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When I Google the units for $E=mc^2$ it says m is mass in kg speed of light is in m/s and energy is in joules. Why does there not need to be some sort of constant? Why does the total energy in joules relate to the speed of light in meters per second. Isn't the second based of how long the earth takes to spin round. If a second was slightly longer would the joule be slightly smaller? If I had 1kg of water, would it have 300 million joules of energy?

Finlay Hutchinson
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    The units can just as well be grams, centimeters per second, and ergs, or many other choices. The units just have to be consistent. The SI units you mention are consistent because one joule is one kilogram times one-meter-per-second squared. – G. Smith Apr 29 '21 at 00:12
  • Isn't the second based of how long the earth takes to spin round. No. Wikipedia: “The second is defined as being equal to the time duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the fundamental unperturbed ground-state of the caesium-133 atom.” – G. Smith Apr 29 '21 at 00:15
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    Why are you asking about $E=mc^2$ and not $F=ma$ or $p=mv$ or $F=GMm/r^2$ or ... ? – G. Smith Apr 29 '21 at 00:26
  • My question was asked poorly and I didn't really get across what I was trying to ask. I have found a post that asks my exact question though. – Finlay Hutchinson Apr 30 '21 at 00:57

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With our current definitions of meter and second, $$c = 299, 792 ,458\frac{m}{s}.$$ Let's say we were to use some other unit of second, let's call it the "zepond" and denote it with $z$ instead of $s$. Let's say one zepond is $2$ seconds, so $$ z = 2 s. $$ Then the speed of light, measured in meters per zepond (instead of meters per second) would be $$ c = 599, 584, 916\frac{m}{z}. $$ Note that this isn't really saying anything. In fact, it just trivially follows from $z = 2s$ as \begin{align} 299, 792 ,458\frac{m}{s} &= 299, 792 ,458\frac{m}{ \tfrac{1}{2} z } \\ &= 2 \times 299, 792 ,458\frac{m}{ z } \\ &= 599, 584, 916\frac{m}{ z }. \end{align}

This is just how units work. $c$ is a physical quantity, and has a meaning no matter what units you use. For instance, we could write it in miles per hour if we want. $$ 299, 792 ,458\frac{m}{ s } = c = 670,616,629 \frac{\rm miles}{\rm hour}. $$ They're still equal to each other. The number is only different because the units are different. So really, your question doesn't have anything to do with $E=mc^2$ in particular, it's just a question about how units work.

user1379857
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