Ed: The Significance of C in E=mc2 Explained

[Mileman10]

Basic Question:

Of the infinite number of other values which could have been the multiplier in E=mc2, it surely cannot be a coincidence that the value of the speed of light squared was the number. So why c?

Ted

 

[Dale]

Hi Mileman10, welcome to PF!

What other combination of dimensionful universal constants could you use to get dimensions of L²/T²?

 

[Mileman10]

Thanks for the reply, but to me, your answer just begs the question.

And, (big disclosure here), I'm just an interested novice, trying to understand things conceptually, if that's possible.

In my college years, I studied Greek at Oxford, and was privileged to play chess once with Paul Dirac. (He won.) He explained basic stellar evolution in an hour, and was so clear and simple even I could get it.

That's what I was hoping for here. So, please, if possible, keep it basic.

Thanks,

Ted

 

[Doc Al]

Thanks for the reply, but to me, your answer just begs the question.

Why do you say that? I think it was a reasonable answer.

Do you realize that the speed of light plays a special role in relativity? Assuming you do, is it so surprising that c appears in many of the equations of relativity, including this one?

 

[Dale]

OK, let me try again.

We are looking for a universal factor which converts a mass into an energy. So just by looking at the units we know that the factor that we are looking for must have units of L²/T². So we look at the universal constants (e.g. c, G, h, etc.) and see how we can compose a factor with the right units of L²/T². There are only a small number of such constants so we quickly find that the only way to do so is to use c².

So the c² should be completely clear just from dimensional considerations. What is not obvious is why it is 1c² instead of 0.5c² or something else. But for some reason nobody ever asks about the subtle question and everyone focuses on the obvious question.

 

[Mileman10]

I appreciate your explanations, but on a very basic level, c is just a very large number, and c squared even more so.

Forgetting desired units for the moment, this large number is most often used in simple discussions of the power of atomic fusion to explain how a pea-sized amount of Uranium could level Hiroshima, for example. The potential energy in even tiny amounts of matter is enormous, by a factor of c2.

Please explain, why it is that particular factor, and not a google, for example.

Ted

 

[Dale]

I appreciate your explanations, but on a very basic level, c is just a very large number, and c squared even more so.

No, c is not just a very large number. In fact, when doing relativity most of the time we use units where c=1. The actual numerical value of c is not very relevant and is completely arbitrary since we can set it to any number we wish simply by choosing our units appropriately.

The important thing about c is not its numerical value, but its units, and the fact that it is frame invariant, finite, and isotropic.

 

[Mileman10]

OK, I think I'm beginning to follow, a little.

So is it still accurate to say that "Energy equals mass times the speed of light squared", or is this technically incorrect?

 

[Dale]

Yes, that is accurate.

 

[Mileman10]

Yes, that is accurate.

Thank you, sir, for your patient help.

Ted

 

[K^2]

Again, one has to be very careful about rest mass vs inertial mass. E = mc² holds if m is the inertial mass, but the symbol m is rarely used for it in relativity. For object in motion, a better equation is E² = p²c² + (mc²)². Here, m is the rest mass of the object, and p is momentum of the object. Note that this equation gives correct energy even for massless particles, such as light.

 

[ChrisJackson]

Basic Question:

Of the infinite number of other values which could have been the multiplier in E=mc2, it surely cannot be a coincidence that the value of the speed of light squared was the number. So why c?

Ted

Piggybacking off Ted's question, why not just say that the equation is E=ML? [ Where L = kinetic energy in light MV^2 with no M]

The equation E= MC^2 is solving for total energy in a quantity of matter and since light is weightless, Kinetic energy in light is simply its velocity, no? What is the difference? Are we really just parsing terms?

The implication is that matter isn’t simply energy like Einstein said. Matter is a specific type of energy, its just light in another form. It is bound up and tied in loops or knots or whatever. I can see why Einstein might not want to make that claim. It sounds outrageous and maybe a little unscientific. Is that the reason we stick to C^2 or am I off base?

