Does born rigidity mean elongation of a body?

[Sreenath Skr]

Consider an unbreakable nail.
A nail of length 10cm moving at 0.866c.
If i stop the nail by hitting its cap, would the tip end countinues to travel?

 

[ghwellsjr]

Consider an unbreakable nail.
A nail of length 10cm moving at 0.866c.
If i stop the nail by hitting its cap, would the tip end countinues to travel?

Yes.

 

[Sreenath Skr]

In bug rivet paradox, we assume a well of 1 metre long in it's own frame and a rivet of length 0.5m in its own frame
Here, assume that the rivet is super strong to withstand compressions but the rivet would eaily broken if it elongates

From the rest frame of the rivet, the well moving at 0.866c would make the well 0.5m and would kill the bug. There's no elongation for the rivet in this case
Now from the rest frame of the well, the rivet is only 0.25m long. After hitting the top of the well, rivet continues to travel forward. Wouldn't that cause the rivet to break?

 

[Nugatory]

Here, assume that the rivet is super strong to withstand compressions but the rivet would eaily broken if it elongates

If you mean that the rivet can experience arbitrarily large compressive forces without breaking or deforming, that's physically impossible. You may, if you wish, assume that the rivet has a very high elastic limit, meaning that no matter how compressed it will always return to its original shape when the compressive force is removed.

The thing to remember about the bug-rivet paradox: if the head of the rivet hits the edge of the well in one frame, then it hits the edge of the well in all frames. If the bug is squashed in one frame, then the bug is squashed in all frames. If you come to a different conclusion, then you have fallen into the paradox-setter's trap and started reasoning from a false premise about the behavior of materials; your "super strong" assumption is equivalent to "there exists a material in which the speed of sound is greater than the speed of light in vacuum", and that premise is false.

 

[Nugatory]

Consider an unbreakable nail.
A nail of length 10cm moving at 0.866c.
If i stop the nail by hitting its cap, would the tip end countinues to travel?

Yes.

And to tie this back to the thread title... In this case the nail is not undergoing Born rigid motion.

 

[ghwellsjr]

In bug rivet paradox, we assume a well of 1 metre long in it's own frame and a rivet of length 0.5m in its own frame
Here, assume that the rivet is super strong to withstand compressions but the rivet would eaily broken if it elongates

From the rest frame of the rivet, the well moving at 0.866c would make the well 0.5m and would kill the bug. There's no elongation for the rivet in this case
Now from the rest frame of the well, the rivet is only 0.25m long. After hitting the top of the well, rivet continues to travel forward. Wouldn't that cause the rivet to break?

Why do you here, and in your title, talk about "elongation"? I have heard of length contraction but never length elongation. I don't understand under what circumstance you are proposing that the rivet (or the nail) elongates. Could you please explain what you mean?

 

[Sreenath Skr]

Okay. I'm sorry about saying invalid assumptions

Okay, we can consider both as diamond
A diamond rivet of length 0.5m & a well of 1m as measured in their own frame

From the rest frame of the rivet, the well is contracted to 0.5m at 0.866c.

At the neck of the rivet, a 10cm long spring is connected. It is thin but absorb most of the impact. The spring completely compresses and kills the bug.
There's no stress on the rivet body but only on its head

Now from the rest frame of the bug, the rivet is only 0.25m. Here the head end stops by shock absorber but the tip must travel into the well
Since the rivet is a brittle object, how can it travel into the bottom of the well without breaking it?

That is what i don't understand

 

[Nugatory]

Okay, we can consider both as diamond
A diamond rivet of length 0.5m & a well of 1m as measured in their own frame

I would say that diamond flows like water under the forces required to bring an object moving at .866c to rest in a distance of one meter, but that would be an enormous understatement. A solid diamond resists those forces the way a puff of smoke resists a bullet.

So the spring is just confusing things here - everything is bending, flexing, changing shape (and the motion is most certainly not Born rigid). Instead, let's go back to the simple case and assume that the rivet is made of some material that is elastic enough that when it finally comes to rest and the various forces acting on it are spent, it will return to its normal shape. (We'll also ignore the fact that the energy released in the collision will produce a multi-megaton explosion - there's a lot of kinetic energy in that moving rivet).

And when we go back to that simple case (well one meter deep and rivet .5 meters long when each is measured in their rest frame, rivet moving at a speed such that from the point of view of the moving rivet the depth of the well is contracted to .5m) you'll get the following results:

1) In the rivet frame, the tip of the rivet squashes the bug. The head of the rivet keeps on moving until it contacts the edge of the well a moment later. Because the two ends of the rivet are moving at very different speeds, the rivet is greatly distorted (not Born rigid!) until all parts of it can come to rest relative to the well.

2) In the bug frame, the head of the rivet contacts the edge of the well first. The tip of the rivet keeps on moving so the bug is squashed a moment later. Because the two ends of the rivet are moving at very different speeds, the rivet is greatly distorted (not Born rigid!) until all parts of it can come to rest relative to the well.

The two frames differ about which event (tip hits, rim hits) happens first - this is the relativity of simultaneity, key to most special relativity "paradoxes" - but not that they both happen.

 

[pervect]

The assumption of a "rigid body" is a very common approximation in Newtonian mechanics, but it works only in the low speed realm even in Newtonian mechanics. Thus it tends to fail very bady as an approximation in relativity.

If you consider a diamond, or any other substance, it will have a speed of sound much less than c at which compressive ways travel through it. A we search shows that the speed of sound in diamond is about 12 km/sec, which is quite high for a speed of sound, but much lower than the speed of light. You can also calculate it through some of the well-known formula under "speed of sound" in the Wikipedia knowing it's bulk modulus and it's density. You can compute the bulk modulus from the young's modulus if you happen to have that instead.

To calculate the exact Newtonian responses of the body in the situations you specify would be difficult, but there is at least one thing that is easy to know beforehand. That is that that mechanical disturbances in matter of any kind are limited to the speed of sound in the matter, much as the speed of light is an absolute limit. It's also worth noting that this speed is a lot lower than "c" for any known material.

To demonstrate this in detail one might write down the (possibly idealized) partial differential equations that represent a continuous deformable media subjected to a force, and show that they satisfy the wave equation. It's easy to show that a shock wave travels at the wave speed, this should be done in many textbooks, though I don't have any specific recommendations. it is additionally true that the wave speed is an upper limit for how fast any disturance can propagate that satsifies the partial differential equations (the wave equations) which are the solution of motion.

It may help to notice that light also obeys a wave equation - its just that it's propagation velocity is much higher than that of any known sort of matter.

As an aside for the diamond case, diamond is very rigid, but it has a low elastic limit. This means it can't deform very much before it shatters, which is the probable fate of the diamond in this circumstance.

There's a related issue in your proposal - the spring design. It might also be interesting to consider the problem in reverse. We don't currently use springs to launch rockets into orbits, or to shoot bullets from guns, but if we time reverse your example problem, you have a spring shooting a diamond payload nearly at the speed of light.

http://en.wikipedia.org/w/index.php?title=Light_gas_gun&oldid=608805490 talks a bit about the physics , the key point is that

The limiting factor on the speed of an airgun, firearm, or light-gas gun is the speed of sound in the working fluid—the air, burning gunpowder, or a light gas. This is essentially because the projectile is accelerated by the pressure difference between its ends, and such a pressure wave cannot propagate any faster than the speed of sound in the medium.

If you want a more realistic experiment, you might consider shooting a diamond moving at your relativistic velocity into a thick steel plate in order to stop it. It will probably penetrate more than the 10cm you specify, and I think you can imagine that the diamond won't survive the process.