General Relativity and Gravitation, Vol. 33, No. 8, 2001 Gravitational Potentials of Triaxial Ellipsoids in Weyl Gravity Ernesto P. Esteban1,2 and Demosthenes Kazanas2 Received July 19, 2000; revised version January 4, 2001 We present analytic expressions for the gravitational potentials associated with triaxial ellipsoids, spheroids, spheres and disks in Weyl gravity. The gravitational potentials of these configurations in Newtonian gravity, i.e. the potentials derived by integration of the Poisson equation Green’s function 1/ | r − r′ | over the volume of the configuration, are well known in the literature. Herein we present the results of the integration of | r − r′ | , the Green’s function associated with the fourth order Laplacian ∇4 of Weyl gravity, over the volume of the configuration to obtain the resulting gravitational potentials within this specific theory. As an application of our calculations, we solve analytically Euler’s equations pertaining to incompressible rotating fluids to show that, as in the case of Newtonian gravity, homogeneous prolate configurations are not allowed within Weyl gravity either. KEY WORDS: Weyl gravity, ellipsoids. 1. INTRODUCTION One of the most attractive features of the standard second order Einstein theory of gravity is that it provides a covariant description of not only the exterior Newtonian gravitational potential but also of the second order Poisson equation as well, which allows the computation of the former for given, arbitrary distributions; in fact, this constitutes one of main reasons for having a second order gravitational theory in the first place. With the observational confirmation of the relativistic corrections to the Newtonian limit that the theory then yields, the overwhelming consensus in the community is that the correct theory of gravity has already been found, at least at the classical level. Despite this 1 Department 2 Laboratory of Physics, University of Puerto Rico, Humacao, PR 00791. High Energy and Astrophysics, NASA/ GSFC. 1281 0001-7701/ 01/ 0800-1281$19.50/ 0  2001 Plenum Publishing Corporation Esteban and Kazanas 1282 consensus (which has so far not been eroded even though the standard NewtonEinstein gravitational theories require the Universe to contain enormous amounts of as yet unestablished non-luminous or dark matter of various forms on different length scales, ranging from massive weakly interacting particle at the scales of galaxies to a cosmological constant at its largest scales), it should be noted that as of today there is in fact no known basic underlying principle which would require a relativistic gravitational theory, or even its weak gravity limit for that matter, to actually be second order. There is thus some value in exploring other candidate covariant equations of motion for the gravitational field to see whether they might also fit observation, so that we can then address basic issues of principle such as the uniqueness of gravitational theory and identify what it is that the data actually mandate. Such a theory is the theory of conformal gravity proposed by H. Weyl and discussed in [1, 2, 3, 4], which is defined by the requirement that the gravitational action be invariant to conformal stretching of the geometry. This requirement leads to a unique action consistent with it, namely ∫ c − 2a d x( − g) / (R ∫ I W c − a d 4 x( − g)1/ 2 Clmn k C lmn k 4 1 2 lm R lm − (Ra a )2 / 3). (1 ) Mannheim & Kazanas have written down and found, among others, also the solution to the static spherically symmetric problem of this theory [1]. The exact solution of this problem, namely the determination of the function B(r) of the line element ds2 c − B(r)d t2 + dr2 / B(r) + r 2 dQ , (2 ) B(r) c 1 − 3bg − b(2 − 3gb)/ r + gr − kr2 , (3 ) is given by where b, g and k are integration constants. While the above expression provides the vacuum spherically symmetric solution, in order to determine the gravitational potentials of realistic sources, one is in need (of at least approximate) solutions associated with sources. In order to achieve this, one needs the analog of Poisson’s equation and its Green’s function solution. One of the surprising features of conformal gravity is the fact that, despite Gravitational Potentials of Triaxial Ellipsoids in Weyl Gravity 1283 the highly non-linear character of its equations (see [1, 2, 4]), at least in the case of the static spherically symmetric problem, there exists a combination of the 00 and 11 components of the gravitational tensor, W mn , which results in a rather simple expression. Namely it was shown that 3(W 00 − W rr )/ B c B′′′′ + 4B′′′ / r c (rB)′′′′ / r c ∇4 B (4 ) This is the equation that plays the role of Poisson’s equation of the second order theories. It should be further noted here that the above expression is an exact relation, while Poisson’s equation obtains only approximately from Einstein’s equations in the weak gravity limit. The obvious problem associated with the 4th order version of Poisson’s equation is that the Green’s function of the ∇4 operator is − | r − r′ | / 8p; thus it is not immediately apparent how could one obtain the Newtonian potential out of the conformal theory. However, as pointed out in [3], integration of this specific Green’s function over an extended source leads to both a Newtonian and a linear potential, precisely as implied by the exact vacuum solution of the theory. Thus, in order to obtain a Newtonian potential in conformal gravity the gravitating sources must by necessity be extended, as it is in fact thought to be the case within the framework of the more popular current string theories. Mannheim & Kazanas [3] have furthermore provided a candidate form for such an extended source which yields for an elementary source a potential of the form V(r) c − b/ r + gr / 2 (5) In the case of weak gravity, one can compute the potential of a source by summing up the potentials of such elementary sources over its entire volume, to presumably obtain a potential of the form of Eq. (3). It is thus of some importance in understanding what is the origin of each such term. The form of metric given by Eq. (3) contains besides the well known Newtonian potential term also linear and quadratic terms. Of these the quadratic term is well known and representsf spherically symmetric solutions with a cosmological constant of magnitude k in Einstein gravity, but it is a term present in the vacuum solutions of the present and other 4th order theories. However, the linear term does not represent any obvious known spherically symmetric solution within Einstein gravity. It has been suggested, both by order of magnitude arguments [1] and also on the basis of fits to observed galactic rotation curves without invoking the presence of dark matter [5], that its presence indicates the regime at which the effects of Weyl gravity become important. A linear term does appear however in some non-spherical (axisymmetric), non-asymptotically flat solutions in general relativity, for instance, in the Esteban and Kazanas 1284 so called magnetized Schwarzschild metric [6]. This electrovac solution of the coupled Einstein-Maxwell equations can be written as follows ds2 c L2 [(1 − 2M / r)d t2 − dr2 / (1 − 2M / r) − r 2 dv 2 ] − L − 2 r 2 sin2 vdJ 2 , (6 ) where L c1+ r 2 B2 sin2 v, 4 (7 ) M is the gravitating source’s, B is the external magnetic field parameter and r, v, J, are “Schwarzschild like” coordinates. This solution is interpreted as describing the exterior space-time of a massive body immersed in an external magnetic field. By expanding the covariant term L2 (1 − 2M / r), in Eq. (6) we can verify the existence of an extra linear term (as well as other terms of higher order) as compared to the standard Newtonian potential. In addition, a Yukawa potential, V Y c GMe − r/ l / r, a general potential consistent with field theory, also seems to suggest the existence of an extra linear term in the gravitational potential over some limited range in r. In fact, assuming r / l << 1, the Yukawa gravitational potential can be written (in a second order approximation) as V Y c GM 冢r − l + l 冣, 1 1 r 2 (8 ) where l is a constant. Before we proceed, it is instructive to discuss the issue of the nature of the linear term in Eq. (3). As noted in references [2, 7], using a coordinate transformation one is able to cast the metric of Eq. (3) in a form conformal to Schwarzschild-de Sitter. Since the theory is conformally invariant it was argued that the linear term will have no consequence on the dynamics. While this may be the case for massless particles, massive particles may still require the specification of a particular conformal frame. It is our view that this issue is still under debate until a well defined physical meaning for this particular term becomes clearer in the future. Another issue with respect to the relevance of the conformal theory in observations was raised in [8]; these authors argued that the matching of the exterior spherically symmetric solution of the Weyl theory to an interior