The complex-valued Maxwell equations considered here are given by: ∇ ⋅ E = 1 ϵ 0 ρ (2.1a) ∇   × E = − i c μ 0   J − ∂ B ∂ t (2.1b) ∇ ⋅ B = i c   μ 0 ρ (2.1c) ∇   × B = μ 0   J + 1 c 2 ∂ E ∂ t (2.1d)

These equations indicate that both electric and magnetic charge exist, and that they are the same entity (items 1, 2 in Table 1). Here E ( B ) is the complex electric (magnetic) field in C 3 , bold font indicates 3-component column vectors, and SI units are assumed. While these equations superficially appear to be similar to Maxwell’s original equations extended as in Eqs. 1.2, they differ in very substantial ways. The electric and magnetic field vectors E and B that appear here have complex-valued components, E = E x 1 + i   E t 1 E x 2 + i   E t 2 E x 3 + i   E t 3 = E x + i   E t B = B x 1 + i   B t 1 B x 2 + i   B t 2 B x 3 + i   B t 3 = B x + i   B t (2.2) where E x = E x 1 E x 2 E x 3 E t = E t 1 E t 2 E t 3 B x = B x 1 B x 2 B x 3 B t = B t 1 B t 2 B t 3 (2.3) are vectors in 3D real-valued space R 3 . These fields are assumed to be functions of location in a spacetime whose points are represented by s = x + i c   t , where x and t are both 3D real-valued vectors and the latter is associated with time. However, the real valued variable t that appears in ∂ t in Eqs. 2.1 remains the familiar clock time–its relation to the vector t is described in Section 3. Vectors E _x and B _x are the usual electric and magnetic fields as they currently appear in Maxwell’s original equations. In contrast, E _t and B _t in Eq. 2.2, both lying in R 3 , indicate that electromagnetic fields have imaginary-valued portions i E _t and i B _t of their components that are unobservable and that are interpreted as extending into time.

It is helpful to introduce some terminology and concepts that facilitate visualization of these fields. Their real field portions E _x and B _x are said to lie in real-valued space, or r-space, that corresponds to familiar and observable 3D space used by the classical Maxwell equations. In contrast, E _t and B _t are taken to exist in a separate temporal space or t-space that is tightly linked to the notion of clock time t. To facilitate visualizing these fields, it helps to think of the complex-valued fields E and B in two different but equivalent ways. First, we can view each of their three field components as lying in the complex (Argand) plane, as shown on the left in Figure 1. Alternatively, we can think of the real and imaginary portions of fields E and B as lying in two 3D spaces, as illustrated on the right in Figure 1. The latter viewpoint is adopted here–it makes the observable vs. unobservable distinction between real-valued spatial and imaginary-valued temporal components explicit. The classical Maxwell equations based on E _x and B _x assume the familiar 3D r-space that is observable, while the imaginary portions E _t and B _t of the complex fields E and B lie in unobservable t-space, which is called “temporal” to emphasize its relationship to familiar clock time t. The formulation of electromagnetism considered here only hypothesizes that electromagnetic fields extend into t-space; it does not assume a priori that matter, charge or any other physical entities extend into t-space.

As with Eqs. 1.2b, 1.2c which include hypothetical magnetic monopoles, Eqs. 2.1b, 2.1c include new terms on their right sides that imply the existence of magnetic charge and current, increasing the underlying symmetry. However, unlike Eqs. 1.2, these terms are purely imaginary, thus explicitly implying the existence of imaginary components in the fields E and B . Further, these terms differ in using ρ and J rather than ρ m and J m as in Eqs. 1.2, and therefore they do not imply the existence of a novel kind of magnetically charged particle.

Eqs. 2.1 also differ from the classic Maxwell’s equations in that the divergence ∇ ⋅ and curl ∇ × operations generalized to complex fields are not the typical operators that one might expect. These non-standard reduction vector operators provide convenient abbreviations whereby vector product operations in a 3D complex space   C 3 are “reduced” to a linear sum of the standard corresponding R 3 operations in r-space and t-space. If C = C x + i   C t and C ′ = C x ′ + i   C t ′ are two arbitrary vectors in   C 3 where C x ,   C t , C x ′ and   C t ′ all lie in R 3 , the reduction dot product ⋅ and cross product × of C and C ′ in   C 3 are defined to be C ⋅   C ′ = C x ⋅ C x ′ + i   C t ⋅   C t ′ (2.4a) C × C ′ = C x × C x ′ + i   C t × C t ′ (2.4b) where the vector products on the right side of these equations are the usual ones in R 3 . The vector product being defined on the left side of each of these equations acts on vectors in   C 3 and, in general, returns a complex number. The complex-valued dot product defined in Eq. 2.4a does not qualify as an inner product, while the complex-valued cross product in Eq. 2.4b avoids the well-known challenges that occur in generalizing the cross product to spaces other than R 3 [9,10]. With this notation, the reduction differential operator ∇ in C 3 is defined as ∇ = ∇ x + i   1 c ∇ t , where ∇ x = ∂ ∂ x 1 ∂ ∂ x 2 ∂ ∂ x 3 ∇ t = ∂ ∂ t 1 ∂ ∂ t 2 ∂ ∂ t 3 (2.5)

The factor 1 c in ∇ occurs because points s = x + i c   t in the underlying spacetime use c to scale t-space dimensions into units of meters, so the t-space components of ∇ are in effect given by ∂ ∂ c t j = 1 c ∂ ∂ t j .

