Axially symmetric static sources: A general framework and some analytical solutions

0.18 MB·English

by  L. Herrera

Axially symmetric static sources: A general framework and some analytical solutions
L. Herrera,∗ A. Di Prisco,† and J. Iba´n˜ez‡
Departamento de F´ısica Te´orica e Historia de la Ciencia, Universidad del Pa´ıs Vasco, Bilbao, Spain
J. Ospino§
Departamento de Matema´tica Aplicada, Universidad de Salamanca, Salamanca, Spain.
(Dated: January 14, 2013)
Weprovideallbasicequationsandconceptsrequiredtocarryoutageneralstudyonaxiallysym-
metric static sources. The Einstein equations and the conservation equations are written down for
ageneralanisotropicstaticfluidendowedwithaxialsymmetry. Thestructurescalarsarecalculated
and the inhomogeneity factors are identified. Finally some exact analytical solutions were found.
One of these solutions describes an incompressible spheroid with isotropic pressure and becomes
thewellknowninteriorSchwarzschildsolutioninthesphericallysymmetriclimit,howeveritcannot
3
be matched smoothly to any Weyl exterior metric. Another family of solutions was found that
1
correspondstoananisotropicfluiddistributionandcaninprinciplebematchedtoaWeylexterior.
0
2
PACSnumbers: 04.40.-b,04.40.Nr,04.40.Dg
n Keywords: RelativisticFluids,nonsphericalsources,interiorsolutions.
a
J

1 I. INTRODUCTION the past. Without pretending to be exhaustive in the
1 revisionof the literature on this problem, let us mention
Observational evidence seems to suggest that de- the pioneering paper by Hernandez Jr. [12], where a
]
c viations from spherical symmetry in compact self- generalmethodforobtainingsolutionsdescribingaxially
q gravitating objects (white dwarfs, neutron stars), are symmetric sourcesis presented. Such a method, or some
r- likely to be incidental rather than basic features of these ofitsmodificationswereusedin[13]-[15],tofindsources
g systems. This explains why spherical symmetry is so of different Weyl spacetimes.
[ commonly assumed in the study of self-gravitating com- This problem has also been considered in [16]-[19].
1 pact objects. However in all these last references the line element has
v However, the situation is not so simple. Indeed been assumed to satisfy the so called Weyl gauge, which
4 (putting aside the evident fact that astrophysical ob- of course severely restricts the family of possible sources
2 jects are generally endowed with angular momentum, (see the next section).
4
and therefore excluding all stationary sources), it is well In this work we present a general description of ax-
2
known that the only regular static and asymptotically ially symmetric sources by deploying all relevant equa-
.
1 flat vacuum spacetime posesing a regular horizon is the tions without resortingto the Weyl gauge,and consider-
0
Schwarzchild solution [1], and all the others Weyl exte- ing the most general matter content consistent with the
3
riorsolutions[2]-[5]exhibit singularitiesin the curvature symmetries of the problem.
1
: invariants(asthe boundaryofthesourceapproachesthe With this purpose in mind it would be useful to in-
v horizon). This in turn implies that, for very compact troduce the so called structure scalars. These form a set
i
X objects, a bifurcation appears between any finite pertur- of scalar functions obtained from the orthogonal split-
bation of Schwarzschild spacetime and any Weyl solu- tingoftheRiemanntensor. Theywereoriginallydefined
r
a tion,evenwhenthelatterischaracterizedbyparameters in the discussion about the structure and evolution of
arbitrarilyclosetothosecorrespondingtosphericalsym- spherically symmetric fluid distributions. Such scalars
metry (see [6]-[11]andreferencesthereinfor adiscussion (fiveinthesphericallysymmetriccase)wereshowntobe
on this point). endowed with distinct physical meaning [20–23] .
From the above comments it should be clear that a In particular they control inhomogeneities in the en-
rigorous description of static axially symmetric sources, ergy density [20], and the evolution of the expansion
including finding exact analytical solutions, is a praise- scalar and the shear tensor [20–23]. Also in the static
worthy endeavour. case all possible anisotropic solutions are determined by
Accordingly, in this work we provide all ingredients two structure scalars [20].
and equations required for such a study. Of course, this Furthermore,the role of electric chargeand cosmolog-
issue has already been considered by several authors in ical constant in structure scalars has also been recently
investigated [24].
Morerecently,suchscalarsandtheirapplicationswere
discussed also in the context of cylindrical [25], [26] and
∗ [email protected];AlsoatU.C.V.,Caracas
† adiprisc@fisica.ciens.ucv.ve;AlsoatU.C.V.,Caracas planar symmetry [27].