one leads either to sources of infinite extent or it is not consistent with the weak energy condition. On the other hand, in [3] Mannheim & Kazanas have given Gravitational Potentials of Triaxial Ellipsoids in Weyl Gravity 1285 an example of a spherically symmetric source whose energy density is positive almost everywhere, while trapping a singularity at the center, which could then provide both a source of finite extent. As of the moment of this writing, it is not apparent, at least to the authors, what the precise form of the stress energy momentum tensor of the elementary particles is and whether it is in fact inconsistent with the structure implied by Weyl gravity. In this paper we take at face value the Green’s function of the ∇4 operator of [3] and provide general analytic expressions which allow one to calculate, in closed form, the gravitational potentials due to a linear term such as that of Eq. (5) inside homogeneous ellipsoids, spheroids, spheres, and heterogeneous flat disks. These are of interest for computing the stability properties of these configurations, using a minimum energy method, as discussed and applied to rotating ellipsoids in Newtonian gravity by Christodoulou [9] and by us within Weyl gravity in a forthcoming publication. We believe that the expressions we derive and our methodology are of interest in domains of astrophysics broader than Weyl gravity, as the associated integrals are also encountered in galactic dynamics [10] (known as the Rosebluth integrals) and in the structure of rotating ellipsoids [11]. The paper is structured as follows: In §2 we provide the general framework for the computation of the linear potential of ellipsoid triaxial configurations that is a generalization of the method used in computing their Newtonian potentials. Then, the general expressions for these potentials are computed in terms of elliptic integrals or general geometric quantities such as those defined and used by Chandrasekhar (1969) [11] in his book “Ellipsoidal Figures of Equilibrium” (hereafter EFE). Specific expressions of these general potentials are also obtained in cases of higher symmetry such as axisymmetric or spherical configurations, as well as disks of specific surface density profiles. Using the expressions obtained in §2 for the linear potentials of ellipsoidal configurations, in §3 we compute the structure of rotating, uniform axisymmetric ellipsoids in this context and compare our results to those of [12], who discussed the existence of such configuration using an alternative more restrictive method. Finally, in §4 our results are summarized and some general conclusions are drawn. 2. POTENTIAL IN THE INTERIOR OF SELF-GRAVITATING HOMOGENEOUS ELLIPSOIDS AND SPHEROIDS To begin with, consider an incompressible (constant density) fluid bounded by a surface of the form y2 h2 z2 c1 + + a2 b2 c2 (9) Esteban and Kazanas 1286 where a, b and c are the ellipsoid’s semiaxes. Also, let P(x, y, z) be the point in which the potential will be calculated and R(y, h , z) will be a random point within the ellipsoid. Following closely the procedure discussed in [13] we can take P(x, y, z) as the origin of a spherical coordinate system (r, f, v). These are related to the coordinates y, h , z associated with the ellipsoid through the following relations y c x + r sin v cos f h c y + r sin v sin f z c z + r cos v (12) r c [(y − x)2 + (h − y)2 + (h − y)2 + (z − z)2 ]1/ 2 . (13) (10) (11) where Thus, the contribution to the gravitational potential due to the linear term in Eq. (5) at point P, V L , is given by ∫ V L c g r dm c gr o p 2p r1 0 0 0 ∫∫ ∫ r 3 sin vdv df dr (14) where r o is the density and g is the dimensionful coefficient appearing in Eq. (5). By substituting Eqs. (10)–(12) into Eq. (9) we obtain the following expression for the coordinate r 1 of the boundary surface of the ellipsoid − B + (B2 − AC)1/ 2 A (15) Ac sin2 v cos2 f cos2 v sin2 v sin2 f + + , a2 b2 c2 (16) Bc x sin v cos f y sin v sin f z cos v + + , a2 b2 c2 (17) Cc x2 y2 z2 + 2 + 2 −1 2 a b c (18) r1 c where The positive root was chosen in the solution of the quadratic equation giving the value of r 1 (Eq. 15), because this parameter must be positive. Gravitational Potentials of Triaxial Ellipsoids in Weyl Gravity 1287 One can easily perform the r-integration in Eq. (14) and then substitute the value of r 1 from Eq. (15). The resulting expression can be simplified significantly noticing that the quantity B changes sign at two diametrically opposite points of the