The Energy - Momentum Equation vs the Energy - Mass Equation

[PeroK]

• In Newton's theory, "mass" and "energy" are names for different physical quantities.
• In SR, the concept of "relativistic mass" visualizes best, via ##E=mc^2## in all inertial frames, that "mass" and "energy" can be regarded as two names for exactly the same physical quantity. That is a major achievement of SR over Newton's theory. According to Occam's razor, either "mass" or "energy" is needless from a physical viewpoint.

To put this in simple terms, PF uses the convention of Lorentz Invariant mass, not relativistic mass. If you want to contribute effectively to the PF relativity forum, then you need to respect that decision and that convention.

Arguing the point for an alternative convention is unproductive (for you and us).

 

[Sagittarius A-Star]

If you want to contribute effectively to the PF relativity forum, then you need to respect that decision and that convention.

Arguing the point for an alternative convention is unproductive (for you and us).

Sorry, it was not my intention to disrespect that decision. Then I take this argument back.

 

[Dale]

It's also important to note that mass is not conserved for composite systems in relativistic physics.

I wouldn’t say it that way. I would say mass is not additive. After all, as you have pointed out mass is the norm of the four-momentum, and the norm of a conserved four-vector is itself conserved.

Or are you speaking of distributed systems in curved spacetime where you have to use the stress-energy tensor instead of the four-momentum? If so, then I would say that the mass is not clearly defined rather than that it is not conserved.

 

[vanhees71]

I meant it in the sense that, e.g., for a scattering process the sum of the invariant masses in the initial state are not necessarily equal to the sum of the masses of the final state, e.g., take ##e^+ + e^- \rightarrow \mu^+ + \mu^-##. Sum of the masses in the in state is ##2 m_e## and in the out state ##2 m_{\mu}##.

What you are referring to is not "conservation of mass" but "conservation of" center-momentum energy, i.e., ##s_{\text{in}}=(\sum p_j^{(\text{in})})^2=s_{\text{out}} = (\sum p_j^{(\text{out})})^2##.

 

[Dale]

the sum of the invariant masses in the initial state are not necessarily equal to the sum of the masses of the final state

Which is because mass is not additive as I said above. i.e. the sum of the invariant masses of the parts is not the invariant mass of the whole

 

[Sagittarius A-Star]

What you are referring to is not "conservation of mass" but "conservation of" center-momentum energy, i.e., ##s_{\text{in}}=(\sum p_j^{(\text{in})})^2=s_{\text{out}} = (\sum p_j^{(\text{out})})^2##.

##p \cdot p=(E/c)^2-\vec{p}^2=m^2 c^2##.

To my understanding, not only the energy in the center-momentum frame is conserved. Overall energy and overall momentum of the composite system are conserved in every intertial frame. Therefore, the rest mass ##M## of the composite system of moving particles with masses ##m_i## is conserved, too.

In the center-momentum frame, the overall energy is: ##E = M c^2##. It follows for the overall rest mass:

##M = \sum_{i} \gamma_i m_i##

 

[etotheipi]

But ##\sqrt{s} = \sqrt{p_{\mu}p^{\mu}} = E_{\text{CoM}} = M##, so saying rest mass is conserved & energy in CoM frame is conserved is a tautology.

 

[vanhees71]

In this sense mass is conserved, but as discussed above for a composite closed system, the so defined invariant mass is not the sum of the invariant masses of the constituents, but ##\sqrt{s}## depends, e.g., on the temperature of a condensed-matter system or the mass of a capacitor depends on the electric field between its plates, etc.

 

[Ibix]

Another way to say it: the "system mass", ##\sqrt{\left(\sum_i p_i\right)^2}## (where ##p_i## is the four momentum of the ##i##th particle), is conserved in a closed system. The total of component masses, ##\sum_i\sqrt{p_i^2}## is not.

 

[Dale]

Agreeing with @Ibix: As far as I know the quantity ##\Sigma |p_i|## has no purpose in physics whereas the quantity ##|\Sigma p_i|## does. I think it makes more sense to name useful quantities rather than useless ones, so I would also say that the system mass is the latter.

 

[vanhees71]

I hope you mean ##\sqrt{s}##. I don't know what ##|p_i|## should be!

