Is the presence of 'c' in E=mc^2 coincidence?

[Shrike15]

Hi

With special relativity I understand that if for example you have one gram of matter and it is converted to pure energy you will get E=0.001*299,792,458^2 joules of energy. However what I ‘am unsure of is; is it merely a coincidence that the value of the speed of light relates to mass/energy conversion or is there some quantum physics theory that explains why the value of the speed of light is related to energy?

Thanks in advance

 

[Calimero]

Hi Shrike15, welcome to PF. No, it is not coincidence. If you write that in units where c=1, you get E=M, so c^2 is there as conversion factor between units of mass and energy.

 

[D H]

Calimero, telling Shrike15 that E=m in some apparently mystical natural units doesn't help explain E=mc².

Shrike15: The answer does not lie in quantum mechanics. It lies in special relativity. The general expression for the relation between energy E, momentum p, and rest mass m₀ in special relativity is

$$E^2 = (pc)^2 + (m_0c^2)^2$$

You can find a derivation of this in almost any sophomore/junior level classical mechanics physics text. Note that for an object at rest (p=0), this reduces to E²=(m₀c²)², or just E=m₀c².

So what's the deal with "natural units"? You have already run into a similar situation with Newton's second law, F=ma. That is not what Newton said. He said that force is proportional to mass times acceleration rather than equal to mass times acceleration. In other words, F=kma. For example, in English units with mass expressed in pounds mass, force in pounds force, and acceleration in feet/second², Newton's second law is

$$F=\frac 1 {32.1740486}\,\frac{\text{lbf}\cdot \text{s}^2}{\text{lbm}\cdot \text{ft}}\, m a$$

This suggests a pedantic version of Newton's second law in SI units,

$$F=1\frac{\text{Newton}\cdot{\text{s}^2}}{\text{kg}\cdot \text{m}}\,m a$$

We don't do that. We instead say that force is a derived unit rather than a fundamental unit. The constant of proportionality is not merely 1. It essentially vanishes: F=ma, period. Similarly, one can define units in such a way that the speed of light c, the gravitational constant G, plus some others, all have a numerical value 1. This raises a question: Is this a convenient trick that makes the math easier, or does it have some deeper meaning? What are the "real" dimensions of the universe (if any)?

 

[Meir Achuz]

Calimero is exactly right.

 

[D H]

Calimero is exactly right.

I would say just the opposite, for three reasons. First off, while answering a high school level question (and this is a high school level question) with an answer that assumes upper undergrad physics knowledge might be technically correct, it does not answer the question in a way that the person asking could possibly understand.

Secondly, telling the person to use natural units does not answer the question. It begs the question: From whence does E=m arise? It doesn't matter if you use SI units or natural units. A good answer would give some inkling as to why E=m (or E=mc²) is correct.

Finally, saying E=m implies the use of relativistic mass, a concept that physics educators are trying to avoid nowadays.

 

[Meir Achuz]

I agree with MA and C.
What does F=ma have to do with the question?

 

[Calimero]

I would say just the opposite, for three reasons. First off, while answering a high school level question (and this is a high school level question) with an answer that assumes upper undergrad physics knowledge might be technically correct, it does not answer the question in a way that the person asking could possibly understand.

I appreciate your stance very much, but this is world-wide public forum, and I can't possibly know if Shrine15 is just high schooler, or physics loving retired salesman with cherry-picked knowledge, so I gave him, for what I thought is most concise answer, and I don't think that it is hard to understand.

Secondly, telling the person to use natural units does not answer the question.

Yes it does, because natural units system automatically incorporate dimensional analysis.

Finally, saying E=m implies the use of relativistic mass, a concept that physics educators are trying to avoid nowadays.

Why $E=m$ implies the use of relativistic mass, and $E=mc^{2}$ do not? This is simply not true. Both rest (Lorentz invariant) mass and relativistic mass can be plugged there.

