Accereration of Relativistic Scaled Objetcs

[Korvan]

I couldn't fall asleep last night because a strange thought popped into my head. What would happen if you had a rod of an extremely long length, say a light year, and gave it a really hard shove at one end? My brain would not net me sleep until I had worked through the problem and the results were a little surprising to myself, but likely obvious to those more familiar with advanced physics. I got a good grasp of special relativity, but I have to admit I could never really get my head around the details of the general theory. To me a tensor is a type of bandage you use when you sprain your ankle.

In my thought experience I considered the case of a light year long rod of a very elastic substance, mass indeterminate. There would be no external gravitational fields to worry about and I would discount gravity generated by the rod itself. I know that the pushing on end of the rod will generate a force and that force will propagate through the rod with a maximum speed of the speed of light. In this case I made the assumption that that it would propagate at the speed of light (more on this later).

So from the time I push on one end or the rod, the far end won't hear about it until a full year later. But what is happening during that year? Since the rod is very elastic (as is all matter to some extent), it will compress as its particles press closer to their neighbors. This compression will be cumulative until the force finally reaches the end. So, lets say I give a really hard shove, hard enough that the the by the time the force reaches the end the rod has compressed to half its length. Once then force reaches the end particles, they are pushed forward opening up space for the particles behind them to expand. This expansion propagates back down the length of the rod and since the rod is uniform, the expansion will take the same amount of time of the compression, namely a year. What follows is a crude diagram of the process...


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The first line shows the rod at the start, the second line shows the rod after one year, and the last line shows the rod after two years. It is interesting to note that during the first year, the far end remained stationary and during the second year it is the near end that remains stationary. At the end of the second year the rod will be moving at a new velocity, all relative to its starting position.

The rod's final velocity can be calculated by looking at the diagram and seeing that the rod moved 1/2 its length over a period of 2 years starting with a velocity of 0, so with Vf = 2d/t it is now moving at 1/2 c. Since it took 2 years to reach this velocity, we can calculate its acceleration to be 2.36 m/s2.

I then worked out a general formula for the acceleration. Starting with a = 2d/t2. d is the length of the rod multiplied by a compression factor which is a constant between 0 and 1 (in our example it was 1/2). So we now have a = 2Cf L/t2. Now t is the twice time it takes for the force to reach the end of the rod, so given the length of the rod L and the speed that the force propagates (c in our example) we get t = 2L/Sp.

The final formula for acceleration is a = 2Cf Sp2 / 4L

Plugging in the numbers from our equation gives us the same answer we worked out before, a = 2.36 m/s2.

Now, from the formula, F = ma, thus a = F/m it seems odd that the formula I worked out for acceleration contains neither mass nor force. However, I did a bit of checking and found out that the speed of which a force propagates in a solid is actually the speed of sound in that solid (limited to the speed of light of course) and the speed of sound in an object is given by the general form, Sp = √C/rho where C is the coefficient of stiffness and rho is the material's density. Now the coefficient of stiffness involves calculating the compression (or expansion) of a material by a given force and density is of course mass/volume. So it seems there is a direct link between my formula for acceleration and a = F/m.

So in the end, that was a lot of math just to tell us what we already knew but I thought it was interesting to look at acceleration in a different way than I was used to. What I thought was really fascinating was that acceleration actually consists of discrete pulses of compress and expansion with the ends of objects remaining stationary with respect to the initial rest frame of the object. This would hold even at non-relativistic lengths, though the pulses would be short in both magnitude and duration that the acceleration would appear to be uniform.

[Wyrm]

So in the end, that was a lot of math just to tell us what we already knew but I thought it was interesting to look at acceleration in a different way than I was used to. What I thought was really fascinating was that acceleration actually consists of discrete pulses of compress and expansion with the ends of objects remaining stationary with respect to the initial rest frame of the object.

I wouldn't go that far as thinking that acceleration consists of "discrete pulses" of compression and expansion. The waves of compression and expansion serve to communicate to the rest of the structure that it has undergone an acceleration at one end. Under steady acceleration, the waves eventually die down and the all parts of the rod are under stress that is constant with time. Thus you can have acceleration that causes no discrete pulses.

