E=MC^2 Mass and Energy, synonymous?

[slickjunt]

Hello everyone,

I just want to throw out a perhaps rehashed idea just to see people's opinions on the matter.
Given Einstein's equation; E=MC^2
If matter could travel at the speed of light it would posses maximum kinetic energy; maximum energy or pure energy is a photon. Theoretically speaking, given light speed is achievable by matter and it reaches a state of maximum kinetic energy, at that current state it would be synonymous with pure energy or a photon. Could we say then matter and energy are equal through the medium of light speed?
This is also backed by the decay of quarks into photon pairs and lesser quarks prompting the idea that fundamental particles are comprised of pure energy/photons. Since a sudden decay or loss of energy produces photons, wouldn't then the matter be converted into pure energy, or is the quark simply made of energy not traveling at light speed?
Perhaps one of the most crucial peices of evidence lies with anti-matter. A positron and an electon collide, then anhilate each other and produce pure energy. Can't we then come to a conclusion that the whole of their existence is pure energy in a state other then traveling at light speed?

tl;dr - Is Energy and Matter synonymous through the medium of light speed?

 

[Popper]

Hello everyone,

I just want to throw out a perhaps rehashed idea just to see people's opinions on the matter.
Given Einstein's equation; E=MC^2
If matter could travel at the speed of light it would posses maximum kinetic energy; maximum energy or pure energy is a photon.

That is incorrect. The term pure energy is a meaningless term used only in science fiction. A photon has energy it is not energy.

Please note that matter which has non-zero proper mass cannot travel at the speed of light.

Theoretically speaking, given light speed is achievable by matter and it reaches a state of maximum kinetic energy, at that current state it would be synonymous with pure energy or a photon.

That is incorrect. Also if by "matter" you're thinking of things which non-zero proper mass then that's wrong also.

Could we say then matter and energy are equal through the medium of light speed?

No. The term matter is not something which is well defined in physics. Its only used loosely. Einstein defined matter as anything which has a non-vanishing stress-energy-momentum tensor.

A quark cannot decay into photon pairs since a quark has charge and photons don't. If such a process occurred then it would, in th very least, violate the principle of conservation of energy.

..and lesser quarks prompting the idea that fundamental particles are comprised of pure energy/photons.

Absolutely not. And what do you mean by the term "lesser quark"? There is no such term used in particle physics.

Is Energy and Matter synonymous through the medium of light speed?

That sentance has no meaning. You're not asking if Energy and Matter are synonymous to something but through something and that's not a meaningful statement. So the answer is "That statement is incorrect."

 

[tom.stoer]

The formula

$$E_0 = mc^2$$

should be understood in terms of the invariant mass or rest mass and in terms of rest energy, i.e. for vanishing momentum p=0. In the general case p≠0 we have

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

Here energy E and the momentum 3-vector p form a 4-vector (E,p) whereas mass m is a scalar.

 

[Popper]

The formula

$$E = mc^2$$

should be unduerstood in terms of the invariant mass or rest mass and in terms of rest energy, i.e. for vanishing momentum p=0.

Note: The appropriate way to write that equation is

$$E_0 = mc^2$$

since E is total energy whereas E₀ is proper energy. Since m is proper mass it follows that $$mc^2$$ is proper energy. If we substituted your expression into

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

then it'd lead to a mistake.

 

[Nugatory]

tl;dr - Is Energy and Matter synonymous through the medium of light speed?

Asking whether matter and energy are the same thing is like asking whether steam and water are the same thing. I can turn one into the other, back and forth all day long. Does that make them the same thing? Depends on the experiment you're doing... If you're going to measure the mass, then there's not a lot of difference between water and steam. If you're going to try floating in it, there's a big difference.

 

[Popper]

Asking whether matter and energy are the same thing is like asking whether steam and water are the same thing. I can turn one into the other, back and forth all day long. Does that make them the same thing? Depends on the experiment you're doing... If you're going to measure the mass, then there's not a lot of difference between water and steam. If you're going to try floating in it, there's a big difference.

