Exploring the Electromagnetic Interactions in the 2T + 3S Dimensions Without Compactification: Implications for a Segmented Space-Time

1. Introduction

Recently, Zahir [1] considered a two-time (characterized by distinct causality speeds c₁ and c₂) and three-space-dimensional Minkowski space (2T + 3S). He derived relativistic coordinate and velocity transformation formulas and expressions for a new effective speed limit $c_e=\sqrt{(c_1{}^2+c_2{}^2/k^2)}$where k is a scale factor connecting the two times t₁ and t₂. If dimensions of t₁, t_2, and any of the space dimensions are denoted by T, t, and L, respectively, then dimensions of c₁, c₂, and k are LT^-1, Lt^-1, and Tt^-1, respectively. We emphasize that although t₁ and t₂ are both “time-like” variables, this time (t₁) is not the same as that time (t₂). Previously, Velev [2] explored the formulations of coordinate and kinematic transformations in a detailed analysis of a flat space-time with extra time dimensions. However, Velev [2] used the same speed of causality c for all time dimensions, including the extra ones. This at least creates one calculational problem – we cannot take the limit for c assigned to the extra times to zero to get back the familiar four-dimensional (i.e., those of Einstein’s Special Theory of Relativity ([3] and references therein)) formulations in the Minkowski ([3] and references therein) space meaningfully. There was no way to distinguish one c from the other [2].

In our familiar space-time structure, the extra time dimensions are not observed, and thus, in all higher-dimensional physics, the idea of compactification has often been considered ([4,5] and references therein). Zahir [1] also discussed compactification in a two-time-dependent Schrödinger-like equation with an infinite square-well potential. After compactifying the extra time dimension on a closed-loop topology with a period matching the Planck time, the solution generated interference among additional quantum states with ultra-short oscillation periods as well. Zahir [6] also considered compactification in another related work on the Klein-Gordon-type equation in 2T + 3S dimensions. Tyagi and Al-Shahristani [7] developed the Dirac equations within Zahir’s paradigm but did not consider compactification of the extra time dimension.

In this paper, we will explore the extension of classical electromagnetic fields in the 2T + 3S dimensions without compactification. We will investigate how Maxwell’s equations are modified, if at all, in the five-dimensional “Lorentz” covariant analysis and its possible implications. We will postulate a bold physical space-time environment defined by two times related to two interactions carried by two massless intermediaries with two distinct speeds of causality. This is exactly the conceptual framework on which Zahir [1] developed his two-time relativity. The outcome will be intriguing as we relate it to experimental results from contemporary physics (see Section 4 and Section 5).

In Section 1, we introduce the concept of Zahir’s 2T paradigm [1] and a bold proposition of the Color Space (CS). In Section 2, the structure of the 2T +3S dimensional space is briefly reviewed. Section 3 is dedicated to developing formulations of classical electrodynamics, modified Maxwell’s equations, and the concept of effective charge in the 2T + 3S dimensions following a covariant approach. In Section 4, we present and extend the concepts of the 2T paradigm in the Color Space, with the massless photon and gluon mediating two interactions that move at two intrinsic speeds of causality. In Section 5, we use the experimental R-values of the e⁺e^- → hadrons collision to derive a possible value of the gluon speed considering the effective charge. Conclusions and discussions are presented in the last Section.

2. Summary of the 2T + 3S Space-Time Structure

As noted earlier, the different speeds of causality for times t₁ and t₂ are given by c₁ and c₂. Thus, the space-time variables in the (2T + 3S) space are ($c_1{t}_1, c_2{t}_2, x, y, z$). We consider Minkowski-like flat space with metric signatures (+, +, -, -, -) such that the invariant space-time interval ds is given by$d{s}^2= c_1{}^{2}d{t}_1{}^2+ c_2{}^{2}d{t}_2{}^2– d{x}^2– d{y}^2– d{z}^2$.