Correct is that that the center-of-momentum energy of a closed system is conserved, and that this is a scalar, because it's defined in a fixed reference frame, which is preferred by the physical situation and thus is well-motivated as a definition for a scalar quantity characterizing an intrinsic property of this system:
$$s=(\sum_i p_i)^2,$$
where the square is meant in the sense of the Minkowski product and ##p_i## are four-vectors. That's the usual definition.

 

[SiennaTheGr8]

Agreeing with @Ibix: As far as I know the quantity ##\Sigma |p_i|## has no purpose in physics whereas the quantity ##|\Sigma p_i|## does. I think it makes more sense to name useful quantities rather than useless ones, so I would also say that the system mass is the latter.

If I may: this is a point in favor of "rest energy" (pedagogically speaking).

Early on, we're taught that mass is the "amount of matter," and it's drilled into us that the "total mass" of a system is the sum of the masses of its constituents. Even the kids that don't pay attention know that.

When learning SR, simply having heard the phrases "amount of matter" and "total mass" is an impediment. They're worse than useless concepts—they are confusing and misleading distractions that years of habit make difficult to purge from thought. Every time they come to mind your understanding takes a hit, and the word "mass" inevitably calls them to mind at first.

"Rest energy" comes with no such baggage. On the contrary, it's only helped by prior contact with the energy concept. If you asked a high-school physics student with no exposure to SR how to calculate the total energy of a system in its rest frame, it wouldn't even occur to them to exclude the kinetic- and potential-energy contributions of the system's constituents. In fact, they'd probably just tell you, "add up the kinetic and potential energy," because they've done it many times. All the right intuitions follow immediately from the additivity and conservation of energy, a concept that the student already understands and which remains valid and crucial in SR. The next step is:

"You know that mass thing we talked about for years? Turns out that it's nothing but this 'rest energy' concept we're introducing (system's total energy in its rest frame). That's what Einstein's ##E_0 = mc^2## tells us. Yes, this renders the very idea of 'total mass' obsolete—what good would it do you to add up just the rest-energies of a system's constituents? it wouldn't give you the system's rest energy (because you've left out kinetic- and potential-energy contributions). So mass isn't additive. Is mass conserved? Well, sure, but only because a closed system's total energy is conserved in any given inertial frame, and the system's rest frame is no exception. So forget about 'total mass' and 'amount of matter,' and demote 'conservation of mass' to a trivial consequence of energy-conservation (but elevate mass's invariance to a fact of primary importance). To stop yourself from falling into old habits, you might find it useful to think of 'rest energy' for a while whenever you encounter the word 'mass,' at least until all of this sinks in. Physicists use the word 'mass,' though."

 

[vanhees71]

Mass is NOT conserved, energy is in special relativity. Mass is NOT additive but energy is.

For one last time: Mass is a fundamentally different quantity in relativistic physics than energy and that's why one must insist to clearly distinguish it also in science teaching. Even kids that don't pay attention should know that within special relativity mass is NOT equivalent to energy as claimed by some popular-science books.

 

[SiennaTheGr8]

Mass is NOT conserved

That depends on how you define "conserved," and there are two competing definitions in the literature. One is "the value doesn't change over time (in a closed system)," and the other is the same as the first but includes the stipulation that the quantity in question be additive. It looks like you're using the second definition, as Landau and Lifshitz do (and which Okun criticized them for). I'm with Okun (and @Dale, it seems) that it's better to explicitly distinguish additivity from conservation.

For one last time: Mass is a fundamentally different quantity in relativistic physics than energy and that's why one must insist to clearly distinguish it also in science teaching. Even kids that don't pay attention should know that within special relativity mass is NOT equivalent to energy as claimed by some popular-science books.

Yes, mass and energy are not the same thing. Mass and rest energy are the same thing, though.

 

[vanhees71]

I thought we agree in this forum to use only the notion of invariant mass as mass and energy as energy, and that's how it's used in contemporary research.

The mass of a composite system is indeed defined by it's energy in its center-of-momentum frame (divided by ##c^2##). This implies that this quantity is not conserved though. Take a piece of iron and heat it up, it increases its mass by ##\Delta q/c^2## though it still consists of the very same iron atoms as before.

