Extra Time Dimension: Deriving Five-Dimensional Relativistic Space-Time Transformations, Kinematics, and Time-Dependent Non-Relativistic Quantum Mechanics with Compactification©

We consider a two-time (characterized by distinct speeds of causality) and three space dimensional Minkowski space and derive relativistic coordinate and velocity transformation formulas and expressions for a new effective speed limit . Extending the ideas of Einstein’s Theory of Special Relativity, concepts of five-velocity and five-momenta are introduced leading to a new formula for the rest energy of a massive object. Based on a non-relativistic limit, a two-time dependent Schrödinger-like equation is developed. Its solution for infinite square-well potential, after compactifying the extra time dimension in a closed loop topology with a period matching the Planck time, generates interference of additional quantum states with an ultra-small period of oscillation, as well. Some cosmological implications of the concept of four-dimensional versus five-dimensional masses are briefly discussed, too.


I. Introduction
The structure of space-time based on Einstein's Theory of Special Relativity (TSR) [1] and the associated four-dimensional concept of Minkowski's space [2] are familiar and well-understood.The underlying framework has one time dimension and three space dimensions (i.e., 1T + 3S space-time structure).Later, Einstein's Theory of General Relativity (TGR) introduced [3] the concept of curved space-time and a new interpretation for the gravitational force -still within the (1T + 3S) fourdimensional framework of space-time.In this paper, we will not consider gravity and thus curved spacetime will remain beyond our discussion (i.e., we will consider only a 'flat' space-time).One of the earliest attempts to introduce extra dimensions (i.e., beyond the 1T + 3S four-dimensional space-time) in physics was the Kaluza-Klein Theory (KKT) that considered a five-dimensional (1T + 4S) space-time with an extra space dimension [4,5].But the time itself remained one-dimensional.The KKT would stand out for exploring two key ideas -1) the unifying idea of the TGR with electro-magnetism and 2) the concept of dimensional compactification (i.e., compactifying the extra space dimension by curling it up into an ultrasmall circular loop).The current highly active field of the String Theory (ST) extended KKT ideas into many more dimensions to unify all fundamental forces including gravity (see [6] and references therein).
In this paper, we will not discuss such theories.We would rather consider only one extra time dimension.Such a five-dimensional space-time with two times has been considered in the literature by several researchers from different perspectives embracing various conceptual domains (for example, see Bars [7] and references therein; also [8][9][10][11][12]).While considering extra time dimensions some authors think that such theories are a live conceptual possibility.If nothing else, they serve to stretch our minds into domains of new physical possibilities (see Weinstein [13,14]).
Our focus in this paper is to explore Lorentz-like space-time transformations (see [15] and references therein) in a five-dimensional space-time with two time and three space (i.e., 2T + 3S) coordinates and the associated kinematics.Consequently, we present an intriguing conceptual framework for understanding the impact of four-dimensional versus five-dimensional masses on physical theories.Detailed work on a five-dimensional (2T + 3S) Lorentz-like transformation is not available in the literature.Our goal is to provide sufficient details so that researchers, especially the fresh entrants in the field and graduate students, can explore the formulations and compare them readily with the familiar four-dimensional Lorentz transformation results [16].We derive the nonrelativistic version matching the Newtonian mechanics.Based on the non-relativistic form and using the familiar operator formulations of quantum mechanics, a two-time dependent Schrödinger-like equation is proposed.Solving a new version of the familiar infinite square-well potential and compactifying the extra time dimension on a periodic topology we get additional secondary quantum levels.A similar example was discussed by others [6] for an extra space dimension and a Kaluza-Klein type [5] compactification technique generated additional quantum levels as well.We present an elaborate analysis and highlight the difference in results in our case.
Conceptual arguments are presented in Section II explaining the meaning of speed of causality considering extra time dimensions.In Section III, we consider a flat space-time with 2T + 3S dimensions and derive the formulas for coordinate transformation between two inertial reference frames in the standard configuration (i.e., uniform velocity is along the x-axis) with all other coordinate axes being parallel to each other.In addition, we obtain formulas for velocity transformations.Expressions for fivevelocity, five-momenta, and energy-momentum relationship -all as an extension of the familiar fourdimensional (i.e., 1T + 3S space) Lorentz transformation are derived in Section IV.In Section V, we derive a non-relativistic approximation of the five-dimensional kinematics and obtain a Schrödinger-like equation in 2T + 1S dimensions for the time-dependent one-dimensional infinite square-well potential.
Solutions and analysis of the quantum mechanical formulations are presented in Section VI.Comments and discussions of some unique outcomes of this research are in Section VII.Finally, Section VIII is dedicated to conclusions.

