On the Null Trajectories in Conformal Weyl Gravity
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On the Null Trajectories in ConformalWeyl Gravity
J. R. Villanueva, a Marco Olivares b a Departamento de F´ısica y Astronom´ıa, Universidad de Valpara´ıso, Gran Breta˜na 1111,Playa Ancha, Valpara´ıso, Chile, a Centro de Astrof´ısica de Valpara´ıso, Gran Breta˜na 1111, Playa Ancha, Valpara´ıso, Chile. b Instituto de F´ısica, Pontificia Universidad Cat´olica de Valpara´ıso,Av. Universidad 330, Curauma, Valpara´ıso , Chile.
E-mail: [email protected], [email protected]
Abstract.
In this work we find analytical solutions to the null geodesics around a black holein the conformal Weyl gravity. Exact expressions for the horizons are found, and they dependon the cosmological constant and the coupling constants of the conformal Weyl gravity. Then,we study the radial motion from the point of view of the proper and coordinate frames, andcompare it with that found in spacetimes of general relativity. The angular motion is alsoexamined qualitatively by means of an effective potential; quantitatively, the equation ofmotion is solved in terms of ℘ -Weierstrass elliptic function. Thus, we find the deflectionangle for photons without using any approximation, which is a novel result for this kind ofgravity. Keywords:
Modified Gravity; Black Holes; Elliptic Functions. Corresponding author. ontents
For nearly a century, the Einstein’s theory of gravitation [1] has been studied from variousaspects, delivering many successes both in theoretical and observational physics, but at thesame time, arising many questions. For example, this theory is not based on any fundamentalprinciple, it is not invariant under conformal transformations, and cannot be described as aquantum field theory. Besides, the observations of the velocity distributions in the vicinityof galaxies is not satisfactory from the point of view of Einstein’s gravitation, leading tointroduction of the dark matter to avoid this problem. Thus, it makes it desirable to find analternative theory of gravity that would repeat the success of Einstein’s theory, but also fixits problems.A possible candidate is the Weyl theory, introduced in 1917 to unify gravity and elec-tricity [2], based on the principle of local invariance of a manifold, endowed with the metric g µν ( x ), under the change g µν ( x ) → Ω ( x ) g µν ( x ) , (1.1)where Ω( x ) is a smooth, strictly positive function. In a series of works, Mannheim andKazanas [3] explored the structure of the fourth-order conformal Weyl gravity (CWG) pro-viding exact solutions to this theory. Thereafter, much research has been made based on thistheory. For example, the study of black holes solutions can be found in [4] for topologicalblack holes; in [5] for static cylindrical black holes; and [6] for AdS and Lifshitz black holes.Recently, spherical solutions for charged Weyl black holes has been found in [7]. Some cos-mological implications of CWG are found in [8], where the authors calculated the abundancesof the primordial light element; also in [9], with ΛCDM model and conformal gravity putface to face by using the γ -ray bursts data, as well as in [10] where the authors explain theproperties of X-ray galaxy clusters without resorting to dark matter. Recently, Mannheim[11] has studied cosmological perturbations showing the first steps for the analysis of thetensor fluctuations in CWG. From the point of view of general relativity test, particularlywith respect to the deflection of light, much research has been performed. In this sense, S.Pireaux presents a series of papers giving account of the critical distances of photons [12], andthe constraints on the linear parameters of the theory [13]. After that, based on the worksby Rindler and Ishak [14–16], in which they show that the deflection of light is influenced bythe cosmological constant, several authors have tried to investigate the influence of the Weylparameter in the deflection of light (see, for example, [18, 19]). Unfortunately, this deduction– 1 –s not entirely exact. Thus, our interest is to study the allowed motion for massless particlesfollowing the Lagrangian formalism showed in [20, 21] to obtain the exact solution for thetrajectories and, consequently, an exact expression to the bending of light in CWG. Let us consider a conformal Weyl gravity. An exact static, spherically symmetric black holesolution is given by [3] d ˜ s = − B (˜ r ) d ˜ t + d ˜ r B (˜ r ) + ˜ r ( dθ + sin θdφ ) , (2.1)where the coordinates are defined in the range −∞ < ˜ t < ∞ , ˜ r ≥
