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Anais da Academia Brasileira de Ciências

Print version ISSN 0001-3765On-line version ISSN 1678-2690

An. Acad. Bras. Ciênc. vol.72 n.2 Rio de Janeiro June 2000 

Dirac's æther in curved spacetime



1Universidade Federal do Rio de Janeiro, Departamento de Astronomia
Observatório do Valongo - 20080-090 Rio de Janeiro, RJ, Brazil
2 Centro Brasileiro de Pesquisas Físicas - 22290-180 Rio de Janeiro, RJ, Brazil

Manuscript received on August 11, 1999; accepted for publication on December 1, 1999;
presented by




Proca's equations for two types of fields in a Dirac's æther with electric conductivity s 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 S 3 or the projective three-space P 3. Simple relations between the range of Proca field l, the Universe radius R, the limit of photon rest mass mg and the conductivity s are written down.
Key words: Dirac's æther, Proca Field, curved spacetime, three-sphere, projective three-space.




The possibility of a nonzero electric conductivity s in cosmic scale (Dirac's æther) has been considered by several authors and in various contexts: Vigier (Vigier 1990), e.g., showed that introducing $\sigma > 0$ in the vacuum is equivalent to attributing a nonzero mass $m_{\gamma} >0$ to the photon. Further study of the relation between s and mg was performed by Kar, Sinha and Roy (Kar et al. 1993), who also discussed possible astrophysical consequences of having nonzero mg. 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 $m_{\gamma} >0$; in the background we assume a curved spacetime together with a constant conductivity $\sigma > 0$. In the next section we present the three existing classes of exact solutions for the field; they depend on the relative values of s, mg 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 $t = const$.



In the static elliptic spacetime we use the cylindrical Schrödinger coordinates $x^{\mu} = (ct; \rho, \phi, \zeta)$ and write the line element

\begin{displaymath} ds^2 = c^2dt^2 - R^2 (d\rho^2 + \sin^2 \rho d\phi^2 + \cos^2 \rho d\zeta^2) \ , \end{displaymath}   (1)

where $R = const$ is the characteristic radius of the three-geometry. We assume a nonstatic four-potential with cylindrical symmetry

\begin{displaymath} \Phi^{\mu} (0; 0, cf(t), 0) \ , \end{displaymath}  (2)

where $f(t)$ is a function to be determined from the field equations; clearly $\Phi^{\mu}$ satisfies the Lorentz gauge, $\partial_{\mu}[(-g)^{1/2} \Phi^{\mu}]=0$. The only surviving independent components of $F_{\mu\nu} = \partial_{\mu}\Phi_{\nu} - \partial_{\nu}\Phi_{\mu}$ are then

\begin{displaymath} F^2_0 = \dot f \ , \qquad F^2_1 = 2 cf \cot \rho \ , \end{displaymath}   (3)

where the overdot means the time $t$ derivative. In the orthonormal basis the nonvanishing components of the E and B fields are

\begin{displaymath} E_{\phi} = -R\dot f \sin \rho \ , \qquad B_{\zeta} = 2f \cos\rho \ . \end{displaymath}   (4)

Proca equations in a conducting medium are

\begin{displaymath} F^{\mu\nu}_{;\mu} + (\kappa/\lambda^2)\Phi^{\nu} = (\sigma/c)u_{\alpha}F^{\nu\alpha} \ , \end{displaymath}   (5)

where $\sigma > 0$ is the electric displacement conductivity, $u_{\alpha} = \delta^0_{\alpha}$ is the four-velocity of the observer, l is the range of the Proca field, and $\kappa = \pm 1$ accounts for two different categories of field. For $\nu = 2$ eq.(5) gives

\begin{displaymath} \ddot f + 2\Gamma \dot f + \gamma f = 0 \ , \end{displaymath}   (6)


\begin{displaymath} \Gamma = \sigma / 2 \ , \qquad \gamma = 4c^2 / R^2 + \kappa c^2 / \lambda^2 \ . \end{displaymath}   (7)