How much energy is in one Kilogram of matter?
E=MC^2 => E =C^2 => E= the amount of energy in light

Summary: The total amount of energy in a particular bit of matter is just a question of how much light is wrapped up in the matter.. The answer is the quantity of matter multiplied by energy of light (C^2)

Is there something I am missing here? If light were the basic building block of matter, it would come as no surprise that C^2 keeps popping up all over the place.

 

[Matterwave]

What's L in that equation of yours?

You are getting a lot of the basics of physics wrong. Try reading some faqs or something. The equation E=mc^2 does not read like "energy is equal to matter times light squared". A statement such as that makes no sense.

 

[Drakkith]

Piggybacking off Ted's question, why not just say that the equation is E=ML? The equation is solving for total energy in a quantity of matter and since light is weightless, its energy reduces to its velocity, no? What is the difference? Are we really just parsing terms?

We don't make a unit equal to C^2 because the value C is a constant and appears in more than one place. It makes it much easier to use it squared or whatnot instead of making a new term in a couple of equations.

Also, E=MC^2 is not useful for light as it has no mass. E² = p²c² + (mc²)² is actually the full equation. We simplify it to E=MC^2 to convert the invariant mass of an object to energy. In the full equation you would still have to square whatever you set C^2 equal to, so we can just use C and square it and it will be easier.

 

[Agerhell]

I think that Mileman10 comes up with a very good question that deserves a better answer. Perhaps someone has a link to the original paper where Einstein is supposed to have come up with this relation?

Regarding the energy containment in a kilogram of matter it is also dependent on the gravitational field. A one kilogram steel ball at rest placed very close to the Schwarzschild radius of a black hole should contain a lot less energy than the same steel ball placed on Earth right?

 

[ChrisJackson]

What's L in that equation of yours?

You are getting a lot of the basics of physics wrong. Try reading some faqs or something. The equation E=mc^2 does not read like "energy is equal to matter times light squared". A statement such as that makes no sense.

Thanks for the response. I didn't say "energy is equal to matter times light squared". I said that it seems like E=MC^2 is similar to Energy = (the amount of matter) ( the amount of energy in light) and that the amount of energy in light is equal to the kinetic energy of light. I guess it would be fair to say that the kinetic energy of light is 0 since Ke = mv^2 would be a large number multiplied by 0.

 

[Matterwave]

c^2 is by no means some measure of "amount of energy in light". The energy of a single photon is given by E=hf where f is the frequency and h is the Planck's constant. It is also E=pc. c^2 doesn't even have units of energy.

All of light's energy is kinetic. It has no rest energy.

 

[questionpost]

Why does it use c? Is it because at that speed matter would have to be equivalent to energy in order to travel that speed?

 

[Matterwave]

Why does it use c? Is it because at that speed matter would have to be equivalent to energy in order to travel that speed?

Matter (with mass) can never move at c. The speed of light, c, is simply a given constant in the theory of relativity. We assume a "maximum speed of information transmission", and call that c. This constant pops up obviously because we have used it as our main axiom. What value you give c depends purely on the choice of units. We just have to posit that such a maximum exists.

Why does such a maximum exist? Because the experiments show that it does. As far as special relativity is concerned, this is a postulate that requires experimental evidence, but is not proved by the theory itself. No theory can prove it's own axioms. That would be tautological.

 

[ChrisJackson]

We don't make a unit equal to C^2 because the value C is a constant and appears in more than one place. It makes it much easier to use it squared or whatnot instead of making a new term in a couple of equations.

Also, E=MC^2 is not useful for light as it has no mass. E² = p²c² + (mc²)² is actually the full equation. We simplify it to E=MC^2 to convert the invariant mass of an object to energy. In the full equation you would still have to square whatever you set C^2 equal to, so we can just use C and square it and it will be easier.

Understood.. E² = p²c² + (mc²)² adjusts for relative inertia of a mass. For the initial intent E=MC^2 was fine since the momentum of the matter in question wasn't significant.

