In virtue of what are energy and momentum additive?

James MC

In STR, the energy of a composite system is the sum of the energies of its parts and the momentum of a composite system is the sum of the momenta of its parts. In every STR text I have seen, these principles are simply introduced without explanation, except to say that they have been experimentally confirmed. But they don't look at all like fundamental laws of the nature. No laws, which make necessary and explicit mention of composite systems, obtained in Newtonian mechanics, and I don't see why STR should be any different. So my question is, in virtue of what are energy and momentum additive in STR? (A related question is: in virtue of what is the energy-momentum 4-vector of the composite simply the sum of the energy-momentum 4-vectors of its parts?)

In trying to answer this myself I have tried three avenues without success.

ATTEMPT ONE: Perhaps the conservation laws of energy and momentum, which are either fundamental (in STR) or deducible from fundamental symmetries via Noether's theorem, might explain these additivity principles? This is because such principles appeal to the sums of energies and momenta of the parts of composites, stating that the sum prior to any interaction (or any point in time) is equal to the sum after the interaction (or after that point in time). But on the face of it, the fact that they appeal to the sums entails little about what properties the composites have. It seems that we cannot derive any contradiction from the conjunction of (i) the claim that the sum of the momenta (for e.g.) of the parts remains the same and (ii) the claim that the momentum of the whole is not the sum of the momenta of the parts.

ATTEMPT TWO: Perhaps there are force composition laws that enable us to deduce them? In Newtonian mechanics we have the fundamental composition of forces law (which doesn't mention composites!), which states that the force acting on particle three given the presence of particle one and particle two is the force that particle one would be exerting on three if they were by themselves plus the force that particle two would be exerting on three were they by themselves. So the two particles together are exerting net force F; so the composite they compose is exerting net force F. (This is useful because from it, together with features of the laws relating mass and force linearly, you can deduce the additivity of mass.) But as far as I can tell, I can't see any force composition laws in STR. In fact, STR textbooks seldom even talk of forces.

ATTEMPT THREE: Perhaps the principles are themselves a priori, as we have no conception of the energies and momentas of composites/wholes other than in terms of the energies and momentas of fundamental parts. Well, that seems like a non-starter for reasons internal to STR. For imagine someone pre-relativity saying that about mass! If the additivity of mass (which is empirically false according to STR) is not a priori than it's hard to see why any of these other additivity principles should be.


Any advice here would be most welcome. Very keen on seeing what people think!

mfb

I think that you mean special relativity (SR) when you talk about STR?

- You can derive global energy/momentum conservation (where the global values are the sum of the individual particle's values) from symmetry (here: translation in spacetime) with Noether's theorem.
- You can derive global energy/momentum conservation (same as above) with Newton's laws of motion.

pervect

In SR, in any given coordinate system, you'll find the total energy E and momentum p of a system (closed or not) by adding up the E and P of its parts. So I'd say that energy and momentum do add.

Mass, however, (by which I mean invariant mass) doesn't add.

Now, if you want to change coordinates, E and p together transform as a 4-vector for any CLOSED system.

It's important that the system be closed - the (E,p) doesn't in general transform as a 4-vector for a non-closed system.

If you're looking for something that always transforms covariantly, even for non-closed systems, what you need is the stress-energy tensor, given that (E,p) isn't a 4-vector for non-closed systems.

The rank 2 stress-energy tensor gives you the total amount of energy E and momentum p in any unit volume, where the unit volume is represented by a vector normal to the volume element, i.e. the four-velocity of the unit volume.

The stress-energy tensor always transforms covariantly as a rank-2 tensor regardless of whether the system is closed or not.

Energy and momentum get considerably more complicated in GR, but the above should suffice for SR.

James MC

Thank you both for responding to my post.


mfb: conservation laws are different from additivity principles. For example, the conservation law of momentum states that if no external forces act on a group of particles, the sum of the momenta of each particle always stays the same; meanwhile the additivity of momentum principle states that the momentum of a whole (composite system) is the sum of the momenta of its parts. These are different and their connection is non-trivial. Thus, while you are correct in saying that conservation of momentum is deducible (via Noether's theorem) from underlying symmetries, we still need a way to get from either the symmetry or the conservaiton law, to the additivity principle (see ATTEMPT ONE discussion).


pervect: your discussion of stress-energy tensors for non-closed systems is helpful and interesting; although I think the key issue here can be discussed entirely in the context of closed systems, so it might conduce to simplicity to stick to the (E, p) 4-vector. Now, I wasn't sure if you were attempting to explain why the (E, p) of a composite must be the sum of the (E, p)'s of its parts by appeal to transformation properties. If you were, how exactly does the argument go?

mfb

I think you misinterpret the relation.
In order to make a statement like "the total momentum stays the same", you need some way to define this "total momentum".

