# Speed of Light, Squared

CarlB
Homework Helper
Swampeast Mike said:
I've always been troubled with the idea of squaring that speed (C ^ 2) unless speed of light to the zero power (C ^ 0) also has meaning at the same time and in the same space; the net effect to an observer fixed in time being 1 = 1 where 0 = 0.

I have no idea why this would trouble you, however, it does turn out that there is a branch of mathematics / physics that does something sort of similar to this. In D. Hestenes' "Geometric Algebra", the dot product and the cross product are combined into a single "multi-vector" operation. Hundreds of physicists and mathematicians use his theory. Hestenes' website is here:
http://modelingnts.la.asu.edu/

Geometric algebra is a subdivision of Clifford algebra. Where the standard physics education meets Clifford algebra is in the Pauli or Dirac gamma matrices, which are examples of geometric algebras defined on the manifold of 3-space and 4-dimensional space-time, respectively.

The reason I'm bringing this up is because the Geometric algebra crosses the usual boundaries of scalars, vectors and tensors. In the GA, for example, one can add a vector to a scalar and get what is called a "multivector". The laws of E&M, for example, can be written with very few symbols in this manner.

I disagree with the poster who said that "c" is a velocity. I believe instead that it is a speed. A velocity has a direction, "c" does not. But if you think of "c" as a velocity, then the conversion that takes you from c to c squared is a dot product.

The other half of a dot product, in Geometric algebra, is a cross product. In standard physics, a cross product takes two vectors and turns them into a "psuedovector". Undergraduates used to be taught that the result of a cross product is a "vector", but in grad school they get taught differently.

Now the operation of squaring a vector and getting back a scalar takes anobject of dimension 1 and turns it into an object of dimension 0. This is mighty odd stuff. To put it back into Dirac gamma matrix form, this gets back to the fact that the square of a gamma matrix is unity.

In the context of the gamma matrices, "unity" really means a 4x4 matrix with ones down the diagonal. That makes sense to me. But Clifford / geometric algebra is written without reference to matrices, and to me the implications of scaling laws with them is odd in the same way that the Swampeast Mike's comment is odd. Two things I do not understand that seem to be for the same reason.

Carl

Carl is of course right with his speed-versus-velocity remark. I made the same error in one of my posts (as a bad excuse I could say that in Dutch the distinction doesn't exist ).
Mortimer said:
Look e.g. at $E=mc^2$ which is usually written more correctly as $E/c=\sqrt{(m_0c)^2+p^2}$. The $c$ on the left part is a scaling factor while the $c$ in the right part is a velocity. (Here it is a velocity if you ask me)
In $t'=\gamma(t-vx/c^2)=\gamma(t-(v/c)(x/c))$, $v/c$ is a ratio of velocities (here it is a speed) while in $x/c$ the $c$ is a correction in the scale of $x$.
Try it. You'll find for most cases either this separation in a velocity and a scaling factor, while in other cases you may find the square to have its roots in a Minkowski equivalent of a "Pythagorean" operation ($A^2=C^2-B^2$ for Minkowski space, equivalent to $C^2=A^2+B^2$ for Euclidean space).
The point I was trying to make still stands, by the way. If anyone can give examples that do not fit the criteria I described, I would be interested.

Last edited:
Dear Mike : There is no physics in this post, or in the OP.

I disagree. This is the most fundamental of physical questions when space and time are relative.

The speed of light (C) is also the speed of perfect conduction.

Would you prefer $$E = \frac{m}{\epsilon_0 \mu_0}$$ instead?

Kino
I like the idea of a scaling factor: E/c = mc, energy divided by c is equivalent to mass multiplied by c.

But perhaps Swampeast Mike, in his original post, mistakenly assumed that the energy was only released when the mass was actually moving at c-squared? That would perhaps explain his confusion...

HallsofIvy
Homework Helper
Swampeast Mike said:
My problem wasn't understanding the "speed of light squared" it was understanding how such works as an individual concept relative to both time and space.

If light can be both a something and a nothing, cannot its' speed be only measurable as a product? a sum? a difference? all? some?

Energy, radiation, light or whatever you want to call it always occurs BOTH WAYS between separated bodies yet we tend to conviently overlook the view of the other observer.

This completely mystifies me! Who said that "light can be both a something and a nothing? In what sense can light be "a nothing"?
Products, sum, differences, etc. are mathematical calculations. They have nothing to do with measuring the speed of light.
Finally, "energy, radiation, light or whatever you want to call it" doesn't "occur", it MOVES from one body to another- while I'm sure light COULD go both ways that is certainly not necessary. Do you have any evidence for this assertion?

HallsofIvy said:
This completely mystifies me! Who said that "light can be both a something and a nothing? In what sense can light be "a nothing"?
Products, sum, differences, etc. are mathematical calculations. They have nothing to do with measuring the speed of light.
Finally, "energy, radiation, light or whatever you want to call it" doesn't "occur", it MOVES from one body to another- while I'm sure light COULD go both ways that is certainly not necessary. Do you have any evidence for this assertion?

Is light not a photon? Is a photon not both a something and a nothing? Or at least has the characteristics of both while being undefinable as either?

Light is a "nothing" in the sense that only nothing can travel at its' speed.

There are many forms of photons--light just happens to be the one that our eyes see. There are infrared photons, radio photons, ultraviolet photons, x-ray photons.

While energy in the form of light might not necessarily move both ways between bodies, energy in some form must. Evidence: energy in the form of heat transferring between bodies--you MUST subtract the energy moving from the cooler to the warmer object.