Let C = C x + i   C t be a continuous differentiable vector field in   C 3 (such as E or B ). Then, following the above, the reduction divergence and curl used in Eqs. 2.1 are defined to be ∇ ⋅ C = ∇ x ⋅ C x + i   1 c ∇ t ⋅ C t ∇ × C = ∇ x × C x + i   1 c ∇ t × C t (2.6a,b) and similarly, the reduction gradient and Laplacian are defined to be ∇ T = ∇ x T x + i   1 c   ∇ t T t ∇ 2 C = ∇ ⋅ ∇ C = ∇ x 2 C x + i   1   c 2   ∇ t 2 C t (2.6cd) where T = T x + i   T t is a continuous differentiable scalar field in C 3 . In the absence of imaginary components, these operations simplify to their usual definitions in 3D real space. It is straightforward but tedious to show that many of the usual relations for ∇ in R 3 , such as ∇ ⋅ ∇ × C = 0 ∇ × ∇ T = 0 ∇ × ∇ × C = ∇ ∇ ⋅ C − ∇ 2 C (2.7abc) continue to hold for ∇ in C 3 as defined above, as can be confirmed by using straightforward algebraic manipulations and well-known identities in R 3 .

The complex-valued Maxwell equations 2.1 introduced here within classical electrodynamics differ very substantially from those described in past work. For instance, previous work based on analogies between Dirac’s equation for the electron and Maxwell’s equations differs in that it uses complex fields E ± i c B and vectors E   i c B   having six components, where E and B are the usual 3D real-valued fields [11,12]. Other recent past work has proposed complex forms of Maxwell’s equations where, unlike here, magnetic charge is often incorporated as ρ = ρ e + i   ρ m and J = J _e + i J _m , with ρ m and J _m being different entities than ρ e and J _e and these are sometimes related to previously proposed magnetic monopoles such as Dirac’s [13–16]. The work done here also differs in a major way from that in [4] where the imaginary field components were interpreted as existing in space rather than in time as is done here. This temporal interpretation is a much more challenging prospect: it involves aligning measurable clock time t with events in t-space, assessing the implications of special relativity, and interpreting the properties of electromagnetic waves propagating through time as well as space. None of this past work has considered the central novel concept proposed here - that electromagnetic fields have components extending into time.

The complex-valued Maxwell’s equations 2.1 exhibit a number of basic properties, including the implication that, unlike the fields, both ρ and J do not have intrinsic imaginary components. If they did, that would make Eqs. 2.1 inconsistent with experimental data. For example, if ρ is replaced by ρ = ρ x + i   ρ t in Eq. 2.1c, this would imply that ∇ x ⋅ B x = − c μ 0 ρ t , which is inconsistent with known observations that ∇ x ⋅ B x = 0 always, so it must be that i ρ t is always zero. Further, the complex valued Eqs. 2.1 continue to imply a continuity equation ∇ ⋅ J = − ∂ ρ ∂ t , as can readily be demonstrated by taking the divergence ∇ ⋅ of both sides of Eq. 2.1d and using relation Eq. 2.7a.

An intriguing implication of allowing electromagnetic fields to have imaginary components is that it permits the existence of magnetic charge while simultaneously explaining why magnetic monopoles have not been detected in numerous past experiments that have searched for them [17]. For theoretical reasons, many physicists continue to believe that magnetic monopoles probably exist in spite of the negative experimental evidence. As a result, a rich variety of possible types of magnetic monopoles have been proposed over the years, including Dirac’s string model [18], ‘t Hooft-Polyakov monopoles [19,20], the Wu-Yang fiber bundle model [21], two-photon models [22,23], and others [24,25]. Eqs. 2.1 represent a novel solution to this issue by implying the existence of a new type of magnetic monopoles. Eq. 2.1c entails that ∇ t ⋅ B t = 1 ϵ 0 ρ and thus indicates that charge serves as a source/sink for radially directed magnetic fields B _t that lie in imaginary-valued t-space. In other words, charges act as both electric and magnetic monopoles, serving as sources (positive electric charges as N poles) and sinks (negative charges as S poles) for unobservable imaginary-valued magnetic fields in t-space. Thus, in the theory developed here, we know that particles carrying a magnetic charge are the same as those carrying an electric charge, that the predicted magnetic monopoles are widespread (every electron, proton, etc.), their existence is consistent with past experimental negative search results, they come in two types, they are stable particles having relatively low mass, they are not dyons [7], they are conserved, etc.

