The Validity of Force and Point Particles in Special Relativity

[PeterDonis]

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 F_{ab} 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).

 

[PeterDonis]

I'm a little confused by these two statements:

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.

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.

 

[atyy]

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 F_{ab} 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:)

 

[PAllen]

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).

 

[atyy]

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?

 

[PAllen]

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?

We get ever more philosophical. I have never considered relativistic mass to be inertial mass (who ever heard or transverse versus longitudinal inertial mass). To me, the temptation to treat relativistic mass as inertial mass comes from using the 'wrong' formula.

However, I get your main point. E could be obsoleted (and I note that Penrose, in all his discussions of EM in "Road to Reality" uses only F, not E or B). Yet most people (myself included) still find E and B useful.

In sum, I'll grant you:

touche.

 

[PeterDonis]

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

Yes. One would be your 4-velocity, the other would be the 4-velocity of the "measuring instrument"; a better term would probably be "test particle", since in general we measure EM fields by watching the acceleration (i.e., the instantaneous change in 4-velocity) of test particles.

 

[juanrga]

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.

Correct, the field theory of gravity treats gravity as a real force associated to a gravitational field. The popular stringy claim that GR is a spin-2 field is a myth (already Wald warns his readers a bit about this myth).

But we can go beyond the limits of field theory toward a fundamental action-at-a-distance theory of gravity.

Two main alternatives are available on the literature: (i) the adaptation to gravity by Schieve and coworkers of the Stuckelberg-Horwitz-Piron theory and (ii) the adaptation to gravity of the action-at-a-distance electrodynamics of Wheeler & Feynman.

Note: I do not discuss superstring-M theory, LQG, and similar nonsenses in quantum gravity literature.

 

[PeterDonis]

Correct, the field theory of gravity treats gravity as a real force associated to a gravitational field. The popular stringy claim that GR is a spin-2 field is a myth (already Wald warns his readers a bit about this myth).

I'm not sure what "myth" you're referring to; can you give a specific reference in Wald? It's known that the theory of a massless spin-2 field on a flat spacetime background yields the exact Einstein-Hilbert Lagrangian; that's what I was referring to as the theory of gravity as a massless spin-2 field. String theory basically claims that the ground state of a closed string loop looks like a massless spin-2 field on distance scales that are large compared to the string size, but you made it clear that you don't consider string theory to be a "modern and satisfactory" theory of gravity, so we can leave that out of the discussion.

Two main alternatives are available on the literature: (i) the adaptation to gravity by Schieve and coworkers of the Stuckelberg-Horwitz-Piron theory and (ii) the adaptation to gravity of the action-at-a-distance electrodynamics of Wheeler & Feynman.

Can you give any references? I'm not familiar with these theoretical efforts, but they sound interesting.

 

[juanrga]

I'm not sure what "myth" you're referring to; can you give a specific reference in Wald? It's known that the theory of a massless spin-2 field on a flat spacetime background yields the exact Einstein-Hilbert Lagrangian; that's what I was referring to as the theory of gravity as a massless spin-2 field. String theory basically claims that the ground state of a closed string loop looks like a massless spin-2 field on distance scales that are large compared to the string size, but you made it clear that you don't consider string theory to be a "modern and satisfactory" theory of gravity, so we can leave that out of the discussion.

Can you give any references? I'm not familiar with these theoretical efforts, but they sound interesting.

Wald explains, in his well-known textbook on general relativity, that the popular claim that GR is equivalent to the theory of a massless spin-2 field on a flat spacetime background makes no sense beyond linearized GR.

I would say more than him, linearized GR is only pseudo-equivalent to the theory of a massless spin-2 field on a flat spacetime background.

An introduction to the classical version of the SHP theory that includes the gravitational counterpart of the original electromagnetic interaction is given in

Classical Relativistic Many-Body Dynamics 1999: Springer. Trump, Matthew A.; Schieve, William C.

