Deflection of light due to rotating mass a comparison among the results of different approaches

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1 Jounal of Physics: Confeence Seies OPEN ACCESS Deflection of light due to otating mass a compaison among the esults of diffeent appoaches Recent citations - Gavitational Theoies nea the Galactic Cente Sanjeev Kalita To cite this aticle: Saani Chakaoty and Asoke Kuma Sen 1 J. Phys.: Conf. Se View the aticle online fo updates and enhancements. This content was downloaded fom IP addess on 17/1/18 at 18:5

2 National Confeence on Contempoay Issues in High Enegy Physics and Cosmology IOP Pulishing Jounal of Physics: Confeence Seies 81 (1) 18 doi:1.188/ /81/1/18 Deflection of light due to otating mass - a compaison among the esults of diffeent appoaches Saani Chakaoty and Asoke kuma Sen Depatment of Physics, Assam Univesity, Silcha, Assam, India saani.cool88@gmail.com Astact. It is known that light gets deflected due to mass. Value of deflection of light fo static ody is not same as it is in the case of otating ody. The deflection fo a static ody entiely depends on the gavitational mass whee in case of a otating ody new tems will e included due to otation. The ending angle of light is also not same in the equatoial plane and non equatoial plane fo otating Ke ody. The light ending angle is also diection of motion dependent i.e. if the motion of the light ay is in the diection of otation, ending angle is geate than the static case and if the ay is in the opposite diection of otation, the ending angle is smalle than the static case in equatoial plane. Thee ae two appoaches to otain the ending angle, null geodesic of photon and change of effective efactive index. In this pape a compaison will e made among the esults of diffeent appoaches. 1. Intoduction Geneal elativity, a theoy of gavitation was developed y Einstein fom 197 and finally fomulated it in Two vey impotant consequences of geneal elativity ae ending of light ay in in pesence of gavitational field and otating mass dag the space time aound them, a phenomenon called fame dagging. The exact solution of Einstein s field equation fo a static, unchaged ody was found y Schwazschild in 1915, fo an unchaged otating ody was solved y Ke in 196 and fo a otating chaged ody was found y Newman which is known as Ke-Newman metic [1, ]. So thee ae thee factos namely mass, angula momentum and chage which can influence the cuvatue of space time. As a diect consequence of it, the amount of deflection of light ay would also e influenced y these factos. Many woks have een done on the deflection of light due to static, otating and chaged otating ody y using two appoaches, null geodesic of photon appoach and effective efactive index of mateial medium appoach. In this pape a compaison will e made among the esults of diffeent appoaches to otain the ending angle fo static and otating mass.. Bending of light due to static mass using null geodesic appoach In the yea of 1911, a pediction was pulished y Einstein fo ending angle of light ay fom a distance souce passing close to the sun at minimum distance R would get deflected with an angle. GM α =, (1) c R Content fom this wok may e used unde the tems of the Ceative Commons Attiution. licence. Any futhe distiution of this wok must maintain attiution to the autho(s) and the title of the wok, jounal citation and DOI. Pulished unde licence y IOP Pulishing Ltd 1

3 National Confeence on Contempoay Issues in High Enegy Physics and Cosmology IOP Pulishing Jounal of Physics: Confeence Seies 81 (1) 18 doi:1.188/ /81/1/18 whee G is gavitational constant, M is the mass of sun and c is the speed of light. The osevational veification of his pediction was made in the yea 1919 duing total sola eclipse. Fo the sun the angle of deflection of light was 1.75 sec. [1, ]. The light deflection due to a static chaged ody was otained y Vihada, Naasimhe & Chite up to second ode tem [] α = GM + G M 15π c C 16 % '+. 1 GM ( GM % + ' + q * - q π /, () & c )* c &,- 8 1 whee q is the chage of the ody, the ending angle fo static ody would e otained y setting q equals zeo α = GM + G M 15π c C 16 % '+ G M () & C Highe ode tems otained y Keeton & Pette [5] α = h π % 'h + 1 & 15π % 'h π % 'h & 6 & 1 65π % 'h π % 'h 6 + Ο(h) 7 () 16 & 6 56 & Whee, m h = =closest appoach and GM m = c In tems of impact paamete, α() = m &+ 15π m m 5 & % & π % m 6 m & + Ο m & % & + 65π % 6 7 m & % (5) An analytical petuation fame wok was developed y Iye & Pette [6] to otain the ending angle of light due static lack hole which is α = h π % 'h + 1 & 15π % 'h π % 'h & 6 & 1 65π % 'h π % 'h 6 + Ο(h) 7 (6) 16 & 6 56 & Equation (6) is simila with equation ().1. Bending of light due to static mass using mateial medium appoach

