Gravitational lensing with f (χ) = χ 3/2 gravity in accordance with astrophysical observations

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1 MNRAS 433, (013 Advane Aess publiation 013 June 18 doi: /mnas/stt75 Gavitational lensing with f (χ = χ 3/ gavity in aodane with astophysial obsevations S. Mendoza, 1 T. Benal, 1 X. Henandez, 1 J. C. Hidalgo 1, andl.a.toes 1 1 Instituto de Astonomía, Univesidad Naional Autónoma de Méxio, AP 70-64, Distito Fedeal 04510, Méxio Depatamento de Físia, Instituto Naional de Investigaiones Nuleaes, La Maquesa Ooyoaa 5750, Méxio Aepted 013 Apil 9. Reeived 013 Apil 5; in oiginal fom 013 Januay 15 1 INTRODUCTION When Einstein intodued his theoy of geneal elativity, an astophysial pedition fo the motion of the planet Meuy (a massive patile though its obit was made (Einstein The seond step was to test geneal elativity though the defletion of light (massless patiles oming fom stas appeaing nea the Sun s limb duing a sola elipse (Dyson, Eddington & Davidson 190. Both obsevations onstituted the fist oheent steps towads the solid foundation of geneal elativity, a theoy apable of desibing gavitation though a oet elativisti desiption. In this sense, any meti theoy of gavity must be ompatible with both kinds of obsevations, the dynamial ones fo massive patiles and the obsevations of the defletion of light fo massless patiles. The oet appoah is extensively desibed in the monogaph by Will (1993 whee it is shown that when woking with the weak field limit of a elativisti theoy of gavity in a stati spheially spae time, the dynamis of massive patiles detemine the funtional fom of the time omponent of the meti, while the defletion of light detemines the fom of the adial one (see also Will 006, and efeenes theein. segio@asto.unam.mx ABSTRACT In this atile, we pefom a seond ode petubation analysis of the gavitational meti theoy of gavity f (χ = χ 3/ developed by Benal et al. We show that the theoy aounts in detail fo two obsevational fats: (1 the phenomenology of flattened otation uves assoiated with the Tully Fishe elation obseved in spial galaxies, and ( the details of obsevations of gavitational lensing in galaxies and goups of galaxies, without the need of any dak matte. We show how all dynamial obsevations on flat otation uves and gavitational lensing an be synthesized in tems of the empiially equied meti oeffiients of any meti theoy of gavity. We onstut the oesponding meti omponents fo the theoy pesented at seond ode in petubation, whih ae shown to be pefetly ompatible with the empiially deived ones. It is also shown that unde the theoy being pesented, in ode to obtain a omplete full ageement with the obsevational esults, a speifi signatue of Riemann s tenso has to be hosen. This signatue oesponds to the one most widely used nowadays in elativity theoy. Also, a omputational pogam, the Meti EXtended-gavity Inopoated though a Compute Algebai System (MEXICAS ode, developed fo its usage in the Compute Algebai System Maxima fo woking out petubations on a meti theoy of gavity, is pesented and made publily available. Key wods: gavitation gavitational lensing: stong gavitational lensing: weak. To ode of magnitude and though a fist petubation analysis, Benal et al. (011b have shown that it is possible to eove flat otation uves and the Tully Fishe elation (i.e. a MONDian-like weak field limit fom a meti theoy of gavity, whih inludes the mass of the system in the gavitational field s ation. Suh limit is of high astophysial elevane at the sales of galaxies, whee Modified Newtonian Dynamis (MOND auately desibes the otation uves of spial galaxies and the Tully Fishe elation without the need of dak matte (see e.g. Milgom 1983; Famaey & MGaugh 01. In this atile, we show the stength of the alulations made by Benal et al. (011b by doing an extensive analysis fom petubation theoy fo a stati spheially symmeti meti and show that in the weak field limit ou esults ae in pefet ageement not only with the Tully Fishe elation, but ae also in exat aodane with obsevations of gavitational lensing ove a wide ange of astophysial sales. Extensions to Einstein s geneal theoy of elativity have been poposed sine the publiation of the theoy itself (see e.g. Shimming & Shmidt Howeve, it has not been until eent times that obsevations at diffeent mass and length sales have onluded that in ode to keep Einstein s field equations valid, unknown dak matte and enegy entities need to be added to the theoy. In this atile, a omplementay appoah is taken whee the existene of these unknown dak entities is not equied. We show C 013 The Authos Published by Oxfod Univesity Pess on behalf of the Royal Astonomial Soiety

2 Gavitational lensing in extended χ 3/ gavity 1803 the theoy built by Benal et al. (011b to be in aodane not only with the vey well established obsevations of the dynamis of massive patiles though the Tully Fishe elation, but also with the dynamis of massless patiles though the bending of light as astophysially obseved. Mendoza & Rosas-Guevaa (007 and Rosas-Guevaa (006 showed fo the fist time that meti theoies of gavity ae apable of poduing moe defletion of light than the one podued by Einstein s geneal elativity. This was done using the meti theoy of gavity onstuted by Sobouti (007. The impliations of this esult invalidated the so-alled no-go theoem fo meti f (R theoies of gavity poposed by Soussa & Woodad (003; Soussa (003. Futhemoe, in this wok, we show that it is possible to explain the obseved gavitational lensing fo galaxies, and goups of galaxies without the need of invoking dak matte. Developments, by see e.g. Capozziello et al. (006, Hováth et al. (01, Nzioki et al. (011, on weak and stong lensing egimes of extended meti theoies of gavity have followed the wok by Mendoza & Rosas-Guevaa (007 but ae not of geneal validity with espet to diffeent astophysial obsevations. Testing any meti theoy of gavity against obsevations an be umbesome. Fom an ation piniple one must deive field equations, whih in piniple, have to be solved fo e.g. in spheially symmeti spae times. The solutions to this lead to meti oeffiients whih in tun, with the use of the geodesi equation, yield obits fo massive and massless patiles, to be then ompaed against astophysial obsevations. These last ae vaied and divese e.g., entifugal equilibium obits at a vaiety of adii, fo systems