The conservation laws say "the added [property] of the particles is constant".
It is convenient to give this sum a name: The total [property].

James MC

Two points:

(i) Your proposal that conservation laws entail additivity principles, faces counterexamples. In particular, mass is conserved, yet non-additive. Hence, you cannot infer additivity principles from conservation laws alone. I'm interested in what you think of this counterexample.

(ii) The argument you give for your proposal is an interesting one, but given the mass counterexample, something must be wrong with it. If I understand you, your arguing that stating the conservation law forces us to quantify over properties of composites (the constant, added property). But that doesn't seem right to me: one can state the conservation laws simply by speaking of the properties of the parts, and saying of those part-properties that their mathematical sum remains constant. That formulation not only does not require one to quantify over composites, let alone composite properties, but it also strikes me as being the more fundamental microphysical formulation.


BTW, there's an interesting debate between two Russian physicists: one thinks the idea of invariant mass is defective simply because mass is conserved yet non-additive. The other (Lev Okun) gives a pretty decisive response; see in particular the top of the second page.
(http://iopscience.iop.org/1063-7869/..._43_12_L09.pdf)

mfb

In closed systems:

Mass (defined as the sum of rest masses of particles) is not conserved. And if you remove the processes which can modify the mass, it is conserved and additive.

Energy (defined as the sum of energy of particles and fields) is conserved. And it is additive.

You just get problems if you switch between both and play around with the words.

I did not make this statement. I just said that there are laws which conserve certain sums of properties.


"Access restricted: you will need to login or make a payment to access the full text of this article."

GeorgeDishman

My view is that you are starting from the wrong point though you identify the solution in the comment above. The key is that the universe seems to be a four-dimensional manifold, therefore for something to exist at other than a fleeting moment, it too must be 4D, thus the 4D energy-momentum vector is what is fundamental. What causes the energy and momenta to combine in the way they do is then just the laws of mathematics which apply to vector components because they are merely components of that 4-vector.

A number of responses have mentioned that masses combine differently, but when you recognise that "mass" is nothing more than a name we give to the magnitude of the 4-vector, whether it is of the individual parts or the aggregate, the same combination method still applies.

I think what brought that home to me was realising that although any individual photon has zero mass, the conceptual aggregate of a pair of photons has non-zero mass. Thus a "box full of light" (with a hypothetical perfectly mirrored interior) weights more than the same box empty.

This of course is an SR/GR view of the world. How the Higgs mechanism of QM relates to that is another matter (and not one I understand).

DaleSpam

Let there be a system composed of 3 parts A, B, and C, with energy Ea, Eb, and Ec respectively. The system energy is given by

E=f(Ea,f(Eb,Ec))=f(f(Ea,Eb),Ec)

Let A and B exchange a quantity of energy dE so that they now have Ea+dE and Eb-dE respectively. Since energy is conserved we have

E=f(Ea+dE,f(Eb-dE,Ec))=f(f(Ea+dE,Eb-dE),Ec)

Is there any other function besides f(x,y)=x+y which can satisfy these properties and is also commutative?

James MC

mfb:
Sorry about the article, I’ve attached it; see the top of the second page for why mass is conserved but not additive. ...If you’re not saying that conservation laws entail additivity principles, and are just saying that there are laws that conserve sums of part-properties (which I agree with), then I’m not sure what your explanation of the (energy and momentum) additivity principles is. My view is that while there are laws conserving the sums of part-properties, it need not be the case that the whole that is composed of those parts has the sum as one of its properties. Mass is the key example. You appear to suggest that I’m equivocating on different notions of ‘mass’ when I say that mass is both conserved and non-additive? I don’t understand, and am curious if you think Lev Okun is making the same mistake (in the attached article). By mass I just mean the property defined by m² = E²/C⁴ - p²/c².

GeorgeDishman:
I think your approach here is very promising, in part because you are trying to propose an explanation of the additivity of momentum and the additivity of energy, in such a way that your explanation is barred from applying equally to mass (which isn’t additive). Thus, you appeal to the vectorial properties of energy and momentum, which cannot be applied to mass because mass is a scalar. I like it! But I don’t understand what the “laws of mathematics” are that entail this result. Momentum is the space component of the (E, p) 4-vector while energy is the time component. Let’s say I have two particles, each with their own (E, p) 4-vectors. Let’s assume they compose a composite that is located where its parts are located (so that its position is a set of spacetime positions with two members) and let’s wonder what the mass, momentum, and energy of the composite is. What can we appeal to? You say that we can get the momentum and energy of the composite by appeal to (i) the 4-vectors of the individual parts together with (ii) some laws of mathematics. If that works, then we can presumably determine the system mass via m² = E²/C⁴ - p²/c². But I don’t understand what (ii) consists in. For example, what rules out the possibility of adding the vectors and multiplying by a constant to get the relevant composite properties? Why not take their cross products? What forces vector addition on us?

