# B E = mc^2, Wikipedia Mass–energy equivalence Clarification

1. Dec 6, 2016

### Albertgauss

Hello,

By far, the easiest explanation of E=mc2, Einstein’s famous mass-energy equation, means that mass is a form of energy and that, mass can be turned into energy (heat, light, other particles, etc) and vice versa. It also seems easiest to say that photons are “pure energy” and are “massless”, based on E = pc.

However, I ran across this on Wikipedia: Mass–energy equivalence: Efficiency.

Although mass cannot be converted to energy, in some reactions matter particles (which contain a form of rest energy) can be destroyed and converted to other types of energy that are more usable and obvious as forms of energy—such as light and energy of motion (heat, etc.). However, the total amount of energy and mass does not change in such a transformation. Even when particles are not destroyed, a certain fraction of the ill-defined "matter" in ordinary objects can be destroyed, and its associated energy liberated and made available as the more dramatic energies of light and heat, even though no identifiable real particles are destroyed, and even though (again) the total energy is unchanged (as also the total mass).

This doesn’t seem correct to me. If the wiki statement above is correct, I would like to understand why. As a counter-example, it seems to me that if two photons (energy) come together and produce a matter-antimatter pair (give that all requirements are met), it seems to me that: before the reaction you had two photons (invariant rest massless) and after the reaction you had two particles with real invariant rest mass, a matter and antimatter particle. The invariant-rest-mass-less photons are gone; in their place are two real particles that have real invariant mass.

Obviously, I have no problem accepting that energy is conserved, but how can invariant rest mass be conserved if there isn’t any invariant rest mass associated with the two photons or annhilation?

Of course, if someone mentions the “mass of a photon”?, it certainly cannot be the same “mass” as the invariant rest mass of a proton, electrons, pion, etc. If you can have a mass of a photon, why is there no such term in E2 = mc2 + pc?

I do feel its splitting hairs to say: “Energy isn’t really created because mass is already a form of energy.” As one article I found said. Then I would just say, “Well, then, what I mean is that the photon mass can be converted into the invariant rest mass of particles and vice versa.”

Also, it seems, that in the article below, the idea that matter and energy can be transformed from one to another via E=mc2] might be conceptually sound. I do not know how experts on this website feel about the validity of the below proposal, but I would be interested to hear what they say. The article below is one-way so far: photons to mass. Of course, if that is true then wouldn't there be a concept to go from mass to photons also?

http://phys.org/news/2014-05-scientists-year-quest.html

Anyway, I am confused.

2. Dec 6, 2016

### Staff: Mentor

A single photon does not have invariant mass, but a system of two or more photons does have invariant mass. Here it is the system as a whole that has mass, not any individual photon.

The annihilation of photons to produce matter particles has long been predicted by physics. There is nothing new there. It's just the reverse of something like an electron-positron annihilation. However, no mass is created or destroyed, as the invariant mass of the system of photons is conserved before and after the annihilation event.

3. Dec 6, 2016

### Staff: Mentor

Stuff like this is the reason that wikipedia is not an acceptable source here.
Conservation of invariant rest mass is an approximation that is valid only when the energies involved in the reaction, divided by $c^2$, are small compared with the rest masses. That's the case for many real-life problems (does anyone care that a 20 kg lead-acid battery is a fraction of a nanogram heavier when fully charged?) but never the case for any interaction that involves photons.

4. Dec 6, 2016

### OCR

If it was... you did read this right at the top, correct ?

5. Dec 6, 2016

### Orodruin

Staff Emeritus
I believe they are talking about the invariant mass of the system, not the sum of the rest masses - or interpreting someone who did. This is the square of the total 4-momentum and conserved since 4-momentum is conserved.

6. Dec 7, 2016

### my2cts

Mass is the energy E/c^2 in the rest frame of a system. The rest frame is the frame in which the total momentum vanishes. If the internal degrees of freedom of a system are relevant during a process or experiment energy is converted into mass. However the concept of mass is not very useful in such cases.
For example if an atom absorbs light and goes from the ground state to an excited state, its mass changes. The new valueof its mass must be used in formulas such as f=ma and 1/2mv^2. On th eother hand, the total energy in the rest frame of the atom plus the light, hence the total "mass", however does not change. It depends on the circumstances which concept of the mass of this system is useful.
The wikipedia article needs a rewrite in my opinion.

Last edited: Dec 7, 2016
7. Dec 7, 2016

### Staff: Mentor

Ah - yes, I think you're reading it right and I was not.

8. Dec 8, 2016

### Staff: Mentor

Wow! I didn't know that. Where should I look to learn more about that?

9. Dec 8, 2016

### Orodruin

Staff Emeritus
The invariant mass of a system is just the square of its total 4-momentum. I am not sure there is much more else to learn, but any introductory SR text should cover it.