‡ [email protected] Asetofdifferentialequationsforsome ofthese scalars
§ [email protected] allows us to identify the inhomogeneity factors.
2
Finally we exhibit two families of solutions. One of (the so called Weyl gauge), which amounts to assume
them corresponds to an incompressible spheroid with that R3+R0 =0.
3 0
isotropicpressure. Itcannotbematchedsmoothlytoany In our notation the Weyl gauge is expressed by
Weyl exterior spacetime. The secondone correspondsto
rsinθ
an anisotropic fluid and in principle is matchable to a D = . (5)
A
Weyl exterior.
Our paper is organized as follows: In the next sec- Let us now provide a full description of the source. In
tion we shall describe the line element corresponding to order to give physical significance to the components of
the most general non–vacuum, axially symmetric static the energy momentum tensor, we shall apply the Bondi
spacetime. Next we provide a full description of the approach[28].
source that is represented by a general anisotropic mat- Thus, following Bondi, let us introduce purely locally
ter. Einstein equations and conservation equations are Minkowski coordinates (τ,x,y,z) (or equivalently, con-
explicitly written down for such a system. We also cal- sider a tetrad field attached to such l.M.f.) by:
culate the electric part of the Weyl tensor (its magnetic
partvanishes)aswellastheelectricandmagneticpartof dτ =Adt; dx=Bdr; dy =Brdθ; dz =Ddφ.
the Riemann tensor. With this information we are able (6)
to obtain all the non–vanishing structure scalars corre- Denoting by a hat the components of the energy–
sponding to our problem. Two differential equations for momentumtensorinsuchlocallydefinedcoordinatesys-
such scalars allow us to identify the inhomogeneity fac- tem, we have that the matter content is given by
tors. The two families of solutions found are described
insectionsV andVI. Asummaryofthe obtainedresults µ 0 0 0
as well as a list of some unsolved issues are presented in 0 P P 0
sectionVII.Finallyanappendixwiththeexpressionsfor Tαβ = 0 Pxx Pxy 0 , (7)
yx yy
the components of the electric Weyl tensor, is included. 0 0 0 P
b  zz 
 
where µ,P ,P ,P ,P denote the energy density
xy xx yy zz
II. THE METRIC AND THE SOURCE and different stresses, respectively, as measured by our
locally defined Minkowskian observer.
Weshallconsiderbounded,staticandaxiallysymmet- Also observe that Pxy = Pyx and, in general Pxx =
6
ric sources. For such a system the most general line ele- Pyy =Pzz.
6
ment may be written in cylindrical coordinates as: Introducing
Vˆ =( 1,0,0,0); Kˆ =(0,1,0,0); Lˆ =(0,0,1,0),
α α α
−
ds2 = A2dt2+B2 (dx1)2+(dx2)2 +D2dφ2, (1) (8)
− we have
where A,B,D are posi(cid:2)tive functions of(cid:3)x1 and x2. We T =(µ+P )Vˆ Vˆ +P η +(P P )Kˆ Kˆ
αβ zz α β zz αβ xx zz α β
number the coordinates x0 =t,x1 =ρ,x2 =z,x3 =φ. −
+(P P )Lˆ Lˆ +2P Kˆ Lˆ , (9)
We shall work in “Weyl spherical coordinates” (r,θ) b yy− zz α β xy (α β)
defined by: where η denotes the Minkowski metric.
αβ
Then transforming back to our coordinates,we obtain
ρ=rsinθ, z =rcosθ. (2) thecomponentsoftheenergymomentumtensorinterms
of the physical variables as defined in the l.M.f.