 

[ChrisJackson]

We don't make a unit equal to C^2 because the value C is a constant and appears in more than one place. It makes it much easier to use it squared or whatnot instead of making a new term in a couple of equations.

Also, E=MC^2 is not useful for light as it has no mass. E² = p²c² + (mc²)² is actually the full equation. We simplify it to E=MC^2 to convert the invariant mass of an object to energy. In the full equation you would still have to square whatever you set C^2 equal to, so we can just use C and square it and it will be easier.

c^2 is by no means some measure of "amount of energy in light". The energy of a single photon is given by E=hf where f is the frequency and h is the Planck's constant. It is also E=pc. c^2 doesn't even have units of energy.

All of light's energy is kinetic. It has no rest energy.

E=hf, this is the answer I was looking for.. thanks

 

[Agerhell]

Matter (with mass) can never move at c. The speed of light, c, is simply a given constant in the theory of relativity. We assume a "maximum speed of information transmission", and call that c. This constant pops up obviously because we have used it as our main axiom. What value you give c depends purely on the choice of units. We just have to posit that such a maximum exists.

Why does such a maximum exist? Because the experiments show that it does. As far as special relativity is concerned, this is a postulate that requires experimental evidence, but is not proved by the theory itself. No theory can prove it's own axioms. That would be tautological.

Now do we really take E=mc² as an axiom or is it possible to derive that relation from the assumption that the energy of a moving object varies with the Lorentz factor? Or from some other assumption?

 

[Drakkith]

Now do we really take E=mc² as an axiom or is it possible to derive that relation from the assumption that the energy of a moving object varies with the Lorentz factor? Or from some other assumption?

Take an electron and annihilate it with a positron and measure the energy of the gamma rays produced. The energy of the photons will equal the rest mass of the two particles + any kinetic energy they had. It can be experimentally verified and has been. The Large Hadron Collider has measured trillions of particle collisions in the last few years, each one is 100% dependent on the validity of the equation. If that isn't verification, then I don't know what is.

 

[Drakkith]

Regarding the energy containment in a kilogram of matter it is also dependent on the gravitational field. A one kilogram steel ball at rest placed very close to the Schwarzschild radius of a black hole should contain a lot less energy than the same steel ball placed on Earth right?

Yes and no. It depends on where you measure each one from. Measuring the energy of each from a local frame near each ball will give you the same energy for each. However if you measure the energy of the ball on Earth from the frame of the black hole, it has far more energy. For example, take the ball and annihilate it with another ball made of antimatter. If you measure the energy of the emitted photons from a frame near the black hole the photons will be extremely blue shifted and have far more energy than they would if you were measuring the event on Earth. The opposite would happen if you did the same thing near the black hole and measured it on Earth.

In it's own frame an objects mass (and therefore the energy you can convert the mass into) does not change no matter the gravitational field it's in.

 

[Agerhell]

Take an electron and annihilate it with a positron and measure the energy of the gamma rays produced. The energy of the photons will equal the rest mass of the two particles + any kinetic energy they had. It can be experimentally verified and has been. The Large Hadron Collider has measured trillions of particle collisions in the last few years, each one is 100% dependent on the validity of the equation. If that isn't verification, then I don't know what is.

Yes absolutely, it is an empirically well proven fact. That is not the question here though, the question is if this fact can be derived from some underlying assumption.

 

[Drakkith]

Yes absolutely, it is an empirically well proven fact. That is not the question here though, the question is if this fact can be derived from some underlying assumption.

Perhaps this will help: http://www.btinternet.com/~j.doyle/SR/Emc2/Deriving.htm

 

[Matterwave]

I didn't say E=mc^2 was the axiom, this is a relation that can be derived from the theory. The axiom is that c=constant is some kind of "universal speed limit".

 

[Agerhell]

Perhaps this will help: http://www.btinternet.com/~j.doyle/SR/Emc2/Deriving.htm

Thank you. I believe that link provides the correct answer to the question Mileman10 posed.