having total masses spanning seveal odes of magnitude, and the obseved sheas and austi positions of gavitational lensing obsevations. Fotunately, we have deived a muh moe diet and genei appoah. Fist, dynamial obsevations egading the amplitudes of galati flat otation uves satisfy a well-known saling with the fouth oot of the total bayoni ontent: the Tully Fishe elation. To seond ode in petubations of the veloity measued in units of the speed of light, this an be shown to imply a definite empiial pesiption fo the time omponent of any meti theoy not equiing dak matte. Seondly, we show that all gavitational lensing obsevations on elliptial and spial galaxies, as well as fo goups of galaxies an be synthesized as the equiement fo the same isothemal total matte distibution as needed to explain the obseved spial otation uves and dynamis about elliptial galaxies, if one assumes Einstein s geneal elativity. Fom studying dietly the lens equation in geneal elativity, this implies a bending angle whih is independent of the impat paamete, and whih sales with the squae oot of the total bayoni mass of a system. It an then be shown that this, in ombination with the empiial time omponent of the meti mentioned above, leads to a fixed empiial pesiption fo the adial omponent, fo any meti theoy not equiing dak matte. Thus, we synthesize all dynamial and gavitational lensing astophysial obsevations at galati and galaxy goup sales, into empiial time and adial meti omponents of a spheially symmeti meti given at seond ode in petubation. It is though ompaing the above to petubed meti oeffiients to the same ode oming fom the meti theoy teated in this pape that we ae able to show its full ompatibility with all elevant dynamial and gavitational lensing astophysial obsevations. The atile is oganized as follows. In Setion, the onept of weak field limit fo a stati spheially symmeti spae time is established and we define the elevant odes of petubation to be used thoughout the atile. In Setion 3, we petub the vauum field equations of the meti theoy built by Benal et al. (011b and show that fo a point mass soue they losely esemble the ones usually adopted in f (R gavity in vauum. Howeve, these equations slightly diffe unde the appoximations of the mass and length sales assoiated with galaxies and goups of galaxies whee gavity is expeted to diffe fom Einstein s geneal elativity in the absene of any dak matte omponent. In Setion 4, we obtain the solution fo the Rii sala up to the seond ode fom the petubed field equations and disuss the impotane of the signatue in the Riemann tenso to yield the oet esults. In Setion 5, we obtain the oeffiients of the meti up to the seond ode in petubation. In Setion 6, we obtain the meti oeffiients up to the seond ode in an empiial way, without efeene to any speifi meti theoy of gavity, using the dynamial phenomenology of galaxies and goups of galaxies and the gavitational lensing podued by these objets. In that setion, we also ompae the meti oeffiients obtained in 5 with those empiially obtained and show full onsisteny. Finally in Setion 7, we disuss ou esults. THE WEAK FIELD LIMIT An exellent aount of petubation theoy applied to meti theoies of gavity (in patiula geneal elativity an be found in the monogaph witten by Will (1993. Moe eently, Capozziello & Stabile (009 have developed a petubation analysis tehnique useful when dealing with lenses in f (R gavity. In this Setion, we define the elevant popeties of the petubation theoy having in mind appliations to the meti theoy developed by Benal et al. (011b. Let us onside a fixed point mass M at the ente of oodinates geneating a gavitational field. Unde these onsideations, the spae time is stati and its spheially symmeti meti g μν is geneated by the inteval ds = g μν dx μ dx ν = g 00 dt + g 11 d d. (1 In the pevious equation and in what follows, Einstein s summation onvention ove epeated indies is used. Geek indies take values 0, 1,, 3 and Latin ones 1,, 3. As suh, in spheial oodinates (x 0, x 1, x, x 3 = (t,, θ, ϕ, whee is the speed of light, t is the time oodinate, the adial one, and θ and ϕ ae the pola and azimuthal angles, espetively. Also, the angula displaement d := dθ + sin θ dϕ. Due to the symmety of the poblem, the unknown funtions g 00 and g 11 ae funtions of the adial oodinate only. Note also that We hoose a ( +,,, signatue fo the spae time meti, whih we maintain thoughout the atile. The adial omponent of the geodesi equations d x α ds + Ɣβλ α dx β dx λ ds ds = 0, ( fo the meti (1 in the weak field limit, i.e. when the speed of light,isgivenby 1 d dt = 1 g11 g 00,, (3 whee the subsipt (, := / denotes the deivative with espet to the adial oodinate. In the above elation, we have assumed that fo the weak field limit ds = dt and sine the veloity v then v i dx 0 /dt with v i := (d/dt, dθ/dt, sin θ dϕ/dt. In the stong limit, both sides of the above equation vanish simultaneously. Thus, the ondition of the ight-hand side of equation (3 vanishing in the v povides a onsisteny hek on the esults

3 1804 S. Mendoza et al. of the following setions, whee a petubative solution to the meti is developed. In this limit, a patile bound to a iula obit about the mass M expeienes a entifugal adial aeleation given by d dt = v, (4 fo a iula o tangential veloity v. The peeding equation is a kinematial elation of geneal validity and does not intodue any patiula assumption of the gavitational theoy. When mateial patiles ae used as test patiles in the weakest limit of the theoy, the meti takes the fom (see e.g. Landau & Lifshitz 1975: g 00 = 1 + φ, g 11 = 1, g =, g 33 = sin θ, (5 fo a Newtonian gavitational potential φ. The above equations ae only used when analysing the motion of mateial patiles when gavity is vey weak (see e.g. Will In ode to demand geate auaies of the theoy and to eove exat esults fo the motion of massless patiles, i.e. to auately desibe the bending of light ays, the following tem in the expansion of g 11 must also be onsideed. Fo a patile on iula motion about the mass M in the weak field limit, the lowest ode of the theoy is obtained when the lefthand side of equation (3 is of the ode of v / andwhenthe ight-hand side is of the ode of φ/. Both ae just odes O(1/ of the theoy, o simply O(. As suh, when lowe o highe ode oetions of the theoy ae intodued we will use the notation O(n fon = 0, 