Physicist Max Tegmark has an interesting discussion on the
physics of a universe with more than one time dimension at , specifically .  In the excerpts
below, n is the number of space dimensions and m the number of time
dimensions, so when he writes m > 1 he means more than one time dimension.
Quoting Tegmark:

: What would reality appear like to an observer in a manifold with more
: than one time-like dimension? Even when m > 1, there is no obvious
: reason why an observer could not, none the less, perceive time as being
: one-dimensional, thereby maintaining the pattern of having "thoughts"
: in a one-dimensional succession that characterizes our own reality
: perception. If the observer is a localized object, it will travel along
: an essentially one-dimensional (time-like) world line through the (n +
: m)-dimensional spacetime manifold. The standard general relativity notion
: of its proper time is perfectly well defined, and we would expect this
: to be the time that it would measure if it had a clock and that it would
: subjectively experience.
: Needless to say, many aspects of the world would none the less appear
: quite different.  For instance, a re-derivation of relativistic mechanics
: for this more general case shows that energy now becomes an m-dimensional
: vector rather than a constant, whose direction determines in which
: of the many time directions the world line will continue, and in the
: non-relativistic limit, this direction is a constant of motion. In
: other words, if two non-relativistic observers that are moving in
: different time directions happen to meet at a point in spacetime, they
: will inevitably drift apart in separate time directions again, unable
: to stay together.
: Another interesting difference, which can be shown by an elegant
: geometrical argument [10], is that particles become less stable when m
: > 1....
: In addition to these two differences, one can concoct seemingly strange
: occurrences involving "backward causation" when m > 1. None the less,
: although such unfamiliar behaviour may appear disturbing, it would seem
: unwarranted to assume that it would prevent any form of observer from
: existing. After all, we must avoid the fallacy of assuming that the
: design of our human bodies is the only one that allows self-awareness....
: There is, however, an additional problem for observers when m > 1,
: which has not been previously emphasized even though the mathematical
: results on which it is based are well known. If an observer is to be
: able to make any use of its self-awareness and information-processing
: abilities, the laws of physics must be such that it can make at least
: some predictions. Specifically, within the framework of a field theory,
: it should, by measuring various nearby field values, be able to compute
: field values at some more distant spacetime points (ones lying along its
: future world line being particularly useful) with non-infinite error
: bars. If this type of well-posed causality were absent, then not only
: would there be no reason for observers to be self-aware, but it would
: appear highly unlikely that information processing systems (such as
: computers and brains) could exist at all.

Tegmark then goes into quite a technical discussion about solving the
equations of physics given various ways of specifying initial values,
the upshot of which is that if m > 1 (i.e. more than one time dimension)
observers would not be able to predict the state in the rest of the
universe from their observations, which would seem to preclude the
existence of observers.  I'm not sure I fully understood this argument.

However the earlier part is quite instructive in giving us a picture of
how a universe could look that had multiple time dimensions.  Any one
entity would still have a single time line, but different ones might
disagree about which direction the future was, and time loops would
be possible.  Personally I think this is a more serious problem than
Tegmark's idea about prediction difficulties, although he seems to gloss
over it as mere unfamiliar behavior.

Nevertheless I think it is instructive to realize that multiple time
dimension universes are a conceptual possibility even if they are unlikely
to contain observers like us.  Tegmark is implicitly writing within the
block universe perspective which is generally adopted by physicists.
Translating this into a "flow of time" view seems quite challenging
and suggests that that viewpoint may not be as flexible in terms of
deep understanding of the notion of time.

Hal Finney

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