I every so often hear claims like:
M-Theory predicts that there are 10 spatial dimensions!
Now I'm not really sure what these claims mean. There are three spatial dimensions that I normally observe so 7 of these dimensions must be different in some way. (Several seconds of researching reveals that the difference is that these dimensions are compact and have very small measure which explains why they are not apparent in every day life)
However there are a ton of dimensions that are not quite like our three spatial dimensions. Time is the one that very clearly springs to mind, but we could also talk about momentum. In fact dimension is pretty frequently used to mean unit (e.g. dimensionless, dimensional analysis).
So there must be some thing relevant about these 7 spatial dimensions that is not relevant with these other dimensions.
My question then is: What is the relevant factor that unites the spatial dimensions not present in other dimensions?
For example if I discovered (or theorized) a new degree of freedom, how would I determine if it corresponded a spatial dimension? What properties would it need to count as spatial?
Allure's answer helps explain the difference between spatial and time dimensions with respects to the Minkowski tensor, but I would like to see a more authoritative answer that would allow me to fully categorize a potential new dimension.
I am looking for a definition that is rather precise rather than intuitive. I have plenty of intuition as to what space is since I live in it and all the sources I can find a couple describe the intuition (e.g. the M-theory Wikipedium). I am curious about a more mathematized definition.
Some notes in response to comments:
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I am already very well aware of what a dimension in general and this question is asking about how we separate dimensions into spatial and non-spatial.
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I already understand compactification. I am not at all confused by the concept and I am not asking for an explanation of compact dimensions.
Best Answer
In this context a spatial "dimension" is not the same as the dimensions of momentum used in dimensional analysis. As you point out yourself, you can observe three dimensions - but you can't observe the dimensions of momentum.
The easiest way to think about these spatial dimensions is to look at it from the point of view of relativity. Ever since special relativity, we've had this equation that puts time and space on an equal footing:
$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$
This is the so-called Minkowski metric. Time ($t$) is different from space ($x, y, z$) because they differ by a minus sign.
The form of this equation obviously suggests a way to extend it to more dimensions. For example say you discover a new dimension, then we simply have:
$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2 \pm da^2$
If your new dimension is a temporal one, then it takes the minus sign, and if it's a spatial one, it takes the positive sign.
This explanation is pretty simplified (the Minkowski metric applies only in empty space for example) but the idea is there.