In virtue of what are energy and momentum additive?

Sam Gralla

To me, the word additive implies a choice of frame. (Otherwise, at what "time" do you add two things?). But once the frame is chosen, the energy-momentum density is just [tex]T_{0\mu}[/tex] where [tex]T_{\mu \nu}[/tex] is the stress-energy of the matter. The energy-momentum in a region is then just the integral over that region of space (in the chosen frame). Integrals are additive: if you draw subregions, the energy-momentum in the region is equal to the sum of the energy-momenta in in the subregions.

Most SR texts only discuss this in the context of particles, but the above is the version for classical smooth matter sources.

GeorgeDishman

Yes. C is an aggregate so we are not implying any new existence here.

Yes, that's the crux.

All perfectly correct.

You can say that physics is the process of modelling reality using maths. In looking at the properties of entities experimentally, we note that vectors are an appropriate model hence vector addition is the logical method. However to expand that as you ask, let's look in a bit more detail.

We you call (E,p) I call V because it is a 4-vector. The components V_t, V_x, V_y and V_z are all of equal status so we can just consider one and the argument applies equally to the others so let's look at V_x, the component of momentum in the x direction.

Consider a block of wood at rest. We shoot a bullet into it in the +x direction and experimentally we note that the block thereafter is moving in the +x direction. It doesn't go in the +y or -y directions because, by symmetry, there is nothing to distinguish one from the other and the only value of y momentum that is preserved under a change of sign is 0. The same applies to the z and t directions so we know the effect of the impact must be limited to motion along the x axis.

Once the block is moving, fire a second bullet at it, also moving in the x direction. The same argument applies so the result must still be in the x direction. An observer moving in the +y direction would also observe changes of velocity ocurring only in the x direction hence there can be no cross terms of V_x appearing in the formula for V_y etc..

That leaves the possibility that the combination might not be linear so think of its expansion as a power series. If the function 'f' in DaleSpam's post was a square then bullets moving in the +x or -x direction would both cause the block to move off in the +x direction. Obviously, by observation, that doesn't happen and by symmetry we can dispose of all even powers for the same reason. If we consider the limit as the speed of the bullet tends to zero, the behaviour must reduce to Newtonian mechanics so the relationship must be linear.

Is that enough to give you an approach? It gets more complex to look at higher powers because momentum in relativity is not a linear function of speed and I've spent longer than I intended on the machine already (the wife has other tasks in mind for me).

pervect

I'm glad someone else is making this point, since I've been rather busy and thus neglecting this thread.

Given that the stress energy tensor is the correct relativistic description for the density of matter, the additive properties of energy and momentum follow directly from the properties of the stress-energy tensor and the notion of simultaneity, as Sam mentions.

I think that the correctness of the stress-energy tensor as a relativistic description for the density of matter is perhaps not obvious to the OP. I suppose it's not particularly "obvious" to me, either, though I've come to accept it.

A useful discussion of why the stress energy tensor is the correct description would focus on issues like covariance, which boils down to the principle that a theory must give self-consistent descriptions of some underlying reality from all different viewpoints or frames. And I suspect this is in fact the underlying issue here - how do we make sure our theories are formulated in a way so that we can change frames at will and get consistent resutls?

Unfortunately, I don't know of any good reference which discusses this issue, and the brief notes I have the time to post don't seem to be enough to get through to anyone who is unfamiliar with the answer.

James MC

Sam Gralla & Pervect,

I'm excited by this suggestion, largely because I have already formulated microphysical proofs for the additivity of (both inertial and gravitational) mass in Newtonian mechanics, by appeal to integral equations for mass-densities. It would be awesome if we could construct analogous proofs in the relativistic context! But I know little about the correct relativistic formulation of the laws for energy-momentum densities, and since reading your posts, I have had a fair bit of difficulty finding a good resource so as to learn about them.

I wonder, if I threw out a little of what I've done in the Newtonian context, perhaps someone might have some thoughts about how it could generalise to relativity theory?

The goal (in the Newtonian context) is to deduce the additivity of mass, purely from microphysical states plus fundamental laws of nature. Crucially, the microphysical state-descriptions make no mention whatsover of composite systems. The fundamental laws could be either Newton's law of gravitation, or F=MA; as the former law is simpler, I'll focus on that, thereby deducing the additivity of gravitational mass.

