--9-Dec-2012, 9:30pm PST **Revised**

## So, what’s with the speed of light these days?

### The Problem

One of the beautiful things about scientific theories is that they all can be challenged at any time. There are no beliefs in science and so what might otherwise be blasphemy is at the heart of the scientific process. In fact, one of the main jobs of physicists is to challenge existing theories to find some fault with them that will then open up new and more accurate theories (not to mention funding). It’s a model building process of continuous refinement.

Recently I had a physics conversation with John Fry, of Fry’s Electronics, and who is also a co-founder of the American Institute of Mathematics. He pointed out to me that no one has ever measured the one-way speed of light. All the measurements of light speed, of any useful accuracy, have so far used a round trip light path, measuring the time it takes for light to go out, be reflected back and arrive at the place where it started.

If a one way measurement has not been done, that would mean that it’s still possible that
some sort of ether effect can make the two directions, out and back, have different light
speeds, which would yet ad up to the usual measurement of the speed of light. Presumably, one
way would be less than c (c is the universal symbol for the speed of light) and the other
would be greater than c so that the net effect is always c. This would seem to be a direct
challenge to Einstein’s special relativity theory. Further down, however, I’m going to rehash
a method of deriving special relativity without any reference to the speed of light. So there
is room to question whether different speeds of light would even change anything in Einstein’s
theory. This kind of light-less derivation has been done many times before^{1,2}, but
I like this particular quick and dirty derivation.

John also told me that this new light speed idea was about to be experimentally tested with a high accuracy measurement of light speed in a one way experiment, but I haven’t heard anything about results yet. And he could not tell me any details yet.

### Special Relativity Primer

Special relativity is special because it doesn’t include things like gravity or acceleration. It only considers what happens to physics when, say, two observers are moving relative to one another at a steady clip. In Einsteins modest quaint explanations he spoke of observers on moving trains and observers on land, while back in Galileo’s day, having no trains running about, Galileo talked of ships moving on steady waters and observers on the shore.

It turns out that Galileo’s statements about space are in fact all we need, to come up with special relativity without any reference to the speed of light. Special relativity is confusing and rather unintuitive because of apparent paradoxes concerning the rate at which time flows. If I’m on the ground looking at my clock and there’s another guy on the train whizzing by looking at his clock, a curious effect can be noticed, especially if the train is going really fast, like near the speed of light. I would see that fellow’s clock as running slower than mine. And likewise, he would observe that my clock is running slower than his! It sounds really strange, but such things can be measured these days. Maybe not for trains but for faster things like planets.

Another curious thing happens to the measurement of the length of things. If I’m on the ground looking at my yardstick, I see it as 36 inches long. If I somehow measure the yardstick belonging to the fellow on the train, I’ll measure it to be shorter the 36 inches. And If he could somehow measure the length of my yardstick, he would measure my yardstick as less than 36 inches. This is all while we’re still moving with respect to each other.

We would not agree either on lengths or times. And, the size of these disparities would depend on how fast we are moving relative to each other, the greater the relative speed, the greater the effects. Strange, but true.

Now here’s where this all comes flowing out of the mind of Galileo, though he didn’t know it, probably due to the lack of modern mathematical methods back then. It’s all based on three almost intuitive observations about the behavior of space and time.

Fist, speed is relative. It doesn’t matter if the guy on the train considers himself as moving or if he thinks I’m out here in the world and the whole world with me on it is whizzing by him while he’s standing still. Same with boats. More importantly, if the train were running really smoothly, there would be no experiment or measurement he could make to detect his own motion. He could look out the windows to see the world whizzing, but he could really never tell whether it’s him or the world that was “really” moving. In the whole universe, there are no objects absolutely at rest. There is only relative motion. Nothing can be considered to be truly at rest or truly moving.

