I The Energy - Momentum Equation vs the Energy - Mass Equation

[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textbooks on theoretical physics everything is manifestly covariant and thus no confusion occurs with concepts that are outdated for more than 100 years ("relativistic mass").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

 

[vanhees71]

I still do not think that one should talk about a norm, and I don't agree with Wikipedia in this case. In mathematics it least the norm has a well defined meaning, and only a proper scalar product (positive definite bilinear form for real or positive definite sequilinear form for complex vector spaces) can induce a norm on vector spaces, but that's semantics.

We agree about the definition of mass of a composite system of non-interacting particles as $\sqrt{s}$ as defined by your first sqrt. We agree also about invariant masses. I don't agree with the use of "relativistic mass". I thought we have an agreement in this forum that we discourage the use of this outdated notion.

Also concerning the additivity of energy one has to be careful (see the example with the non-ideal gas above).

 

[PeroK]

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

I didn't notice it. I was watching the tennis!

 

[Dale]

We agree also about invariant masses. I don't agree with the use of "relativistic mass". I thought we have an agreement in this forum that we discourage the use of this outdated notion.

We do, I am just trying to figure out what notion of mass *you* were referring to saying that it was both NOT conserved and NOT additive. What were you referring to? Please clarify your meaning of "mass" from post 48: https://www.physicsforums.com/threa...-the-energy-mass-equation.993839/post-6397471

 

[vanhees71]

For me mass is always invariant mass. Setting $c=1$ for a composite system it's by definition
$$M^2=(\sum p_i)^2.$$
In scatterings the total energy is conserved but not the sum of the masses of the particles. The consevation of $M=\sqrt{s}$ is however energy conservation when considered in the cm frame. In this sense there's no additional mass-conservation law as in Newtonian physics.

 

[Dale]

For me mass is always invariant mass. Setting $c=1$ for a composite system it's by definition
$$M^2=(\sum p_i)^2.$$

Then your statement of post 48 is incorrect. With that definition of mass (which is in my opinion the best definition) mass IS conserved but it is NOT additive.

In scatterings the total energy is conserved but not the sum of the masses of the particles. In this sense there's no additional mass-conservation law as in Newtonian physics.

Yes but the sum of the masses of the parts is not the mass, per your definition above. The quantity you identified above as the mass IS conserved. So saying that mass is NOT conserved is not correct using your terminology. What would be correct is the following:

Mass IS conserved but it is NOT additive. Because mass is NOT additive the sum of the masses of the parts is NOT conserved.

 

[vanhees71]

Yes, you are right! In SR, there is no additional mass conservation law as an eleventh independent conservation law of the general Noether conserved quantities following from spacetime symmetries but, if you define the mass of a composite system as above, it's subsumed in the energy-conservation law since the so defined mass is nothing than the energy in the center-momentum frame. In a sense, it's superfluous to define this as mass.

Mass is not conserved in the sense that the so defined "mass" is not the sum of the masses of the components. Indeed, mass is not additive. I think this is all semantics. The important thing is the different mathematical nature of mass in Newtonian physics (central charge of the Galilei group) and special-relativistic physics (Casimir operator of the Poincare group).

BTW: energy is also additive only for non-interacting "constituents". That's already so in Newtonian physics.

 

[SiennaTheGr8]

I'd just like to point out that the pre-relativistic notions of the word mass are responsible for the confusion here.

We were all on the same page from the beginning that a system's mass is its rest-frame energy, and yet the idea of "total mass" (sum of constituent masses) somehow slipped into the conversation, even though nobody was talking about it.

"Total mass" is as useless and silly as "total velocity," but nobody would think to sum the velocities of a system's constituents when asked for the system's velocity.

"Rest energy" sidesteps that temptation altogether. It wouldn't even occur to anyone to start summing up constituent rest-energies if asked to give a system's rest energy. "Total rest energy" isn't an idea that would tacitly worm its way into the discussion. It's a bizarre turn of phrase that wouldn't come to mind at all, and that's a good thing.

 

[vanhees71]

Yes, and I think the solution to all this confusion is just to avoid "mass" in relativity whenever you mean "rest energy". Then call it rest energy, because that's what it is, and then it's a scalar and no confusion occurs.

