Are tesseracts and hyperspheres ‘time’? Or is it completely different?
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So lets understand what a dimension is:
The universe is quite apparently geometric. We have a bunch of shapes around. This is poorly extendable to atoms and subatomic particles but lets stick to our macroscopic world. So we have geometry but we want equations. There aren’t many tools involving geometry that’s super useful for understanding physical systems.
So lets turn our geometric arrangements into equations. Thats what coordinate systems are for. And the dimensions are basically stating how many independent coordinates we need to describe a point in our coordinate system. (Precisely we are talking about the minimally required and maximally definable basis vectors. For a physical arrangement you can at most have 3. But for something like an orbit you could use a 2D subspace of the 3D world.)
A coordinate system is a kind of vector space. And vectors are mathematically very well behaved. So with a coordinate system we gain access to more tools.
So can we add a 4th dimension and call it time. Yes and not quite. Its a differential formalism. It’s not about quantifying a geometric arrangement of a thing but geometrically looking at events. But the formalism is different. It’s not just vectors with 4 components we have a buch of extra rules. We are only taking 4D spacetime because of how the formalism works. We introduce four vectors with the 0th time like component and the 3 spacial coordinates. But for example what we call lenght isn’t what linear algebra considers lenght.
So where does the four vector formalism comes from. Coordinate transformations. If we apply classical relativity, lets consider two coordinate systmes K and K’ K is our reference and relative to that K’ is moving at v velocity.
Lets see how we can transform K coordinates to K’ coordinates: If the start at a common origin and lets say that v point along the x axis the origin of K’ moves along the x axis at a rate of v. So a given point like the origin will have K’ coordinates like this: 0-vt. So at t=0 its x=x’=0 but x=0 get more and more negative x’ coordinates. So x’=x-vt and t’=t. As you can see we transom both time like and spacial coordinates. Of course in the Galilean transformation t’=t but that changes in SR.
So if we are doing t->t’ and r->r’ we can write it as a vector with four components and introduce a few additional concepts to tie up the lose ends. But the end result it a formalism that works very well. And you can even analyse things geometrically and construct the tool we call a spacetime diagram.
So yes we are using vectors with 4 components but the mathematics isn’t the same as a 4D linear space that we often think of as a coordinate system.
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