So the fundamental law (describing the grav field) for a single density is:

g(x) = G[itex]\int[/itex]_v [ ρ_i([itex]\xi[/itex])(x - [itex]\xi[/itex]) / |x - [itex]\xi[/itex]|³ ] d [itex]\xi[/itex]

(Sorry it looks a bit ugly, the Latex function didn't appear to like certain embeddings.) Now, the law for multiple densities is key here, it is the superposition of forces, and is legitimate to appeal to as its formulation doesn't mention composites, but only net effects of multiple particles. I have been wondering if relativity theory has an analogous law. At any rate, we now have:

g(x) = G [itex]\sum[/itex][itex]^{N}_{i=1}[/itex] [ [itex]\int[/itex]_v [ ρ_i([itex]\xi[/itex])(x - [itex]\xi[/itex]) / |x - [itex]\xi[/itex]|³ ] d [itex]\xi[/itex] ]

And because the intervals are all over the same interval (the same volumes of space), we can swap the sum and the integral, to get:

g(x) = G [itex]\int[/itex]_v [ [(x - [itex]\xi[/itex]) / |x - [itex]\xi[/itex]|³] [[itex]\sum[/itex][itex]^{N}_{i=1}[/itex] ρ_i([itex]\xi[/itex])] ] d [itex]\xi[/itex]

Now, we can infer the existence of a composite composed of the N particles, located where those N particles are, trivially (composites are no addition to being). We can infer the additivity of the gravitational force from the superposition of forces law. At this stage, we know the composite's position (a set of positions) and its force (g(x)) so now we can use the above equation to determine its mass, which is clearly (in boldface) the required sum.

A similar proof can be made for inertial mass additivity.

I'm wondering what the integral equations for energy-momentum densities look like in relativity theory, and whether anything like the microphysical proof of mass additivity above can be worked out for energy/momentum additivity in the relativisitic context. (Note that we need not get into GR, and can remain in the SR context - I just used gravitational mass here due to its simplicity.)

Any thoughts, hints, or references, would be most welcome!

James MC

GeorgeDishman,

To summarise our progress so far:

(i) We've constructed an argument with numbered premises (1)-(5), and a conclusion (6), which purports to derive composite energy, momentum, and mass, and we agree that premise (3) is all that's left to defend.

(ii) Your defence of (3) appeals to DaleSpam's proof, which I have raised two objections to. Objection one points out that the proof assumes the part-whole premise that composite energy is some function f of the energies of the components and of nothing else. Objection two argues that the proof leaves open alternative possibilities for how energies combine.

(iii) In response to objection one, you have tried to clarify crucial features of the fundamental energy-momentum 4-vectors. In particular, you have argued that because 'energy' is a merely a component of a 4-vector, it enjoys a kind of independence from other properties, an independence that entails that f cannot be a function of other properties. Thus, the example I appealed to to help clarify my objection - composite mass - is a function of momentum and energy (properties other than mass). But you argue that this is no problem once we see that 'mass' is (composite or otherwise) just a name for the magnitude of the four vector, and so of course it is going to be a function of properties other than itself (such as the four-vector components). That's a pretty cool argument - thanks!

(iv) In response to objection two (which grants that you are right about (iii)), you try to argue that f(x, y) = f(x + y) by eliminating alternatives. For example, you produce arguments attempting to eliminate the possibilities of f(x, y) = f((x+y)c) and f(x, y) = f(x*y), where * is a cross-product, based on a thought experiment with two bullets and a block. I like the strategy.

Some comments:

As you note, the cross product possibility is problematic. One thought I had here was that if the energy of just one part of a composite system is 0, then via f(x*y), the composite must have zero energy. This can perhaps be made to conflict with our composition premise. Perhaps we can appeal to a priori principles such as "If a part of C transfers some energy to D then C transfers some energy to D", which would be violated if C's energy is determined by cross product and one of C's parts has zero energy.