$E=mc^{2}$ either means $E=m_{0}c^{2}$ for an object at rest, or $E=m_{rel}c^{2}$ when the object is moving.


So, E=m can be taken to simply mean: the rest-energy of a particle, measured in Planck units of energy, equals the rest-mass of a particle, measured in Planck units of mass.

 

[D H]

I appreciate your stance very much, but this is world-wide public forum, and I can't possibly know if Shrine15 is just high schooler, or physics loving retired salesman with cherry-picked knowledge, so I gave him, for what I thought is most concise answer, and I don't think that it is hard to understand.

It doesn't matter whether Shrine15 is "just high schooler, or physics loving retired salesman with cherry-picked knowledge." It should be obvious that an answer in the form of natural units is not helping. Think about it. Someone who had the requisite knowledge to know what natural units are and what they mean would not have asked that question.

And saying E=m (with c=1) does not answer the question. It instead begs the question. Whether you are using SI units or natural units, energy-mass equivalence is a derived result. There was no mention of that in your post. How does saying E=m possibly answer the OP's question?

The OP asked about a connection with quantum mechanics. The first thing you should have done was to correct the misperception, saying that it instead stems from the field of special relativity. The next thing you should have done was to say that energy-mass equivalence is a direct consequence of special relativity, and maybe given some inkling of the derivation. Did you do either of those?The people who post questions at this site span an immense range of knowledge. Some are kids in middle school who don't know much of anything or parents of those kids who know even less but are earnestly trying to help their children. Give an answer that requires even high school-level understanding to such people and you are not helping. Telling high schoolers who are struggling with algebra-based physics that the answer lies in calculus is no help either. On the other hand, when a PhD candidate who is a bit confused about some fine point of Lie algebras, fire away with the math.You need to tailor your answer to the audience. So how to gauge the audience? The nature of the question often gives big clues regarding the level of knowledge of the person asking the question. In this case, asking whether E=mc² is a mere coincidence and whether it arises from quantum mechanics gives a big clue that the person asking the question either has not even taken freshman physics or took it so long ago that the memory is long gone.

 

[PhilDSP]

Although Einstein apparently was the first to publicize or essentially describe the equation:

$$\Delta E = \Delta mc^2$$

The equation

$$E = \alpha mc^2$$ where $$\alpha$$ is a constant

was known much, much earlier and had been derived directly from Maxwell's Equations by J. J. Thompson among others who did so later. So one could presume the mass-energy relationship is more basic than SR if you are of the opinion that the Maxwell Equations are more fundamental than space-time symmetry or other SR assumptions. But it may be just as often regarded in SR circles that space-time symmetry is more fundamental than Maxwell's equations.

 

[soothsayer]

In this case, asking whether E=mc² is a mere coincidence and whether it arises from quantum mechanics gives a big clue that the person asking the question either has not even taken freshman physics or took it so long ago that the memory is long gone.

No, I think what Shrike was asking was rather cutting-edge.

Yes, obviously, E=mc² comes from special relativity, we all know that. What he's asking is, what the heck does c have to do with it? Why is it that the speed of light somehow relates mass to energy? You could bust out all the derivation known to man, and it wouldn't answer the question. The question more has to do with the nature of mass and the universe that I don't think we fully understand yet. Shrike mentioned Quantum Mechanics because I think he was really trying to get at the root of the relationship, it seems very likely that there could be some sort of atomic explanation for the existence of a c^2 term in relating energy and mass. You kinda got at it when you said

Is this a convenient trick that makes the math easier, or does it have some deeper meaning? What are the "real" dimensions of the universe (if any)?

So maybe E=mc² gets at some even deeper relationship or property of our universe we haven't figured out yet. I think that was what Shrike was musing with his question.

 

[Phrak]

First, matter does not convert to energy.

This is a persistent error. Where did it come from?

 

[pervect]

If we stick to standard units for the time being, then we know from dimensional analysis to convert Mass to Energy we need to multiply by some velocity^2, i.e. E is proportional to m * v^2.