Galileo made a clever bet with some sailors that some say was actually taken up. He asked the sailors what would happen if one of them, on a moving ship, dropped a cannonball from the mast onto the deck (aside from smashing the deck). Would the cannonball fall parallel to the mast and hit the deck directly below the point of release, or would the motion of the boat sort of slide the deck out from directly below the cannonball so that it would fall, say, away from the mast. The result was that Galileo won the bet and the ball falls with respect to the boat and mast, not with respect to a supposedly fixed shore. All the laws of physics on the boat look just the same as they do on shore, and it never matters whether you think of the boat moving or the shore moving.

So that’s the *principle of relativity* If the boat is sailing by me on the shore at
10 knots, and a sailor on the ship rolls a cannonball toward the bow at 5 knots, I will see
the speed of the cannonball as 10 + 5 = 15 knots. Makes sense right? Well in fact they don’t
quite add up like that because of the special relativity that will be derived below, but you
get the idea, they ad up somehow.

As for space, there seems to be *isotropy*,
which is a compact way of saying that space is the same in all directions. If I shoot a gun,
the bullet comes out at muzzle velocity no matter what direction I shoot in. Of course here on
Earth, there are up and down, gravity, and the north magnetic pole, and things like that, but
for special relativity, we’re out in deep space away from anything. All directions in space
are the same. This is a concept concerning space itself and nothing else.

We also have *homogeneity*, which simply means the space is the same everywhere you
go.

That’s basically all you need to know about special relativity. The rest is a lot of algebra, which I’ll get to below. Oh, one more thing. The way I predict what length the other guy’s yardstick will be to me, or what the rate of time passage is in his reference frame, is by using what is called the Lorentz transformation. It’s still just algebra.

### Römer’s Method

But just a quick historical note. The one way speed of light has been measure before, albeit crudely. The diagram at right is

Römer’s method of measuring the speed of light by timing the period of Jupiter’s moon Io from different points in the Earth’s orbit, one point receding from Jupiter and the other approaching Jupiter. Essentially it’s a doppler shift measurement, the kind the cops make of your automobile speed to see how fast you’re going.

While this is a one way measurement of light speed, it’s not accurate enough to see the tiny speed variations that John expects to be detecting. Although, with today’s astronomical instruments, who knows?

### Do We Need Light?

At any rate, I remember a lecture that my physics professor, Marvin Morris at San Jose State, gave 35 years ago in which he derived the necessity for a cosmic speed limit exactly as expressed by the constant c in the Lorentz transformation. Essentially it’s a derivation of the Lorentz transformation starting with just our, of rather Galileo’s, intuitive assumptions about the nature of space, assumptions that could conceivably still be off a bit.

Prof. Morris pointed out that this derivation was not original to him and he didn’t remember where it came from. At any rate, I’m going to present that derivation here in what follows, taking directly from Prof. Morris’ notes as closely as possible. If there are any errors or typos, they will be completely mine.

The point of this derivation of the Lorentz transformation is that it assumes only three things about the nature of space and transformations from one reference frame to another. Namely, the (classical) principle or relativity, and the isotropy and homogeneity of space.

Prof. Morris said that this was what Newton should have been working on all those years ago, since this derivation depends only on classical assumptions about space. Newton had also assumed, however, as everyone did back then, that time flowed in and of itself and always gave the same measure in all reference frames. That turned out to be famously not true because it was based only on intuition born of our necessarily imprecise observations of nature.

Of course, the other assumptions used in this derivation are just as vulnerable as they too are based on our intuition coming from general experience, as I’ve pointed out.

This derivation, then, does not make the time independence assumption. Thus, if we do derive the Lorentz transformation from these three assumptions and then, after John’s experimental results come in, the Lorentz transformation turns out to be wrong, then possibly one or more of our three cherished space intuitions must be somehow modified. Or maybe not.

### The Need for A Cosmic Speed Limit.

To start, the most general way to write the transformation would be

Where the transformation is from the proverbial unprimed frame, S, to the primed frame, S’, and can depend on almost anything, except maybe things like barometric pressure or the color of bananas.

Now, from the principle of relativity we must have the linear system

where the coefficients a_{ij} do not depend on the coordinates themselves
(x, y, z, t).

