Higher Dimensional Time Implications?

[LightningInAJar]

TL;DR Summary

If time were 2 or 3 dimensional what implications are there?

I have read that nothing in math makes time having more than 1 dimension impossible. But if time had 2 dimensions or even 3 dimensions on its own, what would that mean? Could time travel and time travel paradoxes be allowed without destroying the universe or creating branch realities?

I think in 3 dimensional space you can ties your shoes, but in 4 dimensional space a knot can't exist.

Would higher dimensional time allow more that seems forbidden currently?

 

[Dale]

With more than one dimension of time there would be no more distinction between future and past. Closed timelike curves would be possible in flat spacetime. The laws of physics therefore would be acausal.

 

[LightningInAJar]

Time as 1 dimensional is a line? What about that makes time causal? Who says the arrow doesn't go both ways, or that lines can't bipass between future and past?

What about a plane makes physics acasual? Couldn't it be limited to rightward and upward directions so it doesn't circle back around?

 

[DaveC426913]

Time as 1 dimensional is a line? What about that makes time causal? Who says the arrow doesn't go both ways, or that lines can't bipass between future and past?

What about a plane makes physics acasual? Couldn't it be limited to rightward and upward directions so it doesn't circle back around?

Traveling from point A to point C on a 1D spatial line requires passing through midpoint B.
Traveling from point A to point C on a 2D spatial plane allows one to sidestep midpoint B.

The same holds for time dimensions.
Moving from 2020 to 2022 requires passing through 2021.
In 2D time, you could go from 2020 directly to 2022 without passing through 2021.

On second thought, that's a crummy explanation. I'll leave it to the experts.

 

[Ibix]

Time as 1 dimensional is a line?

No. Time (in this sense) is a direction in spacetime, like the x direction on a plane. You cannot draw a smooth curve on the plane that starts pointing in the +x direction and ends up pointing in the -x direction without it pointing in the y direction inbetween. Similarly, you can't draw a line in spacetime that is advancing in the +t direction (as we all do) and end up advancing in the -t direction without pointing in the x, y, or z direction inbetween. And the physics won't let a worldline advancing in time turn into one advancing only in space.

However, if you had two timelike dimensions you could move smoothly from advancing in the +t direction to the -t direction passing through the t' direction. So there would no longer be a distinction between past and future, so there's no causality.

 

[Dale]

Time as 1 dimensional is a line? What about that makes time causal?

The spacetime interval in flat spacetime is $ds^2 = - dt^2 + dx^2 + dy^2 + dz^2$. If you look at the surface formed by all vectors formed with a fixed $ds^2=const.$ then if $0<ds^2$ you have a hyperboloid of one sheet. Any such spacelike vector can be smoothly mapped to any other. In contrast, if $ds^2<0$ you have a hyperboloid of two sheets. We call one sheet the future sheet and the other the past sheet. A future directed timelike vector can be smoothly mapped to any other future directed timelike vector, but it cannot be smoothly mapped to any past directed timelike vector. The future and the past are topologically disconnected sets, unlike with spatial vectors.

If you add a second timelike vector $dS^2 = - dt^2 -du^2 + dx^2 + dy^2 + dz^2$ then timelike vectors with constant $dS^2<0$ form a hyperboloid of one sheet. Then any timelike vector can be smoothly mapped to any other timelike vector. There is then no more distinction between future and past than there is between left and right. You can turn around in time just like you can in space.

What about a plane makes physics acasual?

The future and the past would no longer be distinct sets.

Couldn't it be limited to rightward and upward directions so it doesn't circle back around?

If you did that then you would actually have a single direction of time but just written in a non-orthonormal basis with two timelike basis vectors. It would still be a single time dimension in the right+up direction. You would still have the same hyperboloids.

 

[Hornbein]

I think in 3 dimensional space you can ties your shoes, but in 4 dimensional space a knot can't exist.

The word "knot" is used for two different things. The mathematical knots that can't exist in 4D are closed curves. The practical knot you use to tie your shoes is not a closed curve. Strings are immobilized by friction. Friction knots of strings could exist in any number of dimensions. If the strings are immobilized then it doesn't matter what it might do if they could move freely.

Mathematical knots can exist in N dimensions. They are any closed N-2 dimensional object. In 4D the simplest knot would be a 2-sphere.

 

[Hornbein]

Hawking used what he called "imaginary time." I don't know what it was/is.

 

[Ibix]

Hawking used what he called "imaginary time." I don't know what it was/is.

A bad idea. In Euclidean geometry, the distance between two points is $\Delta s$ where $\Delta s^2=\Delta x^2+\Delta y^2+\Delta z^2$. In Minkowski geometry the "distance" between two events is given by $\Delta s^2=\Delta x^2+\Delta y^2+\Delta z^2-c^2\Delta t^2$ and you can write that last term as $+(ic\Delta t)^2$. It used to be popular to teach this to try to make Minkowski geometry more familiar to students and hide the concept of a metric tensor (Feynman does it in his lectures), but it generalises badly to curved spacetime where you need the metric tensor explicitly so current thinking is that it makes learning general relativity harder. The modern approach is to be upfront about the role of the metric and introduce the importance of geometry in special relativity (and do undergrads the courtesy of assuming they can deal with one minus sign without bursting into tears) before you get to the added fun of the full machinery of pseudo-Riemannian geometry in general relativity.