# How can be proved that one-way speed of light is equal to two-way speed?

Using Special relativity theory, of course. Can Lorentz transformations to "tell" something about it? "Wikipedia's" article: http://en.wikipedia.org/wiki/One-way_speed_of_light .

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How about rotating by 180 degrees? I don't understand what could possibly be different in the 2-way speed, and the term "two way speed" is strange. So what is confusing exactly? –  Ron Maimon Jun 1 '12 at 19:32
@Ron The proposal is that the speed of light might be different on the outbound leg than on the return leg: say $c(\theta) = c/(1 + \epsilon \cos\theta)$. Obviously that is inconsistent with the principle of relativity and a Noetherian origin for the conservation of angular momentum, but it is hard to test experimentally. –  dmckee Jun 1 '12 at 21:43
For the fun of it: Measure period of revolution of a photon at Schwarzshild radius around a black hole –  troyaner Jun 2 '12 at 19:37
@troyaner: 1.5 times the Schwarzschild radius is the circle orbit position. –  Ron Maimon Jun 2 '12 at 20:05
@RonMaimon: sry, correct –  troyaner Jun 6 '12 at 13:53

The big misunderstanding about this "one-way speed of light" thing is that there is something to be proven experimentally. While it is just a matter of convention about

how to synchronize the clocks at the source and the detector

Let me explain with an example. Consider a following experimental setup:

• We have three observers $O$,$A$ and $B$ at rest in our reference frame.
Their positions are $x_O=0, x_A=-1$ and $x_B=1$.
• First they synchronize their clocks following the standard procedure:
1. The $O$ observer sends a signal at $t_O=-2$ both to $A$ and $B$.
2. At the moment of the signal arrival, $A$ and $B$ set their clocks to $t_{A,B}=-1$
3. $A$ and $B$ return those signals and $O$ receives them back at $t_O=0$
4. Clocks are synchronized.
• Now this guys measure something. For example there are two light flashes -- red and green, passing by. And observers are fixing the times of their passage.

Here is the space-time diagram for the process with the "standard" speed of light:

Such a diagram is drawn by the observers when they gather together at a meeting after the experiment was performed. Every observer has a list of events that locally happened at his place. And altogether they recreate "the big picture" of the experiment.

But now the observers have some doubts -- why do they presuppose that the speed of light is the same in both directions? So they agreed to change that and see what happens. Say that right-speed is three times as fast as the left-speed:

The description has changed. But it is just a change in the coordinates. It couldn't possibly affect any experimental predictions by any theory that observers have.

So it is just a convention. And there is nothing to be "proven" experimentally.

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This is true, it is part of the issue regarding this, but the question was to do it experimentally, to determine if the back and forth motion in the interferometer frame is equal. You don't need two frames to check this, and you can determine this using Fizeau methods, or simply using the fact that frequency isn't changed upon reflection. –  Ron Maimon Jun 3 '12 at 18:44
@RonMaimon Well, my whole point is that there is nothing to "do experimentally". It is like asking to "experimentally prove" that there are 60 seconds in a minute. Just a convention... Regarding Fizeau experiments -- I'm pretty sure that those can be reduced to analysis of events in space-time, right? Then my argument holds. –  Kostya Jun 3 '12 at 19:54
I see your point--- the Einstein synchronization makes it obviously a coordinate issue. I was imagining adiabatic synchronization, synchronize two clocks when they are close, then slowly move one clock away to the other side of the room (in relativity, this is the same as Einstein synchronization, but we aren't granting relativity, since this is the whole point). Now ask, relative to adiabatically synchronized clocks, does the light travel with the same speeds back and forth. The methods I gave answer this question. I agree if you define synchronized Einstein's way, it's trivial. –  Ron Maimon Jun 4 '12 at 5:42
+1, BTW, yours is the best answer. I took the problem experimentally seriously, I didn't interpret it as a question about coordinate systems. This is probably the best interpretation, however. –  Ron Maimon Jun 4 '12 at 5:43

### Revision: use a stationary and moving dielectric

My first answer is wrong--- you can easily imagine a model in which light components are independently cancelling without being equal. The best way is by a version of Fizeau's experiment: put a dielectric along the ongoing leg of an interferometer and not the return leg, then shift the same dielectric to the return leg while at the same time removing it from the outgoing leg (you can do this instantly by a small motion of the two beams). You can do this easily with a diamond shaped dielectric, where the back and forth directions cut on opposite tapering sides. Then you slide the diamond back and forth, removing a portion from one leg and adding the same portion to the other, and look for fringes. You won't find any.

If you believe there are mysterious fitzgerald contractions making this experiment work out, you can do Fizeau's experiment--- make the interior of the dielectric a fluid moving in a given direction. If your model is consistent with both dielectrics and Fizeau, at all seasons of the year, then you will be forced to the relativity transformation for the moving frame.

I should point out that nothing prevents you from taking the time in the rest frame of the sun (say) to be the true global time, and then doing boosts by Galilean transformations. This won't preserve the metric, but it will give different laws according to the Galilean transformed metric. In these coordinates, the light travels at different coordinate speeds back and forth, and the cancellations in the two experiments above look like miracle.

### Original experiment that doesn't work

This answer is updated. Originally, I said you should make an equilateral triangle of mirrors in a Michaelson interfermeter, and check for interference fringes as you rotate the apparatus slowly. I assumed that the "back and forth" speed can't cancel on back and forth paths and triangle paths simultaneously, but this is wrong--- in the Galilean boost model, the different components parallel to the velocity do the cancelling. But in more complicated models, you could get an effect here too.

This idea is sort of silly--- the light would have to be conspiratorial for something like this to work. You didn't give a definite model, if you did, it would be easier to rule it out.

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A nice animation about Michelson and Morley's Experiment can be found on: http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/mmexpt6.htm

It clearly shows that the Aether doesn't influence the speed of light if you rotate the whole experiment about 90°. In fact Aether doesn't even exist.

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It can not be proved because they are not equal. It happens that the mean value of light speed, measured by any observer (a body in motion) is always equal to c .

theory that a inquisitive mind should consider reading:
Cosmological Principle and Relativity - Part I
abstract : The absence of an identified consequence at solar system scale of the cosmological space expansion is usually explained considering that space expansion does not affect local anysotropies in matter distribution. This can also be explained considering a time dependent scenario compatible with Relativity Principle, therefore supporting physical laws independent of the time position of the observer. A theory considering such relativist scenario, i.e., generalizing Relativity Principle to position, embodies Cosmological Principle and can be intrinsically able to fit directly both local and cosmic data. In part I it is presented the general framework of such a theory, called Local Relativity (LR), and analysed the space-time structure. Special Relativity space-time is obtained, with no formal conflict with Einstein analysis, but fully solving apparent paradoxes and conceptual difficulties, including the simultaneity concept and the long discussed Sagnac effect. In part II, LR is applied to positional analysis. It is verified the accordance with solar system measurements and with classic cosmic tests, without dark matter or dark energy. Two of the new features obtained in part II are the possibility of a planetary orbital evolution compatible with a null determination for G variation, supporting a warmer scenario for earth (and Mars) past climate, and the possibility of an accelerating component in earth rotation, compatible with the most recent measurements.

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