How do physicists justify using $c^2$ in equations?

Question

Excuse my ignorance but I see physicists use $c^2$ in equations and I was wondering what reasoning they use to justify $c^2$ since, as far as I understand, $c^2$ must be an absurd quantity because there is a speed limit imposed by physicists themselves at $c$. If there is no motion faster than $c$, how do you justify using $c^2$ in an equation?

Answer 1

I don't think people should downvote this question so I will attempt a serious answer.

When scientists deal with physical quantities, they're concerned not just with the number but also what units the number is measured in. So the speed of light, $c$, is $299,792,458$ meters per second ($m/s$). Both parts, the number and the unit, are meaningful and important.

In particular, you do math on both the numbers and the units. So the speed of light squared is $$c^2=299,792,458\space m/s\times299,792,458\space m/s=89,875,517,873,681,764\space m^2/s^2$$

Look carefully at this answer. Sure, $8.99\times10^{16}$ seems like a pretty huge number based on normal everyday experience. But it really doesn't mean anything without the units. In this case the units are meters squared over seconds squared. This is not a unit of speed!

You have to compare apples to apples, or like units to like units. $c^2$ isn't faster - or slower - than $c$. It's just a different thing entirely; an orange.

As a matter of fact, it's not really valid to give it a physical meaning. What is it supposed to mean, square meters per square second? Square meters are area, but what area could this be? And what the heck is a square second? It's just not correct to think of $c^2$ as a really fast speed. Rather it is a mathematical constant that happens to have units of speed-squared.

So it's not a violation of the universal speed limit to use the speed of light squared in an equation.

As to why $c^2$ is used at all: it's basically an extension of the observation that the kinetic energy of a particle is $\frac{1}{2}mv^2$. Notice the units of speed squared there, as well. This is a simplification but should give you the idea.

Answer 2

By using $c^2$, physicists are not implying that anything is moving at that velocity because, as you said, c is the maximum velocity. Note, though, that $c^2$ is just a mathematical term.

A lot of the heavy math in physics doesn't relate to tangible things but rather to mathematical relationships. The example you're probably most familiar with is $E=mc^2$. This doesn't mean that matter moves at the speed of light squared or that it's moving at all. Rather Einstein's famous equation shows that a little bit of rest mass equates to a lot of energy. In atomic bombs, for instance, only a small amount of the bomb's payload is converted to energy but because the mass term is multiplied by the speed of light squared it ends up releasing a LOT of energy.

Someone else can probably provide a much more detailed explanation of where $c^2$ comes from, but I cannot. All I can give you is the above overview.

Answer 3

c² does not represent velocity faster than c since c² does not represent velocity at all, just like a² does not represent length (a denoting the length of a side of a square here). You can see this easily checking the units: c is expressed in m/s while c² is expressed in m²/s².