This is a complicated question and I look forward to reading others responses. In short, I think the answer is no. That equality states an equivalence/equality in a context. Four quarters equals a dollar bill. That does not mean that four quarters is a dollar bill. It means that they are equal/equivalent in a context. My other thought inevitably goes to noether's theorm (which you may not have heard of, but its kind of a big deal). In Noether's theorm a symmetry implies a conserved quantity and a conserved quantity implies a symmetry. But the conservation of mass and the conservation of energy each have the same symmetry implied by them... So with this thinking I might say they are the same. I don't think that is the case though and, as I said above, I'm looking forward to reading some responses.
In units where c=1 they do have the same units. But the full equation is m^2=E^2-p^2, so they are not generally the same. The simplified equation only applies for p=0.
I have a hard time comprehending the theory when I apply c2. Maybe a better question would be how does light apply or how is energy not mass?
Mathematical equations are not definitions. Just because you can write E=MC^{2} does not mean that E and M are the same thing. The idea underlying the math is that whenever an amount of energy equal to E enters or leaves a system, the system gains or loses mass equal to E divided by C^{2}. (Because we can rearrange the equation to M=E/C^{2})
The physics behind E=MC^2 is that mass and energy should be thought of as different sides of the same coin as implied by the post above referencing Noether's theorem. Where you see mass you will find its equivalent energy and vice versa. Depending on the context it might be easier in practice to measure one or the other, but in principle both could always be detected. One important point to note is that E=MC^2 does *not* imply, as is often stated even by many prominent physicists including, believe it or not, Einstein himself, that mass can be *converted* to energy or energy to mass. The classic example to justify this latter false belief is that of a nuclear bomb. The argument is made that the resulting mass of the elements after an explosion is less than that of the elements prior to the explosion. But while the fact of the argument is true, the argument itself is flawed. The reason is that in order to judge whether the mass has changed you need a closed system. The proper thought experiment, then, requires putting a nuclear bomb in a closed (nuclear bomb proof :) ) container. First, measure the mass of the closed system, next explode the bomb, then measure the mass of the closed system again. The mass will be the same. This is exactly what E=MC^2 tells us. In other words, with the bomb exploding within this "black box" we don't "see" all the photons or kinetic energy of the explosion like we would in an open air test; we instead contain these products within a closed system which allows us to measure the mass side of the coin of these constituents along with the resulting elements. When we do this the mass of the system doesn't change, precisely because E=MC^2 tells us mass and energy are two sides of the same coin. When we black-box the bomb, we just see the mass side of the coin. If E=MC^2 told us mass was converted to energy, then the mass of the black box would be less after the explosion.
BTW, in E=MC^2, C^2 is just a constant of proportionality. In other words, one way to look at the equation is to say that the energy side of the coin is proportional to the mass side. This "demystifies" the equation a bit and may illuminate its meaning more clearly. An aside to this is that C is a universal constant associated with spacetime and in a real sense "independent" of light. First and foremost it is the upper limit that any energy or information can travel through space-- photons or otherwise. It just happens to be that light is the easiest way for us to measure this constant, and so this upper limit got tagged early with this moniker. The point here is that its this upper speed limit that is important and not the speed that light itself travels. One way to understand the importance of this distinction is that photons could conceivably have a very tiny rest mass that has yet been undetected and would keep them from traveling at this upper limit; they might be traveling just a tiny bit slower. With this in mind, then, E=MC^2 says that energy and mass are two sides of the same coin and can be determined from each other via a simple constant of proportionality. This constant of proportionality is the square of this universal constant, C-- a property of spacetime. Maybe this doesn't demystify as much as it provides a different way to think about the equation, and decouples the equation with a specific association with the phenomena of light itself. It's time and space with which it is coupled-- the playing field of mass and energy.
Nicely put doesn't make it true. Wherever you find four quarters, you don't find a dollar. Wherever you find mass, you will always find energy, and vice versa.
You do in the USA and when "quarters" is taken to mean 25-cent pieces and "a dollar" is taken to mean a piece of paper that represents 100 cents (and these ARE the normal meaning of those terms in the USA).
I hope somebody can chime in about noether's theorem! From what I remember, noether's theorem states that a symmetry implies one and only one conserved quantity. Also, mass is considered to be conserved in a closed system (contrary to some claims about mass turning into energy). But conservation of mass does not have a unique symmetry... I've read it claimed (i dont recall where) that the symmetry is the time symmetry just like energy... But these statements seem to be in contradiction. We cannot have mass be conserved separately from energy, have a symmetry imply only one conserved quantity and have both energy and mass conservation each associated with the same symmetry.
If x is conserved then any f(x) is also conserved. Mass is a function of E and p and since E and p are conserved then m is conserved also.
Thx for that. It makes sense. But... The mass that is a function of E and p is not the invariant mass is it? Isn't the invariant mass that which is conserved? edit - Also, so the symmetry associated with mass conservation would be both time and translational symmetry then?
Yes, it is the invariant mass: ##m^2 c^2 = E^2/c^2 - p^2## E and p are components of the four-momentum, and m is its norm (neglecting factors of c). Conservation of the four-momentum implies conservation of E, p, and m, but only m is invariant. The other terms are conserved but not invariant. Conservation of the four-momentum is due to space-time translational symmetry (translation in space and time) per Noether's theorem.
That equation is wrong, in general, as it's been wrote to you. No. But a system's mass could be defined as "the energy of that system in a frame of reference where its momentum p is zero". As you see, they are not exactly the same thing, even if there is a relationship between them. Note that the difference is not only "formal": a photon has zero mass but has non-zero energy, for example. -- lightarrow
So this is what I think all this means. When calculating the energy of rest mass you times the mass by C, but you also have to times mass by P and the P would be the same as C so squaring C in e=mc2 doesn't make the final calc higher than it should be. Or mc^2 times mp^2
No. In the full form of Einstein's equation you have e^{2}=m^{2}c^{4}+p^{2}c^{2}. For rest mass, the object isn't moving, so p^{2}c^{2} equals zero, leaving us with E^{2}=m^{2}c^{4}, which simplifies to e=mc^{2}. Mass is never multiplied by momentum.