The concept of mass is different in relativity from Newtonian physics. In Newtonian physics mass is described as a central charge of the Galilei Lie algebra and as such necessarily conserved in addition to the usual Noether conservation laws from Galilei symmetry (energy, momentum, angular momentum, and center-of-mass (sic!) velocity corresponding to the subgroups of time translations, space translations, spatial rotations, and Galilei boosts).

In special relativity there's no extra conservation law for mass. You only have the 10 conservation laws due to symmetry of Minkowski spacetime, the Poincare group. Instead of center of mass you rather have a center of energy.

I don't know what you are referring to concerning Landau and Lifshitz. In his famous 10-volume textbooks on theoretical physics everything is manifestly covariant and thus no confusion occurs with concepts that are outdated for more than 100 years ("relativistic mass").

 

[Dale]

I don't know what |pi| should be!

The (Minkowski) norm of the four-momentum, ##p_i##. I prefer that notation as it is more geometric than algebraic.

 

[SiennaTheGr8]

I thought we agree in this forum to use only the notion of invariant mass as mass and energy as energy, and that's how it's used in contemporary research.

The rest energy concept is covered in virtually every textbook on special relativity I've encountered. It's a fully mainstream term.

I don't know what you are referring to concerning Landau and Lifshitz. In his famous 10-volume textbooks on theoretical physics everything is manifestly covariant and thus no confusion occurs with concepts that are outdated for more than 100 years ("relativistic mass").

In Section 9 of Vol. 2, they explicitly tether conservation to additivity (emphasis mine):

The energy of a body at rest contains, in addition to the rest energies of its constituent particles, the kinetic energy of the particles and the energy of their interactions with one another. In other words, ##mc^2## is not equal to ##\Sigma m_a c^2## (where ##m_a## are the masses of the particles), and so ##m## is not equal to ##\Sigma m_a##. Thus in relativistic mechanics the law of conservation of mass does not hold: the mass of a composite body is not equal to the sum of the masses of its parts. Instead only the law of conservation of energy, in which the rest energies of the particles are included, is valid.

 

[vanhees71]

The (Minkowski) norm of the four-momentum, ##p_i##. I prefer that notation as it is more geometric than algebraic.

There is no Minkowski norm, but I understand what you mean. I'd strongly discourage from using this notation, which I've never seen before, because it can only lead to more confusion as we just see in this thread.

 

[vanhees71]

In Section 9 of Vol. 2, they explicitly tether conservation to additivity (emphasis mine):
The energy of a body at rest contains, in addition to the rest energies of its constituent particles, the kinetic energy of the particles and the energy of their interactions with one another. In other words, is not equal to (where are the masses of the particles), and so is not equal to . Thus in relativistic mechanics the law of conservation of mass does not hold: the mass of a composite body is not equal to the sum of the masses of its parts. Instead only the law of conservation of energy, in which the rest energies of the particles are included, is valid.

Yes, and that's correct. It's not something special of Landau and Lifhitz. It's a fact following from the mathematical structure (i.e., the symmetry properties) of relativistic spacetime.

 

[PeroK]

Yes, mass and energy are not the same thing. Mass and rest energy are the same thing, though.

That's all well and good, but on a practical level there is a risk of overloading the term energy. If we abandon the terminology of invariant mass in favour of rest energy, then we risk overloading the notion of energy. There is the rest energy of each particle, the energy and momentum components of each particle, and the energy of the system in the lab frame and Com frames etc.

Also, it feels more natural to me to think of the mass of a particle as a Lorentz Invariant. Rather than saying the "rest energy of a particle is a Lorentz Inavariant scalar". There is so much new in SR (and then in GR) that coping with a fresh understanding of the term "mass" is a minor problem.

 

[SiennaTheGr8]

Yes, and that's correct. It's not something special of Landau and Lifhitz. It's a fact following from the mathematical structure (i.e., the symmetry properties) of relativistic spacetime.

Some texts use a definition of "conservation" that doesn't rely on additivity. For example, Taylor & Wheeler, chapter 7 (2nd ed.):

A quantity is conserved if it has the same value before and after some encounter or does not change during some interaction. ... The magnitude of total momenergy of a system—the mass of that system—is also conserved in an interaction. On the other hand, the sum of the individual masses of the constituent particles of a system ordinarily is not conserved in a relativistic interaction.

(And Okun too, at least.)

