# B Why is E=MC^2, squared and not cubed to relate to our dimension?

#### Trooper149

Problem Statement: Trying to understand the principles for the equation e=mc2.
Relevant Equations: E=mc2

So just started at a physics A level and so far loving it. I understand this question has indeed been asked before, however for different reasons. I understand the purpose of e=mc2 and I understand the formula. My question is related more the context in which this equation applies? Also, I understand that it is squared to reflect our dimension, however don't we operate in the 3rd dimension, in which case should it not be cubed? Perhaps I have misinterpreted what is being described?

Appreciate any help.

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#### Orodruin

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Problem Statement: Trying to understand the principles for the equation e=mc2.
Relevant Equations: E=mc2

Also, I understand that it is squared to reflect our dimension, however don't we operate in the 3rd dimension, in which case should it not be cubed? Perhaps I have misinterpreted what is being described?
Unfortunately, you have misinterpreted what is being described, yes. What is likely being discussed is dimensional analysis. This has nothing to do with the number of spatial dimensions that we happen to live in. Instead, it has to do with the physical dimensions of different physical quantities. In the SI system, there are seven base units, the ones you will encounter most in relativity is length (L), time (T), and mass (M). In order for a physical equality to hold, both sides must have the same physical dimension. For example, an area has physical dimension of length squared (L^2), which means that you must measure areas in units of that dimension (such as m2). It would not make sense to measure areas in units of time (such as seconds).

For $E = mc^2$ it is the same. Energy has physical dimensions of $\mathsf{M L^2/T^2}$ and mass physical dimensions of $\mathsf M$. The only way the relationship can make sense is if the proportionality constant therefore has a physical dimension of $\mathsf{L^2/T^2}$. The speed of light is a speed and therefore has dimensions of length/time (L/T) and the only way the relationship can make sense is therefore if you have $E = kmc^2$, where $k$ is just a number (you have to determine such numbers through physical arguments, not through dimensional analysis). In this case, $k = 1$.

#### Mister T

Gold Member
Also, I understand that it is squared to reflect our dimension,
No. The power you raise $c$ to has no connection with the number of dimensions.

#### Heikki Tuuri

In newtonian mechanics, in perfectly elastic collisions, mv and 1/2 mv^2 are conserved. The latter is the "energy".

The exponent 2 does not depend on the number of spatial dimensions in the universe. It is the same in a universe which has 100 spatial dimensions.

#### Trooper149

Unfortunately, you have misinterpreted what is being described, yes. What is likely being discussed is dimensional analysis. This has nothing to do with the number of spatial dimensions that we happen to live in. Instead, it has to do with the physical dimensions of different physical quantities. In the SI system, there are seven base units, the ones you will encounter most in relativity is length (L), time (T), and mass (M). In order for a physical equality to hold, both sides must have the same physical dimension. For example, an area has physical dimension of length squared (L^2), which means that you must measure areas in units of that dimension (such as m2). It would not make sense to measure areas in units of time (such as seconds).

For $E = mc^2$ it is the same. Energy has physical dimensions of $\mathsf{M L^2/T^2}$ and mass physical dimensions of $\mathsf M$. The only way the relationship can make sense is if the proportionality constant therefore has a physical dimension of $\mathsf{L^2/T^2}$. The speed of light is a speed and therefore has dimensions of length/time (L/T) and the only way the relationship can make sense is therefore if you have $E = kmc^2$, where $k$ is just a number (you have to determine such numbers through physical arguments, not through dimensional analysis). In this case, $k = 1$.
This is excellent. So quite simply, because energy is measured in joules and joules is squared, they square light in e=mc2 to convert the value?

#### Orodruin

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This is excellent. So quite simply, because energy is measured in joules and joules is squared, they square light in e=mc2 to convert the value?
I do not know what you mean by ”joules is squared”, but if you mean that you must square c to get the appropriate units, then yes, it is the only way in which you can relate these three quantities (up to a multiplicative constant that is dimensionless and must be determined, in this case that constant is one).

Note that this is not saying that such a relationship must exist and be physically meaningful (we had a recent thread with that misconception), just that if it exists, then it must be on that form. This type of argument is a special case of Buckingham’s pi theorem, which is an important result in dimensional analysis.

In general, I find that dimensional analysis - although quite basic and not very difficult or lengthy - is not covered satisfactorily in many physics curricula. I have a section covering the basics in the modelling chapter of my book and there are also many other nice and short texts on the subject. I am not aware of a nice summary on the internet, but admittedly I have not looked very hard for it.

#### haushofer

I agree, Orodruin. I teach students at high school how dimensional analysis eases their lives, with a "what else could the formula be" question central. I find it poorly covered in the textbooks we use at schools.

#### Trooper149

Thank you for the informative replies guys and gals

"Why is E=MC^2, squared and not cubed to relate to our dimension?"

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