Exact gravitational lensing and rotation curve
aa r X i v : . [ a s t r o - ph . GA ] D ec Exact gravitational lensing and rotation curve
G¨unter Scharf and Gerhard Br¨aunlich ∗ Institut f¨ur Theoretische Physik,Universit¨at Z¨urich,Winterthurerstr. 190 , CH-8057 Z¨urich, Switzerland
Abstract
Based on the geodesic equation in a static spherically symmetricmetric we discuss the rotation curve and gravitational lensing. Therotation curve determines one function in the metric without assum-ing Einstein’s equations. Then lensing is considered in the weak fieldapproximation of general relativity. From the null geodesics we derivethe lensing equation and corrections to it. ∗ e-mail: [email protected] Introduction
As long as the dark matter problem is open there is a non-zero probabilitythat general relativity might not hold on the scale of galaxies [1-4]. Thereforea direct test on this scale is highly desired. It is the purpose of this paper toshow how such a test is possible, if kinematical and lensing data of the galaxyare available. The idea is the following: the rotation curve determines partof the metric without assuming Einstein’s field equations, only the geodesicequation is used. Then lensing can be calculated on the basis of the weakfield approximation to general relativity and checked for consistency. Incontrast to the usual way of analyzing the data no model for the galaxymust be constructed. This offers the possibility to test the basic physics.We treat lensing by means of the geodesic equation as well. By comput-ing the null geodesics we derive the lensing equation and we find correctionsto it. Even if these corrections were not needed for the analysis of presentday lensing data, they have to be under control for all eventualities. Weonly consider the static spherically symmetric case here in order to makethe argument as simple as possible.
We consider a static spherically symmetric metric which we write in theform ds = e ν c dt − e λ dr − r ( dϑ + sin ϑdϕ ) (2 . ν and λ are functions of r only. We take the coordinates x = ct , x = r , x = ϑ , x = ϕ such that g = e ν , g = − e λ g = − r , g = − r sin ϑ (2 . g = det g µν = − e ν + λ r sin ϑ. (2 . = 12 ν ′ , Γ = 12 ν ′ e ν − λ , Γ = 12 λ ′ , Γ = − re − λ (2 . = − re − λ sin ϑ, Γ = 1 r , Γ = − sin ϑ cos ϑ, Γ = 1 r , Γ = cot ϑ and zero otherwise, the prime denotes the derivative with respect to r always.The geodesic equation is given by d x α ds + Γ αβγ dx β ds dx γ ds = 0 . (2 . θ = π/
2, then we must solve the following threeequations d ctds + ν ′ d ctds drds = 0 (2 . d rds + ν ′ e ν − λ (cid:16) d ctds (cid:17) + λ ′ (cid:16) drds (cid:17) − re − λ (cid:16) dϕds (cid:17) = 0 (2 . d ϕds + 2 r drds dϕds = 0 . (2 . ν we find ∂ds (cid:16) e ν d ctds (cid:17) = 0so that e ν d ctds = const . = ad ctds = ae − ν . (2 . r we get r dϕds = const . = J where J is essentially the conserved angular momentum, hence dϕds = Jr . (2 . λ ) × dr/ds we obtain dds h e λ (cid:16) drds (cid:17) − a e − ν + J r i = 0 . (2 . b . Thenthe resulting differential equation can be written as (cid:16) drds (cid:17) = a e − ( λ + ν ) + e − λ (cid:16) b − J r (cid:17) . (2 . u α = dx α ds . The term unitary indicates that u α has invariant length 1: u = g αβ dx α ds dx β ds = ( ds ) ( ds ) = 1 . (2 . u α is equal to u α = (cid:16) d ctds , drds , , dϕds (cid:17) . (2 . u = − b = 1 (2 . b = − a must be related to the ge-ometry of the geodesics. To see this we consider the streamlines r = r ( ϕ ).Deviding (2.12) by J = r dϕ/ds we obtain (cid:16) r drdϕ (cid:17) = a J e − ( λ + ν ) + e − λ (cid:16) bJ − r (cid:17) . (2 . w ( ϕ ) = 1 r ( ϕ ) , (2 . (cid:16) dwdϕ (cid:17) = a J e − ( λ + ν ) + e − λ (cid:16) bJ − w (cid:17) . (2 . r : e − ( λ + ν ) = 1 + O ( r − ) , e − λ = 1 − r s r + O ( r − ) . r s = 2 GMc (2 . /r wehave (cid:16) dwdϕ (cid:17) + w = a + bJ − r S w (cid:16) bJ − w (cid:17) . (2 . w = 1 r = 1 p (1 + e cos ϕ ) , (2 . p and e are parameter and eccentricity of the ellipse. p is connectedwith the non-relativistic angular momentum ˜ J by˜ J p = GM. (2 . (cid:16) d ˜ wdϕ (cid:17) + ˜ w = e − p + 2 p ˜ w. (2 . − r S bJ = 2 p . By (2.22) and ˜ J = cJ this gives b = − a + bJ = e − p = − p ˜ a , where ˜ a is the big half-axis of the ellipse, we obtain by (2.22) a = 1 − GMc ˜ a = 1 − r S a . (2 . a is connected with the big half-axis of the Kepler ellipse.The 3-velocity ~v which is measured by astronomers is defined as ~v = (cid:16) dx dt , dx dt , dx dt (cid:17) . (2 . dsdt = ca e ν (2 . ~v = ~u (cid:16) dsdt (cid:17) , (2 . ~u is the spatial part in (2.13). Since our metric is diagonal it issimply given by − ~u = X j =1 g jj dx j ds dx j ds = 1 − a e − ν . (2 . ~v = c (cid:16) e ν − e ν a (cid:17) . (2 . r ≫ r S .Assuming circular motion (˜ a = r ) and using 1 /a = 1 + GM/c r we find ~v → c (cid:16) − ν ( r ) − GMc r + O ( r − ) (cid:17) . (2 . ν = − r s /r we arrive at ~v → GMr , (2 . M is the total mass (normal plus dark). This agrees with Newtonian dynam-ics (Kepler’s third law). Summing up, the relation between observationalquantities and theory is very direct. The rotation curve v ( r ) gives the metricfunction ν ( r ) by solving the quadratic equation (2.29) e ν = a (cid:16) ± s − a v c (cid:17) . (2 . v ≪ c and r ≫ r s this simplifies to e ν = 12 ± s − v c . (2 . Null geodesics and lensing
In the case of null geodesics describing light rays the integration constant b in the geodesic equation (2.16) must be 01 r (cid:16) drdϕ (cid:17) = e − λ (cid:16) a J e − ν − r (cid:17) . (3 . e − ν = 1 − U ( r ) c , e − λ = 1 + 2 U ( r ) c ≈ e ν , (3 . U ( r ) is the gravitational potential. The latter can be obtained fromthe rotation velocity according to (2.32) or (2.33). Expanding the squareroot in (2.33) for v ≪ c we get the very simple result U ( r ) = − v c , (3 . d = Ja (3 . (cid:16) drdϕ (cid:17) = r d (cid:16) − U c (cid:17) − (cid:16) Uc (cid:17) r . (3 . d becomes clear when we consider the trivial solution for U = 0: r = d sin ϕ . It describes a straight line with distance d from the origin in polar coor-dinates (fig.1). After inversion the equation (3.5) can simply be solved byquadrature: dϕdr = ± r q r d (1 − U c ) − − Uc (3 . U ( r ) = − GMr (3 . ϕ ( r ): ϕ ( r ) − ϕ = ± d r Z r dr p r − r ( d + r s ) + rr s d , (3 . r s (2.19). To reduce thisintegral to Legendre’s normal form we need the four zeros a , a , a , a ofthe quartic under the square root. We have the following four real roots a = d (cid:16)s ε − ε (cid:17) , a = εd, (3 . a = 0 , a = d (cid:16) − s ε − ε (cid:17) , (3 . ε = r s d (3 . ε : a = d (1 − ε ε , a = εd, a = 0 , a = − d (1 + ε ε . O ( ε ). The integral (3.8) is an incomplete elliptic integral of the firstkind F (Φ , k ) where the parameter k is given by k = ( a − a )( a − a )( a − a )( a − a ) = 2 ε (1 − ε ) (3 . M ,polar axis goes from M to the observer (see fig.). We integrate (3.8) fromthe apex r to infinity which gives us the deflection angle ϕ ∞ − π/
2. Theapex is defined by the condition drdϕ = 0which gives r = a . Then we obtain ϕ ∞ − π µdF (Φ ∞ , k ) , (3 . µ is equal to µ = 2 p ( a − a )( a − a ) = 2 d (cid:16) − ε ε (cid:17) (3 . ∞ follows fromsin Φ ∞ = a − a a − a = 12 (cid:16) ε − ε (cid:17) = 12 (1 − cos(2Φ ∞ )) (3 . k asfollows F (Φ , k ) = Φ + k −