II. The Case for Two Time Dimensions with Different Speeds of Causality
In the 1T + 3S dimension, the relativistic coordinate transformations, as we get from Einstein's TSR, have been obtained in the literature from a set of postulates [16].In some of these sets of postulates, the constancy of the speed of light was not required (see [17] and references therein) but in the end, they needed a special speed, say V, that required to be 1) the maximum speed possible, 2) is tied to a particle whose rest mass is zero and, 3) is needed to be the same in all inertial frames.From empirical considerations, V eventually becomes the same as the speed c of light (i.e., photons having zero rest mass).Speed of light c enters the Lorentz transformation formulas through a ratio 0 / vc  = and Lorentz factor (here, v is the uniform speed of the inertial reference frame).As nothing can move faster than c, it has implications for the cause-and-effect relationship in all modes of interactions.Thus, c in TSR is not just the speed of light; it can be termed the speed of causality [18].
Velev [19] explores the formulations of coordinate and kinematic transformations in a detailed analysis of a flat space-time with extra time dimensions.However, Velev [19] 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 limit for the extra time's c to zero to get back the familiar four-dimensional (i.e., those of Einstein's TSR results) formulations as there is no way to distinguish one c from the other.
At the current energy level of the universe, we do not see the extra time dimension in any experiment.
Therefore, if we think of any theory of extra time dimension it has to be about the structure of spacetime at an early stage of the universe and that extra time dimension is compactified, so we do not see it now.Thus, we can conceptualize each of the time dimensions to be due to different interactions mediated by distinct massless particles moving with distinct speeds in the expanded space-time.
In such an environment, let us try to formulate the "modified" TSR for space-time through the following "gedanken" scenarios.Let us denote the interactions as interactions A and B.
First, we turn off interaction A and consider a space-time structure with 1T + 3S dimensions where interaction is carried by a massless particle moving with speed c1 which is the speed of causality.
Therefore, the space-time transformations (think TSR) between inertial reference frames will be the Lorentz transformations [16] with 01 / vc  = . Then, we turn off interaction A and turn on interaction B which will be mediated by a massless particle moving with speed c2 that will be the new speed of causality (it does not have to be equal to c1 because it does not know interaction A as we turned it off).
Therefore, now the 1T + 3S dimensional space-time transformations between inertial reference frames will be Lorentz transformations with . Finally, we turn on both interactions A and B carried by respective massless particles.Now, the plausible space-time structure has to be 2T + 3S dimensional where time t1 will be "influenced" by the speed of causality c1 and time t2 will be "influenced" by the speed of causality c2.
The five-dimensional flat space-time will comprise two time dimensions each having, in general, distinct speeds of causality and three space dimensions.This is the conceptual foundation of the extra time dimensional world being considered in this paper.In the standard model of elementary particles (see Salam [20] p. 45 -50 for a lucid explanation) there was a phase in the early universe when electromagnetic and weak interactions were carried by massless photons and W/Z bosons.However, the theory of electroweak unification is formulated in 1T + 3S dimensions, and at high energies when the gauge symmetry was unbroken massless photons and weak bosons cannot have different velocities.
Therefore, we will refrain from identifying c1 and c2 with the speeds of massless photons and any other known interaction at all.They are just two different speeds of causality associated with time dimensions t1 and t2 respectively.To be more specific, we call them "speeds of causality in isolation" implying that c1 is the speed of causality associated with an interaction mediated by a massless particle in the absence of any other "similar" interaction.Likewise, c2 is the speed of causality associated with an interaction mediated by a massless particle in the absence of any other "similar" interaction.When both are present then we conceptualize an expanded space-time with two time dimensions in a Minkowski-like formulation having two distinct speeds of causality plus a three-dimensional space.We show in this paper that the resulting relativistic formulations involve an effective speed of causality ce that is a combination of both c1 and c2.It will be shown that it would be possible to travel at speeds greater than c1 or c2, but not greater than ce.
Velev [19] derived the space-time transformation formulations and considered associated kinematics in (2T + 3S) space-time assuming the same speed of causality (i.e., the same numerical value) for both time dimensions.Consequently, Velev [19]  .He provided detailed results related to coordinate transformations, velocity, energy-momentum transformations, etc. between two inertial frames of reference.It is a natural extension of the familiar Lorentz transformation of four-dimensional Minkowski space exemplifying Einstein's TSR.He examined the causal structure of space-time showing that particles moving in multidimensional time are as stable as particles moving in one-dimensional time if certain conditions are met.Here in this paper, we consider a more general scenario -a fivedimensional space-time (i.e., two time and three space dimensions) but each time dimension has a unique speed of causality not necessarily having the same numeral value.More specifically, we make a more general assumption such that the speed of causality for time t1 and t2 are different.Thus, the space-time variables in the (2T + 3S) space are ( cc we emphasize that although t1 and t2 are both "time-like" variables, this time (t1) is not the same as that time (t2).We denote the dimensions of t1 and t2 by T and t respectively and the dimension of x, y, and z by L. So, the dimensions of c1 and c2 are LT -1 and Lt -1 respectively.Throughout the paper, we keep explicitly the variables c1 and c2 in all expressions without assigning a value of unity (i.e., will not use the natural system of units).This makes it possible to set c2 = 0 to obtain the four-dimensional results (i.e., those of Einstein's TSR) for the sake of comparison and for checking consistencies as needed.