1,,... meaning O(0, O( 1, O(,..., espetively. Having in mind futhe astophysial appliations (of motion of mateial patiles and bending of light massless patiles, we expand the meti g μν about a flat Minkowski meti η μν := diag(1, 1, 1, 1 up to the seond ode in time and adial position in suh a way that g 00 = g ( g( 00 = 1 + g( 00 + O(4, g 11 = g ( g( 11 = 1 + g( 11 + O(4, g = g (0 =, g 33 = g sin θ, (6 whee the supesipt (p denotes the ode O(p at whih a patiula quantity is appoximated. Fom equations (6 it follows that the ontavaiant meti omponents ae given by g 00 = g 00(0 + g 00( = O(4, g 11 = g 11(0 + g 11( = O(4, g = g (0 = 1/, g 33 = g / sin θ. (7 In fat, to the lowest ode of petubation, we need to find the time 00 and adial g( 11 meti omponents up to the seond ode to ompae with the astophysial obsevations of mateial patiles and bending of light (Will 1993, 006. Note that in keeping with the assumption of spheial symmety fo the matte onfiguations to be studied, we onside no petubations on the angula tems of the meti. This assumption thus limits the appliability of all ou following esults to systems not fa fom spheial symmety, e.g. the spheoidal elliptial galaxies about whih gavitational lenses ae often deteted. 3 FIELD EQUATIONS Fo the ase of a point-mass soue geneating a gavitational field, Benal et al. (011b have poposed an extended gavitational field s ation in the meti appoah given by 3 S f = 16πGL f (χ g d 4 x, (8 M fo any abitay dimensionless funtion f (χ of the dimensionless Rii sala: χ := L MR, (9 whee R is the standad Rii sala and L M = ζg 1/ l1/ M, (10 is a length sale with g := GM ( GM 1/, l M :=, (11 a 0 with l M the mass-length sale of the system defined by Mendoza et al. (011, a 0 := ms is Milgom s aeleation onstant (see e.g. Famaey & MGaugh 01, and efeenes theein and ζ is a oupling onstant of ode one whih has to be alibated though astophysial obsevations. This f (χ theoy was onstuted though the inlusion of a 0 as a fundamental physial onstant, whih has been shown to be of astophysial and osmologial elevane (see e.g. Henandez et al. 010; Benal et al. 011a,b; Mendoza et al. 011; Henandez & Jiménez 01; Henandez, Jiménez & Allen 01; Mendoza 01; Caanza, Mendoza & Toes 013. Equation (8 is undestood as a patiula ase of a fulle fomulation whee the details of the mass distibution appea inside of the ation integal, in suh a way that fo a fixed point mass, the esult is the ation (8, as will be moe fully disussed towads the end of this setion. Following the desiption of Benal et al. (011b, the matte ation takes its odinay fom: S m = 1 L m g d 4 x, (1 with L m the Lagangian matte density of the system. The null vaiation of the omplete ation, i.e. δ (S f + S m = 0, with espet to the meti g μν yields the following field equations: f (χχ μν 1 f (χg ( μν L M μ ν g μν f (χ = 8πGL M T 4 μν, (13 whee the Laplae Beltami opeato has been witten as := μ μ, the pime denotes the deivative with espet to the agument and the enegy momentum tenso T μν is defined though the standad elation δsm = (1/T αβ δg αβ. Also, in equation (13, the dimensionless Rii tenso is defined as χ μν := L M R μν, (14 whee R μν is the standad Rii tenso. The tae of equations (13 is f (χ χ f (χ + 3L M f (χ = 8πGL M 4 T, (15 whee T := T α α.

4 Gavitational lensing in extended χ 3/ gavity 1805 To ode of magnitude appoximation, whee d/dχ 1/χ, 1/ and the mass density ρ M/ 3, Benal et al. (011b have shown that the tae (15 equation takes the following fom: χ b (b 3bL χ (b 1 M 8πGML M. (16 3 fo a powe-law fom: f (χ = χ b. (17 As shown by Benal et al. (011b, the thid tem on the left-hand side of equation (15 dominates ove the fist two when the adius of uvatue R R 1/ of spae time is suh that R and so, this oesponds to the egion whee MONDian effets ae expeted to appea. Benal et al. (011b and Mendoza (01 have shown that the funtion f (χ must satisfy the following limits: { χ, when χ 1, f (χ = (18 χ 3/, when χ 1. The limit χ 1 eoves Einstein s geneal elativity and the ondition χ 1 yields a elativisti vesion of MOND. In this last egime, the fist two tems on the ight-hand side of the tae (15 ae smalle than the thid and so (f. Benal et al. 011b f (χ χ f (χ 3L M f (χ, (19 at all odes of appoximation, and so the tae (15 is given by 3L M f (χ = 8πGL M 4 T. (0 Sine we ae inteested in the field podued by a point mass M, then the ight-hand side of equations (13 and (0 ae null away fom the soue and so, the last elation in vauum an be ewitten as f (χ = 0. (1 As shown by Benal et al. (011b, the elation f (χ = χ 3/ yields the oet MONDian non-elativisti limit. Howeve, fo the sake of geneality we will assume in what follows that the funtion f (χ is of powe-law fom 17. In this ase, elation (1 is equivalent to f (R = 0, ( to all odes of appoximation fo a powe-law funtion of the Rii sala f (R = R b. (3 Substitution of the powe-law funtion (17 in the null vaiations of the gavitational field s ation (8 in vauum means that δs f = 3 δ R b g d 4 x = 0, (4 16πG L(b 1 M and so δ R b g d 4 x = 0. (5 This equation gives the same field equations as the null vaiation of the ation fo a standad powe-law meti f (R theoy (3 in vauum. With this in mind, we an follow the standad petubation analysis fo f (R estited by the onstaint equation ( needed to yield the oet MOND-like limit. Sine we ae only inteested in a powe-law desiption of gavity fa away fom geneal elativity (f. equation 18, then in what follows we use the standad f (R field equations fo vauum as desibed by Capozziello & Faaoni (011 fo a powe-law desiption of gavity given by equation (3 with b = 3/, with the onstaint (. To follow thei notation, we wite the field equations (13 in vauum as f (RR μν 1 f (Rg μν + H μν = 0, (6 whee the fouth-ode tems ae gouped into the following quantity: H μν := ( μ ν g μν f (R. (7 The tae of equation (6 is thus given by f (RR f (R + H = 0, (8 with H := H μν g μν = 3 f (R. (9 The mathematial fom of the field s ation (8 inludes the Shwazshild mass (though L M in the desiption of the gavitational field. This is usually not the ase fo the desiption of the gavitational field sine the matte ontent is geneally assumed to appea only in the matte ation (1. Following the emaks by Sobouti (007 and Mendoza & Rosas-Guevaa (007, whee simila onlusions wee eahed, one should expet extensions to the theoy even at the fundamental level of the ation. Fo the