OK, if anyone is so inclined, let's continue the discussion here, since I think we are distracting from the main issues in https://www.physicsforums.com/showthread.php?p=3652601#post3652601 Anyway, may I suggest that the real thing we want to get rid of is not relativistic mass. Rather it is the concept of force and point particle. If we take Maxwell's equations as the reason for special relativity, then we can't really make the Lorentz force law and point particles work, can we? Without those, we don't need relativistic mass. OTOH, since force and point particles are useful in some regime, we keep relativistic mass around as a moral link between the new theories which deal primarily in fields, and the old theories in which point particles and forces were ok. This link is more moral than quantitative, since it has to be generalized to a form applicable to fields, but it is the historical route that indicated what sits on the right-hand side of the Einstein field equation.
I don't follow at all. In all of my experience, relativistic mass makes understanding force in SR enormously harder, because it doesn't substitute in any of the Newtonian force related formulas. Meanwhile, 4-momentum nicely generalizes them. Force in SR is dp/dτ ; it it not F=<relativistic mass> <acceleration [either 3-acceleration or proper acceleration]>. In pre-relativity physics, point particles were just as much an issue if you wanted to be a stickler - what's the density of a point particle? Short of quantum field theory, a point particle is just a practical idealization of a small, sufficiently rigid body, that works really, really well for many problems. Relativistic kinematics is just as well defined and useful as Newtonian. I agree that point charges and EM present difficulties, but these are more the fault of the EM as a field theory not meshing with point charges. This is, of course, is only resolved in QED. I don't see any connection to the utility or lack thereof, of relativistic mass.
Ok, in classical EM and GR one can (should) live without force, ie. define the dynamics in other ways so that force is a useful approximation, but not fundamental. Where actually is the first place in physics that one should get rid of force? Already in Newtonian gravity? How would you motivate the replacement in GR of Newtonian gravitational mass with the stress energy tensor?
Hmm, I don't follow much of this. To my mind, the unique idea of getting rid of gravitational force in GR was that it coupled to the same 'charge' as inertia. Thus, the existence of all other forces was a motivating factor - they all described 'resistance to the force' as inertial mass; this same inertial mass was the source charge for gravity. The geometric approach of GR thus unifies inertia and gravity motivated by considerations of other forces versus gravity. However, I see the likely future of gravitation theory going towards a reversion to the force point of view, to get unification with QM. I don't understand any basis to say forces are obsolete. Perhaps if Einstein's classical field unification had been successful in geometrizing all forces, there would be something to the idea of forces being obsolete. But you know where this effort ended up ....
BTW, I agree if you actually want to calculate anything in SR, relativistic mass is deadly. Funnily, I learnt SR in part from WGV Rosser's SR text (published in the 60s, before I was born) which did not use the relativistic mass, and I was quite brainwashed by it. Then in university the EM course made heavy use of the relativistic mass to my dismay, but I was eventually brainwashed back to the middle. Anyway, I use the relativistic mass mainly to motivate the why the stress-energy tensor is the source of gravity in GR. Of course it's kludgey, since GR is the axiom, not the derivation. But here it is. In SR inertial mass is relativistic mass (not so terrible is we use 3-force and f=dp/dt to tie with the Lorentz force f=q(E+vXB). Relativistic mass is energy, so inertial mass is energy, so by the WEP, gravitational mass is energy. Then we essentially search for the covariant counterpart of Poisson's equation with some form of covariantly conserved energy on the right, and a second derivative on the left. I think forces are obsolete in the most fundamental sense since neither classical GR nor quantum field theory (nor even Schroedinger's equation) has forces. I imagine the most fundamental theory at present is quantum field theory where everything is fields, and we incorporate GR as a spin 2 field. Then by various approximations we get quantum field theory in curved spacetime and classical GR which includes matter fields like Maxwell's equations or perfect fluids. Only in the ray limit of classical GR do we get the geodesic equation, which is the first place a particle and the second derivative of its position appears, and is a kind of force.
I think to have a case, you need to argue that forces really are gone, and not just made more complex by the notion of momentum becoming more complex. And in the case of GR, you have to come up with some very compelling argument that one should dismiss the obvious translation of the notions of momentum and force of SR point particles.
Well, let's say in quantum field theory, is there a notion of inertial mass? Point particles can't exist in GR (unless they are black holes). So point particles only appear as an approximation, before which there are no forces (again, there isn't a fundamental concept of inertial mass).