In these coordinates the line element reads:
T =(µ+P )V V +P g +(P P )K K
αβ zz α β zz αβ xx zz α β
ds2 = A2dt2+B2 dr2+r2dθ2 +D2dφ2, (3) −
+(P P )L L +2P K L , (10)
− yy zz α β xy (α β)
−
It is important not to con(cid:0)found these(cid:1)coordinates with where
Erez–Rosencoordinates (rˆ,θˆ) given by
V =( A,0,0,0); K =(0,B,0,0); L =(0,0,Br,0),
α α α
−
ρ2 =(rˆ2 2mrˆ)sin2θˆ, z =(rˆ m)cosθˆ, (4) (11)
− − Alternativelywemaywritetheenergymomentumtensor
in the “canonical” form:
wheremisaconstanttobeidentifiedwiththemonopole
of the source. T =(µ+P)V V +Pg +Π , (12)
αβ α β αβ αβ
It should be stressed that our line element is defined
by three independent functions, unlike the vacuum case with
where it is always possible to reduce the line element so h
Π =(P P )(K K αβ)
that only two independent metric functions appear. In αβ xx− zz α β − 3
the interior this is not possible in general, though obvi- h
ously one may assume that as an additional restriction +(Pyy−Pzz)(LαLβ − 3αβ)+2PxyK(αLβ),(13)
3
and
P +P +P
P = xx yy zz, h =g +V V . (14)
µν µν ν µ
3
With the above information we can write the Einstein
equations, which read:
1 B′′ D′′ 1 B′ D′ B′ 1 B D B
8πµ= + + ( + ) ( )2+ θθ + θθ ( θ)2 , (15)
−B2 B D r B D − B r2 B D − B
(cid:26) (cid:20) (cid:21)(cid:27)
1 A′B′ A′D′ B′D′ 1 A′ D′ 1 A D A B A D B D
8πP = + + + ( + )+ ( θθ + θθ θ θ + θ θ θ θ) , (16)
xx B2 AB AD BD r A D r2 A D − AB AD − BD
(cid:20) (cid:21)
1 A′′ D′′ A′B′ A′D′ B′D′ 1 A B A D B D
8πP = + + + ( θ θ + θ θ + θ θ) , (17)
yy B2 A D − AB AD − BD r2 AB AD BD
(cid:20) (cid:21)
1 A′′ B′′ B′ 1 A′ B′ 1 A B B
8πP = + ( )2+ ( + )+ θθ + θθ ( θ)2 , (18)
zz B2 A B − B r A B r2 A B − B
(cid:26) (cid:20) (cid:21)(cid:27)
1 1 A′ D′ B A′ D′ B′A B′D 1 A D
8πP = θ θ + θ + + θ + θ + ( θ + θ) , (19)
xy B2 r − A − D B A D B A B D r2 A D
(cid:26) (cid:20) (cid:18) (cid:19) (cid:21) (cid:27)
where prime and subscript θ denote derivatives with re- where the overdot denotes derivative with respect to t,
spect to r and θ respectively. and
Also,thenonvanishingcomponentsoftheconservation
equations Tαβ =0 yield:
;β
µ˙ =0, (20)
A′ B′ D′
P′ + (µ+P )+ (P P )+ (P P )
xx A xx B xx− yy D xx− zz
1 A B D
+ θ +2 θ + θ P +P +P P =0, (21)
r A B D xy xy,θ xx− yy
(cid:20)(cid:18) (cid:19) (cid:21)
A B D A′ B′ D′
P + θ(µ+P )+ θ(P P )+ θ(P P )+r +2 + P +P′ +2P =0. (22)
yy,θ A yy B yy − xx D yy− zz A B D xy xy xy
(cid:20)(cid:18) (cid:19) (cid:21)
Equation (20) is a trivial consequence of the staticity, III. THE STRUCTURE SCALARS
whereas (21) and (22) are the hydrostatic equilibrium
equations.
We shall calculate here the structure scalars for the
static axially symmetric case. For that purpose, let us
firstobtaintheelectricpartoftheWeyltensor(themag-
netic part vanishes identically).
4
The components of the electric Weyl tensor can be ited in the Appendix.
obtained directly from its definition,
E =C VαVβ, (23)
µν µανβ
Equivalently, the electric part of the Weyl tensor may
where C denotes the Weyl tensor. These are exhib- also be written as:
µανβ
1 1
E = (K L +L K )+ K K h + L L h , (24)
αβ 1 α β α β 2 α β αβ 3 α β αβ
E E − 3 E − 3
(cid:18) (cid:19) (cid:18) (cid:19)
where explicitexpressionsfor the three scalars , , Finally, we shallfind the tensorassociatedwith the dou-
1 2 3
E E E
are given in the Appendix. ble dual of Riemann tensor, defined as:
Next, let us calculate the electric part of the Riemann
1
tensor (the magnetic part vanishes identically), which is X =∗ R∗ VγVδ = η ǫρR∗ VγVδ, (31)
αβ αγβδ 2 αγ ǫρβδ
defined by
with R∗ = 1η R ǫρ. Thus, we find
αβγδ 2 ǫργδ αβ
Yρ =VαVµRρ . (25)
β αβµ 1
X =X (K L +K L )+X K K h
αβ TF1 α β β α TF2 α β− 3 αβ
(cid:18) (cid:19)
After some lengthy calculations we find; 1 1
+X L L h + X h , (32)
TF3 α β− 3 αβ 3 T αβ
(cid:18) (cid:19)
where
1
Yαβ =YTF1(KαLβ +KβLα)+YTF2 KαKβ − 3hαβ XT =8πµ, (33)
(cid:18) (cid:19)
1 1
+Y L L h + Y h , (26)
TF3(cid:18) α β− 3 αβ(cid:19) 3 T αβ XTF1 =−(E1+4πPxy), (34)
where
X = [ +4π(P P )], (35)
TF2 − E2 xx− zz
Y =4π(µ+P +P +P ), (27)
T xx yy zz
X = [ +4π(P P )]. (36)
TF3 − E3 yy− zz
The scalars Y , Y , Y ,Y , X , X , X ,
T TF1 TF2 TF3 T TF1 TF2
X , are the structure scalars for our problem.