Central to the issue of symmetry, the complex-valued Maxwell Eqs. 2.1, like Eqs. 1.2, clearly exhibit increased symmetry when compared to the original Maxwell equations, the difference relative to Eqs. 1.2 being that the additional terms in Eqs. 2.1b, 2.1c are imaginary rather than real valued and thus do not contradict existing experimental data. This increase in symmetry is manifest as an electromagnetic duality transformation involving the electric and magnetic fields of the complex-valued Maxwell equations given by   E ′ = c B , B ′ = − 1 c E ,   ρ ′ = i ρ ,   J ′ = i   J (2.8) under which Eqs. 2.1 are invariant. As an example, when this duality transformation is applied to Eq. 2.1a, it gives ∇ ⋅ E ′ = c ∇ ⋅ B = i c 2 μ 0 ρ = i c 2 μ 0 − i ρ ′ = 1 ϵ 0 ρ ′ (2.9) where c 2 = ϵ 0 μ 0 − 1 is used on the last step. Analogous results occur when this transformation is applied to the remaining Eqs. 2.1.

Further, since Eqs. 2.1 involve complex-valued fields, they each represent two sets of equations, one set in r-space (real-valued, observable) and the other set in t-space (imaginary-valued, unobservable). For instance, writing out Eq. 2.1c gives ∇ ⋅ B = ∇ x + i   1 c ∇ t ⋅ B x + i   B t   = ∇ x ⋅ B x + i   1 c ∇ t ⋅ B t = i c   μ 0 ρ . (2.10) and equating the real and imaginary parts of this gives two equations ∇ x ⋅ B x = 0 ∇ t ⋅ B t = 1 ϵ o ρ (2.11a,b) where (omitting the implicit i on both sides of Eq. 2.11b) each is in R 3 , the first involving r-space, the second involving t-space. Applying this procedure to all of the complex-valued Maxwell equations results in four r-space electrodynamics equations given by ∇ x ⋅ E x = 1 ϵ 0 ρ ∇ x × E x = − ∂ B x ∂ t (2.12a,b) ∇ x ⋅ B x = 0 ∇ x × B x =   μ 0   J + 1 c 2 ∂ E x ∂ t (2.12c,d) and four t-space electrodynamics equations given by ∇ t ⋅ E t = 0 ∇ t × E t = − 1 ϵ o   J − c ∂ B t ∂ t (2.13a,b) ∇ t ⋅ B t = 1 ϵ o ρ ∇ t × B t = 1 c ∂ E t ∂ t (2.13c,d)

The first set of these equations (2.12) are the original asymmetric Maxwell equations in r-space where the symbols E _x and B _x represent the familiar electromagnetic fields. These equations show that the complex-valued Maxwell equations truly generalize the originals, and thus that they are consistent with the known experimental results of classical electrodynamics in physically observable r-space. They also do not imply new observable phenomena in r-space that are experimentally absent. The second set of these equations (2.13) describe the imaginary-valued unobservable fields E _t and B _t in t-space. Comparing Eqs. 2.12 to 2.13, it becomes clear that these equations are symmetric with respect to each other if one interchanges the roles of the electric and magnetic fields. This symmetry is manifest by a cross-domain duality transformation given by x ⇒ c   t E x ⇒ c B t B x ⇒ − 1 c E t (2.14a,b,c) that maps the set of equations Eqs. 2.12 into Eqs. 2.13, and vice versa via the inverse transformation. The first rule x ⇒ c   t of this cross-domain transformation implies that ∇ x   ⇒ 1 c ∇ t because this rule indicates that the individual components ∂ ∂ x j of ∇ x transform as ∂ ∂ x j ⇒ ∂ ∂ c t j ⇒ 1 c ∂ ∂ t j . For instance, applied to Eq. 2.12a the cross-domain transformation gives ∇ x ⋅ E x = 1 ϵ 0 ρ   ⇒   1 c ∇ t ⋅ c B t = 1 ϵ 0 ρ   ⇒   ∇ t ⋅ B t = 1 ϵ 0 ρ

The resulting equation is Eq. 2.13c, and similar applications of these transformation rules to the remaining Eqs. 2.12 give the remaining Eqs. 2.13.

How should one interpret the imaginary components of fields E and B that extend into t-space? One possibility is to consider space to be three dimensional, with each dimension being complex-valued (six real-valued dimensions), and time to be an additional separate single dimension. This is what was done in the previous analysis [4], and it represents an approach where time remains formulated in a way that is consistent with most of the existing classical electrodynamics literature. However, such an approach implies a spacetime with a total of seven real-valued dimensions, and a marked asymmetry in the nature of space (three complex dimensions) and time (single real-valued dimension). An alternative possibility, the one considered here, is that t-space is intimately related to time rather than to space. Clock time t is taken as derived from movement through a 3D t-space, motivated by the 3D nature of the imaginary components of the electromagnetic fields which we now accordingly interpret as extending into time. The observable clock time t that we measure is no longer taken to be a dimension of the underlying spacetime because t is derivative: it corresponds to the extent of one’s movement along a trajectory through an unobservable underlying 3D t-space. From this perspective, the imaginary components of the complex electromagnetic fields are viewed as extending into time rather than space, and spacetime has six real-valued dimensions.