III. Space-Time Transformation in 2T + 3S Dimensions (T stands for time and S stands for space)
We consider two times t1 and t2 to represent the first and second times respectively and the speeds of causality are c1 and c2 such that the five-dimensional space-time interval is invariant under transformation between two inertial reference frames K and K'.To be explicit, and ds 2 = ds' 2 .The unprimed (primed) coordinates are defined in K (K').We consider the standard configuration and the motion of K' is along 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 t2 respectively.So, if x0 is the coordinate of the origin of K' at any time, Let us assume that at times t1 and t2 a particle has space coordinates (x, y, z) in K and (x', y', z') in K' corresponding to times t1' and t2'.
To derive the transformations between K and K' we follow Velev [19].Schröder [16] also used this technique to derive the Lorentz transformations in a more general case when the so-called Lorentz boost is in an arbitrary direction (i.e., not necessarily along x) with x, y, and z axes being parallel and coinciding space-time origins.The complete coordinate transformation will be realized by three  The complete transformation between K and K' is obtained by Λ = R -1 LR such that X' = Λ X. X is a column matrix with elements [xi] , i=1,2,3,4,5 and X' is a column matrix with elements [x'i] , i=1,2,3,4,5.
Let us express all trigonometric functions in terms of f1, f2, β and  , and restore coordinate variables (t1, t2, x, y, z) in K and (t1', t2', x' , y' , z' ) in K'.Then, we obtain the final transformation matrix as 11 2 2 00 00 00 0 0 0 1 0 0 0 0 0 1 Finally, the coordinate transformations in our five-dimensional Lorentz-like transformations are, (12) That the above transformation we just derived is correct can be demonstrated by showing that We used Maplesoft TM [21] to do the algebraic computation and the outputs are given in Appendix A. It is worth deriving simplified expressions for the coordinate transformations for comparing with the familiar Lorentz transformation formulas in 1T + 3S dimension.From eq. ( 2) we also have . .
, where . If t1 has a dimension of T and t2 has a dimension of t, then k has a dimension of T.t -1 .We will see how k would help us check dimensional consistencies throughout the text of this paper.The expressions in eq. ( 12) simplify a lot if we use the relations,