Three classes of solutions of (6) exist, depending on the relative values of the constants $\Gamma$ (nonnegative) and $\gamma$ (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 $\Delta = E^2_{\phi} + c^2B^2_{\zeta}$ is independent of $\rho$ only when $R^2\dot f^2 = 4c^2f^2$, which implies that $f(x) \propto \exp (2\epsilon ct/R)$, with $\epsilon = \pm 1$. Three non-equivalent sets of solutions with $\partial\Delta /\partial \rho = 0$ are discussed in the next section, and constraining relations among the quantities $\{\sigma, \lambda, R, c, \kappa, \epsilon\}$ in each set are given.



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 $\zeta$ axis (the axis where $\rho = 0$), and is maximum along the circle $\rho = \pi/2$. Oppositely, the B field is purely longitudinal, is maximum along the $\zeta$ axis and vanishes on the circle $\rho = \pi/2$. 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 $B_{\zeta} = 2C_1 \cos \rho$, and belongs to class $\Gamma^2 > \gamma$ with $C^2 = 0, \gamma = 0, \kappa = -1, \lambda = R/2$.

All non-static solutions are standing Proca waves. Most have exponential damping with increasing time. Nevertheless, in the class $\Gamma^2 > \gamma$, an exception deserves mentioning: when $\gamma <0$, that is $\kappa = -1$ and $\lambda <R/2$ 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 $\Delta = E^2_{\phi} + c^2B^2_{\zeta}$ independent on the location in three-space were encountered: see Table II. Sets a and b both have $\Delta \propto \exp (-4ct/R)$ (damping along the time), and both contain $\lambda\to\infty$, $\sigma = 4c/R$ (a Maxwell field) as a special case. The set c has $\Delta \propto \exp (+4ct/R)$ (increasing along the time). Sets b and c both contain the special case $\lambda = R/\sqrt 8, \sigma =0$ (vanishing conductivity).



A few words seem worthwhile, concerning the physical values of the constants $m_{\gamma}, \lambda, \sigma$ and R. First recall that the mass mg and the range l share the quantum correspondence $m_{\gamma}c = h/\lambda$, where $h = 6.6 \times 10^{-34}$ J s is Planck's constant. Assuming $\lambda \approx R\approx 10^{10}$ l.y. $\approx 10^{26}$ m, then $m_{\gamma} \approx 10^{-68}$ 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 l is presently indiscernible from a Maxwell field. From Table II, and still assuming $\lambda\approx R\approx 10^{26}$ m, one should have $\sigma\approx 10^{-17}$/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 $\Gamma^2 < \gamma$ in Table I, then the resulting frequency would be $\delta \approx 10^{-18}$ Hz; fields with such a slow variation would seem static.



One of us (A.L.O.) wishes to thank Prof. A.O. Caride for calling attention to related work on finite-range theories, Dr. Hirokazu Hori for sending copies of his work on near-field optics and M.F. Carvalho for friendly encouragement and discussions on Dirac's æther.



AHONEN J & ENQVIST K. 1996. Electrical conductivity in the early universe. Phys Lett B 382: 40-44.         [ Links ]

GOLDHABER AS & NIETO MM. 1971. Terrestrial and Extraterrestrial Limits on The Photon Mass. Rev Mod Phys 43: 277-296.         [ Links ]

KAR G, SINHA M & ROY S. 1993. Maxwell Equations, Nonzero Photon Mass, and Conformal Metric Fluctuation. Int J Theor Phys 32: 593-607.         [ Links ]

VIGIER JP. 1990. Evidence For Nonzero Mass Photons Associated With a Vacuum-Induced Dissipative Red-Shift Mechanism. IEEE Transactions on Plasma Science 18: 64-72.         [ Links ]



Correspondence to: A. L. Oliveira

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