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Multiple time dimensions appear to allow the breaking or re-ordering of cause-and-effect in the flow of any one dimension of time. This and conceptual difficulties with multiple physical time dimensions have been raised in modern analytic philosophy.[4]

https://en.wikipedia.org/wiki/Multiple_time_dimensions

I am not sure if I understand the sentence, but it seems to say if time can flow in any dimension then causes and effects can be reordered in any particular way, but how this is possible is not really explained. I don't see how causes and effects can be re-ordered if we allow for multiple time dimensions, can anyone explain?

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    Based on this answer on the physics stack exchange I suspect the reason may have to do with the technical details of what happens when you try to add an extra time dimension in relativistic physics, rather than a matter of any kind of simple conceptual argument that doesn't depend on the physics.
    – Hypnosifl
    Jul 24 at 19:13
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Consider a two-dimensional time, so that you now have a "time plane" rather than a timeline. Mathematically, this means that the two time dimensions are independent.

If you are restricted to only moving forwards in either direction, you will trace a zigzag diagonal across the plane. But such a linear path is mathematically still a line, a one-dimensional path. The maths collapses into a one-dimensional physical time. We actually do this when the equations of physics throw up complex or imaginary time values.

So positing a true second time dimension entails a second, fully independent degree of freedom. You can now wander around your time plane more freely, crossing your original path. Where you make such a crossing, this creates the opportunity for a causal paradox.

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If time has a single dimension like a path or a line, all events/travel must abide by that linear ordering. Adding a dimension would be like dropping 2D plane on top of the 1D line. Any point on the original line could then be gotten to from any other point on the line, without passing through points between them on the line by traveling up into the plane then back down to a different spot. The linear ordering of the line has been undermined.

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    But say you have an xyz space and you imagine both the y and z axis represent time dimensions--if you want to draw a curve that represents something like a world line in this world, and you adopt the rule that as you draw it the y and z coordinates must always be increasing or constant, I think this would be enough to prevent you from drawing a curve that loops back to meet itself at an "earlier" point (i.e. preventing you from drawing a closed timelike curve).
    – Hypnosifl
    Jul 24 at 19:16
  • @Hypnosifl This is tricky because the fundamental laws of physics do not have a preferred time direction. Any directness is partly due to initial conditions, not just the laws. See the Past Hypothesis. For similar reasons, CTC's are allowed by the laws to exist, just not evolve dynamically. They can be there at the big bang already formed then keep existing, but not form dynamically. I put the fact that the fundamental laws alone do not pick out a preferred direction of time as making this simple analogy work. Then there is the question if OP wants to have a physics answer - the url is broad
    – J Kusin
    Jul 24 at 20:50
  • I would say the issue of CTCs is distinct from the issue of time-symmetry--in general relativity, CTCs either require matter/energy that violates the "averaged null energy condition", or universes with certain global properties like Godel's rotating universe solution (which I wrote about here), so if our universe doesn't have those global properties and the ultimate theory of matter/energy doesn't allow violations of the avg. null energy condition, then CTCs could be impossible in our universe despite the time symmetry of the dynamical laws.
    – Hypnosifl
    Jul 24 at 22:30
  • @Hypnosifl What about this "mathematically speaking, the most apparent distinguishing feature of the energy conditions is that they are essentially restrictions on the eigenvalues and eigenvectors of the matter tensor. A more subtle but no less important feature is that they are imposed eventwise, at the level of tangent spaces. Therefore, they have no hope of ruling out objectionable global features, such as closed timelike curves. " en.wikipedia.org/wiki/Energy_condition
    – J Kusin
    Jul 24 at 22:49
  • Yes, there may be arguments saying that it's likely the avg. null energy condition could be violated in quantum gravity, I'm just saying that time-symmetry alone isn't enough to guarantee CTCs are possible.
    – Hypnosifl
    Jul 24 at 23:11

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