Gravitational lensing by spherically symmetric lenses with angular momentum

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1 Astronomy & Astrophysis manusript no. will be inserted by hand later) Gravitational lensing by spherially symmetri lenses with angular momentum M. Sereno,, V.F. Cardone,3 Dipartimento di Sienze Fisihe, Università degli Studi di Napoli Federio II, Via Cinthia, Compl. Univ. Monte S. Angelo, 806 Napoli, Italia Istituto Nazionale di Fisia Nuleare, Sez. Napoli, Via Cinthia, Compl. Univ. Monte S. Angelo, 806 Napoli, Italia 3 Dipartimento di Fisia E.R. Caianiello, Università di Salerno, Via S. Allende, 8408 Baronissi, Salerno, Italia Reeived / Aepted Abstrat. Following Sereno 00a), we disuss the bending of light rays by spherially symmetri lenses with angular momentum. For several astrophysial systems, suh as white dwarfs and galaxies, gravitomagnetism indues a orretion on the defletion angle as large as 0.%. Key words. astrometry stars: rotation osmology: theory gravitational lensing. Introdution Gravitational lensing is one of the best investigated phenomena of gravitation. In the framework of general relativity, its lowest-order preditions have been onfirmed by observative astrophysis on very different sales. On the other hand, the impressive development of tehnial apabilities demands for a full treatment of lensing theory to any order of approximation. The study of higher order perturbative terms is the link between weak and strong regimes of the theory. Mass-energy urrents relative to other masses generate spae-time urvature. This phenomenon, known as intrinsi gravitomagnetism, is a new feature of general relativity and other oneivable alternative metri theories of gravity and annot be dedued by a motion on a stati bakground for a detailed disussion on gravitomagnetism we refer to Ciufolini & Wheeler 995)). In partiular, the effet of the angular momentum of the Send offprint requests to: M. Sereno, sereno@na.infn.it

2 M. Sereno, V.F. Cardone: Gravitational lensing and gravitomagnetism defletor has been studied by several authors Epstein & Shapiro 980; Ibáñez & Martín 98; Ibáñez 983; Dymnikova 986; Gliestein 999; Sereno 00a). One of us Sereno 00a) showed as the gravitomagneti orretion to the lensing quantities an be evaluated in the usual framework of lensing theory see also Capozziello et al. 999), i.e. i) weak field and slow motion approximation for the lens; ii) thin lens hypothesis Shneider et al. 99; Petters et al. 00). In this letter, we onsider the gravitomagneti ontribution to the defletion angle for extended gravitational lenses of astrophysial interest. We disuss spherially symmetri defletors.. The defletion angle As derived in Sereno 00a), the defletion angle, to the order 3,is α) 4G ) d Σ ) v e in l ) R ; ) is the bi-dimensional position vetor in the lens plane, orthogonal to the line of sight l; e in is the inoming light ray diretion; G is the Newton s onstant of gravitation; is the speed of light. Σ is the surfae mass density projeted along the line of sight Σ) ρ,l) dl; ) v e in l is the weighted average, along the line of sight, of the omponent of the veloity v of the mass element orthogonal to the lens plane, v,l) ein ) ρ,l) dl v e in l ). 3) Σ) Let us onsider a spherially symmetri lens that rotates about an arbitrary axis, ˆη, passing through its enter i.e. a main axis of inertia). To speify the orientation of the rotation axis, we need two Euler s angles: α is the angle between ˆη and the -axis; β is the angle between the line of sight ˆl and the line of nodes defined at the intersetion of the l ˆ plane and the equatorial plane i.e., the plane orthogonal to the rotation axis and ontaining the lens enter). Using the axial symmetry about the rotation axis, we find v e in,,l)= ωr)[ os α + sin α os β] ω R) + ω R), 4) where ωr) is the modulus of the angular veloity at a distane R R + R) / from the rotation axis; ˆR that, given the spherial symmetry of the system, an be taken along the line of nodes) and ˆR are the axes on the equatorial plane; ω and ω are the omponents of ω along, respetively, the -andthe -axes. It is R = l os β + sin β, 5) R = l os α sin β + os α os β + sin α. 6)

