Force propagation and length contraction

[birulami]

Consider some material object, more or less rigid, with two ends, A and B, like

A---B

It is at rest at a point in time t_0 in my reference frame. Now I kick it a bit, i.e. I apply some force for a limited amount of time at A in the direction of B. After the kick, the whole object has a speed v in the direction A->B. I reckon that the speed of B, v_B, is never larger than that of A, v_A, before, during and after the whole experiment.

Now I consider how, during application of the force, the force propagates through the object from A to B. The speed of propagation is limited by the speed of light c.

As a consequence it seems that during the force application, there is a small time interval where v_B<v_A. Integration of this delta-speed translates into a reduction of the distance between A and B at the end of the experiment.

If I did not make a mistake, the reduction factor is (1-v/c).

This is a stronger reduction even than the relativistic length contraction factor. How can I get the object longer again to match the relativistic contraction? To me it feels wrong to argue with material dynamics to fix an effect that directly results from a very basic principle.

Ideas?

Harald.

 

[country boy]

Does the object return to inertial motion at the end of the experiment? If so, you need to turn off the acceleration. If not, then length contraction between inertial frames is not a complete description.

 

[pervect]

If you actually kick an object, it will tend to vibrate. So you expect it to contract by more than the steady state amount, then expand, then contract. In real objects, the vibrations will die down, but that requires that one add some dissipative terms to the motion. Your original model is basically appealing to the wave equation, I think, though you didn't describe it in exactly those terms. The wave equation model doesn't include any dissipative terms. So with a model of motion based on the wave equation, the object will vibrate forever.

Also note that the speed of sound in practical materials will be << c (a wave equation is an OK approximation, but the wave velocity is much much lower than c in actual materials).

 

[birulami]

Letting the force travel even slower than c makes things worse, i.e the contraction will be even greater (c smaller => v/c greater => 1-v/c smaller). In particular the limiting case of v->c looks frightening.

Hmm, I'll try to put it into my own words. Feel free to comment further.

In the original post I said that it follows from the setup that

1. $v_B<v_A$ for a limited amount of time, and
2. $v_B\le v_A$ throughout the experiment,

where (1) is a consequence of the limited propagation speed of the force. Consequently we cannot get rid of (1), so (2) must be false, I guess. Consequently there is a time interval where $v_B>v_A$. The additional force required to get B faster than A will likely have to be attributed to material dynamics, i.e. the "spring-like" forces within the material. So I guess this is where your vibration comes in. Only the limiting case where $v_A=c$ after the experiment is puzzling, since $v_B>v_A$ is then certainly out of question.

Thanks,
Harald.

 

[country boy]

A similar experiment can be imagined that does not deal with forces through the material connecting A and B.

A and B are traveling at the same velocity but are separated by a distance. Each has a device that can accelerate and also send or receive light signals. At some time the device at A begins applying an acceleration to A and at the same moment sends a light signal to B. The device at B receives the light signal and immediately begins applying the same acceleration to B. After a while, the device at A stops the acceleration at A and sends a light signal to B. When the device at B receives the signal it stops the acceleration of B. What are the relative velocities and distances during the experiment?

 

[pervect]

Well, I've been busy with other things (esp with the holidays). What I think that should happen is that the Newtonian elastic bar with a force applied to the left end should act just like a transmission line forced with a current.

In the transmission line, V = Z0 I, in the Newtonian elastic bar, it's (dx/dt)(x) = Z0 *F, where dx/dt(x) is the velocity at some point x, and F is the applied force (which acts like the current source).

The equations for the electrical case where I represents current and V represents voltage are:

I = C dV/dt + L $\partial I / \partial x$

We can see that with I being replaced by force, and V by velocity, that C corresponds to a mass, and that L is related to the spring constant.

So I expect the left end of the Newtonian bar with a force acting on it to move at some constant velocity that's proportional to the force, and for this motion to propagate at the speed of sound in the material, just as the voltage propagates as a wave down the transmission line.

It should be possible to confirm this with a standard text - I'm afraid my treatment is very off the cuff here.