Why does E=mc^2?

The other thread about the famous formula got me thinking: Why does E=mc^2 ?

Yes, I appreciate that c is a constant. But what does the speed of light (squared) have to do with the relationship between mass and energy?

This video may not be the greatest way to explain it but it give a quick overview of the concepts involved.

You are asking why mass = energy. One possible thought experiment is: photons have momentum. If an object gives off energy in the form of light, its momentum and kinetic energy will change in an appropriate frame of reference, which means it must have lost some mass. Next step is to work this into a rigorous physical calculation (assuming the appropriate relativistic principles and conservation laws).

I get that mass = energy. But why is c squared the conversion factor?

I don’t think there’s anything special about the value. C just happens to be the conversion factor. And because of the nature of light, it travels at that speed in a vacuum. Think of it as C=√(E/M). Run an experiment where you convert some mass into energy (wear proper safety equipment), measure the quantities, and do the math. You’ll derive C.

To make the units work, the conversion factor has to have the units of velocity-squared. In our universe, c happens to be the appropriate value to be squared, because (for reasons of relativity) c is the natural conversion between space and time, and thus its square is the natural conversion factor between mass and energy.

Here is a short and great explanation/ derivation:

I have some issues with that page. I’m not sure it is correct although it gives the correct outcome. One issue I have is the statement
Force = m x c
Isn’t force = mass x acceleration? Mass x velocity would be a momentum as he states in the line previously

Also, he seems to be looking at the kinetic energy of a mass at the speed of light c, which would be 0.5 x mc^2 so where did his 0.5 end up?.

Oh and on the left side he implicitly uses work = energy and work is force x distance as he points out to end up with f x c but c is a velocity.

I think part of this is he is using all of this as over 1 second to make unit rates but it looks like to me that that is just a way to get the numbers to work out and is not necessarily appropriate in this case.

As a lay person, I think some of the confusion comes from thinking that c is the speed of light in the way that light independently moves at that speed because it wants to or has control over its speed. I think the reality is that light moves as fast at the universe allows it to move, which is at the speed c. It’s like measuring the speed of a floating boat in the river. It’s the speed of the river that we’re really measuring and the boat just happens to be moving at that same speed. So then if you think of c as being a constant related to the universe rather than light, it seems more intuitive that the equation for energy and mass would use a constant which corresponds to a feature of the universe.

It’s complex physics, but basically c is the constant because c is also the speed limit for the (our) universe. Nothing can go faster than the speed of light. Thus, analysis of the physics of particles, mass, motion in time, etc. - yields the formula on the relationship between mass and energy. The paradox (to our Newtonian brains) of relativity is that no matter what frame of reference, what spaceship you travel on, the maximum speed is the same. So to account for the fact that what looks like one ship travelling (2/3)c passing another ship travelling (2/3)c, from the point of view on either of those ships will not look like the other ship recedes at >1c. Time slows down and distance shrinks from the viewpoint of the observer to ensure that their view sees a ship doing less than 1c.

In essence what the formula says is that mass is “frozen” energy, and vice versa, energy is what you get from “unfreezing” mass.

When a nuclear interaction converts an atom of uranium or plutonium to two atoms of smaller elements (plus assorted detrius particles) the difference between the starting and final mass is released as energy. This energy is noticeable if it takes the form of a mushroom cloud, or if it boils water to drive a steam turbine.

“Nothing can go faster than the speed of light” I always like to point out that this isnt true. Nothing can go faster than the speed of light inside of space. But space itself is expanding faster than the speed of light.

c is the speed of light in a vacuum; it will inevitably appear as soon as you try to write down the kinetic energy or the momentum of a moving body:


(here m is the rest mass)

We can get to E = mc2 this way, but someone might wonder where the Lorentz factor (containing c) came from— that is also due to relativistic calculations that assume the speed of light is constant.

This thread is a place to start. In it, there are a number of posts talking about why c shows up in so many places, such as E=mc2). (Unfortunately the old vBulletin markup gets in the way in a few parts.)

Indeed. I would go a step further and say there is nothing of value* on it and it should be stricken from the record.

* Well, there is possibly negative value on it, in that one could easily understand less after reading it.

Unfortunaetly I don’t think there’s a nice snappy justification. Here’s nice proof, but it already assumes E/c = p when m is zero and by using electromagnetic pulses it doesn’t show that you don’t need to reference light to get the relationship.

I hope you won’t consider this a hijack, because to me it is a very basic problem that I have with this equation. Namely, I don’t understand what “c squared” means. By way of analogy, I’ll explain what I’m asking:

When I multiply a length by a length, I get an area.
When I multiply an area by a length, I get a volume.
When I divide mass by volume, I get density.
When I divide distance by time, I get a speed.
When I divide a speed by a time, I get an acceleration.
When I multiply a mass by a velocity, I get a momentum.

All the above concepts make intuitive sense. All you need to do is plug in the numbers. A box that is two high, two wide, and three long, is 12 cubes of whatever units you’re using. Fifty miles in two hours is 25 miles per hour.

But what happens when I multiply a velocity by a velocity? I trust the experts that this is a required part of the formula, but what on earth does it mean to square a velocity?

You get a velocity squared. Which isn’t a type of quantity that’s encountered very often by itself, but what does that matter? We could give it a name, but that wouldn’t really make it any clearer.

When I multiply a force times a distance, I get an energy (Work)
a force is a mass times an acceleration (a mass times a distance over a time squared) so
a force times a distance is a mass times a distance squared over a time squared, so
an Energy is a force times a distance is a mass times a velocity (distance over time) times a velocity = a mass times a velocity squared.

I think you are misreading or misunderstanding. Force is F = mass x acceleration + d/dt(m) x velocity

Since acceleration is zero (the velocity is c and cannot change), the second term comes into play. The d/dt (m) is represented as m since dt is 1 second. It’s the mass “ created” by the force, not the acceleration.

Hope that clears it up.

In your examples, you used full, physically interesting combinations. But here, you have picked out just the c2 part of mc2 and expected it to have an immediate physical interpretation.

Your list of examples could have included classical kinetic energy, which is (one half) mass times velocity squared. If you are okay with that combination giving “energy”, then mc2 should be okay since it follows the same dimensional pattern. But you wouldn’t have targeted just the “velocity squared” part of kinetic energy, as there’s no reason for part of the expression to stand alone in a physically interesting or meaningful way.

A more concocted version: We could encounter an expression where an area is equated to the product L * a * b, with L having dimensions of length and with a and b both having dimensions of (length)1/2. The whole expression is an area. And maybe the product ab has some obvious geometrical interpretation. But there’s no requirement for, say, La (units of (length)3/2) to “mean” anything obvious.

Having said all that, there usually is a way to physically interpret bits and pieces of expressions, although it can be convoluted if forced. In the c case, though, it’s not that crazy: c on its own is a velocity because spacetime has space and time dimensions so the unit conversion factor between those is a velocity; and c2 appears because getting between energy and mass requires two applications of that unit conversion factor, in precisely the same way that connecting “area in inches2” and “area in cm2” requires the conversion factor “c2”=(2.54 cm/in)2.