I don't want this to turn into the "Starts With A Bang Response Blog," but Ethan keeps writing stuff that I feel a need to follow up. This time, it was a post from last week titled Why Does E=mc^{2}?, deriving the World's Most Famous Equation. And while it's solid, physics-wise, and follows something close to the argument Einstein used when he presented the relationship in question back in 1905, I think it's more complicated than it needs to be. There's a much simpler and more elegant way to see what's going on, which I'll sketch out here, the way I do it in public lectures.

In fact, I would argue that asking "Why is *E=mc*^{2}?" is the wrong question to ask. After all, energy has to be equal to *something*, and the question of whether it's described by any particular combination of parameters with the right units is a purely empirical matter (and not the fun kind). The more interesting question, the one that leads simply and directly to the answer, is "Why are we *surprised* that *E=mc*^{2}?"

The answer to that second question traces to our everyday conception of energy, which we typically associate with things that are moving (kinetic energy), or at least have the potential to *start* moving (potential energy). Einstein's famous equation catches us by surprise because it says that even objects at rest have energy, and lots of it.

The source of our surprise also points to the simple resolution to the question. We're surprised by the existence of rest energy because we're thinking about motion the wrong way. When we look at the universe the right way, we see that objects we normally think of as being "at rest" are, in fact, moving, and that motion is the source of the energy. From the correct, relativistic perspective, there's nothign surprising about rest energy. In fact, it's inevitable.

Of course, that's a pretty bold assertion, in need of some unpacking. So, what do I mean when I say we're thinking about motion the wrong way? Well, it goes back to the basic definition of motion, and how we track it.

A stationary observer, say a dog sitting in a window watching the world go by, will track the motion of a passing cat by recording the cat's position at different times. We can make a simple graph to represent this, shown in the figure above, where the dog has set up a nice grid of lines marking units of position (vertical blue lines) and instants of time (horizontal green ones). The black line shows the trajectory of a cat moving from left to right, and the brown line shows the dog remaining where she is.

Now, the theory of relativity takes its name from the simple idea that the laws of physics do not depend on how you're moving, and that includes Maxwell's equations of electromagnetism giving a single value for the speed of light. So we can stick in two red lines representing rays of light moving out from the dog's position, and use them to scale the diagram: light moves one unit of position in one unit of time-- if you like American units, you can measure distance in feet and time in nanoseconds, and this works out pretty nicely.

(If you're paying careful attention, you can count boxes and find that the cat is moving at half the speed of light, which is about right for the local cats when my dog would start barking...)

Since the laws of physics can't depend on how you're moving, the cat is perfectly entitled to view this from her own perspective, but as lazy canine physicists, we'd like to stick to a single diagram, and just replace the gridlines with new lines that represent the cat's perspective. Our first attempt at such a diagram might look like this:

It's easy to add position measurements for the moving cat, who measures all distances relative to herself-- they're the slanting blue lines parallel to the cat's trajectory. And in keeping with classical physics dating back to Newton and even earlier, we keep the dog's time markers, because surely everyone agrees what time it is?

Only, when we do that, we run into a problem with the principle of relativity: that the laws of physics can't depend on how you're moving. If we look at those two rays of light, we see that at the instant of time two dog-units up the diagram, the cat sees the rightward light ray having moved only one-and-a-bit units, while the leftward one has gone three-and-a-half. If we treat the dog's time measurements as universal, the cat sees light moving at different speeds in different directions, which would be a major change in the laws of physics seen by the dog.

To fix this problem, we need to change the definition of time according to the cat. Rather than using the dog's time gridlines, the cat needs her own set, which look like this:

Instants of time according to the cat aren't the horizontal lines they were for the dog, but the slanting green lines on this new diagram. If you count boxes, you can see that this fixes the speed-of-light problem-- in two cat-units of time, both leftward and rightward light rays have moved two cat-units of space. The cost, of course, is that the cat and dog no longer agree about what time it is at different positions in space.

A bunch of physicists noticed this problem in the late 1800's, and Henri Poincaré and Hendrik Antoon Lorentz worked out mathematical ways to deal with it by assigning each observer a "local time." Einstein's great insight of 1905 was that this isn't necessary-- the disagreement between stationary dog and moving cat is inevitable, once you think carefully about what it means to measure times and positions, and how you synchronize clocks at distant locations. This is why the equations we use to move between observations by different observers are called the "Lorentz Transformation," but Einstein gets credit for relativity as a whole.

Returning to the diagram, we see that what the cat sees as an instant of time looks like a mix of space and time to the dog. And vice versa-- a horizontal "instant of time" for the dog is a mix of cat-space and cat-time. One of Einstein's former teachers, Hermann Minkowski, picked up on this, and in a famous 1908 lecture declared that "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." Which is a little overblown if you're not a particle physicist-- in most everyday situations, it's perfectly fine to treat space and time separately-- but is technically correct. Space and time are just different aspects of a single thing, which physicists call "spacetime," because we mostly suck at naming things.

And that brings us around to my preferred form of the question: "Why are we *surprised* that *E=mc*^{2}?" I said earlier that this is because we're thinking about motion incorrectly, and from the above diagrams, we can see what that means. When we say an object is "at rest," we generally mean that it's not moving *through space*. But even stationary objects are moving *through time*, marching forward into the future at one second per second. And as Minkowski pointed out, space and time are different aspects of a single spacetime, so nothing is ever truly at rest. Which means that every object has energy, all the time, and when you work out what that is for an object that's only moving through time, it turns out to be the mass times the speed of light squared.

The simple and elegant answer to "Why does *E=mc*^{2}?", then, is "Because everything is always moving through spacetime." The rest energy that we didn't expect is just due to the time part of that motion, and by looking from the correct perspective, we see that it's not surprising, but inevitable.

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(Credit where due: My presentation of this draws heavily on books by Brian Cox and Jeff Forshaw and Tatsu Takeuchi. Both of those go more into the math than I do, showing why you get the exact combination of quantities that show up in Einstein's equation. If you're going to read just one book on relativity, I'd obviously like it to be the one I wrote, but if you're going to read *three* books on relativity, those are the next two to reach for...)

I'm an Associate Professor in the Department of Physics and Astronomy at Union College, and I write books about science for non-scientists. I have a BA in physics from

…I'm an Associate Professor in the Department of Physics and Astronomy at Union College, and I write books about science for non-scientists. I have a BA in physics from Williams College and a Ph.D. in Chemical Physics from the University of Maryland, College Park (studying laser cooling at the National Institute of Standards and Technology in the lab of Bill Phillips, who shared the 1997 Nobel in Physics). I was a post-doc at Yale, and have been at Union since 2001. My books _How to Teach Physics to Your Dog_ and _How to teach Relativity to Your Dog_ explain modern physics through imaginary conversations with my German Shepherd; _Eureka: Discovering Your Inner Scientist_ (Basic, 2014), explains how we use the process of science in everyday activities, and my latest, _Breakfast With Einstein: The Exotic Physics of Everyday Objects_ (BenBella 2018) explains how quantum phenomena manifest in the course of an ordinary morning. I live in Niskayuna, NY with my wife, Kate Nepveu, our two kids, and Charlie the pupper.