 

[nitsuj]

The equation E=mc^2 does not read like "energy is equal to matter times light squared". A statement such as that makes no sense.

Maybe I'm math illiterate, how else could that equation be "read"?

energy is equivalent to matter.

So why does that statement make no sense?

I see E=mc2 as another example of the symmetry in classical physics, a bit of a truism.

 

[nitsuj]

As far as special relativity is concerned, this is a postulate that requires experimental evidence, but is not proved by the theory itself. No theory can prove it's own axioms. That would be tautological.

Is the entire theory not a mathematical proof of the postulate for c? I don't know math, but it seems like the postulate for c is a bit of a keystone to the theory.

Of course this is different than experimental proof, which you must not have been implying; since a theory is not an experiment.

 

[Dale]

Piggybacking off Ted's question, why not just say that the equation is E=ML? [ Where L = kinetic energy in light MV^2 with no M]

Your units don't work out, E is in units of Joules, M is in units of kg, and if L is kinetic energy then L is in units of Joules. So for units you have J = kg J, which doesn't match.

 

[Dale]

Is the entire theory not a mathematical proof of the postulate for c? I don't know math, but it seems like the postulate for c is a bit of a keystone to the theory.

Of course it is a keystone, but it is a key assumption, not a key proof. In SR there is no proof of the postulate for c, it simply assumes it and goes from there.

 

[Drakkith]

Maybe I'm math illiterate, how else could that equation be "read"?

energy is equivalent to matter.

So why does that statement make no sense?

I see E=mc2 as another example of the symmetry in classical physics, a bit of a truism.

Mathematically the equation simply means that the energy in joules is equal to the mass of something in kilograms times the speed of light squared. It doesn't mean energy is equivalent to matter, it's just a formula to convert mass to energy. Given the definitions of energy and matter the two cannot be "equivalent", as they are totally different things.

 

[Matterwave]

Maybe I'm math illiterate, how else could that equation be "read"?

energy is equivalent to matter.

So why does that statement make no sense?

I see E=mc2 as another example of the symmetry in classical physics, a bit of a truism.

A statement like the one I made makes no sense at all (you can go back and read it and tell me if you think it makes any sense). The equation is a mathematical equation, and not a verbal one. What I was trying to get at with my statement, was that I see a whole lot of people posting things like "Does E=mc2 mean that if matter moves at the speed of light, it turns into energy?" or "does E=mc2 mean that energy is just matter moving at the speed of light squared?" or other non-nonsensical questions like this.

Like Drakkith says, E=mc^2 is simply a way of knowing how much energy is contained within one kilogram of mass at rest.

Also, like Dalespam (lol, I almost wrote that as Dalesperm) says, Special relativity takes the constant c as a postulate. The proof of the postulate can either come from direct tests of the postulate (e.g. the Michelson-Morley experiment), or from testing SR itself (and thereby testing both of its postulates indirectly at the same time), such as with time dilation experiments or particle beam experiments.

 

[nitsuj]

Mathematically the equation simply means that the energy in joules is equal to the mass of something in kilograms times the speed of light squared. It doesn't mean energy is equivalent to matter, it's just a formula to convert mass to energy.

Given the definitions of energy and matter the two cannot be "equivalent", as they are totally different things.

Just a formula? it describes what's observed in physics.

Yes there is a relationship between the two, yes they are "totally different things". The fact they are two "totally different things" and the fact the two are "interchangeable" according to c^2 is a pretty remarkable equivalence.

As if I have to quote wiki for more authority lol here it is
[/PLAIN]
"In this concept, mass is a property of all energy, and energy is a property of all mass, and the two properties are connected by a constant"

I like physics, any definition of mass & energy that excludes mentioning the equivalence between the two is an incomplete definition in my opinion, and would absolutely be two "incomplete" "concepts" from a physics perspective.

 

[nitsuj]

A statement like the one I made makes no sense at all (you can go back and read it and tell me if you think it makes any sense). The equation is a mathematical equation, and not a verbal one.