TF3
Y = 4πP , (28)
TF1 E1− xy
IV. DIFFERENTIAL EQUATIONS FOR THE
STRUCTURE SCALARS AND THE
INHOMOGENEITY FACTORS
Y = 4π(P P ), (29)
TF2 E2− xx− zz
Two differential equations for the Weyl tensor may
be obtained using Bianchi identities [29], [30], they have
been found before for the spherically symmetric and the
cylindricallysymmetriccases(see[25],[31]andreferences
Y = 4π(P P ). (30) therein). Herewecalculatethemforourcase. Weobtain:
TF3 E3− yy− zz
1 2B D B′ D′ 1 B′ 1 4π
E1θ + (2 )′+ E1 θ + θ + + + + = (2µ+P +P +P )′(37)
r 3 E2−E3 r B D E2 B D r −E3 B r 3 xx yy zz
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
A′ A
+4π(µ+P ) +4πP θ,
xx A xyAr
5
1 2B′ D′ 2 B B D 4π
′ + (2 ) + + + E2 θ + E3 θ + θ = (2µ+P +P +P ) (38)
E1 3r E3−E2 θ E1 B D r − Br r B D 3r xx yy zz θ
(cid:18) (cid:19) (cid:18) (cid:19)
A A′
+4π(µ+P ) θ +4πP ,
yy Ar xy A
which, using (27)-(30) and (33)-36), may be written in
terms of structure scalars:
8πµ′ 1 2 1
= X +X (lnB2D) X′ +X (lnBDr)′ + X′ +X (lnBr)′ , (39)
3 −r TF1θ TF1 θ − 3 TF2 TF2 3 TF3 TF3
(cid:20) (cid:21) (cid:20) (cid:21)
(cid:2) (cid:3)
8πµ 1 1 1 2
θ = X +X (lnB) X +X (lnBD) X′ +X (lnB2Dr2)′ . (40)
3r r 3 TF2θ TF2 θ − r 3 TF3θ TF3 θ − TF1 TF1
(cid:20) (cid:21) (cid:20) (cid:21)
(cid:2) (cid:3)
Let us now turn to the inhomogeneity factors. The Using the above in (A.9) we have that
inhomogeneity factors (say Ψ ) are the specific combina-
i
tions ofphysicalandgeometricvariables,suchthattheir 1(0,θ) r, (44)
E ≈
vanishing is a necessary and sufficient condition for the
homogeneity of energy density (i.e. for the vanishing of implying because of (34)
all spatial derivatives of the energy density).
X (0,θ) r. (45)
In the spherically symmetric case it has been shown TF1
≈
that in the absence of dissipation the necessary and suf-
Finallyweshallassumethatthethreestructurescalars
ficient condition for the vanishing of the (invariantly de-
X ,X ,X areanalyticalfunctions(classC∞)in
fined) spatial derivative of the energy density is the van- TF1 TF2 TF3
the neighborhood of r =0.
ishing of the scalar associatedwith the trace free partof
Then, if we assume µ′ = µ = 0. Evaluating (40) in
X (see [20], [31]). θ
αβ the neighborhoodof r 0,since X′ (r =0)is regular,
Weshallnowidentifytheinhomogeneityfactorsinour ≈ TF1
it follows
case.
First, observe that from (39) and (40) it follows at X′ (r =0)=0, (46)
once that X =X =X =0 µ′ =µ =0. In TF1
TF1 TF2 TF3 θ
⇒
order to identify the above scalars as the inhomogeneity
and
factorsweneedto provethatthe inverseis alsotrue (i.e.
µ′F=orµtθh=at0p⇒urpXoTseFw1e=sXhaTllFfi2r=stXesTtFab3l=ish0t)h.e behaviour XTF2θ−2XTF3θ−XTF3cotθ =0, (47)
of different variables in the neighborhood of r 0. We where(41),(43)and(46)havebeenused(inwhatfollows
≈
shall demand that both A and B are regular functions, itis understoodthatallexpressionsareevaluatedatr
≈
and 0).
