spacetime - If mass density curves space-time, then why isn't density (at each $x$, $y$, $z$) considered a dimension in space-time?

From http://science.howstuffworks.com

"Theodor Kaluza theorized that a fourth spatial dimension might link general relativity and electromagnetic theory. But where would it go? Theoretical physicist Oskar Klein later revised the theory, proposing that the fourth dimension was merely curled up, while the other three spatial dimensions are extended."

We know high mass density creates gravitational waves that compress length and time (relative to an observer at lower ambient gravitational waves). Why not include density as a dimension in space-time?

Answer

Because it doesn't add another term to the length element?

One way to understand the dimensionality of space is to examine the nature of the length element. In three dimensional space (and with a Cartesian basis) it is $$ (\Delta s)^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 \;, $$ which you will note has three terms. In four-space it would be $$ (\Delta s)^2 = (\Delta w)^2 +(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 \;. $$

Density doesn't add a term to the length element, instead it effects the coefficients of the sum. Something like $$ (\Delta s)^2 = c_x(\Delta x)^2 + c_y(\Delta y)^2 + c_z(\Delta z)^2 \;, $$ with no guarantee that the $c$'s are unity. That's a different kind of modification to the meaning of space.

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Aside: This kind of definition has a couple of nice features. First, it extends smoothly to include intervals in Minkowski space or the more generalized space of general relativity. Second it allow a clear explanation of why the proposed compact dimensions of string theory don't appear to effect physics at human (or even nuclear) length scales.