3 M. Sereno, V.F. Cardone: Gravitational lensing and gravitomagnetism 3 Let us assume a rigid rotation, ωr) =ω = onst. Wehave v e in l = ω + ω. 7) We an, now, evaluate the integral in Eq.); it is, α,θ) = 4G { M) os θ + I N ) ω os θ ω ) sin θ M> ) ω } ; 8) α,θ) = 4G { M) sin θ + I N ) ω os θ + ω ) sin θ + M> ) ω }. 9) and θ are polar oordinates in the lens plane; M) is the mass of the lens within, M) =π 0 Σ ) d ; 0) M> ) is the lens mass outside, M> ) M ) M); and I N ) =π 0 Σ ) 3 d ) is the momentum of inertia of the mass within about a entral axis. I N ω i is the omponent of the angular momentum along the i -axis. The gravitomagneti orretion onsists of the last two terms in Eqs.8,9), both proportional to some omponents of the angular veloity. Spherial symmetry is broken. In the first ontribution, the angular momentum appears; the seond one is proportional to the mass outside and an be signifiant for lenses with slowly dereasing mass density. 3. The homogeneous sphere The gravitational phenomena onneted to intrinsi gravitomagnetism are generated by mass-energy urrents relative to other masses. The simplest lens model, the point-like Shwarzshild lens, annot produe suh a peuliar effet sine the loal Lorentz invariane on a stati bakground does not aount for the dragging of inertial frames Ciufolini & Wheleer 995). General relativity is a lassial-nonquantized theory where the lassial angular momentum of a partile goes to zero as its size goes to zero. To onsider the gravitomagneti field, we need a further step after the point mass as a lens model, the homogeneous sphere. Let us onsider a homogeneous sphere of radius R and volume density ρ 0.Itis Σ) =ρ 0 R, if R, ) or Σ) = 0 elsewhere; M) =M TOT ) ) 3/, if R, 3) R or M) =M TOT elsewhere, M TOT = 4 3 πr3 ρ 0 ; ) ) / I N ) =IN TOT + ) 3 ) ) 4, if R, 4) R R R

4 4 M. Sereno, V.F. Cardone: Gravitational lensing and gravitomagnetism or I N ) =I TOT N elsewhere, I TOT N = 8 5 πr5 ρ 0 ; For light rays outside the lens, >R, the defletion angle redues to α,θ) = 4G { MTOT { MTOT os θ + ITOT N ω os θ ω )} sin θ, 5) α,θ) = 4G sin θ + ITOT N ω os θ + ω )} sin θ. 6) The gravitomagneti orretion is signifiant if I TOT N M TOT ω = J M TOT > 0 3, 7) where J I N ω is the angular momentum. To have a non-negligible gravitomagneti effet, the angular momentum of the lens has to be non-negligible ompared to the angular momentum of a partile of mass M TOT and veloity in a irular orbit of radius around the rotation axis. Let us onsider a lens rotating about the -axis ω =0,ω = ω) and a light ray in the equatorial plane, θ = 0. The defletion generated by the gravitomagneti field is α GRM = 4G J 3. 8) The sun bends a light ray grazing its limb, = R,by.75 arse. Given its angular momentum, J gm s Allen 983), the gravitomagneti orretion is 0.7 µarse see also Epstein & Shapiro 980)). ) 5/3 For an early type star, J =0 M J M Kraft 967). For M =.4M, R =.R and for a light ray grazing the limb, α GRM 0. milliarse, that is, a orretion of The gravitomagneti field beomes more signifiant for a fast rotating white dwarf, where J 0.GM 3 R Padmanabhan 00).For M M, R 0 R, 6R, α GRM 0.03 arse, that is, a orretion of The isothermal sphere Isothermal spheres ISs) are widely used in astrophysis to model systems on very different sales, from galaxy haloes to lusters of galaxies; also, IS an be adopted to study mirolensing by non-ompat invisible objets in the halo Sazhin et al. 996). Let us onsider an IS with a finite ore radius. The surfae density is Σ IS ) = σ v, 9) G + )/ where σ v is the veloity dispersion. We have, ) ) / M IS ) = πσ v G +, 0) I IS N ) = πσ v 3G 3 ) ) ) / + ) + ; )

