# Distinguishing special relativity from the strange world of classical mechanics

#### Strilanc

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:

$$t = t_0 \frac{\sqrt{1 - p^2 \sin^2 \theta}}{1 - p^2}$$

Which reduces to $$\frac{t_0}{1 - p^2}$$ when the displacement is along the velocity vector, and $$\frac{t_0}{\sqrt{1 - p^2}}$$ 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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#### PeterDonis

Mentor
I found that the predictions it makes are awfully similar to relativity
Huh? Plenty of them are very different. For example:

" If the stands surrounded the baseball field in a circle with a 300 foot radius, then we would witness the following: When a player struck the ball, if you were situated in the N end of the stands, then you would first hear the bat strike the ball and then much later (about 3.4 minutes) see the bat hit the ball."

This is very different from the prediction of relativity, which is that you would see the bat hit the ball virtually instantly (300 nanoseconds after it happened, since light travels at 1 foot per nanosecond), and would hear the bat hit the ball in about 1/3 of a second (since sound in air travels at about 900 feet per second), i.e., after you saw it.

These look an awful lot like the Lorentz factors in relativity
No, they look just different enough to make the difference between classical and relativistic physics clear. In relativity, both round trip times are predicted to be the same (and this is in fact what we see experimentally, as in the Michelson-Morley experiment).

The whole point of the article you linked to is to illustrate how different the predictions of classical and relativistic physics are. I suggest that you re-read it with this in mind.

#### PeterDonis

Mentor

Is special relativity just a really nice parametrization of this classical system with similar properties?
Certainly not!

What kinds of assumptions allow us to separate the two?
The main one is that this assumption of classical physics (from the article) is not valid:

"The measured velocity of light ( denoted c) would depend entirely on the observer’s velocity relative to the aether."

In special relativity, the measured velocity of light is always c. (More precisely, it's always c in any inertial frame, regardless of how that frame is moving relative to other inertial frames.)

#### Strilanc

I don't think I've managed to communicate the ambiguity. I'm saying that, if all of our notions of distance and time ultimately reduce to signals bouncing around, then the classical picture looks a lot more like the relativistic picture.

With that in mind, this:

If the stands surrounded the baseball field in a circle with a 300 foot radius, then we would witness the following: When a player struck the ball, if you were situated in the N end of the stands, then you would first hear the bat strike the ball and then much later (about 3.4 minutes) see the bat hit the ball.
Is not quite right. Later in the essay the author gets to the issue: the sound also can't propagate faster than light. It has to propagate from atom to atom, and those atoms interact via electromagnetism. So you will still see the batter hit the ball before you hear it happening, it's just that there's a large delay. And although someone at the other end of the stands has a tiny delay, the round trip time is still the same and the round trip time is what matters when it comes to cheering at a baseball game (e.g. in an ideal computer network, changing one-way latencies while maintaining round trip time is equivalent to skewing the initial clock synchronization).

More bothersomely, the computation happening in your brain is also affected by the signal delay shenanigans. You will literally think slower. An external observer can see that there's a huge round trip time between you yelling at the batter and them reacting and you seeing their reaction, but to you it just seems like the usual round trip time because it's exactly cancelled by your brain running slower. (I think. Alternatively, the asymmetric delays might just immediately kill you.)

The main one is that this assumption of classical physics (from the article) is not valid:

"The measured velocity of light ( denoted c) would depend entirely on the observer’s velocity relative to the aether."

In special relativity, the measured velocity of light is always c. (More precisely, it's always c in any inertial frame, regardless of how that frame is moving relative to other inertial frames.)
I think what I'm trying to get at is that you might get an observationally equivalent effect by taking into consideration the fact that the observer's measure of time and distance is "made out of round-trip signal delays". Essentially: a mechanism to time dilation and length contraction that's based on changing round-trip delays instead of on a geometric transformation (though frankly the geometric transformation is easier to work with and mathematically more beautiful).

#### PeterDonis

Mentor
if all of our notions of distance and time ultimately reduce to signals bouncing around, then the classical picture looks a lot more like the relativistic picture
No, it doesn't; that's not the point. The point is that, if all of our notions of distance and time ultimately reduce to signals bouncing around, the classical picture is inconsistent. It can't possibly be true.

the sound also can't propagate faster than light. It has to propagate from atom to atom, and those atoms interact via electromagnetism
This isn't an argument that the classical picture must say something different; it's an argument that the classical picture can't possibly be true, because it says inconsistent things. It says the sound will propagate faster than light in some frames, but it also says sound can't propagate faster than light. There's no way to resolve this inconsistency without abandoning classical physics.

you might get an observationally equivalent effect by taking into consideration the fact that the observer's measure of time and distance is "made out of round-trip signal delays". Essentially: a mechanism to time dilation and length contraction that's based on changing round-trip delays instead of on a geometric transformation
The word "instead" here is not correct. You are not describing two possible models, a classical one ("made out of round-trip signal delays") and a relativistic one (based on "a geometric transformation"). You are describing two different ways of looking at the relativistic model: as a model that consistently takes into account signal delays, or as a model that represents everything as spacetime geometry. There is no classical model "made out of round-trip signal delays" that is consistent. That's a key point of the article.

