Four-Momentum: Schutz's Extra Postulate?

[snoopies622]

In section 2.4 of Schutz's, "A First Course in General Relativity", he states that the conservation of four-momentum law

"has the status of an extra postulate...However, like the two fundamental postulates of SR, this one is amply verified by experiment."

Is this correct? I was under the impression that all of SR can be derived from "the two fundamental postulates" and that no more were necessary.

 

[Fredrik]

The proper way to define SR is by writing down a list of axioms about how to interpret the mathematics of Minkowski space as predictions of results of experiments. (See e.g. this post for one of my usual rants about Einstein's postulates). This defines a framework in which you should be able to define theories of matter and interactions*, and those theories should tell you that momentum is conserved.

*) Right now I'm really confused about this, because Meopemuk mentioned the Currie-Jordan-Sudarshan "no interaction theorem" in another thread the other day. It appears to be saying that interactions aren't possible in SR, but surely there can't be anything wrong with just writing down a relativistic version of Newton's second law? (I don't know the answer. Like I said, I'm very confused right now).

 

[atyy]

*) Right now I'm really confused about this, because Meopemuk mentioned the Currie-Jordan-Sudarshan "no interaction theorem" in another thread the other day. It appears to be saying that interactions aren't possible in SR, but surely there can't be anything wrong with just writing down a relativistic version of Newton's second law? (I don't know the answer. Like I said, I'm very confused right now).

I know I said I wasn't going to continue that discussion in this forum, but I can't help myself. I think CJS is usually not applicable because the only "fundamental" relativistic theories we have are quantum field theories. Meopemuk's claim is interesting because it's a quantum mechanics theory, and he cites Weinberg, whom I've always thought was wrong on this, but of course, I'm a biologist and Weinberg isn't ...

 

[bcrowell]

The problem with Schutz's approach is that then you have to have dozens of "postulates." After all, conservation of energy and momentum aren't the only laws of physics that we want to preserve in SR. I think a better way of looking at it is that in addition to the postulates of SR, we assume the correspondence principle. Then I think the vast majority of the nonrelativistic physics has a unique generalization to SR.

 

[Dale]

I think a better way of looking at it is that in addition to the postulates of SR, we assume the correspondence principle. Then I think the vast majority of the nonrelativistic physics has a unique generalization to SR.

I agree with this. I would obtain the conservation of four-momentum by the requirement that the relativistic theory recover the non-relativistic expressions for conservation of energy and momentum in the appropriate limit. I don't know if I would elevate that general approach to the level of a postulate or not, but the approach could certainly be used for more than just this one particular law without requiring a long laundry list of postulates.

 

[atyy]

I think it depends.

Fundamentally, all you need are the 2 postulates (like Fredrik) says, then you have no force law, and just a bunch of dynamics for fields that are Poincare invariant, eg. take Maxwell's equations w/o the Lorentz force law - that's a special relativistic theory, and there is no particle 4-momentum conservation (there are no particles anyway, at least none with definite position and momentum).

However, if we wish to have a force law for particles with definite position and momentum then we need to have a relativistic generalization of Newton's second law of motion, and this requires additional postulates like Schutz, bcrowell, and DaleSpam say. The various possible forms of the additional postulates are discussed in section 6.2 of http://books.google.com.sg/books?id=fUj_LW51GfQC&dq=rindler+relativity&source=gbs_navlinks_s, see especially footnote 1 of p109.

While we are listing additional postulates, I think Fredrik's favourite is the clock one?

 

[snoopies622]

Goodness - so much to think about!

My first question is for Fredrik: I want to be clear - are you saying that

1. Einstein's two postulates imply Minkowski spacetime

and

2. Minkowski spacetime implies conservation of four-momentum

?

edit: I just read what you posted in the "Difference between Theory and Law" thread and now I'm pretty sure your answer to my question would be "no", but I am still having trouble following your reasoning. Is there really ambiguity in the concept of an inertial reference frame?

