The Validity of Force and Point Particles in Special Relativity

In summary, the discussion is about getting rid of the concept of relativistic mass. The author suggests that relativistic mass is a moral link between the new theories which deal primarily in fields, and the old theories in which point particles and forces were ok. They argue that without relativistic mass, force and point particles are not useful in the same way. However, the author thinks that the future of gravitation theory will go back to the force point of view.
  • #1
atyy
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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.
 
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  • #2
atyy said:
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.
 
  • #3
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?
 
  • #4
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 ...
 
  • #5
BTW, I agree if you actually want to calculate anything in SR, relativistic mass is deadly. Funnily, I learned 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.
 
  • #6
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.
 
  • #7
Hurkyl said:
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.

Well, let's say in quantum field theory, is there a notion of inertial mass?

Hurkyl said:
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.

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).
 
  • #8
atyy said:
How would you motivate the replacement in GR of Newtonian gravitational mass with the stress energy tensor?

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.
 
  • #9
atyy said:
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).

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.
 
  • #10
PAllen said:
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 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.

PAllen said:
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.

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?
 
  • #11
atyy said:
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.
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.

atyy said:
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?

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?
 
  • #12
PAllen said:
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, 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 Too is is the field counterpart of the relativistic mass (in flat spacetime), isn't it? So it's a pretty direct kludge.

PAllen said:
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?

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.)
 
  • #13
atyy said:
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 Too is is the field counterpart of the relativistic mass (in flat spacetime), isn't it? So it's a pretty direct kludge.
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.
atyy said:
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.)

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.
 
  • #14
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.
 
  • #15
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?".
 
  • #16
pervect said:
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).
 
  • #18
PAllen said:
(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.
 
  • #19
pervect said:
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.

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.
 
  • #20
PeterDonis said:

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?
 
  • #21
PAllen said:
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.

I think what pervect was referring to is that the "acceleration due to gravity", or "force of gravity", is modeled by the Christoffel symbols, which don't transform as tensors; you can make them all vanish locally at a given event by transforming to a local inertial frame at that event.

But the actual 4-acceleration of a worldline, i.e., the covariant derivative of its tangent vector, does transform as a 4-vector, i.e., a (1, 0) tensor. The 4-acceleration requires contracting the Christoffel symbols with the 4-velocity, which is how it can transform as a tensor even though the Christoffel symbols by themselves don't. (In a local inertial frame, all the Christoffel symbols vanish, but the 4-acceleration still has another term which is the actual partial derivative of the 4-velocity with respect to proper time, and in the local inertial frame that is nonzero for a non-geodesic worldline. In a non-inertial frame, for example a "static" frame using Schwarzschild coordinates around a massive body, the partial derivative vanishes but there the Christoffel symbols don't, and their contraction with the 4-velocity gives the 4-acceleration.)
 
  • #22
atyy said:
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?

Ah, I see. I don't know if Newton-Cartan gravity has been given such a formulation. Obviously you can formulate Newtonian gravity using the Lagrangian formalism, but I don't know if anyone has ever checked to see if such a formulation will lead to the Newton-Cartan formalism by the same route as you can use to get GR from the Einstein-Hilbert action.
 
  • #23
PeterDonis said:
Ah, I see. I don't know if Newton-Cartan gravity has been given such a formulation. Obviously you can formulate Newtonian gravity using the Lagrangian formalism, but I don't know if anyone has ever checked to see if such a formulation will lead to the Newton-Cartan formalism by the same route as you can use to get GR from the Einstein-Hilbert action.

A quick google turns up Goenner, A variational principle for Newton-Cartan theory, whose abstract seems to indicate the answer is "not completely". Unfortunately, this paper is probably not freely available.
 
  • #24
atyy said:
Unfortunately, this paper is probably not freely available.

It isn't. But the abstract does, as you say, imply that the Newtonian case works differently than the GR case.
 
