Force Law Transformations

[GRDixon]

In SRT, Force (F) transforms identically to d(mV)/dt, which can in turn be transformed using the Lorentz transformations and the dependence of m upon speed. This raises the question whether Force in the force laws also transforms the same way among different reference frames. Certainly the Lorentz force does. But what about the other force laws (Hooke’s Law, Nuclear forces, gravitation, etc.)? Should it be a requirement that the force in every force law transform identically to d(mV)/dt? If so, does this suggest that we can learn something about the behavior of moving springs, etc.?

[Bill_K]

Your F transforms in a weird way because of the noncovariant way in which you have chosen to define it. Force is correctly defined as a 4-vector, F = dp/dτ where τ is the proper time and p is the momentum 4-vector, p = (γmv, γmc).

[bcrowell]

Yes, all forces transform the same.

Is your F the four-force, or the three-force? Is your V the four-velocity, or the three-velocity? Is your t coordinate time, or proper time?

The easy way to understand this is to deal exclusively with four-vectors. All four-vectors transform the same. Since all four-forces transform the same, all three-forces (which can be found from the corresponding four-forces) also transform the same.

Relativistic elasticity is hard: http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/SimpleElasticity.html

The connection between the four-force and the three-force is given here: http://en.wikipedia.org/wiki/Four-vectors#Four-force

[pervect]

Well, if you have a conserved momentum, I suppose dp/dtau is a good definition of force. But in general,without a conserved momentum, I'm not sure if gravity even has a definition as a "force".

Most textbooks don't really seem to address the question of whether or not gravity can be defined as a force,they simply point out the way in which GR defines gravity, which is as a curvature of space-time. That's sufficient for the student of GR,but it doesn't always answer the questions of someone who isn't familiar with the subject and is more or less determined to force gravity into the mold of a force, willy-nilly, without really asking if it's a good idea or not.

But it's an interesting question, if it isn't a good idea, why would that be so? Probalby someone somewhere has addresssed the issue,but I've never read anything about it, so a lot of what I've comeup with has been the result of my own experience.

I've got some reasons to believe that curved space-time in general transforms differently than a force, which I've mentioned before - which boils down to curved space-time transforming differently (as a rank-4 tensor) than force does (a simple vector). But it'd be nice to have some papers to see if I've missed something.

[atyy]

Well, if you have a conserved momentum, I suppose dp/dtau is a good definition of force. But in general,without a conserved momentum, I'm not sure if gravity even has a definition as a "force".

How about http://arxiv.org/abs/gr-qc/0411023? I think it recovers GR when spacetime can be covered by harmonic coordinates. Weinberg's old GR text mentions that some other coordinate systems are also allowed, but he doesn't say which ones. Apparently harmonic coordinates can penetrate the event horizon http://prd.aps.org/abstract/PRD/v56/i8/p4775_1. I recently came across the "Weinberg low energy theorem" which apparently even derives the equivalence principle from momentum conservation http://arxiv.org/abs/1007.0435!

[pervect]

I'll take a look at these when I get some free time. Thanks!