After a detailed calculation, Zahir [1] derived the 2T relativistic coordinate transformation for the reference frame K’ moving at a uniform speed relative to reference frame K. He considered the standard configuration, and the motion of K’ is along the x coordinate only, such that at t₁ = 0 and t₂ = 0, the coordinate axes of K and K’ coincide. K’ is moving with uniform velocities v and w defined with respect to times t₁ and t_2, respectively. So, if x₀ is the coordinate of the origin of K’ with respect to K at any time,

$$\begin{gathered} v=d{x}_0/d{t}_1\hspace{0.17em}\hspace{0.17em};\hspace{0.17em}w=d{x}_0/d{t}_2; \\ w={\displaystyle \frac{d{x}_0}{d{t}_2}}={\displaystyle \frac{d{x}_0}{d{t}_1}}.{\displaystyle \frac{d{t}_1}{d{t}_2}}=v.k, where \\ k={\displaystyle \frac{d{t}_1}{d{t}_2}}. \end{gathered}$$

This implies that$t_1=k.t_2$. The expressions for coordinate transformations are given in equation box (1) below. For the full definition and expressions of the five-dimensional proper time, five-velocities, five-momenta, and various invariant relationships, see ref. [1].

$$\begin{gathered} \begin{array}{|c|}\hline x^{\prime }={\displaystyle \frac{x-v{t}_1}{\sqrt{1-\frac{v^2}{c_e{}^2}}}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{t}_1{}^{\prime }={\displaystyle \frac{t_1-\frac{xv}{c_e{}^2}}{\sqrt{1-\frac{v^2}{c_e{}^2}}}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\hspace{0.17em}\hspace{0.17em}{t}_2{}^{\prime }={\displaystyle \frac{t_2-\frac{xw}{k^2{c}_e{}^2}}{\sqrt{1-\frac{w^2}{k^2{c}_e{}^2}}}}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{t_2-\frac{xv}{k{c}_e{}^2}}{\sqrt{1-\frac{v^2}{c_e{}^2}}}}\hspace{0.17em}\\ \\ \begin{array}{c}{c}_e{ }^2=c_1{}^2+c_2{}^2/k^2=c_1{}^2(1+c_2{}^2/c_1{}^2{k}^2)=\hspace{0.17em}{c}_1{}^2(1+{\lambda }^2)\\ \mathrm{where} \lambda \hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{c_2}{c_1\hspace{0.17em}k\hspace{0.17em}}}={\displaystyle \frac{\rho }{k}}; \rho \hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{c_2}{c_1\hspace{0.17em}\hspace{0.17em}}} \mathrm{and} \mathrm{we} \mathrm{will} \mathrm{use} \mathrm{these} \mathrm{variables} \mathrm{later} \mathrm{in} \mathrm{the} \mathrm{text}\end{array}\\ \hline\end{array} \end{gathered}$$

3. Electromagnetic Field Tensors in the 2T + 3S Dimension: The Modified Maxwell’s Equations

3.1. The Electromagnetic Field Tensors

We begin our work with the familiar antisymmetric electromagnetic field tensor $F_{\mu \nu }$as defined in terms of the scalar and vector potentials [8,9], $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$. In the 1T + 3S space-time (i.e., the four-dimensional Minkowski space), the four components of A_µ are φ and -Ᾱ for the time component and three space components, respectively (we use bold notations (or with a bar on top) for three-dimensional vectors). In our 2T + 3S space-time, we need to propose an intuitive ansatz for the second component, A_2, of the five-potential. Note that for the extra time dimension t₂, we assumed $t_2={\displaystyle \frac{1}{k}}\hspace{0.17em}\cdot t_1$. For A_2, we assume $A_2=h.\phi \hspace{0.17em}\mathrm{with} A_1=\phi$ and h is an unknown dimensionless parameter. We will see later what value for h will restore consistency with the familiar form of Maxwell’s equations. Therefore,

$$A_{\mu }\to \hspace{0.17em}{A}_1,A_2,-\overline{A}\hspace{0.17em}\to \phi ,\hspace{0.17em}h\phi ,-A_x,-A_y,-A_z\hspace{0.17em}\mathrm{for} \mu =1,2,3,4,5\hspace{0.17em},\hspace{0.17em}\mathrm{respectively}.$$