The apparently radical notion that time could in some sense be three dimensional initially sounds implausible to some. One dimensional clock time is the basis of the existing laws of classical physics, it is measurable, and it is compatible with our subjective experience of the passage of time (although subjective passage of time has significant differences from passage of clock time [26]). Classical electromagnetism, for example, is founded on the 4D spacetime of special relativity (4-vectors in Minkowski spacetime) having three spatial and one time dimension. However, there have been numerous past proposals in the literature arguing that time may be multi-dimensional based on a remarkably broad range of differing motivations. For example, viewing time as multi-dimensional has been proposed to have advantages in investigating superluminal Lorentz transformations [27], special relativity [28], unification of quantum mechanics and gravity [29], electromagnetism [30], electroweak interactions [31], development of two-time physics [32,33], cosmological modeling [34], Dirac’s quantization condition [35,36], and quantum gravity [37]. Importantly, the one dimensional time that we measure with clocks and experience subjectively does not preclude the possibility that this measure is based on an underlying 3D “temporal space” that is not directly observable.

As described in the Introduction, the current theoretical analysis explores the implications of maximizing the symmetry of Maxwell’s equations without contradicting known experimental findings. From this perspective there are two factors related to symmetry that motivate considering the possibility that time is three dimensional, and that t-space represents time rather than being a part of space. First, assuming that time has three dimensions like space increases symmetry by placing time dimensionality on an equal footing with that of space (Table 1, item 3). It also simplifies the representation of complex spacetime in that, rather than spacetime having three complex spatial dimensions and one time dimension (overall seven real-valued numbers), it simply has three complex spacetime dimensions (6 real-valued numbers). Thus, the dimensionality of spacetime becomes both more symmetric and simpler in the complex-valued Maxwell equations than it would be if one takes t-space to be an aspect of space rather than time as was done in [4]. Second, taking t-space to be the underlying basis of time increases the symmetry of electromagnetic fields in the sense that they extend not only into space but also into time (Table 1, item 4). If one accepts the view of special relativity that space and time truly form an integrated spacetime, it is both puzzling and asymmetric that, as currently conceived, electromagnetic fields only extend into space and not into time.

In considering the possibility that t-space represents the fundamental underlying source of our experience of time, and to lay the groundwork for possible experimental testing of the temporal fields hypothesis (Section 5), we next consider the underlying spacetime implicit in Eqs. 2.1 and characterize how clock time t relates to 3D t-space. As described above, spacetime structure overall is represented here as a 3D complex valued space C 3 , with the real-valued r-space representing familiar 3D physical space, and the imaginary-valued t-space representing a 3D temporal space, the latter being inaccessible to us via direct experiment. Specifically, the occurrence of an event at a location s in   C 3 spacetime is given via cartesian components as s ≡ x 1 + i c t 1 x 2 + i c t 2 x 3 + i c t 3 = x + i c t (3.1) where x = x 1 x 2 x 3 and t = t 1 t 2 t 3 are both vectors in R 3 . Each time component t _j is measured in seconds, so c t _j in Eq. 3.1 is measured in meters, just like x _j . The imaginary portion ic t of s lies in the separate imaginary-valued t-space where the individual quantities t ₁ , t ₂ , and t ₃ are not directly observable. This unobservability of individual t _j values relates to the differences between physical space and time. For example, we can move in any direction in space but are confined to move only “forward” in time, and we can directly perceive events in any direction in space but cannot directly observe events in the future or past. These differences between space and time are widely recognized in physics, psychology, and philosophy, as are that time is poorly understood, that subjective and objective passage of time differ, and that time differs from space [38–42].

In interpreting Eqs. 2.1 and Eq. 3.1 in what follows, it is very important for one to clearly distinguish an entity’s 3D position vector t in t-space from our familiar measurable notion of 1D clock time t. We continue to interpret measured clock time duration d t in the usual way, and thus ∂ t in the complex Maxwell Eqs. 2.1 has the same meaning as in the original Maxwell equations. However, it remains to make explicit how a 1D measurable time duration d t relates to unobservable t-space. In the theory presented here, the measured passage of clock time in an inertial reference frame S is assumed to be linearly proportional to the extent (“distance”) that an entity moves along a 1D trajectory in S’s 3D t-space, just as we associate a 1D distance with the extent that an object moves along a trajectory through S’s 3D r-space.