A presentation of the extension of wheeler and Feynman action-at-a-distance electrodynamics to gravitation is given in

Gravitational interaction in the relational approach 2008: Grav. and Cosm. 14(1), 41—52. Vladimirov, Yu. S.

 

[petm1]

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).

Would it be ok to think of them as wormholes connecting two times, the present and big bang?

 

[PeterDonis]

Wald explains, in his well-known textbook on general relativity, that the popular claim that GR is equivalent to the theory of a massless spin-2 field on a flat spacetime background makes no sense beyond linearized GR.

I'll have to dig out my copy to see what he says; I don't remember reading this (but it's been quite some time since I looked at Wald's textbook). I know that in the Feynman Lectures on Gravitation, which is where I first read about the massless spin-2 field theory, Feynman proves that the Lagrangian of that theory correct to *all* orders (not just linearized order) is the Einstein-Hilbert Lagrangian. (At that point, of course, the flat background is "unobservable"; that actual physical metric is the curved metric. Maybe that's what Wald was referring to.)

Thanks for the other references, I'll look them up.

 

[atyy]

Gravity as spin-2: http://arxiv.org/abs/gr-qc/0411023 , http://arxiv.org/abs/gr-qc/9607039

 

[juanrga]

Gravity as spin-2: http://arxiv.org/abs/gr-qc/0411023 , http://arxiv.org/abs/gr-qc/9607039

Your 04 preprint is a copy of Deser classic article of the 70s. Some mistakes done by Deser in that article are discussed in section 3.3 of

http://arxiv.org/abs/gr-qc/9912003

Your 96 preprint is not even useful

 

[juanrga]

I'll have to dig out my copy to see what he says; I don't remember reading this (but it's been quite some time since I looked at Wald's textbook). I know that in the Feynman Lectures on Gravitation, which is where I first read about the massless spin-2 field theory, Feynman proves that the Lagrangian of that theory correct to *all* orders (not just linearized order) is the Einstein-Hilbert Lagrangian. (At that point, of course, the flat background is "unobservable"; that actual physical metric is the curved metric. Maybe that's what Wald was referring to.)

Thanks for the other references, I'll look them up.

If my memory does not fail, Wald main argument was that the concept «spin 2 field» requires a background metric to be defined precisely and beyond linearized GR, the concept has not meaning (it is a vacuous word as «pertribvb 9-tredx»).

Feynman lectures are completely wrong. He does not proves such stuff. His claim that GR has both a geometrical and a field description are without any support.

A correct treatment shows that GR is different from the field theory, which is not strange the first is a geometric theory the second is physical one. Feynman confounds the curved spacetime metric with the effective metric derived from the field among other basic stuff.

See also http://arxiv.org/abs/gr-qc/0409089 for other myths

 

[PAllen]

Adding to the mix, Deser strongly disputes these critiques, and reasserts his claims in:

http://arxiv.org/abs/0910.2975

 

[PeterDonis]

Plenty to digest, thanks PAllen and juanrga for the links. It looks to me like we might as well let Deser and his critics fight it out; I doubt we're going to settle anything here.

 

[atyy]

Padmanabhan does not dispute gravity=spin 2. His reservations are on the point of uniqueness. I think uniqueness for the geometric point of view was proved only in the 1970s by Lovelock.

 

[juanrga]

Adding to the mix, Deser strongly disputes these critiques, and reasserts his claims in:

http://arxiv.org/abs/0910.2975

He avoids the criticism about (GR ≠ spin-2 field) cited here and only offers a partial dispute with Padmanabhan work also cited here

 

[atyy]

Pitts and Schieve seem to have no disagreement that gravity=spin 2. All the questions are about exactly what assumptions are needed to obtain uniqueness.

 

[pervect]

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.

In the Newtonian concept, in order to get forces to transform via the Galilean transformation, gravity has to be considered to be a force - you can't omit it and keep the tensor nature of forces. Taking the example of standing on the Earth, if your feet were pushing on you that hard, and there was no gravity, you'd be flung into the air. Let me stress again that this is the Newtonian viewpont I'm discussing.

