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To Mike From Somewhere Below.

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Posted by Richard D. Stafford, Ph.D. on July 29, 2002 14:51:52 UTC


I read a post of yours the other day which lead me to compose this document. At the moment, I can't find the post I quoted you from below but I am sure it is there somewhere. Excuse me for posting this at the top.

I think your understanding of Einstein's relativity is just a tad confused.

Quoting a small part of your post:

>>>" 'I say clocks measure what Einstein calls "proper time".' "
You wrote:
"See? Stafford is looking for the luminiferous ether."

I draw a blank here. You raced on ahead of me and possibly it is because you have learned the
standard answer very well. But I am still at square one on how those two connect, and I suspect that's not all bad. By proper time, I understand him to mean "agreed-upon frame" time.
There's nothing ethereal about that. It's rather legalistic, instead. And that's fine too, but it's not clear whether and what proper time is truly "done-to" by light-speed situations.

One gets the impression that you think the word "proper" in the phrase "proper time" is the ordinary adjective defined in the dictionary; it isn't, rather it has a very specific meaning (physicists, like lawyers, tend to mean very specific things by their terms. In any geometry there is a very specific way of defining distances (in fact, a geometry is almost totally defined by the way distance measures are defined). I say almost because I am not an expert on the defining of geometries. Perhaps Paul could add some intelligence on the matter. However, I am quite confident that a description of "how one defines distance" generally tells one what kind of a geometry one is dealing with.

The distance along a path in a geometry is given by integrating dl (an infinitesimal distance along any specified path). How one measures distance is defined by how one obtains dl from the infinitesimal changes in coordinates as the point of integration moves along the path (how dl is obtained from dx, dy, etc., the coordinate changes). The equation which defines that relationship is called the "metric" of the geometry. The "metric" of a Euclidean geometry is given by dl = square root of ( dx squared + dy squared + dz squared). It is, in fact, an expression of the validity of Pythagorean theorem in a Euclidean space.

When Maxwell's equations implied that the speed of light was a fixed constant (if Maxwell's equations were valid, the speed of light could be obtain by static measures in the laboratory without even bringing up the issue of motion - but that's another story for another day), people immediately realized that an accurate measure of the speed of light in every direction was a way of determining your absolute motion (under Newton's relativity - the old velocity addition thing). Michelson and Morley made an attempt to measure the velocity of the earth through the universe and got the answer that it was standing still! After a number of attempts, the final conclusion was that they were absolutely correct: that the idea of determining absolute motion was itself bogus - the speed of light was the same in any reference frame!

Now that idea completely blows the Newtonian relativity relations: velocity addition is WRONG! It was not long before some physicists figured out what the correct relativistic transformations had to be: Hendrick Antoon Lorentz was one of the first and it is because of his work that we still call the relativistic changes in apparent distances "Lorentz Contraction". See:

The correct relativistic transformation equations required to make the speed of light constant are easy to work out and require only ordinary algebra. All one need do is require that all the points on a sphere expanding at the speed of light in one frame transform explicitly into a sphere expanding at the speed of light in the moving coordinate system.

That algebra only requires 6 algebraic steps and can be written down on one sheet of paper. The algebraic steps are written down explicitly at

High school algebra is all one needs to know in order to follow that derivation.

The real question was not "what are the correct relativistic transformations?" but rather, "why should these be the correct relativistic transformations?" Newton had obtained his transformations directly from logic (and the assumption that the universe was properly represented with Euclidean geometry).

Two very important things come out of knowing the correct transformations: 1., Newton assumed all clocks could be set to agree which clearly must be false (t cannot be set equal to t') and 2., observers in frames moving with respect to one another will not agree with each others idea of simultaneity! Non-Euclidean geometry was a big issue in the late 1800's. Many very powerful mathematical techniques were worked out from exactly that perspective; however, today I find that very few physicists are even familiar with the ideas (Hamiltonian Mechanics for example). Einstein had been well trained in geometry and he was the one who realized that the relativistic transformations of Minkowski geometry gave exactly the required transformations to yield those transformations which provide a constant speed of light.

The metric of his representation of Minkowski space was dl = square root of [dx squared + dy squared + dz squared - (cdt) squared]. Minkowski had examined the consequences of an imaginary coordinate. The letter "i" is often used to represent the square root of (-1) and the imaginary coordinate in Einstein's geometry was taken to be "ict" (the "i" creates the all essential minus sign in the metric above). So Einstein's solution to the problem of "why these transformations were required" was they were required because the universe is properly represented by a Minkowski geometry, not a Euclidean geometry.