Let d s be the differential spacetime displacement between two arbitrary infinitesimally separated events s and s ^′ occurring in an inertial reference frame S. Specifically, d s = d x 1 + i c   d t 1 d x 2 + i c   d t 2 d x 3 + i c   d t 3 = d x + i c   d t (3.2) where d x 1 = x 1 ′ − x 1 , d t 1 = t 1 ′ − t 1 , etc., and define d x = d x 1 2 + d x 2 2 + d x 3 2 1 2 (3.3a) d t = d t 1 2 + d t 2 2 + d t 3 2 1 2 (3.3b) to be the distances occurring in r-space and t-space between those two events. (To avoid using numerous parentheses, here and throughout the remainder of this paper, differentiation d is given precedence over raising a quantity to a power, so d t 1 2 is an abbreviation for d t 1 2 , etc.) We designate the measured distance d x between the two events in frame S’s observable r-space in the usual fashion to be d x = d x as defined in Eq. 3.3a. For Eq. 3.3b, we analogously adopt the following temporal correspondence between the measured clock time d t separating the two events in frame S’s t-space and the distance separating the two events in t-space: d t = d t (3.4)

The temporal correspondence of Eq 3.4 is an explicit assertion defining how an increment of familiar clock time d t corresponds to the “distance” d t along a trajectory through S’s t-space, just as d x relates to the distance d x along a trajectory through S’s r-space. ¹

We next define the velocity v of an arbitrary object located at s in an inertial frame S to be v = d s d t = d x d t + i c d t d t = v x + i   v t (3.5) where again t is familiar clock time observed in S and v t = c d t d t (3.6) is an apparent temporal velocity measured in m/s. Note that the quantity d t d t here is the ratio of an infinitesimal displacement d t in S’s t-space occurring during an infinitesimal clock time period d t observed in S. In other words, as conceived in the theory presented here, any physical object is taken to be moving along its worldline in spacetime, not just with its conventional velocity v x in r-space, but also with a velocity v t in the t-space of S. Velocity v t will be discussed further and a second type of temporal velocity designated v τ will be defined in Section 3.4 after first considering a restricted Lorentz transformation.

It is fairly straightforward to extend the standard 4 × 4 Lorentz transformation matrix to a 6D spacetime which, unlike here, incorporates a 3D real-valued time. For example, [28] gives a 6 × 6 transformation matrix Λ = Q R R Q for real-valued 6-vectors x 1 , x 2 , x 3 , c t 1 , c t 2 , c t 3 T where Q = γ 0 0 0 1 0 0 0 1 and R = − γ v x / c 0 0 0 0 0 0 0 0 , assuming appropriate alignment of reference frame axes, expressed in the notation used here. We now specify an analogous transformation L for the   C 3 spacetime used here based on a single 3 × 3 complex-valued matrix L.

To express a restricted Lorentz transformation, consider the perspective of an observer at rest in inertial frame S as an object (clock) at rest in another inertial frame S ∼ moves with constant velocity v = v x + i   v t as measured in S. Let s = x + i c   t be the coordinates of an event observed in S and let s ∼ = x ∼ + i c   t ∼ be the corresponding coordinates of the same event as observed in S ∼ . As is commonly done, orient the r-space axes to be in parallel and let S ∼ move along a shared x 1 x ∼ 1 axis with speed v x , letting t = t ∼ = 0 when the origins are co-located in r-space. Orient the t-space axes analogously so that S ∼ moves along a shared t 1 t ∼ 1 axis with speed v t , letting x = x ∼ = 0 when the origins are co-located in t-space. With this selection of axes, a restricted Lorentz transformation (or boost) between coordinate systems can be expressed as L s = L x + i c   L * t (3.7) where the matrix L is given by L = γ − i β γ 0 0 0 1 0 0 0 1 . (3.8)

Here β = v x / c and γ = 1 − β 2 − 1 2 , and the superscript * indicates the complex conjugate. Note that β is defined in terms of v x as usual.

The restricted Lorentz transformation L in Eq. 3.7 is the same as the Lorentz transformation in 4-vector spacetime under these conditions, as follows. Applied to the coordinates s of an event in S, L gives the coordinates s ∼ = L s of that same event in S ∼ as s ∼ = L x + i c   L * t = γ − i β γ 0 0 0 1 0 0 0 1 x 1 x 2 x 3 + i c γ + i β γ 0 0 0 1 0 0 0 1 t 1 t 2 t 3 (3.9) = γ x 1 − i β γ x 1   + i c γ t 1 − β c γ t 1 x 2 + i c t 2 x 3 + i c t 3 = γ x 1 − β c t 1 x 2 x 3 + i c γ t 1 − β c x 1   t 2 t 3 which is equivalent to x ∼ 1 = γ x 1 − β c t 1   t ∼ 1 = γ   t 1 − β c x 1 (3.10) x ∼ 2 = x 2 t ∼ 2 = t 2 x ∼ 3 = x 3 t ∼ 3 = t 3

These latter six equations are identical to the existing 4-vector Lorentz transformation because t ∼ = t ∼ 1 in the context of the axis orientations selected above. Accordingly, the usual implications of the Lorentz transformation (relativity of simultaneity, length contraction, time dilation, etc.) continue to apply within the complex spacetime considered here.