configuration, terms containing odd powers of B will vanish upon the angular integration. Thus, the expression for V L reduces to VL c gr o 4 p 2p 0 0 ∫∫ (8B4 − 8B2 AC + A2 C 2 ) sin v dv df. A4 (19) We substitute, next, the value of B (Eq. 17) into Eq. (19) above. Two additional ellipsoidal symmetries (v fixed and f r − f; f fixed and v r p − v) similarly guarantee the vanishing of all odd parity terms. Thus, the expression for the linear potential V L reduces to the following form V L c gr o p 2p ∫ ∫ 冢A 0 0 f1 4 + f 2C C2 + 3 A 4A2 冣 sin v dv df (20) where f1 c 2 冢 + z4 x 2 z2 y2 z 2 2 4 2 2 cos v + 6 cos f cos v sin v + 6 cos2 v sin2 v sin2 f c8 a4 c4 b4 c4 x4 y2 x 2 y4 cos4 f sin4 v + 6 4 4 cos2 f sin2 f sin4 v + 8 sin4 f sin4 v 8 a a b b 冣 (21) f 2 c −2 冢 x2 y2 z2 2 2 2 2 cos f sin v + sin f sin v + cos2 v . a4 b4 c4 冣 (22) Again following the methodology employed in [13] for the computation of the Newtonian potential of an ellipsoid, it is convenient to express the appropriate gravitational potential in terms of the quantity W defined as W c ro p 2p 0 0 ∫∫ sin v df dv . A As discussed in EFE [11], the above integral can also be rewritten as (23) Esteban and Kazanas 1288 W c 2pr o I, (24) where I is the ellipsoid’s moment of inertia defined as I c abc ∫ ∞ 0 dv 2bc F(J, k) c D sin J (25) with D c [(a2 + v)(b2 + v)(c2 + v)]1/ 2 . Also, J c Arccos(c/ a), k c ((a2 − b2 )/ (a2 − c2 ))1/ 2 , and F(J, k) is the incomplete elliptic integral of the first kind. The case of the Newtonian potential computation, though it follows the same steps it is far simpler since its general form (the equation analogous to that of Eq. (20) here) involves only terms proportional to A − 2 , in contrast to the present case which involves also terms proportional to A − 3 and A − 4 . To perform this more complicated calculation we define below the auxiliary expression W I , which is expressed in terms of the derivatives of the quantity W defined above. Thus W I c a3 ∂W ∂W ∂W + b3 + c3 ∂a ∂b ∂c (26) Making use of expressions given in EFE, W I can be rewritten as W I c 2pr o [a2 (b2 + c2 )A1 + b2 (a2 + c2 )A2 + c2 (a2 + b2 )A3 ], (27) where A1 c abc A3 c abc ∫ ∫ ∞ 0 ∞ 0 [ 冣[ ] 冢 冣 F(J, k) − E(J, k) k 2p sin3 J 冢 (b/ c) sin J − E(J, k) k 2p sin3 J dv bc c2 D(a2 + v) a2 dv bc c2 D(c2 + v) a2 , (28) ] , (29) with k 2p c 1 − k 2 , and where E(J, k) is the incomplete elliptical integral of the second kind. Also, A2 can be expressed in terms of A1 , A3 from the relation A1 + A2 + A3 c 2 . Using this new algebraic expression, W I , and its derivatives, one can now recast the form of the linear potential of the ellipsoid given by Eq. (20) in the general form Gravitational Potentials of Triaxial Ellipsoids in Weyl Gravity 1289 V L c V I + V I I + V III (30) where the quantities V I , V I I , V III correspond to each of the terms in the integrand of Eq. (20) and can be expressed in terms of W I as follows cg [ p 2p 0 0 ∫∫ V I c gr o x4 12a2 + y4 12b2 + z4 12c2 + 冢 4ab f1 sin v df dv A4 ∂2 W I c ∂2 W I b ∂2 W I + + 2 2a ∂a∂c 2a ∂a∂b ∂a 冢 冣 冢 ∂2 W I a ∂2 W I c ∂2 W I + + 2b ∂a∂b 2b ∂b∂c ∂b2 冣 冢 ∂2 W I a ∂2 W I b ∂2 W I + + 2c ∂a∂c 2c ∂b∂c ∂c2 冣 x 2 y2 ∂2 W I y2 z2 ∂2 W I z2 x 2 ∂2 W I + + ∂a∂b 4bc ∂b∂c 4ca ∂c∂a V I I c gr o c− p 2p 0 0 ∫∫ C g 4 冢 冣] (31) f 2C sin v df dv A3 x 2 ∂W I y2 ∂W I z2 ∂W I + + a ∂a b ∂b c ∂c 冣, (32) 1 sin v df dv gC 2 W I , c 2 A 8 (33) and V III c 1 2 C gr 0 4 p 2p 0 0 ∫∫ Substituting the expression for W I (Eq. 27) into Eqs. (31)–(33), after some term rearranging, V L can be written as V L c pr o g[C1 x 4 + C2 y4 + C3 z4 + C4 x 2 y2 + C5 x 2 z2 + C6 y2 z2 + C7 x 2 + C8 y2 + C9 z2 + C10 ], where (34) Esteban and Kazanas 1290 C1 c − C2 c − b2 6a 2 3c 2 a2 冢1 − c2 冣 − 12a 冢1 + 2 3b 2 a2 冣A 1 + b2 e 2 A3 12a2 3 + c3 ∂A3 cb2 2 ∂A3 bc2 ∂A1 b3 2 A3 − + − e e 6a2 ∂c 6a2 3 ∂c 6a2 ∂b 6a2 3 ∂b + c2 3a − b2 c 2 ∂2 A3 bc2 2 ∂2 A1 b3 2 ∂2 A3 + e3 − e2 e ∂a∂c 12a 12a ∂a∂b 12a 3 ∂a∂b + c2 2 ∂2 A1 b2 2 ∂2 A3 − e2 e ∂a2 6 6 3 ∂a2 a2 6b 2 b2 2a 2 冢 1+ c2 a2 冢 1+ 冣 冣 ∂A1 e2 ∂A3 c 3 2 ∂2 A 1 − b2 3 e23 + e 3a ∂a ∂a 12a 2 ∂a∂c c2 12b2 + 冢 1+ (35) 3a 2 b2 冣 A1 + a2 12b2 冢 1+ 3c 2 b2 − c3 ∂A1 a2 c ∂A3 c2 − + 6b2 ∂c 6b2 ∂c 3b − a2 3b + a2 c2 2 ∂2 A1 a 2 2 ∂2 A 3 ac2 ∂A1 a3 ∂A3 − − − e e 2 3 ∂b2 ∂b2 6b 2 6 6b2 ∂a 6b2 ∂a + a3 c2 2 ∂2 A1 a3 2 ∂2 A3 − e e 2 ∂a∂b 12b3 12b 3 ∂a∂b C3 c − c2 冢1 + 2b 冣 2 b2 6c 2 冢 1− 冢 1+ a2 2b 2 冣 冣A 3 ∂A1 ∂b ∂A3 a2 c 3 ∂2 A3 a2 c3 3 ∂2 A1 − e + e2 3 ∂b ∂b∂c 12b 2 ∂b∂c 12b 3a 2 c2 冣 − (36) 1 a2 2 e A − 1 12c2 12c2 2 冢 1+ + a3 2 ∂A1 a2 + e2 ∂c 3c 3c − a2 b2 2 ∂2 A3 ∂A1 