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#### Strilanc

I'm having difficulty seeing where we disagree except on the specifics of what "classical" means.

I think I'll write up a simulation of the classical picture I have in mind, try to build widgets in it that detect their speed, and report back what I find.

#### PeterDonis

Mentor
the specifics of what "classical" means
I'm using the meaning that was used in the article you linked to, which basically amounts to: physics based on the assumptions the article labels as the "classical" ones.

I'll write up a simulation of the classical picture I have in mind, try to build widgets in it that detect their speed, and report back what I find
What assumptions will you use? If they're the same as the ones given in the article, you won't be able to model them consistently; again, that's a key point the article makes. (If you disagree, where do you think the article goes wrong?)

#### Strilanc

What assumptions will you use? If they're the same as the ones given in the article, you won't be able to model them consistently; again, that's a key point the article makes. (If you disagree, where do you think the article goes wrong?)
I'm going to have a set of point-objects with fixed velocities that can broadcast a message and fire a handler when they received a broadcast message. Broadcasted messages expand circularly at a rate of c with respect to an aether. Handlers get fired as the message horizon passes over receiving objects. I'm going to give all the objects a speed less than c (again with respect to the aether).

From that base I'll try to make things like clocks and such. My expectation is that all interactions "X received message Y before message Z" will play out the same when I Lorentz transform the system.

I'll try to come up with some way for the objects to push each other, that doesn't immediately introduce the ability to detect speed. Preferably without wave functions...

I think the assumption the article makes that I'm violating is that forces and impacts and other interactions continue to propagate normally as you approach c. I want all the interactions to propagate at c with respect to the aether, so you can't really push someone past c: you just find that it's harder and harder to catch up with them and nudge them a bit more.

#### PeterDonis

Mentor
My expectation is that all interactions "X received message Y before message Z" will play out the same when I Lorentz transform the system.
Why are you Lorentz transforming if you're trying to make a "classical" model? In classical physics, the proper transformation is a Galilean transformation, not a Lorentz transformation.

#### Strilanc

Why are you Lorentz transforming if you're trying to make a "classical" model? In classical physics, the proper transformation is a Galilean transformation, not a Lorentz transformation.
I expect that the Galilean transform will preserve the interaction history if I apply it to everything including the aether, and the Lorentz transform will preserve the interaction history if I apply it to everything except the aether. I guess I'll see.

#### PeterDonis

Mentor
I expect that the Galilean transform will preserve the interaction history if I apply it to everything including the aether
A Galilean transformation leaves the time ordering of all events invariant, so yes, I agree with this.

the Lorentz transform will preserve the interaction history if I apply it to everything except the aether
No; a Lorentz transformation only preserves the time ordering of timelike and null separated events. It does not preserve the time ordering of spacelike separated events. So any "interaction history" that includes spacelike separated events will not have its time ordering preserved. If an "interaction history" just means the ordering of events as experienced by a single observer, and you adopt the relativistic assumption that observers can only travel on timelike worldlines, then a Lorentz transformation will preserve all interaction histories.

Note that I didn't say a single thing about the aether in the above. I'm not sure what an "interaction history of the aether" would even mean, but a valid Lorentz transformation will treat all events the way I described above, whether they are events belonging to the aether or not.

#### Strilanc

No; a Lorentz transformation only preserves the time ordering of timelike and null separated events. It does not preserve the time ordering of spacelike separated events. So any "interaction history" that includes spacelike separated events will not have its time ordering preserved. If an "interaction history" just means the ordering of events as experienced by a single observer, and you adopt the relativistic assumption that observers can only travel on timelike worldlines, then a Lorentz transformation will preserve all interaction histories.
Sorry, I was unclear. By interaction history I mean the partial ordering of "X saw Y before Z". I agree that the Lorentz transform can wreck any global ordering involving spacelike separated events.

#### s_luke52

"The measured velocity of light ( denoted c) would depend entirely on the observer’s velocity relative to the aether."
Exactly. Everything is relative to the state of the aether in which it exists, including the rate at which the clocks tick which are used to determine the speed of light. That is why the speed of light is always determined to be c.

When you take an atomic clock to the top of a mountain it ticks faster due to the change in the state of the aether in which it exists.

#### Nugatory

Mentor
Exactly. Everything is relative to the state of the aether in which it exists, including the rate at which the clocks tick which are used to determine the speed of light. That is why the speed of light is always determined to be c.

When you take an atomic clock to the top of a mountain it ticks faster due to the change in the state of the aether in which it exists.
It's been a long and twisty path, but it's taking us to Lorentz ether theory: https://www.physicsforums.com/threads/what-is-the-pfs-policy-on-lorentz-ether-theory-and-block-universe.772224/ [Broken]

Thread closed, but as always it can be reopened if someone makes a case by PM that there is more to say.

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