Now, dimensional analysis can't tall us that this factor is c^2 - it could be 2c^2, pi c^2, or any number * c^2. Or there might be some other fundamental velocity than 'c' that's needed. But certainly E=mc^2 isn't a particularly surprising expression.

E = mc^2 originally came from analysis that showed E^2 - (pc)^2 = constant, as was previously shown. However, the annhillation of equal amounts of matter and anti-matter into energy serves as a graphic and direct demonstration of the truth of the equation, IMO anyway.

 

[Domenico Mico]

No, I think what Shrike was asking was rather cutting-edge.

Yes, obviously, E=mc² comes from special relativity, we all know that. What he's asking is, what the heck does c have to do with it? Why is it that the speed of light somehow relates mass to energy? You could bust out all the derivation known to man, and it wouldn't answer the question. The question more has to do with the nature of mass and the universe that I don't think we fully understand yet. Shrike mentioned Quantum Mechanics because I think he was really trying to get at the root of the relationship, it seems very likely that there could be some sort of atomic explanation for the existence of a c^2 term in relating energy and mass. You kinda got at it when you said


So maybe E=mc² gets at some even deeper relationship or property of our universe we haven't figured out yet. I think that was what Shrike was musing with his question.

Shrike15 posed a very profound question and I was pleased to see that Soothsayer got the message. In order to answer the question with some degree of pertinence, we must first establish, and possibly discuss, what is the role of “c” when it is a term of a linear equation and what it is its role when the same is stuck in an equation such as the one we are talking about. In this last case “c” is operating in a nonlinear field, that is to say: a field where the energy per unit volume becomes asymptotically weaker as we move out and away from the nucleus.
In other words “c” in the linear optic equation t = nx/c where “t” is the transit time of a light ray through a given substance “x” and “n” is the refraction index of that substance, or in the little story of the photon: wavelength lambda = h/p = hc/E is not the same “c” seen in E = mc^2.
If, for some unknown reason, we want them to carry the same symbol “c”, we should at least acknowledge that the linearity tag seen in the first two equations is not applicable to the last one. Good God, inside the nucleus of an atom there is no room to do it. Over and above, the physical function of “c” is totally different from the one carried out in the other two equations.
I am with you Soothsayer, but I wouldn’t inconvenience the Universe. I think the whole thing can be worked out locally and now by Science and scientists alike if their mind ticks the same way it does in the young brain of Shrike15 who just proved he can generate mighty brain-waves.

 

[Shrike15]

Firstly I thank everyone for their replies,

I can't possibly know if Shrine15 is just high schooler, or physics loving retired salesman with cherry-picked knowledge.

To set the record straight I' am a second year engineering student.

I would say just the opposite, for three reasons. First off, while answering a high school level question (and this is a high school level question) with an answer that assumes upper undergrad physics knowledge might be technically correct, it does not answer the question in a way that the person asking could possibly understand.

D H you have assumed way too much about me and my post, when you didn't even understand the question in the first place.

I was pleased to see that both 'Soothsayer' and 'Domenico Mico' understood what I was asking.

I' am sorry if my question wasn't clear enough but I think 'Domenico Mico' basically got it when he said; "In other words “c” in the linear optic equation t = nx/c where “t” is the transit time of a light ray through a given substance “x” and “n” is the refraction index of that substance, or in the little story of the photon: wavelength lambda = h/p = hc/E is not the same “c” seen in E = mc^2." That is to say; why does the speed of a photon have a connection to the potential energy of the matter within the nucleus of the atom?

It seems thus far no one has been able to answer this question, I hope that someday we can as this will further increase our understanding of the nature of the universe. However I still encourage people to discuss this question with any thoughts they may have.

 

[Phrak]

It seems thus far no one has been able to answer this question, I hope that someday we can as this will further increase our understanding of the nature of the universe. However I still encourage people to discuss this question with any thoughts they may have.

