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. Esteban also acknowledges the support given by a NASA fellowship
(1999–2000) to visit NASA/ Goddard where most of this work was carried out.
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