The natural generalization of the standard spacetime interval in the theory presented here is now shown to be invariant under a Lorentz transformation. Let s and s ′ be the coordinates of two infinitesimally separated events in a reference frame S and let d s be as defined earlier (Eq. 3.2). Define the spacetime interval d s 2 associated with s and s ′ to be d s 2 = c 2 d t 1 2 + c 2 d t 2 2 + c 2 d t 3 2 − d x 1 2 − d x 2 2 − d x 3 2 = c 2 d t 2 − d x 2 = c 2 d t 2 − d x 2 (3.11) where again, d x j = x j ′ − x j and d t j = t j ′ − t j , and the last two equalities follow from Eqs. (3.3), (3.4). Then the interval d s ∼ 2 in another inertial frame S ∼ for the Lorentz transformed vectors s ∼ and s ∼ ′ of s and s ′ , respectively, is d s ∼ 2 = c 2   d t ∼ 2 − d x ∼ 2 (3.12) = c 2 γ t 1 ′ − β c x 1 ′ − γ t 1 − β c x 1 2 + c 2 d t 2 2 + c 2 d t 3 2   −   γ x 1 ′ − β c t 1 ′ − γ x 1 − β c t 1 2 − d x 2 2 − d x 3 2 = c 2 d t 1 2 + d t 2 2 + d t 3 2 − d x 1 2 + d x 2 2 + d x 3 2 = c 2   d t 2 − d x 2

It follows that d s ∼ 2 = d s 2 , so d s 2 is invariant under the Lorentz transformation of Eq. 3.7, analogous to the invariance of the interval in standard 4-vector spacetime.

We now consider further the velocity and speed with which any physical entity such as a particle is moving through the t-space of inertial frame S. First, note that while we can observe the individual components of v x , we cannot directly observe the individual components of velocity v t since the latter are based on the unobservable (imaginary-valued) components of t . However, we can determine the apparent speed v t with which an object is moving through t-space. Recalling the definition v t = c d t d t of Eq. 3.6, the temporal correspondence d t = d t of Eq. 3.4 implies that the apparent speed v t with which the particle is moving through t-space is given by v t = v t = c d t d t = c d t d t = c (3.13) regardless of the particle’s speed v x = v x in r-space. This makes sense in that an observer at rest in frame S’s r-space measures the difference in clock times (and hence t-space separation, according to Eq 3.4) of two events in S traversed by the particle using synchronized resting clocks located in S’s r-space at those two events. In other words, from the viewpoint of observers at rest in S the moving particle is going through time at the same rate as an observer. However, Eq. 3.13 fails to capture the rate at which time is actually passing from the viewpoint of the moving particle (time dilation).

Accordingly, we now define a particle’s veridical temporal velocity v τ that differs from its apparent velocity v t , and that facilitates identifying potential experimentally testable predictions of the temporal fields hypothesis (Section 5). Let frame S ∼ be the proper inertial reference frame for a particle moving through frame S’s r-space, so t ∼ (designated henceforth as τ) is the proper clock time measured by the moving particle at rest in S ∼ , and t ∼ (designated as τ) is the particle’s position in S ∼ 's t-space. The veridical temporal velocity for the particle is defined to be v τ = c d τ d t (3.14) where dτ here is a differential displacement in S ∼ 's 3D t-space while dt is a real-valued clock time increment in the frame S. It follows that the veridical speed with which the particle is viewed as moving by an observer in S is given by v τ = v τ = c d τ d t = c d τ d t = c γ (3.15) where the last equality follows because, according to the Lorentz transformation of Eq. 3.7, we have d t = γ d τ , so d τ d t = 1 / γ . The speed v τ tells one the amount of proper time d τ that passes for the moving particle during an amount of clock time d t that passes for a resting observer in frame S. Stated more informally, speed v τ represents how rapidly the moving particle is aging from the viewpoint of an observer at rest in S’s r-space.

Now consider an object such as a clock moving at an arbitrary but fixed velocity v _x through the r-space of inertial reference frame S. Let this moving clock generate two events located at s and s ′ (e.g., the clock leaves a mark in r-space at two different times) that are separated by distance d x and time d t as measured in S. Then the interval between the two events is given by d s 2 = c 2 d t 2 − d x 2 as recorded in S. However, in a reference frame S ∼ in which the clock is at rest, the corresponding interval is given by d s ∼ 2 = c 2 d t ∼ 2 − d x ∼ 2 = c 2 d t ∼ 2 = c 2 d τ 2 since S ∼ is the proper reference frame for the moving clock so d x ∼ 2 = 0 . By the invariance of the interval (Eq. 3.12), these two quantities d s 2 and d s ∼ 2 must be equal, so c 2 d τ 2 = c 2 d t 2 − d x 2 , from which algebraic manipulations give d x d t 2 + c d τ d t   2 = c 2 . (3.16)

Here the first term on the left is v x 2 , the squared speed at which the clock is moving through the r-space of S, and the second term is v τ 2 by Eq. 3.15. Substituting the quantities v x and v τ into Eq. 3.16 gives the following universal speed constraint v x 2 + v τ 2 = c 2 (3.17) for the theory presented here. While we cannot in general directly observe the individual components of v τ , if we know an object’s speed v x through frame S’s r-space, we can easily determine the object’s veridical speed v τ based on Eq. 3.17. There is a well-known analogous result in standard 4-vector special relativity, e.g., [43].

