# Why c squared?

I would imagine this topic has come up before, but, why c squared? As in Einstein’s equation E = m c squared. Is c squared the EXACT constant that solves this equation, or just a rough estimate? If it is exact, how in the world did he come up with c squared as the value?

Because that is just how this universe is. I wasn’t a guess.

This is neither a proof nor a derivation but it has to be c[sup]2[/sup] in order for the product with mass to work out as energy.

From Newton’s equation F = ma. Using English units force is lb. and acceleration is ft/sec[sup]2[/sup] so the units for mass is lb*sec[sup]2[/sup]/ft.

multiplying lbsec[sup]2[/sup]/ft by ft[sup]2[/sup]/sec[sup]2[/sup] tives lfft which is the unit for energy.

c is a constant–specifically, the speed of light in a vacuum–and figured handily into a number of equations pertaining to electromagnetism long before Einstein’s most famous equation. We could say that E=mK, with K=c[sup]2[/sup] and sometimes we do this with constants in order to normalize (i.e. make them cancel out or occur with integer multiples rather than fractions) them with respect to other constants, as is done with Planck’s Constant and the Dirac Constant, but as it happens the quantity c happily normalizes a lot of other relations, especially in relativity, so physicists just learn to cope with the squared term all over the place, the poor slobs.

Check out the last two posts of [post=6114384]this[/post] thread for a more extensive explaination as to “Why E=mc[sup]2[/sup]?”

Stranger

multiplying lbsec[sup]2[/sup]/ft by ft[sup]2[/sup]/sec[sup]2[/sup] gives lfft which is the unit for energy.

Yahbut that only explains why you’d have to multiply by a speed squared, not why that particular speed squared.

OK. I said that I was only showing that it had to be squared.

c is the constant because of the Fitzgerald ratio 1/sqrt(1 - v[sup]2[/sup]/c[sup]2[/sup]). This can be written as (1 - v[sup]2[/sup]/c[sup]2[/sup])[sup]-1/2[/sup]

The latter form can be expanded by the Binomial theorem and for velocities that a low compared to c the result is:

(1 - v[sup]2[/sup]/c[sup]2[/sup])[sup]-1/2[/sup] = 1 + v[sup]2[/sup]/2c[sup]2[/sup] to a close approximation.

The increase in mass because of velocity as discovered by Lorenz is the zero velocity mass multiplied by the Fitzgerald ratio. For low velocities this is:

m[sub]1[/sub] = m[sub]0[/sub](1 + v[sup]2[/sup]/2c[sup]2[/sup])

multiply out the right side and subtract m[sub]0[/sub] from both sides:

m[sub]1[/sub] - m[sub]0[/sub] = m[sub]0[/sub]v[sup]2[/sup]/2c[sup]2[/sup]

the left hand side is the change in mass with velocity which I’ll call just plain m and the right hand side is the kinetic energy divided by c[sup]2[/sup].

m = e/c[sup]2[/sup] or e = mc[sup]2[/sup]

There are probably more advanced ways to make the equivalence but this one is good enough for me.

Got a question. Did the OP really want to know “why c[sup]2[/sup]” or was the post just a way to kill 2 minutes during a dull day?

There’s no complete answer to the OP that doesn’t involve a whole lot of math. Presuming that the OP isn’t really interested in learning all that (if I’m wrong, I recommend pursuing a Bachelor of Science in physics, rather than querying message boards), then this is the most basic thing I can come up with. The whole of Special Relativity can be derived from the assumption that the speed of a beam of light in vacuum will be the same–c--when measured from any reference frame, relative to that reference frame. (I’m not going to get into the reasons for making this assumption.) This means that if I aim a flashlight at a person moving towards me at half of this constant speed, I’ll measure the speed of the beam moving away from me at c, and so my measurement of the relative speed of that person and the beam will be 1.5 c. But according to his measurements, the beam is moving toward him at a relative speed of c, and since I’m moving toward him (from his point of view) at 0.5 c, the light beam is moving relative to me at 0.5 c.

Obviously, the value of c is pretty important, given this assumption, and it figures in the equations of Special and General Relativity at every point … The exact value of c isn’t necessarily important to the theory (although it will, of course, be needed to determine practical implications). It just stands for the speed of light in a vacuum, whatever that speed is.

One can answer, because that’s the only speed in the Universe. Any other speed can only be a relative speed, in some reference frame (and the Universe doesn’t specify any special reference frame to use). But c is c, no matter how you look at it.

Personally, I like the derrivation from Maxwell’s equations. No understanding of relativity is required.

Yes, I like that site too.

I was just wondering why the speed of light had anything to do with relationship between mass and energy.