ase of a geneal matte distibution it is not evident what path to follow. As explained by Caanza et al. (013 and Mendoza (01, fo systems with a high degee of symmety (suh as the Fiedmann Lemaîte Robetson Walke FLRW univese o a spheially symmeti distibution of matte the ation may be postulated as S f = 3 f (χ g d 4 16πG L x, (30 M whee the mass-enegy is given by (see e.g. Misne, Thone & Wheele 1973 M( = 4π 0 ρ( d. (31 Fo the ase of the FLRW univese, the uppe limit of the pevious integal is taken as the Hubble adius (f. Mendoza 01; Caanza et al The onnetion between the ation 30 and the f (R, T theoy desibed by Hako et al. (011 is then evident though the identifiation F (R, T := f (χ L. (3 M The field equations then follow though the full fomal vaiation of the ation with espet to both R and T (see e.g. Hako et al. 011; Mendoza 01. The geneal desiption of the gavitational theoy is by no means omplete, and futhe investigation needs to be aied out in this dietion. We only mentioned one possible genealization of the simple point mass desiption by Benal et al. (011b fo ompleteness. In any ase, the lensing phenomena we ae inteested in ou suffiiently fa away fom the matte distibution poduing it, that these an be oetly desibed as point mass soues. In what follows, the sign onvention used in the definition of the Riemann tenso beomes a elevant point. As disussed in Appendix A, the solutions to the diffeential field equations of any f (R theoy of gavity geatly depend on the signatue hosen fo Riemann s tenso. Two diffeent hoies of signatue bifuate on the solution spae, a popety whih does not appea in Einstein s geneal elativity. This is not supising as it mios the analogous

5 1806 S. Mendoza et al. unfolding of the meti and Palatini appoahes in f (R gavity, whih does not appea in Einstein s f (R = R theoy (see e.g. Olmo 011. Thoughout the atile, we selet a patiula banh of solutions given by the nowadays almost standad definition of Riemann s tenso in equation (A3. In dealing with some of the umbesome algebai manipulations that a petubation to an f (R theoy of gavity pesents, we have used the Compute Algeba System (CAS Maxima to failitate the omputations. The Meti EXtended-gavity Inopoated though a Compute Algebai System (MEXICAS ode (opyight of TB, SM and LAT and liensed with a GNU Publi Liense Vesion 3 we wote fo this is desibed in Appendix B and an be downloaded fom Futhemoe, development on the teatment of the field equations by the MEXICAS ode is desibed in Appendix C. Fo the ase of a stati spheially symmeti spae time (1 it follows that { ( H μν = f R,μν Ɣμν 1 R, g μν g, 11 + ( g11 ln g, R, + g 11 R, ]} f { R,μ R,ν g μν g 11 R,}, (33 and ( H = 3f g, 11 + ( g11 ln g ] R,, + g 11 R, + 3f g 11 R,. (34 Unde the assumption of spheial symmety, the angula tems of the meti ae not petubed and so { ] 1/ g = sin θ g( 11 + O(4}, (35 then, by using the fat that ln ( g = (, g, / g, itfollows that ln ( g =, + 1 ] 00, g( 11, + O(4. (36 Sine Rii s sala depends on the meti omponents and thei deivatives up to the seond ode with espet to the oodinates, it follows it an only have a non-null seond and highe petubation odes, i.e. R = R ( + R (4 + O(6. (37 The fat that R (0 = 0 is onsistent with the flatness of spae time assumption at the lowest zeoth ode of petubation. The expession fo the seond ode omponent of Rii s sala fom the meti omponents (6 is given by ] R ( = 11, + g( 11 00, g( 00,. (38 The global minus sign that appeas on the ight-hand side of equation (38 fo Rii s sala R ( at seond petubation ode diffes fom that epoted by Capozziello, Stabile & Toisi (007, Capozziello & Stabile (009. As mentioned above, and disussed in Appendix A, this fat ous due to the hoie of signs in the definition of Riemann s tenso. The patiula hoie used thoughout the atile is the one given by equation (A3 and so, ou solutions lie in a diffeent banh as the one epoted by those authos. two tems on the left-hand side of the tae equation is O(b. On the othe hand, diet inspetion of the ight-hand side of equation (34 esults in the fat that the lowest ode of H is O(b. Indeed, the last tem of the ight-hand side of this equation is f g 11 R, and so, to the lowest ode of petubation of elations (7 and (37, this means that H ontains tems of the fom R (b 3 R, ( and so, H is of the ode of O(b. This analysis indiates that to the lowest ode the tae equation to onside is H (b = 3 f (b (R = 0. (39 This esult is onsistent with elation ( to the lowest ode of appoximation and is in pefet ageement with the petubative study pefomed by Benal et al. (011b. Note also that this is the only independent equation at this ode. Diet substitution of equations (3 and (37 into the last equation leads to (ln H (b = 3b(b 1R (b g 11(0 ] (0 g, R(, + R, ( + 3b(b 1(b R (b 3 g 11(0 R, ( = 0. (40 Substitution of expessions (7 and (36 in the pevious equation leads to the following diffeential equation fo Rii s sala at ode O(: ] R ( R(, + R, ( + (b R, ( = 0, (41 whih an be witten in a moe suitable fom as ] ln R (, + (b ln R (] =,,. (4 The solution of the pevious equation is ( ] A 1/(b 1 R ( ( = (b 1 + B, (43 whee A and B ae onstants of integation. Fa away fom the ental mass, spae time is flat and so Rii s sala must vanish at lage distanes fom the oigin. This means that the onstant B = 0andso R ( ( = (b 1 A ] 1/(b 1. (44 As explained by Benal et al. (011b, the ase b = 3/ yields a MOND-like weak field limit and so, substituting b = 3/ in elation (44 yields: R ( ( = ˆR, (45 whee ˆR := A /4. This is exatly the same esult as the one obtained by Benal et al. (011b. As these authos have shown, this esult yields a MONDian-like behaviou fo the gavitational field in the limit l M g. Fo this patiula ase, the lowest ode of appoximation of the theoy is O(1, whih has a highe elevane as ompaed to the ode O( of standad geneal elativity fo whih b = 1. Using vey geneal aguments, the authos also showed that the onstant ˆR g /l M and so, ˆR is popotional to the squae oot of the mass of the ental objet. In ode to alulate ˆR fom petubation analysis we need to find the expessions fo the meti at ode O( of appoximation. 4 LOWEST ORDER SOLUTION Let us now alulate the ode of the tae equation (8 using elations (3 and (37. On the one hand, the lowest ode of the fist 5 f (χ = χ 3/ METRIC COMPONENTS Let us now solve the field equations at the next ode O(b of appoximation. At this ode, we expet the meti omponents 00,

