Are tesseracts and hyperspheres ‘time’? Or is it completely different?
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Here are some analogies I linked to time:
You can lift up something out of a 2D drawing in the third dimension. Can something in our world be moved to the past/future?
A dot on a circle can move in one dimension, and if it does so far enough, it comes back to the starting point because the space is curved. On earth you can move in two dimensions curved, move far enough and you come back to the starting point. In 3D space, does time allow you to go back to the starting point?
I think there’s a misconception that time is the 4th dimension. Its merely an added dimension to the 3rd so that we can perceive space time.
An object in 3D space can be marked by a set of coordinates (x, y, z). Imagine a red ball in (3, 4, 5). Now a few seconds later, it moves away and we have a blue ball in the same space (3, 4, 5).
This is confusing, so we add a 4th “dimension” called time to help us separate the two in space time.
(x, y, z, t)
So when t=1, red ball was (3, 4, 5, 1).
Then when t=2, let’s say red ball goes to (4, 4, 4, 2) and blue ball is in (3, 4, 5, 2)
This is a simple way to view space time.
Time is not a spatial dimension, the way it’s usually written out is “3+1” dimensions as a shorthand for “3 spatial + 1 temporal dimensions.” So no, a tesseract, aka a hypercube, is the extension of a cube into four *spatial* dimensions.
> You can lift up something out of a 2D drawing in the third dimension. Can something in our world be moved to the past/future?
I assume you mean that you can turn a drawing of a square into a drawing of a cube, something like that? What’s happening there isn’t really a change from a 2D drawing to a 3D drawing, it’s changing your drawing into a 2D representation of a 3D object. That has no implications for time.
> A dot on a circle can move in one dimension, and if it does so far enough, it comes back to the starting point because the space is curved.
It may be that our universe is like that, in 3 spatial dimensions, but those are *spatial* dimensions. Without proceeding faster than light, you will never arrive before you left, and FTL is forbidden,
> On earth you can move in two dimensions curved, move far enough and you come back to the starting point. In 3D space, does time allow you to go back to the starting point?
No, but if our 3D world is embedded in a 4 or more spatial dimensions then from “outside” of the 3D structure looking in, you would potentially be able to treat time as another axis you could travel on.
Time is not quite the same as a fourth spatial dimension.
The most common way to think about time being a fourth dimension is through the mechanism of something called *Minkowski space*, a mechanism first invented to help describe Lorentz contraction, a phenomenon that got folded into Special Relativity.
In Minkowski space, there are three “real” dimensions, being the three spatial dimensions, and an “imaginary” dimension, time. It’s called that because when you calculate distances, you sum the squares of the three spatial dimensions (like usual), but you actually **subtract** the time component. This means that in a Minkowski space, if two events are 1 light-second away from each other in space, and one second away from each other in time, the “distance” between them is zero!
Here, one interpretation of “distance” is as a sort of a measure of how easily light can pass between things – if the distance between objects is zero or negative, then light can pass between them such that one of these things can affect the other; distance being greater than zero means that they’re too far apart for light to pass between them in time.
Because of this “negative dimension”, hypercubes don’t work the way you’d want them to, and “hyperspheres” actually take the form of hyperboloids.
I’ll address your other specific questions.
>You can lift up something out of a 2D drawing in the third dimension. Can something in our world be moved to the past/future?
Yes, this is called “the passage of time”, whereby all objects tend to exist across an entire interval of time, even as they are often located in only a single point in space at any moment, so that an object’s “trajectory” in Minkowski space tends to be a 1 dimensional-object (called a “world line”).
>A dot on a circle can move in one dimension, and if it does so far enough, it comes back to the starting point because the space is curved. On earth you can move in two dimensions curved, move far enough and you come back to the starting point. In 3D space, does time allow you to go back to the starting point?
No, because time doesn’t curve that way.
What you’re describing is how things move on a surface of *positive curvature*; it is not a requirement that a space work that way. In fact, if you neglect the time component, 3D space by itself does not seem to work that way; instead, if you keep going, you’ll just…keep going. Space-time, as it happens, actually has *negative* curvature, so this will tend not to happen.
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.
The main thing that time has going for it which the spatial dimensions do not is that time *seems* to be asymmetrical. That is, there are certain processes which can happen going forward in time but not going backwards.
For example, if you drop a bouncy ball on the ground and watch it bounce, each time it bounces it will bounce a little bit less high. That’s because as it’s moving through the air and hitting the ground, some of the ball’s energy is constantly being lost to the environment. Now, it would be really really weird if a bouncy ball was sitting motionless on the ground, then spontaneously started bouncing, higher and higher until it landed in your hand.
That observation is captured in the 2nd law of thermodynamics which states that entropy in the universe is constantly increasing. What that means is that energy in the universe tends to spread out over time, rather than concentrating in one place. It’s natural for the ball to spread its energy into the environment but not the other way around. Since entropy only increases as time goes forward, moving forward in time does not have the same rules as moving backward in time, which is very unlike the spatial dimensions, where moving in any direction is pretty much the same.
No, time is its own dimension. We know time is unique because it only flows in one direction, whereas you can travel either direction in either dimension of space. This is what makes it a spacial dimension.
People called time “the 4th dimension” but our universe has 3 spatial dimensions and 1 time dimension, that’s where you get the title of 4th.
String theory (which isn’t experimentally proven, so it should be the string hypothesis) suggests 10 spacial dimensions and 1 time dimension, but physics is moving away from this idea.
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