# I Accelerated Triangle and Length Contraction

#### Foppe Hoekstra

Neil does not see his triangle as length contracted. He sees that the ground is stretched and does not see it as length contracted. Thus the triangle is shorter than the dots.

Note that any explanation for why the ground is stretched will involve a global description of the planet. Neil will have to adopt a non-inertial frame or else regard most of the planet as moving - either way, he can't use simple inertial frame results to do it.
I still don't see why Neil "sees that the ground is stretched". Neither do I see why Neil should adopt a non-inertial frame. The planet can move as it likes, but as long as Neil stays at one point on this planet, to Neil every (relevant) thing is as inertial as can be and, moreover, as motionless as can be. So no contraction, no time dilation and no stretching.

To Michael however, things are indeed non-inertial, as the planet does not take up linear speed, but rotational speed. Some people will probably say that SRT is therefor not applicable here, but that would make the whole Ehrenfest-paradox disputable.

#### PeterDonis

Mentor
I still don't see why Neil "sees that the ground is stretched".
Because it is impossible to rigidly spin up a body from non-rotating to rotating. The Herglotz-Noether theorem forbids it. So the shape of the ground must change during the spin-up process. If we assume that the ground (i.e., the planet) started out unstressed when it wasn't rotating, then it will be under stress when rotating, and the stress will be due to the change in shape during the spin-up process. The reason Neil still sees the dots matching the triangle (if indeed he does--see my next post) is that the triangle is assumed to undergo the same stretching process that the ground does.

Neither do I see why Neil should adopt a non-inertial frame.
Neil can adopt any frame he likes. What he cannot do is treat himself as being at rest in an inertial frame, because he isn't; he has nonzero proper acceleration.

To Michael however, things are indeed non-inertial, as the planet does not take up linear speed, but rotational speed.
The planet's rotation has nothing whatever to do with whether things are "non-inertial" to Michael; like Neil, Michael can use any frame he likes. Michael, however, unlike Neil, is at rest in an inertial frame (at least, if we ignore the planet's gravity).

Some people will probably say that SRT is therefor not applicable here
It is if we ignore the planet's gravity.

#### PeterDonis

Mentor
The east-west-leg of the loose isosceles triangle however, is not subjected to Ehrenfest
If this is the case, then the triangle will not match the dots for Neil either, because the ground will be stretched during the spin-up process but the triangle will not.

#### PeterDonis

Mentor
Both observers are making a statement about where the vertices of the triangle are "at the same time". But the observers are moving relative to one another so because of relativity of simutaneity they are not describing one and the same situation.
I actually don't think relativity of simultaneity is relevant here. The entire setup is stationary (once the spin-up process is complete), so the "distances between worldlines" (heuristically speaking) don't change with time. The numerical values of those "distances between worldlines" will be frame-dependent, but the fact that, for example, the worldlines of the dots do or don't match up exactly with the worldlines of the triangle's vertices, will not.

#### Ibix

Science Advisor
I still don't see why Neil "sees that the ground is stretched".
Because it actually is. This isn't a coordinate effect. It's easy to see why it must be stretched working from Michael's perspective: imagine a ruler nailed to the ground by a single nail in its middle. From Michael's perspective it has a high linear speed, so is length contracted. Now imagine a line of rulers nailed to the ground in the same manner so that, when the planet is at rest, each ruler just touches the next one and they go all the way around the equator. When the planet is spun up, the rulers are length contracted. There are only two options: either they still touch their neighbours or they don't. If they touch, then the circumference of the planet must have decreased - but that would imply that the planet's radius has decreased. If they do not touch, then the planet's surface must have stretched so that the stretched-and-length-contracted length is longer than the sum of the length-contracted lengths of the rulers.. Assuming that the planet's radius remained constant, the stretch is the only possibility.

Whether the rulers touch or not is frame independent - everybody must agree on it. So Neil must also agree that the planet's surface has stretched. His explanation, though, is a lot more complicated unless he just adopts Michaels' frame.
Neither do I see why Neil should adopt a non-inertial frame. The planet can move as it likes, but as long as Neil stays at one point on this planet, to Neil every (relevant) thing is as inertial as can be and, moreover, as motionless as can be.
Neil will be feeling some impressive centrifugal forces, so no, he'll be fully aware that he's not moving inertially. He may adopt an inertial frame in which he is instantaneously at rest - but the vast bulk of the planet is in high-velocity motion in this frame, and under high acceleration. And when you try to explain why the planet is stretched, you cannot neglect the rest of the planet, since it's its bulk behaviour that's important here.
To Michael however, things are indeed non-inertial
Whether or not something is moving inertially is an invariant fact - you just strap an accelerometer to it and look to see if the needle is at zero or not. You seem to me to be confusing "at rest in my (possibly non-inertial) frame" with "moving inertially". These are two very different things, in general.
Some people will probably say that SRT is therefor not applicable here
Some people would say that, but either they were writing in the 1920s or they haven't updated their terminology since then. Special relativity applies to all situations where gravity is not significant. Doing special relativity in non-inertial coordinates is complicated and needs a lot more complicated mathematical tools that get further developed into general relativity - but it is still special relativity.

