Based on the geodesic equation in a static spherically symmetric metric we discuss the rotation curve and gravitational lensing. The rotation curve determines one function in the metric without assuming Einstein's equations. Then lensing is considered in the weak field approximation of general relativity. From the null geodesics we derive the lensing equation. The gravitational potential U(r) which determines the lensing is directly give by the rotation curve U(r)=−v 2(r). This allows to test general relativity on the scale of galaxies where dark matter is relevan

Bräunlich, Gerhard

Scharf, Günter

RERO DOC Digital Library

Gen Relativ Gravit (2011) 43:143–154DOI 10.1007/s10714-010-1077-2RESEARCH ARTICLEGravitational lensing and rotation curveGerhard Bräunlich · Günter ScharfReceived: 15 March 2010 / Accepted: 12 August 2010 / Published online: 24 August 2010© Springer Science+Business Media, LLC 2010Abstract Based on the geodesic equation in a static spherically symmetric metricwe discuss the rotation curve and gravitational lensing. The rotation curve determinesone function in the metric without assuming Einstein’s equations. Then lensing is con-sidered in the weak field approximation of general relativity. From the null geodesicswe derive the lensing equation. The gravitational potential U (r) which determinesthe lensing is directly give by the rotation curve U (r) = −v2(r). This allows to testgeneral relativity on the scale of galaxies where dark matter is relevant.Keywords Gravitational lensing · Geodesic flow1 IntroductionAs long as the dark matter problem is open there is a non-zero probability that generalrelativity might not hold on the scale of galaxies [1–5]. Therefore a direct test on thisscale is highly desired. It is the purpose of this paper to show how such a test is possible,if kinematical and lensing data of the galaxy are available. The idea is the following:the rotation curve determines the time-time component of the metric without assumingEinstein’s field equations, only the geodesic equation is used. Assuming now for thelensing the weak field approximation to general relativity where the metric is givenin terms of the Newtonian gravitational potential U (r) we find the extremely simplerelationU (r) = −v2(r). (1.1)G. Bräunlich · G. Scharf (B)Institut für Theoretische Physik, Universität Zürich, Winterthurerstr. 190,8057 Zürich, Switzerlande-mail: scharf@physik.unizh.ch123144 G. Bräunlich, G. ScharfIn contrast to the usual way of analyzing the data we get this result without assuminga matter model for the galaxy. This offers the possibility to test the basic physics in asituation where dark matter is important.We treat lensing by means of the geodesic equation as well. By computing thenull geodesics we derive the lensing equation. The most important part of the lightpath is the neighborhood of the deflector where the curvature is maximal. In standardlens theory the light path is approximated by two straight rays with a kink, but thisapproximation is particularly bad at the apex. Therefore, it is important to calculatethe exact geodesics.If the circular velocity v(r) can be measured in the vicinity of the apex, then fromthe above relation (1.1) an expansion of U (r) of the formU (r) = C0 + C1r+ C2r2(1.2)can be obtained. That is all what is needed to calculate the lensing observables (seeSect. 3 below). Comparison with observations then tests general relativity in a model-independent way (see Sect. 4).2 Geodesic flow and rotation curveWe consider a static spherically symmetric metric which we write in the formds2 = eνc2dt2 − eλdr2 − r2(dϑ2 + sin2 ϑdϕ2)(2.1)where ν and λ are functions of r only. We take the coordinates x0 = ct, x1 = r,x2 = ϑ, x3 = ϕ such thatg00 = eν, g11 = −eλg22 = −r2, g33 = −r2 sin2 ϑ (2.2)and zero otherwise. The components with upper indices are the inverse of this. Thedeterminant comes out to beg = detgμν = −eν+λr4 sin2 ϑ. (2.3)Let us recall the Christoffel symbols for the metric (2.1) from the appendix of [4]010 =12ν′, 100 =12ν′eν−λ, 111 =12λ′, 122 = −re−λ (2.4)133 = −re−λ sin2 