 

[atyy]

(i) Principle of Relativity ---> Galilean or Lorentzian relativity

(ii) Finite maximum velocity for transmission of signals ---> Lorentzian relativity

(iii) Write a Lagrangian consistent with Lorentzian relativity, by Noether's theorem some function of the Lagrangian will be conserved, and by definition that quantity is the energy/momentum of the field. This can be done for any Lagrangian consistent with Lorentzian relativity, so although you will use additional axioms to pick a particular Lagrangian, the existence of a conserved quantity that can be defined to be the energy/momentum of the field is not dependent on what the additional axioms are. So the first two postulates are enough if you have no particles with definite position and momentum.

Take a look at Eq 1.39 - Eq 1.43 in David Tong's http://www.damtp.cam.ac.uk/user/tong/qft.html

 

[snoopies622]

I think a better way of looking at it is that in addition to the postulates of SR, we assume the correspondence principle. Then I think the vast majority of the nonrelativistic physics has a unique generalization to SR.

I would obtain the conservation of four-momentum by the requirement that the relativistic theory recover the non-relativistic expressions for conservation of energy and momentum in the appropriate limit.

The full quotation from Schutz is,

"This law has the status of an extra postulate, since it is only one of many whose nonrelativistic limit is correct. However, like the two fundamental postulates of SR, this one is amply verified by experiment."

Apparently in this case the generalization is not unique and the conservation law therefore not obtainable only by assuming the correspondence principle along with Einstein's original postulates. I am beginning to agree that some other postulate is necessary. Or to put it another way, if the conservation law can be derived using only Einstein's postulates, I've never seen it done and I certainly don't know how to do it.

 

[bcrowell]

"This law has the status of an extra postulate, since it is only one of many whose nonrelativistic limit is correct. However, like the two fundamental postulates of SR, this one is amply verified by experiment."

I guess in this case the generalization is not unique.

Interesting. Does he suggest what the others are?

 

[snoopies622]

Interesting. Does he suggest what the others are?

I'm afraid not. Maybe he's just guessing.

 

[bcrowell]

I'm afraid not. Maybe he's just guessing.

He's probably discovered a truly marvelous proof of this, which his book is too narrow to contain.

 

[meopemuk]

In the Wigner-Dirac-Weinberg approach to (quantum) relativistic physics (see S. Weinberg, "The quantum theory of fields" vol. 1) there is no need for any additional postulate to guarantee the conservation of the energy-momentum 4-vector.

The generator of time translations is the Hamiltonian H (the operator of energy). Therefore, energy conservation follows from the trivial fact that the Hamiltonian commutes with itself [H,H]=0 (in classical theory, the commutator should replaced with the Poisson bracket).

The conservation of the total momentum P (which is the generator of space translations) follows from the structure of the Poincare Lie algebra, in which H and P commute [H,P]=0. The same is true for the conservation of the total angular momentum J, because [H,J]=0.

The time evolution of any dynamical variable is determined by its commutator with the full Hamiltonian. From this you can obtain all 3 Newton's laws of dynamics.

Eugene.

 

[Fredrik]

I am still having trouble following your reasoning. Is there really ambiguity in the concept of an inertial reference frame?

There's no ambiguity in Newtonian mechanics, but we can't use that definition, because then the theory would be Newtonian mechanics. So all we know at the start of the "derivation" is what an inertial frame isn't. I suppose the name "inertial frame" suggests that our new inertial frames have something in common with Galilean inertial frames, because otherwise they wouldn't deserve the name "inertial frame", but a suggestion isn't a fact.

I really don't understand why so many people are asking me this. I'm not trying to be rude here, but I have to say that I think it should be immediately obvious to anyone who has seen a few mathematical proofs that Einstein's postulates aren't mathematical axioms.

 

[meopemuk]

There's no ambiguity in Newtonian mechanics, but we can't use that definition, because then the theory would be Newtonian mechanics. So all we know at the start of the "derivation" is what an inertial frame isn't. I suppose the name "inertial frame" suggests that our new inertial frames have something in common with Galilean inertial frames, because otherwise they wouldn't deserve the name "inertial frame", but a suggestion isn't a fact.

I really don't understand why so many people are asking me this. I'm not trying to be rude here, but I have to say that I think it should be immediately obvious to anyone who has seen a few mathematical proofs that Einstein's postulates aren't mathematical axioms.

Fredrik, I am still puzzled. Why do you keep talking about difficulties with the definition of inertial frames? Imagine a person holding three mutually perpendicular sticks and wearing a watch on his wrist. Why is it not a good model of an inertial reference frame?

