Size Limit of Accelerated Rigid Body in Irrotational Born Rigid Motion

[pervect]

I was thinking - and reading a bit - about the size limit on accelerated frames, and there is an interesting and relevant result I found.

If we rephrase the question from "is there a size limit on an accelerated frame" to "is there a size limit on an accelerated body in irrottational born rigid motion", it is known that the answer is yes, there is a limit. This was pointed out by Born in 1909, according to wiki;

https://en.wikipedia.org/w/index.php?title=Born_rigidity&oldid=961398833

Already Born (1909) pointed out that a rigid body in translational motion has a maximal spatial extension depending on its acceleration, given by the relation ...

The limit is that the proper acceleration must be less than c^2/R, where R is the radius of a sphere in which the body is located. I couldn't tell from the wiki article where the proper acceleration was measured, presumably at the center of the sphere - it will vary.

So, as long as we demand that our accelerated frame have the property that objects "at rest in the frame" maintain a constant distance from each other, then there IS a known limit on the size of an accelerated frame.

 

[Dale]

That makes sense. I bet it is related to the distance to the Rindler horizon.

 

[PeterDonis]

The limit is that the proper acceleration must be less than c^2/R, where R is the radius of a sphere in which the body is located.

Actually, Born didn't get this quite correct.

The correct statement is that, for the case of linear acceleration, if we imagine an observer traveling along with a particular point in the body (think of it as an atom if you like), the body can only extend below that observer for less than a distance $R = c^2 / a$, where $a$ is the observer's proper acceleration. This is just one way of describing that the observer has a Rindler horizon.

However, there is no limit in principle on how far above the observer the body can extend.

 

[Ibix]

However, there is no limit in principle on how far above the observer the body can extend.

I wondered if he was considering acceleration of constant magnitude although not necessarily direction. In that case, Born is correct I think.

 

[PeterDonis]

I wondered if he was considering acceleration of constant magnitude although not necessarily direction.

The Born reference in the Wikipedia article is given in the section on irrotational motions (class A in the Herglotz-Noether classification), which are in a single linear direction with no change. (The proper acceleration associated with such motion is not "constant" since different points in the object that are separated along the direction of acceleration will have different proper accelerations. Also, even the proper acceleration of a single point in the object does not have to be constant in magnitude, as long as it doesn't change direction.)

Any rigid motion involving change of direction would be a class B motion, which has different properties.

 

[BiGyElLoWhAt]

I don't understand the orientation. The differentiation between above and below. Is there absolute orientation in space time?

 

[Ibix]

I don't understand the orientation. The differentiation between above and below. Is there absolute orientation in space time?

No. But you have an orientation if you are under power. And Born rigid motion means that if you feel an acceleration $a$, an object $c^2/a$ below you would need infinite acceleration to keep up, using your rulers to define "keeping up".

 

[BiGyElLoWhAt]

No. But you have an orientation if you are under power. And Born rigid motion means that if you feel an acceleration $a$, an object $c^2/a$ below you would need infinite acceleration to keep up, using your rulers to define "keeping up".

I guess my question is what's the difference between "an object $c^2/a$" above vs "an object $c^2/a$" below

 

[Ibix]

I guess my question is what's the difference between "an object $c^2/a$" above vs "an object $c^2/a$" below

Born rigid motion means that rulers you carry with you have constant length in their instantaneous rest frame. That turns out to mean that (for constant linear acceleration) every point on your ruler has to follow a hyperbolic path through spacetime, all of which have a common focus. Designate that focus $x=0$ and it turns out that the proper acceleration experienced by the bit of the ruler at $x$ is $c^2/x$. So the proper acceleration decreases "upwards" along the ruler, and diverges as $x$ goes to zero. This latter is closely analogous to the event horizon of a black hole and is called the Rindler horizon.

TLDR: proper acceleration decreases upwards in Born rigid motion. So $c^2/a$ above has no particular issue but $c^2/a$ below cannot accelerate enough to keep up.

 

[Ibix]

Any rigid motion involving change of direction would be a class B motion, which has different properties.

Had to think about this, but it's Wigner rotation, isn't it? If I accelerate in my +x direction then stop accelerating then accelerate in my +y direction then stop accelerating again, my final inertial frame can be related to the original one by the composition of two non-parallel boosts.

 

[vanhees71]

Usually you define as "rotation free local reference frame" one that for an observer (described by his time-like worldline) such a frame is given by Fermi-Walker transporting an arbitrary tetrad (with the four-velocity of the observer being the time-like member of this basis of course) along the observer's worldline. If this world line is not always in the same direction the spatial members of the tetrad nevertheless rotate with respect to the initial tetrad, because the composition of rotation free boosts in different boost direction are not rotation free against a fixed reference tetrad. That's what's mathematically behind the Wigner rotation and Thomas precession. For SR, see Sect. 1.8 in

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf

 

[DrGreg]

I don't understand the orientation. The differentiation between above and below. Is there absolute orientation in space time?

Just in case it's not clear from the answers already given, in this context "up" means "in the direction of the acceleration", and "down" is in the opposite direction. It's not absolute, it's relative to the accelerating body.

 

[PeterDonis]

it's Wigner rotation, isn't it?

If you mean Thomas precession, yes, that comes into play whenever there is acceleration in different directions.