Why E=mc² is wrong

There is perhaps only one equation in all of physics that almost everyone knows, and that equation is E equals m c squared. When you break it down, it's quite surprising that this particular equation is so well known. After all, he makes a rather strange claim: that energy is the same as matter. I mean, that's what that equals sign says. So since the equation is so familiar, it's probably worth asking the question: is it correct? And the answer will probably surprise you. This is not the case, or, perhaps more accurately, the equation is a special case. To use it correctly you need to know more.

Now I don't want my email inbox to explode because I said E equals m c squared and it's

wrong

. And you shouldn't think I'm saying something controversial here. Anyone who masters special relativity will agree. The problem is not the theory, but simply how the well-understood theory of relativity is represented in popular culture. So if E equals m c squared is

wrong

, what is right? It turns out that there are two more precise but similar equations. Let me tell you about both. The first one looks quite similar. It's E equals gamma times m times c squared. I made another video where I talked about relativistic mass, which introduced this term gamma, and I made another video that talked specifically about gamma.

You can see them if you want to learn more about it. However, I can give you the highlights. Gamma is a ubiquitous term in relativity. It is defined as one over the square root of the quantity one minus v squared divided by c squared. V is the speed of an object relative to you and c is the speed of light. If v is zero, then gamma is just one. So that's your first idea: E equals m c squared is only correct if an object is not moving relative to you. And, as the speed of an object increases towards c, the gamma increases more and more and the energy of the object increases.

Note that mass does not increase, but energy does. This is particularly important to realize, because I don't know how many times people tell me that since energy and mass are equal, then a highly energetic object is also massive, at least not in the way that most of these people mean. It's simply not true. So get that misconception out of your head. There is another aspect of this equation where people are at least a little wrong. If you keep the energy constant, but reduce the mass of the object, then the gamma must increase. And if you take the mass to zero, then the velocity must become the speed of light, since the gamma must become infinite.

This is because you start getting into mathematical tricks, where infinity times zero equals a constant. So that works in calculus, but invoking infinities in physics is usually a very bad idea. Therefore, this equation does not really apply to photons, which are massless particles of light. In fact, and this is an important point, this equation only applies to particles with mass and speeds below that of light. Now, there is another equation that applies universally to both massless and massless particles, and that equation is here. It's E squared equals p times c, all squared, plus m c squared, again all squared.

The meaning of the symbols is e is energy, m is mass, which some people call mass at rest, but it is actually the only value of mass: c is the speed of light and p is momentum, which is a measurement of the motion of a particle. Let's see what happens in the case that a particle is not moving, which means setting the momentum to zero. Ready, you get the well-known equation E equals m c squared. Now, let's set the mass of a particle to zero, which of course describes a photon. We see here that the equation becomes E equals p multiplied by c, which is the correct relationship between the energy and momentum of a photon.

That's pretty elegant too. Furthermore, this equation demonstrates precisely why using the equation E equals m c squared for photons is simply nonsense. If you do that, little children will laugh at you. So don't do that. By the way, there is one additional point I would like to highlight. Let's go back to the other equation that relates energy to mass: E is equal to gamma m c squared. Yes, that. Commenters on previous videos asked for a physical meaning for gamma and at least that's how I see it. Remember that m c squared is the energy of a particle at rest.

On the contrary, E is its energy while moving. Therefore, gamma is simply the ratio between the total energy of a particle and its energy when it is not in motion. Since particle physics experiments often measure the energy of a particle, I find this particular physics implication of gamma to be the most useful. That's all! Einstein's most famous equation is true, but it is actually a special case of equations that apply more generally. And now you know how to do it the right way. Relativity is truly amazing, there is no doubt about it. And it is misused very often.

I hope this video has helped you. If so, please like, subscribe, share and tell all your friends - we have to increase our numbers! And don't hesitate to comment. We want to hear what you have to say. See you next time and remember: physics is everything.