pervect said:
Now that I know that you are calling the metric coefficients guv "gravitational potentials", some of your remarks make more sense. But I should note that this usage is not standard in any of my textbooks, and I would imagine it would confuse most readers as much as it confused me.
You're kidding me! Almost all of my GR texts call them that as did Einstein. For example: The following texts refer to g
uv as "gravitational potentials" -
Gravitation and Spacetime, Ohanian & Ruffini, WW Norton n& Co., (1994)
Introducing Einstein’s Relativity, D’Inverno, Oxford Univ. Press, (1992)
Basic Relativity, Mould, Springer Verlag, (1994)
I also recall seeing it in MTW and in Wald but I can't locate it at the moment
Since we have a perfectly good name for the metric coefficients already ("metric coefficients"), and since this usage causes *no* confusion, I would like to suggest that we continue to call metric coefficients metric coefficients.
I've never heard them called that. They are called the components of the metric. But you're free to call them what you like. But when you start to discuss gravitational potentials in GR then you're talking about g
uv whether you want to call them that or not.
Since the energy of a particle will in general depend on the path it takes, I don't see how to define "the energy of the particle by virtue of its position".
The energy of a particle is a function of velocity, position and rest mass. The functionality of position is what I mean by potential energy. I did not say that you can separate these energies into individual pieces. Let me quote Ohanian, page 157
P_0 = \simeq \frac{m}{\sqrt{1-v^2}} - \frac{1}{2}\frac{m}{(1-v^2)^{3/2}} h_{\mu\nu}\frac{dx^{\mu}}{dt}\frac{dx^{\nu}}{dt}+mh_{0\alpha}u^{\alpha}
The first term on the right side is of the form of the usual rest-mass and kinetic energy; the other terms represent gravitational interaction (potential energy).
Oh, weak fields. Sure, one can define potential energy for weak fields, the first thought that comes to mind is to use PPN. But I thought we were talking about GR, not the weak-field version therof.
You understand that by weak I do not mean Newtonian, right? Why do you think that the weak field approximation is not part of general relativity. I don't see any need to make such a distinction myself. I understand that you think that they are different but I don't see them that way.
In any case there are cases where, even in strong gravitational fields, the gravitational force is given in terms of the gravitational potential as
\bold G = -m\nabla \Phi
where m = inertial mass (aka relativistic mass). See derivation (and meaning of phi = gravitational potential) at
http://www.geocities.com/physics_world/gr/grav_force.htm
Its my turn to ask you something - A particle in a gravitational field has energy P
0 where
P is the 4-momentum of the particle. Do you think that a particle in a gravitational field has rest energy? If so then do you think that the rest energy is part of the energy P
0? Its a mixture of these energies which one cannot separate into nifty pieces. Notice that Carlip said that potential energy is not
well defined. He did not say it does not exist or that it is totally meaningless.
By the way, I posted a web page quoting that part of D'Inverno on this metric = potentials in case you don't have that text. See
http://www.geocities.com/physics_world/gr_potential.htm
As you can see, the term "potential" is used here in a similar way to its used in EM where the magnetic field is the curl (which also involves derivatives) of something called the magnetic vector potential. The force on a charged particle can therefore be written in terms of the derivatives of potentials, i.e. the Coulomb potential and the magnetic vector potential. That is the reason for calling the components of the metric "potentials".
Pete