 

[SiennaTheGr8]

Yes, that's exactly my point. But what could one possibly mean by (invariant) "mass" other than "rest energy"?

 

[Dale]

I prefer the term “invariant mass” over “rest energy”:

1) Many systems have no rest frame
2) Many systems have non-inertial rest frames
3) The quantity can be determined in any frame using that frame’s data
4) The quantity is invariant
5) Calling it rest energy gives a false sense that it is additive like energy

The term “center of momentum frame energy” resolves several of those but I am lazy and prefer to type 4 characters over 31. So my strong preference is simply to use the term “mass” as a default and “invariant mass” when I am concerned that it may be ambiguous.

 

[vanhees71]

I couldn't agree more.

6) Calling it rest energy were a misnomer for a(n asymptotic) free photon state, because it's never at rest because its invariant mass is 0.

 

[SiennaTheGr8]

I agree that "invariant" (or perhaps "proper") is a better adjective than "rest." I only use "rest energy" instead of "invariant energy" or "proper energy" because that's the term for $mc^2$ that I've typically seen in the literature.

5) Calling it rest energy gives a false sense that it is additive like energy

Interesting. I always felt the opposite: calling it "[descriptor] energy" gives (me) the right sense that, like "kinetic energy" and "potential energy," it's just one category of energy-contribution that must be accounted for when reckoning a system's (total) energy.

 

[Dale]

I agree that "invariant" (or perhaps "proper") is a better adjective than "rest." I only use "rest energy" instead of "invariant energy" or "proper energy" because that's the term for mc2 that I've typically seen in the literature.

Since energy is not invariant the term "invariant energy" would be highly confusing. I guess that people understand other oxymorons so it could be adopted eventually, but it is jarring to me.

I always felt the opposite: calling it "[descriptor] energy" gives (me) the right sense that, like "kinetic energy" and "potential energy," it's just one category of energy-contribution that must be accounted for when reckoning a system's (total) energy.

Additivity isn't about the categories of energy, it is about the energy of the parts and the whole. For a system with non-interacting parts, the kinetic energy of the system is the sum of the kinetic energies of its parts, and the potential energy of the system is the sum of the potential energies of its parts. But even with non-interacting parts the rest energy of a system is not the sum of the rest energies of its parts.

 

[vanhees71]

Energy is only additive in that sense if the interaction between the constituents can be neglected. I'd not emphasize additivity too much. It's important though in thermodynamics/statistical physics when you find a way to handle a many-body system in some (often tricky) sense as an ideal gas. Then the total energy is indeed the sum over the energy of the constituents (particles, molecules, light quanta or, quasi particles). Already for a real gas the energy is not additive anymore in this sense.

After all this discussion, my conclusion is that the important difference between Newtonian and special relativistic physics is that as analyzed in terms of the space-time symmetry groups/Lie algebras, in Newtonian physics there's an additional conservation law for mass (i.e., there are 11 conservation laws from the physical realization of Galilei symmetry, i.e., a non-trivial central extension of the classical Galilei group, rather than 10 conservation laws from the physical realization of the Poincare group, which has no non-trivial central extensions), while there's none such additional conservation law in special relativistic physics.

Indeed, as @Dale said in the previous posting, the important difference between energy and mass is that the former is a temporal component of a four-vector (energy-momentum four vector) of a closed system while mass is a scalar and associated with the energy in the center-momentum frame of this closed system. Since one can always calculate everything in the center-momentum frame, and there the invariant mass is just the total energy of the system, its conservation is just energy conservation when considered in this preferred (necessarily always inertial!) reference frame of a closed system.

