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

"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?

## closed as unclear what you're asking by ACuriousMind♦, Jon Custer, Gert, user36790, John RennieOct 1 '16 at 5:49

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• I don't understand what you mean by "include density as a dimension in space-time". – ACuriousMind Sep 30 '16 at 22:30

## 1 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.

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.

• Thanks for your answer. If you don't mind I'd like to ask another. Say you have a point "A" in space, and another point "B" in space 10 light years away in relatively flat space with regard to gravitational waves. Then you place a neutron star directly in the middle of point A and B. Now the distance between point A and point B is greater than 10 light years because of time dilation. Is that right? – Paul Oct 1 '16 at 0:29
• I'm not trying to solve a particular problem, just trying to get my head around GR, length contraction and time dilation. It just seems to me (a layman) that if mass density created a change in distance (due to gravitational waves), that it would need to be counted as a dimension. Alright I'm done asking questions, my head hurts. – Paul Oct 1 '16 at 0:33
• To understand GR you need to be thinking about space-time, not just space. And to understand light in GR you need to be thinking about interval. I'm told that most of the weak effects are due to changes in the time-like component rather than the space-time ones. – dmckee Oct 1 '16 at 0:36