ϑ, 212 =1r, 233 = − sin ϑ cos ϑ, 313 =1r, 323 = cot ϑ123Gravitational lensing and rotation curve 145and zero otherwise, the prime denotes the derivative with respect to r always. Thegeodesic equation is given byd2xαds2+ αβγdxβdsdxγds= 0. (2.5)The origin of our reference frame is in the center of the galaxy. We consider geodesicsin the plane θ = π/2, then we must solve the following three equationsd2ctds2+ ν′ d ctdsdrds= 0 (2.6)d2rds2+ ν′2eν−λ(d ctds)2+ λ′2(drds)2− re−λ(dϕds)2= 0 (2.7)d2ϕds2+ 2rdrdsdϕds= 0. (2.8)Multiplying (2.6) by exp ν we find∂ds(eνd ctds)= 0so thateνd ctds= const. = ad ctds= ae−ν . (2.9)Next multiplying (2.8) by r2 we getr2dϕds= const. = Jwhere J is essentially the conserved angular momentum, hencedϕds= Jr2. (2.10)Finally, substituting (2.9) and (2.10) into (2.7) and multiplying by 2(exp λ)×dr/dswe obtaindds[eλ(drds)2− a2e−ν + J2r2]= 0. (2.11)123146 G. Bräunlich, G. ScharfConsequently, the square bracket is equal to another constant = b. Then the resultingdifferential equation can be written as(drds)2= a2e−(λ+ν) + e−λ(b − J2r2). (2.12)To obtain the connection with the rotation curve which is an important astronomicalobservable, we remember the definition of the unitary 4-velocityuα = dxαds.The term unitary indicates that uα has invariant length 1:u2 = gαβ dxαdsdxβds= (ds)2(ds)2= 1. (2.13)In our case uα is equal touα =(d ctds,drds, 0,dϕds). (2.14)Using (2.9), (2.10) and (2.12) we easily see thatu2 = −b = 1 (2.15)which by (2.13) fixes the constant of integration b = −1.Clearly the last constant of integration a2 must be related to the geometry of thegeodesics. To see this we consider the streamlines r = r(ϕ). Dividing (2.12) byJ = r2dϕ/ds we obtain(1r2drdϕ)2= a2J 2e−(λ+ν) + e−λ(bJ 2− 1r2). (2.16)Introducing the variablew(ϕ) = 1r(ϕ), (2.17)we write the equation in the form(dwdϕ)2= a2J 2e−(λ+ν) + e−λ(bJ 2− w2). (2.18)123Gravitational lensing and rotation curve 147To compare this equation with Newtonian dynamics we use the expansion of the metricfor large r :e−(λ+ν) = 1 + O(r−2), e−λ = 1 − rsr+ O(r−2).Herers = 2G Mc2(2.19)is the Schwarzschild radius in case of a point mass. Then to order 1/r we have(dwdϕ)2+ w2 = a2 + bJ 2− rSw(bJ 2− w2). (2.20)In Newtonian mechanics the bounded streamlines are ellipsesw˜ = 1r= 1p(1 + e cos ϕ), (2.21)where p and e are parameter and eccentricity of the ellipse. p is connected with thenon-relativistic angular momentum J˜ byJ˜ 2p= G M. (2.22)The Newtonian equation which corresponds to (2.20) now reads(dw˜dϕ)2+ w˜2 = e2 − 1p2+ 2pw˜. (2.23)Comparing the coefficients in (2.20) and (2.23) we first find−rS bJ 2 =2p.By (2.22) and J˜ = cJ this gives b = −1 in agreement with (2.15). Secondly, froma2 + bJ 2= e2 − 1p2= − 1pa˜,where a˜ is the big half-axis of the ellipse, we obtain by (2.22)a2 = 1 − G Mc2a˜= 1 − rS2a˜. (2.24)This shows that a2 is connected with the big half-axis of the Kepler ellipse.123148 G. Bräunlich, G. ScharfThe 3-velocity v which is measured by astronomers is defined asv =(dx1dt,dx2dt,dx3dt). (2.25)Usingdsdt= caeν (2.26)we can calculatev2 = u2(dsdt)2, (2.27)where u2 is the spatial part in (2.13). Since our metric is diagonal it is simply givenby− u2 =3∑j=1g j jdx jdsdx jds= 1 − a2e−ν . (2.28)By (2.26) we now get the desired velocity squaredv2 = c2(eν − e2νa2). (2.29)As a check we determine the asymptotic behavior for large r  rS . Assumingcircular motion (a˜ = r ) and usinga2 = 1 − G M/c2r (2.30)we findv2 = c2(−ν(r) − G Mc2r+ O(r−2)).Since ν = −rs/r we arrive atv2 = G Mr, (2.31)M is the total mass (normal plus dark). This agrees with Newtonian dynamics(Kepler’s third law). Summing up, the relation between observational quantities and123Gravitational lensing and rotation curve 149theory is very direct. The rotation curve v(r) gives the metric function ν(r) by solvingthe quadratic equation (2.29)eν = a22⎛⎝1 ±√1 − 4a2v2c2⎞⎠ . (2.32)Using (2.30) and the Newtonian value (2.31) we havea2 = 1 − v2c2. (2.33)Then for velocities v  c we obtain from (2.32) the simple resulteν = 1 − 2v2c2. (2.34)We see that we must take the plus sign in (2.32).3 Null geodesics and lensingIn the case of null geodesics describing light rays the integration constant b in thegeodesic equation (2.16) must be 01r4(drdϕ)2= e−λ(a2J 2e−ν − 1r2). (3.1)In the lensing problem one uses the weak field approximation to general relativitye−ν = 1 − 2U (r)c2, e−λ = 1 + 2U (r)c2≈ eν, (3.2)where U (r) is the gravitational potential. The latter can be obtained from the rotationvelocity according to (2.34)U (r) = −v2. (3.3)Introducing the quantityd = Ja(3.4)in (3.1), the following first order equation remains to be solved(drdϕ)2= r4d2−(1 + 2Uc2)r2. (3.5)123150 G. Bräunlich, G. ScharfFig. 