It's of course a very delicate issue to discuss open composite systems. Even a covariant formulation of total energy and momentum as a four-vector is not unique in such a case. This lead to an age-old famous debate about the infamous factor-4/3 problem in the radiation-reaction problem for charged point-particle-like bodies. For that issue, see the very illuminating discussion in Chpt. 16 of Jackson's Classical Electrodynamics. It's amazing that there is still so much debate about this since the entire problem was analyzed completely by von Laue in 1911. The naive expression for the total four-momentum of a continuous system (like continuum mechanics or fields like the em. field)
$$P^{\mu}=\int_{\mathbb{R}^3} \mathrm{d}^3 x T^{\mu 0}(t,\vec{x})$$
is a four vector only if the local conservation law
$$\partial_{\nu} T^{\mu \nu}=0$$
holds, and then $P^{\mu}$ is also conserved (i.e., time-independent), i.e., it's only a proper four-momentum if the system is closed (concerning the exchange of energy and momentum).

To define in a covariant way energy and momentum of an open composite system one has to choose an appropriate preferred reference frame and then transform the energy and momentum from this inertial frame to an arbitrary other inertial frame or write the corresponding integral over the entire spatial volume as observed in the preferred frame in a manifestly covariant way. Then one still has to be careful how to interpret the so defined energy-momentum four-vector in each specific case. All this is nicely discussed by Jackson.

 

[Sagittarius A-Star]

If one would follow the usual naming scheme of the 4-position vector (norm = "space-time distance", which equals for $v<c$ to "proper time $\tau$", then regarding the 4-momentum one would say:

Norm =: "energy-momentum magnitude", it equals for $v<c$ to "proper energy $E_0$".​

Then "invariant mass" is only an agreed alias name for the (invariant) "energy-momentum magnitude".

If a bug is walking on a bathroom scale, the scale displays its (non-invariant) energy $E = \gamma * E_0$. If the bug stops, the scale displays its energy in its CoM frame (="proper energy $E_0$", which equals in this case to the invariant "energy-momentum magnitude" alias "invariant mass").

 

[SiennaTheGr8]

Additivity isn't about the categories of energy, it is about the energy of the parts and the whole.

Seems like a distinction without a difference (you get the same total either way you choose to look at it, precisely because energy is additive).

My point was really that before I learned SR, I was already quite accustomed to summing kinetic- and potential-energy contributions to get a system's total energy. For me, the "rest energy" concept fit nicely into that scheme of things, and absolutely didn't give me the wrong idea that a system's rest energy should be the sum of its constituents' rest energies. The opposite, in fact, and it was my prior experience with the word "mass" that was throwing me off.

Different strokes, clearly.

For a system with non-interacting parts, the kinetic energy of the system is the sum of the kinetic energies of its parts

I guess it depends on what you mean by "a system with non-interacting parts," but I don't think that's accurate. Any system has zero kinetic energy in its center-of-momentum frame, regardless of the kinetic energies of its constituents. The sum of kinetic energies alone isn't generally an interesting quantity.

 

[weirdoguy]

The sum of kinetic energies alone isn't generally an interesting quantity.

Well, as long as thermodynamics is not an interesting part of physics...

 

[Dale]

I don't think that's accurate. Any system has zero kinetic energy in its center-of-momentum frame, regardless of the kinetic energies of its constituents.

Consider a rotating disk. In the center of momentum inertial frame it has non-zero KE. The KE of the disk is equal to the sum of the KE of the various parts of the disk. The rest energy of the disk is greater than the sum of the rest energies of the various parts of the disk. Hence the KE is additive, the rest energy is not.

 

[SiennaTheGr8]

Well, as long as thermodynamics is not an interesting part of physics...

And the sum of a system's constituents' masses is interesting in the Newtonian limit, but that's not what we're talking about. By "generally" I meant "in the general case in SR."

Consider a rotating disk. In the center of momentum inertial frame it has non-zero KE. The KE of the disk is equal to the sum of the KE of the various parts of the disk. The rest energy of the disk is greater than the sum of the rest energies of the various parts of the disk. Hence the KE is additive, the rest energy is not.

The definition of relativistic kinetic energy I had in mind is $(\gamma - 1) E_0$, where $\gamma$ is the system's COM-frame's Lorentz factor (relative to some observer's inertial frame), and $E_0$ is the system's rest energy (with the caveat that $E_0$ is not a straightforward quantity to define for an open composite system, related to points @vanhees71 raised above). By that definition, the rotating disk has zero kinetic energy in its center-of-momentum frame, period. Of course, the disk has more total energy when rotating, but as that's all attributable to the kinetic energies of its constituents (in the COM frame), I'd include it in the system's "proper energy" (i.e., its mass).