 

[SiennaTheGr8]

That's all well and good, but on a practical level there is a risk of overloading the term energy. If we abandon the terminology of invariant mass in favour of rest energy, then we risk overloading the notion of energy. There is the rest energy of each particle, the energy and momentum components of each particle, and the energy of the system in the lab frame and Com frames etc.

Also, it feels more natural to me to think of the mass of a particle as a Lorentz Invariant. Rather than saying the "rest energy of a particle is a Lorentz Inavariant scalar". There is so much new in SR (and then in GR) that coping with a fresh understanding of the term "mass" is a minor problem.

I think the "term-overloading" is a good counterargument. (And to be clear: my argument is only a pedagogical one, and I don't expect everyone to agree with it.)

As to the fresh understanding of the term "mass," I can only say that it was more than a minor problem for me. What made things click was actually using ##E_0## instead of ##m## for a while.

 

[Ibix]

That's all well and good, but on a practical level there is a risk of overloading the term energy.

I agree with this.

(Rest/invariant) mass and rest energy are the same thing - the modulus of the four-momentum. Relativistic mass and total energy are the same thing - the time-like component of the four-momentum. A naming scheme that uses mass for one and energy for the other seems sensible, and the idea of mass being frame invariant while energy isn't matches my feel for what those words mean.

I take @SiennaTheGr8's point about invariant mass having rather different properties in relativity from Newtonian mass, but I'd suggest solving that problem by religiously referring to invariant mass as invariant mass (and never just "mass") in introductory relativity.

 

[vanhees71]

Some texts use a definition of "conservation" that doesn't rely on additivity. For example, Taylor & Wheeler, chapter 7 (2nd ed.):
(And Okun too, at least.)

Sigh. Of course, a conserved quantity doesn't need to be additive. I never claimed such a thing.

It is, however, a mathematical fact that in Newtonian physics mass is both additive and a strictly conserved quantity (following from the structure of the underlying symmetry group of Newtonian spacetime), while in special relativity there is no additional generally valid conservation law (following from the structure of the underlying symmetry group of Minkowski spacetime).

 

[SiennaTheGr8]

Sigh. Of course, a conserved quantity doesn't need to be additive. I never claimed such a thing.

Sorry if I misinterpreted.

Anyway, here is the passage I had in mind where Okun criticizes Landau and Lifshitz on this (minor) point, in case you or anyone else is curious: https://books.google.com/books?id=OjgTS12V0VUC&pg=PA295

 

[vanhees71]

It's not clear to me, what the criticism is here. One thing is nevertheless mathematically inevitable: There's no analogue of Newtonian mass conservation in special relativity. I don't think that Okun implies that there's a mass-conservation law in relativistic physics though you can interpret the last sentence in the above quoted passage 8.10.

 

[Sagittarius A-Star]

It's not clear to me, what the criticism is here. One thing is nevertheless mathematically inevitable: There's no analogue of Newtonian mass conservation in special relativity. I don't think that Okun implies that there's a mass-conservation law in relativistic physics though you can interpret the last sentence in the above quoted passage 8.10.

In the first sentence of passage 8.10, Okun says (differently to Landau and Lifshitz):

Is mass conserved? With E and P conserved, the mass M of the system (a set) of particles, defined by the formula ##M^2 = E^2 - \mathbf {P}^2##, must be conserved as well.

He does not mention in passage 8.10 an explicit additional mass-conservation law. To my understanding, that is not needed, because the 4-momentum can/shall be regarded as one physical quantity. As for all 4-vectors, the norm equals in case of velocities ##<c## the proper value of the 1st component.

But I also think, in the specific use case with electrons and muons from your posting #39, it is impossible, to measure in a lab directly the invariant mass of that system.

 

[vanhees71]

But the center-momentum energy ##\sqrt{s}## is also not conserved. Just take some condensed-matter system and heat it up. The center-momentum energy changes by ##\Delta Q/c^2##, where ##\Delta Q## is the transferred heat energy.

Of course you can measure ##\sqrt{s}##. You just need the particle momenta in the incoming channel and then calculate their energies from the on-shell conditions, ##E_j=c \sqrt{m_j^2 c^2+\vec{p}^2}## and then you get ##s=(p_1+p_2)^2=E_{\text{cm}}^2/c^2##.

