An Electro-Gravitational Induction Effect

An Electro-Gravitational Induction Effect Attay Kremer1 1 Open University of Israel - Ramat Aviv Campus December 27, 2018 Abstract In this paper, we show that in general relativity the gravitational field induces an electromagnetic field. This is done by considering the general solution of the homogeneous Maxwell equations in a gravitational field. The solution of these equations involves an ambiguity in the choice of path. This ambiguity is resolved by the derivation and solution of the path of a particle in curved spacetime under the impression of the force of the induced field. This induction effect shows that in the context of general relativity the gravitational and electromagnetic forces are dynamically linked. 1 Introduction In classical electromagnetism one of the most interesting and historically significant effects is the induction effect discovered by Michael Faraday in 1831. The effect is significant mainly because of its importance in the unification of electromagnetism achieved by James Clerk Maxwell. The induction of a magnetic field by a changing electric field (and the induction of an electric field by a magnetic field) shows the two fields to be dynamically linked, thus allowing for a unified mathematical description as the electroamgnetic field. Maxwell’s equation written in modern tensor notation are ∂µ F µν = µ0 J ν . (1) Here we wish to show that the formalism of general relativity admits an analogous effect in which the gravitational field induces an electromagnetic field. Together with the fact that the electromagnetic field carries with it energy, and affects the relevant energy stress tensor, we see that the gravitational field and the electromagnetic field are dynamically linked as well. The induction of an electromagnetic field by the gravitational field occurs because of the connection between the gravitational field and the metric of space-time. Just as the geodesics are different from flat space geodesics, so is the electromagnetic field different from its flat space counterpart. 1 The influence of the gravitational field on the electromagnetic in scales comparable to the Compton scale is considered in Rosquist[4]. Ruffini et al[2] consider the induction of electromagnetic radiation by motion in an electromagnetic field, and Plebanski[3] and Fucci et al[1] consider the influence of gravitation on the behaviour of electroamgnetic radiation, in the classical and quantum cases respectively. Here, however, we wish to consider the induction of a field (which, in some cases may be static) at classical scales and in the classical case. 2 Induction effect We now wish to show that gravitational field induces, under specific circumstances, an electromagnetic field. By Maxwell’s equations in a gravitational field ∇µ F µν = µ0 J ν (2) . We to see that the existence of a non zero gravitational potential, in this case, non-Minkowskian metric, leads – even when no charge is present – to a variable electromagnetic field. The covariant derivative of the second rank tensor is given by σβ ∇ξ F αβ = ∂ξ F αβ + Γα + Γβξσ F ασ ξσ F (3) σβ ∂ξ F αβ = −Γα − Γβξσ F ασ ξσ F (4) Thus, we write This equation is somewhat analogous to the geodesic equation. We see that the covariant vacuum state for the electromagnetic field is different from that of the Euclidean vacuum state. Equation (4) maybe solved most practically for a particular solution, just as is done for the geodesic equation. Further more, this equation maybe viewed as the equation governing the induction of an electromagnetic field by a gravitational field. With this in mind, we’d like to write a more general solution to the equation so that we may derive a general equation including both the effects of the gravitational field, the electromagnetic field induced by it. To this end, we multiply by δxξ and integrate along a path, so we get Z [α αβ αβ F = κ − Γξσ F β]σ δxξ (5) . where καβ is the tensor representing the intial values of the electric and magnetic fields. It is defined by i Einitial = κ0i 2 (6) Bkinitial = ǫijk κij (7) in natural units, so that c = 1. Iteratively putting the lefthand side into the right hand side of (5), we get the following general solution for F αβ . The Iteration is as follows