# E=mc^2, why squared?

• B
Hi everyone, I'm not a physicist, so please excuse my ignorance. In E=mc2, why is c squared? And why is the value exactly to the power of 2? Why not 2.1, or 2.438?
Thanks in advance anyone who can help

Orodruin
Staff Emeritus
Homework Helper
Gold Member
Anything else would be dimensionally inconsistent, ie, ##mc^{2.4}## would not describe an energy.

Dale
Anything else would be dimensionally inconsistent, ie, ##mc^{2.4}## would not describe an energy.
Thank you very much for your extremely prompt reply! I only partially understand though, sorry. Doesn't the equation for General relativity contain a c to the power of 4? Why 4 in one case, and 2 in the other? Once again, I hope my ignorance is not too cringe-inducing!!

Ibix
2020 Award
Because you need a ##c^4## in the Einstein field equations to make the units work out. Same reason, different equation.

Because you need a ##c^4## in the Einstein field equations to make the units work out. Same reason, different equation.
Thanks, you guys are awesome!

Anything else would be dimensionally inconsistent, ie, ##mc^{2.4}## would not describe an energy.
Because you need a ##c^4## in the Einstein field equations to make the units work out. Same reason, different equation.
Yeah but doesn't the c just kind of pop right out when you derive the Lorentz transformation equations? And from that every factor of c just falls into place with whatever relativistic kinematic relation you are deriving, doesn't it? I guess that wouldn't be much of an explanation, though.

Ibix
2020 Award
Yes, the factors of c drop out of the Lorentz transforms. And that is how you show that it is ##mc^2## and not ##mc^2/2## or something. But the underlying reason for there being a c there, instead of some other velocity, is that it (or rather, its square) is functioning as a unit conversion factor between units of mass and units of energy.

Nugatory
Mentor
Yeah but doesn't the c just kind of pop right out when you derive the Lorentz transformation equations? And from that every factor of c just falls into place with whatever relativistic kinematic relation you are deriving, doesn't it? I guess that wouldn't be much of an explanation, though.
It's not, but the same can be said of the dimensional consistency answer as well - it has to be that way, but it's not much of an explanation.

The difficulty here isn't with the answer, it's with the question - it is very hard to give satisfactory answers to "why?" questions. We often hear people saying that their "why?" is in search of a deeper understanding, but usually (and this thread is no exception) the right question is not "why?" but "how does this work?".

russ_watters, Dale and Ibix
My only discomfort with the "unit" explanation is that it applies to the old kinetic energy equation, too. Why v2 in T= (1/2)mv2? I know you can derive this result using basic kinematic equations, but that's no more satisfying than the "c drops out of the Lorentz transformation equations," I don't think.

So here's an off the wall follow up question: what about the wave equation that comes out of Maxwell's equations? You get c from that, too, and if I'm not mistaken from the permeability/permittivity constants (not to get too carried away with specific numerical values due to unit choices, but speaking of the physical features themselves). So could an explanation be given going back that far?

Referring to this:
https://en.wikipedia.org/wiki/Electromagnetic_wave_equation
Maxwell: "The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws."

So I'm wondering if it could be pushed back further to something more fundamental like the laws of electromagnetism.

In other words, would something like this be valid and also of some explanatory value? Special relativity is an extension of electromagnetic theory, and it so happens that the laws of electromagnetic theory are compatible with the Lorentz transformation equations, which when applied to mechanics yields results like E = mc2.

Of course then you'd have to explain what the Lorentz transformation equations are, and at that point you'd probably have to get into some math (unfortunately for the curious who are not mathematically inclined). I guess one way or another any lay explanation is going to be over the heads of most people. But I think there might be some value in connecting it to something everyone has some sort of familiarity with, like electricity and magnetism.

Just my uniformed opinion, of course. ;)

But in any case, as for the OP's question, why not fractional powers or some other power, I suppose the unit explanation is fine. Of course things can always be taken deeper.

ZapperZ
Staff Emeritus
Thanks, you guys are awesome!
Just out of curiosity, the kinetic energy is often written as E = 1/2 mv2. Why doesn't this also elicit a question from you, considering it also has a square of the speed?

Zz.

russ_watters
In the original paper Einstein wrote it as M=e/C squared. If C is a constant in our universe, it seems like you could really just state some new letter as "the speed of light squared" instead, but I imagine there is some mathy reason for the C squared to be in there. Of course since C is a speed it really can be broken down into feet per second or some such unit of measure - and units of time are always relative to some reference point such as vibrations of a cesium atom - the paper is about inertia really. "If a body gives off the energy L in the form of radiation, its mass diminishes by L/c squared ." I would think this indicates a body losing mass would follow a log scale re: energy loss? Sort of like how spacetime curves into a black hole gradually at first but increasingly as you move into spagettification world.

https://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf

vanhees71
Gold Member
There is no other way to really understand special relativity than to learn the math. In fact Einstein's paper of 1905,

http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Einstein_1905_relativity.pdf

as far as the non-electrodynamics part is concerned is pretty readable and demands only the minimal mathematics needed. So, how far can you get understanding that paper?

The famous formula ##E=mc^2## is not in this paper, but on a very short addendum,

https://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf

It's also pretty little math demanding, and even more important, other than the most popular-science explanations, it exactly describes the correct physical meaning of the the famous formula and not some misleading gibberish about "relativistic mass".