If I didn't say so above then let me say this now: No. Energy is not the same thing s mass. They are not synonymous.

There was an article pointing this out. I'll see if I can dig it up.

 

[tom.stoer]

Note: The appropriate way to write that equation is

$$E_0 = mc^2$$.

Thanks for the hint; you are right, of course; I corrected my post.

 

[DrStupid]

Energy is not the same thing s mass. They are not synonymous.

E/c² is what Newton called mass (the factor between momentum and velocity). Rest energy is equivalent to rest mass and total energy is equivalent to the so called relativistic mass.

 

[Popper]

E/c² is what Newton called mass (the factor between momentum and velocity). Rest energy is equivalent to rest mass and total energy is equivalent to the so called relativistic mass.

That is a common misconception. (relativistic) mass is defined as the quantity M such that p = Mv is conserved. Energy cannot be defined. It's one of those things which defy definition. Loosely speaking we can say that energy is a bookkeeping system such that the total energy of a closed system is conserved and which has the dimensions of kg*m²s^-2. It can be shown that a body has the abiligy to loose energy, for example by emitting em raduation, of the amount W then the mass of the body reduces by the amount W/c². This is the meaning of E = mc². Just because two things are proportional doesn't mean that they are the same thing. For example; a photon of energy E is related to its frequency f by E = hf where h is Planck's constant. This doesn't mean that frequency is equivalent to or the same thing as energy or mass.

I have an article somewhere in my filing cabinet. If your or anyone else would like to read it then please let me know and I'll try to make it available.

 

[jtbell]

Talking about physics in English (or any other natural language) can be confusing, especially in situations like this one. Different people attach different nuances of meaning to words like "synonymous", "equivalent", "same", and even "is." It doesn't help that on this forum we have many non-native English speakers, and many native English speakers who aren't super-precise in their use of English (and even they sometimes disagree on nuances of meaning!).

I personally like to use the phrase "corresponds to" in connection with mass and energy in relativity: an invariant mass a.k.a. "rest mass" m corresponds to a certain amount of "rest energy" E₀ given by E₀ = mc².

 

[Bill_K]

That is a common misconception. (relativistic) mass is defined as the quantity M such that p = Mv is conserved. Energy cannot be defined. It's one of those things which defy definition. Loosely speaking we can say that energy is a bookkeeping system such that the total energy of a closed system is conserved and which has the dimensions of kg*m2s-2.

Can't we make a similar statement for momentum? At this level of reasoning, it's just a "thing which is conserved."

I disagree with this anyway. Both energy and momentum are perfectly well defined, as components of the stress-energy tensor, which is the source of the gravitational field. Gravity defines what we mean by energy (and momentum).

 

[Dale]

That is incorrect. The term pure energy is a meaningless term used only in science fiction. A photon has energy it is not energy.

I want to "second" this comment.

Photons have energy, momentum, spin/polarization, etc. In fact, in some sense photons are energy maximally "co-mingled" with momentum.

 

[DrStupid]

Just because two things are proportional doesn't mean that they are the same thing.

Just because two things are equivalent doesn't mean that they are the same thing. For the case that this is some kind of language confusion as described by jtbell please refer to

http://en.wikipedia.org/wiki/Mass–energy_equivalence

 

[Naty1]

s Energy and Matter synonymous through the medium of light speed?

Light speed squared is the conversion factor between energy and matter' light speed is the conversion between energy and momentum...as per Tom.Stoer's post

 

[Naty1]

Slick:

s Energy and Matter synonymous through the medium of light speed?... Could we say then matter and energy are equal through the medium of light speed?

Those statements are sure close...

Light speed squared is the conversion factor between energy and matter' light speed is the conversion between energy and momentum...as per Tom.Stoer's post...