6 Gavitational lensing in extended χ 3/ gavity and Rii s sala R(4 to play a elevant ole in the desiption of the gavitational field. In fat, the field equations (6 at this ode ae given by br (b 1 R μν ( 1 R(b g μν (0 + H(b μν = 0, (46 whee H μν (b = ( μ ν g μν f (b (R. (47 The omplete H μν (b fom equation (33 is witten in Appendix C. Now, fom equation ( it follows that the Laplae Beltami opeato applied to f (R must be zeo at all petubation odes. In patiula f (b = 0. With this ondition, the field equations (46 simplify geatly and an be witten as br (b 1 R μν ( 1 R(b g μν (0 b(b 1 { R (b R μν (4 ] Ɣ1(0 μν R(4, Ɣμν 1( R(, + (b R (b 3 R (4 R μν ( ]} Ɣ1(0 μν R(, b(b 1(b R (b 3 R,μ ( R(4,ν ] = 0. (48 + (b 3R (b 4 R (4 R,μ ( R(,ν Diet substitution of the following Chistoffel symbols Ɣ 1(0 00 = 0, Ɣ 1( 00 = 1 g11(0 00,, (49 and elations (6 and (7 in the 00 omponent of equation (48 leads to br (b 1 R ( 00 1 R(b + 1 b(b 1g( 00, R(b R, ( = 0. (50 If we now substitute b = 3/, expession (45 and the value of Rii s tenso at O( of appoximation: = g( R ( 00, + g( 00, 00, (51 into equation (50, we obtain the following diffeential equation fo 00 : 00, + 3g( 00, + ˆR = 0, (5 3 and so ( 00 ( = ˆR 3 ln + k 1, (53 whee k 1 and ae onstants of integation. By substitution of this esult in equation (38 and using equation (45 we get the following diffeential equation fo 11 : 11, + g( 11 + k 1 + ˆR = 0, (54 3 with solution: 11 ( = k 1 + k ˆR 3, (55 whee k is a onstant of integation. 6 METRIC COEFFICIENTS FROM ASTRONOMICAL OBSERVATIONS In this setion, we deive the onstaints well established by astophysial phenomenology of asymptotially flat galati otation uves satisfying the Tully Fishe elation, and the umulative gavitational lensing obsevations fo elliptial and spial galaxies and galaxy goups, imply fo the meti oeffiients fo stati, spheially symmeti spae times fo any meti theoy of gavity whee dak matte is not equied. To begin with, let us take the adial omponent (3 of the geodesi equations ( in the weakest limit of the theoy. In this limit, the otation uve fo test patiles bound to a iula obit about a mass M with iula veloity v( given by equation (4 is v ( = 1 g11 g 00,. (56 Exept fo the inne egions of spial galaxies, v( an be well appoximated by a onstant whih sales with the fouth oot of the total bayoni mass M b of the spial galaxy in question, as desibed by the Tully Fishe empiial elation (see e.g. Milgom 008; Famaey & MGaugh 01 v = (GM b a 0 1/4. (57 In fat, it is fom numeous obsevations of galati otation uves and total bayoni mass estimates, that the onstant a 0 has been alibated (see e.g. Famaey & MGaugh 01, and efeenes theein. We now substitute equations (6 and (7 to ode O( of appoximation and elation (57 in equation (56 to obtain the following diffeential equation fo 00 : 00, = ( v (GM b a 0 1/ =, (58 having as solution 00 ( = ( v ln ( = (GM ba 0 1/ ( ln = ( g ln, (59 l M whee is a sale adius whih, fom the point of view only of the flat otation uves of galaxies and the Tully Fishe elation, emains abitay. We theefoe see that a neessay and suffiient ondition in any meti elativisti theoy of gavity, whee all obsevational onstaints of galati otation uves ae satisfied without invoking dak matte, is that 00 must satisfy the pevious empiially deived elation. Compaing the theoetial meti oeffiient 00 given by (53 (obtained fom petubation theoy fo f (χ = χ 3/ and the empiial one (59 (obtained fom the phenomenology of flat otation uves and the Tully Fishe elation, give the following values fo the integation onstants needed in equation (53: k 1 = 0, ˆR = 6 g /l M. (60 In this ase, the gavitational potential φ fom equation (5 takes the fom: ( ( φ = v ln = (GM b a 0 1/ ln, (61 whih yields a adial MONDian aeleation: a = φ = (GM ba 0 1/. (6 Thus, in the v/ 1 limit, the f (χ = χ 3/ pesented is seen to agee with the obseved phenomenology of the obseved galati otation uves in the absene of dak matte, as aleady shown by Benal et al. (011b.

7 1808 S. Mendoza et al. The g 11 meti oeffiient will be obtained fom gavitational lensing phenomenology. We begin fom the geneal deviation angle equation witten fom the point of view of an obseve at infinity, the astonome deteting the gavitational lens in question (see e.g. Weinbeg 197; Shneide, Ehles & Falo 199; Keeton & Pettes 005: g 00(g 11(] 1/ d β = i (/ i g 00 ( i g 00 ( ] π, (63 1/ whee i is the losest appoah to the ental mass M, and it is elated to the impat paamete b though the elation i = b g 00 ( i. Ove the last few yeas, it has beome lea that the omplete phenomenology of gavitational lensing, at the level of extensive massive elliptial galaxies (see e.g. Koopmans et al. 006; Gavazzi et al. 007; Banabè et al. 011, galaxy goups (see e.g. Moe et al. 01, lustes of galaxies (see e.g. Limousin et al. 007; Newman et al. 009 and moe eently spial galaxies (see e.g. Dutton et al. 011; Suyu et al. 01 an be auately modelled using total matte distibutions having isothemal pofiles, when teating the poblem fom the point of view of Einstein s geneal elativity. All these obsevations show that the dak matte haloes needed to explain gavitational lensing unde Einstein s geneal elativity obey the same Tully Fishe saling with total bayoni mass as the ones needed to explain the obseved otation uves of spial galaxies. This means that fo a given total bayoni mass, spial and elliptial galaxies and goups of galaxies equie dak matte haloes having the same physial popeties to explain the obsevations; fom kinematis of otation uves in the fome ase to gavitational lensing in the latte one (Dutton et al. 011; Suyu et al. 01. Unde Einstein s geneal elativity the majoity of these isothemal matte distibution, patiulaly at lage adii, must be omposed of a hypothetial dak matte. Fo a stati spheially symmeti total matte distibution M T, sine assuming the validity of Einstein s geneal elativity Shwazshild s meti holds, and theefoe g 00S = 1/g 11S,we obtain: g 00S = 1 g = 1 GM T( ( v = 1. (64 The subsipt S identifies the oeffiients of the Shwazshild meti, and M T ( = v /G efes to the hypothetial isothemal total matte distibution (f. Binney & Temaine 008 needed to explain the obseved lensing, when assuming geneal elativity. Fom this it follows that the dak matte hypothesis povides a self-onsistent intepetation of obseved