This guy, Albert Einstein, in a diminutive and little known text, The Meaning of Relativity, made an attempt in pages 43 through 47. I'm afraid there is no online and free text available that I could find. But you might pick it up for a few dollars and change.

------------------------

And, by the way, this nonsense about mass changing to energy really is nonsense. This comes about because people such as Pervect, whom I greatly respect, innocently say things like

However, the annihilation of equal amounts of matter and anti-matter into energy...​

It's not wrong; It just leads the recipient to the wrong conclusion. The 'mass' is still there after annihilation, and before annihilation the energy was in the massive particles all along. In this case, the 'mass' is just the vector magnitude of [E, cp].

If mass were converted to energy then we would have this equation:

E + mc^2 = constant, where energy and mass can slosh back and forth as long as their sum is conserved.

This is wrong. We have

E = mc^2.

This is an equivalence relationship; Not a conversion from one thing to another.(Rest) Mass and energy only seem different because there are two different ways of measuring the same stuff. For mass we use a scale or an accelerometer. For energy we have to take a few measurements such as force and distance and deduce the amount of energy. The constant, c is the units conversion factor to change one type of instruments measures into the other.

This occurs in quantum mechanics as well: E=h_bar ω. Energy is the same stuff as frequency, expressed and measured in different units.

 

[Calimero]

Planck mass: $$m_{p}=\sqrt{\frac{\hbar c}{G}}$$

Planck energy: $$E_{p}=\sqrt{\frac{\hbar c^{5}}{G}}$$


$$\frac{E_{p}}{m_{p}}=\frac{\sqrt{\frac{\hbar c^{5}}{G}}}{\sqrt{\frac{\hbar c}{G}}}=c^{2}$$


$$E_{p}=m_{p}c^{2}$$





What is the question here? Why the physical quantities and their corresponding units (both base and derived) are defined the way they are, or what?

 

[bobc2]

Perhaps we have a clue in the more fundamental aspect of nature that the 4-dimensional photon world line (or world ribbon--for string buffs) always bisects the angle between an observer's world line and his instantaneous 3-D cross-section of the universe--that along with the curious motion of observers along their 4-D world lines. And of course we've just arbitrarily assigned units so that we can report that all observers move along their world lines at the speed, c.

But, again, the more fundamental aspect is the observed ratio of X1 displacement to X4 displacement for the photon, for all observers. It's the ratio dX1/dX4 = 1 that's significant--the units leading to c were arbitrarily chosen.

Beyond that, we have other aspects of nature's 4-dimensional structure, i.e., the interesting consistent and symmetric patterns in the weaving of a 4-dimensional fabric that manifests aspects (prompts our perceptions?) such as mass and 4-D momentum vectors. One more skeptical of man's attempts to understand nature might have put it, ...a 4-dimensional fabric of which man's stubborn perceptions lead him to invent words like mass and energy.

 

[PhilDSP]

In other words “c” in the linear optic equation t = nx/c where “t” is the transit time of a light ray through a given substance “x” and “n” is the refraction index of that substance, or in the little story of the photon: wavelength lambda = h/p = hc/E is not the same “c” seen in E = mc^2.

Yes, this is heading towards the mathematical model that J. J. Thompson discovered (or developed) which as I said earlier comes straight out of Maxwell's Equations. What seems to be happening with regard to the photon in the case you proposed (but simplified to approach a single electron) is that the refraction index increases closer to the electron. The photon must decrease its velocity if it is to be synchronized with any exchange of EM energy or EM wave fluctuations. This effective lowering of velocity from the value of c is described by the Ewald-Oseen Extinction Theorem.
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Let's take the proper exposition of the equation:

$$\Delta E = \Delta mc^2$$

And form

$$\Delta c^2 = \Delta (E / m)$$

Then

$$\Delta c = \Delta (E / m)^-2 = \Delta nx/t$$

 

[PhilDSP]

Sorry, maybe the problem with editing tex symbols is due to my browser? Attempting to correct the tex symbols only regenerates the old version with the new version layered at the bottom. I'm using the Preview button to verify any changes.