  • #25
As PAllen pointed out, the point particle concpet is not related to forces. One can always use the 3-force density in Newtonian physics, eg. the Navier-Stokes equations. So in the following "force" always means "force density".

OK, is the following the correct hierachy of concepts?

Newtonian mechanics and gravity - Galilean inertial frames, 3-force, inertial mass

Classical special relativity - 3-force and inertial mass no longer fundamental, instead we have 4-force and invariant mass (which is not the same as inertial mass). The inertial mass is defined via the 3-force and is now velocity dependent. Neither the 3-force nor inertial mass are fundamental, just as E and B fields are not fundamental (I would especially like commentary on this in the light of Bell's "Lorentzian pedagogy" where we say everything can be done in one Lorentz inertial frame).

Quantum special relativity - no concept of force at all. The dynamics are entirely given by Hamiltonian or Lagrangian formalisms.

General relativity - neither 3-force nor 4-force are fundamental, the dynamics is given by the Einstein field equations and equations of state such as Maxwell's equations. 4-force and the geodesic equation are derived concepts.

Quantum general relativity - no concept of force at all, same formalism as quantum special relativity, with gravity being a spin-2 field.
 
  • #26
atyy said:
As PAllen pointed out, the point particle concpet is not related to forces. One can always use the 3-force density in Newtonian physics, eg. the Navier-Stokes equations. So in the following "force" always means "force density".

OK, is the following the correct hierachy of concepts?

Newtonian mechanics and gravity - Galilean inertial frames, 3-force, inertial mass

Classical special relativity - 3-force and inertial mass no longer fundamental, instead we have 4-force and invariant mass (which is not the same as inertial mass). The inertial mass is defined via the 3-force and is now velocity dependent. Neither the 3-force nor inertial mass are fundamental, just as E and B fields are not fundamental (I would especially like commentary on this in the light of Bell's "Lorentzian pedagogy" where we say everything can be done in one Lorentz inertial frame).

Quantum special relativity - no concept of force at all. The dynamics are entirely given by Hamiltonian or Lagrangian formalisms.

General relativity - neither 3-force nor 4-force are fundamental, the dynamics is given by the Einstein field equations and equations of state such as Maxwell's equations. 4-force and the geodesic equation are derived concepts.

Quantum general relativity - no concept of force at all, same formalism as quantum special relativity, with gravity being a spin-2 field.

I'd go along with this. It is certainly a very reasonable point of view, IMO.
 
  • #27
atyy said:
Newtonian mechanics and gravity - Galilean inertial frames, 3-force, inertial mass

Classical special relativity - 3-force and inertial mass no longer fundamental, instead we have 4-force and invariant mass (which is not the same as inertial mass). The inertial mass is defined via the 3-force and is now velocity dependent. Neither the 3-force nor inertial mass are fundamental, just as E and B fields are not fundamental.

Quantum special relativity - no concept of force at all. The dynamics are entirely given by Hamiltonian or Lagrangian formalisms.

General relativity - neither 3-force nor 4-force are fundamental, the dynamics is given by the Einstein field equations and equations of state such as Maxwell's equations. 4-force and the geodesic equation are derived concepts.

Quantum general relativity - no concept of force at all, same formalism as quantum special relativity, with gravity being a spin-2 field.

These all look OK to me.

atyy said:
(I would especially like commentary on this in the light of Bell's "Lorentzian pedagogy" where we say everything can be done in one Lorentz inertial frame)

You can do everything in one inertial frame, but which one you pick will depend on your own state of motion. And we don't fully control our state of motion. For example, the Earth orbits the Sun, meaning that, from the point of view of, say, an inertial frame in which the Sun is at rest, we are constantly changing our state of motion. (Also, of course, we sitting on the surface of the Earth are not really in an inertial frame to begin with, since we feel acceleration.) So from a fundamental physics point of view, we can't privilege any specific inertial frame.
 
