Contribution of the cosmological constant to the relativistic bending of light revisited

Transcription

1 PHYSICAL REVIEW D 76, (007) Contribution of the cosmological constant to the relativistic bending of light revisited Wolfgang Rindler and Mustaha Ishak* Deartment of Physics, The University of Texas at Dallas, Richardson, Texas 75083, USA (Received 8 December 006; ublished 30 August 007) We study the effect of the cosmological constant on the bending of light by a concentrated sherically symmetric mass. Contrarily to revious claims, we show that, when the Schwarzschild-de Sitter geometry is taken into account, does indeed contribute to the bending. DOI: /PhysRevD PACS numbers: Sf, 98.6.Sb, Es I. INTRODUCTION In the ongoing effort to understand the nature of the dark energy that would be resonsible for accelerating the exansion of our Universe, it is of interest to investigate how the various candidates for this role differ in other observable effects. Review aers on the cosmic acceleration roblem and the dark energy associated with it can be found, e.g., in [1 8] and references therein. One of the rime candidates is the cosmological constant. From this oint of view, various authors have studied the ossible contribution of to the local bending of light. Even though such an effect would be many orders of magnitude too small to be measured with resently available instruments, it might, in rincile, constitute one of the distinguishing characteristics of. Yet there seems to be a generally uncontested ercetion in the literature that the basic light-bending effect of a concentrated sherically symmetric mass, as originally obtained by Einstein, is the same by a fortuitous canceling of terms whether or not one includes a cosmological term in the general-relativistic field equations. The urose of our aer is to rove the contrary. The argument for the noninfluence of was aarently first made in [9] and has been remade and reaffirmed by other authors; see for examle [10 14]. The common basis of their arguments is that, in the Schwarzschild-de Sitter (SdS) metric (first derived by Kottler [15]), which alies when is included, nevertheless dros out of the exact r, differential equation for a light ath (null geodesic). Hence the integrated orbital r, relation for a light ath is also the same with or without. And we agree with that. But the differential equation and its integral are only half the story. The other half is the metric itself, which determines the actual observations that can be made on the r, orbit equation. When that is taken into account, a quite different icture emerges: does contribute to the observed bending of light. In fact, as intuition would suggest, since a ositive effectively counteracts gravity, a ositive diminishes the classical Einstein bending of light, as we shall show. *mishak@utdallas.edu II. THE GEOMETRY AND THE BENDING OF LIGHT The metric we shall be concerned with here (as were the other authors) is the above-mentioned SdS metric ds rdt r 1 dr r d sin d (1) where r 1 m r r 3 ; () and where, in the resently used relativistic units (c G 1), m is the mass of the central object. In the limiting case 0 the metric (1) reduces to the Schwarzschild metric; in the other limiting case, m 0, it reduces to the static form of the metric of de Sitter sacetime. As is well known [e.g. Eq. (11.18) in [16]], the satial equatorial coordinate lane = (like all other such central lanes ) in Schwarzschild sacetime, having the -metric dl 1 m 1dr r d ; (3) r has an intrinsic geometry identical to that of the so-called Flamm araboloid of revolution [17], whose equation is z 8mr m (4) in Euclidean 3-sace referred to cylindrical olar coordinates r; ; z. (It is the surface labeled 3 in our Fig. 1.) The imortance of the central lanes lies in the fact that every orbit by symmetry lies in one of them. The central lanes of SdS sacetime naturally have a different intrinsic geometry. Theirs is determined by the - metric dl 1 m r 1dr r d : (5) r 3 Near the central mass (where m=r dominates over r )we get essentially the Flamm geometry. Far from the central mass (where r dominates over m=r) we get the geometry of a shere of radius a 3=: =007=76(4)=043006(5) The American Physical Society

