define length? I could go out on that road and set up markers at regular interval, but after returning to my observation point, the lengths between markers appear to have changed. The ones farther away appear much smaller. Did they change? I don't really know. Next, I have to have a clock of some sort in order to measure the amount of time it takes a car to go from one marker to another. What makes a suitable clock? I don't know. Even if I had one, there is the problem of the difference in time between the moment the car arrived at a particular marker and the time I observed that event. The lag would depend on the speed of light and the distance between my observation point and the particular marker. Both of these quantities are problematic.

71. Since all my difficulties pop up even before I can start doing your thought experiment, I can't even get started, much less conduct the experiment and draw any conclusions.

72. That's where I have to stop for today. I hope I have been clear enough about my difficulties so that you can either help me clear them up, or you can see that I am not in a position to follow your arguments. Let me know what you think.

Warm regards,

Paul

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