The middle term of the last equation should have an exponent of 1/2 or be enclosed by the square root operator.

 

[HallsofIvy]

Sorry, maybe the problem with editing tex symbols is due to my browser? Attempting to correct the tex symbols only regenerates the old version with the new version layered at the bottom. I'm using the Preview button to verify any changes.

The middle term of the last equation should have an exponent of 1/2 or be enclosed by the square root operator.

Yes, that's a known bug with this particular implementation of LaTex. Just pressing the "refresh" button on your browser should correct that.

 

[D H]

To set the record straight I' am a second year engineering student.

D H you have assumed way too much about me and my post, when you didn't even understand the question in the first place.

That you are a second year engineering student fits right in with what I said. As a sophomore you have only had a rudimentary introduction to Maxwell's equations. If you major in electrical engineering or engineering physics, you will unlearn and relearn them multiple times during your college career. Unless you are an engineering physics major, you almost certainly have not taken the sophomore/junior level classical mechanics class in which mass-energy equivalence is derived from the precepts of special relativity. You definitely have not studied quantum electrodynamics, because that is typically taught at the graduate level. It is quantum electrodynamics where the connection between Maxwell's equations, quantum mechanics, and special relativity is fully developed.

Special relativity does not explain why the speed of light is the same to all observers. The constancy of the speed of light is instead an axiom of special relativity. The only other axiom (aka assumption) of special relativity is that the laws of physics are the same to all inertial observers. Mass-energy equivalence derives from these two assumptions.


Where did those assumptions come from? What motivated them? The principle of relativity, that the laws of physics are independent of the observer's speed, is an old principle that goes back to Galileo. The constancy of the speed of light is a much newer concept. It is a consequence of Maxwell's equations, which were published in 1861 and 1862.

Physics was in turmoil during the latter half of the 19th century. The 200+ year old Newtonian mechanics and the nascent theory of electrodynamics appeared to be in direct conflict with one another. The best minds of the time worked very hard to resolve this problem. It was Einstein who resolved the conflict in the clearest and simplest way with his theory of special relativity.


What is the connection to photons? The answer to this question is simple: The connection is not in special relativity. Special relativity is a classical theory of physics rather than a quantum theory. Special relativity does not discuss photons per se. It does however discuss Maxwell's equations. In fact, the second half of Einstein's 1905 paper on special relativity is devoted to Maxwell's equations. In that paper Einstein re-derives Maxwell's equations from the perspective of special relativity.

Special relativity necessarily had to be equivalent to Newtonian mechanics in those domains where Newtonian mechanics had been well-tested experimentally. The same concept applied to quantum mechanics. In particular, to be consistent with known physics, quantum mechanics either had to have all photons move at c or it had to explain the discrepancy. Since quantum mechanics posits (i.e. assumes) that photons do indeed move at c, so there was no discrepancy to explain. Note that this is an after-the-fact development of quantum mechanics. Quantum mechanics largely followed the development of relativity. In fact, it was not until 40 to 50 years after Einstein developed special relativity that physicists were able to fully reconcile quantum mechanics, special relativity, and electrodynamics.

One key consequence of photons moving at c: Photons necessarily must be massless. If photons have a non-zero intrinsic mass, no matter how small, they could move at c. Numerous tests of this have been made, and all have been consistent with the hypothesis of zero intrinsic mass.

 

[dj cornbread]

First, matter does not convert to energy.

This is a persistent error. Where did it come from?

A persistent truth, matter is made of energy (well as far as I am led to believe by the existence of the nuclear bomb and like dh said about this being a back to jr high school thing) ( Yes, King George, VA used to have a quite advanced science's department throughout middle/high school (more emphasis on astronomy,and bio)... tho I wish they could've taught me to spell better)