This would hold even at non-relativistic lengths, though the pulses would be short in both magnitude and duration that the acceleration would appear to be uniform.

Only if the material is much stiffer than the accelertion you wish to put it under — that is, when the acceleration is reasonably small in comparison to the speed of sound in the material. The point at which mundane materials behave like jello is actually quite far below relativistic regime.

[Kuroneko]

In this case I made the assumption that that it would propagate at the speed of light (more on this later).

Realistically, it would be much less, and your experiment would form a shock wave that would break the rod. Actually, there is a theoretical problem with the (im)pulsed idea that restricted their strength: if the rod length is L, then the endpoint acceleration must be bounded by c²/(2L), which is 4.75 m/s² for a light-year long rod, because otherwise, the endpoints will be causally disconnected due to a Rindler (event) horizon.

So, lets say I give a really hard shove, hard enough that the the by the time the force reaches the end the rod has compressed to half its length. Once then force reaches the end particles, they are pushed forward opening up space for the particles behind them to expand.

Alright. You seem to be using this as a limiting case--the kick was just hard enough to get there, at which point the rod stops compressing and all the energy of the kick is stored as elastic potential.

Despite the fact that it's very dubious for high compressions, treating Hooke's law literally gives an elastic energy density
[1] U = (1/2)kε²
per uncompressed volume, where k is Young's modulus, and ε is the strain. By the dominant energy condition, the magnitude of the stress should be bounded by the energy density for the speed of sound to be no larger than the speed of light:
[2] -kε ≤ (ρc² + U)/(1+ε)
where ρ is the (uncompressed) rest mass density. This will allow you to give a bound for the allowed Young's modulus for you rod, and hence how much energy your rod is able to store:
[3] k ≤ -2ρc²/[ε(2+3ε)] = 8ρc²,
the last equality following from ε = -1/2. For a rod right at the dominant limit, this gives an elastic potential energy equal to the rest mass energy, which means the final velocity will be γ = 2, i.e., v/c = sqrt(3)/2, if it eventually becomes all kinetic.

There's an obvious problem with the above model if the rod is compressed a bit more--but then Hooke's law is very unrealistic here, since according to it, it's possible to compress the rod to zero size by applying finite pressure.

What I thought was really fascinating was that acceleration actually consists of discrete pulses of compress and expansion with the ends of objects remaining stationary with respect to the initial rest frame of the object.

Sort of, but not quite how you're thinking. Classically, one can view them as pulses, but with a continuous spectrum. That's a very general property of fields: the underlying mechanism is the electromagnetic field, which can be decomposed through Fourier's trick in that manner. Even an electrostatic field can be thought of as the superposition of electromagnetic waves 'caused' by disappearance and reappearance of the charges, in the instantaneous limit. More toward your situation, a settled, constant push can be similarly decomposed.

[Korvan]

Thanks for the replies. I have to admit I'm a little out of my depth as I didn't progress past 2nd year physics before I switched majors to computer science. The one good thing out of this is I have a better grasp of the scope of things that I don't really understand. Hopefully I'll have a more restful sleep tonight!

[Kuroneko]

The validity of the dominant energy condition is not obvious. But other than that, the rest is conceptually straightforward: you compressed the rod until it was half its original length, so the energy provided is as the work it takes to do so. Hooke's law is a generic model for "springy things" that assumes that force is proportional to displacement: F = -kx. Then the work done is the integral of this, W = (1/2)kx², and this should be the stored elastic potential energy.

Now, this k is not the Young's modulus in the above calculation, since the strain ε is a dimensionless measure of the deformation (-1<ε<0 for compressions), but it is proportional to it, so except for unit conversion, it's virtually the same thing. So if you simply assume DEC, [2] follows directly: stress magnitude is bounded by energy density, which is (rest mass + elastic potential)/volume, and the compression multiplies the volume by 1+ε. The rest is just algebra.

It's unrealistic to assume Hooke's law in this extreme case, for reasons already covered, but without some model the problem is undetermined, because the energy provided would not be known. In terms of engineering, we've basically assumed that the yield strength of the rod is infinite. Maybe something else too (I'm not sure, but an engineer could probably make a laundry list of things wrong with rod).