What I was trying to get at with my statement, was that I see a whole lot of people posting things like "Does E=mc2 mean that if matter moves at the speed of light, it turns into energy?" or "does E=mc2 mean that energy is just matter moving at the speed of light squared?" or other non-nonsensical questions like this.

Like Drakkith says, E=mc^2 is simply a way of knowing how much energy is contained within one kilogram of mass at rest.

Also, like Dalespam (lol, I almost wrote that as Dalesperm) says, Special relativity takes the constant c as a postulate. The proof of the postulate can either come from direct tests of the postulate (e.g. the Michelson-Morley experiment), or from testing SR itself (and thereby testing both of its postulates indirectly at the same time), such as with time dilation experiments or particle beam experiments.

Isn't there entire books written on e=mc^2.

I don't disagree with you or Dalespam regarding the theory not proving the postulate of c experimentally. You kinda raised my point, that because c is a "keystone" to the theory, it can be tested indirectly, if results match predictions, pretty safe to start having confidence in the postulate. So much so most scoff at contradictory results, cerntainly there are even recent examples.

Ah, any test of SR is a test of the postulate, right?

 

[nitsuj]

A statement like the one I made makes no sense at all (you can go back and read it and tell me if you think it makes any sense).

Wait, no pun intended but is; equivalence = equivalent = equal = "=" not true?

Here is the comment and I re read it and yea I still makes sense of it.

"energy is equal to matter times light squared"

Rephrased to "there is an equivalence between energy and mass by a factor of c^2." anymore palatable?

 

[Matterwave]

You can't use an arithmetic operation (multiplication) on non-arithmetic objects (physical objects like matter and light).

You can only use them on numbers. Quantities. Mass, measured in some units, and the speed of light, measured in compatible units.

If you want to interpret my sentence semantically as a mathematical statement, you can, but it just leads to needless confusion, and you'll get people asking the type of nonsensical questions I raised in my last post.

 

[nitsuj]

You can't use an arithmetic operation (multiplication) on non-arithmetic objects (physical objects like matter and light).

You can only use them on numbers. Quantities. Mass, measured in some units, and the speed of light, measured in compatible units.

If you want to interpret my sentence semantically as a mathematical statement, you can, but it just leads to needless confusion, and you'll get people asking the type of nonsensical questions I raised in my last post.

I disagree, understanding the concepts is what is important in discussing (understanding) physics, and in turn the math. Otherwise it's just numbers and arithmetic operators.

 

[Drakkith]

Just a formula? it describes what's observed in physics.

Yes there is a relationship between the two, yes they are "totally different things". The fact they are two "totally different things" and the fact the two are "interchangeable" according to c^2 is a pretty remarkable equivalence.

As if I have to quote wiki for more authority lol here it is
[/PLAIN]
"In this concept, mass is a property of all energy, and energy is a property of all mass, and the two properties are connected by a constant"

I like physics, any definition of mass & energy that excludes mentioning the equivalence between the two is an incomplete definition in my opinion, and would absolutely be two "incomplete" "concepts" from a physics perspective.

You have been using "Matter" in place of "Mass" in your posts. Perhaps this is the source of the confusion. My post you quoted was specifically talking about matter not being equal to energy, as it is not. E=MC^2 requires mass, not matter.

Here is the comment and I re read it and yea I still makes sense of it.

"energy is equal to matter times light squared"

See, here it is again. Please use Mass instead or you will confuse people.

 

[meBigGuy]

Seems like everyone here missed the whole point of the OP. What is it about the speed of light that make *IT* (well, IT squared, anyway) the constant for the relationship between mass and energy? Does it stem from the basic postulate that the speed of light is a universal maximum, a constant, and is independent of the reference frame?

 

[Drakkith]

Seems like everyone here missed the whole point of the OP. What is it about the speed of light that make *IT* (well, IT squared, anyway) the constant for the relationship between mass and energy? Does it stem from the basic postulate that the speed of light is a universal maximum, a constant, and is independent of the reference frame?

See post #26.

