B Center of momentum and mass energy equivalence

[PeterDonis]

I was simply referring to the quantity $m$ that appears in Newtonian equations. That indeed turns out to be nothing but a measure of how much energy a system has as measured in its rest frame: $m = E_0 / c^2$.

No, it doesn't; the Newtonian $m$ is not the same as the invariant mass $m$ in relativity, which is what "energy measured in the system's rest frame" equates to. The simplest way to see this is to note that the Newtonian $m$ is additive, whereas the invariant mass $m$ in relativity is not (as @Mister T has just been pointing out).

 

[SiennaTheGr8]

No, it doesn't; the Newtonian $m$ is not the same as the invariant mass $m$ in relativity, which is what "energy measured in the system's rest frame" equates to. The simplest way to see this is to note that the Newtonian $m$ is additive, whereas the invariant mass $m$ in relativity is not (as @Mister T has just been pointing out).

By "Newtonian" I meant the approximations that special relativity reduces to when $\gamma \approx 1$. See my prior post with approximately-equal signs and the "(for $\beta \ll 1$)" tags.

I really don't think I'm going out on a limb here: Einstein showed that the quantity $m$ was nothing but a measure of a system's energy in its rest frame, and indeed that $m$ is only approximately additive in the classical limit.

 

[PeterDonis]

By "Newtonian" I meant the approximations that special relativity reduces to when $\gamma \approx 1$.

But @sweet springs , who you responded to, is making claims that are not limited to that approximation--at least they don't seem to me to be. And once we go outside that approximation, the claims are simply false. And the Newtonian intuitions that he is relying on are not limited to that approximation.

 

[SiennaTheGr8]

Word.

 

[Herman Trivilino]

No, it doesn't; the Newtonian $m$ is not the same as the invariant mass $m$ in relativity, which is what "energy measured in the system's rest frame" equates to.

But when we measure the mass $m$ of an object we are measuring the rest frame energy.

The simplest way to see this is to note that the Newtonian $m$ is additive, whereas the invariant mass $m$ in relativity is not.

I guess I'm not understanding your point. Certainly it doesn't have the same properties, but I don't see how that makes it a different thing. The emergence of the concept of rest energy and its equivalence to mass didn't change the way we measure mass. It changes the fact that we can't add up the masses of the constituents of a composite body to determine its mass, but that's not a change in the way we measure mass.

 

[PeterDonis]

when we measure the mass $m$ of an object we are measuring the rest frame energy.

If $m$ refers to the invariant mass $m$ in relativity, of course this is true. But in Newtonian physics, the mass $m$ appears in at least two places: the second law $F = ma$, and the law of gravity $F = G m M / r^2$. Neither of those $m$'s corresponds to "rest frame energy"; in Newtonian physics, the energy of an object in its rest frame is zero, because the mass $m$ isn't energy.

Bear in mind that I am making the points I'm making, in this particular thread, because of the misconceptions @sweet springs has been expressing. I understand that there are plenty of other issues involved.

I guess I'm not understanding your point.

In Newtonian physics, if I combine two objects with masses $m_1$ and $m_2$ into a single composite object with mass $M$, then I must have $M = m_1 + m_2$. In relativity, invariant mass (or "rest frame energy", if you insist on using that term) doesn't work that way. So the two symbols can't be referring to the same concept. The concept that Newtonian physics is referring to with the symbol $m$, as I've said, doesn't have a direct counterpart in relativity at all. You have to reinterpret the symbol in some way to have it correspond to any well-defined relativistic concept.

Again, please bear in mind what I said above about the reasons for the points I'm making in this particular thread.

The emergence of the concept of rest energy and its equivalence to mass didn't change the way we measure mass.

But it does change the physical interpretation of those measurements.

 

[SiennaTheGr8]

@Mister T

I think that @PeterDonis is making a distinction between Newtonian physics (where $\vec f = m \vec a$) and the classical limit of special relativity that "corresponds" to Newtonian physics (where $\vec f \approx m \vec a$ in the case that $\gamma \approx 1$).

 

[sweet springs]

Thanks for a lot of teachings.

You're assuming that "time", "length", and "energy" are somehow fundamental concepts. They aren't. That's one of the lessons of relativity.

For an example Planck units shows that time, length and mass are three fundamental quantities. We can choose mass or energy. Choosing the both is redundant.

 

[SiennaTheGr8]

Thanks for a lot of teachings.

For an example Planck units shows that time, length and mass are three fundamental quantities. We can choose mass or energy. Choosing the both is redundant.

For the Planck mass, we can choose mass or rest energy. (The term "energy" by itself signifies total energy—that is, the sum of rest energy and kinetic energy.)

 

[Herman Trivilino]

If $m$ refers to the invariant mass $m$ in relativity, of course this is true.