The difficulty with your statements is that the usual description is something like "Mass–energy equivalence is a consequence of special relativity."...Most would not quibble with that because they recognize it... but words such as 'synonymous' and 'equal' suggest you may mean something different. In general it's a good idea to find a description in common use that you like and use it. That way it's easier to communicate you intent.

 

[Dale]

Energy cannot be defined. It's one of those things which defy definition.

No, energy can be defined. Most textbooks will have a definition of energy.

Different theories have different definitions of energy (although wherever both theories apply the definitions always agree), and energy is frame-variant, but neither of those points imply that energy cannot be defined.

 

[Popper]

No, energy can be defined. Most textbooks will have a definition of energy.

Most textbooks disagree with Feynman and I go with Feynman. From The Feynman Lectures on Physics, Vol I - III, Feynman, Leighton, and Sands, Addison Wesley, (1963)(1989).

It is important to realize that in physics today, we have no knowledge of what energy is. We do not have a picture that energy comes in little blobs of a definite amount. It is not that way. However, there are formulas for calculating some numerical quantity, and we add it all together it gives “28” - always the same number. It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas.

I agree with Feynman. Those textbooks you mentioned attempt to define energy as the ability to do work. That's too vague to have meaning. A moving particle can do work. That could be taken to mean that p = mv is the energy of a particle since its moving and can do work.

What Feynman said is similar to what can be found it Newtonian Mechanics by A.P. French, The MIT Introductory Physics Series. From page 376-368

The above remarks do not really define energy. No matter. It is worth recalling once more the opinion that H.A. Kramers expressed: "The most important thin and most fruitful concepts are those to which it is impossible to attach a well-defined meaning." The clue to the immense value of energy as a concept lies in its transformation. It is conserved - that is the point. Although we may not be able to define energy in general, that does not mean that is only a vague, qualitative idea.

I hold fast to what Feynman and French have argued.

 

[Popper]

Can't we make a similar statement for momentum? At this level of reasoning, it's just a "thing which is conserved."

No. Mass is defined so that the quantity mv is conserved. The quantiry p = mv is defined to be the momentum of a particle.

 

[Naty1]

It's one of those things which defy definition.

Maybe you are thinking it defies understanding??...Energy can be a slippery concept and exactly how energy, matter, gravity, and everything else we see around us is related is not so clear. Early in the universe it is believed they WERE all one entity...but in a high energy unstable state which spontaneously decayed into the apparently separate components we now observe.

As Dalespam implies, we have excellent definitions of the component entities and many of their relationships are captured in the Standard Model of Particle Physics. These represent our observations of how these entities behave. But it seems we have more to learn about their origins and deepest relationships.

 

[WannabeNewton]

Different theories have different definitions of energy (although wherever both theories apply the definitions always agree), and energy is frame-variant, but neither of those points imply that energy cannot be defined.

This sums it up brilliantly. The same goes for momentum as well to an extent.

 

[Dale]

I hold fast to what Feynman and French have argued.

I don't think that either of those quotes support your position. I think that you are reading more into them than they intended.

First, the Feynman quote. When he says, "It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas", that is true, but does not imply as you claim that it cannot even be defined. In fact, he explicitly mentions "there are formulas for calculating some numerical quantity, and we add it all together it gives '28' - always the same number". If energy couldn't be defined then there wouldn't be any definite formulas for calculating it and we would never even be able to get a definite number, like 28.

French may be closer to supporting you, but it certainly isn't convincing from the quote. I don't have the book, so I don't know what "The above remarks" refer to. Furthermore, the comment "we may not be able to define energy in general" doesn't mean that it isn't well defined in non-general cases. French is likely referring to the well-known problem of defining a global total energy in general relativity. But that does not mean that energy cannot be rigorously defined in a wide variety of specific cases, just that the definition is not completely general for all possible situations.