The universal speed constraint indicates that any object is never at rest in the spacetime of an inertial reference frame, consistent with our experience of constantly moving through time even when we are at rest in a reference frame’s r-space. It further implies that there is an upper limit of c on the speed v x that any object can have in r-space, as is well known, and also on the speed v τ that any object can have in t-space, i.e., that 0 ≤ v x ≤ c and 0 ≤ v τ ≤ c . Conceptually, the universal speed constraint Eq. 3.17 can be visualized as caricatured in Figure 2 which plots v x and v τ against each other. According to the universal speed constraint, any object “at rest” in inertial frame S having speed v x = 0 in S’s r-space must have an associated veridical temporal speed v τ = c (Figure 2A). At the other extreme, according to the universal speed constraint, photons traveling through frame S’s r-space with speed v x = c must have an associated veridical temporal speed v τ = 0 (Figure 2B), consistent with relativistic time dilation effects in the limit as v x → c. In the general situation of a particle moving at an intermediate speed v x in S’s r-space with 0 < v x < c, the particle has a speed of v τ = c/γ (Figure 2C), also consistent with time dilation in special relativity.

The first experimental approach involves looking for effects of forces exerted on charged particles by the fields B _t and E _t that exist in imaginary-valued t-space. Much of what we know experimentally about the familiar fields E _x and B _x is based on the effects that they exert on matter in r-space. This suggests that we can test the temporal fields hypothesis by analogously looking for effects in time, i.e., in t-space rather than in r-space, on charged matter resulting from forces due to B _t and E _t. For this, we need to first characterize what those forces would be.

In classical electrodynamics, Maxwell’s equations are complemented by the Lorentz force law F x = q e E x + v x × B x describing the force F x in r-space due to fields E _x and B _x acting on a particle having electrical charge q _e and moving with velocity v _x through r-space. When hypothesized magnetic charge is considered in the literature as in Eq. 1.2, this law is often extended to be F x = q e E x + v x × B x + q m B x − 1 c 2 v x × E x (5.1) where forces due to magnetic charge q _m are included [2,3]. This extension is derived from the classic Lorentz force law based on an electromagnetic duality transformation for Eqs. 1.2. Here we proceed analogously, applying the cross-domain duality transformation Eqs. 2.14 to the classic Lorentz force law, which produces F = F x + i F t = q E x + v x × B x + i c q B t − 1 c 2 v t × E t (5.2)

The quantity v x = d x d t in the traditional Lorentz force law has been mapped by the cross-domain transformation into d c t d t = c d t d t = v t in the rightmost part of Eq. 5.2. Unlike Eq. 5.1, F in Eq. 5.2 involves forces in both r-space and t-space, q replaces both q _e and q _m , and the forces due to fields B _t and E _t are seen to be purely imaginary valued, extending solely through t-space. Thus, a particle with charge q would be expected to accelerate in r-space according to the traditional Lorentz force law, altering v x precisely as we observe, because there is no direct influence on a particle’s movement in r-space due to the B _t and E _t fields. Further, only B _t and E _t would act on a particle’s movement in t-space. For example, a charge q at rest at the origin in r-space produces a field E x = 1 4 π ϵ o q x 2 x ^ in electrostatics underlying Coulomb’s Law, where x ^ is a unit vector in the direction of x . Applying the cross-domain duality transformation Eqs. 2.14 to this indicates that B t =   μ o 4 π c q t 2 t ^ (5.3) is a t-space magnetic analog to Coulomb’s law in electrostatics, where t ^ is a unit vector in the direction of another charge in t-space. Thus, in t-space particles of opposite magnetic charge would attract one another, and those with the same magnetic charge would repel one another. These considerations suggest using experimental tests like the following.

Equations Eqs. 5.2, 5.3 predict that under specific circumstances one would observe time dilation affecting the decay rate of unstable charged particles at rest in r-space. In other words, unlike the time dilation effects predicted by special relativity for charged particles moving at relativistic speeds, we are now considering time dilation affecting unstable particles at rest in the r-space of an observer’s reference frame, something predicted to occur by the complex-valued but not by the classic Maxwell equations. As an example, consider a small and very thin hollow spherical shell at rest in the r-space of an observer’s reference frame, as sketched in Figure 4. At an initial time t _i let this shell have fixed, embedded positively charged particles q ⁺ in it that are unstable and spontaneously decay with a known half-life when they are at rest (e.g., an ionized radioactive isotope). Suppose that at time t _a a much larger amount q ^- of mobile negatively charged stable particles (e.g., electrons) is added to the sphere temporarily, being removed at time t _b . Ignoring transient effects at times t _a and t _b , Eq. 5.3 implies that following time t _b , the dominating negative charge that was present during the period t _a to t _b would result in a strong resultant radial field B _t in t-space pointing towards the shell, as illustrated in Figure 4. By Eq. 5.2, this field would exert a substantial force of c q ⁺ B _t on the positively charged particles remaining on the shell following time t _b that would reduce their speed through t-space, and thus reduce the passage of their proper time. This would be manifest experimentally by an apparently increased half-life with slower decay of the unstable positively charged particles between times t _b and t _f . The control experiment for comparison would be the same procedure except without the temporary addition of charge q ^- to the sphere during time period t _a to t _b .