Indeed it does. When we measure the mass of something that is indeed what we are doing. And that is all we've ever done when we measured mass. It's just that until Einstein came along we didn't know that that was what we were doing.

Attributing mass as the agent of gravity or the resistance to acceleration were all of course thought to be proper uses of mass, but we now know better.

My mass is 105 kg. It would be a mistake to attribute those erroneous properties to that mass, but it's not a mistake to say that my mass is 105 kg. It's just as true now as it would have been in the late 1800's when that standard for measuring mass was adopted.

 

[PeterDonis]

Planck units shows that time, length and mass are three fundamental quantities

No, it doesn't. You can't determine physics by choosing units. Planck units choose those three as fundamental for human convenience, that's all.

 

[PeterDonis]

it's not a mistake to say that my mass is 105 kg. It's just as true now as it would have been in the late 1800's when that standard for measuring mass was adopted.

It's just as true, provided that you adopt the interpretation of the word "mass" that we get from relativity. But nobody knew that interpretation in the late 1800's. Ordinary language words don't have unchanging meanings. If someone in the late 1800's said their mass was 105 kg, they meant by that something different than what you mean by it. We know that because you, today, can give a precise meaning to your usage of the word "mass" by pointing to a precise mathematical theory that didn't even exist in the late 1800's. If you asked a person in the late 1800's to give a precise meaning to their usage of the word "mass", the only precise mathematical theory they could point to was Newtonian mechanics.

What you are saying is not incorrect; but it doesn't address what I see as the confusion that @sweet springs is expressing, which I tried to respond to in post #44.

 

[SiennaTheGr8]

But there was certainly something mysterious about "mass" before Einstein. Here is how Mach put it (in poor translation):

As soon therefore as we, our attention being drawn to the fact by experience, have perceived in bodies the existence of a special property determinative of accelerations, our task with regard to it ends with the recognition and unequivocal designation of this fact. Beyond the recognition of this fact we shall not get, and every venture beyond it will only be productive of obscurity. All uneasiness will vanish when once we have made clear to ourselves that in the concept of mass no theory of any kind whatever is contained, but simply a fact of experience. The concept has hitherto held good. It is very improbable, but not impossible, that it will be shaken in the future, just as the conception of a constant quantity of heat, which also rested on experience, was modified by new experiences.

source: https://books.google.com/books?id=4OE2AAAAMAAJ&pg=PA221

I think it's fair to say that Einstein did what Mach called the "very improbable"—namely, he found "theory ... in the concept of mass." He showed that every time we weigh something, we are in fact measuring its Energieinhalt ("energy content"), or, in modern parlance, its Ruheenergie ("rest energy").

Of course, in an ontological sense ("what the hell is it, really?"), there's something mysterious about energy, too. But better one mystery than two.

 

[vanhees71]

You're assuming that "time", "length", and "energy" are somehow fundamental concepts. They aren't. That's one of the lessons of relativity.

This is also a bit far-fetched. In my opinion time and length are indeed the fundamental concepts of relativity, although I'd rather call it spacetime from the very beginning. You need concepts of space and time to formulate physics, and to a large extent the spacetime structure determines also the form of the physical laws. The notion of time implies a causality structure, i.e., spacetime must in some way enable an order or time in the sense of causality, and that's the fundamental "arrow of time", implied on the laws of physics from the very beginning.

Energy, however, is indeed a derived concept. In Newtonian and special relativistic physics it's defined as a generator of time evolution and of the time-translation transformation, implying that it's conserved on a fundamental level. In GR that's a bit more problematic as the century-long discussion about the concept of energy of the gravitational field shows. It can be defined only in a local sense, and it's no longer conserved on a fundamental level, but that's a discussion for another thread, if needed.

 

[PeterDonis]

In my opinion time and length are indeed the fundamental concepts of relativity, although I'd rather call it spacetime from the very beginning.

Spacetime is a fundamental concept, yes. But "time" and "length" are not; they're frame-dependent.

The notion of time implies a causality structure

No, the Lorentzian geometry of spacetime implies a causality structure.

spacetime must in some way enable an order or time in the sense of causality, and that's the fundamental "arrow of time", implied on the laws of physics from the very beginning.

There is no arrow of time in any of the fundamental laws, unless you want to count the lack of T symmetry of weak interactions; but none of our ordinary experience of time and the arrow of time depends on weak interactions. The standard view, as I understand it, is that our perception of an arrow of time arises from the second law of thermodynamics--or, to put it another way, from the fact that our observable universe started out in a very low entropy state. In other words, it's a matter of the initial conditions of the particular solution of the laws in which we live, not the laws themselves.