Though that was a very compelling reason, he had one very deep problem: the Minkowski geometry (together with a little logic) only provides the transformation between coordinate systems which are moving with respect to one another at a constant velocity (it can not be used to deduce the transformations when the coordinate systems are changing velocity (accelerating). That is exactly the reason it is called the "Special Theory of Relativity"; it is not a "General" solution to the problem. Newton's relativity (although wrong) was "General". Einstein eventually came up with a "General Theory of Relativity" which removed that shortcoming but that issue is to a great extent beside the point. Explaining General Relativity to a layman is not an easy thing.

The single most important part of "General Relativity" was the fact that it reduced gravity to a pseudo force: that is, Einstein had discovered a geometry which could explain gravity as a consequence of geometry. In order to understand the significance of that achievement, one needs to understand pseudo forces themselves. The issue is that forces cause acceleration (Newton's old F=ma). For a moment, let us digress to Newton's universe. Newton showed that rather simple phenomena could appear complex if you chose the wrong coordinate system (wrong from Newton's perspective). If you were in what Newton called an inertial coordinate system, free objects would follow straight lines. If you were in a "non-inertial" coordinate system, free objects would follow paths which were not straight implying they were undergoing acceleration: i.e., there would be an apparent force on the object (a force satisfying F=ma). Since these forces would vanish if you transformed the physics to an inertial coordinate system, physicists called them "pseudo" forces.

Now pseudo forces have one very unique characteristic: the force is always exactly proportional to the mass of the object. This must be so as the real cause of the apparent acceleration is the coordinate system and, since it has nothing at all to do with any characteristics of the object being observed, the mass of the object must vanish from the physics. Clearly if F=ma, the apparent Force being applied to the object must be proportional to the mass so that mass can be divided out.

Newton's General Relativity allowed us to calculate the pseudo forces due to many different types of coordinate transformations. Forces due to a simple accelerating coordinate system such as a rocket (apparent g's the astronaut feels); forces due to simple rotation around a point (centrifugal force) or the common coriolis force which controls winds due to storms and is due to the rotation of the earth. Of course, physicists were well aware of the fact that gravity had that unique characteristic that the force was always exactly proportional to the mass. For this reason, many searched very hard for a geometry which would make gravity a pseudo force. They all failed. In particular, a man named Maupertuis proved analytically that no such geometry existed. The proof essentially stopped investigation into the issue. The physics community conceded that gravity could not possibly be a pseudo force.

It turns out that Maupertuis had not considered the possibility of an imaginary coordinate. When Einstein looked at the issue from the perspective of his "space-time" he found that he was able to create a geometry which did yield gravity as a pseudo force. The fact that he was able to do this gave his perspective extremely powerful support. In fact, I have a text book which claims that Einstein's four dimensional "space-time" geometry is the only geometry which can make gravity a pseudo force. I would like to point out that a short time ago (historically speaking) Maupertuis proved there were none and now the physics community says that there is only one (at least I haven't heard anyone say that they have proved it). I have not heard of anyone investigating the issue.

Now that I have essentially pointed out the central issues of Einstein's relativity, let's go back to some subtle aspects of his theory. First, it is clear from the experimental evidence that there exists no physical phenomena which can lead to a determination of any "absolute" motion. So Einstein's theory is based on the idea that there exists no "preferred" coordinate system. Now that idea is, on the surface, ridiculous! All physicists use a very specific "preferred" coordinate system all the time! The "preferred" coordinate system is the coordinate system in which they or their laboratory are at rest (on occasion they choose a coordinate system at rest with respect to some given object or entity). One should not say there is no "preferred" coordinate system but rather that the "preferred" coordinate system is a function of the problem they are trying to solve.

The issue here is that they want to state the physical laws in a manner such that they appear the same in any coordinate system. The term used to state that proposition is "covariant". They are interested in stating the laws in a form which varies in exactly the same way that the coordinate system varies: pick any coordinate system you wish and the laws can be stated in exactly the same form in that coordinate system as they are in any other. What is significant here is that one is not talking about "any" coordinate system. One is talking about a coordinate system which is moving in any manner relative to some other coordinate system. The fundamental nature (the metric) of the two coordinate frames is assumed to be identical.

This brings us back to that issue, "is the Minkowski type geometry the only geometry which will naturally yield the correct relativistic transformations?" Or, actually even more important, is the Minkowski metric essential to the problem of making gravity a pseudo force? It turns out that the answer to both those questions is NO! I personally have discovered another geometry which yields exactly those same results. Once one sees that alternated geometry, it is quite evident why that geometry has been overlooked for over a hundred years: it has been overlooked because everyone is totally brainwashed to think that clocks measure time.