Out of 25 elements of the antisymmetric tensor$F_{\mu \nu }$, 10 are independent. We recall the following notations before proceeding to compute the 10 elements presented in Table 1.

$$\begin{array}{l}{x}^1=c_1{t}_1,x^2=c_2{t}_2,x^3=x,x^4=y,x^5=z\\ {\partial }_{\mu }={\displaystyle \frac{\partial }{\partial x^{\mu }}};{\displaystyle \frac{\partial }{c_2\partial t_2}}=k{\displaystyle \frac{\partial }{c_2\partial t_1}}={\displaystyle \frac{k}{\rho }}{\displaystyle \frac{\partial }{c_1\partial t_1}}\hspace{0.17em}\hspace{0.17em}\mathrm{using}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{d{t}_1}{d{t}_2}}=k\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\hspace{0.17em}{c}_2=\rho c_1\end{array}$$

While deriving the tensor elements, our inputs are the components of A_µ, and we will define the traditional relations $\overline{\mathrm{E}} = -{\partial }_1 \overline{\mathrm{A}}-\overline{\nabla }\phi \hspace{0.17em}\hspace{0.17em}$and$\overline{\mathrm{B}}\hspace{0.17em}=\hspace{0.17em}\overline{\nabla }\times \overline{\mathrm{A}}$. We could have used any other symbols in place of E and B (say, P and Q), and the resulting equations would involve P and Q. However, at the end, we will need to identify them as E and B to make the equations consistent with the familiar Maxwell’s equations. F_µν can be represented in the matrix form (µ and ν are row and column indices respectively),

$$F_{\mu \nu }=\left[\begin{array}{ccccc}{F}_{11}& F_{12}& F_{13}& F_{14}& F_{15}\\ F_{21}& F_{22}& F_{23}& F_{24}& F_{25}\\ F_{31}& F_{32}& F_{33}& F_{34}& F_{35}\\ F_{41}& F_{42}& F_{43}& F_{44}& F_{45}\\ F_{51}& F_{52}& F_{53}& F_{54}& F_{55}\end{array}\right]$$

(1)

Later, we also need F^µν = ${\partial }^{\mu }{A}^{\nu }-{\partial }^{\nu }{A}^{\mu }$. It can also be derived directly from F_µν as follows,

$$F^{\alpha \beta }=g^{\alpha \nu }{F}_{\nu \delta }{g}^{\delta \beta }\hspace{0.17em}(\mathrm{summation} \mathrm{over} \mathrm{repeated} \mathrm{indices} \mathrm{is} \mathrm{understood}).$$

F^µν can also be represented in matrix form after doing the matrix multiplications using (2) and

$$g^{\alpha \nu }=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& -1& 0& 0\\ 0& 0& 0& -1& 0\\ 0& 0& 0& 0& -1\end{array}\right]$$

Finally, we get

$$\begin{gathered} F^{\mu \nu }=\left[\begin{array}{ccccc}{F}^{11}& F^{12}& F^{13}& F^{14}& F^{15}\\ F^{21}& F^{22}& F^{23}& F^{24}& F^{25}\\ F^{31}& F^{32}& F^{33}& F^{34}& F^{35}\\ F^{41}& F^{42}& F^{43}& F^{44}& F^{45}\\ F^{51}& F^{52}& F^{53}& F^{54}& F^{55}\end{array}\right] \\ =\left[\begin{array}{ccccc}{F}_{11}& F_{12}& -F_{13}& -F_{14}& -F_{15}\\ F_{21}& F_{22}& -F_{23}& -F_{24}& -F_{25}\\ -F_{31}& -F_{32}& F_{33}& F_{34}& F_{35}\\ -F_{41}& -F_{42}& F_{43}& F_{44}& F_{45}\\ -F_{51}& -F_{52}& F_{53}& F_{54}& F_{55}\end{array}\right] \end{gathered}$$

Restoring all values, we get the explicit expressions for the relevant elements of F^µν in Table 2.