But I still worry about the other possibility, and I'm not sure I fully understood your thought experiment. You said that if we think of f as a power series (or a square?) then bullets moving in the +x or -x direction would both cause the block to move off in the +x direction. Sorry I didn't understand this. There's no problem if bullets moving in the +x direction would cause the block to move in the +x direction. Was the problem that it would entail that the block would move off into the -x direction? Or did you mean to say move off of the x direction? In general, I wasn't sure whether the two bullets were playing the role of a two-particle composite system such that you were trying to deduce composite properties from the behaviour of two bullets on separate occasions? I also found it noteworthy that while you say that you're appealing to DaleSpam's proof, you aren't explicitly appealing to commutativity and conservation as he did. -Sorry about the time-consuming nature of the machine, no need to respond, but if you have further thoughts that want to be expressed on here then they would be most welcome!

James MC

DaleSpam,

Fair enough. Googling the theorem seems to consistently show that the theorem concerns kinetic energy (see e.g. the definition here: http://www.dfcd.net/articles/workenergy.pdf), but I'm happy to take your word for it that it also applies to rest energy.

In my first post in this round (three posts up) I asked whether there's anything in relativity theory analogous to the superposition of forces law; that is, an additivity law that can be formulated without appeal to systems/composites. Such laws are very useful in determining part-whole additivity principles, at least given the demanding notion of "determining" I'm assuming. I'm wondering if this is what you had in mind?

If we go back to your earlier argument when you first introduced the theorem, you stated this:

"If A has energy Ea and B has energy Eb then by the work-energy theorem A can do work Ea on an object and B can do work Eb on the same object. The work done on the object by the system of A and B is therefore Ea+Eb so the energy of the system is Ea+Eb."

So if I understand you, you're saying that the theorem entails that in some circumstances, all of the energy that an object has, can be "transferred" or "manifested" or "exhibited" (not sure what the right word is) in terms of work done. In that case, we can appeal to the a priori principle that if two parts that compose C do net work Ea+Eb on D, then C does work Ea+Eb on D, and then by the w/e-theorem applied to C, C must have energy Ea+Eb.

OK, if you're right that the theorem applies to rest energy just as much as kinetic, then I'm liking it! It does seem very similar to the superposition of forces principle mentioned three posts above. That's because work done and force can be specified independently of composites as relational properties among parts, but also allows us to move a priori from the relational properties of parts to intrinsic properties of wholes. Very nice.

One objection would be that this proof is limited to parts that expend all their energy on work (again, not sure if "expend" is the right word). So the result doesn't quite get us to energy additivity. It only gets us to energy additivity for parts that are expending all their respective energies on work. What about parts that are not? Please let me know if you have further thoughts!

Sam Gralla

Hi James,

What your equations describe is the superposition principle for the Newtonian gravitational field: adding the fields together gives you the correct field for the combined sources. This will be satisfied by linear equations, like Maxwell's equation of electromagnetism, but not by non-linear equations, like Einstein's equation of general relativity.

So, you won't be able to get an analog of your calculation for relativistic gravity, but you can do so for electromagnetism. Your integrals will be much more complicated, because the "Green's function" for Maxwell's equation is much more complicated than that of the Poisson equation (describing Newtonian gravity), but everything is still linear, provided you are happy choosing the "retarded solution". Sorry for not explaining these concepts, but I thought I'd just give you the buzzwords to point you in the right direction.

-Sam

DaleSpam

The object doing the work with the net force may have any kind of energy. Because the force is a net force it increases the KE of the object on which the work is being done.

I am sure that you are both sufficiently smart and sufficiently motivated to prove that as a corollary.

James MC

Hey Sam,

Thanks for the electromagnetism tip. I've noticed the similarity between Newton's law of gravitation and Coulomb's law. So I figure a similar proof would apply. Thus, with a bit of help from wiki (http://en.wikipedia.org/wiki/Classic...ectric_field_E) I figure it would go a little something like this:

We have the fundamental law for charge densities:

E(r) = [itex]\frac{1}{4\pi εo}[/itex] [itex]\sum[/itex][itex]^{N}_{i=1}[/itex] ∫ [itex]\frac{\rho(r\acute{})(r - r\acute{})}{|r - r\acute{}| ^{3}}[/itex] d³r[itex]\acute{}[/itex]

And because the integrals are all over the same interval (the same volumes of space), we can swap the sum and the integral to get:

E(r) = [itex]\frac{1}{4\pi εo}[/itex] ∫ [itex]\frac{(r - r\acute{})}{|r - r\acute{}| ^{3}}[/itex] [itex]\sum[/itex][itex]^{N}_{i=1}[/itex] ρ(r[itex]\acute{}[/itex]) d³r[itex]\acute{}[/itex]

Now the rest of the proof is analogous to classical gravity: We can infer the existence of a composite composed of the N particles, located where those N particles are trivially (composites are no addition to being). We can infer the additivity of the charges from the superposition of charges law. At this stage, we know the composite's position (a set of positions) and the electric field it is determining (E(r)) so now we can use the above equation to determine its charge, which is (in boldface) the required sum.