I missed this point. I have several times given my own 'eclectic' motivation for this. You start with invariant mass in SR. This shows, for a system of particles, the invariant mass includes contribution from KE of individual particles. Further, you can quantitatively build from invariant mass to the stress energy tensor of a dust of non-interacting particles of low enough total mass to ignore self gravitation. In another thread, I gave references to relevant sections of MTW providing the components of this heuristic. From here, you note that invariant mass is really just total energy in the COM frame. This motivates that all forms of energy must be included in stress-energy tensor. I admit I've never thought through an elementary motivation for pressure contributions. I find my motivating approach much better than one relying on relativistic mass.
And where does relativistic mass help here? Point particles are an approximation in Newtonian physics, just as much as SR and GR. They are a useful approximation for all. For solar system dynamics, even Jupiter can generally be treated as a point particle.
I wouldn't worry about pressure. Those get included automatically once we want energy-momentum to be conserved for fields. The heuristic through the invariant mass is clever, but is this mass local? The stress-energy is distinguished by its locality. No, relativistic mass does not help. The point is to get rid of it by getting rid of forces. Is there a formulation of Newtonian gravity where there are no forces?
I summarized too much. The full treatment uses invariant mass to motivate a reasonable guess about tensor for the dust, using only SR arguments. Then, you solve that this guess is correct - the ADM mass it implies (under the given, stringent, limiting assumptions) equals the invariant mass of the dust. You actually arrive at a correct stress energy tensor in GR for the given limiting case. Ok, I see. Well, I can't argue that it is necessary to use forces in GR, only convenient. Otherwise, what else is there besides numerical relativity for every problem?
Ok, thanks, I shall have to read MTW for what sounds like an interesting argument. But is this all just to avoid the relativistic mass? I mean T_{oo} is is the field counterpart of the relativistic mass (in flat spacetime), isn't it? So it's a pretty direct kludge. I'd say it's actually necessary not to use forces in GR, except as an approximation. OK, basic question. In SR, we take 4-force F=dP/dτ. Are there any known 4-forces for the LHS of the equation apart from electromagnetism? (I mean like in Newtonian mechanics, friction is a force that sits on the LHS of Newton's 2nd law, and is consistent with all 3 laws of dynamics.)
You won't find my argument in the form given. It uses various results from disparate sections of MTW. The reason I like my approach is, indeed, partly to avoid relativistic mass; but also to end up showing that 'collective' motion of the dust does not contribute to ADM mass; only COM total energy = invariant mass (in this limiting case) contributes. Sure, also in GR. Rocket thrust, idealized springs, what have you. There is, e.g. the photon rocket solution in GR. If you go deep enough, you need some fundamental theory for these. But at the level you use point particles and kinematics, you can introduce all sorts of useful approximate forces.
A concrete example of the utility of 4-force and rest mass in GR: We had a whole thread here recently discussing how much thrust (force) a rocket would need to 'hold position' while a boosted black hole went by, versus hold position near a stationary black hole. Barring the ambiguities of defining a stationary world line for a non-stationary metric, this becomes computation of proper acceleration on a world line or simply computing F=dp/dτ, along the world line and noting the maximum value for the non-stationary case; noting, of course, that p is simply rest mass times 4-velocity. So, rest mass, 4-force, proper acceleration all useful here. No place even to consider relativistic mass for this GR problem.