phenomenology: the same dak matte haloes, whih ae equied to explain the obseved otation uves, have been solved fo by analysing extensive lensing obsevations. Fom equation (64 it follows that fo isothemal total matte haloes unde Einstein s geneal elativity, the meti oeffiient g 00S does not depend on the adial oodinate. We an see this by using the empiial Tully Fishe elation (57 between the veloity and the total bayoni mass in the last identity above. Thus, the oeffiient (64 an then be taken outside of the integal (63 of the deviation angle, whee fo the Shwazshild meti and isothemal total matte haloes we now obtain d β = 1 (v/ ] 1/ i (/ i 1 ] π. (65 1/ The above adial integal yields π/ and we obtain the obseved bending angle as π β = 1 (v/ ] π = π π. (66 1/ 1 (GM b a 0 1/ / ] 1/ We see that the well-established empiial esult of lensing obsevations yielding isothemal total dak matte haloes unde the standad theoy is stitly the obsevation of onstant bending angles whih do not depend on the impat paamete, saling with the obseved bayoni total masses as indiated above. Now, sine (v/ is of the ode of O(, we an wite equation (66 as ( v (GM b a 0 1/ β = π = π = π g l M. (67 The above equation summaizes all empiial esults of gavitational lensing at galati and galaxy goup sales: the bending angle does not depend on the impat paamete and sales with the squae oot of the total bayoni mass. This last equation gives a lea illustation of the link between the dynamis and the spae time uvatue effets indued by the pesene of an obseved bayoni mass. We an now use the esult of equation (67 to onstain the meti oeffiient g 11 fo any meti theoy of gavity, seeking an auate desiption of the obseved gavitational lensing phenomena without the intodution of any hypothetial dak matte. To do this, let us etun to the geneal lensing equation (63, and ask that the esults obtained unde the Shwazshild meti with isothemal total matte haloes math those unde any meti theoy of gavity, at all impat paametes and fo any total bayoni masses: 1 (β + π = 1 + ( v ] i (/ i 1 ] 1/ g 00(g 11(] 1/ d = i (/ i g 00 ( i g 00 ( ] 1/, (68 at O( of appoximation fom equations (63 and (65. Let us eaange integal (68 in suh a way that { ( v ] i (/ i 1 ] 1/ } g 00 (g 11 (] 1/ (/ i g 00 ( i g 00 ( ] d = 0. (69 1/ Sine the esult must hold fo all impat paametes, the integand of the above equation must be equal to zeo and so ( v ] (/ i 1 = g 00 (g 11 ( (/ i g 00 ( i g 00 (. (70 Appoximating the pevious elation to ode O(, it follows that the meti oeffiient g 11 is given by ( v ] (/i g 00 ( i /g 00 (] 1 g 11 ( = 1 +. (71 (/ i 1 Fom a mathematial point of view, sine the ontibution to the integal in the lensing equation (63 is fully dominated by the egion i, and given the vey mild adial dependene of the empiial g 00 tem, we an take g 00 ( i g 00 ( in the above equation to yield: ( v g 11 ( = 1 = 1 (GM ba 0 1/ d = 1 g l M. (7 Thus, any meti theoy of gavity whee g 11 mathes the above expession in the egime whee gavitational lenses ae obseved

8 Gavitational lensing in extended χ 3/ gavity 1809 will auately epodue all the obseved lensing phenomenology, with M b the total bayoni mass of the objet in question (galaxies o goup of galaxies, and no hypothetial dak matte assumed to exist. Equations (59 and (7 give empiial mathematial elations fo the meti oeffiients at petubation ode O( whih epodue all obseved otation veloity and gavitational lensing phenomenology, without the inlusion of any dak matte omponent. Notie that the mass dependene of the seond tem on the ighthand side in expession (7 fo the meti oeffiient g 11 is the same as the fato in expession (59 fo g 00. This last was obtained fo a igoously flat otation uve in aodane with the Tully Fishe elation. This shows that the atio g /l M of the two impotant haateisti lengths of the extended meti theoy of gavity poposed by Benal et al. (011b is the deteminant dimensionless measue of deviations fom flat spae time at galati sales, exatly as expeted fom the dimensional analysis in Henandez (01. The meti oeffiient g 11 in equation (7 an be dietly ompaed to the esults fo the f (χ = χ 3/ meti theoy of Benal et al. (011b obtained in equation (55 with the inlusion of the esults of equation (60. This means that the hoie of the integation onstant k = 0, (73 makes these expessions fo the meti omponent g 11 idential. Use of the mathematial appoximation A x 1 + x ln A to wite the following expessions fo the full empiial meti oeffiients gives: g ( g /l M ln ( / ( / g/l M, (74 g 11 1 ( g /l M e g/l M. (75 We note that all the appoximations used in this setion intodue an eo seveal odes of magnitude smalle than the intinsi obsevational unetainties in the empiial elations used. Theefoe, all of the expessions given an be onsideed as stitly equivalent in egads to the auate modelling of astophysial otation uves and gavitational lensing data. 7 DISCUSSION By onstuting the weak field limit of the meti f (χ = χ 3/ theoy of gavity developed by Benal et al. (011b, we have shown that it is possible to explain both the dynamis of massive patiles and the defletion of light by obseved astonomial systems suh as elliptial galaxies, spial galaxies and goups of galaxies. Reently, the same meti theoy of gavity was shown to be oheent also with the expansion dynamis of the obseved Univese (Mendoza 01; Caanza et al This is an expeted esult fom a theoy of gavity onstuted though astonomial obsevations: it must be oheent at all sales. The egime of Einstein s geneal elativity is by no means violated, sine the appliations developed in this atile ( l M lie fa away fom the mass and length sales assoiated with the ones of Einstein s geneal elativity ( l M (see e.g. Mendoza 01. The esults of this atile wee onstuted using a stati spheially symmeti meti with the time and adial omponents petubed up to ode O( of appoximation. This wok genealizes the one of Benal et al. (011b in whih the adial meti omponent was assumed up to ode O(0 only and so, infomation on the hoie of signatue of the Riemann tenso was lost (see Appendix A. Suh infomation is vey impotant while woking with fouth-ode meti theoies of gavity. We mention again the temendous impotane of a oet hoie fo the signatue of the Riemann tenso as desibed in Appendix A. The hoie (A3, and only that hoie, used in this atile yields esults in ageement with astonomial