#### Strilanc

Science Advisor

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I was recently reading the strange world of classical mechanics. It prompted me to calculate some round trip times for things moving near the speed of light (classically, with an aether). I found that the predictions it makes are awfully similar to relativity, and I can't think of an experiment to distinguish them.

I figured out that, if two objects moving at a fraction p of the speed of light c, and they are displaced by a distance such that at rest a signal would take time t_0 to make a round trip, and the angle of their displacement w.r.t. their velocity is theta, then the round-trip time t when moving is:

[tex]t = t_0 \frac{\sqrt{1 - p^2 \sin^2 \theta}}{1 - p^2}[/tex]

Which reduces to [tex]\frac{t_0}{1 - p^2}[/tex] when the displacement is along the velocity vector, and [tex]\frac{t_0}{\sqrt{1 - p^2}}[/tex] when the displacement is perpendicular to the velocity vector.

These look an awful lot like the Lorentz factors in relativity, except the time dilation and length contraction are getting mixed together instead of being nicely separated.

Now my first instinct for how to experimentally separate this classical system from relativity was basically just to have three objects arranged in an L and bounce signals back and forth. Basically an interferometer. If the interferometer is moving, the signal will propagate at different speeds along each leg (whereas relativity has them moving at equal speeds). The problem is that, in order to setup and hold the legs at a fixed distance while you accelerate, you need to be bouncing signals back and forth and that *also* gets affected and I'm not sure if it all cancels out or not.

A similar test would be to place four objects in a square and accelerate them through another square of the same size. If the objects were dynamically getting closer whenever the round trip time increased, and further when it decreased (i.e. they try to stay some fixed time t apart) then the accelerating square would presumably shrink and pass within the other instead of around it. But then how are the objects measuring time? Does that get skewed in a way that cancels it all out?

I guess what I'm asking is:

- Is

- What kinds of assumptions allow us to separate the two? For example, a clock not based on signal propagation times would probably allow it. The ability to send a directional signal instead of a broadcast might also do it, since at speed you would "miss".

I figured out that, if two objects moving at a fraction p of the speed of light c, and they are displaced by a distance such that at rest a signal would take time t_0 to make a round trip, and the angle of their displacement w.r.t. their velocity is theta, then the round-trip time t when moving is:

[tex]t = t_0 \frac{\sqrt{1 - p^2 \sin^2 \theta}}{1 - p^2}[/tex]

Which reduces to [tex]\frac{t_0}{1 - p^2}[/tex] when the displacement is along the velocity vector, and [tex]\frac{t_0}{\sqrt{1 - p^2}}[/tex] when the displacement is perpendicular to the velocity vector.

These look an awful lot like the Lorentz factors in relativity, except the time dilation and length contraction are getting mixed together instead of being nicely separated.

Now my first instinct for how to experimentally separate this classical system from relativity was basically just to have three objects arranged in an L and bounce signals back and forth. Basically an interferometer. If the interferometer is moving, the signal will propagate at different speeds along each leg (whereas relativity has them moving at equal speeds). The problem is that, in order to setup and hold the legs at a fixed distance while you accelerate, you need to be bouncing signals back and forth and that *also* gets affected and I'm not sure if it all cancels out or not.

A similar test would be to place four objects in a square and accelerate them through another square of the same size. If the objects were dynamically getting closer whenever the round trip time increased, and further when it decreased (i.e. they try to stay some fixed time t apart) then the accelerating square would presumably shrink and pass within the other instead of around it. But then how are the objects measuring time? Does that get skewed in a way that cancels it all out?

I guess what I'm asking is:

- Is

*special*relativity just a really nice parametrization of this classical system with similar properties?- What kinds of assumptions allow us to separate the two? For example, a clock not based on signal propagation times would probably allow it. The ability to send a directional signal instead of a broadcast might also do it, since at speed you would "miss".

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