4 National Confeence on Contempoay Issues in High Enegy Physics and Cosmology IOP Pulishing Jounal of Physics: Confeence Seies 81 (1) 18 doi:1.188/ /81/1/18 Fishack & Feeman otained the ending angle of light up to fist ode tem fo static ody using effective efactive index of mateial medium appoach [] ( ϕ) 1 (1 + γ ) GM = (7) c Fo γ =1 this expession is simila to Einsteins expession of ending angle. Moe exact expession fo gavitational deflection of light had een given y Sen (1) [9] using the mateial medium appoach and the deflection angle fo static mass was otained as g ) Δφ = & - + x 1 + * % ( x (1 x). dx (8) g ( g ) % g ( g ),+ '( /+ Whee x = g / and =closest appoach and fo weak field this expession educe to standad fom 1 GM Δϕ = (9) c It can e said that the esults ae simila fo oth null geodesic of photon and effective efactive index of mateial medium appoach.. Bending angle of light due to otating mass. In geneal elativity the space time aound a ody is descied y Ke metic. Accoding to this metic a otating ody dag the space-time aound it and evey oject coming close to the otating ody entained to paticipate in its otation. This effect is called fame dagging. The otational fame dagging effect was fist deived fom the theoy of geneal elativity y two Austian physicists Josef Lense and Hans Thiing [1,], which was known as Lense-Thiing effect. The deflection poduced in pesence of a otating lack hole explicitly depends on diection of light. Compaed to the zeo spin Schwazschild case, the ending angle was geate fo diect oit and smalle fo etogade oits. Exact ending angle fo otating lack hole in the equatoial plane otained y Iye & Hansen is [7, 8]. α() = m &+ 15π sa & m & πsa + a & m & + % % 65π 19sa + 85π 6 16 a sa & m & % (1) whee s is +1 fo diect and -1 fo etogade motion and j a = Mc and j is the angula momentum. Azami, Keeton & Pette [1, 11] have shown that fo off equatoial light ay ending angle has two components. One is in the equatoial plane and anothe is pependicula to the equatoial plane. Equatoial component is

5 National Confeence on Contempoay Issues in High Enegy Physics and Cosmology IOP Pulishing Jounal of Physics: Confeence Seies 81 (1) 18 doi:1.188/ /81/1/18 α() = m &+ 15π sa & m & πsa + a & m & + % % 65π 19sa + 85π 6 16 a sa & m & % (11) This expession in equation (11) is simila to the expession otained y Iye and Hansen [7] shown in equation (1). Both expession educe to Schwazschild seies if a is set to zeo.. Conclusions Fom aove discussion thee points ae clea that whateve is the method (null geodesic of photon o effective efactive index of mateial medium) the expession of light deflection is same in case of static gavitational mass. Equatoial component of quasi-equatoial ending angle is simila to equatoial ending angle and fo zeo spin it educes to the ending angle expession fo static ody. Spin does not have any contiution to the fist ode tem. Refeences [1] Landau L D and Lifshitz E M 198 The Classical Theoy of Fields volume th edition Buttewoth-Heinemann [] Weineg S 197 Pinciples and Applications of the Geneal Theoy of Relativity John Wiely & Sons Inc. [] Fishack E and Feeman B S 198 Phys. Rev. D 1 [] Vihada K S, Naasimha D and Chite S M 1998 Aston. Astophys [5] Keeton C R and Pettes A O 5 Phys. Rev. D 7 16 [6] Iye S V and Pettes A O 7 Gen. Relativ. Gavit [7] Iye S V and Hansen E C 9 Phys. Rev. D 8 1 [8] Iye S V and Hansen E C 9 axiv:g-qc/98.85 [9] Sen A K 1 Astophysics 5 [1] Aazami A B, Keeton C R and Pettes A O 11 J. Math. Phys [11] Aazami A B, Keeton C R and Pettes A O 11 J. Math. Phys