Interior solution for the Kerr metric

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by  J. L. Hernandez-Pastora

Interior solution for the Kerr metric
J. L. Hernandez-Pastora∗
Departamento de Matematica Aplicada and Instituto Universitario de Fisica
Fundamental y Matematicas, Universidad de Salamanca, Salamanca, Spain
L. Herrera†
Escuela de F´ısica, Facultad de Ciencias, Universidad Central de Venezuela, Caracas 1050,
Venezuela and Instituto Universitario de F´ısica Fundamental y Matema´ticas,
Universidad de Salamanca 37007, Salamanca, Spain
(Dated: January 10, 2017)
A,recentlypresented,generalproceduretofindstaticandaxiallysymmetric,interiorsolutionsto
the Einstein equations, is extended to the stationary case, and applied to find an interior solution
for the Kerr metric. The solution, which is generated by an anisotropic fluid, verifies the energy
7
conditionsforawiderangeofvaluesoftheparameters,andmatchessmoothlytotheKerrsolution,
1
thereby representing a globally regular model describing a non spherical and rotating source of
0
2 gravitational field. In the spherically symmetric limit, our model converges to the well known
incompressible perfect fluid solution.The key stone of our approach is based on an ansatz allowing
n
to define the interior metric in terms of the exterior metric functions evaluated at the boundary
a
source. The physical variables of the energy-momentum tensor are calculated explicitly, as well as
J
thegeometry of thesource in terms of therelativistic multipole moments.
9
PACSnumbers: 04.20.Cv,04.20.Dw,97.60.Lf,04.80.Cc
]
c
q
- I. INTRODUCTION II. THE GLOBAL MODEL OF A
r SELF-GRAVITATING STATIONARY SOURCE
g
[
A. The exterior metric
1
v Since the discovery of the Kerr metric [1] there have
8 beenmanyattemptstofindaphysicallymeaningfulmat- The line element for a vacuum stationary and axi-
9 ter distribution that could serve as its source (see [2–17] ally symmetric space–time, in Weyl canonical coordi-
0 and references therein). However, no satisfactory solu- nates may be written as :
2
0 tion has yet been found to this problem. ds2E =−e2ψ(dt−wdφ)2+e−2ψ+2Γ(dρ2+dz2)+e−2ψρ2dφ2,
. (1)
1
where ψ = ψ(ρ,z) , Γ = Γ(ρ,z) and w = w(ρ,z) are
0
7 Itisthepurposeofthiswork,toprovideaninteriorso- functions of their arguments.
1 lutiontoEinsteinequations,satisfyingalltheusualphys- Forvacuumspace–times,Einstein’sfieldequationsim-
: ical conditions, and smoothly matched on the boundary ply for the metric functions
v
surfaceofthefluiddistribution,totheKerrmetric. With
Xi this aim we shall generalize a procedure, recently pro- f(f,ρρ+ρ−1f,ρ+f,zz)−f,2ρ−f,2z+ρ−2f4(w,2ρ+w,2z)=0,
(2)
r posed to find sources of the Weyl metrics [18], to the
a
stationary case.
f(w +ρ−1w +w )+2f w +2f w =0, (3)
,ρρ ,ρ ,zz ,ρ ,ρ ,z ,z
with f e2ψ and
As a particular example we shall find a source for the ≡
1 1
Kerr metric which consists in an anisotropic fluid, satis- Γ = ρf−2(f2 f2) ρ−1f2(w2 w2)
fying the Darmois matching conditions on the boundary ,ρ 4 ,ρ− ,z − 4 ,ρ− ,z
surface of the matter distribution, thereby excluding the Γ = 1ρf−2f f 1ρ−1f2w w . (4)
,z ,ρ ,z ,ρ ,z
presenceofthinshells,andexhibitingawellbehaviourof 2 − 2
allphysicalvariables,forawiderangeofvaluesofthepa- Noticethat(2),(3)arepreciselytheintegrabilitycondi-
rameters of the solution. These include values which are tion of (4), that is: given any ψ and w satisfying (2),(3),
commonly assumed in realistic models of rotating neu- a function Γ satisfying (4) always exists.
