Can an accelerating rod's length decrease faster than the speed of light?

[qetuol]

hi all!

So let's consider this situation:

I am in inertial system and i have a standing rod with length L_o . Then there is a constant force F accelerating this rod. Will the rod's length shorten? Can i use the equation of length-contraction in case velocity is not constant (of course the velocity of the rod can be calculated before)?

Now imagine that i don't have a rod but 2 separate bodies, distance between them is L_o. The SAME force applies to both of the bodies. Will the distance (space) between them shorten when they are accelerated? Consider that bodies can be connected with rod so the case is the same described above, is not it?

And the final conclusion. What is the velocity of the shortening? I figured out that the velocity of shortening can be greater than the lightspeed. Note, the velocity of the shortening is dependent on the initial length (distance in case of 2 bodies). So if the L_o is enough great, the velocity of the shortening can be arbitrary large even greater than the lightspeed.

(english isn't my first language so if you don't understand something ask and i will enlighten you :D)

 

[Steely Dan]

hi all!

So let's consider this situation:

I am in inertial system and i have a standing rod with length L_o . Then there is a constant force F accelerating this rod. Will the rod's length shorten? Can i use the equation of length-contraction in case velocity is not constant (of course the velocity of the rod can be calculated before)?

Now imagine that i don't have a rod but 2 separate bodies, distance between them is L_o. The SAME force applies to both of the bodies. Will the distance (space) between them shorten when they are accelerated? Consider that bodies can be connected with rod so the case is the same described above, is not it?

Relativity doesn't mean that anything actually shortens. It just means that observers in other reference frames will measure it differently. But if you're in a spaceship and your ship accelerates to the speed of light, the Lorentz contraction will have no effect; your ship will stay the same size (by your measurement) the whole time.

And the final conclusion. What is the velocity of the shortening? I figured out that the velocity of shortening can be greater than the lightspeed. Note, the velocity of the shortening is dependent on the initial length (distance in case of 2 bodies). So if the L_o is enough great, the velocity of the shortening can be arbitrary large even greater than the lightspeed.

Then you did your math wrong. Even if the rod did shorten, there's no way it could shorten "faster" than the speed of light, since that requires information travel at superluminal speed. If you've figured it out, pack your bags for Stockholm.

 

[qetuol]

Relativity doesn't mean that anything actually shortens. It just means that observers in other reference frames will measure it differently. But if you're in a spaceship and your ship accelerates to the speed of light, the Lorentz contraction will have no effect; your ship will stay the same size (by your measurement) the whole time.
Then you did your math wrong. Even if the rod did shorten, there's no way it could shorten "faster" than the speed of light, since that requires information travel at superluminal speed. If you've figured it out, pack your bags for Stockholm.

of course when i say "shortens" i mean "i measure it shortened", its obvious.
i think you have misunderstood me. i do NOT accelerate. my system is inertial. the rod is accelerating. if you need i can put my calculation in. or can you show yours?

 

[Doc Al]

Now imagine that i don't have a rod but 2 separate bodies, distance between them is L_o. The SAME force applies to both of the bodies. Will the distance (space) between them shorten when they are accelerated? Consider that bodies can be connected with rod so the case is the same described above, is not it?

The two bodies will have the same acceleration as measured in your inertial frame, thus they will maintain the same distance apart. (Assuming you've begun their accelerations at the same instant, of course.) This is similar to the "Bell Spaceship Paradox".

Going back to your rod example, let's talk in terms of acceleration instead of force. If somehow you could manage to accelerate each part of the rod identically (with respect to your inertial frame), then the rod will not "shorten" as seen by you. (Of course, you'll end up ripping the rod to bits.)

 

[Fredrik]

I am in inertial system and i have a standing rod with length L_o . Then there is a constant force F accelerating this rod. Will the rod's length shorten? Can i use the equation of length-contraction in case velocity is not constant (of course the velocity of the rod can be calculated before)?

It seems to me that you should be able to do that, but Demystifier came to another conclusion in this paper. I might just have to read it some time.

Now imagine that i don't have a rod but 2 separate bodies, distance between them is L_o. The SAME force applies to both of the bodies. Will the distance (space) between them shorten when they are accelerated?

No. Those two bodies can only accelerate differently if there's something fundamentally different about their starting positions. If space is homogeneous (and it definitely is in special relativity), then they will accelerate the same way, so that the distance between them (in the frame where they were both at rest before the force was applied) will stay the same.

Consider that bodies can be connected with rod so the case is the same described above, is not it?

No, it's not. The rod is shrinking. Its front isn't accelerating as much as its rear, so it isn't the same. By the way, you're entering the territory of "Bell's spaceship paradox". See one of the many threads about that in this forum.

And the final conclusion. What is the velocity of the shortening? I figured out that the velocity of shortening can be greater than the lightspeed.

You mean that the time derivative of the length is >c. I think you're right.

 

[Steely Dan]

of course when i say "shortens" i mean "i measure it shortened", its obvious.
i think you have misunderstood me. i do NOT accelerate. my system is inertial. the rod is accelerating. if you need i can put my calculation in. or can you show yours?

I meant that you could never measure a shortening faster than the speed of light in the proper length reference frame. I don't really know if you could measure it to be faster than c in a different reference frame; I suspect not but I didn't do the math.