Derived in ten minutes while I was on the toilet:

Edit: 11-16-16

There was a slight error in the set-up of the center of mass calculation. Light appears to move an effective mass from -L/2 to L/2-l, the starting and stopping points of light. Here’s the reformulation to capture that.

Time spent on this derivation: ten minutes on the toilet last night and ten minutes before breakfast writing the correction (yeah, that’s what I get for going really fast).

Don’t thank me, this is Einstein’s calculation.

Edit: 11-18-17

As I’m still thinking about this post, I figure it might be beneficial to flesh out the reason that it was written. This post was used as a response to a comment on another blog… if you want to back up a statement you made about something and somebody is accusing you of not providing evidence, most people provide citations, net links, references, etc. In this particular case, the argument was about a piece of math and I was being accused of lying about said piece of math by someone who clearly likes to believe he knows everything without actually knowing practically anything. Yes, skeptics are guilty of Dunning-Kruger, just like everyone else (This is an unfair statement and I apologize for it.) What better way to slam the textbook in someone’s face than to actually work the problem? If you want the final word on what Einstein said about something, quote Einstein’s work! And so, a piece of Einstein’s work is posted above.

The argument in question started with a fellow suggesting to me that mass-energy equivalence can be derived but not proven with classical physics. I beg to differ; energy is a classical concept, from Thermo, E&M and classical mechanics… all three! You don’t need relativity or quantum mechanics to justify statements about how energy works; measurements of kinematics and force are sufficient to show that energy as a concept works. Mass-energy equivalence arose from Einstein’s notion that the newly completed *classical* field of electromagnetism must be consistent with the older fields of classical mechanics. The equation E=pc is not relativistic: it came directly out of electromagnetism (and, believe me, I’ve been through that calculation too because I didn’t believe it at first.) Imposing that these two fields must be cross-consistent is the origin of mass-energy equivalence…. light carries momentum (by Poynting’s vector and well defined in the electromagnetic stress-energy tensor) and light interacts with mass, therefore conservation of momentum (and consequently conservation of center of mass in absence of external forces) requires that light carry an equivalent of mass in order for forces to add up in a situation where light interacts with matter but no forces interact externally on the system comprised by the light and the matter. Mass-energy equivalence is required by this, no ifs, ands, buts or “yeah, but you didn’t proves…”

Einstein’s thought experiment validating this set-up is an exceptionally elegant one. It’s called “Einstein’s Box.” Everybody loves Schrodinger’s cat-in-a-box… well, Einstein had a box too and this box is older than Schrodinger’s. Einstein’s box is a closed box sitting out in space where it feels no external forces. A flash of light is emitted inside the box from one wall and travels across the box to strike the opposite wall. E&M states that light must carry momentum. If the system has no external forces acting on it, the emission of the light inside the box requires that momentum of the system be conserved, which requires that the box recoils with a momentum equal to that carried by the light, causing the box to move at some velocity consistent with the momentum carried by the light (which turns out to be directly proportional to the energy carried by that light as stated by E=pc). Net momentum of the system must remain zero by conservation of momentum. When the light travels across the box and collides with the opposite wall, the momentum of the light cancels the momentum of the box and the box stops moving. Thing about this is that center of mass, as a consequence of momentum conservation, could not have moved. No forces on the outside of the box.

Center of mass is a damn well classical concept, well worked out in the 1700s and 1800s… and since the box moved, the box’s center of mass moved! But no forces acted on the outside of the system, so the overall center of mass of the system could not have moved. This requires the light to have carried with it a value of mass, taken from the location where the light was emitted and deposited again at the location where the light was absorbed. But light is known not to carry mass since it is a wave-like solution of immaterial fields in the form of Maxwell’s equations. If you set up this situation and work through the calculation, Newtonian mechanics and electromagnetism –nothing more–, this turns out the *classical* requirement that energy and mass have an equivalence in the form of E=mc^2. No quantization or probability relations from quantum mechanics, no frame of reference shifting from relativity, not even delineating that light is some package of photons… this is purely classical. Moreover, energy is a tabular result to begin with: it is not something that is by itself ever *directly* observed and it must always be carried by something else, a field, a heat, a potential, a motion or what have you. The statement that this tabular relationship extends to something else that is technically only indirectly observed, mass, is a proof. And yes, mass is indirect since you can only know mass from weight, which is a force!

If you have concepts of weight, momentum and light together in the same model as expressed by classical physics, mass-energy equivalence is required for self-consistency.

Granted, special relativity quite naturally produces this result as well, but special relativity is not required to produce mass-energy equivalence. Had Einstein not discovered it, someone else damned well would’ve and it would not have required relativity to do –at all!