3.2. Deriving Maxwell’s Equations from the Electromagnetic Field Tensors in the 2T + 3S dimension: The Covariant Approach

First, we derive the homogeneous equations using the covariant form [5,6],

$${\partial }_{\lambda }{F}_{\mu \nu }+{\partial }_{\lambda }{F}_{\mu \nu }+{\partial }_{\lambda }{F}_{\mu \nu }=0$$

Here, λ, µ, and ν are all different, with possible values (1, 2, 3, 4, 5) for each. It is worth noting that the field tensors F_µν are antisymmetric and thus the diagonal elements are zero. There are 60 nonzero equations that can be divided into 10 sets of 6 each. The members of each set are different permutations of the indices and are the same except for an overall sign. Since the right-hand sides of the equations are zeros, the sign does not matter. There are 10 unique equations representing each set.

We denote the sets as

A: {λ, µ, ν} → {1,2,3} implying 1,2,3; 1,3,2; 2,1,3; 2,3,1; 3,1,2; 3,2,1

B: {λ, µ, ν} → {1,2,4} C: {λ, µ, ν} → {1,2,5} D: {λ, µ, ν} → {1,3,4}

E: {λ, µ, ν} → {1,3,5} F: {λ, µ, ν} → {1,4,5} G: {λ, µ, ν} → {2,3,4}

H: {λ, µ, ν} → {2,3,5} I: {λ, µ, ν} → {2,4,5} J: {λ, µ, ν} → {3,4,5}

The details of the calculations are given in Appendix A. We present the results,

Three equations representing A, B, and C are identically zero.

Three equations representing D, E, and F combine to give

$${\displaystyle \frac{1}{c_1}}{\displaystyle \frac{\partial \hspace{0.17em} \overline{\mathrm{B}}}{\partial t_1}}+\nabla \times \hspace{0.17em} \overline{\mathrm{E}}=0$$

Three equations representing G, H, and I combine to confirm

$$\overline{\mathrm{B}}\hspace{0.17em}=\hspace{0.17em}\overline{\nabla }\times \overline{\mathrm{A}}$$

The equation representing J gives,

$$\overline{\nabla }· \overline{\mathrm{B}}=0$$

Here we used the traditional relations,

$$\overline{\mathrm{E}}=-{\partial }_1 \overline{\mathrm{A}}-\overline{\nabla }\phi \hspace{0.17em}\hspace{0.17em}$$

(2)

Next, we explore inhomogeneous equations involving sources. They can be derived from the covariant form,

$${\partial }_{\mu }{F}^{\mu \nu }=4\pi {J^{\prime }}^{\nu }$$

Again, we need to make an ansatz regarding the second component of the charge-current five-vector J’^ν,

$${J^{\prime }}^1={\rho }_0;{J^{\prime }}^2=r{\rho }_0; \overline{\mathrm{J}}={\displaystyle \frac{ \overline{\mathrm{J}}}{c_1}};$$

where r is a dimensionless quantity (like h) that will be determined later. We will use the five-dimensional Lorentz condition ${\partial }_{\mu }{A}^{\mu }=0$to decouple the equations. For a full expression of the Lorentz condition in five dimensions, see Appendices B, C, D.

(1)

