Ohmura’s extended electrodynamics: longitudinal aspects in general relativity

Jiménez and Maroto ((2011) Phys. Rev. D 83, 023514) predicted that free-space, longitudinal electrodynamic waves can propagate in curved space-time, if the Lorenz condition is relaxed. The present work studies this possibility by combining and extending the original theory by Ohmura ((1956) Prog. Theor. Phys. 16, 684) and Woodside’s uniqueness theorem ((2009) Am. J. Phys. 77, 438) to general relativity. Our formulation results in a theory that applies to both the field- (E, B) and potential- (Φ, A) domains. We establish a self-consistent, longitudinal wave-propagation theory for the microscopic longitudinal part of the electric field (EL). We first show that the product of the parameters used previously for the extension of classical electrodynamics can be expressed as a superposition of microscopic displacement modes, which are confined to the energy shell, ∣ω∣ = cq. We then show that nonlinear electrodynamic mixing allows creation of longitudinal waves in the near-field region of a source. A propagator approach gives substantial physical insight into the emission process.


Introduction
Section 3 explains the general-relativistic (GR) EED formulation in 3-vector form [16][17][18][19][20][21]. Jiménez and Maroto (JM) [22] extended the potential theory of general relativity to account for unexplained cosmological phenomena. The JM theory relaxes the Lorenz gauge condition, and consequently predicts a new propagating, longitudinal electric field. Section 4 elucidates the longitudinal (L) field propagation via a so-called microscopic L-displacement field (D L ). Two remarkable properties arise for free-space propagation of the L-displacement field (sections 4. 1, 4.2). First, C λ can be eliminated in favor of D L . Second, C λ is completely determined by a plane-wave, Fourier-integral decomposition (q, ω) of D L , in which only values on the energy shell enter (|ω|=cq). Section 4.3 shows the derivation of a matter-and curvature-driven wave equation for the longitudinal electric field (E L ). No wave propagation occurs for E L in classical electrodynamics (CED), although a non-propagating E L -field exists in the free-space, near-field of a transmitter. E L plays a fundamental role in understanding the photon-localization problem [23].
Section 5 discusses the possibility of L-wave excitation in curved space-time via a propagator formalism (section 5.1) and in the wave-vector-frequency domain (section 5.2). We conclude that CED provides no direct possibilities for L-wave excitation [22]. Near-field electrodynamics provides indirect possibilities, in which case off-energy-shell contributions to the classical current density are needed. Section 6 explains our prediction for launching E L -waves in free, curved, space-time by non-linear optical mixing.
We briefly summarize the results of this paper, as follows. Under special relativity, Gauss' law (∇·E=ρ/ε o ) in microscopic, classical electrodynamics shows that the electric field in charge-free space (ρ=0) is transverse. Thus, only transversely-polarized electromagnetic fields can propagate in vacuum. Extension of Gauss' law to general relativity replaces the usual derivative (∇) by the covariant derivative (∇ COV ≡ {∇ μ }). These operators differ by a so-called gauge term. Gauss' law then has an extra term under general relativity [∇·E=ε o −1 (ρ+ρ CURV )]. The new term (ρ CURV ) is an 'effective' charge density that is associated with the space-time curvature. The electric field now has a longitudinal component (E L ) in charge-free space (ρ=0). We show in sections 3, 4 that the E L -field leads to longitudinal wave propagation when CED is extended to EED. More specifically, relaxation of the Lorenz-gauge condition in the potential formalism allows E L -wave propagation under general relativity.
To include magnetic-monopole electrodynamics in EED, we start from the double-potential formalism [1,4,5]. This approach predicts propagation of a longitudinal magnetic field (B L ). The transverse dynamics (with and without magnetic monopoles) are unaffected by the symmetrized extension of electrodynamics. Consequently, the E L -and B L -waves are not accompanied by magnetic-and electric-field components, respectively. Inclusion of magnetic monopoles in the present form of EED makes the analysis of the canonical particle momentum, angular momentum balance, and photon dynamics quite complicated. Therefore, a detailed EED theory with magnetic monopoles is beyond the scope of the present work.

