A&A 369, 690-693 (2001)
V. A. De Lorenci1 - N. Figueiredo1 - H. H. Fliche2 - M. Novello3
1 - Instituto de Ciências, Escola Federal de Engenharia de Itajubá, Av. BPS 1303 Pinheirinho, 37500-903 Itajubá, MG, Brazil
2 - Université Aix-Marseille III, Laboratoire de Modélisation en Mécanique et Thermodynamique, UPRES EA 2596, Case 322, Avenue Escadrille Normandie-Niemen, 13397 Marseille Cedex 20, France
3 - Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150 Urca, 22290-180 Rio de Janeiro, RJ, Brazil
Received 4 November 1999 / Accepted 5 December 2000
In the context of the static and spherically symmetric solution of a charged compact object, we present the expression for the bending of light in the region just outside the event horizon - the dyadosphere - where vacuum polarization effects are taken into account.
Key words: Black-hole physics - gravitational lensing - relativity
The magnitude of the velocity of light described by effective nonlinear electromagnetic theories depends on the field dynamics. Such dependence implies an effective modification of the flat background metric into a curved one, which is accentuated when gravity processes are taken into account. The most famous examples of these aspects are the well known implications of QED in curved spacetimes. The weak field limit of the complete one loop QED is known as the Euler-Heisenberg Lagrangian (Heisenberg & Euler 1936; Schwinger 1951) which yields several important results such as, the phenomenon of birefringence (Birula & Birula 1970; Adler 1971; De Lorenci et al. 2000), which describes the distinct velocity of light propagation for each polarization direction. There are many works dealing with the applications of nonlinear electrodynamics, especially, concerning its coupling with a gravitational field. In this context, Drummond & Hathrell (1980) showed the possibility of superluminal velocities in certain spacetime configurations. Other interesting cases can be found in Novello et al. (1989); Daniels & Shore (1994), Latorre et al. (1995) and Shore (1996).
Recently, Ruffini (1998) and also Preparata et al. (1998) called our attention to a special region just outside the horizon of charged black holes where the electric field goes beyond its classical limit, implying a situation where effects of vacuum fluctuations should be considered. They called such region the dyadosphere. Assuming the existence of such a region, we could consider the possible consequences for the trajectories of light rays that cross it. In this work, we analyze the consequences for the phenomenon of light bending when vacuum fluctuation effects are taken into account. Under such conditions, the paths of light do not follow the usual geodesic of the gravitational field, so it is necessary to consider the effects of the modified QED vacuum (Drummond & Hathrell 1980; Dittrich & Gies 1998; De Lorenci et al. 2000; Novello et al. 2000).
In Sect. 2 we perform the coupling between nonlinear electrodynamics and gravity. We calculate the correction for the Reissner-Nordstron metric from the first contribution of the weak field limit of one loop QED. In Sect. 3 we present the light cone conditions for the case of wave propagation in the Reissner-Nordstron spacetime modified by QED vacuum polarization effects - the dyadosphere region. Finally, in Sect. 4, we derive the field equations for such a situation and evaluate its contribution to the bending of light. Some comments on lensing effects are presented in the Appendix.
The Einstein gravitational field equation is given by
Let us consider a class of theories defined by the general
Lagrangian L = L(F), where
energy momentum tensor has the form
Since (13) is an approximation, it cannot be applied to the cases for which . Thus the contribution of the last term in this equation is negligible for two reasons: is very small and .
In the next sections, however, it is shown that, due to the non-linearity of the electrodynamics in the dyadosphere, there will be an additional correction which is comparable to the Reissner-Nordstron factor in terms of the radial variable.
One might want to investigate the effect of this metric on the gravitational lensing of light propagating in the vicinity of a charged compact object. In the Appendix we apply the formalism developed by Frittelli et al. (2000) to the background spacetime defined by Eq. (5).
In the case of nonlinear electrodynamics, e.g.,
Euler-Heisenberg effective theory,
the wave propagation will suffer a correction
due to vacuum polarization effects.
Such correction is usually presented in terms of a
light cone condition, which in our case, is given by
Condition (14) can be presented in
a more appealing form as a slight
modification of the background geometry
Contraction of Eq. (17) with
Taking the Lagrangian (9),
and setting the only non-zero component of electromagnetic tensor
F01 = f(r), it follows that
In this work we investigate the bending of light in the dyadosphere, a region just outside an event horizon of a non-rotating charged black hole, in which the electromagnetic field exceeds the critical value predicted by Heisenberg and Euler. In such region the propagation of light is affected not only by the gravitational field, but also by the modified QED vacuum. We show that by considering effects due to QED, the modified metric in the first order of approximation is given by Eq. (29).
We also show that the contribution from this effect appears in a significative order in terms of the radial variable. Indeed, it is of the same order of the charged term that arises from Reissner-Nordstron solution. Equation (36) shows that the correction term depends on both the ratio h0/l0 and the charge Q. Thus, for photons with the same frequency propagating in the dyadosphere, the effect will be stronger for those whose trajectory is closer to the center of attraction.
Since we obtained a contribution coming from QED of the same order of magnitude as the classical Reissner-Nordstrom charge term, an interesting extension of this work could be studying gravitational lensing effects for the metric presented in Eq. (29). It is also worth analysing the effects arising from the magnetic field due to the rotation of a charged compact object.
This work was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Comité Français d'Évaluation de la Coopération Universitaire avec le Brésil (COFECUB).
Following this formalism, the exact gravitational lens equation in the framework of the metric (5) is given by
In the above equation, is the angular position of the source with respect to the direction defined by the observer and the lensing object (the optical axis) and is the inverse radial distance. It follows that and , where r0 and are, respectively, the position of the observer and the minimum distance between the light path and the lens. As usual, the origin of the coordinate system is the position of the lens.
A generic point in the light path is given in spherical coordinates by
Applying Eq. (A.1) to the function A(r)derived in (13) it follows that there is no significant contribution from QED to the lensing effect for the same reasons discussed at the end of Sect. 2.