 

[Sagittarius A-Star]

But the center-momentum energy ##\sqrt{s}## is also not conserved. Just take some condensed-matter system and heat it up. The center-momentum energy changes by ##\Delta Q/c^2##, where ##\Delta Q## is the transferred heat energy.

That would not be a closed system. You add energy from outside the system.

Of course you can measure ##\sqrt{s}##. You just need the particle momenta in the incoming channel and then calculate their energies from the on-shell conditions, ##E_j=c \sqrt{m_j^2 c^2+\vec{p}^2}## and then you get ##s=(p_1+p_2)^2=E_{\text{cm}}^2/c^2##.

Therefore, I wrote "directly". I read from Okun:

As for the mass of a system of free particles, it is simply their total energy (divided by c²) in a frame in which their total momentum is equal to zero. The value of this mass is limited only by conservation of energy and momentum, like in the case of two photons in the decay of positronium. As a rule we are unable to measure the inertia or gravity of such a system, but the self-consistency of the relativity theory guarantees that it must behave as mass

Source:
http://phys.sunmarket.com/rus/about/virtual/mtg-lomonosov-13/PDF/23.08.07/Morning/Okun.pdf

 

[vanhees71]

I agree with Okun in his definition of "mass" (though I prefer to use the Mandelstam variables in collision theory; I don't need the notion of "mass" of the total system at all here). I don't know, what he means by to measure inertia to begin with, but of course you can measure the center-momentum energy of two particles in a collision experiment as described above. Gravity is measureable in principle as any field by using test particles. I don't understand, what Okun wants to say with the last sentence in the quote above.

 

[Sagittarius A-Star]

I don't know, what he means by to measure inertia to begin with

I think he means, that you cannot put a system of free particles simply onto a bathroom scale to measure it's inertial mass. You could do it for example with a block of iron, which was heated up before, to show, that also the kinetic energy of the oscillating iron atoms contributes to the inertial mass of the iron block (if the bathroom scale would be sensitive enough).

 

[vanhees71]

One should also note that energy is not necessarily additive. It's only additive in the limit when mutual interactions of the constituents of a composite system can be neglected as, e.g., for a dilute gas which is with good approximation an ideal gas. Nevertheless the energy of a closed system is conserved.

 

[Sagittarius A-Star]

One should also note that energy is not necessarily additive. It's only additive in the limit when mutual interactions of the constituents of a composite system can be neglected as, e.g., for a dilute gas which is with good approximation an ideal gas.

Yes. Let me formulate it more precisely as I understand it: Energy is additive for an ideal box, filled with an ideal gas. A statement, if energy is additive, is only meaningful possible, if all parts of the energy can be uniquely assigned to one of each of the constituents. That is usually not possible for non-ideal gas.

 

[Dale]

There is no Minkowski norm, but I understand what you mean. I'd strongly discourage from using this notation, which I've never seen before, because it can only lead to more confusion as we just see in this thread.

There is a Minkowski norm. Maybe you haven't encountered this terminology before but it is described several places on Wikipedia so it is at least not completely unknown to others
https://en.wikipedia.org/wiki/Minkowski_norm
https://en.wikipedia.org/wiki/Minkowski_space
https://en.wikipedia.org/wiki/Minkowski_space#Norm_and_reversed_Cauchy_inequality
https://en.wikipedia.org/wiki/Four-momentum#Minkowski_norm

Nevertheless, for this thread I will avoid it to avoid unnecessary disputes. I like the notation because it emphasizes the geometry, and because it is easier to write in LaTeX. But I will go the long route in this thread.

Mass is NOT conserved, energy is in special relativity. Mass is NOT additive but energy is.

I am not sure what concept of mass you are referring to here.
##\sqrt{(\Sigma p_{\mu})(\Sigma p^{\mu})}##, the system invariant mass, IS conserved and is NOT additive
##\Sigma(\sqrt{p_{\mu}p^{\mu}})##, the sum of the invariant masses, is NOT conserved and IS additive
##\Sigma p^0##, the relativistic mass, IS conserved and IS additive

Could you clarify which concept of mass you refer to that is NOT conserved and is NOT additive?

 

[Sagittarius A-Star]

@PeroK : Did you notice? It was not me, who wrote this.

##\Sigma p^0##, the relativistic mass