a2 b ∂A3 a2 b + 2 + 2 e22 e3 2 2 6c 6c ∂c ∂b ∂b 6 c 冢 1+ b2 2c 2 冣 3b 2 c2 冣A ∂A3 a2 2 ∂2 A1 − e ∂c 6 2 ∂c2 3 Gravitational Potentials of Triaxial Ellipsoids in Weyl Gravity + a 2 b 2 ∂2 A 1 a2 b3 2 ∂2 A3 a3 ∂2 A1 e2 − + 2 e22 e3 3 12c ∂b∂c ∂b∂c ∂a 12b 6c + a3 ∂A3 a3 2 ∂2 A1 a3 b2 2 + − e2 e 2 ∂a ∂a∂c 6c 12c 12c 3 c2 2 1 e A1 − 2b 2 2 2 C4 c e21 − c2 b2 冢 1+ 冣 A3 + − b 2 ∂A3 ac2 − e3 2 ∂b 2b 2 + ac2 2 ∂2 A1 ab 2 ∂2 A3 − e2 e 2 3 ∂a∂b ∂a∂b 2b C5 c − b2 c2 + + C6 c 1+ 冢 c2 a2 c 2 1+ 冢 b2 a2 a 2 冢1 + c 冣 冣 冣 b2 2 a2 2 1 e − c2 1 2 冢 1+ a2 a2 b2 冢1 + a 冣 2 + 1 2 冢 1+ 冣 A1 + + a2 c 2 ∂2 A1 a2 b 2 e2 e − 2b 2c 3 ∂b∂c 2 冢 1+ b2 a2 冢 ∂A1 a − 2 ∂a 冣 ∂A1 ∂b 冣 c2 冢1 + a 冣 2 ∂A3 ∂a (38) 冣 A1 − 冢 1 2 1+ a2 2 c e A3 − 2 2c 2 3 冢 ∂A3 a2 2 ∂A1 a2 b + + 2 e2 c ∂c ∂b 2b c2 冢1 + b 冣 b2 a2 1+ b2 c2 冣A 3 ∂A3 ac 2 ∂2 A1 ab2 2 ∂2 A3 + − e e2 ∂a ∂a∂b 2c 3 ∂a∂c 2 a2 2c c2 2 b2 a2 c2 2b (37) ∂A3 b2 2 ∂A3 a 2 ∂A1 − + e e 2 2 ∂a ∂c 2c 3 ∂c − C7 c b2 e21 + 1291 A1 − 1+ a2 b2 冣 c2 冢1 + b 冣 2 (39) ∂A1 ∂c ∂A3 ∂b (40) b2 2 ac2 2 ∂A1 ab2 2 ∂A3 − , e 3 A3 + e2 e ∂a 2 2 2 3 ∂a (41) Esteban and Kazanas 1292 C8 c a2 e21 − C9 c − c2 2 a2 2 c2 a2 2 ∂A1 a2 b 2 ∂A3 − , e2 e e2 A1 − e3 A3 + 2 3 ∂b ∂b 2 2 2b (42) a2 2 ca2 2 ∂A1 a2 b2 2 ∂A3 b2 a2 2 a2 2 − , e e A − e A + e e + 1 3 2 3 2 1 ∂c 2 2 2 2c 3 ∂c c2 (43) C10 c b2 a2 2 冢 1+ c2 a2 冣 + c2 a2 2 a2 b2 2 e2 A1 − e3 A3 , 4 4 (44) with e21 c 1 − c2 , a2 (45) e22 c 1 − b2 , a2 (46) e23 c 1 − c2 , b2 (47) Although Eqs. (35)–(44) look somewhat cumbersome, they are the most general equations associated with the linear potentials inside ellipsoids, spheroids, spheres, and flat disks. However, one can use some of the expressions given in EFE to eliminate the derivatives of Ai ’s in favor of multiple index symbols like Aij and Aijk , and also the definitions of the latter to eventually obtain expressions for the Ci ’s above involving only Ai ’s. With theuse of these conventions the expressions for the Ci ’s given above simplify to the following C1 c a2 b2 (A1 − A2 ) + a2 c2 (A1 − A3 ) + b2 c2 (2 − 3A1 ) , 12(a2 − b2 )(a2 − c2 (48) C2 c a2 b2 (A2 − A1 ) + b2 c2 (A2 − A3 ) + a2 c2 (2 − 3A2 ) , 12(b2 − a2 )(b2 − c2 ) (49) C3 c a2 b2 (2 − 3A3 ) + a2 c2 (A3 − A1 ) + b2 c2 (A3 − A2 ) , 12(c2 − a2 )(c2 − b2 (50) Gravitational Potentials of Triaxial Ellipsoids in Weyl Gravity 1293 C4 c b2 A2 − a2 A1 , 2(a2 − b2 ) (51) C5 c c2 A3 − a2 A1 , 2(a2 − c2 ) (52) C6 c A3 c2 − A2 b2 , 2(b2 − c2 ) (53) C7 c b2 A2 + A3 c2 , 2 (54) C8 c a2 A1 + c2 A3 , 2 (55) C9 c a2 A1 + b2 A2 , 2 (56) a2 b2 (2 − A3 ) + a2 c2 (2 − A2 ) + b2 c2 (2 − A1 ) . 4 (57) C10 c In passing, we note that in Newtonian physics the gravitational potential inside a homogeneous ellipsoid V N can be written EFE as V N c pGr o (I − A1 x 2 − A2 y2 − A3 z2 ). It is instructive to compare this result to the general form of potential due to the linear term (Eq. (34)), in order to get a feeling for the changes induced by the different dimensionality of the Green’s function and also appreciate the ensuing complication. The total gravitational potential due to both terms in Eq. (5) will then be given by V N + V L . The above potential can now be used to obtain that corresponding to specific cases. Thus the prolate (b c c) spheroid potential can also be obtained from Eqs. (35)–(44). One should note however that, while some of the coefficients of this specific figure (let us call them Cip ) namely C1p , C4p , C5p , C7p , C8p , C9p , C10p , can be obtained directly from Eqs. (48)–(57), simply on setting b c c, the remaining cannot, as the denominator of the corresponding expression vanishes. For these cases one should revert to the more general expressions Eqs. (35)–(44), in order to obtain these coefficients. The coefficients Cip then read, C 1p c 1 3 冢1 − 2 A 冣 冢 e C 2p c C 3p c 3 1 3 1 8e21 2 1 冣 −1 , 冢1 − 2 A 冣 − 16 , 3 A3 3 (58) (59) Esteban and Kazanas 1294 −1 e21 C 4p c C 5p c 冢1 − 2 A 冣 − 3 3 A3 , 2 (60) C 6p c 2 C 2p , (61) C7p c c2 A3 , (62) [ C 8p c C 9p c a 2 1 − 冢 C10p c a2 c2 1 − ] A3 (1 + e21 ) , 2 冣 A3 2 e . 2 1 (63) (64) Thus, the gravitational potential (due to the linear term) inside a prolate spheroid (V p ) can be written as V p c pr o g[C1p x 4 + C2p r 4 + C4p x 2 r 2 + C7p x 2 + C8p r 2 + C10p ], (65) where r c ( y2 + z2 )1/ 2 . An analogous procedure can then yield the Ci ’s corresponding to an oblate spheroid Cio setting a c b. These are C 1o c C 2o c − C 3o c A1 (1 − e21 ) (3 − 4e21 ), + 16e21 8e21 A1 1 − , 2 2 e1 3e21 C 4o c 2 C 1o , C 5o c C 6o c (67) (68) (1 − e12 ) A1 − (3 − 2e21 ), e21 2e21 (69) a2 A1 (2e21 − 1), 2 (70) C 7o c C 8o c c 2 + C9o c a2 A1 , C10o c a2 c2 + (66) (71) a4 2 e A1 . 