 

[nitsuj]

You have been using "Matter" in place of "Mass" in your posts. Perhaps this is the source of the confusion. My post you quoted was specifically talking about matter not being equal to energy, as it is not. E=MC^2 requires mass, not matter.
See, here it is again. Please use Mass instead or you will confuse people.

I had quoted someone else there, but it was in defense of the comment.

So yes I did read Matter as Mass, drakkith, you are absolutely right. Sorry for my confusion. If I had realized that, we wouldn't have had this back 'n forth.

And thanks for your attention.

I agree with you guys that matter is not equivalent to energy...or mass, for that matter.

 

[Naty1]

Mileman:

So why c?

no intuitive logic would necessarily reveal this value until you use Einstein's relativity as a model. Then it flows as in the post #26 derivation...so we start with a theory and many results flow from it. We confirm which of those match experimental observations...and if they do, voila we have a viable theory. If not, we have to change our theory...so 'ether' was a start, but 'relativity' better fit observations...so we use that.

It's akin to asking..."Why are Planck time, [or energy and length] what they are?"

Wikipedia says it this way:

The Planck time is the unique combination of the gravitational constant G, the relativity constant c, and the quantum constant h, to produce a constant with units of time.

Without such a framework, you could go on guessing from among an infinite variety of values for a long time. In fact it's quite likely that such a quantized view of things would never even be thought about without some experimental observations combined with some theory. It sure would not flow easily from relativity.

If you subscribe to the big bang model, apparently all these apparently disparate pieces [light,mass,energy,space,time,etc] somehow originated from a unified origin...So far we have a bunch of piecepart models that reveal some things we observe around us...but not all, not what caused the bang, for example. Hence dark matter and dark energy, for example, haven't unambiguously appeared in any 'logical theory'...we just discovered them experimentally in the 1990's! And we haven't yet figured out how relativity and quantum mechanics can be 'unified'...so pieces appear to be missing.

 

[Dale]

What is it about the speed of light that make *IT* (well, IT squared, anyway) the constant for the relationship between mass and energy?

What else could it be? There is no other way to get a constant with units of L^2/T^2 from any of the other universal constants. So it has to be some multiple of c^2. The only other possibility would be for there to be no conversion between mass and energy.

 

[harrylin]

Seems like everyone here missed the whole point of the OP. What is it about the speed of light that make *IT* (well, IT squared, anyway) the constant for the relationship between mass and energy? Does it stem from the basic postulate that the speed of light is a universal maximum, a constant, and is independent of the reference frame?

It may be related: c is a property of space, and thus c² the same.
Somehow, the inertial property of energy is determined by that constant - and it would be great if someone physically (not just mathematically) would understand the "somehow".

 

[-Physician]

That formula is wrong anyway, Einstein wasn't that smart than sciencists today, even though if Einstein wouldn't make this mistake sciencists wouldn't have work to do. So:
$E≠mc^2$,
$E^2=m^2 c^4 + p^2 c^2$.

 

[harrylin]

That formula is wrong anyway, Einstein wasn't that smart than sciencists today, even though if Einstein wouldn't make this mistake sciencists wouldn't have work to do. So:
$E≠mc^2$,
$E^2=m^2 c^4 + p^2 c^2$.

Formula's are never on themselves, they relate to certain conditions and definitions. Both equations are correct for their application.

 

[-Physician]

Formula's are never on themselves, they relate to certain conditions and definitions. Both equations are correct for their application.

Einstein defined:
The velocity of light is the fastest one. His formula says: $E=mc^2$, but that theory gone down, so that formula wouldn't stay for any application, if it would then it would be wrong.

 

[Doc Al]

That formula is wrong anyway, Einstein wasn't that smart than sciencists today, even though if Einstein wouldn't make this mistake sciencists wouldn't have work to do. So:
$E≠mc^2$,
$E^2=m^2 c^4 + p^2 c^2$.

Both equations are due to Einstein. The first reflects the rest energy of a massive particle. They are hardly 'wrong'.