obsevations. In othe wods, astonomial obsevations fix the oet (and unique hoie of signatue fo Riemann s tenso. This is an impotant esult, sine othewise solutions fom the othe banh appea whih ae not in aodane with astonomial obsevations. Table 1 summaizes ou main esults. It is impotant to emphasize that the empiial values of the meti omponents 00 and g( 11 do not depend on any gavitational theoy and as suh, they epesent funtions that any suessful theoy of gavity (suh as the one used in this atile needs to math. Notie that obsevationally, independent empiial onstaints fixed the / g /l M fatos in g 00 and g 11 to be equal; it is enouaging that the fomal mathematial petubation teatment of the theoy poposed also yields idential ˆR/3 fatos in the expessions fo g 00 and g 11. If this wee not the ase, even given the ompatible funtional foms of empiial and theoetial meti oeffiients, the f (χ = χ 3/ poposal would have been ejeted. An impotant fat aises fom the usage of the f (χ meti theoy of gavity and not the f (R fomalism. Although losely elated to eah othe fo a powe-law funtion (3 and a mass point soue, the oet dimensional appoah f (χ intodues mass and length sales that, as shown by Benal et al. (011b, need to be inopoated into the gavitational field ation. Although the field equations in vauum fo both f (χ andf (R unde a powe-law epesentation yield the same field equations (sine the mass M geneating the gavitational field is a onstant, f (R gavity is not apable of epoduing the uial lensing obsevations as it laks a uial onstaint equation (. The gavitational theoy f (χ = χ 3/ is able to do so sine unde this appoah the oet limit whee MONDian-like effets ae expeted yield the onstaint equation (1 o (. Notie howeve that both f (R andf (χ with the Table 1. The table shows the esults obtained fo the meti omponents 00 and g( 11 fo a stati spheial symmeti spae time in sales of galaxies and galaxy goups obtained empiially fom astonomial obsevations of these systems and the ones pedited by the meti f (χ = χ 3/ theoy of gavity of Benal et al. (011b. A good meti theoy of gavity must be suh that it onveges to the infeed values pesented in the table. The theoy f (χ = χ 3/ is in pefet ageement with the obseved meti omponents. The dimensionless atio fomed by the quotient of the gavitational adius g to the mass-length sale l M (see equation 11 is the deteminant dimensionless quantity of the poblem. Sine the meti omponents detemine the gavitational potential of the system, the length is undetemined. Howeve, sine the natual length sale of the system is l M one an always assume = l M, whih also ensues no sign hange in the potential in equation (61 ove the domain of appliability > l M. Meti oeffiient Obsevations ( g l M ln g l M (Tully Fishe (lensing ( ˆR 3 ln + k 1 k 1 + k ˆR 3 Theoy f (χ = χ 3/ ˆR = 6 g /l M k 1 = 0 ˆR = 6 g /l M k = 0

9 1810 S. Mendoza et al. appopiate hoie of Riemann s tenso (A3 ae able to epodue the flat otation uves of galaxies and the Tully Fishe elation. In an effot to genealize and look fo a fundamental basis to an f (χ theoy of gavity, Caanza et al. (013 and Mendoza (01 have shown that these meti theoies ae equivalent to the f (R, T onstution of Hako et al. (011. These authos have also shown that the patiula theoy f (χ = χ 3/ is in exellent ageement with osmologial obsevations of SNIa. An f (χ theoy of gavity satisfying the limits of equation (18 implies that gavity is no longe sale invaiant. In fat, peise gavity tests have been pefomed only at stong egimes of Einstein s gavity, whee χ 1, and so the involved aeleations of test patiles ae suh that a a 0 (see e.g. Will 006. In exatly the opposite egime, whee χ 1, whee the involved aeleations of test patiles ae suh that a a 0,gavity diffes fom Einstein s geneal elativity. The taditional appoah of assuming Einstein s geneal elativity to be valid at all sales means that unknown dak entities ae needed to explain vaious astophysial obsevations. This atile heavily einfoes many othes (Henandez et al. 010, 01; Benal et al. 011a,b; Mendoza et al. 011; Henandez & Jiménez 01; Mendoza 01; Caanza et al. 013 that show how astophysial and osmologial obsevations an be aounted fo without assuming the existene of dak entities and extending gavity so as to be non-sale invaiant. ACKNOWLEDGEMENTS The authos aknowledge the input of an anonymous efeee, helpful in eahing a leae final vesion of the atile. This wok was suppoted by thee DGAPA-UNAM gants (PAPIIT IN , IN and IN The authos TB, XH, JCH, SM and LAT aknowledge eonomi suppot fom CONACyT: 0759, 5006, 51009, 6344 and The authos thank the kind assistane povided by Cosimo Stonaiolo fo finding the analyti solution (43 of the diffeential equation (41. REFERENCES Banabè M., Czoske O., Koopmans L. V. E., Teu T., Bolton A. S., 011, MNRAS, 415, 15 Benal T., Capozziello S., Cistofano G., de Lauentis M., 011a, Mod. Phys. Lett. A, 6, 677 Benal T., Capozziello S., Hidalgo J. C., Mendoza S., 011b, Eu. Phys. J. C, 71, 1794 Binney J., Temaine S., 008, Galati Dynamis, nd edn. Pineton Univ. Pess, Pineton, NJ Capozziello S., Faaoni V., 011, Beyond Einstein Gavity. Spinge, Belin Capozziello S., Stabile A., 009, Classial Quantum Gavity, 6, Capozziello S., Cadone V. F., Toisi A., 006, Phys. Rev. D, 73, Capozziello S., Stabile A., Toisi A., 007, Phys. Rev. D, 76, Caanza D. A., Mendoza S., Toes L. A., 013, Eu. Phys. J. C, 73, 8 Caoll S., 004, Spaetime and Geomety: An Intodution to Geneal Relativity. Addison Wesley, San Faniso Dutton A. A. et al., 011, MNRAS, 417, 161 Dyson F. W., Eddington A. S., Davidson C., 190, Phil. Tans. R. So. A, 0, 91 Einstein A., 1916, Ann. Phys. Lpz., 354, 769 Famaey B., MGaugh S., 01, Living Rev. Relativ., 15, 10 Gavazzi R., Teu T., Rhodes J. D., Koopmans L. V. E., Bolton A. 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Sobouti Y., 007, A&A, 464, 91 Soussa M., 003, in Dumahez J., Giaud-Héault Y., Tân Thanh Vân J., eds, Po. 15th Renontes de Blois: Physial Cosmology: New Results in Cosmology and the Coheene of the Standad Model Soussa M. E., Woodad R. P., 003, Classial Quantum Gavity, 0, 737 Suyu S. H. et al., 01, ApJ, 750, 10 Toth V., 005, pepint (axiv:s/ Weinbeg S., 197, Gavitation and Cosmology: Piniples and Appliations of the Geneal Theoy of Relativity. Wiley, New Yok Will C. M., 1993, Theoy and Expeiment in Gavitational Physis. Cambidge Univ. Pess, Cambidge Will C. M., 006, Living Rev. Relativ., 9, 3 APPENDIX A: COMMENTS ABOUT THE SIGN CONVENTION IN RIEMANN S TENSOR In the study of the gavitational field equations, the link between the uvatue of spae time and the matte ontent is a key fat. All the infomation egading the uvatue of spae time is ontained in the Riemann uvatue tenso R α βηθ, whih is a funtion of the fist and seond deivatives of the meti. Fom a puely mathematial point of view, the Riemann tenso an be obtained fom the Commutato of ovaiant deivatives (Caoll 004: μ, ν ]V ρ = R ρ σμν V σ, (A1 fo any veto field V α. Fom a geometodynamial point of view, the uvatue tenso is onstuted though the hange A μ in a veto A μ afte being displaed about any infinitesimal losed ontou (Landau & Lifshitz 1975: A μ = Ɣ λ μν A λdx ν. By the use of