2 1 (72) From Eq. (31) and Eqs. (63)–(69) the oblate spheroid potential V o can be now written as Gravitational Potentials of Triaxial Ellipsoids in Weyl Gravity 1295 V o c pr o g[C1o r 4o + C3o z4 + C5o z2 r 2o + C7o r 2o + C9o z2 + C10o ], (73) f where r o c x 2 + y2 . Next, we focus our attention on obtaining the gravitational potential for a homogeneous self-gravitating sphere (V st ). To this purpose we set first, a c b c c (except in the derivatives) in Eqs. (35)–(44). Thus, 冢 ∂A1s 1 + ∂a 3 C 1s c 1 a (1 − A1s ) + 2 3 C 2s c 1 a (A1s − A3s − 1) + 2 3 + C 3s c 1 3 冢 冢 冢 ∂A1s ∂A1s + ∂c ∂b C4s c − A3s + a 冢 冢 ∂A3s 1 + ∂c 3 冢 冣冣 , ∂A3s ∂A3s + ∂b ∂a (75) 冣 冣, (76) ∂A3s ∂A1s ∂A3s − − ∂b ∂a ∂a 冣, (77) ∂A1s ∂A3s + ∂c ∂a 冣, (78) ∂A1s ∂A3s ∂A3s + − ∂c ∂c ∂b 冣, (79) C 5 s c − 2 + A1 s + A3 s + a C6s c − A1s − a 冢 (74) ∂A3s ∂A1s + ∂b ∂b ∂A1s ∂A3s ∂A1s ∂A3s + + + ∂c ∂c ∂a ∂a 1 a (1 − A3s ) + 2 3 冣冣 冢 C7s c aA1s , (80) C8s c a2 (2 − A1s − A3s ), (81) C9s c a2 A3s , (82) C10s c a4 , (83) where the subscript “s” is added in Eqs. (74)–(83) to indicate that the Ai ’s and its derivatives must be calculated for the sphere. A straightforward calculation allows us to write Eqs. (74)–(83) as follows Esteban and Kazanas 1296 C1s c C2s c C3s c − 1/ 15, C4s c C5s c C6s c − 2/ 15, C7s c C8s c C9s c (2/ 3)a2 , C10s c a4 , (84) (85) (86) (87) Then, Eqs. (84)–(87) are substituted into Eq. (34) to obtain [ V st c pr o g − + 1 2 (x 4 + y4 + z4 ) − (x 2 y2 + y2 z2 + y2 z2 ) 15 15 ] 2 2 2 a (x + y2 + z2 ) + a4 . 3 (88) The correctness of the above expression, can be verified in this highly symmetric case by computing the potential inside and outside a spherical shell and integrating. The geometry associated with this undertaking is the same to that given in [14] in a similar calculation in Newtonian physics. At a point P exterior to the shell, the linear potential (V os ), can be calculated as follows ∫ ∫ V os c g rr o dv′ c r o rr′ 2 sin h dr′ dh df. (89) where dv′ is a differential of volume inside the shell, r, r′ , and R are respectively the distances between dv′ and P, dv′ and the shell’s center, and the shell’s center and P. Further, dh is the angle between r′ and R, and a and b, are the shell’s outer and inner radii. Next, for a given r′ we differentiate the expression r 2 c r′ 2 + R2 − 2r′ R cos h , (90) r dr c sin h dh . r′ R (91) to obtain Then, Eq. (91) is substituted into Eq. (89) and therefore V os c 2pgr o R ∫ a b r′ dr′ ∫ R + r′ R − r′ dr. (92) Gravitational Potentials of Triaxial Ellipsoids in Weyl Gravity 1297 A direct integration of Eq. (92) gives the linear potential at P. Thus, V os c 4pgr [ ] (a3 − b3 ) 1 (a5 − b5 ) . R+ 3 15R (93) To calculate the contribution of the linear potential inside the shell (V is ) we just change the inferior limit (R − r′ − > r′ − R) in Eq. (92). We obtain (for R < b) 冢 V is c pgr o a4 − b4 + 冣 2 2 2 R (a − b2 ) , 3 (94) where R c (x 2 + y2 + z2 )1/ 2 . For points within the shell (b < R < a) we use the previous results (Eqs. (90)–(91)) to calculate the potential in the interior of the spherical shell V b < R < a . Thus, V b < R < a c pgr o [ 4 4 (R3 − b3 )R + (R5 − b5 ) 3 15R + a4 − R4 + ] 2R2 2 (a − R2 ) . 3 (95) As usual, by setting b r 0 in Eq. (95) we obtain the potential (V s ) inside a homogeneous solid sphere. Thus 冢 V s c pgr o a4 + 冣 2 2 2 1 4 a R − R , 3 15 (96) which is as expected the same as Eq. (88). Up to this point, we have obtained the gravitational potential (due to an extra linear term) inside tri-axial self-gravitating structures. However, since many galaxies are also modeled by flat disks, it is also of interest to obtain the potential inside these two dimensional objects. To obtain the potential inside a flat’s disk we first set z c 0 in Eq. (34). Then we collapse the homogeneous solid ellipsoid onto the x-y plane by taking r o r ∞ and c r 0 such that r o c c j o / 2, where j o is the disk’s central density. Note that the disk’s surface density j is given by Esteban and Kazanas 1298 冢 j c jo 1 − x2 y2 − 2 2 a b 冣 1/ 2 . (97) Thus, from Eq. (34), the disk’s potential (V D ) is written as follows V D c pj o [c1d x 4 + c2d y4 + c4d x 2 y2 + c7d x 2 + c8d y2 + c10d ], (98) The coefficients of the above equation can be obtained from the general expressions of the corresponding Eqs. (48)–(57) by letting c r 0 after one has replaced A3 by 2 − A1 − A2 . Taking this limit is possible because the expressions for A1 , A2 are proportional to the product abc, so one can let c r 0 after one has formed the product j o c 2r o c in these expressions. On letting c r 0, the expressions for A1 , A2 as given in EFE can be now expressed in terms of the complete elliptic integrals E(x) and F(x). After these substitutions the coefficients cid of Eq. (98) above take the form c 1d c (l − 1)j o [2(l − 1)F(l) − 2(l − 2)E(l)], 12bl 2 c 2d c jo [(l − 2)E(l) + 2(1 − l)F(l)], 12bl 2 (100) c 4d c (1 − l)j o [2E(l) + (l − 2)F(l)], 2bl 2 (101) c 7d c bj o [E(l) − (1 − l)F(l)], 2l (102) c 8d c bj o [F(l) − E(l)], 2l (103) b3 j o E(l), 4(1 − l) (104) c10d c (99) where l c e22 and where, as mentioned above, F(l) and E(l) are the complete elliptic integrals of the first and second kind, respectively. The Newtonian gravitational potentials for the non-axisymmetric disks considered above have been computed in the literature and can be found in [15]. Using the potentials for the general non-axisymmetric ellipsoids given above, one can easily investigate the existence of homogeneous, self-gravitating prolate or oblate spheroids within the context of Weyl gravity. This we do in the next section. Gravitational Potentials of Triaxial Ellipsoids in Weyl Gravity 1299 3. STEADILY ROTATING PERFECT FLUIDS IN WEYL GRAVITY The hydrodynamic equations for a perfect fluid in time independent flow have the form (see e.g. [16]) (rua ub + Pd ab ), b c rV , a , (rua ), a c 0, (105) (106) where r is the density, P the pressure, u the fluid three-velocity and V is the gravitational potential. A comma denotes a partial derivative, and the summation convention is used. These can be recognized respectively as the Euler and continuity equations. For the particular case of an incompressible fluid (r c constant) rotating with a given angular velocity q about the x-axis, the continuity equation is identically satisfied, while the Euler equations take the form P, x c rV , x − r q y + P, y c rV , g − r q2 z + P, z c rV , z (107) 2 (108) (109) Using the expression for the gravitational potential obtained earlier (Eq. 34), one can readily integrate Eq. (107). Imposing additionally the boundary condition P c 0 at the ellipsoid’s surface to eliminate the integration constant, the ellipsoid’s pressure takes the following form P c pr o 2 g[C1 x 4 + C4 x 2 y2 + C5 x 2 z2 + C7 x 2 − a2 (1 − y2 / b2 − z/ c2 ) . (C1 a2 (1 − y2 / b2 − z2 / c2 ) + C4 y2 + C5 z2 + C7 )]. (110) With the expressions for the gravitational potential and the pressure known throughout the ellipsoid, one can substitute their values in Eq. (108) to obtain the following expression for the angular velocity q 2 c 2gpr o 冢 +2 − 冢 + −2 [冢 2 a2 a2 2 C − a C + C7 − C8 1 4 b2 b2 冣 a2 a4 C 1 − C 2 + 2 C 4 y2 4 b b 冣 ] a4 a4 a2 C + C + C5 − C6 z2 . 1 4 b2 c2 c2 c2 冣 (111) Esteban and Kazanas 1300 In an analogous manner, using Eqs. (107) and (109) we obtain the following (additional) expression for the angular velocity q 2 c 2gpr o [ 冢2 c 冢 + −2 冢 +2 − a4 C 1 − a2 C 5 − 4 a2 C7 − C9 c2 冣 a4 a2 a2 C + C + C 5 − C 6 y2 1 4 b2 c2 c2 b2 冣 a2 a4 C5 z2 C − C + 1 3 c2 c4 冣 ] (112) The compatibility of these two independent expressions for q 2 yields a geometric constraint between a, b, c appropriate to a rotating incompressible fluid under the action of a linear potential. An identical analysis within Newtonian gravity yields the well known geometric constraint of Jacobi ellipsoids (EFE, chapter 6, Eq. 4). We shall explore this further in a future publication. In the present we would simply like to indicate that in the case of an axisymmetric prolate configuration (b c c) the two expressions for q 2 are indeed identical, yielding the following expression for the prolate spheroid’s angular velocity as a function of its meridional eccentricity e1 q 2p c gpr o + + + [ a2 e51 冢 − 5e 1 + 22 3 e − e51 3 1 冢 1 (1 + e 1 ) (5 − 9e21 + 3e41 + e61 ) log (1 − e1 ) 2 r2 − 1) e51 (e21 冢− 冣冣 35 25 3 e1 + e − e51 4 3 1 冢 1 (1 + e1 ) (35 − 45e21 + 9e41 + e61 ) log (1 − e 1 ) 8 冣] (113) To study oblate spheroids (a c b) choose now the z-axis as the rotational axis. In this case, the Euler equations (Eq. 105) read P, z c rV , z , (114) − r q x + P, x c rV , x , (115) (116) 2 − r q y + P, y c rV , y , 2 Gravitational Potentials of Triaxial Ellipsoids in Weyl Gravity 1301 Using the same procedure as above we obtain the following expression for its pressure P c gpr 2o [C3 z4 + C5 x 2 y2 + C6 y2 z2 + C9 z2 ) − c2 (1 − x 2 / a2 − y2 / b2 ) . (C3 c2 (1 − x 2 / a2 − y2 / b2 ) + C5 x 2 + C6 y2 + C9 )]. (117) and the following two expressions for its angular velocity [ q 2 c 2gpr o 2 c4 c2 2 C − c C − C + C9 3 5 7 a2 a2 冢 + 2 − C1 − 冢 + −2 c2 c2 C3 + 2 C5 x 2 2 a a 冣 ] c4 c2 c2 C − C + C + C 6 y2 , 3 4 5 a2 b2 b2 a2 冣 (118) and q 2 c 2gpr o [冢 冢 + −2 冢 2 c4 c2 2 C − c C − C + C9 3 6 8 b2 b2 冣 c4 c2 c2 C3 − C4 + 2 C5 + 2 C6 x 2 2 2 a b b a 冣 + 2 − C2 − ] c4 c2 C + C 6 y2 . 