F αβ =κ αβ − Z [α Γξσ κβ]σ δxξ − Z Z Z + [α Z Z [α [β] Γξσ Γξ1 σ1 κσ]σ1 δxξ1 δxξ [β] [σ] Γξσ Γξ1 σ1 Γξ2 σ2 κσ1 ]σ2 δxξ2 δxξ1 δxξ Z Z Z [α [β] [σ] + Γξσ Γξ1 σ1 Γξ2 σ2 Γ[σ1 ] F σ2 ]σ3 δxξ3 δxξ2 δxξ1 δxξ (8) where the anti-commutation of indices is used such that [α [β] β σσ1 − Γσξ1 σ1 F βσ1 ) Γξσ Γξ1 σ1 F σ]σ1 = Γα ξσ (Γξ1 σ1 F σσ1 − Γσξ1 σ1 F ασ1 ) − Γβξσ (Γα ξ 1 σ1 F (9) . That is to say, that γ [i γ [j] γ k] is just γ [i Gj]k (10) Gjk = γ [j γ k] (11) with . We see that the solution given by the limit of this process is F αβ = Σ∞ n=0 Z ··· Z [α [β] [σ ] [σ n−3 n−2 σn−1 ]σn Γξσ Γξ1 σ1 . . . Γξn−1 δxξn . . . δxξ σn−1 Γξn σn κ (12) Taking a first order approximation of (12), using the second term on the right hand side of (8), we have 2∂ [µ F ν]τ ≈ k νσ [g ατ (∂ µ gασ + ∂ µ ∂[σ xα] ) + (gασ + ∂[σ xα] )∂ µ g ατ ] − k τ ξ [g αν (∂ µ gαξ + ∂ µ ∂[ξ xα] ) + (gαξ + ∂[ξ xα] )∂ µ g αν ] + k τ ξ [g αν (∂ ν gαξ + ∂ ν ∂[ξ xα] ) + (gαξ + ∂[ξ xα] )∂ ν g αν ] − k νσ [g ατ (∂ ν gασ + ∂ ν ∂[σ xα] ) + (gασ + ∂[σ xα] )∂ ν g ατ ] 3 (13) so that, in general, the electromagnetic field is path dependent. The path dependence of the induced field is built into the algorithm used to solve (4). This means that the solution (12) is not unique: integrating along different paths xξ would yield different fields. Since F µν is integrated along a path, the solution is given only once a path is specified. To resolve this ambiguity in (12), we wish to write a differential equation for the path. The equation for the motion of a particle in a gravitational and electromagnetic fields in General Relativity is given by the following, and will be referred to as the geodesic-Lorentz equation[7]. q ǫα F gαβ ẋβ − Γǫµν ẋµ ẋν m ẍǫ = (14) . Putting equation (12) into the geodesic-Lorentz equation, we get a integrodifferential equation for the path by the particle. Thus, we may – in principle – calculate the path traced by a particle under a gravitational field and the electromagnetic field induced by it. In general we write ẍǫ = q gαβ ẋβ Σ∞ n=0 m Z ··· Z [ǫ [σ [α] ] [σ n−3 n−2 σn−1 ]σn ξ Γξσ Γξ1 σ1 . . . Γξn−1 ẋ . . . ẋξn ds . . . ds σn−1 Γξn σn κ − Γǫµν ẋµ ẋν (15) . This equation is quite complicated and I am unaware of methods to approach it in this form. In order to solve it we use an iterative series based on the unperturbed geodesics of the metric. That is to say, we look at the physically significant path of the geodesic xξ0 , and calculate the electromagnetic field along it. This is used as the first term in right hand side of (15), where the path solved for now is the next step in the iteration x¨1 ǫ = q gαβ x˙1 β Σ∞ n=0 m Z ··· Z [ǫ [α] [σ ] [σ n−3 n−2 σn−1 ]σn Γξσ Γξ1 σ1 . . . Γξn−1 x˙0 ξ . . . x˙0 ξn ds . . . ds σn−1 Γξn σn κ − Γǫµν x˙1 µ x˙1 ν . (16) Each equation in the iteration may be solved with the usual methods applied to the geodesic-Lorentz equation since F0ǫα does not involve xǫ1 . This approach is most practical and helps to calculate the geodesic under the influence of the induced field to the required accuracy. Unlike the solution (12), this doesn’t provide us with a formal solution for the path. In order to get a full solution need a general solution for the usual geodesic-Lorentz equation. The combination of a general solution and the iterative approximation for (15) form a recursive relation which would yield a general form n-th step of the approximation. Taking this expression with n going to infinity would yield a formal solution of (15). With this in mind, a general solution for the geodesic-Lorentz equation is not known. In order to give a resolution for the ambiguity in the solution we solve 4 equation (15) in harmonic coordinates by use of an iterative methods similar to that used for (4). In harmonic coordinates we have g µν Γα µν = 0. (17) g µν ẋµ ẋν (18) We multiply both sides by so that (14) becomes q ǫα F gαβ ẋβ . m Integrating with respect to proper time, we get Z q ǫα ǫ ǫ ẋ = δ + F gαβ ẋβ ds. m ẍǫ = (19) (20) Recursively putting the right hand side into the left hand side of (20), we have ẋǫ = Σ∞ n=0 Z ··· Z ( q n ) gαβ . . . gαn βn F ǫα F βα1 . . . F βn−1 αn δ βn ds . . . ds m Z q = Pexp[ δ β gαβ F ǫα ds] m (21) where P is a path ordering operator defined in the usual way[6]. Assuming that the higher order Γ terms