Understanding Potential and Potential Energy in Relativistic Electrodynamics

[pmb_phy]

I'm curious. How many of you know what potential is and potential energy is and what role it plays in relativistic electrodynamics. How many of you know the difference between 4-momentum and generalized 4-momentum. Do you know what the time component of each of these 4-vectors are?

Thanks

Pete

 

[pervect]

I'd answer a qualified yes to the first question - I would assume you're talking about the 4-version of the magnetic vector potential A if you started talking about 4-potentials in electrostatics, the 4-vector that generates the Faraday tensor when you take it's exterior derivative.

But it's possible that you'd be talking about something completely different the way things have been going :-)

As far as generalized momentum goes, I'd answer qualified no. Usually a generalized momentum is derived from a Lagrangian, so I'd guess that that's what you were doing, but I don't think I've seen people talk about generalized 4-momenta.

 

[pmb_phy]

I'd answer a qualified yes to the first question - I would assume you're talking about the 4-version of the magnetic vector potential A if you started talking about 4-potentials in electrostatics, the 4-vector that generates the Faraday tensor when you take it's exterior derivative.

Yes and no. There is a relativistic generalized potential which appears in the Lagrangian. Off the top of my head (and check me on this) this is U = qA*u - q*Phi. The Phi is the Coulomb potential. V = q*Phi is the potential energy of the charged particle in an EM field. The total energy, W, is an integral of motion and has the value

W = K + E_o + V

where V = potential energy.

As far as generalized momentum goes, I'd answer qualified no. Usually a generalized momentum is derived from a Lagrangian, so I'd guess that that's what you were doing, but I don't think I've seen people talk about generalized 4-momenta.

Its well worth your time to check into this. I recommend Goldstein's "Classical Mechanics - 3rd Ed."

I think this is an interesting subject myself. I've been thinking about it more lately and will be making new web pages to describe/discuss it (better than trying to post it all here in Latex ) But I have to wait until my modem comes. Its being shipped by UPS and will be here within a week. I'll be back more then and can get more detailed then too.

One interesting notion is the idea of gauge transformation. E.g. if you take the make the following gauge transformation A -> A + grad Psi, then you have to change Phi to Phi -> Phi - @Psi/@t (@ = partial derivative). Psi is a function of space and time. Then the electric field

E = -grad Phi - @A/@t

becomes

E = -grad (Phi - @Psi/@t) - @(A + grad Psi)/@t

which simplifies to

E = -grad Phi - @A/@t

and therefore the field remains unchanged as it must. But if you make this gauge transformation in the Lagrangian then what you're really doing is adding d(Psi)/dt to the original Lagrangian. Since this is the time derivative of a function of space and time it can be deleted from the Lagrangian since both L and L' = L + d(Psi)/dt yield the exact same equations of motion. However the time component of the generalized 4-momentum changes from W to W + @Psi/@t. In such a case the time component of the generalized 4-momentum is no longer the total energy. Interesting stuff.


Happy Thanksgiving by the way!

Pete

 

[pmb_phy]

Okay. I worked through what I had to regarding potentials and gauge transformations. Its all online now at

http://www.geocities.com/physics_world/em/gauge_trans.htm

Pete

 

[speeding electron]

Nice work there pmb phy! Thanks.

 

[pmb_phy]

Nice work there pmb phy! Thanks.

You're most welcome. Please call me Pete.

I plan on making many more. During the worst part of my convelesance (i.e. back surgery) I had plenty of time to think about some of the more tricky things.


That one topic was in response to someone (whom shall remain nameless since its from another forum/newsgroup) who claimed that potential and potential energy is meaningless in special relativity. I used the Coulomb potential as an example since its the time component of a 4-vector, i.e. the 4-potential. That person's response was to claim that since the time component can be transformed away by a gauge transformation then it was meaningless. I found that impossible to believe and it seemed like a vastly mislead comment. It seemed to me that such a claim was meant to imply that since the Phi can be transformed away that what it does also goes away by such a transformation. However as those calculations show - the gauge transformation can null out the time component of the 4-potential but it does nothing to the total energy and hence it does nothing to the potential energy.

I later realized that I ended up answering two homework questions from Goldstein's Classical Mechanics - 3rd Ed text.

So it was a useful exercise. Glad you liked it. There's a bunch of stuff I want to write up that can't be found in texts so that's how I'll be spending my time during the rest of my convelesance.

Pete