Special relativistic EED
We begin with the covariant notation that is used. The contravariant 4-potential is {A μ }=(Φ/c, A), for μ=0-3. Bold symbols denote 3-vectors. Here, Φ and A are the usual scalar and vector potentials, respectively. The speed of light in vacuum is c=(ε o μ o ) −1/2 ; ε o and μ o are the vacuum permittivity and permeability, respectively. The contravariant 4-current is {J μ }=(cρ, J), where ρ and J are the microscopic electric charge and current densities, respectively. The covariant 4-derivative is {∂ μ }=(c −1 ∂/∂t, ∇), where t is time. Indices are raised and lowered, using a metric signature of (−, +, +, +). The covariant metric tensor is {g μν } with a determinant of g<0, which is shared by the contravariant metric tensor, {g μν }. Summation over repeated lower and upper indices is implicit throughout this work. The wave operator (d'Alembertian) has the form, ∂ μ ∂ μ =∇ 2 −c −2 ∂ 2 /∂t 2 ≡,, as denoted by the box symbol. SI units are used throughout this paper.

Extended covariant Lagrangian density
The particle component of the Lagrangian density is omitted in the subsequent description, because it is of no importance in this theoretical formulation. The extended Lagrangian density that we use is: The field (F) Lagrangian density with parameter (λ) is: The field-matter interaction Lagrangian density is: For λ=1, F  reduces to the well-known (not extended) covariant field Lagrangian density: An alternative covariant electrodynamic formulation uses the Fermi (F) Lagrangian density [24], instead of : Equations (4) and (5) are equivalent, because they differ only by a 4-divergence. In equation (5), F μν =∂ μ A ν −∂ ν A μ is the μν-th component of the Maxwell field tensor. A form often used in extended electrodynamics is: Equation (6) reduces to the Fermi Lagrangian density for λ=1, making equations (2) and (6) equivalent. The generic form of the Euler-Lagrange equation is: Application of equation (7) to equation (1) for F  from equation (2) yields: However, all covariant gauges have the following form: Here, K is a constant that is independent of space and time. When equation (9) holds, then equation (8) reduces to the classical set (ν=0-3) of covariant wave equations. The well-known Lorenz gauge uses K=0.

Generalized uniqueness theorem
The extension to classical theory is the second (4-gradient) term in equation (8). This extension affects only the wave equations for the scalar potential and the irrotational part of the vector potential. The Helmholtz theorem guarantees the unique decomposition of any 3-vector field (V) into longitudinal (L) [rotation or curl free] and transverse (T) [divergence free] components, V=V L +V T . Consequently, equation (8) yields the following wave equations: The new scalar term (C≡∂ μ A μ ) in equations (11), (12) is: The extension does not affect the magnetic field, B=∇×A T , as is sometimes indicated in the literature. Indeed, equation (10) shows that transverse dynamics satisfies classical electrodynamics (CED). The T-L decomposition is not relativistically invariant. However, T-and L-dynamics is not mixed in a transformation between inertial frames in relative motion. Rather, E T and B are transformed together, while E L and J L mix separately.
Previous uniqueness theorems in Minkowski and pseudo-Riemann space involve C(r, t), but not λ [13,15]. The uniqueness theorems are useful in EED, but the connection to λ has been obscure until now. We clarify the role of λ by rewriting equations (11), (12) as: Equations (14), (15) have new terms on the right-hand side (RHS), which are: The longitudinal part of the electric field (E L ) is: (13); and use of equation (14) to further simplify the expression. The result is: A longitudinal extension of Gauss' law can be obtained by taking the divergence of equation (18); ; use of the definition of C from equation (13); and application of equation (15) to simplify the expression. The result is: Equations (19), (20) are analogous to Woodside's uniqueness theorem for extended electrodynamics, which assumes only Minkowski space [15]. The definitions of C in equation (13) and E L in equation (18) are uniquely specified via the source terms, J L l and ρ λ , through the hyperbolic wave equations (14), (15). This result holds even if J L l and ρ λ depend on C and E L , from the definitions in equations (16), (17). A dependence on E L arises (e.g., in linear response) when J L is proportional to E L . This generalized uniqueness theorem includes the factor, 1−λ, as expected.
Further simplification is possible by introduction of: (20) can then be rewritten in the form: The wave equations for A L in equation (14) and Φ in equation (15) can then be expressed in terms of C λ : The divergence of equation (22) added to c −2 ∂/∂t on equation (23) is: Classical charge conservation requires the RHS of equation (26) to be zero, leaving the left-hand side (LHS) zero as well. Since C λ is simply scaled by λ, the C-wave equation is: is a source free wave equation for C (and also C λ ). As shown previously, a gauge transformation with a gauge function (Λ) creates the bridge from the old (C OLD =C) to the new (C NEW =λC) C in equation (21), requiring the condition: The effective charge (ρ λ ) and current (J L l ) densities also satisfy a continuity equation: To obtain equation (29), we apply μ o ∇· to equation (16), and add the result to (ε o c −2 )∂/∂t as applied to equation (17). We then employ equation (27) plus classical charge conservation, ∇·J L +∂ρ/∂t=0. [16][17][18][19][20][21] showed that Maxwell's equations can be extended to general relativity (GR) in 3-vector notation. The extension in [21] uses so-called generalized microscopic polarization and magnetization fields. The resultant equations are form-identical to the macroscopic Maxwell's equations, but with a very different physical interpretation, as discussed next.

