The Planck length and the constancy of the speed of light in five dimensional space parametrized with two time coordinates

In relativity and quantum field theory, the vacuum speed of light is assumed to be constant; the range of validity of general relativity is determined by the Planck length. However, there has been no convincing theory explaining the constancy of the light speed. In this paper, we assume a five dimensional spacetime with three spatial dimensions and two local time coordinates giving us a hint about the constancy of the speed of light. By decomposing the five dimensional spacetime vector into four-dimensional vectors for each time dimension and by minimizing the resulting action, for a certain class of additional time dimensions, we observe the existence of a minimal length scale, which we identify as the Planck scale. We derive an expression for the speed of light as a function of space and time and observe the constancy of the vacuum speed of light in the observable universe.


Introduction
Since Maxwell's theory of classical electrodynamics [1], it has been known that all electromagnetic waves travel with the speed of light. In 1864 and 1881 experiments performed by Michelson and Morley [2,3] gave a hint that electromagnetic waves travel equally fast in all inertial systems. This result was confirmed by many more experiments [4,5,6,7,8] proofing the constancy of the speed of light within the validity of the laboratory setups.
When Albert Einstein derived special relativity [9], he postulated that the speed of light be constant, and he used this assumption as a key ingredient for special relativity. Consequently, he derived that the speed of light is the upper velocity limit, be it for particles or for information. As such, the limitation of velocities ensures causality; vice versa, information sent with velocities above the speed of light could eventually harm causality.
However, up to now, there have not been sufficiently convincing explanations why the speed of light is constant and why it has the value which it has. Loop quantum gravity, for example, dictates that the velocity of a photon is not defined to be constant, but has a value depending on its frequency [10].
Indeed, there have been suggestions that its value might vary with the age of the universe and that it might not have been constant in the early stages of the universe. Albrecht and Magueijo [11] show that the cosmological evolution equations together with a variable speed of light might solve the horizon, flatness and cosmological constant problem and together with cosmological perturbations the homogeneity and isotropy problem. Deriglazov and Ramírez [12,13,14] observed a discrepancy between the speed of light and the critical speed in the theories of spinning particles on curved and electromagnetic backgrounds. Additionally they noticed that the constancy of the speed of light is closely related with the self-consistent definition of the three-acceleration in general relativity.
The necessity of deriving a theory for quantum gravity resides from the problem that general relativity loses its validity at small length scales [15,16] and that, for example, quantum electrodynamics stops being a self-consistent theory if gravitational effects [17] are added to the theory. The length scale of these effects is in the order of approximately 10 −35 m and was already introduced by Planck in 1899 [18] after the discovery of the Planck's constant when he realized that he could derive a unit system depending on the gravitational constant, speed of light, Planck's constant, Boltzmann's constant and Coulomb's constant only.
Recently Faizal [19] and Pramanik et al. [20] used the Planck length to investigate the deformed Heisenberg algebra. Based on this algebra Faizal investigated the deformation of the Wheeler-DeWitt Equation and thus showed that the big bang singularity gets obsolete. Additionally, Faizal et al. [21] adopted the idea of a minimum length scale or equivalently of a minimum time scale and showed that this leads to corrections to all quantum mechanical systems by the deformed Heisenberg algebra and thus to a discrete spectrum for time.
One of the attempts to unify general relativity with the quantum description of the microscopic cosmos has led to string theory [22,23,24,25]. One of its main features is the existence of 10 or 26 space dimensions which is larger than the commonly experienced three space dimensions [26]. These extra dimensions are assumed to be compactified and are consequently too small to be observed.
On the contrary, there have not been many approaches to a second or more time dimensions. Tegmark [27] summarized that a universe with a large second time dimension cannot contain observers, like us humans, because of the lack of causality. Hence, as for the additional space dimensions in string theory, extra time dimensions need to be compactified in case of their existence.
Multitemporal spacetime dimensions have for example been discussed by Bars and Kounnas [28,29]; they consider two time dimensions and construct actions for interacting p-branes within two dimensions. They show that after a phase transition the additional time dimension becomes part of the compactified universe. Additionally, they present a new Kaluza-Klein like dimensional reduction mechanism and propose an action for a string in two time dimensions. Due to new gauge symmetries, they observe that quantum constraints are consistent only in spacetime dimensions with signature (25,2) or (26,2) for a bosonic string or (9,2) or (10,2) in the supersymmetric case.
Chen [30] interprets two extra time dimensions as quantum hidden variables and shows that non-local properties of quantum physics or that the de Broglie wave length are natural consequences of the existence of two additional time dimensions.
We here now suggest the existence of a compactified second time dimension and subsequently derive the existence of a smallest length scale, i.e. the Planck length, and explain the constancy of the speed of light in our observable universe. Our derivations also suggest that the speed of light varied in the early universe. In section 2 we derive the Lagrangian for a five dimensional space time parametrized with two local coordinates of the surface. On the basis of this Lagrangian we calculate the equations of motion in section 3 and derive the existence of the Planck length as well as of the constancy of the speed of light in the observable universe in sections 4 and 5. We finally conclude in section 6 2 The action in five dimensional space with two local time coordinates In order to study the effect of five dimensional space parametrized by two time coordinates on the speed of light and on the Planck length, we choose a (2, 3) spacetime vector extended by an additional timelike coordinate r · f (γτ /Λ). τ is the second time parameter, r ∈ R describes the size of the second time dimension and γ is the characteristic velocity, thus the equivalent of c. f describes the shape of the second time dimension and Λ ∈ R is a normalization parameter such that γτ Λ is dimensionless. As stated in [30], the additional time dimension has to be small compared to the first time dimension constraining r.
If we define with η ∈ (0, 1), we can decompose x µ into x µ = x µ t + x µ τ . Inspired by the Nambu Goto action [31,32] which is a two dimensional integral over time and the surface of a string, we define the action as As for standard electrodynamics, we make the ansatz for S where ds t and ds τ describe the infinitesimal line elements along t and τ .
It is (applying Einstein's sum convention) where we useċ = dc/dt andẋ = dx/dt. Note that in the limit c = const. and η → 1, (10) becomes ds t = √ c 2 −ẋ 2 dt which is the line element of a free particle in one time dimension with a constant speed of light.