Now, the thing that doubly made me angry about this conversation is that it was with a fellow who absolutely craved physicist street cred: he name dropped Arxiv and seemed to want to chase around details. Sadly, his whole argument ultimately amounted to insulting someone and not backing up his ability to absolutely *know* what he was claiming to know. Does it matter that you don’t believe my statement if you aren’t competent to evaluate the field in question? Not at all: such a person has no place at the table to start with. This is why it’s possible for a Nobel Laureate to descend in to crankery… just because you have a big prize doesn’t mean you are always equally competent at everything! I’m guessing the guy was a surgeon given the ego and the blog, but if he was a physicist, I’m very disappointed. A physicist who doesn’t know Einstein’s box is a travesty. I’m not the greatest physicist that ever lived, but I work at it and I know what I’m talking about… where I have gaps, I do my best to admit it.

edit: 11-18-17

(Statement redacted. It was an unfairly insulting comment)

edit: 11-20-17

As this is still nagging at me, one further thought. What I consider the last statement of the conversation before it simply became obvious trolling, the fellow accused me of not including “a variable speed for light” in my calculation. In Einstein’s calculation, the speed of light is given the constant “c.” This is a constant which comes with a caveat; “c” is the speed of light when it is not passing through anything material, the speed of light in a vacuum. This distinction is important because light can travel at lower speeds when it’s passing through a material. This situation is well-handled by E&M and is considered a “solved effect” by Relativity, whose postulates include the explicit notion that E&M simply be true everywhere. The constant “c” is the maximum possible speed that light can travel, but it will travel at lower speeds in a medium with an index of refraction greater than 1, where permittivity and permeability might have values other than their vacuum values, which has the wonderful result of making lenses possible in glasses and microscopes. In my lavatory derivation above, a little screw up on my part is that I didn’t clobber the reader over the head with the constancy of the value “c,” I said “a box in zero gravity” and I said light travels at “c,” but I didn’t say “this is definitely all in a vacuum” which I probably should have. If index of refraction is “n”… the velocity of light in a medium with that refractive index is v = c/n. There are other ways to encode refractive index which allow for more sophisticated optical behaviors, but everything in that line is completely out of the pail for the argument in question, and drawing attention to it is simply chaff intended to shift the focus of the argument.

Light can travel at speeds lower than “c,” but “c” itself is so far found to be invariant. Moreover, the fact that light can travel at speeds other than “c” does not change the Einstein’s box derivation, which is set in explicit conditions where light would travel at “c.” Somebody who doesn’t know this isn’t a physicist (11-20-17: I’ll moderate this it’s unfair and was too angry.)

Also, as an aside, I mention above that Special Relativity can produce E = mc^2. Thinking about it, but not running through the calculations, I think this is actually backward; E=mc^2 is sort of needed first before it shows up in Special Rel. Einstein made some amazing leaps.

Edit: 11-20-17

As an added extra, here is a derivation of E=pc from the stress-energy and electromagnetic power continuity equations. These were written a few years ago, but I had the good sense to scan them:

The E=pc derivation begins on the second page above. The first page is the end of the continuity equation derivation. I’ll neglect that. No relativity here, just pure E&M. There are a couple pieces in here that I don’t remember so well and I need to think about to decide if they’re correct. The first page is included to show clearly the relation between force and the stress-energy tensor divergence.

Edit: 11-21-17

I’ve spent some time thinking about the form Narad put forward in the comments.

First of all, we have to be really sure of what is meant by “p” on the left side of this equation. My first reading of it was as “momentum,” but I’m realizing that it isn’t, and this may be leading to some misunderstandings about what is meant by E=pc. The thing in the middle is average poynting vector divided by speed of light… Poynting vector has units of Watts/meter^2 and speed of light has units of meters/sec, which works out to Newtons/m^2, or force per area, which is pressure, not momentum. The thing on the right is actually in units of energy… permittivity times peak E-field^2 over 2, which is just a form of electromagnetic energy, in units of Joules. For a literal reading of the equation above, unit analysis put me at momentum = pressure = energy, which is not right (apple can’t equal orange can’t equal pear). If I take “p” as pressure rather than momentum, the left side makes sense, but the right side still doesn’t quite work.

It’s a nice try, all the elements are there. It has energy and momentum can be massaged out of it. I think the route being taken here is to try to use the form of a plane wave to figure out the momentum based on the pressure and specifically for a plane wave form of the poynting vector, or else the peak E-field intensity wouldn’t be needed.

