 # Multiple time dimensions

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

## Physics

### Relation to special relativity

Special relativity describes spacetime as a manifold whose metric tensor has a negative eigenvalue. This corresponds to the existence of a "timelike" direction. A modified metric with multiple negative eigenvalues would correspondingly imply a number of such timelike directions, but there is no consensus regarding the possible relationships of these extra "times" to time as conventionally understood.

If the special theory of relativity is 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.

The metric signature is then

$(\underbrace {+,\cdots ,+} _{k},\underbrace {-,\cdots ,-} _{n})$ (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:

$t'_{\sigma }=\sum _{\theta =1}^{k}\left(\delta _{\sigma \theta }t_{\theta }+{\frac {c^{2}}{v_{\sigma }v_{\theta }}}\beta ^{2}(\zeta -1)t_{\theta }\right)-{\frac {1}{v_{\sigma }}}\beta ^{2}\zeta x_{k+1},$ $x'_{k+1}=-c^{2}\beta ^{2}\zeta \sum _{\theta =1}^{k}{\frac {t_{\theta }}{v_{\theta }}}+\zeta x_{k+1},$ $x'_{\lambda }=x_{\lambda },$ 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, ..., tk; $\beta ={\frac {1}{\sqrt {\sum _{\mu =1}^{k}{\frac {c^{2}}{v_{\mu }^{2}}}}}};$ $\zeta ={\frac {1}{\sqrt {1-\beta ^{2}}}};$ σ = 1, 2, ..., k; λ = k+2, k+3, ..., k+n. Here δσθ is the Kronecker delta. These transformations are generalization of the Lorentz boost in a fixed space direction (xk+1) in the field of the multidimensional time and multidimensional space.

Denoting ${\frac {dx_{\eta }}{dt_{\sigma }}}=V_{\sigma \eta }$ 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, consider only one spatial dimension x3 and the two time dimensions x1 and x2. (E. g., x1 = ct1, x2 = ct2, x3 = x.) Assuming that in point O, having coordinates x1 = 0, x2 = 0, x3 = 0, there has been an event E. Further assuming 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.

### Connection to the Planck length and the speed of light

The motion of a test particle may be described by coordinate

$x^{\mu }={\begin{pmatrix}ct\\r\cdot f\left({\frac {\gamma \tau }{\Lambda }}\right)\\\mathbf {x} \end{pmatrix}}$ which is the canonical (1,3) spacetime vector $(ct,\mathbf {x} )^{T}$ with $x\in \mathbb {R} ^{3}$ extended by an additional timelike coordinate $r\cdot f(\gamma \tau /\Lambda )$ . $\tau$ is then a second time parameter, $r\in \mathbb {R}$ describes the size of the second time dimension and $\gamma$ is the characteristic velocity, thus the equivalent of $c$ . $f$ describes the shape of the second time dimension and $\Lambda \in \mathbb {R}$ is a normalization parameter such that $\gamma \tau /\Lambda$ is dimensionless. Decomposing $x^{\mu }=x_{t}^{\mu }+x_{\tau }^{\mu }$ with

$x_{t}^{\mu }={\begin{pmatrix}ct\\0\\\eta \mathbf {x} \end{pmatrix}};\ x_{\tau }^{\mu }={\begin{pmatrix}0\\r\cdot f\left({\frac {\gamma \tau }{\Lambda }}\right)\\(1-\eta )\mathbf {x} \end{pmatrix}},\eta \in (0,1)$ and using the metric $(+,+,-,-,-)$ , the Lagrangian becomes

$L(x,{\dot {x}},x^{\prime },t,\tau )={\frac {r}{\Lambda }}{\sqrt {{\dot {c}}^{2}t^{2}+c^{2}-\eta ^{2}{\dot {\mathbf {x} }}^{2}+2{\dot {c}}ct}}{\sqrt {(\gamma ^{\prime 2}\tau ^{2}+\gamma ^{2}+2\gamma \gamma ^{\prime }\tau )\left(\left.{\frac {df}{dz}}\right|_{z={\frac {\gamma \tau }{\Lambda }}}\right)^{2}-(1-\eta )^{2}\mathbf {x} ^{\prime 2}}}.$ Applying the Euler-Lagrange Equations

${\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {x}}_{i}}}+{\frac {d}{d\tau }}{\frac {\partial L}{\partial x_{i}^{\prime \ }}}-{\frac {\partial L}{\partial x_{i}}}=0$ the existence of the Planck length and the constancy of the speed of light can be derived.[citation needed]

As a consequence of this model it has been suggested that the speed of light may not have been constant in the early universe.

### Speculative approaches

Speculative theories with more than one dimension of time have been explored in physics. The additional dimensions may be similar to conventional time, compactified like the additional spatial dimensions in string theory or components of a complex time. Complex time is two-dimensional, comprising one real time dimension and one imaginary time dimension, moving time from the positive real number line into the complex plane.

The special orthogonal group, SO(10,2), representing the GUT spin group of the extended supersymmetry structure of M-theory, led to the systematic development of "two-time physics" by Itzhak Bars.

In F-theory the possibility of one or two compactified additional time dimensions is not ruled out.

Walter Craig and Steven Weinstein have shown the existence of a well-posed initial value problem for the ultrahyperbolic equation (a wave equation in more than one time dimension). This demonstrated that initial data on a mixed (spacelike and timelike) hypersurface, obeying a particular nonlocal constraint, evolves deterministically in the remaining time dimension.

5D spacetime theory has been developed by Paul Wesson and colleagues. It extends Kaluza–Klein theory in order to account for matter along with time and space. They found that while the universe has a big-bang singularity in conventional 4D spacetime, it may be smooth in 5D space.

Introducing complex time into Minkowski spacetime allows a generalization of Kaluza–Klein theory. The approach is being developed by the Statistics Online Computational Resource (SOCR) group at the University of Michigan, who refer to complex time as "kime" and to the modified spacetime model as "space-kime". An advantage of this approach is said to be that it enables spacetime inference and data-driven analytics based on the extension of longitudinal data (e.g., time-series) to timesurfaces over the 5D spacetime manifold, which is complete and solves many of the problems of time.

## Philosophy

Multiple time dimensions appear to allow the breaking or re-ordering of cause-and-effect in the flow of any one dimension of time. This and conceptual difficulties with multiple physical time dimensions have been raised in modern analytic philosophy.

As a solution to the problem of the subjective passage of time, J. W. Dunne proposed an infinite hierarchy of time dimensions, inhabited by a similar hierarchy of levels of consciousness. Dunne suggested that, in the context of a "block" spacetime as modelled by General Relativity, a second dimension of time was needed in order to measure the speed of one's progress along one's own timeline. This in turn required a level of the conscious self existing at the second level of time. But the same arguments then applied to this new level, requiring a third level, and so on in an infinite regress. At the end of the regress was a "superlative general observer" who existed in eternity. He published his theory in relation to precognitive dreams in his 1927 book An Experiment with Time and went on to explore its relevance to contemporary physics in The Serial Universe (1934). His infinite regress was criticised as logically flawed and unnecessary, although writers such as J. B. Priestley acknowledged the possibility of his second time dimension.