# Rotation in four dimensions

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So, most people think of rotation in 3D with respect to an axis, which is like rotating each 2D plane perpendicular to the axis. But relativity says we live in a 4D universe, although time is a rather special dimension.

Anyhow, I have several questions about rotation. What would each of the following look like?:

*Rotation in the real world. How is the time dimension affected?

*Rotation in 4 normal dimensions (called $R^4$?)

*Rotation in spacetime, along the time axis.

*Rotation in spacetime, as if it were $R^4$

I'm not really sure what I'm looking for, or even if there is anything to find. Maybe I'll have an idea when I see the answers...

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So, most people think of rotation in 3D with respect to an axis, which is like rotating each 2D plane perpendicular to the axis. But relativity says we live in a 4D universe, although time is a rather special dimension.

Anyhow, I have several questions about rotation. What would each of the following look like?:

1. *Rotation in the real world. How is the time dimension affected?

2. *Rotation in 4 normal dimensions (called $R^4$?)

3. *Rotation in spacetime, along the time axis.

4. *Rotation in spacetime, as if it were $R^4$

I'm not really sure what I'm looking for, or even if there is anything to find. Maybe I'll have an idea when I see the answers...

1. No effect on the axis, but the further from the axis the more time slows down.

2. 3. 4. Hmmm...wonder if any of these resemble "quantum spin" in any way?

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Some hints as how you could proceed;

1) The rotation group in 4-d is $O(4)$.

2) The Lorentz group $O(3,1)$ is very "similar" to $SO(4)$ mod some signs.

In both of these cases, you could investigate them explicitly by using a matrix representation.

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Is there any proof that the universe consists of only 3 spatial and one temporal dimension, or could there be more?

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String and M-theory suggest there should be more. So, yes there could be more.

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Does that include the possibility of more temporal dimensions?

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you would need to invoke a 5`th dimension in which to rotate it in/through.

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Temporal means that the metric on the "enlarged" space can be of the form (at a point) $diag\{-1, -1, \cdots 1,1, \cdots 1\}$. Yes, this is possible.

I don't like to say a dimension is temporal or spacial, the distinction comes from the metric and not the directions themselves.

I don't know much about multi-time physics, but things from sting theory suggest this could be the case.

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• 2 months later...
So, most people think of rotation in 3D with respect to an axis, which is like rotating each 2D plane perpendicular to the axis. But relativity says we live in a 4D universe, although time is a rather special dimension.

Anyhow, I have several questions about rotation. What would each of the following look like?:

*Rotation in the real world. How is the time dimension affected?

*Rotation in 4 normal dimensions (called $R^4$?)

*Rotation in spacetime, along the time axis.

*Rotation in spacetime, as if it were $R^4$

I'm not really sure what I'm looking for, or even if there is anything to find. Maybe I'll have an idea when I see the answers...

Well, first you might want to know what a four dimensional object looks like. If we consider a dice for the moment, we know it is three dimensional because it has six sides, from mere observation.

But if something like a dice was cast into four dimensions, it would look different. Its ends would be elongated, and it sides stretched out.

Thinking in five dimensions is even stranger. An object in five dimensions is almost impossible to imagine, but if something moved through the fifth dimension, it would shrink to the size of a superstring, and then back to normal size. Not only that, but if something did move in this dimension, it would quickly end up where it had started!

My impressions of the sixth dimension (also hypothetical alongside the fifth and some consider also the fourth), is that from inside that dimension (usually envisioned as being compactified), can look very similar to the three dimensional space. This is why it is hypothesized that entire galaxies could exist in the sixth dimension.

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So, most people think of rotation in 3D with respect to an axis, which is like rotating each 2D plane perpendicular to the axis. But relativity says we live in a 4D universe, although time is a rather special dimension.

Its not quite right to say that the 4D universe was something that relativity told us. We can go back to Newtonian physics and describe physics there in terms of spacetime as well. As far as teh universe being "4D" its misleading to think of it as such. There are 3 spatial dimensions and 1 time dimension. But space and time have very different physical meanings.

*Rotation in the real world. How is the time dimension affected?

*Rotation in 4 normal dimensions (called $R^4$?)

*Rotation in spacetime, along the time axis.

*Rotation in spacetime, as if it were $R^4$

Please elaborate what is being rotated. Thanks.

Pete

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4-D is simply a 3-D or (x,y,z) grid of length with a stop watch or clock for time. It is a movie and not a still picture. It amounts to (x,y,z) in a movie. If you wanted to express a movie with a single photo you stretched out the roll of movie film. We can also cut the stretched out roll, into separate frames, and stack them to see the movement in the original (x,y-z), allowing position to substitute for movie time. Any of these will work mathematically.

The (x,y,z) in a movie, is the way the sensory systems will see it. But it is not easy to draw a movie on a piece of paper. We need to stretch it out or stack the frames. The last is often the nebulous image we get of space-time where time and space appear to blend into a single (x,y,z) axis with something extra.

To be honest this just popped into my head, but it seems to cover the bases with a simple visual.

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Its not quite right to say that the 4D universe was something that relativity told us. We can go back to Newtonian physics and describe physics there in terms of spacetime as well.

That's true, but I'm pretty sure that the spacetime in a Newtonian would be 4D, whereas in relativity it is 3D with one time dimension. That is, the time dimension in Newtonian physics would be completely separate from the space dimensions, whereas in relativity they are intertwined.

As far as teh universe being "4D" its misleading to think of it as such. There are 3 spatial dimensions and 1 time dimension. But space and time have very different physical meanings.

I know, but what I don't know is what is so different about the time dimension. Hence, I was curious as to how things like rotations would be different in 4D and 3+1D.

Please elaborate what is being rotated. Thanks.

You can rotate whatever would be easy, but clear. Perhaps a vector, or a rectangular prism. Whatever you want, just don't choose a sphere I suppose

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I don't see how time can be rotated since "rotation" involves an object changing constantly in reference to an x,y and z axis but it takes time for an object to rotate. In this case, time is continuing on (or happening) as it always does.

I struggle with the comprehension of space/time fabric so bear with me here.

But what would a rotating object look like in rotating time.

Would it just not move at all? Or even be perceivable? Or exist?

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That's true, but I'm pretty sure that the spacetime in a Newtonian would be 4D, whereas in relativity it is 3D with one time dimension. That is, the time dimension in Newtonian physics would be completely separate from the space dimensions, whereas in relativity they are intertwined.

It makes no difference in which theory one defines spacetime because it is not theory dependant. Spacetime is the collection of events and consists of one time coordinate and three spatial coordinates making a total of four coordinates. Since the number of coordinates neccesary to uniquely define a point is the dimension of the space at hand it follows that spacetime is 4-dimensional. Its the stucture added to the spacetime which determines what theory one is considering.

I know, but what I don't know is what is so different about the time dimension. Hence, I was curious as to how things like rotations would be different in 4D and 3+1D.

The difference between dimensions has to do with the physical meaning of the dimension itself. Space has a different physical meaning than time. Only in certain respects are they treated mathematically on the same footing. If you were to look at two Minkowski coordinate systems super imposed on each other then the time axis is rotatred with respect to the space axis. This doesn't happen with a Galilean transformation. However the spatial axis of the new coordinate system is also rotated with respect to the old one when both the Lorentz transformation and the Galilean transformation are used.

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We use (x,y,z,t) by convention. We could also define space-time with polar coordinates plus time, but this is not as convenient to use. Although, the curvature of space-time seems to suggest using polar space-time.

What is interesting, a polar coordinate system addresses rotation easier but leads to an interesting question. Does relativity create an angle dilation and does this have a physical meaning?

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