Relativity, June 9, 2001

I have always been facinated by relativity. When I was in High School I read a book called "Relativity for the Millions" which started my though processes on this subject.

I suppose it is only fair to point out that Relativity was not that much of a stretch for me because I find it difficult to assign one set of facts to account for everything. That is, in High School I would have considered myself a total success if I could have proposed two entirely contradictory theories to explain the Universe at the same time, both of which would be plausable, and both validated by experimental analysis. Sounds like Relativity and Quantum Mechanics.

With this in mind, you should be able to understand why I could criticize a theory, and support it at the same time.

For years I've been contemplating what happens with a rotating object in a Special Relativistic space. I've seen statements that indicate there is a problem here because if an object is at a great distance, say one light year, from the center of the rotation (which we will say is once per second) then the object would be going quite a bit faster than the speed of light. In particular, so the book says, it would be traveling at 2 * PI light years per second, something that Special Relativity doesn't allow.

However, I looked at it differently. The analysis above assumes that the space is a Newtonian space. But this is a Relativistic space. Therefore the distance arround relative to the stopped observer would be 2 * PI * r, but the distance relative to the moving observer would be 2 * PI * r * sqrt(1 - (v/c)^2), which is less. This would mean actual speed would be v = 2 * PI * r * sqrt(1 - (v/c)^2) / t, where t is the time of rotation, r is the radius, v is the speed, and c is the speed of light. Given any t and r this could be solved for v, and the resulting v would be less than the speed of light. (Even in our example!) I'll leave it to the reader to have the fun of solving this equation.

But how can this be? Either space is bent, or there must be some contraction in the direction horizontal to the direction of motion.

Space could be bent arround the point of rotation so things that are farther away are, in the final analysis, much closer than thought due to the rotaion. Much as, if one is on the north pole the distance to the equator is 2 * PI * r / 4 = Pi * r / 2, where r is the radius of the Earth. A flat space would say that the radius at the equator would, therefore, be 2 * PI * PI * r / 2 = PI ^ 2 * r. However, the real radius is 2 * PI * r, so the curvature of the earth eliminates (PI - 2) * PI * r of the distance. Space could do the same thing around a rotating object.

I will postulate that it doesn't make any difference which model you use.

Ulimately the theory of relativity became necessary because of two observations. In the early 19th century physicist were pointing telescopes at the sky and saw something. In particular they saw double stars, where the plane of the stars was more or less parallel the line from earth to the star. This contradicts Newton because Newton said the speed of the light should be equal to the speed of the star plus that of light. Since the two stars were rotating arround each other in a Newtonian space one would expect the light to be totally mixed up by the time it reached Earth so that the star would appear as a steady, rather uninteresting, brightness. However, the rotation could be seen.

Scientist, therefore, concluded that light must be travelling at a constant speed regardless of the speed of the source. Since the only thing people had seen with this property was sound, and other waves, the conclusion was that light was not a particle, but rather was a wave traveling at a constant speed relative to an "ether" which was postulated to be fill all of space.

By the late 19th century this was taken as fact. Therefore, someone got the idea that it would be interesting to see how fast Earth was traveling through the "ether." Hence the famous Michelson-Morley experiment (which is the second observation).

Michelson and Morley designed an experiment which was to be a race. They made some equipment with two arms at 90 degrees to each other. Each arm was the same length with a mirror at the end. There was a beam splitter in the center, and a mechanism for seeing both beams coming back at the same time. The two beams were allowed to interfer with each other. Therefore, they could see very accurately which the relative speed of the two beams.

The theory was the since it was known that the beams actually traveled at the same speed any difference in appearent speed must be due to the speed of the Earth through the ether.

The discovered the Earth was the center of the Universe. That is, there was no motion. To eliminate the possibility that the Earth may have, in its very complex motion, stopped for the instant of the experiment, it was repeated at various times, with the Earth in various parts of its orbit. Still the Earth looked stopped relative to the ether.