12 sin 2Φ) + O ( k ) (3 . ϕ ∞ = − ε + ε (cid:16) π − (cid:17) . (3 . ε = r s /d is Einstein’s result.Next we want to derive the lens equation. In this problem the observeris not at infinity but in a finite distance D d from the lens. The light sourceis at a distance D ds at the other side of the lens and an amount η off theoptical axis (see fig.); we use the same notation as in [5]. We have nowto compute the null geodesics from the source at distance D ds through theapex r = r , ϑ = π/ D d . Then the polar angle˜ β = ηD ds of the source follows from π − ˜ β = µd [ F (Φ ds , k ) + F (Φ d , k )] . (3 . d is given bysin Φ d = a − a a − a D d − a D d − a == 12 (1 − cos(2Φ d )) = 12 h ε − Θ(1 − ε ) + Θ ε i (3 . d/D d is the angle under which the observer sees the source. Thenfrom (3.17) we obtain F (Φ d , k ) = π ε (cid:16)
12 + π (cid:17) − Θ2 − ε Θ4 + ε (cid:16) − π (cid:17) . (3 . (Φ ds , k ) is given by the same formula with Θ substituted by α = d/D ds .Now we find from (3.19) − ˜ β = − Θ − α + 2 ε + ε (cid:16) − π (cid:17) + O ( ε ) . (3 . β = ˜ β D ds D s , Θ = r s D d ε , α = Θ D d D ds . (3 . β = Θ − D ds D s r s D d Θ − (cid:16) − π (cid:17) D ds D s (cid:16) r s D d Θ (cid:17) (3 . ϕ ( r ) − π ± d r Z r drr p r (1 − u ) − (1 + u ) d , (3 . u ( r ) = 2 c U ( r ) . (3 . r is the apex and it is important to note that only the potential valuesfor r ≥ r contribute. For u = 0 the trivial lens equation β = Θ comes out,this follows from (3.24) for r s = 0. For u = 0 but | u | ≪ u ( r ) in the vicinity of r = r . For thispurpose we use the beginning of the multipole expansion u ( r ) = c + c r + c r . (3 . r because we need r ≥ r only.With the three terms in (3.27) we get an elliptic integral of the first kindagain: ϕ ( r ) − π ± d r Z r dr p G ( r ) = ± µd q − c F (Φ , k ) . (3 . G ( r ) = (1 − c ) r − c c r − ( d + c d + c +2 c c ) r − ( c d +2 c c ) r − c d − c , (3 . µ and the parameter k are the same as before (3.13) (3.15).The four zeros of G ( r ) are obtained by solving the two quadratic equations1 + u ( r ) = 0 , r (1 − u ( r )) − d = 0 . This leads to a = c − c ) + s d + c − c + c − c ) a = − c c ) + s c c ) − c c a = − c c ) − s c c ) − c c (3 . a = c − c ) − s d + c − c + c − c ) . Then the exact lens equation is contained in the analogous equation to (3.19) π − ˜ β = µd q − c [ F (Φ ds , k ) + F (Φ d , k )] . (3 . d , Φ ds are given by the same formula (3.20) as before. Theappropriate expansion of the lens equation (3.31) depends on the particularvalues c , c , c in (3.27).Regarding applications of our results one must replace the euclideandistances by angular diameter distances as usual. Galaxies with joint lensingand dynamical data can be found in the Sloan Lens ACS Survey (SLACS)11nd its follow-up project [8]. Unfortunately, until today only one systemSDSSJ 2321-097 has been analyzed in detail. This is an early-type ellipticgalaxy which cannot be approximated by a spherically symmetric metric.So we must extend our model-independent analysis the the elliptical case orhope that the astronomers come up with a E0 lens galaxy. References [1] M. Milgrom, MNRAS (2001) 126[2] P.D. Mannheim, ApJ (1992) 429, astro-ph/0505266v2 (2005)[3] P. Horava, Phys. Rev. D 79:084008 (2009)[4] G. Scharf, From massive gravity to modified general relativity,Gen.Relativ.Gravit.(2009) DOI 10.1007/s10714-009-0864-0[5] A. Erdelyi et al., Higher transcendental functions, McGraw-Hill BookCo., Inc., New York,N.Y. 1953[6] P. Schneider, C. Kochanek, J. Wambsganss, Gravitational lensing:strong, weak and micro, Springer Verlag, Berlin Heidelberg 2006[7] P. Schneider, J. Ehlers, E.E. Falco, Gravitational lenses, Springer Verlag,Berlin Heidelberg 1992[8] O. Czoske, M. Barnab`e, L.V.E. Koopmans, T. Treu, A.S. Bolton, MN-RAS384