General relativistic CED in 3-vector form
The manifestly covariant form of the inhomogeneous GR field equation is: The GR charge (ρ GR ) and current (J GR ) densities are: Here, ρ CURV and J CURV are contributions from the space-time curvature (CURV).

GR longitudinal waves
We next introduce a so-called longitudinal displacement field by the definition: The longitudinal component of the GR-polarization density (P L ) has the following properties: The curvature part of the polarization density is: provided that the GR metric tensor is known. The curvature polarization density is discussed further in section 5.5.

Elimination of C λ in favor of D L
The vector fields in this work (generically denoted as F) all can be represented as Fourier integrals: with an inverse transformation: The position vector is r; ω is the angular frequency; and q is the wave vector with an amplitude, q. The gradient (∇) and partial-time derivative (∂/∂t) operators transform into multiplication by iq and -iω in the (ω, q)-domain. Hence, equations (46), (47) become: Both D L and q are in the longitudinal direction, q q q .
= / D L can then be rewritten as D L =q D L · resulting in: In the space-time (r, t) domain, the solution for C(r, t) is: Here, δ(K) is the Dirac delta function. A factor of π appears in the first line of equation (56), arising from our choice of D L scaling. Then, the factor of ½ in the second line of equation (56) is convenient, when adding a term and its complex conjugate. Integration of equation (56) over ω then yields: Substitution of q→-q in the last term of equation (57) gives: Here, 'c.c.' denotes the complex conjugate. Equation (60) shows that C λ (r, t) is completely determined by the value of D L (q, ω) on the energy shell, |ω|=cq. C λ (r, t) is then composed of plane waves with a wave vector (q), propagating at the vacuum speed of light. Equation (54) also implies that C λ satisfies equation (26) with the RHS equal to zero for each plane-wave mode.  As discussed elsewhere [23], the first-order differential equation, equation (66), plays a significant role in understanding the fundamental limitations for spatial photon localization. The presence of the additional nonzero term, ∂C λ /∂t, in the EED form of Gauss' law, equation (38), prevents the reduction of equation (64) to firstorder form.

Longitudinal wave emission
We introduce a longitudinal curvature (CURV) displacement field to investigate the electrodynamic excitation of longitudinal waves in curved space-time: The longitudinal polarization density is related to the microscopic charge and L-current densities via ρ=-∇·P L and J L =∂P L /∂t. These relations can be substituted into equation (70) to obtain the form: We take the current density distribution (J) to be localized to a finite, space-time volume, V. J and J L are related in a spatially, non-local manner [23]. More specifically, the space-time volume that is associated with the J L is V L , which is different from (larger than) V with both volumes evaluated at the same time. The general solution to equation (71) is then given by: The scalar (Huygens) propagator is given by: ( ) originates in electrodynamic sources that are located outside V L . We assume that the dynamics of the external sources are unaffected by the radiation form the sources inside V L . Neither V nor V L has a sharp boundary. Usually, the microscopic current density exhibits an exponential fall-off in the surface region due to the quantum-mechanical, wave-function spill-off, whereas the longitudinal part has a slower algebraic falloff in general. In either case, the integration limits may extend to ±∞.
We next focus on the inhomogeneous part of equation (72): Propagator formalism in the (q, ω)-domain Section 4.2 showed that the CED extension only affects the electrodynamics on the light-energy shells, ω=±cq.
In this view, equation (50) can be rewritten as: If ω≠±cq, then the second line of equation (76) is zero, yielding P q D q , , .
L CURV In the (q, ω)domain, we obtain P L (q, ω)=J L (q, ω)/(−iω), since J L =∂P L /∂t. This last equation can be substituted into equation (76) to obtain an enlightening form: where the last term in equation (77) is: The extension of CED appears in equation (77) as an additional contribution to the longitudinal current density. Such an interpretation is reasonable, because C λ =λC relates to the 4-potential in equations (13) and (21). This relation in turn leads to the longitudinal component of electric field in equation (18). Since E L (q, ω)=J L (q, ω)/(iε o ω) in CED, then EED would naturally have an extension in the propagator description.
In the (q, ω)-domain, equation (74) takes the algebraic form: (74) is a folding integral. The scalar propagator is then given by: where q=|ω|/c, as the vacuum wave-number of light. The singularities of the propagator at ω=±cq correspond to the singularity at R=0 in the (r, t)-domain [equation (73)]. Contour integration over a complex ω-plane is often used in CED. The poles are located on the real axis at ω=±cq. These poles are encircled in the upper or lower half-plane, depending on the specific application.