Something was wrong. Michelson and Morley made the observation that this could happen if Newton was abandoned for a moment, and it was assumed that there was a distortion of time and space caused by the motion.

In particular, they saw their setup. If one assumed the speed of light was constant with respect to any frame of reference, then if their experiment was viewed from two frames of reference, one stopped and one travelling at speed v, then (if the length of the arm was d) to the stopped observer the time taken to complete the horizontal traversal must t0 = 2 * d / c. The moving observer would see the light going out (at the speed of light) and traversing the distance in time t1 = (2 * sqrt(d^2 + (v * t1 / 2)^2) / c. Therefore, c^2 * t1^2 / 4 = d^2 + v^2 * t1^2 / 4 or t1^2 ( c^2 - v^2) = 4 * d^2 = t0^2 * c^2. Or t1^2 (1 - (v/c)^2) = t0^2. Therefore, t1 = t0 / sqrt(1 - (v/c)^2). Looking the at the arm that is parallel the direction of travel the stopped observer would see the same thing, t0 = 2 * d0 / c. The moving observer would see t1 = 2 * d1 / c. Therefore, d0 = t0 * c / 2, and d1 = t1 * c / 2. Therefore, d1 = t0 / sqrt (1 - (v/c)^2) * c / 2 = (t0 * c / 2) / sqrt(1 - (v/c)^2) = d0 / sqrt(1 - (v/c)^2).

Hm, this is not the formulas I remember, since time slows down as speed increases, as everyone knows [just as everyone know light was a wave in the ether]. But there is no contradiction because the only difference is who is considered stopped, and who is moving, be relativity says I can do it either way.

At this point I am inclined to get a bit philosophical. If everything motion in the Universe were to stop for 10,000 years would you notice. I think there is no experiment that could be devised to test this. In fact, I would say this is really without meaning, that is, time is defined in terms of motion so to say everything stops is an oxymoron. Similarly, if everything in the universe were to suddenly double in size could you detect it. As above, I think this is an oxymoron, size is all relative. If the speed of everything were to double, would you notice it. Another oxymoron.

The fact is that motion gives the relationship between the time dimension and the physical dimensions, that and that alone. Perhaps c is constant not because it is due to some physical law that we may or may not be aware of, but because it is defined to be constant.

I think an understaning of this philosophical point is needed to understand Relativity.

Back to Michelson-Morley. In coming up with the equations I made an assumption. I assumed that the distance horizontal to the direction of motion was a constant. Why should I assume that? Suppose time is constant and the differences are caused by a change in the horizontal distance. In this case, d1 equal the horizontal distance as perceived by a moving observer, and d0 is the distance as perceived by a stopped observer. Then d1 = d0 * sqrt(1 - (v/c)^2), time is the same, and the distance along the line of travel is the same.

Now we visit the twin paradox. There are two people who are twins. A stays home, and B visits a nearby star. Therefore, according to Einstein, when the twins are reunited the traveling twin will be much younger than the twin that stayed home. Yet, since everything is relative, the twin that traveled may be the one we look at as being stopped, and the one that stayed home as traveling---that is, home went a long way away, then returned. In the second scenario the stay at home twin should be much younger. Which is true. (I have seen books that try to explain this away by saying it is the twin that moved relative to the fixed stars that is younger. A nuevo ether.)

I was thinking this morning when I put my hands together. The angle between them was about 100 degrees. Then I looked at this from the viewpoint of my left hand, that is, such that my left hand was perpendicular to my line of sight. My right hand looked very narrow. Similarly, when the right hand was perpendicular, my left hand looked very narrow. In "Reality" the hands are about the same size. This is similar to the twin paradox. Perhaps each twin sees the other as having time running slower, but when they get together they find out that their time is the same. In fact, perhaps they find that, while the traveling twin never went faster than the speed of light he might return much sooner than would be expected, even faster than light. Something to think about.