The approach in the E=pc derivation I posted above is *really* different. My starting point is with a classical structure called the Electromagnetic stress-energy tensor and with a second structure which is conservation of power given energy flux. (Wikipedia actually kind of pissed me off about this: they want to masturbate over the four-dimensional relativistic version, but wouldn’t provide me a clean on-line citation for the classical version shown above; the form given here is the same as it appears in Jackson E&M) The first equation is a consequence of the Lorentz force law (F = qE+qvxB) where the system has electromagnetic waves, but is sealed so that there is no net force… the equation says that the change in Poynting vector per unit time is equal to the divergence of the electromagnetic stress-energy tensor, all of which is in units of force or change of momentum with time. The second equation is a consequence of Power=current*voltage, believe it or not, and just says that the change in energy density in the system is equal to the divergence of the Poynting vector, all in units of power. These structures make no real initial assumptions about the form that the electromagnetic fields are taking, they speak only of change of momentum per time and change of energy density given energy flux and are derived directly from application of Maxwell’s laws.

The first step is to take the stress-energy continuity relation and to hold it as change in Poynting vector with time is equal to change in momentum density with time by direct application of Newtonian force. You end up with an expression that says that Poynting vector is equal to momentum density times speed of light squared.

The second step is to throw this Poynting vector relation into the power equation so that you get a relation that says that the momentum flux out of a volume of space is equal to the change of energy density with time. This gives you a “momentum current vector” equation, which is analogous to the relationship between electrical current “I” and current vector “J.”

I next establish a momentum current, basically just a beam of light with no specific frequency or field configuration. You could write this as white light in a Fourier composition. A set of very simple manipulations gets you to a relation that directly says that energy density is equal to momentum density times speed of light. Integrate out the density and you get E=pc directly. Please note, this set-up is explicitly agnostic on the idea of photons since it depends on a mixture of frequencies to produce a constant envelope of plane waves with constant momentum density distributed everywhere and therefore does not require quantum mechanics to work. I can’t claim this work is Einstein’s because I didn’t follow anyone to make it… this is me using Jacksonian E&M technique to prove E=pc for myself, all using classical physics.

With E=pc in hand by these means, the classical derivation of E=mc^2 is pretty much a shoe-in. Again, I used no quantum and no relativity. If I could do this, the geniuses at the turn of the century got it faster;-)

Edit: 11-22-17

I must’ve done something wrong with the Latex, it doesn’t seem to want to render in the body of my post; I’m still looking into whether I need to get the plugin…

Further, I figured out what was wrong with the unit analysis I did above… the right side of that equation is energy density (J/m^3) rather than energy (J)… and since J =N*m, J/m^3 is N/m^2…. the equation above is all in units of light pressure. To get to E = pc in approximate form in the plane wave, you just need to sub in the relation for momentum *density* per Poynting vector S = pc^2, then cancel the density by integrating over volume.

One additional thing about the Einstein’s box derivation that is important; it works in a classical framework. What I’ve provided above, then, is E=mc^2 as a classical equation, which is really torturing the point that it was “proven.” I’ve been thinking about whether or not I was doing this right since the whole discussion started and the derivation is only consistent from the standpoint that there are no effects included taking into account the potential relativistic characteristics of the box as it moves. I’m sorry about that, Narad. The derivation above would be insufficient from a modern physics standpoint because the box would undergo length contractions and dilations as it moves. To be perfectly honest, this nagged at me a tiny bit as I wrote the derivation, but maybe not as much as it should have… I drew the box as strictly “before” and “after” so that I ended up looking at the system only when it is located in the inertial frame of reference. That would call into question the nature of the boost pushing it into motion. I was assuming that the completely undisclosed relativistics located between the end-points were sufficient to conspire that the end-points be right! And, that’s an open end since length contraction would place the wall of the box in a different location depending on the frame… throwing off the whole calculation.

(For the people at home, here is something very important about how I designed to write this blog. I leave my edits visible so that the progression of my thinking is clear… one of the hardest, most human aspects of working in sciences is facing the fact that nobody is always right about everything. I think that being a good scientist is not about being right all the time, but about changing your mind when it’s important to do so. And, it’s about admitting when someone else was right, sometimes very publicly! Are you smart if you’re unwilling to abandon a sinking ship? I think not. Smart is being able to turn the steering wheel and to grow when its necessary to do so –especially when it effects your pride. I think this is the difference between arguing loudly and arguing productively.)

Here is the derivation converting the light pressure equation Narad offered into E=pc…

Hopefully that ties up all the loose ends! (Don’t be surprised to see me back here playing with a relativistic E=mc^2 proof at some point.)