Anyway, a simple example of what I really had in mind is the COM frame of a system that consists of two electrons moving in opposite directions at identical speeds. The system's kinetic energy in this frame is zero, which is obviously not the sum of its constituents' kinetic energies.

 

[Dale]

And the sum of a system's constituents' masses is interesting in the Newtonian limit, but that's not what we're talking about. By "generally" I meant "in the general case in SR."
The definition of relativistic kinetic energy I had in mind is $(\gamma - 1) E_0$, where $\gamma$ is the system's COM-frame's Lorentz factor (relative to some observer's inertial frame), and $E_0$ is the system's rest energy (with the caveat that $E_0$ is not a straightforward quantity to define for an open composite system, related to points @vanhees71 raised above). By that definition, the rotating disk has zero kinetic energy in its center-of-momentum frame, period. Of course, the disk has more total energy when rotating, but as that's all attributable to the kinetic energies of its constituents (in the COM frame), I'd include it in the system's "proper energy" (i.e., its mass).

Anyway, a simple example of what I really had in mind is the COM frame of a system that consists of two electrons moving in opposite directions at identical speeds. The system's kinetic energy in this frame is zero, which is obviously not the sum of its constituents' kinetic energies.

OK, but that seems like a pretty odd definition of KE. With that (to me very strange) definition of KE I do see how you would not consider KE additive.

So I guess the disagreement is semantic, but I certainly think my semantics are better. I don’t accept your definition for KE at all.

 

[SiennaTheGr8]

Leaving aside the non-rotational rotational [edited] case—how would you define the kinetic energy of the two-electron system I mentioned? Would you just define it as the sum of the particles' KE?

 

[Sagittarius A-Star]

According to Wikipedia, both definitions are possible.

Definition 1:

A system of bodies ... The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains.

Definition 2:

When discussing movements of a macroscopic body, the kinetic energy referred to is usually that of the macroscopic movement only. However all internal energies of all types contribute to body's mass, inertia, and total energy.

Source:
https://en.wikipedia.org/wiki/Kinetic_energy#Kinetic_energy_of_systems

 

[vanhees71]

In manybody physics Definition 2 is the usual one. Kinetic energy is a very inconvenient quantity, because it's not nicely transforming under Lorentz transformations. That's why one always includes the "rest energies" so that together with momentum one gets a four-vector.

 

[SiennaTheGr8]

I might call Definition 1 "the kinetic energy in a system" for short in some contexts, but never "the kinetic energy of a system"—otherwise I'd have to say that the cup of coffee sitting on the desk next to me has non-zero kinetic energy!

Speaking of my coffee, is it not a universally accepted consequence of SR that it will lose mass as it cools (because its constituents will have less kinetic energy)? The situation with the rotating disk seems similar. It might be useful sometimes to calculate how much more energy a disk has when it's rotating than when it isn't, and you could call that the "relativistic rotational kinetic energy," but either way it contributes to the disk's total energy in its COM frame (aka its mass), doesn't it?

 

[vanhees71]

Of course, it is a universally accepted consequence of SR that your coffee looses mass when it cools.

In general, all quantities referring to intrinsic properties of a material of whatever kind is defined by scalar quantities. Intrinsic properties are all properties needed to characterize this system in its center-momentum frame. E.g., in fluid dynamics, i.e., a liquid, gas, or plasma close to local equilibrium all the intrinsic quantities are defined in the rest frame of each fluid cell, i.e., the usual thermodynamical quantities like internal-energy density, enthalpy density, entropy density, conserved-charge densities (electric charge, baryon number, strangeness, isospin,...) pressure, temperature and chemical potentials associated with the conserved charges.