Also, says the un-named guru, without loss of generality, all the axes and origins of both the prime and unprimed frame can coincide at t = 0 = t’. The reference frames are coincident to start with at time zero. The primed frame is moving along the x axis at velocity v.

With this choice of reference frames, y’ is independent of t. Also, when t = 0, y’ is independent of x and y and is therefore always independent of x and y. So, we gotta have

And similarly for z’,

From the isotropy of space, we must have a_{13} = a_{12} and any
x dependence on y and z can be only a function of the distance from the origin,

which for non-trivial values of y and z means a_{12} = 0. So we now have

which so far looks encouraging. Now consider the inverse of the above, namely

The inverse transform must necessarily have the same form as the forward transform
(as v → -v) from isotropy and the fact that if I do an experiment in my reference frame,
then transform the results to another reference frame, and then use those results to
transform back to my own reference frame, I must end up with the same experiment! That
means that the coefficients of x’ and y’ in the equation for x must vanish. Which
means that a_{42} = a_{43} = 0 for the transform to be non-trivial.

Also, from isotropy a_{22}(v) = a_{22}(-v) (that is, the value of
a_{22} you would use in the inverse transform is the same one used in the forward
transform). And from the above inverse we also have

so that

When v = 0, a_{22} = 1, and also we have a_{33} = 1. And we
find the transform so far to be

Consider a point x’ = constant, we have then

so,

Now, the inversion with the y and z coefficients equal to zero and
a_{22} = a_{33} = 1 is

Consider a spherical cow - just kidding - consider a point x = constant. So

So

and by using the previous result we will find that a_{44} = a_{11}.
Now we have a transform that looks like

where we’ve also dropped the subscript “_{11}” on three of the
a_{11} coefficients. We are half way there. For the above matrix, the
determinant is

The inversion, for x is now

Remembering that the inverse transform (-v) must have the same form as the forward transform (v), and remembering the inverse transform above for x’, we get

Now, make a simultaneous measurement in the unprimed frame, S, of a length, like two ends of a stick, located at rest in the primed frame, S’

But t_{2} = t_{1} so that

From isotropy, the length of a stick does not depend on the sign of v, a(v) = a(-v). So

We now know everything except for a.

Write down the matrix for a transform from S → S’ at velocity v1, and multiply
by the matrix for a transform from S’ → S’’ at velocity v_{2}. The product is the
transform from S → S’’. Note in particular that the resulting matrix elements
a_{11} and a_{44} must be equal in all cases.

Since a_{11} = a_{44} in the resultant transform (S → S’’), we must have

Noting the previous expression above for a_{41} , and substituting in
the last expression, we get, after a bit of algebra

Now, this must be true for any arbitrary velocities v_{1} and v_{2} ,
and therefore for the expressions to be equal, they must be a constant.
Call this constant 1/c^{2} . After some algebra again, we get finally

So, there must be a constant velocity c , a maximum velocity, that depends only on our three starting postulates of time and space. At least, that is, if our intuitions about space and time are correct.

### Thots

Now why is it that the speed of light is this same as this constant c, notwithstanding the result of John’s experiment? Or maybe it isn't. Come to think of it, who has ever determined the value of c without reference to light? What would have to be done is to accurately measure the length and time changes to high precision at extremely high speeds. And let's say that light speed and c turn out to be the same. Would that be a coincidence? There is a similar result in mechanics where the inertial mass turns out to be, as closely as we can measure, the same as the gravitational mass. I don’t think it’s far fetched to say that these are not coincidences but rather point to a deeper yet undiscovered theory of space, time and mass. This is my intuition telling me that I don’t think John’s idea of different values for light speed will pan out. But, I’m certainly no expert.

1. Lights out on Einstein's relativity. By: Buchanan, Mark, New Scientist, 02624079, 11/1/2008, Vol. 199, Issue 2680

2. The Theory Of Relativity - Galileo’s Child, Mitchell J. Feigenbaum, The Rockefeller University, May 25, 2008, http://arxiv.org/abs/0806.1234v1