###### By Don Lincoln, Fermilab

## Our best and most modern theory of motion is Albert Einstein’s theory of special relativity. It’s called special relativity because it doesn’t include gravity. It’s Einstein’s theory of general relativity that includes gravity. Nonetheless, we can’t use the famous Einstein equation most of the time. Why? Read on to find out.

### Einstein’s Theory of Special Relativity

One doesn’t have to have studied any physics at all to have heard about the most famous of all of Albert Einstein’s equations, *E = mc ^{2}*.

*E*stands for energy,

*m*is mass, and

*c*is the speed of light. What it means is pretty straightforward. On one side of the equation, we have energy and the other side we have mass times a constant. The equal sign basically says that energy and mass are the same up to a constant. It’s really not very different from saying that pounds and ounces are the same when you apply a conversion factor, or grams and kilograms.

However, in general, *E = mc ^{2}* is wrong, or at least incomplete. That equation is a special case that applies only for particles that a) have mass and b) are stationary. The equation is totally wrong for moving particles or for ones that have no mass. So that’s the first shocker, which is that we can’t use the famous Einstein equation most of the time.

It turns out that there is a more general version of Einstein’s equation that works for all particles—massive and massless, moving and stationary. This equation looks similar to the famous one, but it adds momentum, which one can more or less think of as moving energy.

### Adding Momentum

Momentum is mass times velocity and is true for slow moving particles. It’s not, however, strictly true for quickly moving particles. Just think of momentum as a measure of the moving energy of an object, with the symbol *p*.

So, what is the more correct version of Einstein’s equation? Well, the easiest way get to it is to start with the original famous equation *E equals m c-squared* and square both sides. We then get * E^{2} = mc^{2}*, all squared. Now we add to the mass side a term that includes the momentum and that term is

*p c*all squared. Thus, the most correct Einstein equation of motion is the grand E

^{2}=(mc

^{2})

^{2}+(pc)

^{2}, all squared. Now that is a delightful equation.

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### New Set of Units

The easiest way to simplify it, is to pick a new set of units. And the most simplifying set of units in this case is to define all velocities as a fraction of the speed of light. Travel at 30,000 kilometers per second, or 18,600 miles per second? That’s 10% of the speed of light, so the velocity is 0.1.

A 150,000 kilometers per second or 93,000 miles per second would be 50% the speed of light, or a velocity of 0.5. And, of course, if one travels at 300,000 kilometers per second, or 186,000 miles per second, they are travelling at 100% the speed of light and their velocity is 1.

So, in this unit system, if we are travelling at the speed of light, which is *c* in the equation, then *c* is equal to 1. And that super-simplifies Einstein’s equation. We get the very simple equation *E ^{2} = p^{2} + m^{2}*.

### Mass

All of the other parameters, too, then have to get different units, and it’s a bit of a mess at some level. But it’s what particle physicists do to simplify the equations. But, accepting this simplifying choice illustrates the key points. We start by doing a smidge of algebra and rewrite the equation as *m ^{2} = E^{2} − p^{2}*.

So, now that we have the simpler equation, where do we go with that? Well, we then move on to the meaning of mass. In most textbooks, this particular mass is what is called the rest mass, which is to say that it is the mass one would measure if it were sitting stationary next to them. And yet, why should the mass change as velocity changes? Well, many physics textbooks tell us that mass changes with velocity. That’s one of the things about relativity that is used to explain why we can’t go faster than light.

Needless to say, we could have a fascinating conversation about exactly what changes when an object moves at high speeds, but, it is certainly a credible position. However, the bottom line remains, that, the rest mass never changes. It’s immune to all of the weirdness of special relativity.

### Common Questions about Theory of Special Relativity

**Q: To which particles does the equation***E = mc*^{2}**apply?**In general, *E = mc ^{2}* is wrong, or at least incomplete. That equation is a special case that applies only for particles that a) have mass and b) are stationary. The equation is totally wrong for moving particles or for ones that have no mass.

**Q: Which of the Albert Einstein’s equation works for all particles?**There is a more general version of Albert Einstein’s equation that works for all particles—massive and massless, moving and stationary. This equation looks similar to the famous one, but it adds momentum,

**Q: Which is the most correct Einstein equation of motion**?The most correct Einstein equation of motion is the grand * E^{2}=(mc^{2})^{2}+(pc)^{2}*, all squared.