Swampeast Mike said:
Is light not a photon? Is a photon not both a something and a nothing? Or at least has the characteristics of both while being undefinable as either?
Perhaps you are thinking of the fact that light (and everything) has characteristics of both particles and waves, but both particles and waves are something.

Swampeast Mike said:
Light is a "nothing" in the sense that only nothing can travel at its' speed.
Light doesn't have mass, but that still doesn't mean it's nothing. If it was nothing there would be no reason to say it travels and we wouldn't be talking about it. It doesn't exhibit any characteristics of "a nothing" because a nothing doesn't have characteristics to exhibit.

russ_watters
Mentor
There is an irony here, Mike - it seems to me that you are trying to argue against some philosophical aspect of light, but its you who is injecting that philosophical aspect by saying things that are factually wrong about the nature of light! Bizarre. Anyway, if you stopped looking for philosophical contradictions for a little while and learned what is actually known about the nature of light, I think these issues you are having would drop away.

Maybe that's overly optomistic though: to say that light is "nothing" is inexplicably bad science and philosophy. I can't fathom why you would suggest such a thing.

Last edited:
Really not trying to ask philosophic questions and I apologize if my vocabulary or nomenclature is failing me. If I am offering utter falsehoods, please let me know as that is not my desire.

Rather hard for me to consider a massless bundle of energy as being an object (or thing) in conventional terms. While you can certainly tell that I'm not a trained physicist, I have been studying deeply and intently for years before posing these questions.

Unless there has been some recent breakthrough of which I am unaware, there are still MANY unanswered questions about the nature of energy transfer between objects.

Am working on a book regarding hydronic (hot water) space heating for structures and much of it is devoted to the proportional delivery and production of heat. As I research and experiment, I find myself getting deeper and deeper into physics so that I can at least begin to understand the underlying principles involved.

Everything I have posted at this forum is related in some way to heating systems. A recent boiler change in my home produced some very strange anomolies of easily measurable magnitude.

Seeming simple questions like how radiation passes through window glass find (in my opinion) poor answers. I ask if a visible light photon passes through such or if it is absorbed on one surface and propagated out the other. Am told that it doesn't matter because the photons are indistinguishable. Only the apparent result matters, not the process? When we're talking about things that do not necessarily conform to our sense of time and space how (when the process is uncertain) can we be sure that the observable result is the only result?

I don't have a philosophic problem with light–I have a problem understanding photons.

Again, please tell me if I am making factual errors when I say that:

1) Light is a form of energy that travels via photons–just like radio photons, IR photons, etc. Our eyes just happen to be sensitive to energy in the frequency range of visible light.

2) Energy moves between bodies in waves–waves that vary in both amplitude and frequency.

3) This energy travels in discreet bundles at an immutable velocity regardless of how the waveform changes in our perception of length with amplitude and frequency.

4) This energy is always moving both ways between objects.

4) In some (and perhaps all) cases this waveform is not simple–it is composed of two distinct waves in direct opposition (at right angles to) one another.

4) We are immersed in a sea of energy that does not conform to our sense of time and distance.

russ_watters
Mentor
Swampeast Mike said:
Rather hard for me to consider a massless bundle of energy as being an object (or thing) in conventional terms.
No one ever said light was an object, just that it is something, and not nothing. You're being really loose with your word usage here.
When we're talking about things that do not necessarily conform to our sense of time and space how (when the process is uncertain) can we be sure that the observable result is the only result?
That's a philosophical question (what is science?). The answer is that in order for science to work, we cannot randomly assume the existence of things for which we have no data. Thus your question of what goes on with radiation through a window is moot: if the energy measured on one side is the same as that measured on the other, then there can be only one conclusion: the photons are interchangeable.

As for "conform[ing] to our sense of time and space", you're looking at science backwards: you must conform your "sense" to the "observable result," not the other way around. If your "sense" doesn't match the observable result, then its wrong, period.
I don't have a philosophic problem with light–I have a problem understanding photons.
You have a problem with both - you don't understand light, and you don't understand how science works. As a result, you're making things up as you go along and not really listening to people who are trying to teach you.
Again, please tell me if I am making factual errors when I say that:
Ok...
1. Correct.
2. Not specific enough. If you're talking about em radiation (light, radio waves, etc), saying it is a wave is an incomplete (at best) explanation of what a "photon" is.
4. Again, if you mean em radiation, essentially yes - objects radiate energy as a function of their temperature.
5. Your characterization of light as waves is inaccurate/incomplete. That statement is pretty much meaningless.
6. The nature of light conforms to my "sense" just fine. Again, if it doesn't conform to your "sense", you need to change your "sense" to conform to it. Human knowledge is all learned. Apparently, you have learned incorrectly (not terribly surprising, since in high school, they teach a version so watered down it gives people incorrect impressions of what is really going on).

"To every action there is an equal and opposite reaction"

E = M (C^2)

Where is the opposite when time and space are relative across the equivalence?

If time and space were the same, the first atomic bomb would have unified the universe.

russ_watters
Mentor
Mike, that entire post is gibberish. To use my word-of-the-week, its meaningless word-salad.

jtbell
Mentor
Swampeast Mike said:
"To every action there is an equal and opposite reaction"

E = M (C^2)

Where is the opposite when time and space are relative across the equivalence?

"Action" and "reaction" in your statement of Newton's Third Law of Motion refer specifically to the forces (as in F = ma) that objects exert on each other. You are generalizing those words into a context where the statement has no physical meaning.

A more explicitly restricted statement of the Third Law is: "If object A exerts a force on object B, then object B exerts a force on object A which is equal in magnitude and opposite in direction."

Janus
Staff Emeritus