Multiple time dimensions

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, ..., 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

(ds_k,n)² = (cdt₁)² + ... + (cdt_k)² − (dx_k+1)² − … − (dx_k+n)².

The metric signature is then

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

or

${\displaystyle (\underbrace {-,\cdots ,-} _{k},\underbrace {+,\cdots ,+} _{n})}$ (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:[4]

${\displaystyle t'_{\sigma }=\sum _{\theta =1}^{k}\left(\delta _{\sigma \theta }t_{\theta }+{\frac {c^{2}}{v_{\sigma }v_{\theta }}}\beta ^{2}(\zeta -1)t_{\theta }\right)-{\frac {1}{v_{\sigma }}}\beta ^{2}\zeta x_{k+1},}$

${\displaystyle x'_{k+1}=-c^{2}\beta ^{2}\zeta \sum _{\theta =1}^{k}{\frac {t_{\theta }}{v_{\theta }}}+\zeta x_{k+1},}$

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

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

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

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

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

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 ${\displaystyle \Delta T={\sqrt {(\Delta t_{1})^{2}+(\Delta t_{2})^{2}}}\geq 0}$ 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.[4]

The motion of a test particle in a spacetime with a second time dimension may be described by coordinate

${\displaystyle x^{\mu }={\begin{pmatrix}ct\\r\cdot f\left({\frac {\gamma \tau }{\Lambda }}\right)\\\mathbf {x} \end{pmatrix}}}$

which is the canonical (1,3) spacetime vector ${\displaystyle (ct,\mathbf {x} )^{T}}$ with ${\displaystyle x\in \mathbb {R} ^{3}}$ extended by an additional timelike coordinate ${\displaystyle r\cdot f(\gamma \tau /\Lambda )}$. ${\displaystyle \tau }$ is the second time parameter corresponding to ${\displaystyle t}$, ${\displaystyle r\in \mathbb {R} }$ describes the size of the second time dimension and ${\displaystyle \gamma }$ is the characteristic velocity equivalent to ${\displaystyle c}$. ${\displaystyle f}$ describes the shape of the second time dimension and ${\displaystyle \Lambda \in \mathbb {R} }$ is a normalization parameter such that ${\displaystyle \gamma \tau /\Lambda }$ is dimensionless.

Decomposing ${\displaystyle x^{\mu }=x_{t}^{\mu }+x_{\tau }^{\mu }}$ with

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

and using the metric ${\displaystyle (+,+,-,-,-)}$, the Lagrangian becomes

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

Applying the Euler-Lagrange Equations

${\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {x}}_{i}}}+{\frac {d}{d\tau }}{\frac {\partial L}{\partial x_{i}^{\prime \ }}}-{\frac {\partial L}{\partial x_{i}}}=0}$

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.[5]