For the electromagnetic part of the paper, I recommend a modern textbook using modern vector notation for the Maxwell equations etc. Einstein's notation, writing out everything in components, is very inconvenient for modern eyes. A bit later Einstein himself adopted the very efficient Ricci notation, adding his "summation convention" to make it even more efficient :-).

Nugatory
Mentor
"If a body gives off the energy L in the form of radiation, its mass diminishes by L/c squared ." I would think this indicates a body losing mass would follow a log scale re: energy loss?
It depends on the rate at which the radiation is emitted, which depends on what's causing the emission. It's easy to construct systems in which the rate of radiation is pretty much constant for a time and then suddenly drops to zero - for example a lead-acid automobile battery connected to an incandescent light bulb. (It's also a good exercise to calculate the amount of mass lost in this example and compare with the total mass of the system; this will show why we usually focus on the rate of energy emission instead of the mass change).

Sort of like how spacetime curves into a black hole gradually at first but increasingly as you move into spagettification world.
Not really - Different equations, different curves.

Dale
Mentor
2020 Award
You get c from that, too, and if I'm not mistaken from the permeability/permittivity constants (not to get too carried away with specific numerical values due to unit choices, but speaking of the physical features themselves).
The vacuum permittivity and permeability are not physical features. They are artifacts of the SI unit system, and don't even exist in some other systems.

It's just basic dimensional analysis. Energy = work = force X distance. Force = mass X acceleration. The units of acceleration are distance over time squared. So the units of mechanical energy have to be mass X distance2 / time2, or mass X velocity2. This shows up most directly in the equations KE = mv2/2 and E = mc2, but it's true everywhere else too if you convert the units properly.

The vacuum permittivity and permeability are not physical features. They are artifacts of the SI unit system, and don't even exist in some other systems.
Why were they contrived? I was under the impression that permeability was, roughly speaking, how much a material gains magnetization within a magnetic field, and that the permeability of free space was the same thing but in a vacuum. As I understood it, while it isn't any particular number that still is tied to physical reality. Did I miss the boat on that?

There is no other way to really understand special relativity than to learn the math. In fact Einstein's paper of 1905,

http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Einstein_1905_relativity.pdf

as far as the non-electrodynamics part is concerned is pretty readable and demands only the minimal mathematics needed. So, how far can you get understanding that paper?

The famous formula ##E=mc^2## is not in this paper, but on a very short addendum,

https://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf

It's also pretty little math demanding, and even more important, other than the most popular-science explanations, it exactly describes the correct physical meaning of the the famous formula and not some misleading gibberish about "relativistic mass".

For the electromagnetic part of the paper, I recommend a modern textbook using modern vector notation for the Maxwell equations etc. Einstein's notation, writing out everything in components, is very inconvenient for modern eyes. A bit later Einstein himself adopted the very efficient Ricci notation, adding his "summation convention" to make it even more efficient :-).
Thank you vanhees71, that was very considerate of you. I haven't got time to read and digest it just yet, as I am immersed in a PhD, but I do hope to try and educate myself to become a physicist, and I gratefully receive anything you more learned minds can teach me

Dale
Mentor
2020 Award
Why were they contrived? I was under the impression that permeability was, roughly speaking, how much a material gains magnetization within a magnetic field, and that the permeability of free space was the same thing but in a vacuum. As I understood it, while it isn't any particular number that still is tied to physical reality. Did I miss the boat on that?
So the vacuum permeability is not a measured constant in SI units. The BIPM, the committee which defines the SI, got together and took a vote and decided to set the value of the vacuum permeability to be exactly ##\mu_0=4\pi \; 10^{-7} \; N/A^2##. Similarly for the vacuum permittivity. The BIPM is free to set them to exact values precisely because they are in charge of the SI and they are artifacts of the SI unit system, not features of nature. Basically, they define the ampere to be the quantity of current that makes that value exactly true.

In contrast, in Lorentz-Heaviside units ##\mu_0=1##, a dimensionless value. Although there is no governing body for those units, if a group of people can make a convention to give the vacuum permeability an exact value in SI units then a different group of people can make a different convention to give it a different exact value in their units.

Just out of curiosity, the kinetic energy is often written as E = 1/2 mv2. Why doesn't this also elicit a question from you, considering it also has a square of the speed?

Zz.
Hi ZapperZ,
I feel it would be discourteous of me not to respond to your post. How do I answer your question? I don't even know where to start. I can't help the feeling that it is not a genuine desire on your part to understand my motivations, but simply (and I hope I have very seriously misjudged you), an attempt by you to highlight my complete and utter ignorance of physics to the whole forum. Even though I have already clearly and abundantly emphasised that intellectual shortcoming of mine in my opening sentence, and in a subsequent post. But I will try and answer your question in total honesty. First of all, the idea wasn't even mine, but was suggested to me by a friend, so I just innocently posted the question on this forum, without even considering any parallels the equation may have had with any other equation in physics. So I have come around in a full circle, and can only plead total ignorance of physics for not asking you about kinetic energy first, before jumping in with special relativity. I trust your curiosity is now sated.

Nugatory
Mentor
And on that not, we can close the thread.