meopemuk said:
The 2nd postulate of special theory explicitly refers to the behavior of light or photons (the speed of light is the same in all frames). It does not say anything about other particles - protons, electrons, etc. It doesn't say anything about particles interacting with each other. So, in fact, if one wants to apply special relativity universally to all physical systems, one should introduce a 3rd postulate, which would say that all kinematical relations established in SR (Minkowksi space-time, 4-tensor calculus, etc.) are universally applicable to all physical systems without exceptions.
I agree, but this is a weakness of Einstein's postulates,
not of special relativity. Einstein's postulates are ill-defined and incomplete. They can't be taken as the starting point of a rigorous proof of anything, and they can't be taken as the definition of special relativity. They should be viewed as loosely stated guidelines that can help us
guess what mathematical model we should use in a new (in 1905) theory of space and time.
That mathematical model is Minkowski space, and the only definition of special relativity that makes sense is the one I've been advocating in this forum. SR is defined by a set of postulates that tell us how to interpret mathematical statements as predictions about the results of experiments. Unfortunately, I have never seen a complete list of the postulates that are needed to define SR (remarkably, no physics book I've seen has even attempted to define what "special relativity" is), and I haven't worked it out myself. Well, not completely anyway, but I can tell you that the two most obvious postulates are:
1. Physical events are represented by points in Minkowski space. (Note that this implies that the motion of a classical object can be represented by a set of curves in Minkowski space, and that this suggests that we define a classical particle to be "a physical system whose motion can be represented by exactly one such curve").
2. A clock measures the proper time of the curve in Minkowski space that represents its motion.
It's clear that we also need to postulate something about measurements of length, but this is more difficult because of Lorentz contraction. There might be more than one OK way to do it, and one of them seems to be to postulate that lengths are measured by rulers, and that a solid object (like a ruler) that's accelerated gently undergoes Born rigid motion (which guarantees that if we slowly change its velocity, its measurements before and after the acceleration will be consistent with the Lorentz contraction formula).
Unfortunately, this postulate is inappropriate when we formulate theories of matter in this framework (as described in my previous post). It must be possible to possible to use those theories to
prove the Born rigidity of an accelerating solid. However, that result isn't going to be a testable prediction of the theory unless we use something other than solid objects to perform length measurements.
If we can't use solids, it seems natural to use light instead. We can use radar to define distance. If we emit a signal at time t
1 (as measured by a clock attached to the radar device) and detect the signal coming back at time t
2, then the distance to the reflection event can be defined as (t
2-t
1)/2. But of course this only works as well as we'd like if the radar isn't accelerating. So we can take the third postulate to be that "lengths are measured by radar devices that move on geodesics",
or that "
infinitesimal lengths are measured by radar devices".
Hey, I learned something.
Some of these things weren't perfectly clear to me before, in particular the reason why the postulate about length measurements should use radar instead of rulers.
meopemuk said:
The point made by Brown and Pooley is valid and non-trivial. They basically say this: if special relativity is correct, and if our dynamical theories indeed agree with special relativity, then we should be able to *calculate* the length contraction and time dilation effects within our dynamical theories, and the result should agree exactly with SR predictions.
For example, we could take a relativistic theory of inter-atomic forces and first calculate the equilibrium length of a rod in the reference frame at rest. Then repeat the same calculation in a moving reference frame. If everything is consistent, we should obtain *exactly* the length contraction predicted (but not explained) by SR. Unfortunately, as far as I know, there were no such convincing calculations for the length contraction and time dilation effects.
I'm not sure I understand what you're saying here. Are the "relativistic" "dynamical" theories you're talking about formulated in the framework of special relativity or not? (See my previous post for a clarification of what I mean by the "framework"). If they are, then this is all very trivial. One of the things that such a theory of inter-atomic forces would tell us is that the length of the moving rod can be calculated by calculating its rest length and doing a Lorentz transformation.
Actually, it seems equally trivial if you're talking about theories that aren't formulated in this framework (like a theory with Lorentz invariance and a spacetime that has a preferred frame). If the theory is Lorentz invariant, then it's only a matter of calculating its rest length and doing a Lorentz transformation.
Note also that these authors aren't criticizing those theories on the grounds that they can't be used for these calculations. They are criticizing the
standard formulation of SR on the grounds that it doesn't
explain why the moving rod has a different length than it did when it was stationary. (At least that's my interpretation, but maybe I read it too quickly).
I think you should be talking about a rod that's initially at rest (in some inertial frame) and then gently accelerated for a while, so that its velocity (in the same frame) when the acceleration period is over is v. Maybe that's what you meant. I would interpret what you said as involving two rods of the same material that have been moving at constant but different velocities forever. If the situation that concerns you is one rod moving at different velocities at different times, then I feel that what I said earlier in this post answers it pretty well. A theory of matter formulated in the framework of (a properly defined version of) SR
does predict the Born rigidity that implies that if you change the velocity of a rod, its new measured length will be consistent with the Lorentz contraction formula.
Of course it doesn't! It's not a theory's job to explain itself. Its job is to predict the results of experiments.