2 WOLFGANG RINDLER AND MUSTAPHA ISHAK PHYSICAL REVIEW D 76, (007) FIG. 1. Schwarzschild and Schwarzschild-de Sitter geometries. 3 is the Flamm araboloid reresentation of a central coordinate lane in Schwarzschild sacetime; is the corresonding surface in Schwarzschild-de Sitter sacetime; 1 is an auxiliary lane with an r, grah, L 1, of the orbit equation (9). The curves L and L 3 are the vertical rojections of L 1 onto and 3, and reresent the true satial curvature of the orbits. dl 1 r 1 dr a r d : (6) [See Eq. (16.18) in [16] with r a, and Fig there, where corresonds to the de Sitter r.] Hence the surface of revolution that relaces the Flamm araboloid in the case of SdS sacetime is a combination of a Flamm araboloid and a shere, as shown in Fig. 1. At r 3= we have a coordinate singularity in the metric (actually the de Sitter horizon) just as there is a coordinate singularity at r m (the Schwarzschild horizon). But these singularities need not concern us here, since the region of interest lies in between. Figure 1 is a useful icture to have in mind. Here we have lotted, first, on a flat r, lane 1, the grah of a tyical hoton orbit L 1 as given by the r, relativistic orbit equation, from which, by general agreement, is absent. Exactly the same r, relation holds on the Flamm araboloid 3 and on the SdS shere. Hence the hoton orbits L 3 in the Schwarzschild sacetime, and L in the SdS sacetime, corresond to the vertical rojections of L 1 onto the surfaces 3 and, resectively. In the case of Schwarzschild sacetime, where 3 becomes flat at infinity, the asymtotes of L 3 are identical to those of L 1.So the total deflection angle is the same as that of L 1 in the flat sace 1, and indeed it is usually so evaluated. But for SdS sacetime the situation is more comlicated. SdS sacetime in its static reresentation does not become flat at infinity. As r aroaches the value 3=, L climbs u the SdS shere to the horizon, and angle measurements differ severely from the corresonding ones in 1 and 3. We therefore investigate next how affects deflection measurements in the finite region between the two horizons. As is shown in many textbooks [see, for examle, Eq. (14.4) in [16] with h 0, as justified before Eq. (11.6) there], the orbital equation for light in SdS sacetime is d u d u 3mu ; u 1=r; (7) without aroximation and in site of the resence of in the SdS metric. This equation is the same as, e.g., Eq. (17) in [9] and Eq. () in [10]. The orbit that is usually discussed is a small erturbation of the undeflected straight line in flat sace, r sin R (8) (see Fig. ). This first aroximation to Eq. (7) is then substituted into the comaratively small relativistic correction term 3mu, and the resulting linear equation for u solved in the usual way. Thus one obtains [Eq. (11.64) in [16]] 1 sin 3m u r R R 1 1 : 3 cos (9) It is this orbit that we shall take as L 1 in Fig. 1 and as the relevant r, relation in both Schwarzschild and SdS sacetime. Its arameter R is related to the hysically meaningful area distance r 0 of closest aroach by 1 1 r 0 R m R : (10) Other authors, see for examle [18,19], used the imact arameter b to discuss the bending of light in Schwarzschild sacetime, but the SdS sacetime is not asymtotically flat and one needs to define other arameters and constants of motion such as R. In Schwarzschild sacetime, for the asymtote of the orbit, we can let r!1in (9) and, corresondingly,! 1 (small). Thus we find at once that 1 m=r. So the total deflection, defined as the angle between the asymtotes (both in 1 and 3 ), is 4m=R, as usual. But in SdS sacetime, r!1makes no sense. What we can measure here are the various angles that the hoton orbit sin FIG.. The orbital ma. This is a lane grah of the orbit equation (9) and coincides with 1 in Fig. 1. The one-sided deflection angle is