tron stars and white dwarfs. We can write the line element above, in Erez-Rosen
[19], or standard Schwarzschild–type coordinates r,y
cosθ or in spheroidalprolate coordinates x r−{M,y≡
} { ≡ M }
[20]:
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] ρ2 =r(r 2M)(1 y2) , z =(r M)y, (5)
− − −
2
where M is a constant which will be identified later. may be writen as:
In terms of the above coordinates the line element (1)
ds2 = e2ψ(r,y)(dt wdφ)2+e−2ψ+2[Γ(r,y)−Γs]dr2+e−2[ψ−ψs]+2[Γ(r,y)−Γs]r2dθ2+e−2[ψ−ψs]r2sin2θdφ2, (6)
E − −
where ψs and Γs are the metric functions correspond- the continuity of the metric functions and the continuity
ing to the Schwarzschildsolution, namely, of the first derivatives ∂ g , ∂ g , ∂ g , producing:
r tt r θθ r φφ
ψs = 21ln r−r2M Γs =−12ln (r−r(Mr)22−My)2M2 , aaΣsΣ ==ψψΣΣs ,, a(a′Σs)=′Σψ=Σ′(ψ, s)g′ΣΣ, =ΓΣ , gΣ′ =Γ′Σ,
(cid:18) (cid:19) (cid:20) − (cid:21)
(7) gs =Γs , (gs)′ =(Γs)′ ,
Σ Σ Σ Σ
where the parameter M is easily identified as the Ω =w , Ω′ =w′ , (10)
Schwarzschildmass. Σ Σ Σ Σ
where prime denotes partial derivative with respect to r
and subscript Σ indicates that the quantity is evaluated
B. The interior metric ontheboundarysurface. Itisimportanttokeepinmind
that we are using global coordinates r,θ on both sides
{ }
Weshallnowassumefortheinterioraxiallysymmetric of the boundary.
line element: Thus,our line element(8) matchessmoothly with any
stationary exterior (6), provided conditions (10) are sat-
ds2 = e2aˆZ(r)2(dt Ωdφ)2+ e2gˆ−2aˆdr2+e2gˆ−2aˆr2dθ2 isfied.
I − − A(r) In the particular case when we want to match our in-
+ e−2aˆr2sin2θdφ2, (8) terior with the Schwarzschild exterior (the static limit),
thenψ =ψs andΓ=Γs,andthesourceisaperfectfluid
with if aˆ=gˆ=Ω=0.
Weshallnowseehowthefieldequationsconstrainfur-
aˆ a(r,θ) as(r) , gˆ g(r,θ) gs(r,θ), (9) ther our possible interiors.
≡ − ≡ −
where Ω = Ω(r,θ), and as(r) and gs(r,θ) are functions
that, on the boundary surface, equal the metric func- D. The field equations and constraints
tionscorrespondingtotheSchwarzschildsolution(7),i.e.
as(rΣ)=ψΣs and gs(rΣ)=ΓsΣ. Also, A(r)≡1−pr2 and Let us first analyse the well known case when the in-
3 1
terior is spherically symmetric, then aˆ = gˆ= 0, and the
Z A(r ) A(r), where p is an arbitrary con-
Σ
≡ 2 − 2 physical variables are obtained from the field equations
stant and the boundary surface of the source is defined
p p for a perfect fluid, the result is well known and reads (in
by r =r =const.
Σ relativistic units)
The case w = 0, gˆ = aˆ = 0, corresponds to a spher-
ically symmetric distribution, more specifically, to the 3p
T0 µ = ,
wellknownincompressible(homogeneousenergydensity) − 0 ≡ 8π
perfect fluid sphere, and hence the matching of (8) with √A √A
2M T1 =T2 =T3 P = µ − Σ , (11)
the Schwarzschild solution implies p= rΣ3 . The simple 1 2 3 ≡ 3√AΣ−√A!
condition w =0 recovers,of course, the static case.
2m(r) 2Mr2
Itshouldbenoticedthatforsimplicityweconsiderhere with A = 1 =1 pr2 =1 , where µ
− r − − r3
only matching surfaces of the form r = r = const, of Σ
Σ and P denote the energy density and the isotropic pres-
course more general surfaces with axial symmetry could
surerespectively,andforthemassfunctionm(r)wehave
however be considered as well.