Well, if you were Einstein you wouldn’t wonder, you’d figure it out.

Qualitatively, think of it this way: once “c” becomes a universal constant, it shows up all over the place. In Newtonian mechanics, “c” is just any old number, like the speed of sound or the speed of a wave on the ocean. It doesn’t have any special meaning, and only shows up in optical equations. But in order for “c” to be invarient in all reference frames, it has to creep into equations that we intuitively think have nothing to do with light.

I can’t think of any other way to explain it w/o getting bogged down in lots of math.

Do you know what is meant by “the speed of light in a vacuum is invarient in all inertial reference frames”? That is how “c” gets tangled up in mass and energy.

Did anyone else listen to “This American Life” on Sunday? One of the stories was about this autodidactic crank that was convinced Einstein was wrong and he could prove it. Is this what prompted the op?

I heard that. It was sort of sad to listen to this guy go on and on about this Ivory tower of complex math keeping the physicists from this fundamental truth. Then listening to some of the conversations between him and the physicist discussing how important it was to get the units to match on both sides of the equation and realize that it wasn’t the complex math that was the problem but the simple high school math that was the problem.

The speed of light in a vacuum is invariant in all inertial reference frames, therefore it must have some relationship with the ratio of energy to mass? Somehow, this doesn’t seem obvious to me.

It it not an obvious consequence of the constancy of the speed of light in vacuo, which is why Einstein’s peers were blind to it until he pointed it out in the fourth of his revolutionary 1905 papers (centenary this year).

Special relativity, of which E=mc[sup]2[/sup] is but a part, takes two facts:[ul][li]Nobody is really standing still, and the laws of physics are symmetrical for all such reference frames.[/li]The speed of light in vacuo is constant.[/ul]…and explores the consequences of these facts. One of these consequences, via some mathemtical manipulation, is E=mc[sup]2[/sup]. The kinetic energy of a body is (1/2)mv[sup]2[/sup], but there must also be a relativistic element to this quantity. ie. E= [symbol]g[/symbol]mc[sup]2[/sup] - mc[sup]2[/sup], which reduces to just mc[sup]2[/sup] for the mass at rest.

I heard TAL Saturday as well, one of my favorite radio shows as well. I didn’t think of his error as a math problem but as a conceptual one, using momentum which may seem more intuitive to some people than kinetic energy. I had trouble with that in my first physics course but had a good instructor who helped me make sense of it all. I have gotten into huge usenet wars with cranks trying to explain so-called over unity electronic devices because they don’t understand how capacitors charge and discharge which I think parallels the momentum vs KE confusoin the guy on the radio show had. I don’t think you can understand any physics until you understand the definitions and relationships of force, power, energy and work and unfortunately that doesn’t seem to be common knowledge.

And furthermore, everything in the universe moves at c through spacetime; even objects at rest (with respect to an inertial reference frame) are moving at velocity c on the time axis. Objects in motion, with respect to the reference frame are moving in time more slowly, owing to relativistic effects at very high velocities. This is why a muon formed in the upper atmosphere by collision of a cosmic ray and with a lifetime of only 2.2 microseconds can last long enough to be detected at the Earth’s surface.

That we commonly refer to c as “the speed of light” gives the incorrect impression that this is its first “function” and therefrom arises the notion that other relationships to c are peculiar or uncanny. c is a fundamental quality of all energy and matter, not just a velocity that the spaceships of science fiction writers aspire to achieve or somehow exceed. That it happens to be the speed at which photons move (with respect to all inertial frames) is a conclusion of Special Relativity but it isn’t the fundamental property of c, any more than a baseball need be described in terms relating a bowling ball because they’re both spherical; it is their common property (being a sphere) that gives both their characteristic nature and functionality.

Stranger

I didn’t say it would be obvious, just “not too surprising”. There is something very fundamental about the speed of light in terms of how the universe “works”. Nothing can go faster than light, so there should be some equations to prevent that. For instance, if we look at Newton’s famous law F=ma from which we derive the equation for kinetic energy E =( mv^2)/2, there is nothing in those equations to prevent v > c. As it turns out, one consequence of the speed of light being invarient is that the mass of an object increases as it approaches the speed of light. In fact, if it could hit the speed of light, it would have infinite mass.

We need to substitute the relativistic mass into Newton’s equation to make it obey relativist laws. Let’s call that new mass m’. It turns out that:

m’ = m/sqrt(1 - v^2/c^2); where m = rest mass (what you measure when you’re in the same inertial reference frame as the object)

what happens when v -> c? You get 0 in the denominator, or (losely) one could say that the mass becomes infinite.

That’s how “c” creeps into the relation between mass and energy (which is pretty much what David Simmons already posted, above).