The Brass Tacks of the Ehrenfest Paradox
Coordinate Transforms for Rotating Reference Frames
Back Story
Wikipedia explains the conundrum in the Ehrenfest paradox thusly.
Imagine a disk of radius R rotating with constant angular velocity . The reference frame is fixed to the stationary centre of the disk. Then the magnitude of the relative velocity of any point in the circumference of the disk is . So the circumference will undergo Lorentz contraction by a factor of .
However, since the radius is perpendicular to the direction of motion, it will not undergo any contraction. So
This is paradoxical, since in accordance with Euclidean geometry, it should be exactly equal to π.
(Image credit: By Geek3 - Own work, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=24038062)
Conventional Wisdom
I won't bore you with the details, but Wikipedia goes on to explain why the general consensus is, the value of 𝜋 is subjective (i.e. observer-dependent.) In other words, the geometry of a merry-go-round as perceived by a passenger is non-Euclidean. To be safe, you should make it a habit of bringing your mathematician when you visit a playground.
The Brass Tacks
The folly in Ehrenfest's argument is, he presumes that the so called "rest" length of an object in a moving reference frame is always the same as its length when it's at rest in a bystander's reference frame. That's only true if the front and back launch (or land) simultaneously in the former context. To quote John Bell when he was trying to educate CERN about his spaceship paradox, when the front and back launch simultaneously in the bystander’s reference frame, each "will have at every moment the same velocity, and so remain displaced one from the other by a fixed distance." The geniuses at CERN (Einstein included) shouted him down and opted for invariant rest length.
If you adopt John Bell's analysis, rest length in the traveller's reference frame depends on the context in which the launch events for the front and back were simultaneous. Here's the Lorentz transformation when the events are simultaneous in the bystander's frame:
You can certainly orchestrate events to conserve rest length, but not unless you know the final cruising speed:
In general, rest length in the traveller's frame is a function of the front/back launch delay in the bystander's frame as well as the final cruising speed:
Formula: (1 - vt) / sqrt(1 - v^2)
To conserve rest length and avoid mechanical stress, you have to launch each point along the object's length progressively in infinitesimally small steps. The bystander will perceive elongation in that case, but the traveller will perceive no change in length and therefore no stress. The bystander perceives no shrinkage (or elongation) if every point launches simultaneously in that context, but the object will be stretched out from its own point of view.
The same is true of the rotating disk. According to a bystander, every point on the circumference launches simultaneously so (as John points out) there can be no shrinkage from that point of view. (We presume the disk is perfectly rigid so it is not distorted by the applied torque.) That being said, the view from the edge is quite different.
Clocks that are attached to a spinning disk are stationary with respect to one another, but the rate at which they run depends on their location with respect to the centre of rotation. The Sagnac experiment confirms that effect and it is entirely consistent with special relativity (regardless of how you define "rest" length.) As such, a laser range finder attached to the disk will report the centre-to-edge distance to be independent of the speed of rotation, but not the edge-to-centre distance. In fact, the latter goes to zero as the tangential velocity approaches light speed. The question is, do we take those results at face value or do we compensate for time dilation to make the radius invariant?
Well, put yourself in the range finder's shoes. It doesn't know that its clock is running slow. As far as it can tell from its vantage point on the edge, the disk gets smaller as it spins up. The rest of the universe wobbles to and fro, but on average, it closes in, too. You can interpret that as subjective 𝜋, but the net effect is, everything is getting closer. For all intents and purposes, radial distances are indeed shrinking.
OK. So let's go fishing with a regular tape measure. We'll stand in the middle and spool it out to the edge to get a baseline measurement. Then we'll stand on the edge and spool it out to the middle. Do we get the same result? Of course. The tape is subject to the same length contraction as the disk and the rest of the universe. The graduations are closer together from that point of view, but a length-contracted yard is still three length-contracted feet. And that's the thing. Does anything actually change if the entire universe shrinks or expands? Well, yes in fact. Let's not forget about Doppler.
The anomalous result from the range finder is an illusion of sorts, but it's a stubbornly persistent one because it has real-world consequences in terms of Doppler shift. As a passenger on the disk, the colour of the laser light, the tape, and the rest of the universe depends on your point of view. However you choose to reckon distances, you'll still have to adjust the tuner on your radio when you move to a different location.