Note that the difference between relativistic and non-relativistic frames is not in their internal setup, but in the structure of the group of transformations between them. The exact group is the Poincare one. When only low-speed frames are considered, then the Galilei group is a decent approximation.

Eugene.

 

[snoopies622]

Is there really ambiguity in the concept of an inertial reference frame?

There's no ambiguity in Newtonian mechanics, but we can't use that definition, because then the theory would be Newtonian mechanics.

To me an inertial frame of reference is one in which objects don't accelerate unless they are pushed or pulled. Are you saying that this is not the case in the inertial reference frames of special relativity?

 

[Fredrik]

Fredrik, I am still puzzled. Why do you keep talking about difficulties with the definition of inertial frames? Imagine a person holding three mutually perpendicular sticks and wearing a watch on his wrist. Why is it not a good model of an inertial reference frame?

A mathematical proof uses a mathematical statement as the starting point. What you just said isn't a mathematical statement. To "derive" Minkowski space from a non-mathematical statement is like trying to define real numbers from what a biologist can tell you about frogs.

That's not to say that we don't need statements like the one you made. We absolutely do. They can be used as loosely stated guidelines that might help us guess what mathematical structure to use in a new theory. And when we have found a mathematical structure that seems to do the job, they can be used to define a theory of physics. The mathematical structure alone can't define a theory. The theory is defined by a set of axioms that tells us how to interpret the mathematics as predictions about results of experiments. Your statement can be used to define at least two different theories of physics. When the mathematical structure is Minkowski spacetime, we get the special relativistic theory of motion in inertial frames (which in this theory are identified with members of the Poincaré group), and when the mathematical structure is Galilean spacetime, we get the non-relativistic theory of motion in inertial frames (which in this theory are identified with members of the Galilei group).

To me an inertial frame of reference is one in which objects don't accelerate unless they are pushed or pulled. Are you saying that this is not the case in the inertial reference frames of special relativity?

When we're done with the definition of SR, and done choosing stuff in the mathematics that can represent the things you're talking about (like "acceleration" or "objects"), then we can show that the theory does say that what you just said. But even a word like "push" is ill-defined until we have chosen what mathematics to use. The claim I'm objecting to is that you can prove that we have to use Minkowski space from statements using concepts that we intend to define properly once we have figured out what mathematical structure to use.

 

[meopemuk]

Your statement can be used to define at least two different theories of physics. When the mathematical structure is Minkowski spacetime, we get the special relativistic theory of motion in inertial frames (which in this theory are identified with members of the Poincaré group), and when the mathematical structure is Galilean spacetime, we get the non-relativistic theory of motion in inertial frames (which in this theory are identified with members of the Galilei group).

I don't think that such mathematical structures as "Minkowski spacetime" or "Galilean spacetime" are needed in physics. In my opinion, they are not just useless, they are misleading. In order to build a complete physical theory you need to know just a few things: (i) a rather vague definition of observer (like the person holding three sticks), (ii) the principle of relativity (the equivalence of different observers), (iii) the postulate that the group of transformations between observers is the Poincare group, and (iv) postulates of quantum mechanics. Then simple logic and math leads you to quantum relativistic physics as described in works of Wigner, Dirac, and Weinberg.

Approximately, you can replace the Poincare group by the Galilei group. Then you physics will be non-relativistic. This is a valid approximation, and it does not depend on the definition of inertial observers at all.

Eugene.

 

[Fredrik]

I don't think that such mathematical structures as "Minkowski spacetime" or "Galilean spacetime" are needed in physics. In my opinion, they are not just useless, they are misleading. In order to build a complete physical theory you need to know just a few things: (i) a rather vague definition of observer (like the person holding three sticks), (ii) the principle of relativity (the equivalence of different observers), (iii) the postulate that the group of transformations between observers is the Poincare group, and (iv) postulates of quantum mechanics. Then simple logic and math leads you to quantum relativistic physics as described in works of Wigner, Dirac, and Weinberg.