Near-field, off-energy-shell, longitudinal dynamics
The propagator formalism in the (q, ω)-domain reveals a 'hidden contribution' in equation (76) that is nonvanishing only for ω=±cq in the extension of CED. When ω≠±cq, the agreement with CED is complete. Consequently, S L (q, ω) in the extended theory (EXT) must be replaced by: The structure of S L (q, ω) is given by rewriting equation (74): ) in equation (85). Thus, we obtain: , , 86 The inhomogeneous solution to equation (71) is then obtained by transformation to the (r, t)-domain and use of equation (66): The partial-time derivative of equation (87) gives equation (35). CED provides no direct excitation of longitudinal waves in curved free-space-time. However, CED provides indirect possibilities. Namely, a longitudinally polarized electric field does exist in the near-field (rim) zone of the source [21]. The rim zone extends over the spatial region of J L (r, t), or equivalently P L (r, t). Section 6 shows an example of how a near-field E L can be used as a source for L-wave in GR.

E
The second term on the RHS of equation (88) can be re-written: The second line of equation (89) uses q→−q; the third line of equation (89) relies on C(r, t) being real so that: Here, 'c.c.' is an abbreviation for complex conjugate. Thus, generation of a current density, t J r, , L l ( ) launches a longitudinal displacement field. When no external field exists, the scalar-propagator description yields: implies that a rapidly varying longitudinal current density produces a strong displacement field. 5.5. E L propagation in curved space-time For the following analysis, we assume that t t J r, L ¶ ¶ l ( )/ is non-zero over a finite space-time interval. After t t J r, L ¶ ¶ l ( )/ is zero, equation (92) then implies: In the absence of gyromagnetic effects [21], we have: The relative dielectric tensor is r e  with Cartesian elements in terms of the contravariant metric tensor, g μν : g g g g g . 6. J L l by nonlinear optical mixing Strong electromagnetic fields are present in many astrophysical processes. For example, longitudinal electricfield oscillations (e.g., plasma oscillations) may dominate, which have a frequency that is a function of the plasma density, and the mass and charge of the oscillating ions and electrons. Moreover, sum-and difference-frequency modes are generated in nonlinear (NL) optics at lowest (second) order. Such modes generate a NL current density (J NL ) at the sum-and difference-frequencies for a non-vanishing susceptibility tensor, q q , ; , , Hence, E L in the source's near-field via the generated J NL can drive L-waves in free, curved space-time. A closely related source is current density fluctuations.

Conclusions
The present work provides insight into extended electrodynamics (EED) that was pioneered by Ohmura [1] and studied recently by others [8][9][10][11][12][13][14][15]. In the Lagrangian formulation, EED is described in terms of the well-known λ-parameter (λ not necessarily one). Previous uniqueness theorems [13][14][15] formulated EED with a non-zero, scalar field, C=∂ μ A μ . In general relativity, EED allows propagation of longitudinal (L) waves in charge-and current-free, space-time regions. L-waves cannot exist in free-space CED. We show that EED: (i) only affects the longitudinal components of the fields and vector potential, and (ii) needs both C and λ to obtain consistency between the longitudinal electric field (E L ) and the potential descriptions (Φ, A L ) descriptions. We prove that the parameter product, λC, is completely determined by a superposition of longitudinal displacement-field modes, all of which are confined to the energy shell, |ω|=cq. This formulation predicts that L-waves can be emitted into free-space from the near-field of a source by non-linear, electrodynamic mixing. A propagator formalism offers substantial physical insight into the L-wave emission process.