  • #28
PeterDonis said:
You can do everything in one inertial frame, but which one you pick will depend on your own state of motion. And we don't fully control our state of motion. For example, the Earth orbits the Sun, meaning that, from the point of view of, say, an inertial frame in which the Sun is at rest, we are constantly changing our state of motion. (Also, of course, we sitting on the surface of the Earth are not really in an inertial frame to begin with, since we feel acceleration.) So from a fundamental physics point of view, we can't privilege any specific inertial frame.

Ok, but is there anything I can't do if I use 3-force, inertial mass and E and B fields in any particular inertial frame? In other words, is the only thing I lose manifest Lorentz covariance, or do I actually lose any physics (including the ability to predict what the physics is like in any other frame, inertial or not)?
 
  • #29
atyy said:
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.

The outdated concept of relativistic mass was abandoned because was a source of many confusion. The mass of an electron is me, which is independent of its velocity.

The problems with point particles in Maxwell theory are not solved by introducing extended distributions of charge (Poincaré stresses). The problem is not in the existence of a force but in the self-interaction associated to the field interaction. Wheeler-Feynman theory (an action-at-a-distance theory) uses forces but solves the difficulties because abandon the interaction through fields.

What you say about field theory is just the inverse. Field theory is the old theory and the modern theories under development agree on that the concept of field is approximated only. Sometimes we use the term «effective theory» to emphasize that field theory is not fundamental.
 
  • #30
PAllen said:
However, I see the likely future of gravitation theory going towards a reversion to the force point of view, to get unification with QM.

This is already the present in research.

All modern and satisfactory theories of gravity consider gravity a real force and consider that the geometrical description given by GR is only valid as a first approximation.
 
  • #31
atyy said:
Ok, but is there anything I can't do if I use 3-force, inertial mass and E and B fields in any particular inertial frame?

I don't think so, but you were talking about what different theories consider "fundamental". Standard SR does not consider anything that is frame-dependent to be "fundamental"; only frame-invariant objects are "fundamental". So, for example, the electromagnetic field tensor [itex]F_{ab}[/itex] would be fundamental, but any particular decomposition into E and B fields would not be, since that is frame-dependent.

Perhaps it's also worth noting that, just as you can calculate everything using a single inertial frame, you can calculate things without using a frame at all. Any number that you can actually measure in an experiment can be expressed as a scalar (PAllen made this point in a recent thread on a similar subject), meaning you can express it without ever having to commit yourself to any specific frame, just write the expression in terms of contractions of vectors and tensors (which you can do in abstract index notation, without ever specifying a particular set of components).
 
  • #32
I'm a little confused by these two statements:

juanrga said:
Field theory is the old theory and the modern theories under development agree on that the concept of field is approximated only. Sometimes we use the term «effective theory» to emphasize that field theory is not fundamental.

juanrga said:
All modern and satisfactory theories of gravity consider gravity a real force and consider that the geometrical description given by GR is only valid as a first approximation.

But the theory of gravity as a "real force" is a field theory; it's the theory of a massless spin-2 field, or the fancier versions given in string theory (is string theory one of the "modern and satisfactory theories of gravity"?), which are also field theories, just not field theories based on point particles. If the original massless spin-2 field theory is only an "effective" theory, are the string theory versions "effective" too?

Other "modern" theories of gravity (I don't know whether you consider them "satisfactory"), such as loop quantum gravity, don't look to me like theories of gravity as a "real force"; they look more like theories of geometry.
 
  • #33
PeterDonis said:
I don't think so, but you were talking about what different theories consider "fundamental". Standard SR does not consider anything that is frame-dependent to be "fundamental"; only frame-invariant objects are "fundamental". So, for example, the electromagnetic field tensor [itex]F_{ab}[/itex] would be fundamental, but any particular decomposition into E and B fields would not be, since that is frame-dependent.

Perhaps it's also worth noting that, just as you can calculate everything using a single inertial frame, you can calculate things without using a frame at all. Any number that you can actually measure in an experiment can be expressed as a scalar (PAllen made this point in a recent thread on a similar subject), meaning you can express it without ever having to commit yourself to any specific frame, just write the expression in terms of contractions of vectors and tensors (which you can do in abstract index notation, without ever specifying a particular set of components).