w v k t k t ==
and we get (see Appendix B), Results of eq. ( 13) also satisfy (see Appendix C), So, we checked that . Thus, k is invariant under the transformation of eq. ( 13).
One can easily derive the length contraction and time dilation formulas. ( Here, , , ,( ) , ,and( ) l l t t t t     have familiar meanings requiring no further explanations as the subject is discussed in detail in all textbooks of TSR [16].Compared to the familiar four-dimensional ; ; Lorentz transformations, the only difference is that c is replaced by ce.If we set 21 = 0 and = c c c , we get the familiar Einstein's TSR formulas.
Just as in traditional Lorentz transformation in the Minkowski space, it is interesting to determine how the velocities transform from the reference frames K to K'.Since we have two different time-like dimensions denoted by variables t1 and t2, there are two velocities for each space dimension x, y, and z.We define them as Vx, Vy, Vz and Wx, Wy, Wz for velocities in reference frame K with respect to t1 and t2 respectively.Similar quantities are defined by primed notations for the reference frame K'.
For reference frame K, For reference frame K' , ; , ; , Using Similar relations exist for the primed velocities defined in K' reference frame.
We use the results from eq. ( 12) and calculate the formulas for velocity transformations.
From eq. ( 12), we get, We get, Restoring expressions for G, H, I, A, B, and C and simplifying further, eq. ( 22) reduces to, Same way, we get transformations of other velocity components as ( 1)( ) ( 1)( ) These expressions agree with those in Velev [19] if one uses c1 = c2 = c.These expressions can be simplified just like the coordinate transformations of eq. ( 13).Alternatively, as is done in the standard text on TSR (see for example [16]), we can take derivatives on both sides of the simplified transformation equations ( 13) and derive the velocity transformation equations easily.(29) We can solve for Vx and Wx and get, (1 / ) and ( 1) Again, it can be checked that by setting c2 = 0 and c1= c, we get the familiar Einstein's TSR formulas.

IV. Proper time, Five velocity, Five momenta, and Relativistic Energy -Momentum Relationship
Define five space-time variables (in 2T + 3S dimensions) And the metric gµν is (+1, +1, -1, -1, -1) such that the displacement s 2 = gµν x µ x ν is given by, ( ) We can define two proper times Explicitly, one type of quantities are, ; Just like two types of five-velocities, we have two types of energy-momentum five vectors. (37) And for the other type, we have, In addition, we observe the following relations,

Also using
we can derive .

Also, and
We have another relation, .
Using the above relations, it is easy to p ( ) can be defined as energy of type 2 corresponding to time component 2

New Expression for Rest-mass Energy and Derivation of the Non-Relativistic Limit
From the previous section, we have (using various relations) Using various identities given above, we can simplify expressions in eq. ( 33) and eq.( 34) as we did in the case of coordinate and velocity transformations and get, where c is the speed of light and m is the four-dimensional mass.
When or , we get or / respectively.
In addition, we have the the approximation, The factor is introduced for the sake of dimensional


, we have the non-relativistic limit for a free particle,


, we have the non-relativistic limit for a free particle, Similar expressions can be obtained using ( ) and( ) EE .However, for the sake of illustration, we focus on Case 1 and derive two time-dependent Schrödinger-like equations (i.e., the one-dimensional infinite square-well potential problem in the 2T+1S dimension).Eventually, the extra time dimension will be compactified.We start with the non-relativistic limit eq. ( 36), px is the non-relativistic linear momentum 0 x mv .The first term in the square bracket is the rest-mass energy that we drop from now on and the second term is the kinetic energy.Adding a potential energy term V(x) we get, Next, by replacing the classical energy and momentum functions with corresponding quantum operators we get the time-dependent Schrödinger-like equation.( ) , ( ) ,