Next, from (39) we obtain
D(r,θ) rsinθ, (41)
≈
2X =X , (48)
TF2 TF3
at r =0. Then from (21) it follows that in the neighbor-
hood of r =0 where(46)andtheregularityoffirstderivativesofstruc-
ture scalars have been used.
P r, P P r, P P r, (42)
xy xx yy xx zz Then, feeding back (48) into (47) we obtain
≈ − ≈ − ≈
and from (19) α
X = , (49)
TF2 sin2/3θ
A (0,θ) r3, B (0,θ) r3, A(0,θ)′ r2.
θ ≈ θ ≈ θ ≈
(43) where α is a constant, implying X =X =0.
TF2 TF3
6
Thus in the neigborhood of r 0 we have X = Under the conditions above (21) and (22) can be inte-
TF1
X′ =X =X =0. ≈ grated to obtain:
TF1 TF2 TF3
Next, taking r-derivative of (40), and evaluating at
r 0 we get X′′ = 0. Continuing this process it ζ
fol≈lows that X(n)TF1∂nXTF1 =0 for any n 0. P +µ0 = A, (55)
TF1 ≡ ∂rn ≥ and
Also, from the r-derivative of (39) evaluated in the
neighborhood of r 0 it follows ξ(r)
≈ P +µ = , (56)
0 A
2X′ =X′ . (50)
XTF2 TF3
where ξ is an arbitrary function of its argument. Using
Feeding this equation back into the r-derivative of (40) boundary conditions (54) in (55) (56) it follows that:
produces
α
A(r ,θ)=const.= , ζ =constant. (57)
β 1 µ
X′ = , (51) 0
TF2 sin2/3θ
Since,asmentionedbefore,oursolutionisconformally
where β is a constant, implying X′ =X′ =0. It is flat, then using = = = 0 and P = 0, in (19)
TF2 TF3 E1 E2 E3 xy
notdifficulttoseethatthisprocedurecanbecontinuedto and (A.6) we obtain
obtain, in the neighborhood of r 0, X(n) = X(n) =
≈ TF1 TF2 A′ A′B A B′ 1
X(n) = 0, for any n 0. Therefore we can continue θ θ θ + =0, (58)
TF3 ≥ A − A B − A B r
analyticallytheirvalueatthecenter,fromwhichweinfer: (cid:18) (cid:19)
X =X =X =0 µ′ =µ =0. (52)
TF1 TF2 TF3 θ
⇔ D′ D′B D B′ 1
θ θ θ + =0. (59)
Thislastresultallowsustoidentifythethreestructure D − D B − D B r
(cid:18) (cid:19)
scalarsX ,X ,X as the inhomogeneityfactors.
TF1 TF2 TF3
We shall next find some explicit analytical solutions. Introducing the auxiliary function A¯(r,θ) defined by
A(r,θ)=A¯(r,θ)B(r,θ), (60)
V. THE INCOMPRESSIBLE, ISOTROPIC
SPHEROID and assuming
We shall now find an analytical solution correspond- D(r,θ)=B(r,θ)rsinθ, (61)
ing to a bounded spheroid with isotropic pressures and
the equations (58) and (59) can be integrated to obtain
homogeneousenergydensity. Fromtheresultsofthepre-
vious section, (28)-(30) and (34)-(36), it is evident that
such a solution is also conformally flat.
A¯(r,θ)=A˜(r)+rχ(θ),
Thus let us assume P = P = P = P, P = 0
xx yy zz xy
and µ=µ =constant 1
0 B(r,θ)= , (62)
Forsimplicity weshallassumethe boundarysurfaceΣ R(r)+rω(θ)
to be defined by the equation:
where A˜, χ, R and ω are arbitrary functions of their
r =r1 =constant. (53) argument.
Next, from (A.7) and (A.8), taking into account (61)
ThentosatisfyDarmoisconditions(continuityofthefirst
and (62) we get
andsecondfundamentalforms)wedemandthatallmet-
ric functions as well as r derivatives, to be continuous
across Σ (see [12]). Obviously θ derivatives of A,B,D χ=acosθ, A˜(r)=αr2+β. (63)
and A′,B′,D′ are continuous too across Σ.
From the above and (16) and (19) it follows that Where a, α and β are constants of integration.
From the above it follows that conformally flat solu-
Σ
P =0. (54) tions are described by the line element
1
ds2 = (αr2+β+arcosθ)2dt2+dr2+r2dθ2+r2sin2(θ)dφ2 . (64)
[R(r)+rω(θ)]2 −
(cid:2) (cid:3)
Next, from the condition P P =0 and the equa- tions (17)-(18) we get
yy zz
−
ω(θ)=bcosθ, (65)
7
andfromtheconditionP P =0andtheequations (16)-(19) we obtain
xx yy
−
R(r)=γr2+δ, (66)
where b, γ and δ are constants of integration.