ν = 1

The corresponding equation is,

$$\begin{array}{l}{\partial }_{\mu }{F}^{\mu 1}=4\pi {J^{\prime }}^1\\ {\partial }_1{F}^{11}+{\partial }_2{F}^{21}+{\partial }_3{F}^{31}+{\partial }_4{F}^{41}+{\partial }_5{F}^{51}=4\pi {\rho }_0\\ \mathrm{Using} \mathrm{the} \mathrm{expressions} \mathrm{in} \mathrm{Table} 2, \mathrm{and} {\displaystyle \frac{\partial }{c_2\partial t_2}}=k{\displaystyle \frac{\partial }{c_2\partial t_1}}={\displaystyle \frac{k}{\rho }}{\displaystyle \frac{\partial }{c_1\partial t_1}}\\ -{\displaystyle \frac{1}{c_2}}{\displaystyle \frac{\partial }{\partial t_2}}[(h-{\displaystyle \frac{k}{\rho }}){\displaystyle \frac{\partial \phi }{c_1\partial t_1}}]+{\displaystyle \frac{\partial E_x}{\partial x}}+{\displaystyle \frac{\partial E_y}{\partial y}}+{\displaystyle \frac{\partial E_z}{\partial z}}=4\pi {\rho }_0\\ -(h-{\displaystyle \frac{k}{\rho }}){\displaystyle \frac{k}{\rho }}{\displaystyle \frac{\partial \phi }{c_1{}^2\partial t_1{}^2}}+\overline{\nabla }·\mathrm{E}=4\pi {\rho }_0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\end{array}$$

(3)

(2)

ν =2,

$$\begin{array}{l}{\partial }_{\mu }{F}^{\mu 2}=4\pi {J^{\prime }}^2\\ {\partial }_1{F}^{12}+{\partial }_2{F}^{22}+{\partial }_3{F}^{32}+{\partial }_4{F}^{42}+{\partial }_5{F}^{52}=4\pi r{\rho }_0\\ \mathrm{Using} \mathrm{the} \mathrm{expressions} \mathrm{in} \mathrm{Table} 2, \mathrm{simplifying} \mathrm{and} \mathrm{regrouping} \mathrm{terms}, \mathrm{we} \mathrm{get} (\mathrm{see} \mathrm{Appendix} \mathrm{B}),\\ h{\square }_5\phi -{\displaystyle \frac{k}{\rho }}{\displaystyle \frac{\partial }{c_1\partial t_1}}({\partial }_{\mu }{A}^{\mu })=4\pi r{\rho }_0\\ \mathrm{where} {\square }_5\hspace{0.17em}\mathrm{is} \mathrm{the} \mathrm{five}-\mathrm{dimensional} \mathrm{d}\text{'} \mathrm{Alembertian} \mathrm{and} \mathrm{is} \mathrm{given} \mathrm{by}, \\ {\square }_5\hspace{0.17em}=\hspace{0.17em}{\partial }_{\mu }{\partial }^{\mu }={\partial }_1{}^2+{\partial }_2{}^2-{\overline{\nabla }}^2\\ \mathrm{Using} \mathrm{the} \mathrm{five}-\mathrm{dimensional} \mathrm{Lorentz} \mathrm{condition} ({\partial }_{\mu }{A}^{\mu }=0), \mathrm{we} \mathrm{get} \mathrm{the} \mathrm{decoupled} \mathrm{equation},\\ {\square }_5\phi =4\pi {\displaystyle \frac{r}{h}}{\rho }_0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\end{array}$$

(4)

We can get an interesting result if we substitute Equation (4) in Equation (3) and use Equation (2), in which

items with a bar on top are three-dimensional vectors corresponding to the space coordinates (3), (4), and (5). See the detailed calculations in Appendix C, and we get,

$$-{\partial }_1({\partial }_{\mu }{A}^{\mu })=4\pi {\rho }_0(1-{\displaystyle \frac{r}{h}})$$

Using the Lorentz condition ${\partial }_{\mu }{A}^{\mu }=0$, we get,

$r=h$ which simplifies Equation (4) to

$${\square }_5\phi =4\pi {\rho }_0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$$

(5)

(3)