1 Null geodesic for lensingHere we have neglected the quadratic term in U in accordance with (3.2). The meaningof d becomes clear when we consider the trivial solution for U = 0:r = dsin ϕ.It describes a straight line with distance d from the origin in polar coordinates (Fig. 1).After inversion the equation (3.5) can simply be solved by quadrature:dϕdr= ±1r√r2d2 ) − 1 − 2Uc2(3.6)The sign herein depends on the branch of the geodesic to be calculated.In case of a point-mass (Schwarzschild) lens we haveU (r) = −G Mr(3.7)and from (3.6) we get an elliptic integral for the polar angle ϕ(r):ϕ(r) − ϕ0 = ±dr∫r0dr√r4 − r2d2 + rrsd2, (3.8)where we have again used the Schwarzschild radius rs (2.19). To reduce this integralto Legendre’s normal form we need the four zeros a1, a2, a3, a4 of the quartic underthe square root. It is convenient to expand everything in powers of ε:ε = rsd(3.9)which is a small parameter. Then we have the following four real rootsa1 = d(1 − ε2), a2 = εd, a3 = 0, a4 = −d(1 + ε2)(3.10)123Gravitational lensing and rotation curve 151up to O(ε2). The integral (3.8) is an incomplete elliptic integral of the first kindF(, k) where the parameter k is given byk2 = (a2 − a3)(a1 − a4)(a1 − a3)(a2 − a4) = 2ε + O(ε2) (3.11)(see [5], vol. II, p. 310).As first application we compute the Einstein deflection angle. The origin of ourcoordinate system is at the mass M , polar axis goes from M to the observer (seeFig. 1). We integrate (3.8) from the apex r0 to infinity which gives us the deflectionangle 2(ϕ∞ − π). The apex is defined by the conditiondrdϕ= 0 (3.12)which givesr0 = a1. (3.13)Then we obtainϕ∞ − π2 = μd F(∞, k), (3.14)where the Jacobian μ is equal toμ = 2√(a3 − a1)(a4 − a2) =2d(1 − ε2)(3.15)The argument ∞ follows fromsin2 ∞ = a4 − a2a4 − a1 =12(1 + 32ε)= 12(1 − cos(2∞)) (3.16)(see [5], vol. II, p. 310). The elliptic integral can be expanded for small k as followsF(, k) =  + k24( − 12sin 2) + O(k4) (3.17)(see [5], vol. II, p. 313). This finally gives Einstein’s resultϕ∞ = π + ε. (3.18)Next we want to derive the lens equation. In this problem the observer is not atinfinity but in a finite distance Dd from the lens. The light source is at a distance Ddsat the other side of the lens and an amount η off the optical axis (see Fig. 1); we use thesame notation as in [6]. We have now to compute the null geodesics from the source123152 G. Bräunlich, G. Scharfat distance Dds through the apex r = r0 to the observer at distance Dd . Then the polarangle β˜ = ηDds of the source follows fromπ − β˜ = μd [F(ds, k) + F(d , k)] . (3.19)Here the angle d is given bysin2 d = a4 − a2a4 − a1Dd − a1Dd − a2= 12(1 − cos(2d)) = 12[1 + 32ε −  + 2ε](3.20)where  = d/Dd is the angle under which the observer sees the source. Then from(3.17) we obtainF(d , k) = π4 + ε(12+ π8)− 2− ε4. (3.21)F(ds, k) is given by the same formula with  substituted by α = d/Dds . Now wefind from (3.19)− β˜ = − − α + 2ε + O(ε2). (3.22)The lens equation is usually written in terms of the anglesβ = β˜ DdsDs,  = rsDdε, α =  DdDds. (3.23)Then (3.22) gives the following lens equationβ =  − 2 DdsDsrsDd(3.24)for the point-mass lens which is the standard result ([7], p. 27).Now we turn to the formulation of the lens equation in an arbitrary sphericallysymmetric metric. From (3.6) we have the following integral for the polar angle of thenull geodesics:ϕ(r) − π2= ±dr∫r0drr√r2 − (1 + u)d2 , (3.25)where we have introduced the dimensionless gravitational potentialu(r) = 2c2U (r). (3.26)123Gravitational lensing