Accelerated Triangle and Length Contraction

In summary, Neil and Michael visit a solid not revolving planet and attach a jet engine to it in order to make it turn around its axis. Two dots are placed (1 meter apart) in line with the engines thrust: pointing out the east-west-leg. A third dot is placed 1 meter north of the first dot, together pointing out the north-south-leg. So the isosceles triangle fits perfectly between the three dots. Then they start the engine, Neil remains on the planet and Michael boards the spaceship to watch the accelerating planet from a stationary place above the surface of the planet. After a while, when the planet has reached a constant revolving speed, Neil checks whether the dots on the surface of the planet
  • #1
Foppe Hoekstra
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TL;DR Summary
Is an isosceles triangle after being accelerated still an isosceles triangle to every observer?
Two astronauts, Neil and Michael, visit a solid not revolving planet. They mount a jet engine on this planet to get it turning around its axis. Before starting the engine they put three dots on the surface with the help of an isosceles triangle, which measures 1 by 1 meter. Two dots are placed (1 meter apart) in line with the engines thrust: pointing out the east-west-leg. A third dot is placed 1 meter north of the first dot, together pointing out the north-south-leg. So the isosceles triangle fits perfectly between the three dots.
Then they start the engine, Neil remains on the planet and Michael boards the spaceship to watch the accelerating planet from a stationary place above the surface of the planet.
After a while, when the planet has reached a constant revolving speed, Neil checks whether the dots on the surface of the planet still correspond with the isosceles triangle, and so they do.
To Michael however, things behave a little different. The surface of the planet has gained a speed in east direction and according to Lorentz transformations it should contract in that direction. But to please Ehrenfest, the surface of the planet will be stretched to compensate the east-west-length contraction (so the radius of the planet will hold). The east-west-leg of the loose isosceles triangle however, is not subjected to Ehrenfest and so it will show length contraction to observer Michael in the stationary spaceship, whereas the east-west-dots still have their 1 meter distance, according to Michael.
So Neil reports to Earth that the triangle still fits the dots, whereas Michael reports to Earth that the triangle does not fit the dots anymore. Who are we to believe?
 
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  • #2
The triangle will not fit the dots.

This is easy to explain from the inertial frame - length contraction applies to both the dots on the planet surface and the triangle, but the surface of the planet cannot contract because the interior of the planet prevents it, so must stretch to compensate. Strain gauges on the surface will show internal stresses, and the dots will be longer than the triangle.

It's rather harder to describe in the rotating frame, partly because defining a rotating frame needs to be done formally. But that the surface is under stress and the triangle isn't remains true. Your description of Neil's measurements is incorrect.

Note that I'm assuming that the triangle isn't attached to the planet in some way that would make it stretch (e.g. a single nail through the middle, not one in each corner).
 
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  • #3
If the planet is very small will the triangle even accelerate along with the accelerating surface? I think for a sensible answer we might need to factor the planet mass/gravity, radius, atmospheric density and the triangle’s drag coefficient and mass...
 
  • #4
metastable said:
If the planet is very small will the triangle even accelerate along with the accelerating surface? I think for a sensible answer we might need to factor the planet mass/gravity, radius, atmospheric density and the triangle’s drag coefficient and mass...
If a realistic planet were spun up it would disintegrate long before relativity became relevant. So this question can assume that the triangle is attached to the surface using the same ridiculously strong material of which the planet is made.

In other words, this is a highly idealised scenario, so no, we do not need to introduce complicating elements to get a "sensible answer".
 
  • #5
Foppe Hoekstra said:
So Neil reports to Earth that the triangle still fits the dots, whereas Michael reports to Earth that the triangle does not fit the dots anymore. Who are we to believe?

What if the planet was small enough and because there was no mechanism, the triangle never accelerated (except for with respect to the surface), and neither did neil, and neil looked down at the triangle very close to the time the sphere had completed an integer number of rotations, so he sees the holes line up. in this case the planet is also rotating from michael’s perspective, so if he looks at a slightly different time as neil and he doesn’t see the holes line up... so they are both correct...
 
