US LHC reader David left an excellent question on my previous post on E=mc^{2} that touches on another important physics topic. To give a proper response I’d like to dedicate another post to the matter. Here’s his question:

What I can’t understand is the constant assertion that mass and energy amount to the same thing in the e=mc^2 equation. My (albeit basic) education tells me that e/m = c^2, so how are they the ’same’?

He goes on to express that it’s not clear what kind of units one is supposed to use to make the famous equation make sense. Great questions, David. This will leads us to another equation that sounds really weird: *c=*1. (Where *c* is the speed of light.) But let’s start with the basics.

In order for our equations to make sense, then they’d better be consistent to matter what units we use. Nature doesn’t care whether we use inches or centimeters. What nature knows, however (and what we had to discover), is that there are constants of proportionality that allow one to measure one quantity in terms of the units of another.

Okay. That last sentence just got really abstract. Let’s start with simple examples from everyday life. (We’ll get to the speed of light at the end, when hopefully things will be crystal clear.)

**What is a unit? E.g. counting calories**

The past few weeks I’ve been doing a lot of my work at Starbucks. I know that one grande iced coffee has something like 80 calories. This sets up a natural conversion between units of food energy (calorie) and units of grande iced coffee:

1 grande iced coffee = 80 calories.

Now I can count calories in terms of grande iced coffees. Let’s say bananas have something like 160 calories. Then I can say that the amount of food energy in a banana is two times the amount of food enregy in iced coffee. Or in short-hand notation,

1 banana = 2 grande iced coffees

Note that in this short-hand notation it has to be clear that the quantities we’re equating are measuring a previously understood quantity; in order for this to make sense we have to say “the food energy in…” before each side of the equals sign. The point is that we can now use grande iced coffees to measure the food energy of other things, like a banana.

This is exactly what we mean by a **unit**: it’s a conversion between counting numbers and ‘dimensionful’ quantities. For example, it would make sense to say a banana has 160 food energy or 2 food energy. It only makes sense to say that a banana has 160* calories* of food energy or a banana ias the food energy of 2 grande iced coffees. But each of these latter expressions is equivalent, they convey the exact same information.

[Technical note: I’m writing ‘food energy’ explicitly here because this represents the energy that can be released by the chemical reactions of digestion. I’m not including things like the ‘matter potential energy’ of the atoms in the food which stays tied up in matter.]

**Algebra of units: converting into useful quantities
**

In our banana = 2 iced coffees equation, one nice feature is that it no longer matters how we actually measured the food energy. We originally used calories because this is what you find on nutrition labels. But this unit doesn’t make any sense to me, I don’t know what it means to ‘spend one calorie.’

My favorite recreational activity is playing basketball. I can look up that for a person of my weight, playing one hour of basketball burns about 500 calories. And just like that we just did another unit conversion:

1 hour basketball = 500 calories.

Since I understand what it means to play an hour of basketball, the natural units to measure the food energy of a banana is in hours of basketball played. You may have already done the calculation in your head.

Now let’s make this a little more formal and do what I call the **algebra of units**. Let’s see how it works:

The trick is to **multiply by 1**. That’s right. Okay, I guess the *real *trick is to write the number 1 in a clever way. Note that one doesn’t have units. Here’s what we do:

Note that each of the quantities in paretheses is just the number 1. I’ve just written 1 in terms of the ratio of two dimensionful things. Where did I get these expressions for 1? Well I took the equation

1 iced coffee = 80 calories

and I divided both sides by “1 iced coffee” to get 1 = (80 food cal)/(1 iced coffee). Then I did the same thing for the equation of basketball hours to calories. It is critically important that we explicitly wrote out the units of each quantity, because now we can simplify the expression on the right hand side.

