Question about the limits of space contraction near light speed

[DarkMattrHole]

Hi all. I have a question about something Nima Arkani-Hamed said in his lecture on space-time about space contraction near light speed. I included a link to the lecture at the point where he refers to contraction of two space ships with a 'cable' between them, they are accelerating towards the speed of light together, and at a point in the space contraction the 'cable' snaps. Is this the point where further contraction is impossible creating a black hole? And does the contraction of space affect the symmetry of the atoms and their electrons in the cable, and then end up squeezed down to the plank length? reference - @ time 35:24

 

[Ibix]

Is this the point where further contraction is impossible creating a black hole?

No. You can make an arbitrarily long cable snap at arbitrarily low acceleration by using a weak enough cable. And you never create a black hole from just speeding things up - if you could, it would imply an absolute sense to "speed", which is contradictory to the foundations of relativity theory. (Edit: You may wish to look up Bell's Spaceships Paradox in this forum.)

And does the contraction of space affect the symmetry of the atoms and their electrons in the cable

Length contraction affects everything, as far as we are aware. Although note that in the rest frame of an atom in the cable nothing happens to it. Length contraction and time dilation are always effects measured by someone else.

and then end up squeezed down to the plank length

The Planck length is not anything special in relativity. There is no "smallest distance" in relativity.

 

[BvU]

It is pure nonsense. Things don't 'shrink'. The cable as 'described' (*) does not snap.

(*) he actually says 'which I don't have time to explain'
The way I pick it up both ships accelerate at exactly the same rate

 

[PeterDonis]

The cable as 'described' (*) does not snap.

As far as I can tell, he is describing the scenario of the Bell Spaceship Paradox. In that scenario, the cable does snap.

 

[Ibix]

It is pure nonsense. Things don't 'shrink'. The cable as 'described' (*) does not snap.'

Yes it does - he's (not quite) describing Bell's spaceship paradox. Or so it seems to me.

 

[PeterDonis]

PF has an Insights article on the Bell Spaceship Paradox:

https://www.physicsforums.com/insights/what-is-the-bell-spaceship-paradox-and-how-is-it-resolved/

 

[BvU]

Thanks for the correction ! Didn't consider this simultaneity issue.
I'm lightly shocked but will recover.

 

[Janus]

Hi all. I have a question about something Nima Arkani-Hamed said in his lecture on space-time about space contraction near light speed. I included a link to the lecture at the point where he refers to contraction of two space ships with a 'cable' between them, they are accelerating towards the speed of light together, and at a point in the space contraction the 'cable' snaps. Is this the point where further contraction is impossible creating a black hole? And does the contraction of space affect the symmetry of the atoms and their electrons in the cable, and then end up squeezed down to the plank length? reference - @ time 35:24

If we are in the inertial frame watching these two ship accelerate past at a constant rate, then the cable is trying to shorten while the distance between the Ships remains constant. If the cable is not strong enough to stop draw the ships together against the force of the ship's engines, then at some point the cable will break.
It's probably easier to see why if see view things from the perspective of the cable. According to it, it is simply trying to maintain a constant length, while the distance between the ships is increasing.( In its frame, the ships are not maintaining the same speed relative to each other, with the trailing ship lagging further and further behind the lead ship) Any physical cable will first start to stretch as the ships move apart, but will eventually reach its limit and snap. So the answer to when it will snap depends on the elasticity and tensile strength of the cable.

 

[DarkMattrHole]

Thanks guys. So the rockets are firing their engines in order to accelerate along the axis of the cable one ship behind the other, and, at the same time using additional (opposing each other) thrusters to hold distance from each other, under the pull of the cable as it (shrinks) pulls them together, with force proportional to the acceleration?

 

[Ibix]

So the rockets are firing their engines in order to accelerate together and at the same time using additional (opposing) thrusters to hold distance from each other

Not sure what you mean. The rockets are accelerating in the same direction with the same proper acceleration. No additional thrusters are really needed - each rocket pilot keeps an eye on their on-board accelerometer and sets the engine thrust to keep the reading at some agreed value. If the string is much lighter than the rockets and doesn't have an absurd tensile strength then this boils down to "turn engine on, sit back".

 

[Nugatory]

Thanks guys. So the rockets are firing their engines in order to accelerate together and at the same time using additional (opposing) thrusters to hold distance from each other, under the pull of the cable as it (shrinks) pulls them together, with force proportional to the acceleration?

No.
The rockets are firing their engines in order to accelerate. The trick to this paradox is hidden in the words "accelerate together" which assume that both ships are changing their speed by the same amount at the same time - and whenever you see the words "at the same time" in any relativity problem, alarm bells should start ringing.

Because of the relativity of simultaneity (if you are not already familiar with it, google for "Einstein train simultaneity" - you absolutely must understand this concept before you can take on anything else in relativity) there is only one frame in which both ships are accelerating together. In Bell's version of the problem, it's the frame in which the earth-bound observer is at rest, both ships take off at the same time and subsequently fly the same acceleration profile, and the distance between the ships remains constant. In this frame, the cable is length-contracting so breaks because it becomes too short while the distance between the ships remains the same.
However, we could also work the problem from some other frame. In these frames the ships are not changing their speed at the same time as they accelerate, so the distance between them is increasing over time. We can always find a frame in which the cable is not length-contracted (just choose the frame in which it is at rest at that particular moment) but in that frame the cable breaks because the distance between the ships is increasing.

So in some frames we conclude that the cable breaks because it shrinks while the separation between the ships remains constant, while in other frames the cable breaks because its length stays the same while the separation between the ships increases. But all frames agree that the cable breaks, agree about what stress gauges attached to the ends of the cables would read, and agree that the cable breaks when that stress exceeds the strength of the cable.