  • #6
In that case Neil would see the triangle length contracted so it did not fit and Michael would see the stretch of the dots cancelling their length contraction and the triangles will coincide.

@metastable - can I suggest that it might be polite, if you intend to ask more tangential questions, to start a new thread and link here? The OP hasn't had a chance to respond to my response yet and you are adding more scenarios that he may not be interested in.
 
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  • #7
Foppe Hoekstra said:
Is an isosceles triangle after being accelerated still an isosceles triangle to every observer?

Do you consider the following question to be equivalent: To an observer at rest relative to a triangle, the triangle is isosceles. To an observer moving relative to the triangle, is the triangle isosceles?
 
  • #8
Ibix said:
Your description of Neil's measurements is incorrect.
Could you be more specific about where my description of Neil’s measurements is incorrect?
To Neil, or more formal to Neil’s frame of reference, nothing is moving: Neil, the dots and the triangle are all accelerated in the same way, so these things should, between them, not experience any relativistic effects. Hence, to Neil both the dots and the triangle should keep exactly the same dimensions.
 
  • #9
Mister T said:
Do you consider the following question to be equivalent: To an observer at rest relative to a triangle, the triangle is isosceles. To an observer moving relative to the triangle, is the triangle isosceles?
I think it is, but please note that the summary does not represent the real question very well (for which I am to blame). What really puzzles me is: how can one and the same situation have to two fundamentally different occurrences?
 
  • #10
Foppe Hoekstra said:
What really puzzles me is: how can one and the same situation have to two fundamentally different occurrences?

But they are not the same situation! In one case the observer is at rest relative to the triangle, and in the other case the observer is moving relative to the triangle. Note that you don't change the triangle just because you happen to be moving relative to it. If each leg is measured to have a length of 1.0 meter in its rest frame, all observers will agree that that is their proper length. Thus proper length is a relativistic invariant, as are all measurements. Like if a voltmeter reads 1.5 volts, all observers will agree that that is the reading on the voltmeter.
 
  • #11
Foppe Hoekstra said:
how can one and the same situation have to two fundamentally different occurrences?
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.
 
  • #12
Foppe Hoekstra said:
To Neil, or more formal to Neil’s frame of reference, nothing is moving: Neil, the dots and the triangle are all accelerated in the same way, so these things should, between them, not experience any relativistic effects
In the inertial frame the ground would be length-contracted because it is moving. But the circumference of the planet cannot change because the radius doesn't change (assuming a near-infinite tensile strength - as noted, any realistic material would actually expand and disintegrate). Therefore the ground must be stretched so that its stretched-and-length-contracted circumference equals its rest circumference.

Neil sees the ground stretched because that's an invariant (you could install strain gauges to detect it and their readings can't be frame variant), but he does not see it length contracted. So he sees the dots elongate. The triangle itself is unstressed, so Neil sees it as normal.
 
  • #13
Mister T said:
But they are not the same situation!
We have one triangle and one set of dots, so we have only one ‘situation’ ‘triangle between dots’. I just find it weird that this one situation can be seen differently by every observer moving at a different speed.
 
  • #14
Ibix said:
In the inertial frame the ground would be length-contracted because it is moving. But the circumference of the planet cannot change because the radius doesn't change (assuming a near-infinite tensile strength - as noted, any realistic material would actually expand and disintegrate). Therefore the ground must be stretched so that its stretched-and-length-contracted circumference equals its rest circumference.

Neil sees the ground stretched because that's an invariant (you could install strain gauges to detect it and their readings can't be frame variant), but he does not see it length contracted. So he sees the dots elongate. The triangle itself is unstressed, so Neil sees it as normal.
But the ground is not moving in respect to Neil. Neither are the dots or the triangle moving in respect to Neil. So how can there be any relativistic effects between these entities?
 
