The possibility that there might be more than one dimension of time has occasionally been discussed in physics and philosophy.
Contents

Physics 1

Philosophy 2

In fiction 3

See also 4

References 5

External links 6
Physics
Special relativity describes spacetime as a manifold whose metric tensor has a negative eigenvalue. This corresponds to the existence of a "timelike" direction. A metric with multiple negative eigenvalues would correspondingly imply several timelike directions, i.e. multiple time dimensions, but there is no consensus regarding the relationship of these extra "times" to time as conventionally understood.
If thе special theory of relativity can be generalized for the case of kdimensional time (t_{1},t_{2},…,t_{k}) and ndimensional space (x_{k+1}, x_{k+2},..., x_{}k+n), then the (k+n)dimensional interval, being invariant, is given by the expression (ds_{k,n})^{2}=(cdt_{1})^{2}+…+(cdt_{k})^{2}−(dx_{k+1})^{2}−…−(dx_{k+n})^{2}. The metric signature will be

(\underbrace{+,\cdots,+}_{k},\underbrace{,\cdots,}_{n})\,  timelike sign convention,

(or (\underbrace{,\cdots,}_{k},\underbrace{+,\cdots,+}_{n})\, spacelike sign convention ).
The transformations between the two inertial frames of reference K and K′, which are in a standard configuration (i.e., transformations without translations and/or rotations of the space axis in the hyperplane of space and/or rotations of the time axis in the hyperplane of time), are given as follows:^{[1]}

t'_{\sigma}=\sum_t_{\theta}+\frac {c^2} {v_{\sigma} v_{\theta}}\beta^2({\zeta}1)t_{\theta}\right)\frac {1} {v_{\sigma} }\beta^2{\zeta}x_{k+1},

x'_{k+1}=c^2\beta^2\zeta\sum_ {v_{\theta}}+{\zeta}x_{k+1},

x'_{\lambda}=x_{\lambda},
where \mathbf{v}_1=(v_1,\underbrace{0,\cdots,0}_{n1})\,\!, \mathbf{v}_2=(v_2,\underbrace{0,\cdots,0}_{n1})\,\!, \mathbf{v}_k=(v_k,\underbrace{0,\cdots,0}_{n1})\,\! are the vectors of the velocities of K′ against K, defined accordingly in relation to the time dimensions t_{1},t_{2},…,t_{k}; \beta = \frac{1}{\sqrt{\sum_}}; \zeta = \frac{1}{\sqrt{1\beta^2}}; σ=1,2,...,k; λ=k+2,k+3,...,k+n. Here δ_{σθ} is the Kronecker delta. These transformations are generalization of the Lorentz boost in a fixed space direction (x_{k+1}) in the field of the multidimensional time and multidimensional space.
Causal structure of a spacetime with two time dimensions and one space dimension
Let us denote: \frac {\mathrm d x_{\eta}} {\mathrm d t_{\sigma}}=V_{\sigma\eta} and \frac {\mathrm d x'_{\eta}} {\mathrm d t'_{\sigma}}=V'_{\sigma\eta}, where σ=1,2,...,k; η=k+1,k+2,...,k+n. The velocityaddition formula is then given by

V'_{\sigma\left(k+1\right)}=\frac {V_{\sigma\left(k+1\right)}\zeta\left(1\beta^2\sum_\right)} {1+\frac {V_{\sigma\left(k+1\right)}} {v_{\sigma}}\beta^2\left(\left(\zeta1\right)\sum_\zeta\right)},

V'_{\sigma \lambda}=\frac {V_{\sigma \lambda}} {1+\frac {V_{\sigma\left(k+1\right)}} {v_{\sigma}}\beta^2\left(\left(\zeta1\right)\sum_\zeta\right)},
where σ=1,2,...,k; λ=k+2,k+3,...,k+n.
For simplicity, let us consider only one spatial dimension x_{3} and the two time dimensions x_{1} and x_{2}. (E. g., x_{1}=ct_{1}, x_{2}=ct_{2}, x_{3}=x.) Let us assume that in point O, having coordinates x_{1}=0, x_{2}=0, x_{3}=0, there has been an event E. Let us further assume that a given interval of time \Delta T=\sqrt{\left(\Delta t_1\right)^2+\left(\Delta t_2\right)^2}\geq 0 has passed since the event E. The causal region, connected to the event E includes the lateral surface of the right circular cone { (x_{1})^{2}+(x_{2})^{2}−(x_{3})^{2}=0 }, the lateral surface of the right circular cylinder { (x_{1})^{2}+(x_{2})^{2}=c^{2}ΔT^{2} } and the inner region bounded by these surfaces, i.e., the causal region includes all points (x_{1},x_{2},x_{3}), for which the conditions

