Thе special theory of relativity can be generalized for the case of k-dimensional time
(t1,t2,…,tk) and n-dimensional space (xk+1, xk+2,..., xk+n),
then the (k+n)-dimensional interval, being invariant,
is given by the expression (dsk,n)2=(cdt1)2+…+(cdtk)2−(dxk+1)2−…−(dxk+n)2.
Theories with more than one dimension of time have sometimes been advanced in physics,
whether as a serious description of reality or just as a curious possibility.
Itzhak Bars's work on "two-time physics", inspired by the SO(10,2) symmetry of the extended
supersymmetry structure of M-theory, is the most recent and systematic development
of the concept. Walter Craig and Steven Weinstein proved the existence of a well-posed
initial value problem for the ultrahyperbolic equation (wave equation in more than one time dimension)
This showed that initial data on a mixed (spacelike and timeline) hypersurface obeying
a particular nonlocal constraint evolves deterministically in the remaining time dimension.