5 M. Sereno, V.F. Cardone: Gravitational lensing and gravitomagnetism 5 When = 0, we have the singular isothermal sphere SIS). Then, M SIS ) = πσ v, ) G IN SIS ) = πσ v 3G 3. 3) Sine the total mass is divergent, we introdue a ut-off radius R. FortheSIS,the defletion angle redues to,θ) =4π,θ) =4π σv ) { os θ + ω [ ) ] os θ + R 3 σv ) { sin θ + ω [ ) ] os θ + R 3 ω } sin θ, 4) 3 + ω } sin θ. 5) 3 The orretion ouples kinematis, through the angular veloity, and geometry, through the ut-off radius. As an be easily seen, the gravitomagneti effet is signifiant when ω R > 0 3 ; 6) In partiular, in the inner regions R), the above equations redue to σv ) { R, θ) =4π os θ Rω }, 7) σv ) { R, θ) =4π sin θ + Rω } ; 8) the orretion derives from the mass outside the onsidered radius. We an model a typial galaxy as a SIS with σ v 00 km s, R < 50 kp and J I N R) ω 0.M kp s, as derived from numerial simulations Vitviska et al. 00). It is, ω R G ) 3 J R ) σ v The gravitomagneti orretion is quite signifiant, inreases with the ordered motion of the stars i.e., with the angular momentum) and dereases with the random proper motions i.e., with the dispersion veloity). 5. Power law models Power law models an be onsidered as a generalization of the IS Shneider et al. 99) and are often adopted to model mass distribution in lusters of galaxies by lensing inversion Sereno 00b). It is ) +p Σ PL =Σ 0 + ) ) p ; 30)

6 6 M. Sereno, V.F. Cardone: Gravitational lensing and gravitomagnetism the slope parameter p determines the softness of the mass profile of the lens. A power law model with p =/ approximates the isothermal sphere at large radius. It is, ) ] p M PL ) =πσ 0 [+, 3) I PL N ) = πσ 0 4 p + p) ) ) p ) ) ) p) + p. 3) 6. Summary and disussion We have investigated the effet of dragging of inertial frames in gravitational lensing for spherially symmetri lenses and a general expression for the defletion angle, to the order 3, has been derived. We have expliitly onsidered isothermal spheres, power law models and the homogeneous sphere. Both for galaxies and white dwarfs, the gravitomagneti orretion an be as large as 0.%. The satellite Hipparos, launhed in 989 by ESA, an measure the position of stars with auray of nearly a milliarse. New generation spae interferometri mission, suh as SIM by NASA sheduled for launh in 009), should greatly improve this auray. Measurements of defletion of eletromagneti waves ould give one of the first experimental evidenes of gravitomagnetism. Referenes Allen C.W., Astrophysial Quantities, 983, The Athlone Press, London Capozziello, S., Lambiase, G. Papini, G. Sarpetta, G., 999, Phys. Lett. A 54,. Ciufolini I., Wheeler J.A., 995, Gravitation and Inertia,Prineton University Press, Prineton Dymnikova I., 986, Relativity in Celestial Mehanis and Astrometry, eds. Kovalevsky J., Brumberg A., 4. Epstein R., Shapiro I., 980, Phys. Rev. D, 947. Glienstein J.F., 999, A&A 343, 05. Ibáñez J., 983, A&A 4, 75. Ibáñez J., Martín J., 98, Phys. Rev. D 6, 384. Kraft R.P., 967, ApJ 50, 55 Padmanabhan T., 00, Theoretial Astrophysis Vol. II, Cambridge University Press, Cambridge Petters A.O., Levine H., Wambsganss J., 00, Singularity Theory and Gravitational Lensing, Birkhäuser, Boston Sazhin M.V., Yagola A.G., Yakubov A.V., 995, Phys. Lett. A 08, 76 Shneider P., J. Ehlers J., Falo E.E., 99, Gravitational Lenses, Springer-Verlag, Berlin) Sereno M., 00a, [astro-ph/00948] Sereno M., 00b, A&A in press; [astro-ph/0090] Vitviska M., Klypin A.,Kravtsov A.V., et al., 00, ApJ, submitted; astro-ph/005349