of (12) are small, we take a first order approximation of (12) to get Z [ǫ F ǫα = k ǫα − Γξσ k α]σ ẋξ ds. (22) Putting (22) into (21), we get q ẋ = Pexp[ δ β gαβ m ǫ Z ds(k ǫα − Z [ǫ Γξσ k α]σ ẋξ ds)] (23) recursively putting (21), and (22) into the right hand side of (23), we obtain q ẋ = Pexp[ δ β gαβ m ǫ Z ǫα ds(k − Z [ǫ dsΓξσ k α]σ Pexp[ q β1 δ g α1 β1 m Z ds(k ξα1 − Z (24) By the same assumption that the higher order Γ terms are small, we may simplify this expression by truncating the argument of the exponential at first order just as we did for the (12). This yields 5 [ξ dsΓξ1 σ1 k α1 ]σ1 . . . ]]. q ẋ = Pexp[ δ β gαβ ( m ǫ Z ǫα k ds − Z Z [ǫ Γξσ k α]σ (δ ξ q + g α1 β 1 δ β1 m Z k ξα1 ds)dsds]. (25) By the same considerations we may approximate the exponent by the first two terms of the sum, thus obtaining the final result for path ẋǫ = δ ǫ + q β δ gαβ ( m Z k ǫα ds − Z Z [ǫ Γξσ k α]σ (δ ξ + q g α β δ β1 m 1 1 Z k ξα1 ds)dsds. (26) Equations (19) and (26) together form a well defined solution of (4) and describe, to first order, the electromagnetic field induced by the curvature of spacetime in the absence of a current. The Energy-stress tensor for the electromagnetic field is given by [5] 1 1 (g αβ Fµα Fνβ − gµν F αβ Fαβ ) (27) 2µ0 4 so that the induced electromagnetic field has an energy content which affects the metric. The metric in turn affects the induced force. This allows the gravitational field to be affected by the electromagnetic field, which would in turn be effect the gravitational field. First, gµν affects (and at times induces) Fµν by Maxwell’s equations in a gravitational field. Then, Fµν would affect (and at times induce) gµν by the Einstein field equations. A solution of the Einstein field equations induces an electromagnetic field, which affects the energy content, and thus the solution. One could, in principle, solve both the Einstein field equations with (27), and (12). This could be done using an iterative approximation, putting the induced field of a solution into a new equation and solving again repeatedly. This process, if convergent, would yield an exact solution of the complicated equations. Tµν = 3 Conclusions In this paper we have showed that in general relativity the gravitational field induces an electromagnetic field. This, in conjunction with the presence of the electromagnetic field in the energy-stress tensor, shows that the electromagnetic field is dynamically linked to the gravitational. This was shown by examining the homogeneous Maxwell’s equations in a gravitational field (4) which is a geodesic-like equation, showing what the electromagnetic field looks like in curved-spacetime. This equation was solved generally to yield (12), expressing the dependence of the induced field the connection. In the solution (12), we found an ambiguity in the choice of path. To resolve this ambiguity we used the expression for the electromagnetic field in the geodesic-Lorentz equation, to obtain an integro-differential equation for the path. The equation for the path was solved in harmonic coordinates under the assumption that the higher order terms of (12) become small to obtain (25). 6 4 Acknowledgements I’d like to thank Lawrence Horwitz for his invaluable advice, his careful readings of each draft, and the comments and suggestions made throughout the writing of this paper. References [1] G. Fucci and I. G. Avramidi. On the Gravitationally Induced Schwinger Mechanism. In K. A. Milton and M. Bordag, editors, Quantum Field Theory Under the Influence of External Conditions (QFEXT09), pages 485–491, April 2010. [2] Mark Johnston, Remo Ruffini, and Frank Zerilli. Gravitationally induced electromagnetic radiation. Physical Review Letters, 31(21):1317, 1973. [3] Jerzy Plebanski. Electromagnetic waves in gravitational fields. Physical Review, 118(5):1396, 1960. [4] K. Rosquist. Gravitationally induced electromagnetism at the Compton scale. Classical and Quantum Gravity, 23:3111–3122, May 2006. [5] Robert M Wald. General relativity. University of Chicago press, 2010. [6] Steven Weinberg. The quantum theory of fields, volume 2. Cambridge university press, 1995. [7] Steven Weinberg. Gravitation and cosmology: principles and applications of the general theory of relativity. Wiley, 2014. 7