10. Dec 8, 2016

### Staff: Mentor

In general, the invariant mass of a system of particles does not equal the sum of the invariant masses of the component particles. In units where c = 1, for two particles: $$m_1 = \sqrt{E_1^2 - |\vec p_1|^2} \\ m_2 = \sqrt{E_2^2 - |\vec p_2|^2} \\ M_{\rm{sys}} = \sqrt{(E_1 + E_2)^2 - |\vec p_1 + \vec p_2|^2}$$ In general, $M_{\rm{sys}} \ne m_1 + m_2$.

11. Dec 8, 2016

### Orodruin

Staff Emeritus
Or more generally:
$M_{\rm inv}^2 = (p_1 + p_2)^2 = p_1^2 + p_2^2 + 2p_1\cdot p_2 = m_1^2 + m_2^2 + 2p_1\cdot p_2 \geq m_1^2 + m_2^2 + 2m_1m_2 = (m_1+m_2)^2$
with equality if and only if the particles are at relative rest.

Can also be done for an arbitrary number of particles
$M^2 = \left(\sum_i p_i\right)^2 = \sum_{ij} p_i \cdot p_j \geq \sum_{ij} m_i m_j = \left(\sum_i m_i\right)^2$

12. Dec 8, 2016

### Albertgauss

Hi all,

Was busy yesterday but I had a few more follow-up questions and responses.

I think the main goal for me here is: can I say with 100 % correctness that:

“Mass can be turned into energy and energy can be turned into mass” via E = mc2.

Also, can I say with 100% correctness that the photon is a massless particle, where mc2 of the photon is zero?

I have always heard the above concept explained this way, even by professors and grad students. If the above statement is not strictly true, how would I modify the wording above to be absolutely true?

I’ve used wiki before as a science source, and usually not too much problems. I thought perhaps a new understanding had possibly emerged that invalidated the concept of “energy to/from mass” but I realize-- in this case--the wiki article here is mostly wrong and that the statement “energy to/from mass” is actually a correct way of understanding E = mc2.

Also, I think I confused the word “invariant”. I realize that on this forum, the phrase “invariant rest mass” means the energy square of the four-vector. E2 = (mc2)2 + (pc)2. If it is meant that the photons in the system have an “invariant rest mass” to mean an invariant energy four-vector, then, yes I agree with that. That seems to be the lefthand side of Orodruin’s equation, M2inv = (p1+p2)2.

But in Orodruin’s equation, p1 and p2 are still momenta, and not photon rest masses (mc2 for photon). Thus, I would still say the photon is a massless particle, and it is pure energy, or pure momentum up to c, E = pc.

The following equation is the usual pair-production/annihilation conservation of energy equation.

2(mc2) = 2hf

Again, I agree that the invariant energy four vector is conserved in all frames (E, p four vector). But the above equation still means that "energy to/from mass" is correct and the photons do not have the same kind of mass for hadrons and leptons). If the photon had the same kind of mass as matter, why is it never written in the equation above, or in Orodruin’s equation in the line where only momenta appear (p1+p2)2?

I agree that I should have paid attention to the wiki disclaimer that the site needs to be edited. Wiki seems to have usually been a good source of physics, so errors like this seem to me be rare. I could be wrong, though, I realize, based on this thread. If the general consensus on this forum is that wiki is not a good source of physics, is there some better website this forum accepts as valid physics in a similar encyclopedia format?

13. Dec 8, 2016

### DrStupid

No, that's not correct. E=mc² means that mass and rest energy are equivalent. A system at rest with the mass m always has the energy mc². Turning mass into energy or vice versa would violate this fundamental relationship as well as conservation of energy.

Yes, that's correct.

Yes, a single photon is massless but a system of two photons may have mass. Mass is not additive.

That applies to the special case that the kinetic energy of the particles is negligible. In the general case the total energy of the system is not equal to the rest energy of the particles.

I don't see any error in this article.

14. Dec 8, 2016

### jbriggs444

No, you cannot say either thing with 100% correctness. If you do, you allow for fallacious reasoning.

E=mc2 is a declaration that the energy of an object at rest is equal to its mass times the square of the speed of light. If you say it without the "at rest" qualification, you leave the statement open to an interpretation that the energy of a thing is always equal to its invariant mass times the square of the speed of light, even if that object is moving. Such an interpretation would be incorrect. For a moving object a correct generalization is $E^2 = m^2c^4 + p^2c^2$ where p is the momentum of the body and m is its invariant mass.

mc2 for a photon is zero. But that is a statement with little or no physical significance. The invariant mass of a photon is zero, yes. But a photon is never at rest. So, for a photon, E=mc2 is always false. For a photon, the more general formula reduces to the correct result: E=pc.

Of course, you cannot assert that E=pc with 100% correctness either unless you make it clear that you are talking about objects whose invariant mass is zero.

Edit: with apologies to @Orodruin for the superfluous "invariant" qualifier on mass. :-)

Last edited: Dec 8, 2016
15. Dec 8, 2016

### Orodruin

Staff Emeritus
No, it is not that mass and energy are different things that can be "turned into each other". Mass is a form of energy, just like kinetic energy is. You would not say "turn kinetic energy into energy" would you?