Finally, the metric of incompressible conformally flat
isotropic fluids can be written as follows.
1
ds2 = (αr2+β+arcosθ)2dt2+dr2+r2dθ2+r2sin2θdφ2 . (67)
(γr2+δ+brcosθ)2 −
(cid:2) (cid:3)
Next, the physical variables can be easily calculated. total mass and the radius of the sphere respectively,and
Thus, using (67) into (15) the energy density reads: the following relationships are satisfied
4π 2γδr3
m= µr¯3 = , (75)
8πµ=12γδ 3b2. (68) 3 (γr2+δ)3
−
To obtain the pressure we shall use (55) and (57),
which produce m(r¯ )=M, (76)
1
αr2+β γr2+δ+brcosθ and
8πP =(3b2 12γδ) 1 1 ,
− (cid:20) − γr12+δ αr2+β+arcosθ((cid:21)69) r2 = β+δ. (77)
1 γ α
where −
At this point it is pertinent to ask the question: to
what specific exterior spacetime can we match smoothly
αr2+β αr2+β
ζ =µ 1 , a= 1 b, (70) our solution? This is a relevant question, since there
0γr2+δ γr2+δ
1 1 are as many different (physically distinguishable) Weyl
solutions as there are different harmonic functions.
in order to satisfy the junction condition (54).
Theanswertothe abovequestionis the following: our
It may be instructive to recover the spherically sym-
solution cannot be matched to any Weyl exterior, even
metric case (the interior Schwarzschildsolution). In this
though it has a surface of vanishing pressure. This is
case we have a=b=0.
so because the first fundamental form is not continuous
Toseehowthiscomesabout,letusperformthetrans-
across the boundary surface.
formation
Indeed from the continuity of g and g components
tt φφ
r
r¯= , θ¯=θ, t¯=t, φ¯=φ, (71) at r =r1, we have:
γr2+δ
αr2+β+ar cosθ
where overbar denotes the usual Schwarzschild coordi- AW(r1,θ)= γr12+δ+br1cosθ , (78)
nates. Then,it isa simple matterto checkthat(67) and 1 1
(69) are identical to the well known expressions charac- and
terizing the interior Schwarzschild solution:
A (r ,θ)=γr2+δ+br cosθ, (79)
W 1 1 1
1 2M 2m(r¯) where AW(r1,θ) denotes the √gtt of any Weyl exterior
gt¯t¯= 4 3(1− r¯ )(1/2)−(1− r¯ )(1/2) , (72) solution (evaluated on the boundary surface). It is a
(cid:20) 1 (cid:21) simplemattertocheckthatthetwoequationsabovecan-
not be satisfied unless a = b = 0 which corresponds to
2m(r¯) the spherically symmetric case. This result is in agree-
g =(1 )−1, (73)
r¯r¯ − r¯ ment with theorems indicating that static, perfect fluid
(isotropic in pressure)sources are spherical (see [32] and
references therein)
(1 2m(r¯))1/2 (1 2M)1/2 Observethattheaboveresultisaconsequenceof(61).
P =µ − r¯ − − r¯1 , (74) Thereforeinorderto find matchablesolutions weshould
"3(1− 2r¯M1 )1/2−(1− 2mr¯(r¯))1/2# relax this condition. In this later case, of course, nei-
ther isotropy of pressure nor homogeneity of the energy
where m(r¯), M and r¯ denote the mass function, the density is preserved.
1
8
The remaining possibility is trying to match on a turn may severely affect some important characteristics
boundary surface given by the equation r=r (θ). How- ofcompactobjects (see [44]–[46] and referencestherein).
1
everthis does notseemto solvethe problemsince inour Besides the magnetic field, local anisotropy of pressure
case A(r ,θ)=constant. may be produced by a variety of physical phenomena
1
(see [47] and references therein).
Thus, let us assume P = = = P P = 0,
xy 1 3 yy zz
E E −
VI. ANISOTROPIC INHOMOGENEOUS although = 0 and P = P . Then from the equa-
2 xx yy
SPHEROIDS tion(19)aEnd6(A.9)weobtai6ntheequations(58)and(59).