The equivalence of gravitational and inertial mass in the Newtonian theory was noticed for a long time. I believe it's accurate to say that this equality between gravitational and inertial masses gave the equivalence principle it's very name, though I don't have a reference.

Now, the equivalence principle is vague enough where you could still take the position that this equivalence is some sort of happy accident, but if you start to believe it's not just a coincidence, I think you're more or less drawn to the position I mentioned. I.e. you start to think that inertial forces are the same gravitational forces, and you already know that the inertial forces don't transform properly, i.e. like tensors. As I mentioned, you also know that gravitational forces need to be included as "real" forces in the Newtonian viewpoint.

I suppose it is also self consistent to think of gravity as being a "curvature of space time" and in no ways a force, which seems to be your suggestion. So then you do get to keep all the other forces as tensors, and you insist that gravity is "not a force".

 

[PAllen]

The equivalence of gravitational and inertial mass in the Newtonian theory was noticed for a long time. I believe it's accurate to say that this equality between gravitational and inertial masses gave the equivalence principle it's very name, though I don't have a reference.

Now, the equivalence principle is vague enough where you could still take the position that this equivalence is some sort of happy accident, but if you start to believe it's not just a coincidence, I think you're more or less drawn to the position I mentioned. I.e. you start to think that inertial forces are the same gravitational forces, and you already know that the inertial forces don't transform properly, i.e. like tensors. As I mentioned, you also know that gravitational forces need to be included as "real" forces. I think you get into quite a muddle, until you take the position that none of the forces were tensors under the most general sorts of coordinate transformations, and you've been making them look like tensors by considering a restricted set of transformations. Which is OK and self-consistent and even traditional, but if you want to open up the playing field to treating all sorts of transformations equivalently, you need to sacrifice the idea that forces are tensors. And you also then sacrifice the notion of inertial frames being "special", but I think this later part is or less a plus, you have one less thing to worry about defining.

I disagree with these two paragraphs.

1) The equivalence of gravitational and inertial mass was noticed long ago. The equivalence principle was Einstein's huge generalization of it: that no local physics of any kind can distinguish free fall from inertial motion in 'empty space' with no significant mass around.

2) The conclusion I draw is almost the opposite of yours. It is that gravity is not a force at all, and there are no such thing as inertial forces. Then, forces are tensors and no coordinate systems are privileged in any way. You never need to worry about inertial frames if you don't want to. In any coordinates, it is unambiguous whether a world line is experiencing force or not.

 

[PeterDonis]

I suppose it is also self consistent to think of gravity as being a "curvature of space time" and in no ways a force, which seems to be your suggestion. So then you do get to keep all the other forces as tensors, and you insist that gravity is "not a force".

This is more or less my position, but my terminology may be a bit different. I would say that tidal gravity is curvature of spacetime. Gravity as whatever it is that makes a rock fall to Earth is not tidal gravity, though it is related to it. But I would agree that the latter type of gravity is still not a force, because the rock falling to Earth feels no force (its 4-acceleration is zero, neglecting air resistance). A force is something that causes a nonzero 4-acceleration; by this definition, yes, all forces are tensors.

 

[juanrga]

Pitts and Schieve seem to have no disagreement that gravity=spin 2. All the questions are about exactly what assumptions are needed to obtain uniqueness.

(GR ≠ spin-2 field)

They avoid the criticism cited here and merely repeat Deser, Feynman, and others mistakes

 

[atyy]

(GR ≠ spin-2 field)

They avoid the criticism cited here and merely repeat Deser, Feynman, and others mistakes

Dang, you are persistent! I thought by citing Schieve, some of whose work you had approved of, I could make you change your mind

 

[PAllen]

He avoids the criticism about (GR ≠ spin-2 field) cited here and only offers a partial dispute with Padmanabhan work also cited here

He doesn't ignore it. He rejects it completely as incorrect. I am not going to try to referee this debate among people who know enormously more than I, but there is no doubt what Deser thinks:

"In summary, I have annotated the steps involved in the non-geometric derivation [1] of
GR from special relativistic field theory as the unique consistent self-interacting system,
(13) extending the initial free massless spin 2."

