# How can one write a line element for non-integer dimensions?

Is there a meaningful way to write a line element $ds^2=....$ for a spacetime with a non-integer number of dimensions?

I would imagine that a line element in a 1+2.n dimensional spacetime might look something like $ds^2=-dt^2+a^2 (dx^2 + dy^2) + b^2 dz^2$ where $dx$ and $dy$ are "regular" spacial dimensions, $dz$ is another direction with fractional dimensionality, and a and b are some functions (which I don't want to specify further since I am working with an unusual line element). I chose to square a and b to ensure that x and y remain space-like.

I will admit, though, that I am guessing at how to describe the line element. (Or, are there any resources or fields of physics/math to look into that can be useful to addressing this question.)

Edit #1: I'd like to add that if I can think of a function (let's call it b(t)) that smoothly transitions from 2-->3 and describes a change in dimensionality, maybe I can write $ds^2=-dt^2+ (dx^2 + dy^2) + (b-2)^2 dz^2$, which would in effect "turn the z dimension on or off". In this case, when b=2, the z dimension is turned off, and when b=3, we have regular Minkowski spacetime.

• Would Mathematics be a better home for this question? Feb 9, 2018 at 16:26
• Quite possibly, I will ask there too.
– Bob
Feb 9, 2018 at 16:31
• For any value of $b$ other than 2 in your second expression, you still have 3+1-dimensional Minkowski space: you can define a new coordinate $\bar{z} = (b-2) z$, and then the metric becomes $ds^2 = -dt^2 + dx^2 + dy^2 + d\bar{z}^2$. So that method won't work. Feb 9, 2018 at 17:26
• @MichaelSeifert: Thanks for the comment. I intend on having b as a function with 2 and 3 as it's limits. When $b=2$ or $b=3$, we would have 2+1 or 3+1 Minkowski space, but when $2<b<3$, we would probably have not have Minkowski space (we would have some other non-isotropic spacetime). If I write b(t) (for example), than I would explicitly have $\overline{z}$ as a function of time also. In changing the original metric $g_{\mu\nu}$ to a new metric $\overline{g}_{\mu\nu}$, wouldn't that still lead to noticeable effects from $T_{\mu\nu}=\frac{-2}{\sqrt{g}}\frac{\delta S_{mat}}{\delta g^{\mu\nu}}$?
– Bob
Feb 9, 2018 at 20:52
• Related: Can a fractal be a manifold? The answer appears to be "no", at least not in the usual sense. And if you don't have a manifold structure on your set, it's difficult to see how you could generalize it to get a metric. Feb 9, 2018 at 21:44