As for non-Euclidean manifolds, if you break the orbit down into a series of small linear segments, the perceived lengths of all the previous segments change whenever the traveller changes heading. The orbit is therefore compressed in the direction of motion at any given time so it’s always elliptical from the satellite’s point of view. But the ellipse rotates with the satellite so the distance to the centre of rotation is invariant. However, the time dilation effect is cumulative so the orbital period is correspondingly smaller.
Again, the push back for this kind of explanation stems from the “informal and non-systematic survey of opinion at CERN”, when John Bell was trying to talk sense about his spaceship paradox. The upshot is, travellers are supposed to shrink from the bystander’s perspective while they are getting underway. It’s a stubbornly persistent delusion.
Having said that, the perceived distance to the centre of rotation is not exactly as illustrated. Ponder this geometry for example:
All of special relativity is based on the false assumption that measurements of intervals in a relativistically moving frame is the same process as measurements in a co-moving (relatively stationary) frame. I call this the WYSIWYG protocol. It is only valid for Newtonian frames of reference, because, like the rest of Newtonian physics, it is only a low speed approximation. The relativistic version is simply the vector dot product, which does reduce to WYSIWYG at very low velocity. However, it is not merely a mathematical convenience that relative velocity can be specified in terms of a rotation angle, v = c sin(angle). This is quite literally the angle between two inertial spacetime frames that are otherwise supposed to be parallel. It is not a coincidence that the Lorentz Transformation is a hyperbolic rotation. The angle that specifies velocity is the gudermannian of the hyperbolic rotation angle. The gudermannian is the included angle between two arbitrary vectors, like a reference in one spacetime plane and an interval in a rotated spacetime plane resulting from relative velocity. No experiment will observe this rotation, because the observable dimensions are all parallel. Point is, this IS the included angle, and the dot product is the product of the magnitudes of two vectors with the cosine of the included angle. From simple geometry, units in the rotated (i.e. moving) frame can only project the cosine fraction of themselves back onto the reference frame. Then ct' = ct cos(angle) and r' = r cos(angle). The vector dot product is commutative, so it doesn't matter which frame is selected as the reference. Continuing with our choice, the two equations are equivalent to ct = ct' sec(angle) and r = r' sec(angle).
ReplyDeleteThis tells us that the stationary observer can only measure real, cosine projections. If the target is in a slow-moving frame, the included angle is essentially zero, and all measurements are 100% of the magnitude of the interval. This applies to hundreds of years of Newtonian physics, as it was all relatively slow. But at relativistic speeds, it tells us to NEVER expect any more than the cosine projection, which literally vanishes at lightspeed. More to the point, it tells us our measurements will be shrunken, because we are incorrectly expecting them to remain constant. The vector dot product tells us that we should always expect cosine projections, appropriate for the relative velocity. Given that v/c = sin(angle), we can express the Lorentz factor in terms of the same angle that defines relative velocity. In this case, γ = sec(angle). So our equations become ct = γct' and r = γr'. These are the Einstein equations of time dilation and length contraction. But NOTHING SHRANK. These are the geometric projections of invariant magnitude vectors. Shadow projections do not have a single unique value, as they are different for every rotation angle.
And that is Einstein's colossal blunder. The assumption that the WYSIWYG protocol of Newtonian physics is valid at relativistic speeds when nothing else of Newton's mechanics is valid at those speeds was the first mistake. Then instead of applying logic, he invented the fantasies of time dilation and length contraction to obtain the results of Lorentz Transformation. All the experiments ever performed confirm the Lorentz Transformation, a hyperbolic rotation, and none require the fictions of the Einstein Interpretation. In other words, what Einstein proved was that if you start with a false assumption, you can still reach a true conclusion, if your argument includes false assertions that cancel out the false assumption. The fictitious properties of the Einstein Interpretation are nothing more than the ordinary properties of geometry. With the understanding that the measurement rule is the vector dot product, there is no further need for special relativity. You may continue to call it relativity because that describes its behavior. But there will be none of the Einstein Interpretation left in it.
Gudermannian. That's a term you don't hear everyday! In any case, I'm not sure what your point is. The Lorentz transformation can certainly be formulated in terms of a hyperbolic angle, but that's just a mathematical convenience. The observables are still time and space. And that's what Einstein was on about - time and space are subjective.
DeleteThis comment has been removed by a blog administrator.
ReplyDeleteDuplicate
Delete