VI. Solving the Infinite Square-Well Potential Problem in 2T + 1S dimension with Compactification of the Extra Time Dimension and Analysis of Results
We consider one-dimensional infinite square-well potential, Such an example is well discussed in the literature as a time-independent Schrödinger equation in 1T + 2S dimension where the extra space dimension is compactified as in Kaluza-Klein theory [4,5] on a circle of radius R (for example, see Zwiebach [6]).The energy eigenvalues are determined by two quantum numbers q and l.
The term involving q defines the familiar quantum effect in an infinite square-well problem, whereas the term involving l is due to the compactification of the extra space dimension.We will not discuss this result further as it is available in the literature.We focus on our example with an extra time dimension that will be compactified on a circle of period T0.The solution will be obtained by following the standard separation of variable techniques.
The time-dependent equation becomes, This implies that both sides will be equal to a constant (this choice leads to simple equations).
We further apply the separation of variable technique to the eq.(57).Let 12 ( , ) X t t = 1 1 2 2 ( ) ( ) T t T t and derive two equations where, .
We assume that time t2 is compactified in the sense that, and T0 is very small, say like Planck time.This gives, For the sake of illustration, we take the positive values.Now, let us consider the x-dimensional equation.
We introduce the variable, Then, the solution that satisfies boundary conditions at x = 0 and x = a is well-known and available in any introductory quantum mechanics textbook (see for example [23]), ( ) sin( ); , 1, 2,...
Energy levels are we get, Or, The complete solution is a linear combination of the products.
n p are arbitrary constants.For the sake of analysis and to highlight the results, we consider a couple of cases as follows.
Case 1: Let p =1 and n = 1, 2 and consider the linear combination of two terms only. . .
All constants are lumped into C(1,1) and C(1,2) and we assume them to be real for the sake of simplicity.The sum of the two solutions gives, (1,1)sin .
This gives the time evolution with respect to time t1.Dependence on t2 did not appear due to the particular choice of the quantum numbers.
All constants are lumped into D1 and D2 and considering them to be real for the sake of simplicity, the sum of the solutions gives, So, the frequency f of oscillation is .We assumed that t2 was compactified such that t2 = t2 + T0.We remember that kT0 has the same dimension as t1 and we can take its value to be equal to Planck time Tp whose value is 5.39 x 10 -44 sec.Thus, / 1 We can plot / p PT against λ and see how it varies as λ (see Figure 1).When λ = 0 (i.e., c2 = 0), we get zero value for a period as it reduces to four-dimensional spacetime (i.e., 1T + 3S) making compactification of the extra time dimension redundant.The smallest time measured so far is about 2.5 x 10 -19 seconds [24].We need to point out that this analysis is based on a specific example that was chosen to demonstrate the overlapping effect of the non-relativistic compactified quantum states and on the compactification method.

Comments and Discussion
We are considering relativistic transformations in a 2T + 3S space with the square of interval given by, If we assume that the right-hand sides of eq. ( 83) and eq.( 84) are equivalent quantities, then we get,  We plotted the mass ratio against λ and see the interesting behavior (Figure 2).In the range 01   , the ratio m0/m is less than 1 and is greater than 1 beyond.Does it imply that even the rest mass of an object depends on the finer details of the dimensional structure of space and time?
On the other hand, if we assume that the four-dimensional and five-dimensional masses are the same, then the rest energies denoted by (Pt) 4 rest and (Pt) The plot of eq. ( 88) will be just as Figure 2 except that the vertical axis will imply The static compactification at the coordinate level has a weakness as it will prevent further expansion of physical theories into the dynamic space (i.e., classical and quantum field theory).The 2T + ö1S dimensional square-well Schrödinger-like equation is a good example.Because of the second time dimension in the formulation of the wave equation, we could compactify the extra time dimension on a circular topology and derive additional secondary quantum energy levels.If the extra time variable is not available, we cannot develop field theoretical formulations as it will not be possible to couple with additional fifth tie components or their corresponding derivatives (e.g., five-dimensional Klein-Gordon field theory or equivalent Abelian gauge theories).

