Dirac ’ s æther in curved spacetime

Proca’s equations for two types of fields in a Dirac’s æther with electric conductivity σ are solved exactly. The Proca electromagnetic fields are assumed with cylindrical symmetry. The background is a static, curved spacetime whose spatial section is homogeneous and has the topology of either the three-sphere S3 or the projective three-space P 3. Simple relations between the range of Proca fieldλ, the Universe radiusR, the limit of photon rest mass mγ and the conductivityσ are written down.


INTRODUCTION
The possibility of a nonzero electric conductivity σ in cosmic scale (Dirac's aether) has been considered by several authors and in various contexts: Vigier (Vigier 1990), e.g., showed that introducing σ > 0 in the vacuum is equivalent to attributing a nonzero mass m γ > 0 to the photon.Further study of the relation between σ and m γ was performed by Kar, Sinha and Roy (Kar et al. 1993), who also discussed possible astrophysical consequences of having nonzero m γ .More recently, Ahonen and Enqvist (Ahonen & Enqvist 1996) studied the electric conductivity in the hot plasma of the early universe.
In this paper we study the time evolution of an electromagnetic field with m γ > 0; in the background we assume a curved spacetime together with a constant conductivity σ > 0. In the next section we present the three existing classes of exact solutions for the field; they depend on the relative values of σ , m γ and the curvature of spacetime as given by a constant radius R. In the last section we describe in some detail a set of solutions in which the quantity E 2 + c 2 B 2 is homogeneous throughout the spacelike hypersurfaces t = const.

EQUATIONS AND SOLUTIONS
In the static elliptic spacetime we use the cylindrical Schrödinger coordinates x µ = (ct; ρ, φ, ζ ) and write the line element where R = const is the characteristic radius of the three-geometry.
We assume a nonstatic four-potential with cylindrical symmetry where f (t) is a function to be determined from the field equations; clearly µ satisfies the Lorentz gauge, ∂ µ [(−g) 1/2 µ ] = 0.The only surviving independent components of where the overdot means the time t derivative.In the orthonormal basis the nonvanishing components of the E and B fields are Proca equations in a conducting medium are where σ > 0 is the electric displacement conductivity, u α = δ 0 α is the four-velocity of the observer, λ is the range of the Proca field, and κ = ±1 accounts for two different categories of field.For ν = 2 eq.( 5) gives where Three classes of solutions of (6) exist, depending on the relative values of the constants (nonnegative) and γ (arbitrary); see Table I, where C 1 and C 2 are integration constants.Solutions in which the field energy is homogeneously distributed in three-space are of particular interest.From eqs.(4) we find that the quantity

Classes
Exact solution of (6

DISCUSSION
As is seen from ( 4), in all solutions the E and B fields are mutually orthogonal and spatially inhomogeneous.The E field is purely azimuthal, vanishes on the ζ axis (the axis where ρ = 0), and is maximum along the circle ρ = π/2.Oppositely, the B field is purely longitudinal, is maximum along the ζ axis and vanishes on the circle ρ = π/2.These expressions for the fields are globally possible whenever the topology of the underlying 3-space is either the simply connected 3-sphere S 3 or the multiply connected real projective 3-space P 3 .No other multiply connected 3-space endowed with the elliptical geometry (e.g. the Poincaré dodecahedron) seems appropriate to globaly accomodate these forms of field.
From Table I we immediately distinguish two static solutions: one is the trivial no-field solution E = B = 0, corresponding to C 1 = C2 = 0; the other is a pure magnetostatic field with E = 0 and B ζ = 2C 1 cos ρ, and belongs to class 2 > γ with All non-static solutions are standing Proca waves.Most have exponential damping with increasing time.Nevertheless, in the class 2 > γ , an exception deserves mentioning: when γ < 0, that is κ = −1 and λ < R/2 in eq.( 7), the potential f (t) and the Proca fields show an exponential growth as time increases.Three sets of non-static solutions with the quantity = E 2 φ + c 2 B 2 ζ independent on the location in three-space were encountered: see Table II.Sets a and b both have ∝ exp(−4ct/R) (damping along the time), and both contain λ → ∞, σ = 4c/R (a Maxwell field) as a special case.The set c has ∝ exp(+4ct/R) (increasing along the time).Sets b and c both contain the special case λ = R/ √ 8, σ = 0 (vanishing conductivity).

TABLE II
Parameters for uniform (t).
A few words seem worthwhile, concerning the physical values of the constants m γ , λ, σ and R.
An. Acad.Bras.Ci., (2000) 72 (2) First recall that the mass m γ and the range λ share the quantum correspondence m γ c = h/λ, where h = 6.6 × 10 −34 J s is Planck's constant.Assuming λ ≈ R ≈ 10 10 l.y.≈ 10 26 m, then m γ ≈ 10 −68 kg, which is fifteen orders of magnitude smaller than the upper limit obtained by experimental techniques (Goldhaber & Nieto 1971); this amounts to saying that a Proca field with that value for the range λ is presently indiscernible from a Maxwell field.From Table II, and still assuming λ ≈ R ≈ 10 26 m, one should have σ ≈ 10 −17 /s for systems with E 2 + c 2 B 2 homogeneous over the 3-space; this value for the conductivity coincides in order of magnitude with that of ref. (Kar et al. 1993), obtained in a different context.To conclude, if we consider the above values for the various constants in the damping harmonic class 2 < γ in Table I, then the resulting frequency would be δ ≈ 10 −18 Hz; fields with such a slow variation would seem static.