10 Gavitational lensing in extended χ 3/ gavity 1811 Stokes theoem it then follows that fo a suffiiently small losed ontou: A μ 1 Rλ μνθ A λ f νθ, (A whee f νθ epesents the infinitesimal aea enlosed by the ontou of the line integal. In this espet, it follows that the Riemann tenso measues the uvatue of spae time (f. Landau & Lifshitz In equations (A1 and (A, the Riemann tenso has been defined as R β μνα := Ɣβ μα,ν Ɣβ μν,α + Ɣβ λν Ɣλ μα Ɣβ λα Ɣλ μν. (A3 If Riemann s tenso is defined by equation (A3, then Rii s tenso is R να := g βμ R βμνα and Rii s sala is R α α. Sine these ae the most used definitions in elativity theoy nowadays, we will efe to these quantities as standad. Howeve, thee is anothe way in whih Riemann s tenso (and Rii s tenso an be defined, usually adopted by mathematiians and by CASs suh as Maxima ( In these ases, the syntaxis is suh that (see e.g. Toth 005 Rμ, ν, α, β] := R β μνα = Ɣβ μν,α Ɣβ μα,ν + Ɣβ λα Ɣλ μν Ɣβ λν Ɣλ μα. (A4 If Riemann s tenso is defined by equation (A4, then Rii s tenso is R να := g βμ R βμνα and Rii s sala is R α α. Although this hoie of signs fo the Riemann and Rii tensos is not vey muh in use these days, some well-known textbooks use them (see e.g. the table of sign onventions at the beginning of efeene Misne et al The CAS Maxima uses the definition (A4 and is suh that R maxima = R standad, (A5 in fee-index notation. As disussed in the table of sign onventions of Misne et al. (1973, geneal elativity an use any of the above definitions (and a few moe simply beause of the lineaity with whih Rii s sala and Rii s tenso appea in Einstein s field equations. This is howeve not the ase in meti f (R theoies of gavity, sine fo example in those theoies, the tae of the field equations is given by (see e.g. Capozziello & Faaoni 011 f (RR f (R + 3 f (R = 8πG T. (A6 4 To highlight the point, let us substitute the powe-law funtion (3 in the pevious equation to obtain (b R b + 3b R b 1 = 8πG T. (A7 4 This equation eflets a uial fat about the hoie of sign in Riemann s tenso. Due to the pesene of the deivative tem f (R = br b 1, depending on the sign onvention of the definition of the Rii sala, thee appeas a sign fato (± b 1 whih is not global to all the tems in the equation. This establishes a bifuation in this lass of solutions of the theoy. Indeed, fo a situation whee f (R = R a + R b o any moe ompliated funtion of R, thee is not (a pioi any indiation of whih onvention in the definition of Riemann s tenso should be used to desibe a patiula physial phenomena. In this atile, we show that, unde the theoy being pesented, the onvention an be settled. The esults pesented in this atile wee obtained with the standad definition of Riemann s tenso in equation (A3. That hoie (and only that one an aount fo both obseved dynamis of massive patiles in spial galaxies though the Tully Fishe elation, and fo the defletion of light obseved in gavitational lenses. An impotant aspet to point out is that the ase f (R = R of Einstein s geneal elativity is fee fom the above ambiguity. This is so beause it is possible to edefine the signatue fo the enegymomentum tenso to eove the same field equations (see e.g. Misne, Thone & Wheele 1973; Hobson, Efstathiou & Lasenby 006. We see fom this esult that pevious woks by Capozziello et al. (007, Capozziello & Stabile (009 have seleted the onvention used by the CAS Maxima in ode to ompute thei esults. In that espet, thei esults lie in anothe banh of the solutions of the field equations. If we would have taken fo example, the definition of Riemann s tenso by Maxima, then the meti oeffiients would have been: 00 = ˆR ln (/9 + A ln( + B and 11 = ˆR ln(/9 + D/ + ( ˆR A/ (whee A, B and D ae onstants. These ae vey diffeent fom the ones obtained in equations (53 and (55 and would have neve epodued the astophysial obsevations teated in this atile. It is only though the oet hoie of signs in the definition of Riemann s tenso, suh as the ones used in the pesent atile and epesented in equation (A3, that the good ageement with the Tully Fishe elation and lensing obsevations an be oetly obtained. APPENDIX B: COMMENTS ABOUT THE MAXIMA CODE In this setion, we give a bief intodution to the ode we wote in the CAS Maxima ( to obtain the field equations. Speifially, we wok with the module tenso (f. Toth 005. The syntax of suh module is that, when invoked, it uns an input intefae to design the fom of the ovaiant meti. The Maxima ode MEXICAS is opyight of TB, SM and LAT, liensed unde a GNU Publi Geneal Liense (GPL, vesion 3 (see an be obtained fom (see the setion about opyight and usage in that web page. Fo the implementation of the ode, we onside a petubative appoah in the paamete ɛ := 1/, suh that the ovaiant omponents of the meti ae given by g 00 = 1 + ɛ 00 + O(4, g 11 = 1 + ɛ 11 + O(4, (B1 whee the angula omponents ae given by the standad expessions fo spheial oodinates as shown in equation (6. With these equations, it is simple to onstut the ontavaiant omponents of the meti: g 00 = 1 ɛ 00 + O(4, g 11 = 1 ɛ 11 + O(4. (B With these onsideations, the meti is eoded in the tenso module. Fom this fat, it is simple to invoke all the quantities equied to onstut the field equations, eithe in geneal elativity o fo any extended meti theoy of gavity. Fo example, in a desiptive way onening the syntaxis of maxima it follows that histof(ms Ɣ λ μν, (B3 and with simila syntaxis fo the Riemann tenso, the Rii tenso and the Rii sala. Due to the fat that the meti has an ode paamete ɛ, all the tensoial quantities involved in the onstution of the field