# I Accelerated Triangle and Length Contraction

#### Foppe Hoekstra

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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#### Ibix

Science Advisor
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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#### metastable

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...

#### Ibix

Science Advisor
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".

#### metastable

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...

#### Ibix

Science Advisor
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.

#### Mister T

Science Advisor
Gold Member
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?

#### Foppe Hoekstra

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.

#### Foppe Hoekstra

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?

#### Mister T

Science Advisor
Gold Member
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.

#### Nugatory

Mentor
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.

#### Ibix

Science Advisor
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.

#### Foppe Hoekstra

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.

#### Foppe Hoekstra

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?

#### Ibix

Science Advisor
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.

#### sysprog

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.

#### Ibix

Science Advisor
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.

#### Ibix

Science Advisor
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.

#### sysprog

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).

#### metastable

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?
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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#### Ibix

Science Advisor
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.

#### Ibix

Science Advisor
From Michael’s observations can he “calculate“ that Neil still sees the triangle as isosceles?
Of course.
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.

#### metastable

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:

"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."

#### Ibix

Science Advisor
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.

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