As for spacetime structure, the geometry of spacetime does not pick out which half of the light cone at any event is the "future" half and which is the "past" half. The most you can get out of spacetime structure is that, assuming the spacetime is time orientable, picking out the "future" and "past" half of the light cone at one event is sufficient to pick it out at every event. But the choice at the one event is still arbitrary; spacetime structure doesn't impose it.

 

[vanhees71]

The point is that we assume an arrow of time by dinstinguishing the past and the future lightcone, even in the parts of physics, where time-reversal symmetry holds (i.e., everything except the weak interaction). The fact that this (large) parts of the physical laws are time-reversal symmetric makes the "direction of time" an additional assumption rather than something that can be derived from the rest of the postulates about the structure of spacetime.

Often you read that the arrow of time only comes in via something like Boltzmann's H theorem, but the only thing that can be proven (from the (weak) principle of detailed balance, i.e., from the unitarity of the S-matrix) is that the "thermodynamical arrow of time" (determined by increasing rather than decreasing entropy) is identical with the fundamentally assumed "causality arrow of time" mentioned above.

 

[PAllen]

I would like to respond generally to one topic in this thread, viz. the relation between mass in relativity versus pre-relativity physics. I will make the case that they are as similar as any other quantities over this transition, e.g. velocity. (By mass, of course, I mean invariant mass).

First, I think Newton's law of gravitation is a red herring because the coincidence of gravitational and inertial mass was completely a coincidence in this theory and was made less of a coincidence by GR.

As for m being inertial mass, this feature is fully preserved in SR as long as one makes the transition from F=mA expressed in 3 vectors to 4 vectors. Similarly, for momentum.

As for mass no longer being additive, we have the same situation for velocities. Though it isn't so commonly done, one may write down e.g. a SR mass addition law similar to what is done for velocities:

M² = m_1² + m₂² + 2 E₁E₂(1-v₁⋅v₂)

 

[vanhees71]

I think within Newtonian physics the equality between inertial and gravitational mass is a purely empirical input, and it was indeed an enigma until Einstein's GR.

Mathematically mass in Newtonian physics is a non-trivial central charge of the Lie algebra of the Galilei group, while in relativistic physics it's a Casimir operator of the Lie algebra of the Poincare group. This is what makes the concept of mass different in the two spacetime models.

 

[nitsuj]

The invariant mass is a fundamentally different concept than energy. The mass is the invariant norm of the four momentum and the energy is the frame variant timelike component of the four momentum.

An individual photon doesn't, but a system of two or more does. This becomes important e.g. In analyzing positron emission tomography.

how so, my understanding is the important thing there is the symmetric emission of photons and simultaneous detection, being able to position the source. Nothing to do with rest or invariant mass of such a system of two opposing photons. It is VERY cool that we see the massive positron decay as two photons moving in opposing directions...AT c!~ lol bye bye mass..oh wait. Ima call it a system so its still massive.

 

[Herman Trivilino]

lol bye bye mass..oh wait. Ima call it a system so its still massive.

Whatever you choose to call it, either its mass changes or it stays the same. Those two choices are mutually exclusive, they can't both be true.

 

[Dale]

the symmetric emission of photons

It is only symmetric if the positron and electron have no net momentum. If they have momentum then the photons are emitted asymmetrically in the detector frame, and the decay does not occur on the line between the two detectors. This leads to blurring in the images, and is one of the fundamental limits to the resolution for some PET tracers.

This is precisely the case where energy and mass are different

 

[Ibix]

It is VERY cool that we see the massive positron decay as two photons moving in opposing directions...AT c!~ lol bye bye mass..oh wait. Ima call it a system so its still massive.

Mass is conserved in a closed system, so the two photons must have mass. You can treat the two photons separately in which case you don't have a closed system. I don't really understand why you find this a problem.

 

[vanhees71]

No, each of the photons is massless. Both photons together have a total invariant mass of $\sqrt{s}$, i.e., the center-mass energy of the colliding electron-positron pair. That's just a definition of the kinematics of scattering theory.

 

[nitsuj]

Mass is conserved in a closed system, so the two photons must have mass. You can treat the two photons separately in which case you don't have a closed system. I don't really understand why you find this a problem.

Because you're calling two photons moving in opposite directions a closed system. I cannot understand how two thing moving at c in opposite directions could thought of as a "closed system"; causally they couldn't be farther apart.

 

[SiennaTheGr8]

Because you're calling two photons moving in opposite directions a closed system. I cannot understand how two thing moving at c in opposite directions could thought of as a "closed system"; causally they couldn't be farther apart.

A "closed" system (aka an "isolated" system) is simply one that doesn't interact in any way with its environment. It's an idealization—no system is truly closed in the real world.

