Center of momentum and mass energy equivalence

[sweet springs]

I agree with you as for system of single elementary particle where mass M and (proper) rest energy Mc^2 are not distinguishable.

Say two photon system e.g. after electron-positron annihilation, has mass M or rest energy Mc^2. In COM each photon has energy Mc^2/2. I do not think each of the two photons has mass M/2 also. This is an example of my point that and energy or strictly four momentum is enough and we do not need concept of mass in addition.

 

[SiennaTheGr8]

We need the concept of [conserved scalar commonly known as "total energy" or just "energy"], and we need the concept of [total energy as measured in a system's rest frame].

We are free to express these two quantities in whatever units we prefer. For much of the twentieth century, both quantities were variously expressed in units of energy ("total energy" / "rest energy") and units of mass ("relativistic mass" / "rest mass"). This was recognized as redundant and confusing, so eventually "relativistic mass" was mostly phased out, and "rest mass" is now usually just called "mass."

Of course, "rest energy" and "mass" are still redundant. For purely conventional reasons, "mass" is generally used. So now we've got a situation where the concepts [total energy] and [total energy as measured in a system's rest frame] are expressed in different units and with different words ("energy" vs. "mass").

This was confusing to me as a beginner, and anecdotal evidence tells me that I'm not alone. At the very least, I think it's beneficial to see some of the equations in special relativity with $E_0$ at some point. For instance, compare these two equations:

$E = \gamma mc^2$

$E = \gamma E_0$

Which of them immediately brings to mind the following equation for time dilation?

$t = \gamma t_0$

(where $t_0$ is proper time). As soon as you see the parallel here, you understand that kinetic energy is a relativistic effect in exactly the same way that time dilation is. That's a nice insight. Gets you thinking, too: yes, the equations are similar, but we've known about kinetic energy (to lowest order) for centuries, whereas differential aging is imperceptible even at what humans naively consider very fast speeds!

Once you understand the physics, none of this matters. And at that point, you can set $c = 1$ anyway, so energy and mass and momentum are all expressed in the same unit after all. This is about semantics and pedagogy (as was the "relativistic mass" debate).

 

[vanhees71]

I agree with you as for system of single elementary particle where mass M and (proper) rest energy Mc^2 are not distinguishable.

Say two photon system e.g. after electron-positron annihilation, has mass M or rest energy Mc^2. In COM each photon has energy Mc^2/2. I do not think each of the two photons has mass M/2 also. This is an example of my point that and energy or strictly four momentum is enough and we do not need concept of mass in addition.

For two photons, in the center-momentum frame you have (with $c=1$)
$$p_1=(|\vec{p}|,\vec{p}), \quad p_2=(|\vec{p}|,-\vec{p}).$$
Each photon has $m_j^2=p_j^2=0$ and the total energy is $\sqrt{s}=2|\vec{p}|$. To analyze this simple kinematics, we need to know that photons are quanta with mass 0. I don't understand, where your problem is or why you must make up one!

 

[DrGreg]

I totally agree that the norm of the four momentum of a particle or a system is a useful idea whether we call it mass, rest energy or invariant energy.

Well, "mass" has four letters whereas "rest energy" has 11 and "invariant energy" has 16, so that's one practical reason to prefer it as the name for this concept.

Also mass is a property of a particle or system, whereas energy depends on both the system and the observer. Mathematically, mass is $\sqrt{|\mathbf{P} \cdot \mathbf{P}|}$, a property of 4-momentum $\mathbf{P}$ alone, whereas energy is $|\mathbf{P} \cdot \mathbf{U}|$, where $\mathbf{U}$ is the 4-velocity of the observer (under the convention $c=1$). I think mass and energy are sufficiently different to justify different names.

Further justification are equations such as $\mathbf{P} = m \mathbf{V}$ or $\mathbf{F} = m \mathbf{A}$ which, as 4-vector equations, look just like the corresponding Newtonian 3-vector equations.

 

[sweet springs]

Other invariants under Lorentz transformation are proper time and proper length. Why we coin the word mass instead of proper energy ?

 

[DrGreg]

Other invariants under Lorentz transformation are proper time and proper length. Why we coin the word mass instead of proper energy ?

Because the word "mass" already existed from Newtonian physics, whereas there's no obvious Newtonian terminology that could be redefined to distinguish between proper time and coordinate time.

 

[sweet springs]

Proper time is a kind of time.
Proper length is a kind of length.
Proper energy is a kind of energy.
We decided to call proper energy mass in 20th century. The word mass still has its traditional meaning in equation of motion and gravity that would cause some confusion or misunderstanding as discussed in this thread and other. So no use of mass but energy is a simple way though I respect the history of physics. Mass is redundant concept. Energy or strictly four momentum is enough.

 

[Dale]

An example where mass cannot be substituted by energy

For example an electron at rest annhiliting with a moving positron in the frame of a PET system. By measuring the energies of the resulting photons you do not know the initial momentum. You must also know the mass as a separate piece of information.

Mass is redundant concept. Energy or strictly four momentum is enough.

