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:
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.