  • #15
Foppe Hoekstra said:
But the ground is not moving in respect to Neil. Neither are the dots or the triangle moving in respect to Neil. So how can there be any relativistic effects between these entities?
There aren't any. Neil does not see either item length contracted; Michael sees both items length contracted. The stretch of the ground is an invariant, not a coordinate effect.
 
  • #16
If and only if the equal legs of the isosceles triangle were moving equally away from or toward the observer their lengths would from the point of view of the observer remain co-equal.
 
  • #17
Ibix said:
There aren't any. Neil does not see either item length contracted; Michael sees both items length contracted. The stretch of the ground is an invariant, not a coordinate effect.
Let me expand a bit on this.

Michael sees the triangle Neil is carrying as length contracted in the direction of motion. He also sees the dots as length contracted, but the ground is stretched and the two effects cancel out - so the dots look like they originally did. Thus the triangle is shorter than the dots.

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.
 
  • #18
sysprog said:
If and only if the equal legs of the isosceles triangle were moving equally away from or toward the observer their lengths would from the point of view of the observer remain co-equal.
Assuming the triangle is unstressed, I agree. However the triangle of dots is stretched so that (according to Michael) the length contraction and actual extension cancel out. The original triangle is unstressed and will no longer be the same shape, as you say.
 
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  • #19
Ibix said:
Assuming the triangle is unstressed, I agree. However the triangle of dots is stretched so that (according to Michael) the length contraction and actual extension cancel out. The original triangle is unstressed and will no longer be the same shape, as you say.
I appreciate your insight; however, I think it's simpler to say that if and only if, from the point of view of the observer, the movement of the (unstressed or evenly stressed) isosceles triangle is exactly perpendicular to the base, the skeles (legs) of the triangle will remain isotensive (of the same length).
 
  • #20
Foppe Hoekstra said:
But the ground is not moving in respect to Neil. Neither are the dots or the triangle moving in respect to Neil. So how can there be any relativistic effects between these entities?
Ibix said:
Michael sees the triangle Neil is carrying as length contracted in the direction of motion.
From Michael’s observations can he “calculate“ that Neil still sees the triangle as isosceles?

If the planet rotates fast enough would relativistic aberration make the light from Neil, the triangle and dots (light emitted at the same time as when they are geometrically beneath michael) invisible or almost invisible to michael?
 
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  • #21
sysprog said:
I think it's simpler to say that if and only if, from the point of view of the observer, the movement of the (unstressed or evenly stressed) isosceles triangle is exactly perpendicular to the base, the skeles (legs) of the triangle will remain isotensive (of the same length).
But this is not entirely relevant to this case. It does apply to the original triangle, but it does not apply to the triangle formed by the dots. The motion of the dots for Michael is parallel to one of the equal length sides, as is the stress. The two effects cancel out for him.
 
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  • #22
  • #23
metastable said:
From Michael’s observations can he “calculate“ that Neil still sees the triangle as isosceles?
Of course.
metastable said:
If the planet rotates fast enough would relativistic aberration make the light from Neil, the triangle and dots (light emitted at the same time as when they are geometrically beneath him) invisible or almost invisible to michael?
The Doppler effect and relativistic beaming would present observational challenges, sure. But we've just spun a planet up to nearly the speed of light and somehow stopped it disintegrating from the centrifugal force. Observational challenges are minor.
 
  • #24
Ibix said:
The Doppler effect and relativistic beaming would present observational challenges, sure. But we've just spun a planet up to nearly the speed of light and somehow stopped it disintegrating from the centrifugal force. Observational challenges are minor.
So if I understand correctly, Neil could be invisible to Michael, for similar reasons as the far jet appearing invisible in this image:
x313.jpg


