# Multiple time dimensions

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

## Physics

Special relativity describes spacetime as a manifold whose metric tensor has a negative eigenvalue. This corresponds to the existence of a "time-like" 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 *k*-dimensional time (*t*_{1}, *t*_{2}, ..., *t*_{k}) and *n*-dimensional space (*x*_{k+1}, *x*_{k+2}, ..., *x*_{k+n}), then the (*k*+*n*)-dimensional interval, being invariant, is given by the expression

- (d
*s*_{k,n})^{2}= (*c*d*t*_{1})^{2}+ ... + (*c*d*t*_{k})^{2}− (d*x*_{k+1})^{2}− … − (d*x*_{k+n})^{2}.

The metric signature will be

- (timelike sign convention)

or

- (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]}

where are the vectors of the velocities of *K*′ against *K*, defined accordingly in relation to the time dimensions *t*_{1}, *t*_{2}, ..., *t*_{k}; *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.

Let us denote and where *σ* = 1, 2, ..., *k*; *η* = *k*+1, *k*+2, ..., *k*+*n*. The velocity-addition formula is then given by

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 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}Δ*T*^{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}| ≤*c*Δ*T*} or - {(
*x*_{1})^{2}+ (*x*_{2})^{2}=*c*^{2}Δ*T*^{2}and |*x*_{3}| ≤*c*Δ*T*} or - {(
*x*_{1})^{2}+ (*x*_{2})^{2}− (*x*_{3})^{2}> 0 and (*x*_{1})^{2}+ (*x*_{2})^{2}<*c*^{2}Δ*T*^{2}}

are fulfilled.^{[1]} 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",^{[2]} 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 (see also F-theory). Walter Craig and Steven Weinstein proved the existence of a well-posed initial value problem for the ultrahyperbolic equation (a wave equation in more than one time dimension).^{[3]} This showed that initial data on a mixed (spacelike and timelike) hypersurface obeying a particular nonlocal constraint evolves deterministically in the remaining time dimension.

## Philosophy

*An Experiment with Time* by J. W. Dunne (1927) describes^{[4]} an ontology with an infinite hierarchy of conscious minds, each with its own dimension of time and able to view events in lower time dimensions from outside.^{[clarification needed]} 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.^{[5]}

John G. Bennett, an English philosopher, theosophist, anthroposophist, and follower of the mystic George Gurdjieff, posited a six-dimensional Universe with the usual three spatial dimensions and three time-like 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 (by Bennett) to be characterised as an ableness-to-be and may be more noticeable in the realm of quantum processes. According to Bennett, 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, while the third time-like dimension hyparxis could allow the theoretical existence of sci-fi possibilities such as time travel, sliding between parallel worlds and faster-than-light travel.

No well-known physicist or cosmologist has endorsed these ideas. While Bennett has put forward some curious speculation, the question of measurement (how one would measure these hypothetical extra time-like dimensions) is left unaddressed, as is how one might falsify his suggestions (which is generally regarded ^{[6]} as the distinguishing feature of science since the work of Karl Popper).

## 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 one-dimensional, straight time during a two-dimensional time»^{[7]} *The Number of the Beast*by Robert A. Heinlein (1980) features a six-dimensional cosmology in which there are three time dimensions, denoted*t*,*tau*(Greek τ) and*teh*(Cyrillic cursive*т*).- The Ware Tetralogy by Rudy Rucker features aliens called Metamartians who "are from a part of the cosmos where time is two-dimensional".
^{[8]} - 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 its advantage when Sonic meets his evil twin Scourge.

## See also

## References

- ↑
^{1.0}^{1.1}Velev, Milen (2012). "Relativistic mechanics in multiple time dimensions".*Physics Essays*.**25**(3): 403–438. Bibcode:2012PhyEs..25..403V. doi:10.4006/0836-1398-25.3.403.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Bars, Itzhak. "Two-Time Physics". Retrieved 8 December 2012.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Craig, Walter; Weinstein, Steven. "On determinism and well-posedness in multiple time dimensions". Proc. R. Soc. A vol. 465 no. 2110 3023-3046 (2008). Retrieved 5 December 2013.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ McDonald, John Q. (15 November 2006). "John's Book Reviews: An Experiment with Time". Retrieved 8 December 2012.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Weinstein, Steven. "Many Times". Foundational Questions Institute. Retrieved 5 December 2013.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Stanford Encyclopedia of Philosophy Entry on Karl Popper
- ↑ Сергей Снегов Кольцо обратного времени / Сост. и авт. вступ. ст. Е. Брандис, В. Дмитревский. — Л.: Лениздат, 1977. — С. 11-270. — 639 с. — 100 000 экз.
- ↑ Rucker, Rudy (25 November 2005). "Notes for Realware" (PDF). Retrieved 8 December 2012.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>

## External links

- Itzhak Bars, Gauge Symmetry in Phase Space, Consequences for Physics and Spacetime, Int. J. Mod. Phys. A
**25**(2010) 5235-5252, arXiv:1004.0688 [hep-th]. - Itzhak Bars, John Terning, Extra dimensions in space and time, New York, Springer, Multiversal journeys series, 2010, ISBN 978-0-38777637-8. DOI 10.1007/978-0-387-77638-5.