But we are still stuck in the classical context, and it is not clear that anything like this can be passed over to the relativistic context (e.g. to Jefimenko's equations?).

This style of proof also applies to inertial mass additivity in Newtonian physics, given that f=ma can be formulated for densities, and has a corresponding force superposition principle.

It is interesting that you say that this cannot be passed over to relativity theory (SR or GR) to deduce the additivity of energy/momentum. Is there really no superposition principles at all? Fair enough we can't add fields (or inertial forces), but is there no other function we can put in front of the equation here, that yields net forces? If there are nothing like these, then it doesn't seem to me that your (and Pervects) earlier solution can work, and we can appeal to the above resources to see why.

In your first post you said: "The energy-momentum in a region is then just the integral over that region of space (in the chosen frame). Integrals are additive: if you draw subregions, the energy-momentum in the region is equal to the sum of the energy-momenta in in the subregions."

Your statement following the colon is the thing we are trying to prove, so we cannot assume it. I am trying to show that SR does not radically differ from Newtonian physics, in that it does NOT postulate fundamental micro-to-macro additivity principles. In order to know whether or not SR postulates fundamental micro-to-macro laws, we start with a purely microphysical description, together with fundamental laws, to see what we can prove. Thus, we might have our energy-momentum densities specified for a couple of distinct infintesimal regions (these are our 'parts'). And we want to use that description, together with the laws, to prove that the energy-momentum of the total region is equal to the sum of the energy-momenta in the two infintesimal sub-regions.

The key point here is that in the three proofs of classical additivity principles above, we needed, as well as the integral equation for densities, something like a force superposition principle. If you're right that nothing of the sort exists in SR, then it is difficult to see how the equations you're appealing to do can do the job in SR.

Pervect, when commenting on your post, said: "the additive properties of energy and momentum follow directly from the properties of the stress-energy tensor and the notion of simultaneity, as Sam mentions."

But if, in the classical case, analogous integral equations for densities alone are insufficient to prove additivity principles (because they also need composition principles), then I don't really see why the same wouldn't also be true in SR.

Further thoughts on this would be most welcome.

Sam Gralla

Hi James,

I wouldn't have called your calculation a derivation of the additivity of momentum, but rather a derivation of the superposition principle for force. I don't see where you've even defined momentum.

You can indeed repeat your derivation for relativistic electromagnetism. It's most straightforward working at the level of the Green's function for the vector potential; see the first two equations below "retarded potential solutions" in http://en.wikipedia.org/wiki/Li%C3%A...hert_potential. I don't see why you can't also use Jemfiko's equations as well. The only property your calculation uses is linearity in the source, and Jemfiko's equations are linear in the source.

Regarding my comment on the additivity of momentum, the fact that integrals are additive (i.e., integrating a function over a region R3 = R1 union R2 is the same the sum of the integrals over R1 and R2 (provided R1 intersection R2 = null)) is just a property of integrals. The momentum in the EM field (given a choice of reference frame) would be integral of E cross B over space. So, the fact that you can find the momentum in R1 and find the momentum in R2 and add them to get the momentum in R3 just boils down to a property of integrals.

Maybe the real property you're interested in is not "additivity of momentum" but some other property?

GeorgeDishman

You seem to be mixing up points a little. Momentum is already the cross product as I said. Energy is the dot product which results in a scalar so when combining two energies, you cannot take a cross product of two scalars.

Suppose f(a,b) includes a term involving a². If one bullet going in the +x direction hits, the block also moves in the +x direction. However, if one bullet travelling in the -x direction hits the block, the block would move in the +x direction since the square always gives a positive result. Only negative powers can appear in the equation otherwise the result would be anisotropic.