I do think it's important for the serious physics student to learn Lagrangian and/or Hamiltonian methods at some point, preferably an early point. I'm also of the opinion that this is a better way to really think of physics than F=ma for the serious student. There's been some effort, in particular by Taylor, to try to teach the principle of least action at a much earlier level,, to bring the power of the Lagrangian and Hamiltonian methods into play as early as possible. I'm not sure how successful or well received this effort has been, though. At the high school level, though, I don't see F=ma going away, so we'll probably continue to have the effect where we teach forces first, then later on say "there's a better way". I am also personally also convinced that forces are not fundamental, in the sense that they transform as tensors when you consider transformation between inertial frames, but not when you consider more general transformations. It seems pretty clear that inertial forces don't/ can't transform as tensors, but the usual practice seems to be to try to exclude inertial forces from being "actual" forces for this very reason. This doesn't seem compatible with the principle of equivalence, though, which suggests that one should consider inertial forces to be just as "real" as any other forces. If we do lump "inertial forces" in with other forces, though, we wind up thinking that the only reason we ever thought forces were tensors in the first place was by giving special philosophical status to "inertial frames", and that when we start considering all frames to be "equal", forces stop becoming tensors. Using Lagrangian or Hamiltonian methods, this demotion of forces to "not really tensors" doesn't seem to me to have any adverse consequences. And I tend to go further, from "it's not a tensor" to "therefore it's not really fundamental" as well. However, while I'm personally pretty much convinced myself of this, I'm not confident enough to say that everyone agrees with me, and I haven't been convinced long enough that I'm sure there's nothing that I've overlooked. I think there would probably be a general agreement that Lagrangian/Hamiltonian methods are more powerful and more fundamental than the F=ma approach, that I wouldn't meet much resistance from this point. I'm a lot less sure that if I started saying "Oh, you know, forces arent' really tensors, they never have been", that I would get general agreement, "Oh yeah, that's welll known, everyone knows that, what took you so long to figure it out?" rather than "Say what?".
I would agree with a large part of this thrust [esp. about Lagrangian/Hamiltonian], but not all. Especially problematic for me is the claim of force being tensorial only considering inertial frames. 4-force is a vector in GR under general diffeomorphism. Of course, it is only well defined for the proverbial 'test body' in GR, but that only limits its utility some; it does not eliminate it or change it to a non-tensor. To me, the geometric interpretation of GR is what encourages treating both gravity and fictitious forces as non-forces; and this encapsulates rather than violates the EP. The analog I see with Newtonian concepts is: curvature replaces / explains tidal forces. Gravitational attraction is not a force, 'fictitious forces' are not forces; there is just inertia and forces (those associated with non-inertial motion, which is an invariant concept, at least for world lines). So, to me, trying to make fictitious force (and gravity!) forces goes against the spirit of the EP, not in line with it. (This is separate from my point of view that quantizing gravity favors moving away from the geometric approach, and treating all interactions in a similar framework).
Actually, you can equally well view quantum field theories like the Standard Model of particle physics as ways of making all interactions look like geometry the way gravity does, by adding "dimensions" to the gauge group on which the theory is defined. Basically it's the same idea as Kaluza-Klein theory: each "point" of spacetime is no longer viewed as a point but a multi-dimensional "internal" space with an internal geometry that is described by a gauge group; for the Standard Model the gauge group is SU(3) x SU(2) x U(1), modulo some technicalities, as described by John Baez here: http://math.ucr.edu/home/baez/week253.html. What we think of as "forces" are then described as connections on subgroups of the gauge group, the same way the "force of gravity" is described by a connection on spacetime; in other words, when you put all the forces together you are looking at geometry on some higher-dimensional space that includes what we normally call "spacetime" as a subspace, plus other subspaces for the other "forces" besides gravity. String theory works along the same lines.
You may be implicitly switching between two views of "inertial forces" here. Strictly speaking, there is a key physical difference between "inertial forces" and "real forces": real forces are actually felt as acceleration; inertial forces are not. This is modeled in differential geometry as the covariant derivative of a worldline: it's zero for a body moving solely due to "inertial forces", but nonzero for a body subject to "real forces". But often when we talk about "inertial forces", we forget that the actual "force" we feel is not due to the inertial force itself; it's due to the real force that is pushing us out of the geodesic path that the inertial force would have us follow. I feel a force sitting here on the surface of the Earth, and speaking loosely I might say this is the "force of gravity": but actually it's not, it's the force of the Earth pushing on me. A rock falling past me is moving due to the "force of gravity", but it feels no force. The principle of equivalence does not require me to say that I and the rock are equivalent; so IMO it doesn't require me to say that inertial forces and "real" forces are equivalent either.
OK, let me clarify my terminology. I'm essentially considering the geodesic equation a "force equation". In full GR, the geodesic equation is not fundamental. It acts on "test particles" which are again not fundamental. So Newton-Cartan as I understand it usually has a geodesic equation and test particles. Is there a Newton-Cartan or Newtonian gravity in which the geodesic equation is not fundamental, ie. something like GR where we write the full action as Einstein-Hilbert plus minimally coupled matter actions and the fundamental degrees of freedom (with respect to which we vary the action) are all fields?