and rotation curve 153Here r0 is the apex and it is important to note that only the potential values for r ≥ r0contribute. For u = 0 the trivial lens equation β =  comes out, this follows from(3.24) for rs = 0. For u = 0 but |u|  1 the modification of the result comes fromthe neighborhood of the apex. It is therefore good enough to expand the potentialu(r) in the vicinity of r = r0. For this purpose we use the beginning of the multipoleexpansionu(r) = c0 + c1r+ c2r2. (3.27)The constant term is necessary in view of the flat rotation curves; note that the potentialhas an absolute normalization in (3.2). It is unimportant that (3.27) breaks down forsmall r because we need r ≥ r0 only.With the three terms in (3.27) we get an elliptic integral of the first kind again:ϕ(r) − π2= ±dr∫r0dr√G(r)= ± μd√1 − c20F(, k). (3.28)Here the quartic is given byG(r) = r4 − (d2 + c0d2)r2 − c1d2r − c2d2, (3.29)the Jacobian μ and the parameter k are the same as before (3.13), (3.15). The fourzeros of G(r) are obtained by solving the two quadratic equations1 + u(r) = 0, r2(1 − u(r)) − d2 = 0.This leads toa1 = c12(1 − c0) +√d2 + c21 − c0 +c214(1 − c0)2a2 = − c12(1 + c0) +√c214(1 + c0)2 −c21 + c0a3 = − c12(1 + c0) −√c214(1 + c0)2 −c21 + c0 (3.30)a4 = c12(1 − c0) −√d2 + c21 − c0 +c214(1 − c0)2 .Then the lens equation is contained in the analogous equation to (3.19)π − β˜ = μd√1 − c20[F(ds, k) + F(d , k)]. (3.31)123154 G. Bräunlich, G. ScharfThe angles d ,ds are given by the same formula (3.20) as before. The appropriateexpansion of the lens equation (3.31) depends on the particular values c0, c1, c2 in(3.27).4 A test of general relativitySuch a test is possible if for a spherical (E0) galaxy rotation curve and lensing data areavailable. From the rotation curve v(r) we get the gravitational potential U (r) accord-ing to (3.3). In practice only three velocity values are needed in order to fit U (r) tothe multipole expansion (3.27). The quality of the fit will be good if some velocityvalue v(r) is measured near the apex r0 of the deflected light ray. Of course one mustreplace the euclidean distances by angular diameter distances as usual. Then knowingthe three parameters c0, c1, c2 the lens equation (3.31) can be solved using the values(3.30) in the elliptic integrals. The calculated lensing data can then be compared withthe observed ones. A discrepancy would be either due to the lack of spherical symme-try or to a failure of the weak field metric (3.2) which is a consequence of Einstein’sequations. The second possibility is interesting in connection with the dark matterproblem [5].Galaxies with joint lensing and dynamical data can be found in the Sloan LensACS Survey (SLACS) and its follow-up project [9]. Unfortunately, until today onlyone system SDSSJ 2321-097 has been analyzed in detail. This is an early-type ellipticgalaxy which cannot be approximated by a spherically symmetric metric. So we mustextend our model-independent analysis to the elliptic case or hope that the astronomerscome up with a E0 lens galaxy.References1. Milgrom, M.: MNRAS 326, 126 (2001)2. Bekenstein, J.D.: Contem. Phys. 47(6), 387 (2006)3. Mannheim, P.D.: ApJ 391 (1992) 429, astro-ph/0505266v2 (2005)4. Horava, P.: Phys. Rev. D 79, 084008 (2009)5. Scharf, G.: From massive gravity to modified general relativity, Gen. Relativ. Gravit. (2009)doi:10.1007/s10714-009-0864-0 and G. Scharf, Quantum Gauge Theories—spin one and two, GoogleBooks, free access (2010)6. Erdelyi, A., et al.: Higher transcendental functions. McGraw-Hill Book Co. Inc., New York (1953)7. Schneider, P., Kochanek, C., Wambsganss, J.: Gravitational lensing: strong, weak and micro. Springer,Berlin (2006)8. Schneider, P., Ehlers, J., Falco, E.E.: Gravitational lenses. Springer, Berlin (1992)9. Czoske, O., Barnabè, M., Koopmans, L.V.E., Treu, T., Bolton, A.S.: MNRAS 384, 987 (2008)123

Gravitational lensing and rotation curve

http://doc.rero.ch/record/319966/files/10714_2010_Article_1077.pdf

Gravitational lensing and rotation curve

Abstract

Similar works

Full text

Available Versions

RERO DOC Digital Library