This is mostly true, but it doesn't have a lot to do with what we've been talking about so far. The claim I'm defending is that you can't derive anything from Einstein's postulates, and that in particular, you can't derive the Poincaré transformation. Now you're suddenly talking about relativistic quantum mechanics, and you have also replaced one of the postulates with something that includes a mathematical version of both postulates and a mathematical definition of an inertial frame. (Inertial frames can simply be identified with the Poincaré transformations. You introduced the Poincaré group in a way that guarantees that there's an invariant speed. And you will eventually have to specify that a "law of physics" is "the same" in all inertial frames if it's a relationship between tensor components). You are clearly not deriving a result from Einstein's postulates. In fact, you're doing precisely the sort of thing I've been saying that you have to do if you're going to do something that resembles a derivation.

Einstein's postulates aren't well-defined, but if we're really nice, we can interpret them as representing a set of well-defined statements, one for each definition of "inertial frame", each definition of "law of physics", each definition of what it means for a law of physics to "be the same" in two inertial frames, and each definition of "light" or "the speed of light". And the closest thing to a derivation that we can do, is to find out which of the well-defined statements are consistent with all the other assumptions we'd like to make.

I strongly disagree with the claim that Minkowski spacetime is useless and misleading. Without it, we'd be stuck with the old fashioned definition of a tensor, which just makes me angry each time I see it. It's just so dumb and awkward compared to the modern definition, that this fact alone is enough to justify the use of Minkowski spacetime. There are lots of other reasons to use it, e.g. the fact that it really helps to understand it when you start studying GR.

If your dislike for Minkowski space comes from a belief that spacetime should be a result of some kind of interactions rather than just a passive stage on which the interactions occur, then I can understand it to some extent, but even if this idea is correct, it's not a reason not to use Minkowski space in classical SR.

Another problem with your approach is that it doesn't include any coordinate systems that aren't inertial frames. I'm wondering if you want to eliminate those specifically because you want to consider particles as more fundamental than fields? (The Unruh effect can be interpreted as saying that the number of particles in a region of space depends on your acceleration, and that makes it hard to think of particles as fundamental). This seems futile, because even if we can eliminate non-inertial frames from SR, we still aren't getting rid of them from GR.

 

[meopemuk]

And you will eventually have to specify that a "law of physics" is "the same" in all inertial frames if it's a relationship between tensor components).

No. Most physical observables (in interacting system) do not have tensor transformation laws. The CJS theorem is just one evidence for this statement.

(The Unruh effect can be interpreted as saying that the number of particles in a region of space depends on your acceleration, and that makes it hard to think of particles as fundamental)

This argument against particles is rather weak. First, there is nothing shocking if different observers see different number of particles. Second, nobody has ever seen the Unruh effect. Perhaps it does not exist after all.

Eugene.

 

[Fredrik]

No. Most physical observables (in interacting system) do not have tensor transformation laws. The CJS theorem is just one evidence for this statement.

I still don't understand what that theorem says. As I've mentioned, I found a book that says that it only applies to action-at-a-distance theories and an article that says it only applies to theories with Lagrangians with at least a 6th order term in c^-1, so I still don't know if it's relevant at all. Does it for example prevent Maxwell's equations in classical SR? (Both of those sources seem to be suggesting that it doesn't).

This argument against particles is rather weak. First, there is nothing shocking if different observers see different number of particles. Second, nobody has ever seen the Unruh effect. Perhaps it does not exist after all.

Maybe, but someone told me that the Unruh effect is essentially the same as Hawking radiation. The radiation just comes from the Rindler horizon instead of from a black hole horizon. So if there's no Unruh effect, there probably is no Hawking radiation either, and (I could be misinformed but) there seems to be several different reasons to think Hawking radiation is real. (No, I wouldn't be able to write down a list of reasons and explain them. I don't know this stuff that well).

You probably would think of these things as an argument against particles if you hadn't decided that the idea of a spacetime manifold is "useless" and "misleading". (I haven't ever heard anyone else say that). Note that Weinberg's chapter 2 is just defining non-interacting particles in a flat spacetime. I think the fact that Poincaré invariance is only local in GR strongly suggests that the particle concept can't even be properly defined on a curved spacetime.

 

[snoopies622]

I have another question for Fredrik: Is it your view that "On the Electrodynamics of Moving Bodies" contains logical flaws? or is its problem simply the ambiguity of the terms used?