DaleSpam:
This is a very interesting argument, but I worry that it begs the question in the way you define f. While your argument does not presuppose that f is an additive function of the energies of the parts, what it does presuppose is that f is a function only of the energies of the parts. This assumption in your argument can be seen to be problematic if we reflect upon mass. The mass of the whole is not merely a (additive or otherwise) function of the parts, it is in fact a function of other properties: the energies and momenta of the parts. (See equation (10) in the attached article.)
I do hope that’s clear, as I’m pretty keen to hear how you would respond. How do you licence your initial assumption that the system energy is a function f of only the energies of the parts and is not a function of other properties too, as is the case with mass?

Attached Files

Okun on Mass.pdf (249.0 KB, 6 views)

GeorgeDishman

Right, mass is treated as either 'emergent' or more accurately nothing more than a name given to the result of a calculation. I am saying that the fundamental property of existence for any particle is the 4-vector. To get the energy, you take the dot product of that with a unit vector in your chosen time direction (in practice the tangent to the worldline of an observing clock) while the momentum is the cross product. "Energy" and "momentum" are thus only components and "mass" is the magnitude of the vector.

The "space part" is already a cross product. If the four vector of an object is V and T is a unit vector parallel to the observer's time axis, using consistent units (c=1):

E = T.V = V_t

p = T*V = (V_x, V_y, V_z)

m² = V_t² - (V_x² + V_y² + V_z²) = E² -|p|²

Right, if your particles are V₁ and V₂ and the composite is V_c then:

V_c = V₁ + V₂

after which the same definitions of energy, momentum and mass given above still apply to V_c.

Note that p is a vector so it is the square of the magnitude but yes, that is the same as the fully expanded versions above.

What forces us to use vector addition is the fact that they have conserved values in different directions, thus they are vectors. Noether's theorem applies for each of the components independently so the vector components must be conserved individually so the constant must be 1. See DaleSpam's proof for that part.

mfb

@James MC: The argument given in the paper is that they calculate the invariant energy in the center of mass system. This energy depends on the direction of particles. But the system itself depends on the direction as well!

Consider this nonrelativistic system: Two cars with mass 1 ton each travel in opposite directions with 10m/s. You can calculate their kinetic energy in the center of mass system (the system of the street), it is 2*1/2*mv^2=100kJ.
Now consider two cars which travel in the same direction. The center of mass system is now moving with the cars, so the total kinetic energy in this system is 0.

Does this imply that kinetic energy is not additive?
No. It just implies that it depends on the inertial system. And that you should not switch between two of them and expect to get the same results.

If you stay in one system, the energy is additive, and unless you create new particles (which is possible) the sum of masses is constant as well. The total invariant mass of the system is not additive, but this has the issue mentioned above.

DaleSpam

Well, my argument doesn't apply to mass at all since if A and B exchange a quantity of mass dM they do not have masses Ma+dM and Mb-dM respectively. So I don't see why a weakness of an argument based on masses would be relevant.

I just thought of one other thing, the work energy theorem. 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 and thus by the work energy theorem the energy of the system is Ea+Eb.

James MC

DaleSpam,

We need to distinguish two claims: (i) the energy of the whole is some function f of the energies of the parts and only of the energies of the parts, and: (ii) if (i) then f is additive (in virtue of energy conservation and the commutativity of f). I was trying to say that explaining energy additivity requires an explanation of both (i) and (ii) whereas you have only given an explanation of (ii), so far as I can tell. The mass example was not necessary to point this out. I only appealed to it to show that in general, explaining (i) is non-trivial given that sometimes physical properties of composites that are not merely functions of how those properties (and only those properties) are instantiated among the parts.

Let me try to put this in another way. Let’s say we know nothing about the empirical world until we are given a microphysical description, which includes descriptions of microphysical states (making no mention of composites) as well as any SR laws of nature. The question is what we can infer from this a priori, and whether we can infer from this description alone, that energy is additive. We can model possible a priori inferences from the microphysical description (1.) as material conditionals (2.)-(4.), as follows:

1. Microphysical description: there are 3 particles A, B and C, with energies Ea, Eb and Ec. The laws of SR obtain.
2. If (1) then there is a composite D composed of A, B and C.
3. If (1)&(2) then D has energy E that is some function f(x, y) of the energies of its parts and of nothing else.
4. If (1)-(3) then because D’s energy is conserved, then f(x, y) can be nothing other than f(x + y).
5. Energy is additive (from (1) and (4)).

I’m saying that while you have argued for (4), an actual explanation of energy additivity requires an argument for (2) and (3) as well. I think (2) is trivial and knowable a priori: it just involves grouping A, B, and C in thought and slapping a label on the group ‘D’. But how do we know (3)? Determining composite-properties from part-properties is non-trivial, and again, I think mass is a nice illustration of this. What are your thoughts?