Exactly. Hit the block with one bullet then hit it with another. The block has the combined momentum of the two impacts. What can we deduce about the constraints that must apply to the formula for combining them? Energy is another component of the 4-vector, just in a different direction, so the rules must be the same.

I didn't want to appear to be taking credit for his argument when I felt I was just recasting it a little.

James MC

Hi Sam,

Thanks for the link. I'll study these electromagnetism equations and hopefully come up with an analogous argument for them.

You're right that I never (in post 21) gave a derivation of the additivity of momentum. Although I also wouldn't have called it a derivation of the superposition principle of force - that principle was presupposed as an initial premise in the derivation. It was meant to be a derivation of the additivity of (gravitational) mass in Newtonian mechanics.

The basic idea was that we start with a microphysical force law for multiple densities. We look to the left of the integral (where the superposition of forces is explicitly stated) and we infer the additivity of forces from the superposition of forces. We then look to the rest of the equation, and notice that the law treats masses and forces as linearly proportional in such a way that if forces are additive then so are masses.

This sort of strategy more generally allows us to prove Newtonian additivity of both gravitational mass and inertial mass, as well as, it seems, the classical and relativistic additivity of charge.

Now, what I'm looking for are analogous or quasi-analogous proofs for the additivity principles in relativity theory, whatever they are. It turns out, there is no such principle for mass: mass is non-additive in relativity theory. Nonetheless, momentum and energy are additive. That's one reason why I'm focusing on momentum (and energy) in the relativistic context.

Although in general I'm interested in explaining composite mass from microphysical premises (premises that don't mention composites) in the demanding sense of explanation I'm assuming (roughly: apriori entailment). But to explain why composites have the masses they do in relativity theory, from microphysical principles, you first need to know composite energy and momentum. This is why such additivity principles are important to me in the relativistic context.

Regarding a proof for the additivity of relativistic energy and momentum, you said:

This is helpful. If what you are saying is correct, then it looks like your argument generalises to Newtonian additivity of (gravitational) mass, does it not? At the very least I can see how it applies to the additivity of gravitational forces. So consider the argument I gave above (post 21). If I've understood your strategy then it seems we can at least prove force additivity (or the additivity/superposition of gravitational potentials). So imagine we have two mass densities ρ₁(x) and ρ₂(x), and each of them determine the following gravitational potentials for position x:

g(x)₁ = [itex]\int[/itex][itex]_{v}[/itex] [ [itex]\frac{\rho_{1}(\xi)(x - \xi)}{|x - \xi|^{3}}[/itex] ] d([itex]\xi[/itex])

g(x)₂ = [itex]\int[/itex][itex]_{v}[/itex] [ [itex]\frac{\rho_{2}(\xi)(x - \xi)}{|x - \xi|^{3}}[/itex] ] d([itex]\xi[/itex])

And now, and this is I think the key move of your proof, we assume the following as an a priori principle about the composite density ρ_C:

ρ_C(x) = ρ₁(x) [itex]\bigcup[/itex] ρ₂(x)

This allows us to show that:

g(x)_C = [itex]\int[/itex][itex]_{v}[/itex] [ [itex]\frac{\rho_{C}(\xi)(x - \xi)}{|x - \xi|^{3}}[/itex] ] d([itex]\xi[/itex]) = g(x)₁ + g(x)₂

Hence we've proved the additivity of forces (or at least, the superposition principle, which itself a priori entails the additivity of forces).

So, before trying to apply your strategy to relativistic integrals, I wanted to check that I've actually understood your strategy using integrals I better understand. Have I understood?

If I have understood, then my task now is to determine the form of the equation that has momentum density to the left of the equals sign (and then the form of the equation that has the energy density to the left of the equals sign). You said that for momentum the equation is E cross B. But isn't that for an electromagnetic wave? Where E is an electric field and B is a magnetic field? (http://scienceworld.wolfram.com/phys...umDensity.html)

A couple of things. Firstly, I don't see how that relates to your first post when you said I should use the energy-momentum tensor. Can we still do that? I still don't know the correct formulation of the integral equation for densities, but I've at least found what appears to be the analogue for point particles, here. In general, so that I could relate this back to mass, it would be nice to use an expression for energy and momentum density that stays away from electromagnetism (unless you think that is by far the simplest method?). I wondered if we couldn't just go for an integral formulation of E = [m²+p²]^1/2
Thanks for your help!