{ (x_{1})^{2}+(x_{2})^{2}−(x_{3})^{2}=0 and x_{3}≤cΔT } or

{ (x_{1})^{2}+(x_{2})^{2}=c^{2}ΔT^{2} and x_{3}≤cΔT } or

{ (x_{1})^{2}+(x_{2})^{2}−(x_{3})^{2}>0 and (x_{1})^{2}+(x_{2})^{2}<c^{2}ΔT^{2} }
are fulfilled.^{[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 "twotime physics",^{[3]} inspired by the SO(10,2) symmetry of the extended supersymmetry structure of Mtheory, is the most recent and systematic development of the concept (see also Ftheory). Walter Craig and Steven Weinstein proved the existence of a wellposed initial value problem for the ultrahyperbolic equation (wave equation in more than one time dimension).^{[4]} This showed that initial data on a mixed (spacelike and timeline) hypersurface obeying a particular nonlocal constraint evolves deterministically in the remaining time dimension.
Philosophy
An Experiment with Time by J.W. Dunne (1927) describes^{[5]} an ontology in which there is an infinite hierarchy of conscious minds, each with its own dimension of time and able to view events in lower time dimensions from outside. His theory was often criticised as exhibiting an unnecessary infinite regress.
The conceptual possibility of multiple time dimensions has also been raised in modern analytic philosophy.^{[6]}
The English philosopher John G. Bennett posited a sixdimensional Universe with the usual three spatial dimensions and three timelike dimensions that he called time, eternity and hyparxis. Time is the sequential chronological time that we are familiar with. The hypertime dimensions called eternity and hyparxis are said to have distinctive properties of their own. Eternity could be considered cosmological time or timeless time. Hyparxis is supposed to be characterised as an ablenesstobe and may be more noticeable in the realm of quantum processes.
The conjunction of the two dimensions of time and eternity could form a hypothetical basis for a Multiverse cosmology with parallel universes existing across a plane of vast possibilities. The third timelike dimension hyparxis could allow for the theoretical existence of scifi possibilities such as time travel, sliding between parallel worlds and fasterthanlight travel.
While Bennett has put forward some curious speculation, his ideas stop at some subjective aspects of the perception of time which does not place them on a fully scientific basis. The question of measurement, how one would measure these hypothetical extra timelike dimensions, is left unaddressed.
In fiction

In the final novel of the trilogy Humans as Gods «The Reverse Time Loop» (1977), Sergey Snegov puts into the mouth of the protagonist words: «This is my idea  to break out of the onedimensional, straight time during a twodimensional time»^{[7]}

In Diane Duane's Star Trek novel, The Wounded Sky, the Hamalki physicist K't'lk states that time has three dimensions, called "inception," "duration," and "termination."

The comic series Sonic The Hedgehog uses this theory to it's advantage when Sonic meets his evil twin Scourge.
See also
References

^ Velev, Milen (2012). "Relativistic mechanics in multiple time dimensions". Physics Essays 25 (3): 403–438.

^ Velev, Milen (2012). "Relativistic mechanics in multiple time dimensions". Physics Essays 25 (3): 403–438.

^ Bars, Itzhak. "TwoTime Physics". Retrieved 8 December 2012.

^ Craig, Walter; Weinstein, Steven. "On determinism and wellposedness in multiple time dimensions". Proc. R. Soc. A vol. 465 no. 2110 30233046 (2008). Retrieved 5 December 2013.

^ McDonald, John Q. (15 November 2006). "John's Book Reviews: An Experiment with Time". Retrieved 8 December 2012.

^ Weinstein, Steven. "Many Times". Foundational Questions Institute. Retrieved 5 December 2013.

^ Сергей Снегов Кольцо обратного времени / Сост. и авт. вступ. ст. Е. Брандис, В. Дмитревский. — Л.: Лениздат, 1977. — С. 11270. — 639 с. — 100 000 экз.

^ Rucker, Rudy (25 November 2005). "Notes for Realware". Retrieved 8 December 2012.
External links

Itzhak Bars, Gauge Symmetry in Phase Space, Consequences for Physics and Spacetime, Int.J.Mod.Phys. A25 (2010) 52355252, arXiv:1004.0688 [hepth].

Itzhak Bars, John Terning, Extra dimensions in space and time, New York, Springer, Multiversal journeys series, 2010, ISBN 9780387776388. DOI 10.1007/9780387776385.


Concepts
in time




Theories
of time



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