(p.s. I wasn’t sure how the work energy theorem helps with energy additivity because the work energy theorem concerns kinetic energy whereas energy additivity concerns total energy. Please let me know if I’m wrong about that.)

James MC

GeorgeDishman,

Thanks for the helpful explanations of the equations. I want to frame this in a particular way (in argument form) so that it is easier to discuss specific aspects of the overall argument. Let’s imagine an ideal reasoner who knows nothing about the empirical world, is given a microphysical description of two particles, their respective (E, p)’s, and whatever fundamental SR laws we like. We then ask the reasoner whether she can a priori infer composite systems and their energies, momenta and masses. The reasoner might write out the microphysical description (1) and then express her inferences as a set of material conditionals (2)-(5) as follows:

1. [Insert description of microphysical state plus fundamental laws]
2. If (1) then there is a composite C composed of (E, p)₁ and (E, p)₂.
3. If (1)&(2) then C has (E, p)_c = (E, p)₁ + (E, p)₂
4. If (1)-(3) then C has total energy, momentum and mass, as determined by the three equations you specified.
5. If (1)-(4) then energy and momentum are additive while mass obeys eqn 10 (from the attached article).
6. Therefore, energy and momentum are additive while mass obeys eqn 10.

Premise (2) is trivial as it simply involves our considering two objects at once and slapping a label ‘C’ on their union, and asserting that C exists in virtue of 1 and 2 existing. Premise (3) is the key premise, let’s come back to it. Premise (4) just invokes the equations you mentioned—and we are just applying them to C’s (E, p). Premise (5) just compares the calculated values of the wholes with the values of the parts given in (1). Finally, the conclusion in (6) is logically entailed by (1) and (5).

OK—I hope this structure is helpful in that it makes transparent the sort of explanation I’m after and allows us to refer back to elements of the explanation by appeal to the premise numbers. Now, premise (3) is the key here and I don’t quite understand where it is coming from. In particular, why is it not simply presupposing the additivity principles that we are setting out to derive? What resources from (1) is it appealing to?

You say that premise (2) is forced upon us because (E, p)'s "have conserved values in different directions, thus they are vectors". I would love to see this unpacked a bit more, it's not entirely clear to me what your suggesting here.

One way to defend premise (2) would be to (i) argue that C has some (E, p) that is some function f of the (E, p)'s of its parts, and try to use the resources in (1) to rule out all possibilities other than that (E, p)_C = (E, p)₁ + (E, p)₂. You have done this for one case, that is, you have ruled out the alternative possibility that (E, p)_C is the addition of the (E, p)'s of parts multiplied by some constant > 1. To this you say “Noether’s theorem applies for each of the components independently so the vector components must be conserved individually so the constant must be 1. See DaleSpam’s proof for that part.” Again, I would love to see this fleshed out a bit. Notice that DaleSpam's proof presupposed that the part-to-whole function f was only a function of the energies of the parts, as opposed to a function of the energies of the parts multiplied by some contant. Or at least, that's what I argued above.

James MC

mfb,

You might be right that I’m getting muddled with respect to the fact that the (invariant) mass of a system is calculated by appeal to a privileged frame (that isn’t quite the system’s rest frame (as such a thing is not well defined for a thing with moving parts), but is something “near enough”, such as its centre of momentum frame, or centre of mass frame).

So we have three equations relating important physical properties of composites to properties of their component parts. Energy additivity, momentum additivity, and mass non-additivity (see equations (3) and (4) in the attached for two versions of the mass equation).

Now, I take it that the result we get for the former two (additivity) equations is going to be dependent on our frame (given that the E_i's and p_i's are dependent on our frame). But is it different in the case of equations (3) and (4) in the attached? That is, do these equations yield the same result no matter what frame we are in? And why is that?

In equation (4), does the subtraction of momentum from total energy amount to getting rid of all frame-variant properties, so that we are left with the frame invariant property of rest mass (or at least, rest mass over c²)? And is this what you mean when you say that "the argument given in the paper is that they calculate the invariant energy (mass?) in the center or mass system (center of momentum frame?)"?

Relatedly, I don't really get what Okun means when he says that the total mass is dependent on the angle between then momenta of the parts.

The reason I ask all this: In the above posts I continually appeal to the assumption the mass of the whole is determined by properties other than the mass of its parts. And I appeal to equations (3) and (4) in defence of this. But you're making me think this might be mistaken. Though I'm not sure.

Sorry if these questions seem a bit naive, but I think you've hit upon some gaps in my understanding here.

Attached Files

equations for additivity discussion.pdf (75.9 KB, 4 views)

DaleSpam

All energy can do work, not just KE. The work energy theorem argument is conclusive.