E.g., an ideal fluid is usually described by the internal energy density (including "rest energy") and pressure via the energy-momentum tensor (west-coast convention)
$$T^{\mu \nu}=(u+p) u^{\mu} u^{\nu}-p \eta^{\mu \nu}.$$
Here $u$ is the internal energy in local rest frame of the fluid cell (LRF), $p$ the pressure (also measured in this LRF) and $u^{\mu}$ the four-velocity field with $u_{\mu} u^{\mu}=1$. The equations of motion (relativistic Fluid equation) is given by energy conservation, i.e.,
$$\partial_{\mu} T^{\mu \nu}=0.$$
There's no additional mass-conservation equation as in non-relativistic fluid dynamics.

In addition you have for any conserved charge $Q$ the corresponding current
$$j_Q^{\mu}=Q n u^{\mu}$$
and an equation for its conservation,
$$\partial_{\mu} j_Q^{\mu}.$$
In addition you need an equation of state to close the system of equations, where $n$ is something like a "net-particle number density" ("particles minus anti-particles" in the fluid cell).

You can of course also split the internal energy in a "invariant-mass density" $\mu$ and the rest $\tilde{u}$. This is advantageous when you want to derive the Newtonian limit. But as stressed already for several times there's no additional conservation law for the total mass!

 

[Sagittarius A-Star]

6) Calling it rest energy were a misnomer for a(n asymptotic) free photon state, because it's never at rest because its invariant mass is 0.

Would it be correct and a good idea, to give it the symbolic expression $\parallel \mathbf {P}\parallel$ and call it "energy-momentum magnitude"?

 

[Dale]

how would you define the kinetic energy of the two-electron system I mentioned? Would you just define it as the sum of the particles' KE?

Yes.

 

[vanhees71]

Would it be correct and a good idea, to give it the symbolic expression $\parallel \mathbf {P}\parallel$ and call it "energy-momentum magnitude"?

No, I'd never ever abuse the mathematically well defined definition of a norm. A norm on a vector space is a map $\|\cdot \|:V \rightarrow \mathbb{R}$ fullfilling the conditions

Positive definiteness: $\|\vec{v} \| \geq 0$ and $\|\vec{v}\|=0 \Leftrightarrow \vec{v}=0$.
Homogeneity: $\|\lambda \vec{v} \|=|\lambda| \|\vec{v} \|$.
Triangle inequality: ##\| \vec{v}_1 + \vec{v}_2 \| \leq \|\vec{v}_1 \| + |\vec{v}_2|.

It is easy to show that for a scalar product (a positive definite bi- (for real vector spaces) or sesqui- (for comoplex vector space) linear form) induces a norm in the usual way
$$\|\vec{v} \|=\sqrt{ (\vec{v},\vec{v})}.$$
This obviously does not work for any more general fundamental form, which is not positive definite. The Minkowski product, which is a fundamental form of signature (1,3) or (3,1) on $\mathbb{R}^4$, cannot induce a norm.

I would simply stick to the modern conventions and call it invariant mass defined by
$$M^2=P_{\mu} P^{\mu}/c^2=s/c^2,$$
where $P^{\mu}$ is the total four-momentum (of a closed system).

 

[Sagittarius A-Star]

I would simply stick to the modern conventions and call it invariant mass defined by
$$M^2=P_{\mu} P^{\mu}/c^2=s/c^2,$$
where $P^{\mu}$ is the total four-momentum (of a closed system).

That's a possibility. But I want to find out, if a simple symbol exists, that indicates intuitively, that it stands for the (context-dependent Euklidian or pseudo-Euklidian) magnitude of the vector, which the symbol $m$, without an additional explanation, doesn't. And of course, I want to avoid "ict" as a possible workaround.

A good example from the 4-position is $\Delta s$. It is intuitively regarded as some kind of "distance".

 

[vanhees71]

[EDIT: corrected confusing typos below: the Minkowski fundamental form is not a metric but a pseudo-metric; and it's $\mathrm{d}s^2$ rather than $\mathrm{d}s$.]

The Minkowski pseudo-metric does not induce a metric, because it's not positive definite. Why do you want to introduce totally useless and confusing ideas?

I guess you mean
$$\mathrm{d}s^2 = g_{\mu \nu} \mathrm{d} x^{\mu} \mathrm{d} x^{\nu}?$$
That's not a distance (squared), because it's not positive definite.