This is just simplifying fractions. We have “iced coffee” in the numerator and “iced coffee” in the denominator. So we can cancel out these units. Note that we have to leave the numbers, we’re just cancelling the unit “iced coffee.” Similarly, we can cancel the units of “calories” from the numerator and denominator. What we are left with is

1 banana = (2 x 80 / 500) hours of basketball

Doing the arithmetic we find that in order to burn off the food energy of one banana I have to play .32 hours of basketball, or about 19 minutes. (Until roughly half-time… which would be a good time to snack on another banana.)

Along the way we note that we’ve made a conversion from banana to hours. But I know that bananas are different from hours… so is this statement crazy? *No* — as long as we know that we mean “the enegy in a banana” and “the energy expended playing an hours of basketball.” This is at the heart of understanding the units in E=mc^{2}.

**The formal statement of what we’re doing
**

Now that we’ve given a tangible example, let’s explain once again what we’ve been doing using high-falutin’ fancy-pants language.

We’ve used equations that relate fixed numbers of one unit to fixed numbers of another unit. In particular, we’ve defined conversion factors. In the above example these conversion factors were just the number 1 written in fancy ways that combine units. The point is that these conversion factors are *constants*. If they weren’t constants, then they don’t make sense. For example, maybe I don’t just order a grande iced coffee. Maybe depending on how I feel I’ll order a smaller or larger sized up, or maybe I’ll have it with milk. In this case the number of calories in what I called an ‘iced coffee’ is *not *constant because there are more parameters. One would have to be more specific when defining the conversion factor so that it really is a constant.

The lesson to take home is this: dimensionful **constants allow us to convert between units**.

**The speed of light is constant
**

One of the great experimental discoveries in all of science is the fact that the speed of light [in vacuum] is constant. This is the basis for special relativity. For our present discussion however, the point is that now we have a dimensionful constant which we can use to convert units.

In units that I remember, the speed of light is given by

*c =* 300 000 000 meters / second.

This tells us that we can write out an equality

[the distance travelled by light in] 1 second = 300 000 000 meters.

Now this looks like our silly “1 banana = *x* hours of basketball” statement, but it does have a clear meaning. We can change units. In fact, this gives us a natural definition for **lightsecond**:

1 lightsecond =the distance travelled by light in 1 second = 300 000 meters.

In this way a lightsecond (or lightyear, etc.) is both a measurement of time and distance since we’re using the speed of light (a constant) as a conversion. In these units physicists like to say that

the speed of light, *c* = 1.

This seems like a weird statement, but it’s really just saying that light in vacuum travels at the speed “1 lightsecond per second.” In any real particle physics calculation we always write things in units where the speed of light is 1 since this makes our equations much simpler (just look at the original post and see how even those equations simplify.) If we want to convert back into useful units we can always insert the appropriate factors of 1 = *c = *300 000 000 meters / second, just like we did using the ‘algebra of units’ above.

**The meaning of E=mc ^{2}, redux**

So hopefully this makes the meaning of E=mc^{2} a little more transparent. In fact, I would write this as E=m. The factors of *c* are just there to convert into normal units. I think David wanted me to write something out explicitly as an example, so let me consider the energy associated with the mass of the proton. I can look up

proton mass, *m* = 1.7 x 10^{-27} kg

speed of light*, c* = 300 000 000 meters/second.

Then the right-hand side of E=mc^{2} tells us

mc^{2} = 1.7 x 10^{-27} kg x (300 000 000 meters/second)^{2}

= 5.02 x 10^{-19} kg(m/s)^{2}

I’d like to write this into something like joules, so I’d better look up the appropriate conversion from kilograms, meters, seconds into joules:

1 J = 1 kg (m/s)^{2}

So the conversion of units (‘unit algebra’) is very easy — we can just swap the kg(m/s)^{2} for J using 1 = J/[kg(m/s)^{2}]. And Voila: we discover that the energy associated with the proton mass is about 5 x 10^{-19} Joules. (I know this as “approximately 1 GeV.”)

Hope that helps! Thanks for the great question.

– Flip