 # Units of Measurement and E=M

The c in the famous equation is the speed of light, typically in kilometers per second. If the unit employed is, instead, lightyears per year, c squared is one times one, or one. Thus, E=M. Which is basically what the equation is saying, that mass and energy are equivalent.

To which column is this replying?

No … if we square 1 light-year per year we have 1 light-year squared per year squared … multiplying 1 kilogram by 1 light-year squared per year squared gives 1 kilogram light-year squared per year squared. This is fine as long as we remember that there would be 8 x 10[sup]31[/sup] Joules (kg m[sup]2[/sup] s[sup]-2[/sup]) in 1 kilogram light-year squared per year squared.

Hi Murfyn and welcome to the Straight Dope Message Board!

In this particular forum, we generally try to provide a link to the column in question, so everyone is on the same page. No harm no foul though.

Here’s the link to the SDSAB article: “In E=mc2, what units of measurement was Einstein using?”

The thing about E=Mc^2, at least for me, is how much energy is equal to a small amount of mass. I remember being a kid and finally understanding the ramifications: energy = mass times the speed of light squared?!?!. Suddenly nuclear power (and bombs) made sense.

Hm, I’d forgotten about that Staff Report. To the answer therein, I’d add that the units used for mass and speed determine the units for energy. If you input mass in kilograms and speed in meters per second, then you’ll get energy in units of kgm^2/s^2. And it happens that, since physicists often use those units, that unit of energy shows up quite often: So often that we give it its own name, and call it a joule. Similarly, if you input mass in grams and speed in centimeters per second, you’ll get energy in units of gcm^2/s^2, and that’s a unit that shows up quite a lot, too, and so it also gets its own name, the erg. By the same token, if you input mass in slugs and speed in rods per fortnight, then you’ll get energy in units of slug*rod^2/fortnight^2. That’s also a perfectly valid unit of energy, but it’s one that nobody ever uses except pedagogues making a rhetorical point, and so nobody has ever bothered to give that unit its own name.

If we can express mass in units of energy it would convert into, can we also express energy in units of mass it would convert back into? Like \$200 billion for a kilogram of electric service …

Isn’t this basically what particle physicists are doing when they talk about Electron Volts?

No, that’s the other way around.

And we can measure energy in mass units. For instance, the recently-detected gravitational wave had an energy of three solar masses.

FYI, all the exponents in that column have reverted to numbers, really screwing up the relationships.

Anyway, no, that’s fallacious. The use of “canonical units” is useful in computation. In orbital mechanics, if we use “astronomical units” and “years” then we can eliminate the pesky constant of gravitation: it comes to “1” in our calculations, so we can set it aside.

But it’s still there! The n-m^2 / kg^2 still hangs around, and is vital to the actual physics. All we’re doing is simplifying the equations by choosing easy-to-use constants.

A year isn’t “equal” to an astronomical unit, even if, for earth, Kepler’s third equation says, when Gm1m2 = 1, that they are “the same.”

But there is a more fundamental sense in which c really is equal to 1, in a way that AUs or years aren’t.

Or, arguably, i.

The use of units of energy to describe mass or units of mass to describe energy baesd on four-momentum is always unambiguous. Or in other words the use of different units for energy and mass is rather artificial.

You could reduce the argument that c=1 to the fact that when constructing Minkwoski space from the Lorentz transformations c acts as the unit length/norm and the norm is always a real number.

Then again, if you use c = i, then you don’t even need to use the Minkowski metric. I can see the logic.

Perhaps it’d be best to just say that the magnitude of c is 1.

If you still want the mass in kg but want c to be light years per year you have, in effect multiplied one side of the equation by 1/90,000,000,000. Basic algebra requires you multiple the other side by the same factor or change the energy unit of measure:

e/90,000,000,000 = m is true but doesn’t have the same elegance.

No, you haven’t multiplied it by 1/90,000,000,000. You seem to think that c = 300,000. It’s not.

E ≠ M

E bears a proportional relationship to M. There is a linear relationship between E and M that accounts for the scaling difference between the two.

While the relationship can be written without any units, it is understood that the units chosen must be internally consistent, or else additional unit correction factors must be used.

Now getting that there is a relationship in which mass and energy are the same “stuff” at some fundamental level is a good insight, but you can’t drop the c[sup]2[/sup], any more than you can drop any other unit conversion factor.

3 inches ≠ 3 cm, even though L = L. Only one of those values at most accurately describes L.

c is defined as the speed of light in a vacuum, which has different values in different systems of units, just like an object’s length will have a different value for different units chosen to express it.

Certain systems of units can set c=1, but that is simply chosen to make the calculations simpler. It does not remove the relationship of the speed of light from the equation.

Of course 3 inches ≠ 3 cm. That’s not remotely analogous to saying that c = 1. Saying c = 1 does not mean that 1 m = 1 s; it means that 3e8 m = 1 s, just like 2.54 cm = 1 in.