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 "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.

Causal structure of a space-time with two time dimensions and one space dimension

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.

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 (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). This showed that initial data on a mixed (spacelike and timelike) hypersurface obeying a particular nonlocal constraint evolves deterministically in the remaining time dimension.

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.

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

Philosophy

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.