If energy cannot be defined then there is no way to know if it is conserved or if any of the other expressions containing E are correct.

Those textbooks you mentioned attempt to define energy as the ability to do work. That's too vague to have meaning. A moving particle can do work. That could be taken to mean that p = mv is the energy of a particle since its moving and can do work.

No. This certainly doesn't follow. Momentum is not equal to the amount of work that a particle has the ability to do (e.g. even a stationary particle has the ability to do work even though it has no momentum).

In any case, there are other possible definitions of energy besides that one. E.g. the conserved quantity corresponding to time translation symmetry in the Lagrangian. One nice thing about that definition is that it makes it clear that it isn't general, since there clearly are Lagrangians without time translation symmetry.

 

[Popper]

I don't think that either of those quotes support your position. I think that you are reading more into them than they intended.

I disagree. But then again that's the problem with quoting a text. You only get part of the argument and not the entire arguement. You really need to read the whole thing to understand the quote I posted.

First, the Feynman quote. When he says, "It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas", that is true, but does not imply as you claim that it cannot even be defined.

I strongly disagree. That’s your personal opinion of what Feynman meant amd I strongly disagree with your interpretation. You agree that he’s saying "we have no knowledge of what energy is." and then you assert that while we have no knowledge of what energy is we can define it.

Let me give you an idea of this "energy is the ability to do work." Some texts use that to define energy while other texts use it to define potential energy. But those are different things.

To me the statement “we have no knowledge of what energy is” cannot be taken to mean that while we don’t know what it is we can define it.

In fact, he explicitly mentions "there are formulas for calculating some numerical quantity, and we add it all together it gives '28' - always the same number". If energy couldn't be defined then there wouldn't be any definite formulas for calculating it and we would never even be able to get a definite number, like 28.

That's correct. Feynman goes through a lot of effort to explain that we can define various kinds of energy he concludes that energy itself is something unknown to us and to me that means that we can't define it. If you disagree then so be it. We'll agree to disagree. But I recommend that you read the entire section of Feynman in which that's found.

This is the danger of reading only a portion of his entire section on the topic. We don’t know what energy is but that doesn’t mean that we can’t fine expressions for various forms of energy such that the total is conserved.

If energy cannot be defined then there is no way to know if it is conserved or if any of the other expressions containing E are correct.

I disagree. As French explains that while “we may not be able to define energy, that doesn’t mean that it is only a vague, qualitative idea. We have set up quantitative measures of various kinds of energy:…”

No. This certainly doesn't follow. Momentum is not equal to the amount of work that a particle has the ability to do, and even a stationary particle has the ability to do work even though it has no momentum.

I believe that you missed my point. I didn't say that energy is the amount of work done. The definition is that something that has energy can do work. Although I don't see how that can be applied to a neutrino at rest and isolated from everything else.

The statement “energy is the ability to do work” cannot be taken as a definition of energy because it does not tell you what it is or how its measured or anything about how to write down a quantitativ eexpression for it.

One cannot use the "ability to do work" to write down a quantitative expression for it. Energy by virtue of motion does not tell you that this form of energy is mv^2/2 and not mv.

In any case, there are other possible definitions of energy besides that one. E.g. the conserved quantity corresponding to time translation symmetry in the Lagrangian.

They knew what they were writing about when they made those comments. The quantity one uses in Lagrangian mechanics for energy is not a definition of energy, it is an equality. Recall that you have to be given the Lagrangian to calculate the energy but you have to know what the forms of energy in the Lagrangian are before you calculate the total energy. Heck. If you knew the various forms in the system then you don't need the lagrangian to define it. Then there's the possibility that the system does not have a time translation symmetry such as when the system is exchanging energy with its environment and that leads to a time dependant Lagrangian and thus a non-conserved energy whereas energy by definition be conserved.

Do you have the Feynman lectures? Note what he said “It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas.”

At this point I'm bowing out of this debate since I've said all that I'd want to say on this subject.