ν = 3, 4, 5

$$\begin{array}{l}{\partial }_{\mu }{F}^{\mu 3}=4\pi {J^{\prime }}^3\\ {\partial }_1{F}^{13}+{\partial }_2{F}^{23}+{\partial }_3{F}^{33}+{\partial }_4{F}^{43}+{\partial }_5{F}^{53}=4\pi {\displaystyle \frac{J^3}{c_1}}=4\pi {\displaystyle \frac{J_x}{c_1}}\\ \mathrm{Using} \mathrm{the} \mathrm{expressions} \mathrm{in} \mathrm{Table} 2,\\ -{\displaystyle \frac{\partial E_x}{c_1\partial t_1}}+{\displaystyle \frac{\partial }{c_2\partial t_2}}[{\displaystyle \frac{k}{\rho }}{\displaystyle \frac{\partial A_x}{c_1\partial t_1}}+h{\displaystyle \frac{\partial \phi }{\partial x}}]+{\displaystyle \frac{\partial B_z}{\partial y}}-{\displaystyle \frac{\partial B_y}{\partial z}}=4\pi {\displaystyle \frac{J_x}{c_1}}\\ \mathrm{Combining} \mathrm{the} \mathrm{results} \mathrm{for} \nu =4 \mathrm{and} 5, \mathrm{and} \mathrm{using} {\displaystyle \frac{\partial }{c_2\partial t_2}}=k{\displaystyle \frac{\partial }{c_2\partial t_1}}={\displaystyle \frac{k}{\rho }}{\displaystyle \frac{\partial }{c_1\partial t_1}}\\ {\displaystyle \frac{k}{\rho }}{\displaystyle \frac{\partial }{c_1\partial t_1}}[{\displaystyle \frac{k}{\rho }}{\displaystyle \frac{\partial \overline{\mathrm{A}}}{c_1\partial t_1}}+h\overline{\nabla }\phi ]+\overline{\nabla }\times \overline{\mathrm{B}}-{\displaystyle \frac{\partial \overline{\mathrm{E}}}{c_1\partial t_1}}=4\pi {\displaystyle \frac{ \overline{\mathrm{J}}}{c_1}}\end{array}$$

Using $\overline{\mathrm{E}} = -{\partial }_1 \overline{\mathrm{A}}-\overline{\nabla }\phi$ in the equation above, we get,

$$\overline{\nabla }\times \overline{\mathrm{B}}-(1+{\displaystyle \frac{k^2}{{\rho }^2}}){\displaystyle \frac{\partial \overline{\mathrm{E}}}{c_1\partial t_1}}+(h-{\displaystyle \frac{k}{\rho }}){\displaystyle \frac{k}{\rho }}{\displaystyle \frac{\partial }{c_1\partial t_1}}(\overline{\nabla }\phi )=4\pi {\displaystyle \frac{ \overline{\mathrm{J}}}{c_1}}$$

(6)

In Equation (6), we substitute $\overline{\mathrm{B}}\hspace{0.17em}=\hspace{0.17em}\overline{\nabla }\times \overline{\mathrm{A}}$and expand $\overline{\nabla }\times \overline{\nabla }\times \overline{\mathrm{A}}$as usual.

Using Equation (2) and $\mathrm{the} \mathrm{five}-\mathrm{dimensional} \mathrm{Lorentz} \mathrm{condition} ({\partial }_{\mu }{A}^{\mu }=0)$, we get,

$${\square }_5 \overline{\mathrm{A}}=4\pi {\displaystyle \frac{ \overline{\mathrm{J}}}{c_1}}\hspace{0.17em}\hspace{0.17em}$$

(7)

It is a decoupled equation as in four-dimensional electrodynamics. See Appendix D for the detailed calculation. We can also derive the continuity equation by applying $\overline{\nabla }·\hspace{0.17em}\hspace{0.17em}$on both sides of Equation (6), and using Equation (3) and Equation (5) and simplifying the tedious calculations (see Appendix E),

we get,

$$\begin{array}{l}{\partial }_1{\rho }_0+{\partial }_2(h{\rho }_0)\hspace{0.17em}+\overline{\nabla }·({\displaystyle \frac{ \overline{\mathrm{J}}}{c_1}})=0\\ {\partial }_{\mu }{J^{\prime }}^{\mu }=0\hspace{0.17em}\hspace{0.17em}(\mathrm{remember} h=r;\hspace{0.17em}{J^{\prime }}^1={\rho }_{{}_0};{J^{\prime }}^2=r{\rho }_0{}_{{}_0}; \mathrm{J} \prime ¯={\displaystyle \frac{ \overline{\mathrm{J}}}{c_1}})\end{array}$$

