The Energy - Momentum Equation vs the Energy - Mass Equation

[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 I thought we have an agreement in this forum that we discourage the use of this outdated notion.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 I thought we have an agreement in this forum that we discourage the use of this outdated notion.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.

 

[Sagittarius A-Star]

I might call Definition 1 "the kinetic energy in a system" for short in some contexts, but never "the kinetic energy of a system"

That disagrees to the formulation of Wikipedia, but I think, your formulation is the systematic one.

 

[Sagittarius A-Star]

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

I don't want to introduce something new, I investigate, if such a thing already exists.

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

And what about calling it spacetime interval ?

 

[Dale]

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

I'd never ever abuse the mathematically well defined definition of a norm

Many others would. It is accepted terminology, and why not? It is a perfectly reasonable generalization of a norm to manifolds with mixed signature.

We generalize many things in pseudo Riemannian geometry by relaxing some requirement of other geometries. So this is commonly done. That you personally prefer not to generalize the norm does not make it invalid.

Instead of explaining that the Minkowski norm violates the standard definition of a norm (since that is obvious for any generalization of any concept) why don’t you argue where such a definition can cause confusion or something?

The Minkowski metric does not induce a metric

Hmm

 

[vanhees71]

You got me. It's of course the Minkowski pseudo-metric.

I've never seen any scientific paper or textbook introducing a "norm" in this sense. The first time I've seen it is from you pointing to the Wikipedia article. Is this "norm" then imaginary for space-like vectors (in the west-coast convention) or for time-like vectors (in the east-coat convention). Why do you think it's useful to introduce an unneeded concept against well-established mathematical definitions in use for decades/centuries?

 

[Dale]

Why do you think it's useful to introduce an unneeded concept against well-established mathematical definitions in use for decades/centuries?

I am glad you asked. The reason why is because it highlights the geometry. IMO, the key concept of relativity is the generalization of geometry to spacetime.

All generalizations inevitably go against some established definitions, so that in itself is not particularly problematic. The question is if the insight gained by the generalization is valuable. My experience is that it is valuable.

For example, people who understand the geometry never have trouble with the twin paradox. People who understand the geometry can move to GR more easily. People who understand the geometry can see the link between charge density and current or between energy and momentum easier. Etc.