 

[Popper]

E/c² is what Newton called mass (the factor between momentum and velocity). Rest energy is equivalent to rest mass and total energy is equivalent to the so called relativistic mass.

I recommend reading the following article

On the Meaning of E = mc² by Mendel Sachs, International Journal of Theoretical Physics 8, 377-383 (1973)

I have it if you or anyone else wants to read it. I can't post much more today. Bad acid indigestion.

 

[Dale]

We don’t know what energy is but that doesn’t mean that we can’t fine expressions for various forms of energy such that the total is conserved.

You are contradicting yourself here. If you have a definite expression for energy then energy is obviously defined. At a minimum, you can take the expression as a definition for that particular system. If a quantity is not defined then a number cannot be assigned to it.

You cannot have it both ways. If, as you claim, Feynman and French intended to convey that energy was fundamentally and always completely undefinable then you can NEVER find an expression for it nor can you ever claim that it is conserved.

I urge you to reconsider your interpretation, I don't believe that it makes sense. I think that it makes much more sense to conclude that they were referring to the well-known fact that energy is not always defined in general, not that they were claiming that energy is never defined.

PS The fact is that neither of the quotes you produced explicitly state your claim "energy is one of those things that defy definition". So you are assuming that they meant something that they didn't state. Furthermore, you recognize that the way you have chosen to construe their words is not in keeping with most standard textbooks. I think that is a mistake.

 

[PeterDonis]

Mass is defined so that the quantity mv is conserved. The quantiry p = mv is defined to be the momentum of a particle.

I don't think this is correct without some qualification. It's correct if we define m to be the relativistic mass, but that's not the only possible definition for mass, and it's not even a favored one nowadays, to the best of my understanding; most physicists now mean rest mass when they use the term "mass".

 

[HomogenousCow]

Restricting ourselves to SR, is the rest energy actually meaningful in anyway?
It is not like we are working in GR, where the total energy matters nor are we working in QM where massive particles can decay into massless ones.

 

[PeterDonis]

Restricting ourselves to SR, is the rest energy actually meaningful in anyway?

Sure, it's the invariant length of the object's 4-momentum.

It is not like we are working in GR, where the total energy matters

Total energy is not the same as rest energy anyway, so I'm not sure how this is relevant. Also, the definition of rest energy that I gave above is valid in GR too.

nor are we working in QM where massive particles can decay into massless ones.

You can model this perfectly well in SR without having to go into the quantum mechanical details. In fact this is exactly what was done before the quantum mechanical details were fully understood.

 

[jtbell]

Restricting ourselves to SR, is the rest energy actually meaningful in anyway?

A particle's rest energy tells you how much energy (at most) it can "deliver" to another particle in a decay or inelastic collision process, even if it (the first particle) is at rest to begin with. Example: an electron and positron come together with negligible kinetic energy and annihilate. The outgoing photons have total energy equal to the sum of the rest energies of the electron and positron, and of course have no rest energy themselves.

 

[HomogenousCow]

I find the particle decay example given to laymen riddled with issues, first of all when we calculate these things we make the assumption that they can happen beforehand and calculate the probability using peturbation theory, this has the issue that one could dream up any manner of transition and go ahead with the calculations, this is unelegant and rather ad-hoc as we are not actually solving some initial value problem and looking at the time evolution of the system.

 

[DrStupid]

I recommend reading the following article

On the Meaning of E = mc² by Mendel Sachs, International Journal of Theoretical Physics 8, 377-383 (1973)

As I red Einstein's original article I know what it means. It is the relationship between rest energy and rest mass. But as I wrote before there is the same relationship between total energy and relativistic mass.

 

[tom.stoer]

I've already indicated that there is one equation for ...

the general case p≠0

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

Here (total) energy E and the momentum 3-vector p form a 4-vector (E,p), whereas (rest) mass m is a scalar.

