Time appears to be one-dimensional. But does it have to be that way?

For example, here is a simulation of a ball bouncing in two dimensions of time, `t`

and _{x}`t`

:_{y}

The ball's velocity is an orientation (tangent plane or normal vector) in `(t`

space, and its acceleration (from gravity) is the local curvature of the surface._{x}, t_{y}, h)

But suppose you built a digital circuit in a universe with two time dimensions. Specifically, a synchronous digital circuit, whose discrete state `S`

is a function of two discrete time variables, `i`

and _{x}`i`

:_{y}

State changes are driven by a pair of two-dimensional clocks:

As the system crosses over a clock edge, it transitions according to a pair of rules:

`S`

= f_{ix,iy}_{x}(`S`

)_{ix-1,iy}

`S`

= f_{ix,iy}_{y}(`S`

)_{ix,iy-1}

Such state transitions only avoid a race condition at the intersection of rising edges if `f`

._{x} · f_{y} = f_{y} · f_{x}

Well, that's possible. For example, the following state transitions satisfy the above consistency criterion:

S | f_{x}(S) |
f_{y}(S) |
---|---|---|

A | B | C |

B | A | D |

C | D | A |

D | C | B |

The problem is, if `f`

, then any sequence_{x} · f_{y} = f_{y} · f_{x}

`f`

_{x} · ... · f_{y} · ... · f_{y} · ... · f_{x}

is just

`f`

._{x} · ... · f_{x} (n times) · f_{y} · ... · f_{y} (m times)

That is, there is no interaction between the two time dimensions. The system is, for all intents and purposes, two independent systems, each with one-dimensional time. And, if you lived in such a universe, you would never know, because each version of "you" would have no way of knowing about the other.

And so it would be just like the world we know.