3 CONTRIBUTION OF THE COSMOLOGICAL CONSTANT TO... PHYSICAL REVIEW D 76, (007) 0 m 1 m R R makes with successive coordinate lanes const. (See Fig..) Some interesting oints were discussed in [0] and it has been argued there that one should consider the turning oint r 0 (oint of closest aroach) and not the constant of motion b in the null geodesic differential equation. Then, since r 0 is not affected by, and by virtue of the null geodesic equation, should not contribute to the bending of light. As exlained in the Introduction, this and other arguments were based on the null geodesic differential equation; however, as we demonstrate in the next section, the contribution of to the bending angle comes from the sacetime metric itself [1], indeendently from what one will use for the arametrization of the null geodesic differential equation (an indeendent argument is also discussed in the note []). III. RESULTS AND DISCUSSION In order to calculate the bending angle, we use the invariant formula for the cosine of the angle between two coordinate directions d and as shown in Fig. (whose roof is immediate by going to locally Euclidean coordinates): g ij d i j cos : (11) g ij d i d j 1= g ij i j 1= For our urose the relevant g ij is the -metric of, namely (5). Then g 11 r 1 1 m r 1; g r : r 3 (1) Next, we differentiate (9) and multily by r, to find dr d mr R sinr cos Ar; : (13) R Then, if we call the direction of the orbit d and that of the coordinate line const, we have d dr; d A; 1d d < 0; r; 0 1; 0r: Substituted into (11), these values yield cos or, more conveniently, (14) jaj A rr 1= (15) tan sec 1 1= r1= r : (16) jaj The one-sided bending angle is given by. Let us calculate 0 when 0. By(9), this occurs when r R =m, and consequently jaj R 3 =4m. Equation (16) then yields for the (small) angle 0 R4 4m : (17) This is the formula of most astrohysical significance. Twice 0 is the total bending of a light ray by a massive object, if both source and observer are far from that object. In Schwarzschild sace we would simly let r!1 in (16) to get this angle. In SdS sace, on the other hand, r cannot exceed its horizon value 3=, and even that value is unrealistic. The only other intrinsically characterized r value for this urose is that at 0. As Eq. (17) shows, at that oint the classical Einstein one-sided bending angle m=r has already been reached, to first order, when 0 [we recall here that R is related to r 0 by (10) and, to first order, the Einstein angle has the same exression in terms of R or r 0 ]. And beyond that oint we find ourselves in the very extensive, essentially flat, region of transition between Schwarzschild and de Sitter geometry, in which no further significant bending takes lace (note that this holds only for articular values of m, R, and ). It is in that region that both the source and the observer may be assumed to be situated, and where the observer measures the hysical bending angle 0 directly as the angle between the aarent and the undisturbed osition of the source. Note, from (17), that a ositive diminishes the bending angle, as exected. For comleteness sake it is of interest also to look at bending angles occurring at values other than zero, and their connection with observations. As an examle, we calculate when 45. To sufficient accuracy, (9) is then satisfied by r R. With that, and assuming m=r and R to be small, we find successively from (), (13), and (16) m R R A m 1 R 1= 1 m 1= m R 3 R 1 R R 3 ; tan 1 m R R 3 : (18) The actual (small) bending angle at 45 is. For this, we find, from (18), m R R 6 ; (19) where we used tan tan 1=1 tan and tan 1. Once again there is the exected negative contribution from. The observational situation in the general case is as follows. The observer is at the intersection of the ray coming from the distant source and a radius having coordinate direction. Clearly such an observer can measure directly: it is the visually determined angle from the center of the lens to the aarent osition of the star being lensed. ;

4 WOLFGANG RINDLER AND MUSTAPHA ISHAK PHYSICAL REVIEW D 76, (007) In rincile, is also measurable, but it requires the inut of a second observer far away who has determined the total bending angle 0. Then 0, where is the angle from the center of the lens to the undisturbed osition of the star. We rovided here a brief rescrition for observations; however, a full study on how to ut our results into an observational context is ursued and will be resented in a follow-u aer. We will also include there second order erturbations to the geodesic equation in order to match the higher order terms in Eq. (17). Of course, the contribution of to the bending of light is very small. We know from cosmology that is of the order of cm. With that, the ratio of the two terms on the right-hand side of (17), in the case of a ray grazing the limb of the