Understanding this relation, and especially understanding the difference between the 0-component of a 4-vector and a scalar, is key for SR. Everything else like rest energy or relativistic mass is nothing else but an inflation of definitions; neither do they explain anything, nor do they provide additional insights.

The topic of this thread on equivalence of mass and energy shows clearly that we do not need additional definitions, but precise definitions, namely rest mass vs. relativistic mass and rest energy vs. total energy. At the end of the discussion the conclusion is that two definitions (rest mass and total energy) are sufficient and that everything else (rest energy and relativistic mass) is redundant, so my conclusion is that we should have less (but precise) definitions.Of course one can define E₀=mc², M(p)=E/c² and even p₀=mc and many other things. Everybody is allowed to do that, but then discussions are more about numerous definitions instead of physics.

 

[DrStupid]

At the end of the discussion the conclusion is that two definitions (rest mass and total energy) are sufficient and that everything else (rest energy and relativistic mass) is redundant, so my conclusion is that we should have less (but precise) definitions.

Radius and perimeter of a circle are also redundant but that does not necessarily mean that we should abolish one of them. If you want to use rest mass and total energy only you can do that. If someone else wants to use rest energy and relativistic mass he can do it as well. It's just a matter of taste.

 

[tom.stoer]

If someone else wants to use rest energy and relativistic mass he can do it as well. It's just a matter of taste.

Agreed. But many questions and confusion regarding SR a due to these two concepts, not to physics. If you use a concept that does not solve anything but create confusion you better think about the concept. Einstein an others did.

Einstein: "It is not good to introduce the concept of the mass M ... of a moving body for which no clear definition can be given. It is better to introduce no other mass concept than the ’rest mass’ m. Instead of introducing M it is better to mention the expression for the momentum and energy of a body in motion."

Taylor & Wheeler: "The concept of "relativistic mass" is subject to misunderstanding. That's why we don't use it. First, it applies the name mass - belonging to the magnitude of a 4-vector - to a very different concept, the time component of a 4-vector. Second, it makes increase of energy of an object with velocity or momentum appear to be connected with some change in internal structure of the object. In reality, the increase of energy with velocity originates not in the object but in the geometric properties of spacetime itself."

 

[Dale]

Even though you have bowed out of the debate, I thought that your comments on the Lagrangian-based definition of energy deserved specific rebuttal.

The quantity one uses in Lagrangian mechanics for energy is not a definition of energy, it is an equality.

So what? All good scientific definitions are in the form of an equality. This certainly is irrelevant in whether or not something qualifies as a definition.

Recall that you have to be given the Lagrangian to calculate the energy but you have to know what the forms of energy in the Lagrangian are before you calculate the total energy. Heck. If you knew the various forms in the system then you don't need the lagrangian to define it.

Two responses here, first, to me this contradicts your overall point. Not only is energy possible to define, it is so easy to define energy that it is the usual method to determine a Lagrangian in practice. Second, you don't have to know the forms of energy in order to obtain the Lagrangian. The Lagrangian can be determined from the equations of motion. Once you have the equations of motion (e.g. experimentally) then you work backwards to determine the Lagrangian, then you find the time translation symmetry (if any), and define that quantity to be energy.

Then there's the possibility that the system does not have a time translation symmetry such as when the system is exchanging energy with its environment and that leads to a time dependant Lagrangian and thus a non-conserved energy whereas energy by definition be conserved.

And IMO this is one of the strengths of that definition, as I mentioned above. It shows that energy is not a general concept, as I believe was intended by the quotes you posted above.

 

[Popper]

Even though you have bowed out of the debate, I thought that your comments on the Lagrangian-based definition of energy deserved specific rebuttal.

Something dawned on me just now which hadn't occurred to me earlier. Let me think it over and get back to you later on this. I bowed out because I had nothing left to say. At least at that point. However that's changed. Plus there are some new comments I'd like to touch on.