A second potential experimental approach to evaluating the temporal fields hypothesis involves the prediction that the imaginary components of electromagnetic waves propagate through time (t-space). While such waves are not directly observable according to the theory presented here, under special circumstances they may be detectable indirectly because of the universal speed constraint.

The standard Maxwell’s equations are often re-expressed for use inside of matter by introducing an electric displacement vector D and an auxiliary magnetic vector H that capture the macroscopic effects of polarization and magnetization. Taking a similar approach here for the complex-valued Maxwell equations, this would correspond in a homogeneous linear medium to using complex-valued D = ϵ E and H = 1 μ B where ϵ > ϵ o and μ > μ o are the medium’s permittivity and permeability, respectively. In the absence of free charge and free current, the complex-valued Maxwell equations inside the medium become ∇ ⋅ E = 0 ∇ × E = − ∂ B ∂ t (5.4a,b) ∇ ⋅ B = 0 ∇ × B = ϵ μ ∂ E ∂ t (5.4c,d) from which one derives ∇ 2 E = ϵ μ ∂ 2 E ∂ t 2 and ∇ 2 B = ϵ μ ∂ 2 B ∂ t 2 (5.5a,b) as complex-valued wave equations. These are the same as the vacuum wave equations 4.1 except that 1 c 2 = ϵ o μ o has been replaced by ϵ μ .

Equating the real and imaginary parts of these equations gives two sets of wave equations, ∇ x 2 E x = ϵ μ ∂ 2 E x ∂ t 2 and ∇ x 2 B x = ϵ μ ∂ 2 B x ∂ t 2 (5.6a,b) and ∇ t 2 E t = ϵ μ c 2 ∂ 2 E t ∂ t 2 and ∇ t 2 B t = ϵ μ c 2 ∂ 2 B t ∂ t 2 (5.7a,b) in r-space and imaginary-valued t-space, respectively. It follows from an analysis similar to that of Sect. 4 that wave speed inside the medium is v x = v t = 1 ϵ μ = c n (5.8) where n = ϵ μ ϵ o μ o is the medium’s index of refraction.

Consider the observable portion of a planar electromagnetic wave that is initially traveling through vacuum with speed v x = c in the r-space of an inertial reference frame S. According to the universal speed constraint Eq. 3.17, the photons in that wave are traveling at speed v τ = 0 through S’s t-space and thus are not aging. When this wave is normally incident upon a dielectric material at rest in S that is substantially transparent at the wave’s frequency, the transmitted portion of the wave’s speed through the dielectric decreases to v x = c / n . In this situation the universal speed constraint implies that for the photons comprising the wave, v τ 2 = c 2 − v x 2 = c 2 − 1   n 2 c 2 = c 2 1 − 1   n 2 > 0 (5.9) must hold, and thus these photons in the wave inside the dielectric are aging, unlike with a wave moving through a vacuum, but it is unclear how this could be experimentally verified.

On the other hand, consider a portion of a wave initiated inside of a sphere of dielectric material that is traveling inside the dielectric in the same direction through t-space as the dielectric material. By Eq. 5.8, the photons in this portion of the wave would have a speed of v τ = c / n . It follows from Eq. 3.17 that for these photons, v x 2 = c 2 − v τ 2 = c 2 − 1   n 2   c 2 = c 2 1 − 1   n 2 (5.10) must hold. Thus, unlike in a vacuum t-space, photons comprising this portion of the wave would “spill over” into r-space, and thus they would be potentially detectable as they exit the dielectric. These considerations suggest experimental tests such as the following.

Consider a solid sphere of homogeneous linear dielectric material having a refractive index of n > 1 at rest in reference frame S (Figure 5). Let there be a very short burst of electromagnetic radiation at the center of the sphere at a frequency at which the dielectric is largely transparent. In r-space, the observable part of the wave rapidly weakens due to attenuation and to transmission outside of the block at the boundaries (Figure 5, left side). For example, for light in a typical glass sphere ( n ≈ 1.5 ) surrounded by air/vacuum, only 4% of the wave’s energy remains in the sphere after just the first reflection. The imaginary-valued portion of the wave inside the dielectric sphere that is moving in the same direction in t-space as the block (see Figure 3) does not directly experience losses from transmission outside the block because it does not encounter these r-space boundaries (Figure 5, right side). However, because the photons in this part of the wave have speed v τ = c n < c inside the dielectric, by the universal speed constraint and by Eq. 5.10 they must also have speed v x = c 1 − 1   n 2 1 / 2 > 0 there. Thus, the temporal fields hypothesis predicts that these persistent r-space electromagnetic waves will be transmitted through the block’s boundaries for a substantially longer time period beyond what would be predicted by the classic Maxwell equations. Such r-space waves, although perhaps quite weak, should be detectable by a nearby observer at rest in r-space. To be maximally informative, variations of such an experiment could be done using materials with different n values, waves of different frequencies (e.g., from ELF to visible), etc.