Next, introducing the auxiliary functions A˜(r,θ) and
In order to find solutions that could be matched to a R(r,θ) defined by
Weyl exterior, we shall relax the condition of isotropy of
pressure and energy density homogeneity. A(r,θ)=A˜(r,θ)B(r,θ)r,
Theintroductionofthepressureanisotropyiswellmo- D(r,θ)=R(r,θ)B(r,θ)r, (80)
tivated frompurely physicalconsiderations. Indeed, it is
wellknownthatcompactobjects(white dwarfsandneu- the equation (58) and (59) can be rewritten as:
tron stars) are endowed with strong magnetic fields (see
[33]–[39] and references therein). On the other hand it A˜′ R′ 1
θ = θ =Br( )′. (81)
has been shown that the effect of such a magnetic field A˜ R Br θ
on a degenerate Fermi gas manifests itself through the
appearance of a strong anisotropy due to the magnetic FromP P = =0andthe equations(A.8)(A.11)
yy zz 3
− E
stresses (see [40]–[43] and references therein). This in we get
R′′ R′ 1 A˜′ 1 A˜ A˜ R
+ ( )+ ( θθ θ θ)=0, (82)
R R r − A˜ r2 A˜ − A˜ R
R′ A˜′ B′ 1 1 A˜ R B R B2 A˜ B
( + + )+ ( θ θ + θ θ +2 θ θθ θθ)=0. (83)
R A˜ B r r2 A˜ R B R B2 − A˜ − B
Inordertofindasimplesolutionthatsatifiesthebound-
ary condition, we choose:
R(r,θ)=sinθ[b(r )+γcosθ], (84)
1
whereα,γ,β anda(r),b(r)arearbitraryconstantsand
then from the equations (81)-(83) we find functions of integration respectively. a(r), α and β have
thedimensionofaninverseoflength,whereasthe others
γ
A˜(r,θ)=α sin2θ b(r ) cosθ+a(r), (85) are dimensionless.
1
2 −
h i
For the above metric, Einstein equations yield the fol-
1
B(r,θ)= , (86) lowing expressions for physical variables.
βr γ sin2θ b(r )cosθ +b(r)
2 − 1
(cid:2) (cid:3)
2b′′ 3(βΓ+b′)2 3(βΛsinθ)2 1 4(βΓ+b′) 4βΣ
8πµ=(βrΓ+b)2 + +
βrΓ+b − (βrΓ+b)2 − (βrΓ+b)2 r βrΓ+b βrΓ+b −
(cid:26) (cid:20) (cid:21)
1 b +4γcosθ
1 1 , (87)
− r2 − Λ
(cid:20) (cid:21)(cid:27)
9
2a′(βΓ+b′) 3(βΓ+b′)2 3(βΛsinθ)2
8πP =(βrΓ+b)2 − + +
xx (βrΓ+b)(αΓ+a) (βrΓ+b)2 (βrΓ+b)2
(cid:26)
1 2a′ 6(βΓ+b′) 4βΣ 2αβΛ2sin2θ
+
r αΓ+a − βrΓ+b − βrΓ+b − (αΓ+a)(βrΓ+b)
(cid:20) (cid:21)
1 2αΣ b +4γcosθ
3+ 1 , (88)
− r2 αΓ+a − Λ
(cid:20) (cid:21)(cid:27)
a′′ 2b′′ 2a′(βΓ+b′) 3(βΓ+b′)2 3(βΛsinθ)2
8πP =8πP =(βrΓ+b)2 + +
yy zz αΓ+a − βrΓ+b − (βrΓ+b)(αΓ+a) (βrΓ+b)2 (βrΓ+b)2
(cid:26)
1 3a′ 4(βΓ+b′) 2βΣ 2αβΛ2sin2θ 1 αΣ
+ + 1+ , (89)
r αΓ+a − βrΓ+b − βrΓ+b − (αΓ+a)(βrΓ+b) r2 αΓ+a
(cid:20) (cid:21) (cid:18) (cid:19)(cid:27)
where order to study the physical relevance of nonsphericity in
the structure of the source it is necessary to match the
γsin2θ
Λ=b +γcosθ, Γ= b cosθ, b(r )=b above metioned solution to a specific Weyl exterior (so
1 1 1 1
2 −
that the arbitrary parameters of the source could be re-
Σ=Λcosθ γsin2θ. (90) lated to the parameters of the exterior metric), but such
−
a task is beyond the scope of this paper.
The equations above describe a wide class of solutions
In general, solutions as the one presented here and
that can be matched to any specific Weyl metric by an
manyothersfoundbyeitheranalyticalornumericalpro-
appropriate choice of functions and constants of integra-
cedurescouldprovideanswerstoimportantquestionsre-
tion. Furthermore physically reasonable models can be
lated to stellar structure, namely:
obtained at least from slight deviations from spherical
symmetry.
Whatisthe limit ofcompactnessofastaticaxially
•
symmetric source?
