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Multiple dimensions of space makes sense. We can understand multiple dimensions of space by lines ,planes,spaces etc. But what does a 2-D time,3-D time etc means. Visualizing time as a line makes sense ,where forward means future,backwards means past and present means the point we are in right now. But a 2-D or 3-D time would like a plane or something else. So what does it means to travel right ,left in time?

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There is so far no indication that multiple dimensions of time describe this universe. However, there have been some strictly mathematical investigations into describing what the laws of physics would look like in a universe with multiple dimensions of time. The mathematical machinery is rather straightforward, but the implications do get complicated fast.

First, let's talk about the math that describes the actual universe. In the absence of gravity, the spacetime metric in standard 3D+1D spacetime is $ds^2=-c^2 dt^2 + dx^2+dy^2+dz^2$. If $ds^2 > 0$ then the spacetime interval is called spacelike and is measured with a ruler. If $ds^2< 0$ then the spacetime interval is called timelike and is measured with a clock, and for convenience we often call the timelike spacetime interval corresponding to a clock measurement "proper time" which is $d\tau^2=-ds^2/c^2>0$.

This metric describes how time and distance relate to each other, and it has been repeatedly confirmed experimentally, so it does seem to be a very accurate description of how our universe works. In this metric there are only two distinctions between space and time. The first is the sign. That simply corresponds to the use of clocks to measure timelike intervals and rulers to measure spacelike intervals. The big difference is the number of time dimensions: there are three spatial dimensions and one time dimension. We describe this as the "signature" of the metric, which is $(-+++)$.

Now, let's look in more detail about the implication of this. First, let's plot a couple of surfaces of constant spacetime interval, one timelike and one spacelike.

Surface of constant spacelike separation Surface of constant timelike separation

The x and y axes are in light years and the t axis is in years. The one on the left is everything that is a spacelike (measured by a ruler) spacetime interval of about 1.7 light years away from the origin. Notice that it forms a hyperboloid of one sheet. If you pick any event on the hyperboloid you can smoothly transform to any other event, e.g. by a rotation. This basically encapsulates the idea that if you and I are facing each other, then your left is my right, but I can rotate around so that we are facing the same direction and then we agree.

The one on the right is everything that is a timelike (measured by a clock) spacetime interval of about 1.7 years away from the origin. Notice that it forms a hyperboloid of two sheets. If you pick any event on the top (future) hyperboloid you can smoothly transform it only to other events on the top hyperboloid, but not to any of the events on the bottom (past) hyperboloid. Future and past are distinct, and no amount of rotating or other transformations will turn future into past. There are a set of events that are 1.7 years in the future and a completely separate set of events that are 1.7 years in the past. So mathematically, the thing that makes time behave like time is that it forms a hyperboloid of one sheet, with a distinct future and past.

Now, let's talk about what happens if we add a dimension of time. Mathematically it is quite simple, the metric just becomes $ds^2=-c^2 dt^2 - c^2 du^2 + dx^2+dy^2+dz^2$. This would be a $(--+++)$ spacetime signature meaning 2 dimensions of time and 3 dimensions of space.

The big impact from this change is that now time starts to look like space. Because there are two dimensions of time, the surface of constant time (1.7 years from the origin) is now a hyperboloid of one sheet. There is no longer any solid distinction between future and past. You can pick any event on the hyperboloid and smoothly rotate or otherwise transform to any other event. So if I am facing you, not only would we disagree about left and right, but we could disagree about future and past. And if I turned to face the same direction then we could agree.

Timelike separation with 2D time

Physically, this would be very weird. Without the clean separation between future and past the causal structure is gone. There would be nothing in such a universe that we would recognize as causality. You could have closed timelike curves (time travel) just as easily as you can now walk in a circle. Although the idea of such a universe being able to contain life that could walk is probably pretty impossible. It would be truly bizarre.

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