r
m(r)= 4π r2T0dr, (12)
C. The matching conditions − 0
Z0
implying
We shall now turn to the matching (Darmois) condi-
tions[21]. Thusthecontinuityofthefirstandthesecond rΣ pr3
M m(r )= 4π r2T0dr = Σ. (13)
fundamental form across the boundary surface implies ≡ Σ − 0 2
Z0
3
This model, which describes the well known incom- spherical case. Thus, for our line element (8) we have
pressible perfect fluid sphere, is further restricted by the the following non vanishing components of the energy–
requirement that the pressure be regular and positive momentum tensor:
everywhere within the fluid distribution, which implies
τ rΣ > 9. Asitisevidentfrom(11)the pressurevan- −T00 = κ(8πµ+pˆzz −E+3δJ++δΩI),
≡ M 4 T1 = κ(8πP pˆ δJ ),
ishes at the boundary surface. 1 − xx− −
Finally, if we impose the strong energy condition P < T22 = κ(8πP +pˆxx+δJ−),
µ,weshouldfurtherrestrictourmodelwiththecondition T3 = κ(8πP pˆ +δΩI+δJ ),
8 3 − zz +
τ > . T3 = κδI, (14)
3 0 −
We shall now proceed to consider the general, non–
κ cosθ gˆ (1 A) Ω Ω′
T2 =gθθT = 2aˆ aˆ′ gˆ′ ,θ + − (2aˆ gˆ ) δ ,θ , (15)
1 12 −r2 ,θ − sinθ − r r√A(3√A √A) ,θ− ,θ − 2r2sin2θ
(cid:20) Σ− (cid:21)
e2aˆ−2gˆ Fromtheexpressionsabove,using(14)-(16)andintro-
with κ , δ e4aˆZ2, Ω denotes derivative of Ω
≡ 8π ≡ ,y ducing the dimensionless parameter s r/rΣ, we can
with respect to y cosθ, and now obtain the explicit expressionsfor t≡he physicalvari-
≡
ables, (with A=1 (2s2)/τ, A =A(s=1)):
Ω 2 Ω′ 2 − Σ
,y
J = − A ,
± 2r2 ± 2rsinθ
(cid:18) (cid:19) (cid:18) (cid:19)
Ω′′A 2Ω′aˆ′A Ω Ω aˆ
,yy ,y ,y
I = + + +2 +
2r2sin2θ r2sin2θ 2r4 r4
(1 A)Ω′ 4√A 3√A
Σ
+ − − ,
2r3sin2θ 3√AΣ √A !
−
aˆ′9√A 4√A
E = 2∆aˆ+(1 A) 2 Σ− +2aˆ′′ gˆ′′ ,
− − " r 3√AΣ √A − #
−
aˆ′ aˆ aˆ cosθ
∆aˆ=aˆ′′+2 + ,θθ + ,θ ,
r r2 r2 sinθ
aˆ2 gˆ′ gˆ cosθ
pˆ = ,θ +aˆ′2+ ,θ +
xx −r2 − r r2 sinθ
aˆ′ √A gˆ′3√A 2√A
+ (1 A) 2 aˆ′2+ Σ− ,
− " r 3√AΣ √A − r 3√AΣ √A #
− −
aˆ2 gˆ′ gˆ
pˆ = ,θ aˆ′2 ,θθ gˆ′′+
zz −r2 − r − − r2 −
aˆ′ √A gˆ′
+ (1 A) 2 +aˆ′2+2 . (16)
− "− r 3√AΣ √A r#
−
κ 6 aˆ2 gˆ gˆ aˆ 2 cosθ
T0 = ,θ A (aˆ )2+(gˆ ) 2(aˆ ) ,θθ +(1 2A) ,s 2(1 3A) ,s + aˆ +aˆ +
− 0 rΣ2 (τ − s2 − ,s ,ss − ,ss − s2 − s − − s s2 (cid:18) ,θθ ,θsinθ(cid:19))
(cid:2) (cid:3)
κδ 3 (Ω )2 A(Ω )2 Ω A 2Ω aˆ A Ω Ω aˆ (1 A)Ω 4√A 3√A
,y ,s ,ss ,s ,s ,yy ,y ,y ,s Σ
+ + +Ω + + +2 + − − ,
rΣ4 (4(cid:18) s4 s2sin2θ(cid:19) "2s2sin2θ s2sin2θ 2s4 s4 2s3sin2θ 3√AΣ−√A !#)
(17)
4
κδ Ω A 2Ω aˆ A Ω Ω aˆ (1 A)Ω 4√A 3√A
T3 = ,ss + ,s ,s + ,yy +2 ,y ,y + − ,s Σ− , (18)
0 −rΣ4 "2s2sin2θ s2sin2θ 2s4 s4 2s3sin2θ 3√AΣ √A !#
−
κ 6 √A √A aˆ2 gˆ cosθ aˆ √A(1 A) gˆ (1 A)(3√A 2√A)
T1 = − Σ + ,θ (aˆ )2A ,θ 2 ,s − ,s − Σ− 1 +