(8)

It is worth noting that, so far, we have derived one condition: r = h. However, if we set,

$$h={\displaystyle \frac{k}{\rho }}$$

the Equations (3) and (6) become more like the familiar four-dimensional ones,

$$\overline{\nabla }· \overline{\mathrm{E}}=4\pi {\rho }_0$$

(9)

$$\overline{\nabla }\times \overline{\mathrm{B}}-(1+{\displaystyle \frac{k^2}{{\rho }^2}}){\displaystyle \frac{\partial \overline{\mathrm{E}}}{c_1\partial t_1}}=4\pi {\displaystyle \frac{ \overline{\mathrm{J}}}{c_1}}$$

(10)

Equation (7) remains unchanged. Let us explore Equation (8).

$$\begin{array}{l}{\partial }_1{\rho }_0+{\partial }_2(h{\rho }_0)\hspace{0.17em}+\overline{\nabla }·({\displaystyle \frac{ \overline{\mathrm{J}}}{c_1}})=0\\ {\displaystyle \frac{\partial {\rho }_0}{\partial t_1}}+{\displaystyle \frac{k^2}{{\rho }^2}}{\displaystyle \frac{\partial {\rho }_0}{\partial t_1}}+\overline{\nabla }· \overline{\mathrm{J}}=\overline{\nabla }· \overline{\mathrm{J}}+\hspace{0.17em}(1+{\displaystyle \frac{k^2}{{\rho }^2}}){\displaystyle \frac{\partial {\rho }_0}{\partial t_1}}=0\end{array}$$

(11)

$$\mathrm{Here} \mathrm{we} \mathrm{used} \mathrm{again}\hspace{0.17em}{\displaystyle \frac{\partial }{c_2\partial t_2}}=k{\displaystyle \frac{\partial }{c_2\partial t_1}}={\displaystyle \frac{k}{\rho }}{\displaystyle \frac{\partial }{c_1\partial t_1}},\hspace{0.17em}\mathrm{and} h={\displaystyle \frac{k}{\rho }}$$

(11)

Let us define

$${\tilde{\rho }}_0={\rho }_0(1+{\displaystyle \frac{k^2}{{\rho }^2}})\hspace{0.17em};\hspace{0.17em} \mathrm{E} ˜¯= \overline{\mathrm{E}}\hspace{0.17em}(1+{\displaystyle \frac{k^2}{{\rho }^2}})\hspace{0.17em};\hspace{0.17em}\tilde{\phi }=\phi (1+{\displaystyle \frac{k^2}{{\rho }^2}})\hspace{0.17em}$$

Then, the Equations (5), (9), (10), and (11) can be rewritten as,

${\square }_5\tilde{\phi }=4\pi {\tilde{\rho }}_0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$	$\overline{\nabla }· \mathrm{E} ˜¯=4\pi {\tilde{\rho }}_0$
$\overline{\nabla }\times \mathrm{B} ¯-{\displaystyle \frac{\partial \mathrm{E} ˜¯}{c_1\partial t_1}}=4\pi {\displaystyle \frac{ \mathrm{J} ¯}{c_1}}$	$\overline{\nabla }· \overline{\mathrm{J}}+\hspace{0.17em}{\displaystyle \frac{\partial {\tilde{\rho }}_0}{\partial t_1}}=0$

