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/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.

Hence, you cannot infer additivity principles from conservation laws alone.

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]

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?

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

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).

 

[Dale]

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 license 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?

 

[GeorgeDishman]

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. ... 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).

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.

Momentum is the space component of the (E, p) 4-vector while energy is the time component. ... Why not take their cross products?

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

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.

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.

If that works, then we can presumably determine the system mass via m² = E²/C⁴ - p²/c².

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.

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?

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.

 

[Dale]

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 license 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?

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.

 

[Dale]

(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.)

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

 

[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 $$T_{0\mu}$$ where $$T_{\mu \nu}$$ 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]

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.

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

Premise (3) is the key premise, let’s come back to it.

Yes, that's the crux.

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).

All perfectly correct.

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 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.

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. ... 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.

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]

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 $$T_{0\mu}$$ where $$T_{\mu \nu}$$ 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.

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$\int$_v [ ρ_i($\xi$)(x - $\xi$) / |x - $\xi$|³ ] d $\xi$

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

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

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

 

[Dale]

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),

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.

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!

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/Classical_electrodynamics#The_electric_field_E) I figure it would go a little something like this:

We have the fundamental law for charge densities:

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

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

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/Liénard–Wiechert_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]

... (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.

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.

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?

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 traveling 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.

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?

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 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.

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:

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

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

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

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) $\bigcup$ ρ₂(x)

This allows us to show that:

g(x)_C = $\int$$_{v}$ [ $\frac{\rho_{C}(\xi)(x - \xi)}{|x - \xi|^{3}}$ ] d($\xi$) = 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/physics/MomentumDensity.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!