Yes. I realize you were addressing the poetic aspects of my question. I just wanted to check that there wasn't an underlying technical issue I was missing.

Now I just need to find a way to make the E field a scalar:)
 
  • #34
atyy said:
Yes. I realize you were addressing the poetic aspects of my question. I just wanted to check that there wasn't an underlying technical issue I was missing.

Now I just need to find a way to make the E field a scalar:)

Ah, but if you try to measure the E field there are two things to consider: the EM field and an instrument. The world line of the instrument (e.g. its motion in your chosen frame) interacts with the field, producing a measurement consisting of one or more scalars. The instrument may well measure magnetic field strength due to its motion in your coordinates (in which, say, the E/M field is pure coulomb).
 
  • #35
PAllen said:
Ah, but if you try to measure the E field there are two things to consider: the EM field and an instrument. The world line of the instrument (e.g. its motion in your chosen frame) interacts with the field, producing a measurement consisting of one or more scalars. The instrument may well measure magnetic field strength due to its motion in your coordinates (in which, say, the E/M field is pure coulomb).

Yes, I think I need at least two vectors to slot into the Faraday tensor to make a scalar.

BTW, now that we established that the E field is as real or as fake as the inertial mass, why is the latter considered so much more pedagogically harmful than the former?
 
<h2>1. What is the concept of force in special relativity?</h2><p>In special relativity, force is defined as the rate of change of momentum of an object with respect to time. It is a fundamental concept that explains the motion of objects in the presence of other objects or fields.</p><h2>2. How do point particles fit into the theory of special relativity?</h2><p>In special relativity, point particles are considered to be objects with no physical size or extension. They are used as a simplification to understand the behavior of particles at very small scales, such as in particle accelerators.</p><h2>3. Is the concept of force applicable to point particles in special relativity?</h2><p>Yes, the concept of force is applicable to point particles in special relativity. Even though they have no physical size, point particles still have mass and can experience forces, such as gravitational or electromagnetic forces.</p><h2>4. How does special relativity explain the validity of force and point particles?</h2><p>Special relativity explains the validity of force and point particles by incorporating them into its mathematical framework. The theory accounts for the effects of time dilation and length contraction, which are necessary to accurately describe the behavior of objects moving at high speeds.</p><h2>5. Are there any limitations to the validity of force and point particles in special relativity?</h2><p>There are some limitations to the validity of force and point particles in special relativity. For example, the theory does not account for the effects of quantum mechanics, which are necessary to describe the behavior of particles at very small scales. Additionally, some phenomena, such as black holes, require the use of general relativity to fully understand their behavior.</p>

1. What is the concept of force in special relativity?

In special relativity, force is defined as the rate of change of momentum of an object with respect to time. It is a fundamental concept that explains the motion of objects in the presence of other objects or fields.

2. How do point particles fit into the theory of special relativity?

In special relativity, point particles are considered to be objects with no physical size or extension. They are used as a simplification to understand the behavior of particles at very small scales, such as in particle accelerators.

3. Is the concept of force applicable to point particles in special relativity?

Yes, the concept of force is applicable to point particles in special relativity. Even though they have no physical size, point particles still have mass and can experience forces, such as gravitational or electromagnetic forces.

4. How does special relativity explain the validity of force and point particles?

Special relativity explains the validity of force and point particles by incorporating them into its mathematical framework. The theory accounts for the effects of time dilation and length contraction, which are necessary to accurately describe the behavior of objects moving at high speeds.

5. Are there any limitations to the validity of force and point particles in special relativity?

There are some limitations to the validity of force and point particles in special relativity. For example, the theory does not account for the effects of quantum mechanics, which are necessary to describe the behavior of particles at very small scales. Additionally, some phenomena, such as black holes, require the use of general relativity to fully understand their behavior.

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