In short, Deser remains convinced: GR=spin 2 field theory.

Truth is not determined by poll, but I would predict that Deser's position would handily win a poll of relevant experts. For example, I have seen a number of comments by Professor Steven Carlip endorsing this point of view.

 

[pervect]

I disagree with these two paragraphs.

1) The equivalence of gravitational and inertial mass was noticed long ago. The equivalence principle was Einstein's huge generalization of it: that no local physics of any kind can distinguish free fall from inertial motion in 'empty space' with no significant mass around.

2) The conclusion I draw is almost the opposite of yours. It is that gravity is not a force at all, and there are no such thing as inertial forces. Then, forces are tensors and no coordinate systems are privileged in any way. You never need to worry about inertial frames if you don't want to. In any coordinates, it is unambiguous whether a world line is experiencing force or not.

I was thinking about the topic as I wrote it You managed to respond to one of the earlier/earliest versions. You make some good points, I think. It's interesting that the final version I came up with (which was before I read this criticisms) almost seems as if I read your criticism (but at the time I finished my original, I didn't see the criticisms yet).

I'll agree that it is possible to view gravity as not being a force, but rather due to the curvature of the space-time. This way one gets to keep the traditional structure of "forces" mostly intact, having only to sacrifice the original Newtonian idea of gravity being a force with the replacement idea that "it's not really a force".

Then one can explain that curved space-time transforms as a tensor, but it's a rank 4 tensor, i.e. the Riemann, thus it has more degrees of freedom than any "force" concept allows.

I still don't think this approach is really in the true spirit of "general covariance", because one still has to single out inertial frames as being "special - and one has to be careful of what sort of transformations are allowed as well. But it's a reasonable way of looking at things nonetheless.

 

[juanrga]

Dang, you are persistent! I thought by citing Schieve, some of whose work you had approved of, I could make you change your mind

Then that is a strong difference between you and me. I do not cite authors, I cite specific works .

And as many others I am well aware that the same author can be completely right regarding some topics and completely wrong regarding others.

 

[juanrga]

He doesn't ignore it. He rejects it completely as incorrect. I am not going to try to referee this debate among people who know enormously more than I, but there is no doubt what Deser thinks:

"In summary, I have annotated the steps involved in the non-geometric derivation [1] of
GR from special relativistic field theory as the unique consistent self-interacting system,
(13) extending the initial free massless spin 2."

In short, Deser remains convinced: GR=spin 2 field theory.

Truth is not determined by poll, but I would predict that Deser's position would handily win a poll of relevant experts. For example, I have seen a number of comments by Professor Steven Carlip endorsing this point of view.

Deser only partially answers Padmanabhan work.

Deser completely ignores the rest of criticism to his derivation (nowhere Deser replies the criticism to his 'derivation' done in the section 3.3 of http://arxiv.org/abs/gr-qc/9912003

This reference shows conclusively that (GR ≠ spin 2 field theory).

Effectively, truth is not determined by poll, neither by appeal to authority. If you or some other want to rely on what some 'big' names say, you are welcomed. I prefer to check by myself the literature

 

[atyy]

Deser only partially answers Padmanabhan work.

Deser completely ignores the rest of criticism to his derivation (nowhere Deser replies the criticism to his 'derivation' done in the section 3.3 of http://arxiv.org/abs/gr-qc/9912003

This reference shows conclusively that (GR ≠ spin 2 field theory).

Effectively, truth is not determined by poll, neither by appeal to authority. If you or some other want to rely on what some 'big' names say, you are welcomed. I prefer to check by myself the literature

The reference only claims to show it for quantum GR in very strong gravity (eg. near spacetime singularities), for which it is agreed that GR may not be spin 2.