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#### Ibix

Science Advisor
The reason Neil still sees the dots matching the triangle is that the triangle is assumed to undergo the same stretching process that the ground does.
This isn't my understanding of the scenario. I think the triangle template is carried by Neil and is unstressed - it's just a set-square used to draw the dots and is not attached to the planet's surface. The dots, of course are attached to the surface and are stressed. Thus from Neil's perspective the triangle template is as it always was and the triangle of dots is stretched. From Michael's perspective, the triangle template is length contracted, while the triangle of dots is length contracted and stretched in such a way that the measured length equals the rest length.

I'm assuming that the radius of the planet does not change.

#### PeterDonis

Mentor
This isn't my understanding of the scenario
As the scenario is presented in the OP, it is inconsistent. The OP claims that Neil will see the triangle still match the dots, which implies that the triangle must stretch. But it also claims that Michael will see the triangle length contracted, which implies that the triangle does not stretch.

Either choice would be consistent by itself; but only one of the two can be true of a single scenario.

#### metastable

Can I say the triangle became stretched due to the normal force when it was rested on the ground? How do we calculate the reduction in normal force on the triangle from centrifugal forces once the planet is up to speed?

#### PeterDonis

Mentor
Can I say the triangle became stretched due to the normal force when it was rested on the ground?
No, because the normal force points in the wrong direction--it points vertically upward, but the stretching is horizontal.

#### metastable

No, because the normal force points in the wrong direction--it points vertically upward, but the stretching is horizontal.
Suppose the planet is a prolate spheroid such that when it rotates it becomes a sphere through centrifugal forces... when Neil descends from orbit with a "pristine" isoscelese, he sets it on the surface...

Assuming the planet isn't spinning too quickly, will the corners of the triangle sag slightly or not from the differential applied forces-- the triangle only be supported by a minuscule point in the middle (or from the side if he's holding)?

If so when the points are drawn on the surface, wouldn't they no longer have the same distance between them as the "holes" in the triangle originally had when they were still "pristine" in orbit?

Above a certain angular speed wouldn't centrifugal forces cause Neil to lift off the surface (and possibly bounce) if he wasn't holding something, causing the triangle to stretch in the opposite direction as it would under the normal force if Neil is using any force to hold it to the surface?

#### PeterDonis

Mentor
Suppose the planet is an oblate spheroid such that when it rotates it becomes a sphere through centrifugal forces
An object that is oblate before it starts rotating will get more oblate as it rotates, not less.

More generally, I don't see the point of piling on complications to the scenario, since all they do is distract and obfuscate the primary issues that the OP was raising.

#### metastable

Yes, thank you I corrected it to prolate.

#### Ibix

Science Advisor
As the scenario is presented in the OP, it is inconsistent. The OP claims that Neil will see the triangle still match the dots, which implies that the triangle must stretch. But it also claims that Michael will see the triangle length contracted, which implies that the triangle does not stretch.

Either choice would be consistent by itself; but only one of the two can be true of a single scenario.
The triangle is specified to be loose. So, to my mind, the OP is describing the scenario where the triangle does not stretch and incorrectly describing Neil's observations.

Possible I'm reading more into your writing than you intended, but it seemed to me you were entertaining the possibility that the loose triangle could stretch. I've been answering on the basis that it wouldn't, and still can't see a reason it would. But you've been known to correct my understanding on occasion...

#### Foppe Hoekstra

As the scenario is presented in the OP, it is inconsistent. The OP claims that Neil will see the triangle still match the dots, which implies that the triangle must stretch. But it also claims that Michael will see the triangle length contracted, which implies that the triangle does not stretch.
Actually I did not intent to “claim” the matching, but more assumed that it would match. From my previous question (Why is the null-result of M&M experiment considered as proof for RT) I learned (from Nugatory) that “the experiments are at rest relative to the apparatus, so they find no length contraction.”. Hence, I also assumed that the stretching of the dots would not take place in Neil’s perspective.
But as you try to tell me now, I should understand that here an influence of ‘higher’ order is involved.

#### PeterDonis

Mentor
it seemed to me you were entertaining the possibility that the loose triangle could stretch
No, I wasn't. If it is clearly specified that the triangle is loose, then it won't stretch and the description of what Neil will see in the OP is incorrect.

#### Ibix

Science Advisor
But as you try to tell me now, I should understand that here an influence of ‘higher’ order is involved.
I wouldn't call it an influence of a higher order. It's just that there is an awful lot of mass bound together and undergoing ferocious acceleration, and you can't neglect the effects of that by using an inertial frame (or any other way). In the Michelson-Morley case nothing is accelerating so there are no forces distorting anything.

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