# General definition of spatial dimension?

• I
Gold Member
The question is not to distinguish space from time, but in general, what distinguishes a spatial dimension from other types of dimensions? For example, Hilbert space has an infinite number of dimensions, but they are not spatial; string theories add extra spatial dimensions. Is there a rigorous definition for spatial dimensions? Otherwise put: what makes the extra dimensions in string theories spatial?

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.Scott
Homework Helper
As I understand it, the basic models use only one time dimension. All other dimensions are f(t).
Often, "spacial dimensions" beyond the first three are described as "flattened" to the point where macroscopic travel through them is meaningless.

Gold Member
Thanks for the reply, .Scott, but I think you missed the point of my question. (By the way, when you said "flattened", did you mean "compactified", i.e, curled up, i.e., periodic with a small period?) A dimension of a structure (to avoid the use of "space" in two meanings in this post) is basically one of the smallest number of independent variables used to specify any element in the set of the structure. One variable would be, for example, the electric charge, but that is not a spatial dimension. There are new independent variables introduced into the equations of string theory, but why do we label them "spatial"?

Dr. Courtney
Gold Member
Units of length.

Gold Member
Dr. Courtney, if I understand your cryptic answer, you are saying that any vector space that has a metric defined on it (and hence units of length) should be called "spatial" (in the sense of spacetime)? But according to that, the infinite dimensions of the Hilbert space used in quantum mechanics would be called spatial -- but they are not.

Dr. Courtney
Gold Member
Dr. Courtney, if I understand your cryptic answer, you are saying that any vector space that has a metric defined on it (and hence units of length) should be called "spatial" (in the sense of spacetime)? But according to that, the infinite dimensions of the Hilbert space used in quantum mechanics would be called spatial -- but they are not.
By units of length, I mean a literal length - meters in the SI system. Usually, the vectors in Hilbert spate in quantum mechanics are wave functions (or a normalized sum of wave functions that form an orthonormal basis). Wave functions in QM do not have units of length. See:

https://physics.stackexchange.com/q...the-3-dimensional-position-space-wavefunction

• A spatial dimension is a direction that is not a vector sum of any other directions (i.e. a 90º angle for example).
Categorizing spatial and temporal dimensions is actually a carry-over from classical physics that refers to Euclidean space. However, Minkowski space treat space and time dimensions alike. We still think of spatial and temporal dimensions separately because we can move freely through the spatial dimensions, but seem plastered down to the temporal dimension. However, suppose there were 5 or 6 dimensional beings; they could potentially walk up their sidewalk into our future, turn around and walk down their sidewalk into our past.
Still, like other dimensions, time is represented as a 90º angle to x, y, & z. You see this depicted in light-cone diagrams that we use to illustrate causality of events.

By units of length, I mean a literal length - meters in the SI system.
I don't think that is enough for the OP. In relativity the time dimension can also use units of length or you could also use seconds as a unit for both time and space dimensions. There is a difference between time and spatial dimensions in that the signature of the manifold is ( - + + +). I don't know how to generalise that difference though.

Cheers

• Mister T
Gold Member
We still think of spatial and temporal dimensions separately because we can move freely through the spatial dimensions, but seem plastered down to the temporal dimension.
Note that you can change your position, but you can't change your clock-reading. Put another way, if you're not happy with our position you can change it by moving, but if you are happy with it you can keep it by not moving. If you're not happy with your clock-reading, all you can do is wait while it changes in a predictable and consistent manner, if you are happy with it you can't keep it because it will change.

Could this be the answer to the OP's question? A spatial dimension is a dimension that's measured in units of length because that's what changes as a result of your motion. I'm aware of the circularity, that this is the definition of what it means to move. But I think this is a circularity that must be present when defining any dimension. Like time is the thing we measure with a clock, and a clock is the thing we use to measure time.

Gold Member
Sorry for the delay in answering.
I am not sure I quite understand Mr. T's response: if you move, you change several quantities -- any operator which is not invariant under translation, for example. As well, when you move, you are changing your coordinates in spacetime. Or am I missing something here?
Then, I thought the idea of dimensional analysis (or, do the maths and see what units come out in the wash) seemed nice, but then cosmik debris threw a spoke in that reasoning.
So I am still a little unsure....

Mister T
Gold Member
As well, when you move, you are changing your coordinates in spacetime.
Yes but note that you can, for example, keep your position coordinates the same by simply not moving. But you cannot do such a thing with your time coordinate.

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