sun, is 10 8 :1. In this resect, though it is also hoelessly small, the contribution of to the advance of the erihelion of Mercury [see Eq. (14.5) in [16]] is suerior: it could be as much as of the total. However, one would exect this to be different at very large scales (e.g. clusters and suerclusters of galaxies), and this will be exlored in detail in a follow-u aer. In conclusion, the resent aer rovides a long overdue correction to revious works on whether or not the cosmological constant affects the bending of light by a concentrated sherically symmetric mass. We showed that, when the geometry of the Schwarzschild-de Sitter sacetime is taken into account, indeed contributes to the light bending. ACKNOWLEDGMENTS M. I. acknowledges artial suort from the Hoblitzelle Foundation. [1] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989). [] M. S. Turner, Phys. Re. 333, 619 (000). [3] V. Sahni and A. Starobinsky, Int. J. Mod. Phys. D 9, 373 (000). [4] S. M. Carroll, Living Rev. Relativity 4, 1 (001). [5] T. Padmanabhan, Phys. Re. 380, 35 (003). [6] P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (003). [7] A. Uadhye, M. Ishak, and P. J. Steinhardt, Phys. Rev. D 7, (005). [8] A. Albrecht et al., arxiv:astro-h/ [9] N. J. Islam, Phys. Lett. 97A, 39 (1983). [10] W. H. C. Freire, V. B. Bezerra, and J. A. S. Lima, Gen. Relativ. Gravit. 33, 1407 (001). [11] A. W. Kerr, J. C. Hauck, and B. Mashhoon, Classical Quantum Gravity 0, 77 (003). [1] V. Kagramanova, J. Kunz, and C. Lammerzahl, Phys. Lett. B 634, 465 (006). [13] F. Finelli, M. Galaverni, and A. Gruuso, Phys. Rev. D 75, (007). [14] M. Sereno and Ph. Jetzer, Phys. Rev. D 73, (006). [15] F. Kottler, Ann. Phys. (Leizig) 361, 401 (1918). [16] W. Rindler, Relativity: Secial, General, and Cosmological (Oxford University Press, New York, 006), nd ed. [17] L. Flamm, Phys. Z. 17, 448 (1916). [18] R. Wald, General Relativity (University of Chicago Press, Chicago, 1984). [19] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973). [0] K. Lake, Phys. Rev. D 65, (00). [1] It is the metric that measures the distances in the sacetime. [] Our results are indeendent but consistent with the following argument. The focusing of rays of light can be described by the Ricci focusing [3 5] R R ab k a k b (0) and the Weyl focusing (u to a hase factor) F C aibj k a k b t i t j ; (1) where R ab is the Ricci tensor, C aibj is the Weyl tensor, k a and t i are null vectors, and we followed here the notation of [4,5]. Following [4], we verified for the Schwarzschild-de Sitter metric (1) that R 0 and F 3mh r 5 () [in agreement with Eqs. (7) and (35) in [4]]. In Schwarzschild sacetime, h is the imact arameter, but in Schwarzschild-de Sitter sacetime h is the constant of motion J=E where J and E are, resectively, the momentum and the energy of the hoton. The relation between the oint of closest aroach r 0 and the constant of motion h (or b in other notations) is given by, e.g. [1,18], r m 0 1 r 0 h r 0 3 : (3) For Schwarzschild sacetime, the solution r 0 h of this relation is given on. 145 of [18]. The solution can be immediately exanded to include and is given by r 0 q 3 h 1 cos 3 arccos 3m s 3 ; (4) h in agreement with footnote [9] in [13] and, for 0, it reduces to Eq. (6.3.37) that was derived for Schwarzschild sacetime in [18]. Now, if one chooses to exress the Weyl focusing fully in terms of the oint of closest aroach r 0, then using (3) in order to substitute for h into () gives r

5 CONTRIBUTION OF THE COSMOLOGICAL CONSTANT TO... PHYSICAL REVIEW D 76, (007) 3m r 3 1 m r 0 1: (5) 0 r 0 3 F 3mh r 5 0 So we see that does aear in the final result. Similarly, if one chooses to exress the focusing fully in terms of h, then using (4) in order to eliminate r 0 from () givesf as a function of m, h, and. So again the final exression for the Weyl focusing has in it. In summary, regardless of choosing r 0 or h to exress the Weyl focusing in the Schwarzschild-de Sitter sacetime, does aear in the final exression for the focusing. This result suorts the conclusion that contributes to the bending of light in this sacetime. [3] R. K. Sachs, Proc. R. Soc. A 64, 309 (1961). [4] C. C. Dyer, Mon. Not. R. Astron. Soc. 180, 31 (1977). [5] P. Schneider, J. Ehlers, and E. E. Falco, Gravitational Lenses (Sringer-Verlag, Berlin, 199)