0, 0 ≤ θ ≤ π and0 ≤ φ ≤ π , and the lapse function, B (˜ r ), is given by B (˜ r ) = 1 − β (2 − β ˜ γ )˜ r − β ˜ γ + ˜ γ ˜ r − ˜ k ˜ r . (2.2)Here β , ˜ k and ˜ γ are positive constants associated to the central mass, cosmological constantand the measurements of the departure of the Weyl theory from the Einstein - de Sitter,respectively. Clearly, taking the limit ˜ γ = 0 recovers the Schwarzschild - de Sitter (SdS) caseso that we can identify β = M [22]. Is more convenient to work with dimensionless constantsappearing in the lapse function, by making the following identifications d ˜ sβ → ds, ˜ tβ → t, ˜ rβ → r, β ˜ γ → γ, β ˜ k → k. We obtain ds = − B ( r ) dt + dr B ( r ) + r ( dθ + sin θ dφ ) , (2.3) B ( r ) = 1 − (2 − γ ) r − γ + γr − kr , (2.4)such that the characteristic polynomial of CWG can be written as p ( r ) = − rB ( r ) k = r − γk r − (1 − γ ) k r + (2 − γ ) k . (2.5)The zeros of the polynomial p ( r ) give us the locations of the horizons (if there are any). Inorder to study the nature of its roots, we perform the following change of variable r = x + γ k (2.6)which yields to p ( x ) = x − η x − η (2.7)where the coefficient are given by η = 1 k (cid:18) − γ + γ k (cid:19) , (2.8) η = − k (cid:18) − γ − γ (cid:18) − γk (cid:19) − γ k (cid:19) , (2.9)– 2 –nd the cubic discriminant is ∆ c = 27 η − η . Therefore, there are three options for theroots. If ∆ c <
0, there is one real negative root and a complex pair of the roots, and thisrepresents a naked singularity; if ∆ c >
0, there are three different real roots, two positiveand one negative, which looks similar to the SdS spacetime with an event and cosmologicalhorizons; finally, if ∆ c = 0, there are three real roots, a negative one plus a degeneratepositive root, and this represents the extreme case. Assuming that γ is small, one couldcheck that the coefficient η is negative, and therefore the discriminant of the polynomial ispositive, ∆ c >
0. Denoting r w = γ k , R = r η , ϕ = 13 arccos (cid:18) | η | R (cid:19) , (2.10)we can find the expression for the event horizon, r + , the cosmological horizon, r ++ , and thenegative root (without physical meaning), r n , and they have the form r + = r w + R ϕ − √ ϕ ) , (2.11) r ++ = r w + R cos ϕ, (2.12) r n = r w − R ϕ + √ ϕ ) . (2.13)Again, the SdS case [23] is recuperated for zero Weyl parameter ( γ = 0).The motion of photons in this geometry can be determined using the standard La-grangian procedure for geodesic [21]. In fact, the Lagrangian for the metric (2.1) is givenby 2 L = − B ( r ) ˙ t + ˙ r B ( r ) + r ( ˙ θ + sin θ ˙ φ ) ≡ , (2.14)where the dot denotes a derivative with respect to the affine parameter τ along the geodesic.The equations of motion are obtained from˙Π q − ∂ L ∂q = 0 , (2.15)where Π q = ∂ L /∂ ˙ q are the conjugate momenta to the coordinate q . Since ( t, φ ) are cycliccoordinates, its corresponding conjugate momenta are conserved, thereforeΠ t = − B ( r ) ˙ t = −√ E, and Π φ = r sin θ ˙ φ = L, (2.16)where E and L are integration constants dimensionless associated to each of them. Further-more, these two constants of motion allows us to define the impact parameter by the relation b ≡ L √ E . Without lack of generality we consider that the motion is developed in the invariantplane θ = π/
2, in which case our set of differential equations are (cid:18) drdτ (cid:19) = E (cid:18) − V ( r ) E (cid:19) , (2.17) (cid:18) drdt (cid:19) = B ( r ) (cid:18) − V ( r ) E (cid:19) , (2.18) (cid:18) drdφ (cid:19) = r b (cid:18) − V ( r ) E (cid:19) , (2.19)– 3 – V r ( ) b c b d b f r + r ++ b b ∞ r c r d Figure 1 . This plot shows the conformal effective potential for photons in Weyl’s gravity with thevalues of parameters k = γ = 0 .