So let us return to that Minkowski metric and examine it very carefully. This gets me back to the issue of "proper" time. The metric of our Minkowski geometry tells us how distances are measured in that geometry. Einstein calls dl (obtained from dl = square root of [dx squared + dy squared + dz squared - (cdt) squared]) the invariant interval. He calls it the "invariant" interval because the value of dl is not a function of the motion of the coordinate system: i.e., dl obtained from a path written in terms of x,y,z, and t is exactly the same as dl obtained from x',y',z' and t'. It is a very important "covariant" variable.

Now Einstein's space is a complex space containing both real and imaginary coordinates. From that it should be obvious that, depending on the details of the path being specified, dl may be either real or imaginary. Look at the metric: if dt is zero, dl must be real and if dx,dy and dz are zero, dl must be imaginary. Since dl is a covariant variable dependent only on the specified path (i.e., it has the same value independent of the observer's preferred coordinate system) the fact that it is real or imaginary divides all paths into two different types: they are either space like paths (dl is real) or they are time like paths (dl is imaginary).

Which case you are dealing with depends on the specifics of the problem you are trying to solve. If one is working with "time like" paths, it is the standard convention in physics to divide the invariant interval by ic and work with the variable usually called tau. Since all we have done is divide the invariant interval by a constant (which is the same in all coordinate systems - the speed of light is "c" in all coordinate systems) the resultant tau is just as covariant as the invariant interval (in fact it is actually the same thing). The conventional name used to reference that variable, tau, is "proper time".

As I have said elsewhere, almost any entity which is of interest to scientists may be regarded as a clock. That is to say, in most interesting entities, there is some behavior associated with that entity which changes in time. Now, clearly, if the laws of physics are covariant (independent of the coordinate system) then the description of that behavior must be describable in a form which is covariant (independent of the coordinate system). Furthermore, any entity which can be used as a clock measures time in its own rest frame (I think that one is by your definition guys). But what is its rest frame? Isn't that the frame attached to the path where dx, dy, and dz all turn out to be zero? Along that path, d(tau) is exactly dt as measured by the clock. Since tau is a covariant variable, it follows as the night the day that all clocks measure exactly the covariant variable tau completely independent of the coordinate system! And tau is not time! Tau is equal to time only in the rest frame of the clock.

You can look at this from another perspective. Consider a standard digital clock as seen by several different observers moving with respect to one another. The reading on the clock passes through a number of progressive readings 12:00, 12:01,12:02, ... etc. Now each observer may attach different times to those readings but they can all log down the readings! The readings themselves are a direct function of the covariant laws of physics. Again, leading once more to the conclusion that clocks do not measure time.

So my complaint with Einstein's geometry is very simple: he uses time as one of his coordinates when what clocks measure is trajectory path lengths in his geometry. That simply strikes me as a rather foolish thing to do; but of course, that is no more than an opinion and it clearly holds no water with the scientific community.

But there are a couple of other issues related to that that I think you guys ought to think about. First, when we go to measure an object, we need to bring our ruler up to the object and align the zero on the ruler to one end and then take the reading at the other end (we don't generally like to have our ruler very far away from the object being measured - we would prefer to have our ruler in direct contact with that object) . If the object is moving, it is very important that we take these two readings simultaneously. Now right there we have a problem. If we compare our results with an observer of exactly that procedure who is in a different coordinate system moving with respect to us, he will say we did not take our two measurements simultaneously. So clearly, measurements made with the ruler are not covariant. Notice that, when it comes to measuring a moving object, anyone who is rational will hold their ruler against the moving object and take both readings (usually at different times) essentially in the frame of reference of the object being measured. If one does that, then that measure is also covariant!

Now, let us instead look at some dynamic phenomena which takes place. We want to measure the time between two different states in some object. Again, if we wish to make this measurement accurately, we should try our best to keep our measuring device in contact with the entity we are measuring. So we take our clock (our reference device) and set it to zero when the first state occurs. Keeping it in contact with the object being measured, we should take the reading on the clock the second state occurs. Our observer described above will agree 100% with the consequence of our measurement. The difference between the distance measure above and the time measure discussed here is the fact that we can keep our ruler in contact with a moving object during the measurement because (in our frame) the two readings take place simultaneously. However, we cannot keep our clock in contact with a moving object without changing it's position.

That should stir up enough controversy. I have already written beyond the attention span of most of you.

Have fun -- Dick

PS - I am not looking for the "luminiferous ether."

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