The individual photons are massless, the two-photon system is not. It has an invariant mass of $2E_1E_2(1-\cos\theta)$, where $\theta$ is the angle between the photons.

Note: The "rest" in "rest mass" is superfluous. Nobody talks about any other mass nowadays.

What do you mean by "kind of mass"? Mass, like energy, is a property of a system, not a substance of its own.

16. Dec 8, 2016

### Staff: Mentor

No. What you can say is that if a system with no momentum has energy, then it also has mass. And vice versa.

I wouldn't say that. It can't be pure energy since it has momentum too. And it also has spin. There is no good reason to call it "pure energy" and a lot of good reasons to not.

But you are right that it is massless.

17. Dec 11, 2016

### Albertgauss

First, I thought the article might be wrong based on reply 4. But then, based on reply 13, it seems the Wikipedia article might be correct after all. Anyone to break the tie?

I understand now that the invariant mass includes other forms of energy. Heat and light within a system contribute to the invariant mass.

One quick question: is rest mass the same as invariant mass? What is the difference between the two, if they are different? It seems, as I’ve thought about it and re-read about it, the rest mass can have included in it other forms of energy not associated with mass (gas in a box weighs more due to the thermal motion of the gas particles), but this is also what the invariant mass does, as everything I reread seems to imply.

I also don’t have invariant mass being destroyed, defined in replies 10,11, 14 and 15. I understand that it is m2=E2+p2. (squares). In a reaction, the invariant mass and energy would be conserved, “missing energy or invariant mass” would mean products of a reaction escaped into the environment and left the system in which the reaction took place.

I’m fine with “mass not being converted into energy”, as mass iis a property of the particle, as mentioned in reply 15. Yes, I agree with that, I would never say “turn kinetic energy into energy” since kinetic energy is already a form of energy. But could it be said instead that

“The energy of particle at rest can be converted into other forms of energy”?

From classical physics, or energy in general, energy is allowed to change forms. Can the energy associated with mass change forms into other forms of energy also? For example, when you drop a ball off a building, (classical) its gravitational potential energy is converted into kinetic energy. The gravitational potential energy is gone just before the particle hits the ground, and is now replaced with kinetic energy that wasn’t there when you let the ball go. The total energy of the system is conserved, but the potential energy has changed forms into the kinetic energy.

Likewise, from classical physics, if you push a box along the floor and let the box go to glide on its own, the box will eventually come to a stop on the floor. The kinetic energy you gave it is gone, reappearing as heat from the friction. It seems here that kinetic energy is gone that was there just when you pushed the box, and heat has appeared which was not there before. Total energy would be conserved, but the kinetic energy would have been converted to the heat released with friction.

If the energy associated with a particle’s mass is a form of energy, how is it that this energy cannot be converted to other types of energy?

So for particle annihilation, the invariant mass is conserved, and I agree that the combined photons have invariant mass for both of them. For this situation, I won’t let the particle and anti-particle to have any extra kinetic energy above the energy associated with their rest mass (in p = 0 frame) when they collide, or at least, very little. The rest mass (p = 0 frame) of each particle forbids the particles to move at C. Also, when the particles collide, the particle and anti-particle disappear, they are gone, they do not exist anymore. Two photons replace the particles that weren’t there before the reaction took place. The energy associated with the photons requires the photons to move at speed “c” and have energy only associated with momentum (reply 16). The photons do not have individual invariant mass (as in reply 14), but they combine (not additive) for their system invariant mass to be conserved. But it seems that energy has, in fact, changed forms. There was an individual, p=0 rest mass energy associated with the existence of each matter particle that is gone now (since the particles are gone), and replacing it are individual energy associated with momentum of the photons, which did not exist previously. It seems that the energy associated with (p =0) rest mass of particles has changed into an energy associated with the momentum of photons. These seem to be different kinds of energy changing forms. I’m not sure why these last statements are wrong.

This also came from the wiki article. I do not know if this is correct or not.

“In theory, it should be possible to destroy matter and convert all of the rest-energy associated with matter into heat and light (which would of course have the same invariant mass), but none of the theoretically known methods are practical. One way to convert all the energy within matter into usable energy is to annihilate matter with antimatter.”

Is it possible to say that the energy associated with a particle’s rest mass can be converted to other forms of energy? Not the mass itself, but the energy associated with having mass?

Sorry for the long post and replies, but I am trying sincerely to understand these concepts.

18. Dec 11, 2016

### DrStupid

Yes and today it is just called mass.

Because it is conserved. Asked yourself if the total energy of a system can be converted into other types of energy.

19. Dec 11, 2016

### Staff: Mentor

Yes, this happens in any spontaneous decay of a particle. For example, in $\Lambda^0 \rightarrow p + \pi^-$, part of the energy associated with the $\Lambda^0$ mass is converted to kinetic energy of the $p$ and the $\pi^-$. $$m_\Lambda c^2 = m_p c^2 + m_\pi c^2 + K_p + K_\pi$$ in the reference frame in which the $\Lambda^0$ is at rest initially.