If we integrate

$$\overline{\nabla }· \mathrm{J} ¯+\hspace{0.17em}{\displaystyle \frac{\partial {\tilde{\rho }}_0}{\partial t_1}}=0$$

over the space volume, we get,

$$\begin{array}{l}{\displaystyle \int \overline{\nabla }· \mathrm{J} ¯\hspace{0.17em}dv}\hspace{0.17em}=\hspace{0.17em}-\hspace{0.17em}{\displaystyle \frac{\partial }{\partial t_1}}{\displaystyle \int {\tilde{\rho }}_0\hspace{0.17em}dv=-(1+\frac{k^2}{{\rho }^2})}{\displaystyle \frac{\partial }{\partial t_1}}{\displaystyle \int {\rho }_0\hspace{0.17em}dv}\\ {\displaystyle \oint \mathrm{J} ¯·\hspace{0.17em}d\overline{s}}\hspace{0.17em}=-(1+{\displaystyle \frac{k^2}{{\rho }^2}}){\displaystyle \frac{\partial q}{\partial t_1}}=-{\displaystyle \frac{\partial \tilde{q}}{\partial t_1}}\\ \mathrm{where} \\ \tilde{q}=q(1+{\displaystyle \frac{k^2}{{\rho }^2}})\end{array}$$

In summary, the new formulations are very similar to the corresponding Maxwell’s equations in four dimensions. The effect of the extra time dimension (albeit subject to the various ansatzes) is that it gives us an effective electric charge,

$$\tilde{q}=q(1+{\displaystyle \frac{k^2}{{\rho }^2}})\hspace{0.17em}$$

(12)

In addition, the wave equations, the continuity equation, and the Lorentz condition are extended to five dimensions. We will use these theoretical results in the context of a new, intriguing proposal regarding two-time physics with different speeds of causality. We present it later in Section 4 and Section 5.

4. Discussions and Conclusions

In the preceding sections of this paper, we explored a new formulation for classical electrodynamics in the 2T + 3S dimensions. The five-dimensional Maxwell’s equations are very similar to the ones in 1T + 3S dimensions. In the two-time paradigm, the wave equations, the continuity equation, and the Lorentz (gauge) condition naturally emerge as five-dimensional. In addition, we found an effective charge $\tilde{q}$. However, it is worth noting that these formulations were derived based on a couple of ansatzes – one for the extra potential A₂ and one for the extra component of the charge-current. Then, we extended the concept of two-time with distinct speeds of causality to the hadronic Color-Space (CS) populated by color-charged particles - quarks and gluons. We made a bold proposition that the extra time dimension is tied to a massless gluon moving at an intrinsic speed c₂, distinct from c₁, the speed of light. Next, we explored the R–values of e⁺e^- → hadrons collisions in terms of the effective charge. The experimental values of R are found to exceed the theoretical ones even with the QCD corrections. This led us to an estimate of the gluon speed as a multiple (>1) of the speed of light. We presented the analysis as an illustration of the proposed concept. Further investigation of data over an extended range of the center-of-mass energy of the collider experiments is required to confirm the new conceptual space-time configuration in the 2T + 3S dimensions. Since we propose that the 2T paradigm applies to the CS, we expect that any estimate of λ would be very small from experiments probing outside the CS.

While Minkowski space-time is flat, in General Relativity [25], it is curved due to the presence of matter and energy. What we are proposing here is a new concept in which flat space-time may be affected by the presence of color through the addition of an extra time dimension with a distinct speed of causality, identified with the speed of gluons in Color-Space (CS).

The CS is already enigmatic because of the confinement phenomenon and its conceptualization as a two-time paradigm characterized by the speed of gluons (c₂) and the speed of light (c₁) (with $c_2\ne c_1$) may add an extra feature to it. It can be interpreted as a kind of segmentation of space-time - the Color Space and the one exterior to it. We may enquire whether there can be a space-time phase transition from one to the other. To find an answer, we need further theoretical developments to determine whether the phase transition is of Type I or Type II and what the order parameter is. This will raise a new question about whether any latent energy is involved. Whether this energy can provide any clues about the cosmological evolution of the Universe is unknown. After all, matter was formed in the early stage of the Universe via hadronization when quarks (and gluons) became confined within the Color-Space. Our conceptual proposition and the preliminary results from the R-value analysis may even influence how we interpret data from heavy-ion collisions to explain the dynamics of the quark-gluon plasma (QGP).