Gravitomagnetic Effects in the Kerr-Newman Spacetime

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1 Advaned Studies in Theoretial Physis Vol. 10, 2016, no. 2, HIKARI Ltd, Gravitomagneti Effets in the Kerr-Newman Spaetime A. Barros Centro de Desenvolvimento Sustentável do Semiárido Universidade Federal de Campina Grande, CEP Sumé, PB, Brazil Copyright 2016 A. Barros. This artile is distributed under the Creative Commons Attribution Liense, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited. Abstrat In this work we onsider gravitomagneti effets in the ontext of the Kerr-Newman solution of the General Relativity theory. Firstly, the gravitoeletri and gravitomagneti fields are defined with the aid of the expression of the gravitational fore, whih is a Lorentz-type fore. Then, as an appliation, we study the frame dragging effet, the light defletion and the gravitomagneti time delay, exhibiting the eletri harge ontribution in eah ase and omparing the results obtained with those predited in the Kerr spaetime. PACS: q, Nx Keywords: Gravitomagnetism, Kerr-Newman spaetime, Weak field approximation 1 Introdution In the ontext of the General Relativity theory, using the weak field approximation and onsidering a material soure with low rotating motion, we an work in the gravitoeletromagneti framework, a formal analogy with the eletrodynamis. In this analogy, the rotation of a mass reates the gravitomagneti field, while the rest mass only generates the gravitoeletri field. Consequently,

2 82 A. Barros one defines the gravitomagneti field in the setting of the Kerr metri, whih desribes the urved spaetime geometry around a rotating mass [12]. Indeed, the rotation of the Earth produes a gravitomagneti field that auses a preession in gyrosopes orbiting around the planet. This effet, alled the Lense-Thirring effet, was verified by the GP-B experiment with an auray of 19% [6]. In reent years, several aspets of gravitomagnetism have been studied, taking into aount effets produed by the gravitational field of rotating astronomial soures [13, 2, 10]. In this paper, we onsider the spaetime of a rotating, eletrially harged body, whih an represent a Kerr-Newman blak hole [1, 11]. Initially, we define the gravitoeletri and gravitomagneti fields in the Kerr-Newman spaetime, exhibiting their dependene in relation to the eletri harge of the soure. To this end, we alulate the Lorentz-type gravitational fore produed by the entral body. Thereafter, we investigate gravitomagneti effets as frame dragging and gravitomagneti time delay; also, we examine if the gravitomagneti field ontributes to the light defletion. The results obtained are ompared with those predited in the Kerr solution ontext. The gravitoeletri and gravitomagneti fields are defined in the next setion and the expressions are utilized in Setion 3 to investigate the frame dragging and light defletion. Then, in Setion 4, the study of gravitomagneti time delay is developed. Finally, Setion 5 is devoted to our onlusions. 2 Gravitoeletri and gravitomagneti fields Let us onsider the Kerr-Newman metri, that desribes the gravitational field of a entral mass M rotating with angular momentum and eletri harge Q. The line element of Kerr-Newman, taking into aount: i the weak field approximation onditions GM 1 and GQ2 1, 1 4 r 2 where G is the gravitational onstant and is the speed of light in free spae, and ii a loalized and slowly rotating soure that satisfies the relation 1, 2 Mr is given by [4] ds 2 = 1 2GM + GQ2 dt GM GQ2 dr 2 4 r 2 4 r 2 +r 2 dθ 2 + sen 2 θdϕ 2 4G 1 Q2 senθ 3 r 2 2 Mr r senθdϕ dt. 3

3 Gravitomagneti effets in the Kerr-Newman spaetime 83 It is useful to introdue the transformation r = r 1 + GM GQ2. Thus, 4 4 r 2 we obtain the expression of equation 3 in isotropi oordinates: ds 2 = 1 2GM + GQ2 dt GM 4 r 2 GQ2 2 4 r 2 [dr 2 + r 2 dθ 2 + sen 2 θdϕ 2 ] 4G 1 Q2 senθ 3 r 2 2 Mr rsenθdϕ dt. 4 In this way, for a Cartesian-like oordinate system x µ = t, r with r = x, y, z and µ = 0, 1, 2, 3, we have the line element ds 2 = 1 2GM + GQ2 dt GM 4 r 2 GQ2 δ 2 4 r 2 i dx i dx 4 1 Q2 2 Mr being A 0 = G r r and = ẑ. 3 The equation 5 an be written as ds 2 = 1 2Φ 1 dt 2 + A 0 d x dt, Φ 2 δ i dx i dx 4 A d x dt, 6 where we define Φ 1 = GM r GQ2 2 2, 7 Φ 2 = GM GQ2 r 4, 8 2 A = 1 Q2 A Mr To first order in Φ 1, Φ 2 and A, the Lagrangian for the motion of a test partile of mass m is 1/2 L = mds/dt = m 1 v2 + mγφ 1 + mγ v2 Φ 2 2 2m γ v A, 10 where γ = 1/ 1 v 2 /. Considering that in the presene of a weak gravitational field the partile has a small veloity v 2 / 1 [9], one finds 1/2 L = m 1 v2 + mφ 1 2m v A. 11

