Austin0:
Austin0 said:
We are talking about systems accelerating along a linear path. No rotation. No systems moving at angles relative to the direction of motion.
Limited to this context do you think moving mass around before or behind the line would shift the CoM relative to the direction of travel?
Yes. Suppose, for example, that our system is composed of two subsystems, a front segment and a rear segment. The segments are initially at rest, and are identical in length and mass in the original rest frame. That means your imaginary line would be drawn precisely at the boundary between the two segments.
Now apply a forward impulse to the rear edge of the rear segment. The rear segment will compress, but until a time equal to the length of the rear segment divided by the speed of light has elapsed, the front edge of the rear segment will not move. So we have the rear segment compressed, meaning the CoM of the rear segment (and hence of the whole system) moves forward, but no mass crosses your imaginary line.
Austin0 said:
You have just given a description of Born acceleration. WHich is itself basically a definition of coordinate events.
The events described are exactly those of the essential Lorentz contraction and lead to exactly the same world lines.
SPECIFICALLY: the fundamental postulates propose that internal proper distances remain the same in inertial frames. ANd that the Lorentz contraction is regarded as a purely kinematic effect.
No. The worldlines of each segment of the ship are accelerated worldlines, not inertial worldlines. The MCIF at each event along those worldlines is an inertial frame, but the ship segments are at rest in each MCIF only momentarily. So the fact that "internal proper distances remain the same in inertial frames" is irrelevant, since that only applies to objects moving on inertial worldlines.
You are correct that the distance between any two adjacent segments (i.e., between appropriate events on their worldlines), which remains constant in each MCIF along the worldlines, will appear to be increasingly Lorentz contracted in the "initial" frame, the one in which Born rigid acceleration begins at time t = 0. This is indeed a "kinematic" effect. However, the physics that determines which worldlines each segment follows is *not* just "kinematics"; it requires knowledge of forces and accelerations, which are genuine physical quantities. See further comments below.
Austin0 said:
SO on the basis of this and the clock hypothesis, you could plot the worldline of the back of a normally propulsed ship, simply on the basis of constant proper acceleration.
You could then plot the front of the ship on the basis of instantaneous Lorentz contraction
and would then have a pair of worldlines that were indistinguishable from Born worldlines
Correct?
Not really. You are ignoring the internal forces between the parts of the ship. Remember how I answered your question about simple rear thrust and Born acceleration:
You're correct here too, in the sense that simple rear thrust could lead to a steady-state equilibrium in which the object as a whole was undergoing Born rigid acceleration; but in that equilibrium state, there would be internal stresses in all parts of the ship, in addition to the rear thrust.
So the object would only be undergoing Born rigid acceleration *after* all the oscillations of internal forces had damped out, and the ship had settled into a steady-state equilibrium. In that equilibrium state, the *net* force on the rear end of the ship would be *less* than the rocket thrust applied there, because there would also be a compression force from the next segment forward that would act towards the rear, in the opposite direction from the thrust. So the *final* proper acceleration of the rear segment would be *less* that the proper acceleration you would calculate just from the rocket thrust alone.
When I gave the proof that Born rigid acceleration preserves internal distances, I ignored the complications above. To make my specification consistent with those complications, we would simply define the "initial" reference frame, the one in which each segment starts accelerating at time t = 0 in accordance with the precise profile required for Born rigid acceleration, to be the MCIF of the ship at the instant when all the internal oscillations are gone and the steady-state equilibrium has just been reached. The ship as a whole would be momentarily at rest in that frame, and the accelerations on each segment of the ship (due to internal forces, plus the rocket thrust at the rear end only) would be in just the right profile to produce Born rigid acceleration from then on. In essence, we would be ignoring the details of how the ship got to the steady-state equilibrium and just focusing on the equilbrium state itself.
If we do focus just on the equilibrium state, then you're basically correct that we can calculate what all the worldlines have to look like for Born rigid acceleration based on relativistic kinematics. However, doing that can lead to confusion if we forget that we were originally led to calculate those worldlines by specific *physical* assumptions, so our calculations are only relevant in specific physical situations where those assumptions are applicable.
Austin0 said:
Based on these worldlines it would inevitably appear as if there had to be unequal acceleration at the front and the back. But this does not neccessarily have any physical meaning.
Wrong. The acceleration of a worldline is simply the derivative of its 4-velocity with respect to proper time:
A = \frac{dU}{d \tau}
This is an invariant, the same in all reference frames, so it's a bona fide physical quantity. (An "inertial" worldline is just one for which A = 0.) For the specific worldlines I gave for Born rigid acceleration for each segment, you can compute this acceleration, and verify that it is equal to x_s for each segment (i.e., the original x-coordinate of the segment in the "initial" frame).
Austin0 said:
So aside from the description, the Born hypothesis is making a statement about the nature of the Lorentz contraction and is proposing that it will not take place through equal acceleration throughout a system , which will result in the opposite result . Catastrophic expansion.
You're correct that equal *proper* acceleration of all parts of a system will result in expansion, since equal *proper* acceleration of all parts is enough to specify the worldline of each part, and from there it's all kinematics, as I've said before. However, this doesn't mean that the Born hypothesis is just a statement about kinematics. I've already addressed this somewhat in my comments above, but let me restate them in more detail:
-- The original motivation for working out the specification of Born rigid acceleration was to determine what kind of accelerated motion would result in unchanging proper distances between the parts of a system.
-- The reason for wanting to know *that* was the *physical* assumption that internal stresses between the parts of a system depend on the proper distances between those parts. If that weren't the case, nobody would care about Born rigid acceleration; we'd be discussing some other specification (whatever one turned out to keep internal stresses constant).