VII. CONCLUSIONS
How is the above limit related to (influenced by)
•
some exterior parameters such as the quadrupole
We have established the general framework to carry
moment of the source?
out a systematic analysis of general static axially sym-
metric sources. By “general” we mean that the Weyl
How are, intrinsically nonspherical physical vari-
gaugewasnotassumedandthematterdescriptionisthe •
ables ( e.g. P ), related to multipole moments
xy
mostgeneralcompatiblewithaxialsymmetryandstatic-
(higher than monopole)?
ity.
Thus, we started from the most general line element
Are specific Weyl exteriors related to specific
andconsideredageneralanisotropicfluidassourceofthe •
sources?
exterior Weyl spacetime. Relevant equations were then
written down and structure scalars were calculated.
We have seen that the three structure scalars associ-
atedwith the tracefree partofthe tensor X define the ACKNOWLEDGMENTS
αβ
inhomogeneity factors.
We have found an exact analytical solution represent- L.H. thanks Departamento de F´ısica Teo´rica e Histo-
ing a spheroid of isotropic pressure with homogeneous ria de la Ciencia, Universidad del Pa´ıs Vasco for finan-
energy density. In the spherically symmetric limit our cialsupportandhospitality. ADP andJ.O.acknowledge
solution becomes the well known Schwarzschild interior hospitality of the Departamento de F´ısica Teo´rica e His-
solution. Such an interior cannot be matched (except in toria de la Ciencia, Universidad del Pa´ıs Vasco. This
the spherically symmetric case) to any Weyl exterior. workwaspartially supportedby the SpanishMinistry of
Matchable solutionscanbe found by relaxingthe con- Science and Innovation (grant FIS2010-15492) and UFI
ditionsofisotropyofpressureanddensityinhomogeneity. 11/55 program of the Universidad del Pa´ıs Vasco. J.O.
The anisotropy of pressure was also justified on physical acknowledges financial support from the Spanish Min-
grounds. An example was given in the last section. In istry of Science and Innovation (grant FIS2009-07238).
10
Appendix: Expression for the components of the
electric Weyl tensor
The nonvanishing components as calculated from (23)
are:
1 2A′′ B′′ D′′ 3A′B′ A′D′ B′ 3B′D′ 1 D′ B′ A′
E = +( )2+ + (2 )
11 6 A − B − D − AB − AD B BD r D − B − A
(cid:20) (cid:21)
1 A B 2D 3A B A D B 3B D
+ θθ θθ + θθ + θ θ θ θ +( θ)2 θ θ , (A.1)
6r2 − A − B D AB − AD B − BD
(cid:20) (cid:21)
r2 A′′ B′′ 2D′′ 3A′B′ A′D′ B′ 3B′D′ 1 D′ B′ 2A′
E = + + ( )2+ + ( + )
22 − 6 A B − D − AB AD − B BD r D B − A
(cid:20) (cid:21)
1 2A B D 3A B A D B 3B D
θθ + θθ + θθ + θ θ + θ θ ( θ)2 θ θ , (A.2)
− 6 − A B D AB AD − B − BD
(cid:20) (cid:21)
D2 A′′ 2B′′ D′′ 2A′D′ B′ 1 D′ 2B′ A′
E = + +2( )2+ ( + )
33 −6B2 A − B D − AD B r D − B A
(cid:20) (cid:21)
D2 A 2B D 2A D B
θθ θθ + θθ θ θ +2( θ)2 , (A.3)
− 6B2r2 A − B D − AD B
(cid:20) (cid:21)
1 A′ D′ B D′ A′B B′A D B′ 1 A D
E = θ θ + θ θ θ + θ θ θ . (A.4)
12 2 A − D B D − AB − AB D B − r A − D
(cid:20) (cid:18) (cid:19)(cid:21)
Thesecomponentsarenotindependentsincetheysatisfy For the three scalars , , we obtain:
1 2 3
E E E
the relationship:
1 B2
E + E + E =0. (A.5)
11 r2 22 D2 33
1 1 A′ D′ B A′ D′B B′A D B′ 1 D A
= ( θ θ θ + θ θ + θ )+ ( θ θ) , (A.6)
E1 2B2 r A − D − B A D B − B A D B r2 D − A
(cid:20) (cid:21)
1 A′′ B′′ A′B′ A′D′ B′ B′D′ 1 B′ D′
= + + + ( )2 + ( )
E2 −2B2 − A B AB AD − B − BD r B − D
(cid:20) (cid:21)
1 B D A B A D B B D
θθ θθ θ θ + θ θ ( θ)2+ θ θ , (A.7)
− 2B2r2 B − D − AB AD − B BD
(cid:20) (cid:21)