4 84 A. Barros This equation is analogous to the eletromagneti ase [7], so that the gravitational fore F = d p /dt, with p = γm v, takes the form F = m E 2m v B, 12 where the gravitoeletri field E and the gravitomagneti field B are expressed by GM E = Φ1 = GQ2 r, 13 r 2 3 [ B = A = B 0 Q2 G 4 r r 2 ], 14 2 Mr r 3 being B 0 = A 0 = G [ 3 r r ] r 3 15 the usual gravitomagneti field of a rotating mass [4]. If = 0, we have that the ontribution for gravitational effets is only due to the gravitoeletri field, given by 13, orresponding to the Reissner-Nordström spaetime [3]. On the other hand, equation 14 expliitly shows the dependene of the gravitomagneti field with the eletrial harge of the rotating mass. 3 Frame dragging and light defletion Now, onsidering 14, we an obtain the following expression to the angular veloity of preession of gyrosopes relative to distant stars [4] [ B Ω = = Ω 0 Q2 G 4 r r 2 ], 16 2 Mr 3 with B 0 Ω 0 =. This preession is equivalent to a dragging of inertial frames aused by the gravitomagneti field. The result obtained shows the eletri harge ontribution to the gravitational effet of frame dragging [14]. An expression for the light defletion angle by a Kerr-Newman mass in the equatorial plane, whih onsiders higher order terms of M, and Q 2, was reently alulated by Chakraborty and Sen [3, equation 34]. From their formula, we an apply the approximations of weak field and slow rotation of the soure. In this ase, the defletion angle stays α = 1 [3π 4 Φ 1R 3π 8Φ 2 R], 17

5 Gravitomagneti effets in the Kerr-Newman spaetime 85 where Φ 1 R and Φ 2 R are the gravitoeletri potentials defined by 7 and 8, being R the distane of losest approah. Therefore, taking into aount the approximations adopted, the gravitomagneti field does not influene the light defletion phenomenon. 4 Gravitomagneti time delay Let us onsider a ray of eletromagneti radiation that propagates from a point : t 1, r 1 to a point P 2 : t 2, r 2 in the spaetime given by 6, where g µν = η µν + h µν and η µν = diag 1, 1, 1, 1 is the Minkowski metri. The total time spent by the ray in the path is [5] t2 t 1 dt = 1 d r h µν k µ k ν dl, 18 where k µ = 1, ˆk, ˆk is the onstant unit propagation vetor of the signal and dl = d r = δ i dx i dx 1/2 denotes the Eulidean length element along the straight line that oins to P 2. In turn, the gravitational time delay is defined as = 1 2 and the gravitomagneti time delay is expressed by Using 6, we have B = 1 B = 1 h µν k µ k ν dl 19 with kdl = d r. One an now write that B = 2 3 h 0i k 0 k i dl. 20 [ 2 ] 1 Q2 A d r, 21 Mr A 0 d r + Q2 P2 A 0 5 M r d r. 22 The above equation exhibits expliitly the dependene of the gravitomagneti time delay with the eletri harge. On the other hand, if Q = 0 we reover the known expression in the Kerr spaetime [5]. It is interesting to note that the gravitomagneti time delay ould have a notieable partiipation in the gravitational lensing delay of extragalati soures, so that should be onsidered in the analysis of observational data [8].

6 86 A. Barros 5 Conlusion We study the gravitoeletromagneti formalism in the ontext of the Kerr- Newman spaetime. In this sense, the expression of the gravitational fore was obtained and, then, we define the gravitomagneti field, whih inludes the eletri harge ontribution for gravitational effets. In sequene, we examine some gravitomagneti effets, suh as frame dragging and gravitomagneti time delay, exhibiting the eletri harge terms in eah ase. In the partiular ase of light defletion, we verify that only gravitoeletri potentials affet the defletion angle. Referenes [1] T. Adamo and E. T. Newman, The Kerr-Newman metri: a review, Sholarpedia, , [2] D. Bini, L. Iorio and D. Giordano, Orbital effets due to gravitational indution, Gen. Rel. Grav., , [3] S. Chakraborty and A. K. Sen, Light defletion due to a harged, rotating body, Class. Quantum Grav., , [4] I. Ciufolini and J. A. Wheeler, Gravitation and Inertia, Prineton University Press, Prineton, [5] I. Ciufolini, S. Kopeikin, B. Mashhoon and F. Rii, On the gravitomagneti time delay, Phys. Lett. A, , [6] C. W. F. Everitt, D. B. DeBra, B. W. Parkinson et al, Gravity Probe B: final results of a spae experiment to test General Relativity, Phys. Rev. Lett., , [7] J. D. Jakson, Classial Eletrodynamis, Wiley, New York, [8] S. Kopeikin and B. Mashhoon, Gravitomagneti effets in the propagation of eletromagneti waves in variable gravitational fields of arbitrary-moving and spinning bodies, Phys. Rev. D, ,

7 Gravitomagneti effets in the Kerr-Newman spaetime 87 [9] L. Landau and E. Lifshitz, The Classial Theory of Fields, Elsevier Butterworth-Heinemann, Oxford, [10] P. Landry and E. Poisson, Gravitomagneti response of an irrotational body to an applied tidal field, Phys. Rev. D, , [11] W. Lin and C. Jiang, Exat and unique metri for Kerr-Newman blak hole in harmoni oordinates, Phys. Rev. D, , [12] B. Mashhoon, in The Measurement of Gravitomagnetism: A Challenging Enterprise, edited by L. Iorio, Nova Siene, New York, 2007, [arxiv:gr-q/ ] [13] B. Mashhoon, Time-varying gravitomagnetism, Class. Quantum Grav., , [14] B. M. Mirza, Does harge ontribute to the frame dragging of spaetime?, Int. J. Theor. Phys., , Reeived: January 7, 2016; Published: February 19, 2016