3 b4 b2 冣 (119) As it is the case with the prolate spheroids the above two equations are identical for a c b, yielding the following expression for the angular velocity q in terms of the meridional eccentricity e1     q 2o c gpr o c2 − 8 4 Arcsin e1 5 + (5 − 6e21 ) g + 2 + 4 3 e1 3 e1 e51 1 − e21   冢 1 105e1 − 215e31 + 118e51 − 8e71 12e51 g − 3 1 − e21 (35 − 60e21 + 24e41 )Arcsin e1 r 2o +  冣 ] (120) 1302 Esteban and Kazanas It should be noted that the expressions for q 2 (113) and (120) agree with the corresponding expressions given in [12] appropriate to the linear part of the corresponding potentials. One can now investigate the existence of prolate and oblate spheroids within Weyl gravity by investigating the sign of q 2 in Eqs. (113) and (120) for e1 in the range 0 < e1 < 1. Before doing so, care should be taken that the sign of the gravitational potential is properly taken into account. [12] used the following convention for the potential V(r) c b/ r − gr, with this quantity being positive. This convention therefore detrmines the sign of the constant g relative to that of the Newtonian constant b. Using the more standard convention that the gravitational potential is negative, i.e. that V(r) c − b/ r + gr will necessary yield a change in the overall signs of Eqs. (113) and (120). With this convention, one can now see that Eq. (113) yields q 2 < 0, while Eq. (120) yields q 2 > 0, indicating, as concluded in [12] that prolate spheroids, which do not exist within Newtonian gravity, cannot exist within Weyl gravity either. 4. CONCLUSIONS In this work we have provided closed form solutions of the gravitational potential inside homogeneous ellipsoids, spheroids, spheres and heterogeneous flat disks in the framework of Weyl gravity; for that we computed the integrals of the Green’s function associated with the ∇4 operator of the static spherically symmetric equations of this theory over the volume of the above configurations. We expect that these expressions will be of interest to researchers who would like to further study the nuances of this or similar theories. We believe that our results are also of interest from the methodological point of view, first for providing such closed form expressions and second for indicating a general framework for obtaining closed form expressions for similar quantities. In this respect our method supplements and completes that discussed in [12] where the expression for the forces within the framework of the same theory were obtained. One would think that computation of the forces would be sufficient in the determination of the properties of configurations such as postulated in the present as well in the work of [12]. However, this is not the case, in that the potentials are more general quantities, absolutely necessary for the study of the stability of such configurations using minimum energy arguments, a treatment which we undertake in a companion work. We have also used these general expressions to provide the same potentials for configurations of higher symmetry, i.e. axisymmetric prolate and oblate spheroids, thin inhomogeneous disks such as those studied in [15] in the context of Newtonian gravity. Finally we also provided the expressions of these potentials for spheres by setting all axes of the ellipsoid equal, a procedure which serves as a test for the correctness of our results, since in this last case the poten- Gravitational Potentials of Triaxial Ellipsoids in Weyl Gravity 1303 tials can be easily computed from first principles. With respect to this last point, i.e. the correctness of the expressions provided in this paper, we have checked by explicit differentiation that the expression ∇4 V yields indeed − 8pgr as it should indeed be the case. ACKNOWLEDGMENTS The authors would like to thank D. Christodoulou and N. K. Spyrou for stimulating and helpful discussions. E. P. Esteban also wishes to thank his students T. Maldonado, L. Roldán, D. Rodrı́guez and J. Santiago for developing numerical and symbolic programs to check most of the equations presented in an early version of this paper. E. P. 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