In some contexts—thermodynamics especially—it's common to distinguish between a "closed" system (no mass leaves or enters, though energy might) and an "isolated" system (no mass or energy leaves or enters; i.e., the system doesn't interact with its environment at all). But in special relativity this distinction becomes less useful ($E_0 = mc^2$), and so the terms are often used interchangeably to mean the latter.

Note that whether a system is closed has nothing at all to do with how far apart its constituents are. For many purposes our galaxy is well-approximated as a closed system.

 

[Herman Trivilino]

Because you're calling two photons moving in opposite directions a closed system. I cannot understand how two thing moving at c in opposite directions could thought of as a "closed system"; causally they couldn't be farther apart.

An imaginary boundary separates a system from its environment.

The system is anything within the boundary. If you choose it to be an electron-positron pair about to collide then that's what it is. It doesn't cease to be a system after the interaction. Moreover, if nothing crosses the boundary then the system is said to be closed. The laws of physics describe this interaction and in one of those descriptions the mass of the system doesn't change during the interaction as long as nothing crosses the boundary.

 

[PAllen]

Because you're calling two photons moving in opposite directions a closed system. I cannot understand how two thing moving at c in opposite directions could thought of as a "closed system"; causally they couldn't be farther apart.

Consider two equal energy photons moving towards each other. If high enough energy, they can (very rarely) interact producing a positron+electron pair. Or, they can both be absorbed by a body. In either case, the invariant mass of the system remains constant. The e/p pair would have the same invariant mass the photons did before. The absorbing body would increase in mass by the invariant mass of the two photons. If it makes sense to attribute mass to the photon pair moving towards each other, do you really want to say this mass disappears if they pass each other without interacting?

 

[Dale]

I cannot understand how two thing moving at c in opposite directions could thought of as a "closed system"; causally they couldn't be farther apart.

As @Mister T mentioned above, they can clearly be called a system simply by arbitrarily choosing your system boundaries to include them. Whether they are a closed system depends on if momentum or energy are crossing the system boundary. It does not matter if the interior of the system is interacting with itself.

Suppose that you start with the initial electron and positron, and suppose further that you draw your system boundaries very far away. Suppose further that there is nothing else inside the boundaries and that no energy or momentum cross the boundary.

Then, it should be clear that the system has some well-defined energy and momentum and therefore some mass. Now, since those are conserved then they must remain constant whether or not the electron and positron anhilate. If they do not anhilate then the mass is attributed to the positron and electron system since that is the only thing inside the boundary. If they do anhilate then the mass is attributed to the two photon system since that is the only thing inside the boundary.

 

[Herman Trivilino]

@nitsuj Perhaps you are thinking of a bound system, such as a neutral Hydrogen atom, where the nucleus and the orbital electron are bound by an attractive electrical force. There's an interaction bonding the constituents. But a system need not be bound. The constituents of a system need not be interacting. If you ionize that neutral Hydrogen atom by removing the electron so that it's very far from the nucleus you can still have the same system. Of course, you would need to transfer enough energy to the system to perform that ionization, but when you do that you increase the mass of the system by the amount of energy that you transfer.

The fact that a pair of oppositely moving photons are not interacting, and are indeed causally disconnected as you say, has no bearing on the claim that they constitute a system. If it did then the ionized atom and the free electron wouldn't constitute a system. Think about that nucleus and that electron moving apart even though they are so far from each other that their interaction is negligible. The kinetic energy that each has relative to their center of momentum contributes to the mass of the system, if it didn't energy, momentum, and mass wouldn't be conserved! Note that $m^2=E^2-p^2$; therefore if any two of those three are conserved, so is the third.

 

[Ibix]

Because you're calling two photons moving in opposite directions a closed system. I cannot understand how two thing moving at c in opposite directions could thought of as a "closed system"; causally they couldn't be farther apart.

In addition to all the comments that have come before, it's also worth noting that the photons do not need to be traveling in opposite directions. Merely not traveling exactly parallel is enough.

Edit: Of course, two photons traveling in opposite directions is two photons just traveling not quite parallel, as described in some other frame.

 

[nitsuj]

Ironic that I just bought a tiny veil of tritium to light the inside of my watch...I went to read about the type of radiation and came here to ask will it be firing off these symmetric photons..I see there's no love for my "it's things moving apart at c, those two things meaningless to each other physically.." lol

 

[Dale]

I see there's no love for my "it's things moving apart at c, those two things meaningless to each other physically.."

Even if they are meaningless to each other they are not meaningless to us

 

[SiennaTheGr8]

Even if they are meaningless to each other they are not meaningless to us

Deep.

 

[nitsuj]

Even if they are meaningless to each other they are not meaningless to us

'nuff said :)

So read more and it doesn't look like tritium makes such photon systems...by does have electron and electron neutrino "pairs" ...very interested to read about the angles they part but have yet to find something on that