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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?

dmckee --- ex-moderator kitten
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Zeynel
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    Welcome to Physics.SE! You'll note that I have formatted your mathematical symbols using LaTeX style markup which is supported on the site with the MathJax engine. – dmckee --- ex-moderator kitten Dec 13 '11 at 21:22
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    Could you be more precise? What is wrong with $c^2$? – Misha Dec 13 '11 at 21:24
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    Note that while $c$ is a velocity, $c^2$ has units $m^2/s^2$ and is therefore not a velocity, so while of course the numerical value of $c^2$ measured in $m^2/s^2$ is much larger than the numerical value of $c$ measured in $m/s$, there is no violation of the "nothing can be faster than light" rule. – Lagerbaer Dec 13 '11 at 21:31
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    Even if it did have units of speed, that would be fine. Nobody said you can't have a number greater than c. – Colin K Dec 13 '11 at 21:38
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    You would have to look at the context to determine if $c^2$ were absurd or not. More often than not, it serves to maintain consistent units. Context is the key. I just don't understand why people down-vote a question?? This kind of trolling is getting to be ridiculous. Added a +1. – Antillar Maximus Dec 13 '11 at 22:28
  • Not that it's relevant to the question, but you can have a velocity greater than $c$, if you're a tachyon. If you are, you're not allowed to slow down to $c$, and the only direction you can travel in time is backward. – Mike Dunlavey Dec 13 '11 at 22:35
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    @AntillarMaximus - questions like this are too full of misconception to have any real use. -1. –  Dec 13 '11 at 23:57
  • @zephyr, personally I object more to the tone of it "$c^2$ must be an absurd quantity ..." when the poster has already professed ignorance on the matter. I don't think that's the way you get people to like you. – dmckee --- ex-moderator kitten Dec 14 '11 at 00:07
  • If the numerical value of c or its square concern you, I suggest you start utilizing more convenient units. Hint: the value of c is 1 lightsecond/second. – Johannes Dec 14 '11 at 01:02
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    @zephyr, the OP started with "excuse my ignorance", what more do you want to see? Pushing an irrelevant answer down the chain is fine (by ++good_answers), but why shoot down someone's curiosity? I see no misconceptions. The guy is merely asking a question. It is for us to dispel misconceptions. – Antillar Maximus Dec 14 '11 at 03:41
  • Yeah, there's nothing wrong with the way he asked the question. C^2 is a big number and he was thinking it was a physical quantity. It's a normal thing to try to relate math to real things. The answers below both point out that math doesn't always do this in a tangible way. This question should not be down voted. – John Dec 14 '11 at 13:58
  • @AntillarMaximus - to interpret c^2 as a velocity requires a fundamental lack of understanding of basic algebra. This is supposed to be a site about physics, not remedial math/science/logic. This question is nothing but useless clutter. –  Dec 14 '11 at 14:32
  • @AntillarMaximus - and as for tone, I agree completely with dmckee. The first few words do not excuse the rest. –  Dec 14 '11 at 14:33
  • @zephyr: This site is indeed about Physics and if I am not mistaken, the objective is to provide the best answer according to your interpretation of the question. It is a bit weird that you are expecting somebody to know the answer to the question they are asking. PS:Delivering an insult with a ":)" is still an insult. – Antillar Maximus Dec 14 '11 at 15:56
  • @AntillarMaximus just because someone has a question, does not mean that question is about physics or suitable to this site. PS: way to miss the point. –  Dec 14 '11 at 16:07
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    If he had just asked the question, I wouldn't have downvoted. But he wrote this question as an implicit accusation that physicists are doing something stupid and absurd intentionally. My downvote was for the rudeness and trolling. – Colin K Dec 14 '11 at 16:10
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    There is a meta question about this which would be a good place to post opinions about this - definitely preferable to holding a discussion in comments. (There is also [chat], of course) I'll probably be deleting the comments here after a couple days or so. – David Z Dec 14 '11 at 19:12
  • In units where $c=1/2$, $c^2$ is less than $c$. – WillO Feb 01 '20 at 04:57
  • If the highest house number on my street is 4000, how can there be a house on my street with 5000 square feet of living space? – WillO Feb 01 '20 at 05:01

3 Answers3

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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.

Mark Beadles
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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.

Vineet Menon
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John
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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.

Adam Zalcman
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