Mass is not redundant with energy. They are separate and distinct concepts.

Of course, they are both parts of the four momentum, but they are different parts. Although the engine and the tires are both parts of a car, they are not redundant and you often must distinguish which one needs to be repaired.

 

[PeterDonis]

Proper time is a kind of time.
Proper length is a kind of length.
Proper energy is a kind of energy.

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

We decided to call proper energy mass in 20th century.

No, we discovered that the concept of "mass" as it's used in Newtonian physics doesn't work when you take relativity into account. (Newtonian physics doesn't have a concept of "proper energy" at all.) So we had to rearrange our concepts.

The word mass still has its traditional meaning in equation of motion and gravity

No, it doesn't. That's one of the key points that people are trying to communicate to you in this thread.

 

[SiennaTheGr8]

We decided to call proper energy mass in 20th century.

I'd say rather that the (somewhat-mysterious) quantity physicists called "mass" before the 20th century turned out to be nothing but proper energy.

On the other hand, the term "relativistic mass" came into fashion early on, so it seems that some physicists may have looked at the mass–energy equivalence in precisely the opposite way, as if what had always been called "energy" had turned out to be nothing more than a kind of extension of the mass concept. This seems backwards to me, but who am I to judge Lewis and Tolman et al.?

The word mass still has its traditional meaning in equation of motion and gravity that would cause some confusion or misunderstanding as discussed in this thread and other.

Only in the Newtonian limit.

That said, I do prefer "rest energy" (or "proper energy") to "mass," as I've made clear. It's only a preference, though I stand by my contention that many beginners could benefit from using $E_0$ sometimes.

 

[PeterDonis]

I'd say rather that the (somewhat-mysterious) quantity physicists called "mass" before the 20th century turned out to be nothing but proper energy.

I don't think the Newtonian concept of "mass", strictly speaking, corresponds to any concept in relativity. Much of the discussion in this thread is illustrating the problems with trying to draw such a correspondence.

 

[Mister T]

Say two photon system e.g. after electron-positron annihilation, has mass M or rest energy Mc^2. In COM each photon has energy Mc^2/2. I do not think each of the two photons has mass M/2 also. This is an example of my point that and energy or strictly four momentum is enough and we do not need concept of mass in addition.

All you illustrate with this example is that mass is not additive. That is the lesson of the Einstein mass-energy equivalence. The thing that we measure when we measure mass, whatever you choose to call it, is not additive. That additive property is part of the Newtonian approximation.

Instead of a two-photon system let's look at a two-electron system. The electrons move away from each other and it's possible to find a frame of reference where the electrons have identical speeds. We call this frame the center-of-momentum frame, but do not confuse this with a point in space that lies midway between the electrons. The relationship between the two is simply that this point is at rest in this frame of reference. This point need not be the origin of the spatial axes in this frame, for example.

Now to my point. Each electron has a mass $m$, the two-electron system has a mass $M$. Note that $M \neq m+m$. But they are equal in the Newtonian approximation. So it's not the very useful concept of mass that we need to reject, it's the often very useful, but also often very wrong, notion that mass is additive.

 

[Mister T]

Say two photon system e.g. after electron-positron annihilation, has mass M or rest energy Mc^2.

The relevant point being that prior to the annihilation the system had the same mass $M$ that it has afterwards.

 

[SiennaTheGr8]

I don't think the Newtonian concept of "mass", strictly speaking, corresponds to any concept in relativity. Much of the discussion in this thread is illustrating the problems with trying to draw such a correspondence.

I think it depends on what you mean by "the Newtonian concept of mass." If you mean a heuristic like "amount of matter" or "the measure of inertia," then I agree with you. These concepts have no straightforward counterparts in relativity. By "measure of inertia," one might reasonably mean rest energy, total energy, or even the matrix that relates the force and acceleration vectors.

But 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: $m = E_0 / c^2$.

 

[SiennaTheGr8]

Further justification are equations such as $\mathbf{P} = m \mathbf{V}$ or $\mathbf{F} = m \mathbf{A}$ which, as 4-vector equations, look just like the corresponding Newtonian 3-vector equations.

Of course, if you use $E_0$ instead of $m$ in the Newtonian approximation, you can still see a correspondence. Using Newton's overdot notation to mean a $ct$-derivative (so $\dot{\vec{\beta}} = \vec a / c^2$), and a little circle to mean a $ct_0$-derivative (so $\mathring{\vec{B}} = \vec A / c^2$, where $\vec{B} = \vec{V} / c$ is the "normalized" four-velocity):

$\vec{p}c \approx E_0 \vec{\beta} \quad$ (for $\beta \ll 1$)

$\vec{f} \approx E_0 \dot{\vec{\beta}} \quad$ (for $\beta \ll 1$)

$\vec{P} = E_0 \vec{B}$

$\vec{F} = E_0 \mathring{\vec{B}}$

(I'd use boldface for the vectors, but it doesn't work for $\beta$.)

 

[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.

 

[Mister T]

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.)

 

[Mister T]

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.

 

[Mister T]

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.