Center of momentum and mass energy equivalence

[Dale]

So do we still need concept of mass in addition to energy ?

Yes. One refers to the norm of the four momentum and the other refers to the timelike component. They are separate concepts and both are needed.

Mass can be applied strictly only to an elementary particle

It applies to anything with a four momentum.

 

[sweet springs]

Yes. One refers to the norm of the four momentum and the other refers to the timelike component. They are separate concepts and both are needed.

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.
This is my preference not to call it mass due to the concerns I posted. Again do we still need a concept of mass in addition to energy or four momentum more strictly ? Does mass survive under Ockam's razor ?

 

[Dale]

Again do we still need a concept of mass in addition to energy or four momentum more strictly ? Does mass survive under Ockam's razor ?

We are just going in circles here. Yes both concepts are needed and yes mass survives Occham's razor.

 

[sweet springs]

Um... An example where mass cannot be substituted by energy or four momentum including its norm could help me. Thanks in advance.

 

[vanhees71]

I don't understand your question. Already to know, which elementary particle you are observing you need to know its mass (and other parameters like electric charge etc.).

 

[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.

 

[Herman Trivilino]

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.

 

[Herman Trivilino]

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.

 

[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.