01, and L = 3. Since γ is small, the spacetime looks similar to theSchwarzschild-de Sitter. Note that r d → r ++ when b → ∞ . where V ( r ) corresponds to the conformal effective potential defined by V ( r ) = B ( r ) b Er = (cid:18) − − γr − γ + γr − kr (cid:19) b Er . (2.20)A typical plot of the effective potential (2.20) is shown in FIG.1. Radial motion corresponds to a trajectory with null angular momentum (or zero impactparameter), and photons are destined to fall toward the event horizon or to the cosmologicalhorizon. From Eq. (2.20) we can see that for radial photons we have V ( r ) = 0, so that eqs.(2.18) and (2.19) become drdτ = ±√ E, (2.21)and drdt = ± B ( r ) , (2.22)respectively, and the sign + ( − ) corresponds to photons falling into the cosmological (event)horizon. Choosing the initial conditions for the photons as r = r i when t = τ = 0, eq. (2.21)yields τ + = r i − r + √ E , (2.23)and τ ++ = r ++ − r i √ E , (2.24)which tell us that, in the proper frame of the photons, they arrive to the event (cosmological)horizon in a finite proper time. On the other hand, a straightforward integration of eq. (2.22)leads to t ( r ) = ± √ k ln "(cid:12)(cid:12)(cid:12)(cid:12) r − r + r i − r + (cid:12)(cid:12)(cid:12)(cid:12) µ (cid:12)(cid:12)(cid:12)(cid:12) r ++ − rr + − r i (cid:12)(cid:12)(cid:12)(cid:12) − µ (cid:12)(cid:12)(cid:12)(cid:12) r − r n r i − r n (cid:12)(cid:12)(cid:12)(cid:12) µ , (2.25)– 4 –here the constants are given by µ = r + √ k ( r ++ − r + )( r + − r n ) ,µ = r ++ √ k ( r ++ − r + )( r ++ − r n ) ,µ = − r n √ k ( r ++ − r n )( r + − r n ) . Furthermore, taking the limit r → r + ( r → r ++ ) in eq. (2.25), it can be shown that themassless particles take an infinite coordinate time to cross the event (cosmological) horizon.This facts result to be common with the spherically symmetric spacetimes of general relativity[20, 21]. In FIG.2, eqs. (2.23, 2.24, 2.25) are plotted. T I M E A X I S r r + r ++ r i t t ττ + + ++ ++ Figure 2 . Plot for the radial motion of photons in CWG. As we can see, in the proper frame, thephotons arrive to the horizon in a finite time, τ + (or τ ++ ) given by (2.23) (or (2.24)), while in thecoordinate frame they arrives in an infinite time. This behaviour is common with the motion ofphotons in static spherical symmetric spacetimes of general relativity. When the angular momentum is non-vanishes, b = 0, then V ( r ) = 0 in eq. (2.20). Afirst observation from this conformal effective potential is that it reaches the maximum at r c = 3 (independently on γ and k ), which is also characteristic for the Schwarzschild’sspacetimes. For sure, the impact parameter has a different value at this distance, giving b c = 27(1 + 3 γ − k ) − , which includes the terms from the cosmological constant, k , andCWG, γ .Next, based on the impact parameter values shown in FIG.1, we present a brief quali-tative description of the allowed angular motions for photons in CWG. • Capture Zone : If 0 < b ≡ b f < b c , photons fall inexorably to one of the two horizons,depending on initial conditions, and its cross section, σ , in these geometry is [24] σ = π b c = 27 π γ − k . (2.26)– 5 – Critical Trajectories : If b = b c , photons can stay in one of the unstable inner circularorbit of radius r c = 3. Therefore, the photons that arrive from the initial distance r i ( r + < r i < r c , or r c < r i < r ++ ) can asymptotically fall to a circle of radius r c . Theproper period in such orbit is T τ = 18 πL , (2.27)which results to be the same as the one in the Schwarzschild case [25], whereas thecoordinate period depends on k and γ as T t = 6 √ π √ γ − k . (2.28) • Deflection Zone . If b c < b = b d < ∞ , the photons come from a finite distance r i ( r + < r i < r c or r c < r i < r ++ ) to a distance r = r d (which is solution of the equation V ( r d ) = E ), then return to one of the two horizons. This photons are deflected. Also,is possible to find the deflection distance, r d , which results to be r d ( b ; γ, k ) = r ̺ "
13 arccos ̺ s ̺ ! + a , (2.29)where, a = γ D , and D is the anomalous impact parameter given by D = b √ k b . (2.30)We also have ̺ = 4 D − γ + γ D ) , (2.31)and ̺ = 4 D
27 ( − − γ D + 2 γ D + 9 γ (9 + D )) . (2.32)Note that, making b → ∞ in (2.30), drives to ̺ = η (c. f. eqs. (2.31), (2.8)), and ̺ = η (c. f. eqs. (2.32), (2.9)), so we obtain the identity r d ( ∞ ; γ, k ) = r ++ .With this in mind, after a brief manipulation, it is possible to integrate out eq. (2.19)and obtain a general solution r ( φ ) = 1 + α ℘ ( φ + ω ) + α , (2.33)where ℘ ( x ) ≡ ℘ ( x ; g , g ) is the ℘ -Weierstrass function with g = 1 /