 

[Fredrik]

I haven't read that article. I don't think he claimed that he was deriving a mathematical theorem, so I don't think I would have a problem with it.

 

[meopemuk]

Note that Weinberg's chapter 2 is just defining non-interacting particles in a flat spacetime.

You should read chapter 3, especially page 119, about interacting particles.

I think the fact that Poincaré invariance is only local in GR strongly suggests that the particle concept can't even be properly defined on a curved spacetime.

If I needed to choose between the concept of particles and the the concept of curved spacetime, I would've chosen the former one. I see particles every day of my life, but I've never seen "curved spacetime".

Eugene.

 

[meopemuk]

I still don't understand what that theorem says. As I've mentioned, I found a book that says that it only applies to action-at-a-distance theories and an article that says it only applies to theories with Lagrangians with at least a 6th order term in c^-1, so I still don't know if it's relevant at all. Does it for example prevent Maxwell's equations in classical SR? (Both of those sources seem to be suggesting that it doesn't).

The Currie-Jordan-Sudarshan theorem applies only to Hamiltonian theories (i.e., theories in which dynamics is constructed as a representation of the Poincare group). Maxwell's electrodynamics does not belong to this class. So, formally, CJS theorem does not apply there. However, the non-Hamiltonian nature of classical electrodynamics is a weakness rather than strength, in my opinion. The conservation laws (of energy, momentum, angular momentum) are very obscure in Maxwell's theory. In fact, it is impossible to define the energy of any system of charged point particles, due to divergences. There are lot of other paradoxes and inconsistencies.

On the other hand, QED and other quantum field theories are basically theories about particles (despite the word "field" being used). The fundamental quantity calculated in QFT is the S-matrix, which is defined as a set of coefficients connecting states of particles with well-defined momenta and spins. According to Weinberg, QFT is constructed in terms of a Poincare group representation. So, CJS theorem applies there. In the traditional formulation of QFT (based on bare particles), it is not clear how to describe interacting particle states. But this problem is successfully resolved in the "dressed particle" approach to QFT.

Eugene.

 

[Fredrik]

Maxwell's theory can be derived from a Lagrangian. Doesn't that imply that there's a Hamiltonian as well? (Sorry if it's a dumb question. The class that was supposed to teach us this stuff used the worst book and the worst teacher I ever had at the university. We hardly learned anything at all from that class, and it's been a long time since then).

 

[meopemuk]

Maxwell's theory can be derived from a Lagrangian. Doesn't that imply that there's a Hamiltonian as well?

In my opinion, the Hamiltonian approach does not work well in Maxwell's theory. First, the energy of the electromagnetic field surrounding a point charge is infinite. This makes any calculation of the energy balance suspicious. Second, there are lot of yet unresolved paradoxes related to the non-conservation of quantities that MUST be conserved in any Hamiltonian theory.

In the Kislev-Vaidman paradox the total energy is not conserved: A. Kislev, L. Vaidman, "Relativistic causality and conservation of energy in classical electromagnetic theory", Am. J. Phys., 70 (2002), 1216. http://www.arxiv.org/abs/physics/0201042

In the Trouton-Noble paradox there is a problem with conservation of the total angular momentum: O. D. Jefimenko, "The Trouton-Noble paradox", J. Phys. A: Math. Gen., 32 (1999), 3755.

In the Cullwick paradox the total momentum is not conserved: E. G. Cullwick, "Electromagnetic momentum and Newton's third law", Nature, 170 (1952), 425.

These (and many other) problems have been discussed in journal articles until this day, but I don't think they got a satisfactory resolution within Maxwell's electrodynamics.

Eugene.

 

[snoopies622]

For those who are interested in my original question, I also have a copy of Lillian Lieber's "The Einstein Theory of Relativity" and she makes the case something like this:

In classical physics, the quantity

$$\Sigma m ( \frac {dx}{dt}, \frac {dy}{dt}, \frac {dz}{dt})$$

for a set of particles is conserved. In relativity, since dt is no longer invariant and ds is, it is instead this quantity

$$\Sigma m ( \frac {dt}{ds}, \frac {dx}{ds}, \frac {dy}{ds}, \frac {dz}{ds})$$

which is conserved.

I don't suppose that argument meets anyone's standard of a proof, but it at least seems very reasonable to me.