Physically, the effect can be interpreted as an object moving from the “false vacuum” (where = 0) to the more stable “true vacuum” (where = v). Gravitationally, it is similar to the more familiar case of moving from the hilltop to the valley. IN the case of the Higg’s field the transformation is accompanied with a “phase change“, which endows mass to some of the particles

G -> H -> … -> SU(3) x SU(2) x U(1) -> SU(3) x U(1).

Neither would I be to upset that what was illucive in terms of energy particle discriptions in Agasa was the continued struggle to describe what mathematics was approaching in terms of these new higher energy particles.

This was the original, and applying to what remains illucive until actual experimentation is just something we do in descrbing the mystery. We understand the nature of the scientists work here.

Some may have other agendas?

And, yeah, I know, I will get hammered, “religion is mostly represented at least publicly by fundamentalists in the US today, yadda yadda yadda.” I get that every time I do this. But I think it very important to emphasize that creationism is not religion. As somebody who’s religious, I want God back, dammit, too! I want to take back religion so that folks like you don’t feel the need to hate it. I’m pissed at the fundamentalists for driving us to that. My way of fighting this is reminding everybody where possible that you don’t have to be a creationist to be religious… and that if you don’t feel the need to stupidly and literally interpret your religious text, perhaps you won’t feel the need to stupidly and literally interpret every metaphor spoken by a scientist.

-Rob

Count Iblis, I agree that if you do an experiment in which the incoming state for our system S is completely isolated from the environment, and the outgoing state for S is just a boosted copy of the ingoing state, then an identical force applied to S for an identical time will lead to an identical change in velocity, in whatever direction the force is applied. That follows from conservation of energy-momentum.

Nevertheless, I believe it’s still possible to have anisotropic effects taking place during the interaction. If you imagine, say, a small square of elastic material subject to forces of 100 N and 101 N across the x-direction, and 200 N and 201 N across the y-direction, it will be subject to a net force of 1 N in both directions, but the different average tensions will give rise to small differences in its accelerations in the two directions. Given some complex anisotropic system, all that the conservation laws guarantee is an identical net effect over the course of the whole interaction, not a constant, direction-independent ratio between force and acceleration, holding moment by moment.

That said, everything I’ve read about protons since first raising this question makes it seem unlikely that anyone will be measuring such an effect for a proton any time this century.

Greg: I had an article in Physics Essays years ago about the problem of accelerating a mass at the end of a long string. The wild west of extended body dynamics in relativity is one of my favorite avocations, and surprisingly it has been argued and disagreed about for years in journals. Ken Nordtvedt wrote about the equivalent gravitational situation earlier, in AJP. His effect, that a mass suspending pulls less on the holding end than the mg measured locally at the mass, didn’t get the attention it deserved (for example, it never AFAIK appeared in writings as a cute implication of GR, like “You could hold up an elephant in earth-style gravity if you had a long enough cord.”) Maybe one reason is, the increasing magnitude of the hyperbolic gravity field just cancels that “red shift” of weight, so you hold things anyway with the mg using g where your hand is! This confuses the issue and got critic Ø Grøn of Norway all worked up, and in error in my view. That guy sure was a booger for me too later, as referee of my own paper.

In any case, I’m sure you appreciate that stressed bodies in motion have a correction to the usual expressions for mass and momentum. (One illustration: if co-moving observes apply forces simultaneously to a rod seen by us in motion, the rod just sits there for them and no velocity increase for us. However, we see the rear force applied earlier. That puts “extra momentum” (f delta t) and energy (f dot v delta t) into the rod. The lateral shear version really explains the infamous “right-angle lever paradox.” When you apply those corrections to moving bodies, everything is supposed to work out (and per fundamental theorems), but it does make a mess when the stuff is accelerating. I finally got things to work out OK after the paper, where I couldn’t solve it at the time.

It should be the case that the anisotropic effects are described by the angular momentum. So, mass and angular momentum are the two relevant parameters that you can extract from the energy-momebtum tensor.

It may be more convenient to look at collisions instead of a steady force. It should be the case that you have conservation of momentum and angular momentum. Any anistropic effects should be a consequence of this…

Another way to look at all this is to think about boosts. [...] if you apply a boost to T, there’s absolutely no guarantee that the change in E will be independent of the direction of the boost.

Sorry, that assertion was dead wrong. The total energy-momentum vector you get by integrating T^{0a} for a closed system will transform like an ordinary 4-vector (Misner, Thorne and Wheeler, page 145, makes this clear), so the change in E will be independent of the direction of the boost.

This doesn’t change the rest of my argument. As I’ve said, I’m not 100% certain about any of this, but it seems clear that tension does modify the effective inertial mass of a composite object, and so anisotropic tension should give rise to an anisotropic effective inertial mass.

As I understand it, the nice simple formula they teach in elementary SR relating four-force, rest mass and four-acceleration, F=ma, is only strictly true for point particles (though in most situations it should be a very good approximation). A continuum object needs to be analysed with a stress-energy tensor, and if that tensor is anisotropic, the force needed to achieve acceleration of the body in different directions will be anisotropic too. (You emphasise the notion of “starting with the object at rest”, but that’s implicit in the whole idea of proper acceleration anyway: you measure proper acceleration in a frame co-moving with the body’s centre-of-mass.)

To take one simple example, suppose you have an elastic string being trailed by a uniformly accelerating body. I’ve analysed this scenario for a simple linear model of elasticity on this page. It’s not hard to show that conservation of energy-momentum, i.e. setting the divergence of the stress-energy tensor of the material to zero, yields the equation:

(1/s) [rho(s)+p(s)] + p’(s) = 0

where s is a spatial coordinate measured orthogonal to the world lines of the Rindler frame for the accelerating body, rho(s) is the proper density of mass-energy in the material (including elastic potential energy), and p(s) is the pressure (which will be negative, as the string will be under tension).

Now -p’(s) gives the net force on an infinitesimal element of the string, and (1/s) gives the acceleration of the hyperbolic world lines in a Rindler frame. So this equation resembles “F=ma”, but instead of rho(s) alone — the proper mass-energy of our element of string — we have rho(s)+p(s). This is due to the fact that the pressure/tension in an accelerating body is changing direction in space-time, and contributing to the momentum density.

However, if we calculate the force/acceleration relationship for an acceleration in a direction in which there is no tension, we will get a different effective inertial mass: just rho(s). So there is an anisotropic response to applied force.

Another example is the case of a rotating ring of elastic material. The total mass-energy of the ring in the centre-of-mass frame will be modified by kinetic and elastic potential energy, and if you compute the force needed to accelerate the ring with a given proper acceleration, a, in a direction orthogonal to the plane of the ring, it will simply be proportional to that total mass-energy. But if you accelerate the ring in the plane of rotation, the constant of proportionality, the effective inertial mass, will be different.

Another way to look at all this is to think about boosts. When you have a point particle, it simply possesses an energy-momentum vector, P, and its total energy as measured by an observer with 4-velocity u is just E=P.u. If you start with a particle at rest, E will initially equal m=|P|, and if you apply a boost to P then E will transform in a very simple way, which will be independent of the direction of the boost.

But when you have a composite system, E is found by integrating T^{00}, the time-time component of the system’s stress-energy tensor T. You can also integrate all four time components, T^{0a}, to get a total energy-momentum vector P. But even if you start with the centre-of-mass of the system at rest (i.e. the total momentum of the body in the observer’s reference frame is zero), if you apply a boost to T, there’s absolutely no guarantee that the change in E will be independent of the direction of the boost.