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 (t₁, t₂, ..., t_k) and n-dimensional space (x_k+1, x_k+2, ..., xk+n), then the (k+n)-dimensional interval, being invariant, is given by the expression

(ds_k,n)² = (cdt₁)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)

(\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 _{{\theta =1}}^{k}\left(\delta _{{\sigma \theta }}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 _{{\theta =1}}^{k}{\frac {t_{\theta }}{v_{\theta }}}+\zeta x_{{k+1}},

x'_{\lambda }=x_{\lambda },

where ${\mathbf {v}}_{1}=(v_{1},\underbrace {0,\cdots ,0}_{{n-1}}),$ ${\mathbf {v}}_{2}=(v_{2},\underbrace {0,\cdots ,0}_{{n-1}}),$ ${\mathbf {v}}_{k}=(v_{k},\underbrace {0,\cdots ,0}_{{n-1}})$ are the vectors of the velocities of K′ against K, defined accordingly in relation to the time dimensions t₁, t₂, ..., t_k; $\beta ={\frac {1}{{\sqrt {\sum _{{\mu =1}}^{k}{\frac {c^{2}}{v_{\mu }^{2}}}}}}};$ $\zeta = \frac{1}{\sqrt{1-\beta^2}};$ 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 space-time with two time dimensions and one space dimension

Let us denote ${\frac {dx_{\eta }}{dt_{\sigma }}}=V_{{\sigma \eta }}$ and ${\frac {dx'_{\eta }}{dt'_{\sigma }}}=V'_{{\sigma \eta }},$ where σ = 1, 2, ..., k; η = k+1, k+2, ..., k+n. The velocity-addition formula is then given by

V'_{{\sigma (k+1)}}={\frac {V_{{\sigma (k+1)}}\zeta \left(1-\beta ^{2}\sum _{{\theta =1}}^{k}{\frac {c^{2}}{v_{\theta }V_{{\theta (k+1)}}}}\right)}{1+{\frac {V_{{\sigma (k+1)}}}{v_{\sigma }}}\beta ^{2}\left((\zeta -1)\sum _{{\theta =1}}^{k}{\frac {c^{2}}{v_{\theta }V_{{\theta (k+1)}}}}-\zeta \right)}},

V'_{{\sigma \lambda }}={\frac {V_{{\sigma \lambda }}}{1+{\frac {V_{{\sigma (k+1)}}}{v_{\sigma }}}\beta ^{2}\left((\zeta -1)\sum _{{\theta =1}}^{k}{\frac {c^{2}}{v_{\theta }V_{{\theta (k+1)}}}}-\zeta \right)}},

where σ = 1, 2, ..., k; λ = k+2, k+3, ..., k+n.

For simplicity, let us consider only one spatial dimension x₃ and the two time dimensions x₁ and x₂. (E. g., x₁ = ct₁, x₂ = ct₂, x₃ = x.) Let us assume that in point O, having coordinates x₁ = 0, x₂ = 0, x₃ = 0, there has been an event E. Let us further assume that a given interval of time $\Delta T={\sqrt {(\Delta t_{1})^{2}+(\Delta t_{2})^{2}}}\geq 0$ has passed since the event E. The causal region connected to the event E includes the lateral surface of the right circular cone {(x₁)² + (x₂)² − (x₃)² = 0}, the lateral surface of the right circular cylinder {(x₁)² + (x₂)² = c²ΔT²} and the inner region bounded by these surfaces, i.e., the causal region includes all points (x₁, x₂, x₃), for which the conditions

{(x₁)² + (x₂)² − (x₃)² = 0 and |x₃| ≤ cΔT} or

{(x₁)² + (x₂)² = c²ΔT² and |x₃| ≤ cΔT} or

{(x₁)² + (x₂)² − (x₃)² > 0 and (x₁)² + (x₂)² < c²ΔT²}

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. 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.

References

1 2 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.
↑ Bars, Itzhak. "Two-Time Physics". Retrieved 8 December 2012.
↑ 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.
↑ 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.
↑ 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.

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.

Philosophy of time

Concepts in time	Time Duration A priori and a posteriori Eternity Event Action Deterministic system Free will Eternal return A-series and B-series Temporal parts Growing block universe Multiple time dimensions	Time Portal

Theories of time	Determinism Temporal finitism Fatalism Indeterminism Compatibilism and incompatibilism Presentism Eternalism Four dimensionalism Endurantism Perdurantism B-Theory of time

Related articles	Metaphysics Etiology Teleology Post hoc ergo propter hoc The Unreality of Time The Singular Universe and the Reality of Time

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