The equation of motions with two time coordinates
As stated in [30], the equation of motion for a Langrangian in two time dimensions is similar to the equations of motion of a string [33], hence d dt where i indexes the i-th space dimension. In the following, we insert (16) into (17) and derive the equations of motion. For each spatial dimension i, it is For simplicity we assume that x ′ is independent of t andẋ is independent of τ . We define and rewrite Eq. (18) as Dividing (21) by F 1 (t)F 2 (τ ) yields Since (23) holds for all t on the left-hand-side and for all τ on the right-handside, both sides have to be equal to a constant Ω i : We now investigate both sides separately. The left-hand-side is thus If we rescale (1 − η) 2 /η 2 Ω i → Ω i , we thus obtain Multiplying (26) Now we can integrate both sides over t and obtain 1 2 where I i,1 is an integration constant. Inserting (19) into (29) gives 1 2 For simplicity, but without loss of generality, we assume a test particle moving in one of the three spatial directions k only, hence Here the second equation is equivalent to x j = −I j,1 /Ω j , i.e. the test particle has a fixed position in dimensions j. The first equation is equivalent tȯ and with another integration constant I k,2 . Similarly, performing the same steps for the right-hand-side of (24), leads to where I k,3 and I k,4 are other integration constants.

The existence of the Planck length
Since the solutions (35) and (36) have to be physical, the arguments of the logarithms must be positive: with the definitions where the upper sign and the usage of z 1 are for the argument of the logarithm in (35) with integration constant I k,1 and the lower sign as well as the usage of z 3 are for the argument of the logarithm in (36) with integration constant I k,3 . We will now prove by contradiction that for the second time dimension, i.e. for the lower sign, it is Let us therefore assume that −8Ω 2 k z 3 + 2Ω k ≥ 0. Squaring which is obviously wrong. Thus −8Ω 2 k z 3 + 2Ω k must be negative; this leads to where we used that a product of two factors is smaller than 0 iff one of the factors is larger than 0 and the other one is smaller than 0. I k,3 is determined by which is the equivalent of (31) for τ . Given the initial conditions x ′ k (τ 0 ) =: x ′ k,0 and x k (t 0 , τ 0 ) =: x k,0 , it is For some functions f and for some parameters {γ, Λ}, i.e. for some particular shapes of the second time dimension, we can assume R/Ω k > 0 and hence, inserting (55) into (52), which implies that for Ω k > 0. Eq. (57) states that there is always a lower limit for any position; thus, the existence of a second time dimension naturally explains the existence of the so-called Planck-length, i.e. a smallest length scale. Hence we identify 1/Ω k being in the order of the Planck scale ℓ P ≈ 10 −35 m, and thus, for the observable universe it is 1/Ω k → 0, or equivalently Ω k → ∞.

The constancy of the speed of light
In this section, we show that the existence of the Planck length ℓ p ∼ 1 Ω k and the subsequent limit Ω k → ∞ implies the constancy of the speed of light for the observable universe.
Inserting x k (t 0 , τ 0 ) = x k,0 into (35) gives Inserting (58) further into (35) and solving for c yields and further, taking the limit Ω → ∞: We now show by induction that for Ω k → ∞ it is c For Ω k → ∞ (33) becomeṡ with derivativeẍ Inserting (62) into (64) then gives Note that the derivative of (61) leads tö which is equivalent toẍ(t) = 2c ∞ · (t − t 0 ) and thus because of (65), and thus from (62) alsȯ 2. Inductive step: We first show by induction that We have already calculatedċ ∞ (61) andc ∞ (66) which are consistent with (69). Now let us assume that (69) is correct. Then it is to easy to see that for odd n and furthermore hence we have proven that (69) is indeed the correct term for the n-th derivative of (59) in the limit Ω k → ∞.
For even n, (69) is equivalent to x (n) (t) = 2c Using (72) and the rule of L'Hôpital in (69), it follows immediately for odd n.
Finally from (64) it follows per induction that As x (n) (t 0 ) = 0 for even n (72) and c Thus we conclude that c and since c (n) (t 0 ) = 0 for all n, it followṡ Thus we conclude that for the observable universe the speed of light is indeed constant.

Conclusions
We have assumed a second time-like dimension with its own characteristic speed and length and the decomposability of any spacetime vector into a vector each for the first and the second time dimension. As a consequence of these assumptions, we have derived two fundamental results: 1. the existence of a smallest length scale which we have identified with the Planck length and 2. the constancy of the vacuum speed of light for the observable universe for particular shapes of the second time dimension. For very small length scales of the present universe, or for the very early universe, we have derived an expression for the speed of light. We see that for both cases, the speed of light is not constant, but depends on space and time.
This is consistent with current results from loop quantum gravity or string theory [34,35] on the non-constancy of light speed.
Finally we here give a hint about the correctness of the assumptions of theories explaining large scale structures of the universe due to a variable speed of light in the early universe [11].