"Figure 5.21: This VLA image of the radio-loud quasar 3C 175 shows the core, an apparently one-sided jet, and two radio lobes with hot spots of comparable flux densities. The jet is intrinsically two sided but relativistic, so Doppler boosting brightens the approaching jet and dims the receding jet. Both lobes and their hot spots are comparably bright and thus are not moving relativistically. Image credit: NRAO/AUI/NSF Investigators: Alan Bridle, David Hough, Colin Lonsdale, Jack Burns, & Robert Laing."
https://www.cv.nrao.edu/~sransom/web/Ch5.html
 
  • #25
metastable said:
So if I understand correctly, Neil could be invisible to Michael, for similar reasons as the far jet appearing invisible in this image
Not really, unless Michael hovers over the pole. If he's got any sense he hovers over the equator, and then half the time Neil is coming towards him. Or he uses a less naive experimental setup than just looking at Neil.
 
  • #26
Ibix said:
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.
 
  • #27
Foppe Hoekstra said:
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.

Foppe Hoekstra said:
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.

Foppe Hoekstra said:
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).

Foppe Hoekstra said:
Some people will probably say that SRT is therefor not applicable here

It is if we ignore the planet's gravity.
 
  • #28
Foppe Hoekstra said:
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.
 
  • #29
Nugatory said:
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.
 
  • #30
Foppe Hoekstra said:
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.
Foppe Hoekstra said:
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.
Foppe Hoekstra said:
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.
Foppe Hoekstra said:
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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  • #31
PeterDonis said:
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.
 
  • #32
Ibix said:
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.
 
  • #33
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?
 
  • #34
metastable said:
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.
 
  • #35
PeterDonis said:
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?
 
<h2>1. What is the Accelerated Triangle?</h2><p>The Accelerated Triangle is a geometric concept used in special relativity to explain the effects of acceleration on space and time. It consists of three sides representing the three dimensions of space (length, width, and height) and a fourth side representing time.</p><h2>2. How does acceleration affect the length of an object?</h2><p>According to the theory of special relativity, as an object accelerates, its length in the direction of motion will appear to contract or shorten from the perspective of an outside observer. This phenomenon is known as length contraction.</p><h2>3. What is the formula for calculating length contraction?</h2><p>The formula for length contraction is L = L<sub>0</sub> * √(1 - v<sup>2</sup>/c<sup>2</sup>), where L is the contracted length, L<sub>0</sub> is the rest length of the object, v is the velocity of the object, and c is the speed of light.</p><h2>4. How does length contraction relate to time dilation?</h2><p>Length contraction and time dilation are two sides of the same coin in special relativity. As an object's length contracts, its time also appears to slow down. This means that an object moving at high speeds will experience both length contraction and time dilation simultaneously.</p><h2>5. Can length contraction be observed in everyday life?</h2><p>Yes, length contraction can be observed in everyday life, but only at extremely high speeds close to the speed of light. For example, particles in a particle accelerator experience length contraction as they approach the speed of light, which is necessary for them to collide and produce high-energy reactions.</p>

1. What is the Accelerated Triangle?

The Accelerated Triangle is a geometric concept used in special relativity to explain the effects of acceleration on space and time. It consists of three sides representing the three dimensions of space (length, width, and height) and a fourth side representing time.

2. How does acceleration affect the length of an object?

According to the theory of special relativity, as an object accelerates, its length in the direction of motion will appear to contract or shorten from the perspective of an outside observer. This phenomenon is known as length contraction.

3. What is the formula for calculating length contraction?

The formula for length contraction is L = L0 * √(1 - v2/c2), where L is the contracted length, L0 is the rest length of the object, v is the velocity of the object, and c is the speed of light.

4. How does length contraction relate to time dilation?

Length contraction and time dilation are two sides of the same coin in special relativity. As an object's length contracts, its time also appears to slow down. This means that an object moving at high speeds will experience both length contraction and time dilation simultaneously.

5. Can length contraction be observed in everyday life?

Yes, length contraction can be observed in everyday life, but only at extremely high speeds close to the speed of light. For example, particles in a particle accelerator experience length contraction as they approach the speed of light, which is necessary for them to collide and produce high-energy reactions.

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