VIII. Conclusions and Future Research
This is an expanded version of the paper [26] where we considered the space-time structure with two time and three space dimensions.Other researchers also investigated the possibility of two-time physics from different considerations.Our conceptual framework for the scientific considerations of this paper refers to the early stage of the universe when all particles of the standard model were massless.
We postulated that the present era (i.e., current energy scale) 1T + 3S dimensional special theory of relativity (TSR) can be extended to a 2T+3S dimensional TSR applicable to a scenario where each time dimension was tied to a distinctly separate speed of causality c1 and c2.That scenario contrasts the present era one with one speed of causality c, the speed of light.In our conceptual framework speeds c1 and c2 are tied to some fundamental interactions mediated by some massless particles that carry information needed for implementing the "cause-and-effect" phenomenon.
We derived formulations for coordinate transformation, velocity transformation, and energymomentum relations in the five-dimensional (i.e., 2T + 3S) spacetime and showed (see Appendices) the usefulness of MappleSoft TM .We found that the maximum speed possible is ( ) 12 / e c c c k =+ (k is defined in the text) which is greater than either c1 or c2 without violating the "new TSR".The term e c is the speed of any particle having non-zero energy but a zero rest mass.The rest energy of any particle is given by eq. ( 43).This can be compared with the four-dimensional TSR expression eq. ( 44).Starting from the five-dimensional energy-momentum relation, we derived the "non-relativistic" limit from which the time-dependent Schrödinger-like equation (TDE) of the non-relativistic quantum mechanics is derived for a 2T + 1S dimension.Our consideration of an extra time dimension was conceptualized at the very early stage of the creation of the universe.We anticipate this to be after the inflationary expansion and during the "reheating" phase when all the standard particles were created, albeit in a mass-less form.However, as in the current energy range, we only see a 1T + 3S space-time configuration the four-dimensional STR works well.Thus, the extra time dimension is expected to be compactified.In the context of the Kaluza-Klein theory, compactification is done on an ultrasmall circular topology (either spatial or temporal dimension).However, we have not explored if space-time can be compactified (or expanded) in other, perhaps more dynamic, ways [27].For example, cosmologists mention "metric expansion" to explain the inflationary phase by conceptualizing the existence of a scalar field called inflaton [28].But how that works mathematically or in theoretical formulations is not known.The Discovery of the Higgs particle confirms theoretical developments including symmetry-breaking in electroweak unification.But we do not know how the Higgs field can play any role in compactifying the extra time dimension.Conceptually, if inflaton can be responsible for metric expansion during the inflationary phase, making the Higgs field responsible for compactifying the extra time dimension may not be too far-fetched in imagination [29].With t1 = k t2, the first term in the numerator drops.We get, successive rotations denoted by five-dimensional rotation matrices R, L, and R -1 .First, R represents a proper rotation in the x1-x2 plane through angle α giving new axes x1R and x2R keeping the other three dimensions (x3, x4, x5) unchanged.This transformation is described by the matrix R,

Figure 1 :
Figure 1: Period of Oscillation in Units of Planck Time vs λ dimensionless quantity.When both masses are four-dimensional and are equal.

Figure 2 :
Figure 2: Ratio of Five-dimensional Mass to Four-dimensional Mass vs. λ of cosmological phenomena, how can we explain the variation of the ratio of the fivedimensional mass to four-dimensional mass or λ and compactification in the context of (rest mass) energy release or absorption spontaneously during the early evolution of the universe?
Let us manually simplify the Maplesoft TM output further.Before we can do that, we write down the mapping of Maplesoft variables with respect to the actual variables used in the text.T = t1 , TT = t2 , X = x , nTP = t'1 , nTTP = t'2 , nXP = x'We use t1 = k t2Transformation of t1With t1 = k t2, the two square root terms in the parenthesis in the numerator, cancel each other and the expression simplifies to This is an interesting result of this paper (see Section VII).The rest energy of the object is given by,

four-dimensional mass.
That's why we got similar results in the previous sections.To be more specific, all the expressions of relativistic transformation formulas reduce to the familiar Lorentz ones as e c in our case.
5rest are different.Here the subscript t is chosen Pt stands for the "time component" of the corresponding four or five momenta.From eq.