12 (again, independenton k and γ ). We also have in that case g ( b ; γ, k ) = 116 (cid:20) − γ + γ + (cid:16) α D (cid:17) (cid:21) , (2.34)and the other parameter is α = 1 − γ , together with ω which is an arbitrary constant ofintegration. Note that different orbits are obtained from (2.33) depending on the value ofthe impact parameter b in (2.34). Thus, the capture trajectory for b < b c is showed in FIG.3,while the allowed critical trajectories for b = b c are shown in FIG.4.– 6 – + r + r Figure 3 . Polar plot for a capture trajectory of photons. Massless particles inevitably fall into oneof the horizons rr + r c r c r + r i Figure 4 . Polar plot for a critical trajectory of photons coming from a distance r i (LEFT: r c < r i We acknowledge stimulating discussion with Olivera Miskovic (PUCV), V´ıctor C´ardenas(UV) and Mario Pedreros (UTA). M.O. was supported by PUCV through the Proyecto DIPostdoctorado 2012. References [1] A. Einstein, Die feldgleichungen der Gravitation , 1915 Sitzung der physikalisch-mathematischenklasse Zur Gravitationstheorie , 1917 Annalen der Phys. ReineInfinitesimalgeometrie , 1918 Math. Zeitschr. Gravitation und Electrizit¨at , 1918Sitz. Ber. Preuss. Ak. Wiss. 465; R. Bach, Zur Weylschen Relativit¨atstheorie und der WeylschenErweiterung des Kr¨ummenstensorsbegriffs , 1921 Math. Zeitschr. Exact Vacuum Solution To Conformal Weyl Gravity AndGalactic Rotation Curves , 1989 Astrophys. J. General Structure Of The Gravitational Equations Of Motion In Conformal Weyl Gravity ,(1991) Astrophys. J. Suppl. Solutions to theReissner-Nordstr¨om, Kerr, and Kerr-Newman problems in fourth-order conformal Weyl gravity ,1991 Phys. Rev. D Newtonian limit of conformalgravity and the lack of necessity of the second order Poisson equation , (1994) Gen. Rel. Grav. Topological black holes in Weyl conformal gravity , 1998 Class. Quantum Grav. Exact Static Cylindrical Solution to Conformal WeylGravity , 2012 Phys. Rev. D – 9 – 6] H. Lu, Y. Pang, C. N. Pope, and J. F. Vazquez-Poritz, AdS and Lifshitz Black Holes inConformal and Einstein-Weyl Gravities , 2012 Phys. Rev. D Spherical Solutions due to the Exterior Geometry of a Charged WeylBlack Hole , 2012 Int. J. Theor. Phys. Primordial nucleosynthesis in conformal Weyl gravity , 1993astro-ph/9311006.[9] A. Diaferio, L. Ostorero and V. F. Cardone, γ -ray bursts as cosmological probes: Λ CDM vs.conformal gravity , 2011 JCAP X-ray clusters of galaxies in conformal gravity , 2009 Mon. Not.Roy. Astron. Soc. Cosmological Perturbations in Conformal Gravity , 2012 Phys. Rev. D Light deflection in Weyl gravity: Critical distances for photon paths. , 2004 Class.Quant. Grav. Light deflection in Weyl gravity: Constraints on the linear parameter , 2004 Class.Quant. Grav. Contribution of the cosmological constant to the relativistic bendingof light revisited , 2007 Phys. Rev. D The Relevance of the Cosmological Constant for Lensing , 2010 Gen.Rel. Grav. More on Lensing by a Cosmological Constant , 2010 Mon.Not. Roy. Astron. Soc. , 2152.[17] F. Finelli, M. Galaverni and A. Gruppuso, Light Bending as a Probe of the Nature of DarkEnergy , 2007 Phys. Rev. D Light bending in thegalactic halo by Rindler-Ishak method , 2010 JCAP Bending of light in conformal Weyl gravity , 2010 Phys. Rev. D The Mathematical Theory of Black Holes , Oxford University Press, NewYork, 1983.[21] N. Cruz, M. Olivares and J. R. Villanueva, The geodesic structure of the Schwarzschild anti-deSitter Black Hole , 2005 Class. Quant. Grav. Motion of charged particles on the Reissner-Nordstr¨om (Anti)-de Sitter blackholes , 2011 Mod. Phys. Lett. A Photons motion in charged anti de Sitter black holes , 2013 Astrophys. Space Sci. , 437-446.[22] A. Edery and M. B. Paranjape, Classical tests for Weyl gravity: Deflection of light and timedelay , 1998 Phys. Rev. D , 024011.[23] M. J. Jaklitsch , C. Hellaby and D. R. Matravers, Particle Motion in the Spherically SymmetricVacuum Solution with Positive Cosmological Constant , 1989 Gen. Rel. Grav. , 941.[24] R. Wald, General Relativity , University of Chicago Press, Chicago, 1984.[25] B. Shutz, A First Course in General Relativity , Cambridge University Press, 1990., Cambridge University Press, 1990.