So the original motivation for studying Born rigid acceleration was physical, not just kinematic. Also:
-- The proper acceleration of each part of a system depends on the *net* force on that part, *not* just the externally applied force. You must also take into account the internal forces between the parts, as we've seen already.
So, as we've seen before, equal proper acceleration does *not* necessarily mean equal rocket thrust. You need physical assumptions to determine *how* equal proper acceleration would be realized (or any other acceleration profile, for that matter).
Austin0 said:
It also seems to place a meaning to lines of simultaneity that I don't quite understand.
The meaning of each of the lines of simultaneity (lines of constant q in Greg Egan's diagram) is just what it seems: each line of constant q is the line of simultaneity for the MCIF of each worldline it crosses, at the point where it crosses that worldline. In other words, all events on a line of constant q will be seen to be simultaneous by observers traveling on any of the accelerated worldlines we've specified (the lines of constant s in Greg Egan's diagram).
What may be confusing you is the fact that all of those lines of simultaneity intersect at the origin of the "initial" frame, x = 0, t = 0. That may seem paradoxical, but it's not: it's just an unusual fact about accelerated worldlines. It does, however, mean that the "pivot point" (x = 0, t = 0 in this case), and the past and future light cone at that point (the lines t = x and t = -x; part of the t = x line appears as a 45-degree dotted line on Greg Egan's diagram) have a special meaning to observers traveling on those accelerated worldlines. Look up "Rindler horizon" for more information (Greg Egan's page has some discussion of this).
Austin0 said:
v_{1-before} > v_{2-after} > v_{3-after} > v_{4-after} > v_{5-after} > v_{6-after} .... > v_{N-after} = 0
...
Perhaps you could explain this . You say there are a finite number of objects among which the momentum can be shared but the momentum can never reach zero in that direction. Well that would seem to neccessitate an infinite number of objects or infinite division of the quantity of momentum .
I don't understand how you're coming up with these conclusions. Let me re-state once more what I've been saying:
-- If the total momentum of a system is p_x before some process, the total momentum of the same system (provided there are no external forces on the system) after the process will also be p_x (we're assuming here that all motion is in the x-direction only).
For example, in the case of N spheres, we have (when sphere #1 is the only one with any momentum before the process):
p_{1-before} = p_{1-after} + p_{2-after} + ... + p_{N-after}
We appear to have agreement on this; but you may not be realizing that when I say "the momentum can never reach zero in that direction", the above, conservation of momentum, is *all* that I'm saying. I'm *not* saying that the momentum of one particular sphere could not be zero; only that the *total* momentum, the sum of the momenta of all the spheres, must remain constant, so if it starts out non-zero, it must end up non-zero (meaning that at least one sphere must have a non-zero momentum after the process).
-- If the original momentum, which was carried by only one sphere before the process, is shared between two or more spheres *after* the process, we will have
p_{1-before} > p_{i-after}
for any value of i from 1 to N (i being the number of any particular sphere). But this does *not* imply that the momenta are strictly decreasing as we go from left to right (i.e., as i increases from 1 to N). Nothing I've said implies that, and in general it won't be true. See below.
-- In the presence of dissipation, the rest mass of any given sphere after the process will be greater than it was before the process. Thus, the velocity of any given sphere after the process will be *less* than it would have been without dissipation (because some of the energy of the process goes into increasing the sphere's rest mass instead of its velocity). So even in a limiting case where virtually all of the original momentum ends up in a single sphere after the process, we will still have
v_{1-before} > v_{i-after}
for any value of i from 1 to N. But this, again, does *not* imply that the velocities will be strictly decreasing as we go from left to right (i.e., as i increases from 1 to N). In fact, since the original momentum was to the right (increasing x-direction), it is easy to see that, since sphere i + 1 is to the right of sphere i, and since one sphere can't move through another, we must have
v_{i-after} <= v_{i+1-after}
for any value of i from 1 to N - 1. If this were not true, we would have, for some i, sphere i moving to the right *faster* than sphere i + 1, which is to its right, *after* spheres i and i + 1 have collided. That's obviously impossible.
Does this help any?
Austin0 said:
Well if any of them were less than a later sphere then there were seem to be a definite violation of conservation of momentum. Energy out of nowhere
Once again, I don't know how you're coming up with this conclusion. Let's consider the simplest interesting case, with three spheres. Before the collisions, sphere #1 has velocity v_0 in the positive x-direction, and spheres #2 and #3 are at rest. After the collisions, sphere #1 has velocity v_1, sphere #2 has velocity v_2, and sphere #3 has velocity v_3 (all velocities are in the x-direction, but we don't yet know anything about their signs). Also, before the collisions, all spheres have the same initial rest mass M; after the collisions, dissipation has increased the rest masses of all the spheres by the same factor k > 1.
The equations for conservation of energy and momentum then look like this, using the fully relativistic formulas (I've eliminated the common factor M in both equations, since it appears in every term):
\gamma_0 + 2 = k \left( \gamma_1 + \gamma_2 + \gamma_3 \right)
\gamma_0 v_0 = k \left( \gamma_1 v_1 + \gamma_2 v_2 + \gamma_3 v_3 \right)
where we have written the relativistic \gamma factors with the same subscripts as their corresponding velocities. Unless I'm misunderstanding you, you're claming that it must be the case that p_{2-after} > p_{3-after}, or
\gamma_2 v_2 > \gamma_3 v_3
or the above conservation equations will be violated. That's not correct; there are plenty of solutions of the above equations for which \gamma_2 v_2 < \gamma_3 v_3. (In fact, if we consider the additional constraint I gave above, that the spheres can't move through each other, we see that we must have v_2 <= v_3, meaning that
\gamma_2 v_2 <= \gamma_3 v_3
must hold.)