1 rΣ2 (τ 3√AΣ √A s2 − ,s − s2 sinθ − s 3√AΣ √A − s " 3√AΣ √A − #)
− − −
κδ 1 (Ω )2 A(Ω )2
,y ,s
, (19)
− r4 4 s4 − s2sin2θ
Σ (cid:26) (cid:18) (cid:19)(cid:27)
κ 6 √A √A aˆ2 gˆ aˆ √A(1 A) gˆ
T3 = − Σ + ,θ +(aˆ )2A+(gˆ )+ ,θθ +2 ,s − ,s(1 2A) +
3 rΣ2 (τ 3√AΣ √A s2 ,s ,ss s2 s 3√AΣ √A − s − )
− −
κδ 1 (Ω )2 A(Ω )2 Ω A 2Ω aˆ A Ω Ω aˆ (1 A)Ω 4√A 3√A
,y ,s ,ss ,s ,s ,yy ,y ,y ,s Σ
+ + +Ω + + +2 + − − ,
rΣ4 (4(cid:18) s4 s2sin2θ(cid:19) "2s2sin2θ s2sin2θ 2s4 s4 2s3sin2θ 3√AΣ−√A !#)
(20)
κ cosθ gˆ 2s(2aˆ gˆ ) κδ Ω Ω
T2 = 2aˆ aˆ gˆ ,θ + ,θ− ,θ + ,θ ,s . (21)
1 −s2r3 ,θ ,s− ,ssinθ − s √τ 2s2(3√τ 2 √τ 2s2) r5 2s4sin2θ
Σ (cid:20) − − − − (cid:21) Σ (cid:20) (cid:21)
Obviouslyforanyspecific(non–spherical)modelweneed and gˆ such that, once the junction conditions (10) are
to provide explicit forms for aˆ, gˆandΩ, howeverevenat satisfied, the angular derivatives of such functions are
this level of generality we can assure that the junction continuous, i.e.: (aˆ ) = (ψ ) and (gˆ ) = (Γ ) , as
,θ Σ ,θ Σ ,θ Σ ,θ Σ
conditions (10) imply (P g T1) =0. well as (Ω ) =(w ) .
rr ≡ rr 1 Σ ,θ Σ ,θ Σ
We shall first proceed to prove the above statement,
and then we shall provide a general procedure to choose r 2M
Σ
Then, using A = − in (14), we obtain for
aˆ and gˆ producing physically meaningful models. Σ r
Σ
It is always possible to choose the metric functions aˆ pˆ +δJ on the boundary surface:
xx −
1 cosθ
(pˆ +δJ ) = (ψˆ )2 +(Γˆ ) +2Mψˆ′ +r (r 2M)ψˆ′2 (r M)Γˆ′ +
xx − Σ r2 − ,θ Σ ,θ Σsinθ Σ Σ Σ− Σ − Σ− Σ
Σ (cid:20)
r e4ψ w˙ 2 r 2M
+ Σ Σ (w′ )2 Σ− , (22)
4(rΣ−2M)sin2θ (cid:18) rΣ2 − Σ rΣ (cid:19)(cid:21)
whereψˆ ψ ψs,Γˆ Γ Γs,anddotalsodenotes partial derivative with respect to θ.
Σ ≡ Σ− Σ Σ ≡ Σ− Σ
5
Taking into account the Einstein’s vacuum equations tives of the metric function w, from the Einstein tensor
(G = 0), we find the following relation for the deriva- component G =0:
αβ 11
re4ψ w˙2 r 2M cosθ M2
− (w′)2 − = (ψ )2+(Γˆ ) +r(r 2M)ψ′2 (r M)Γˆ′,(23)
4(r 2M)sin2θ r2 − r − ,θ ,θ sinθ − − r(r 2M) − −
− (cid:20) (cid:21) −
implying the vanishing of (T1) . the boundary surface, if we take into account that the
1 Σ
In a similar way it can be shown that T2 vanishes on Einstein vacuum equation G =0 produces
1 12
w′w˙ e4ψ =4r(r 2M)ψ˙ψ′ 2(r M)Γˆ˙ r(r 2M)2Γˆ′cosθ, (24)
sin2θ − − − − − sinθ
and from (15), we have the following expression for T2 shall demand:
1
on the boundary
aˆ′ =aˆ =aˆ =aˆ′ =aˆ′ =0
cosθ 0 ,θ0 ,θθ0 ,θ0 ,θθ0
(T12)Σ = 2(ψˆ,θ)ΣψˆΣ′ −Γˆ′Σsinθ + gˆ0′ =gˆ,θ0 =gˆ,θθ0=gˆ,′θ0 =gˆ,′θθ0=0
rΣ−M (Γˆ ) + 2M (ψ ) + gˆ0′′ =gˆ,′θ′0 =0, (27)
,θ Σ ,θ Σ
− r (r 2M) r (r 2M)
Σ Σ Σ Σ
− −
w′ (w ) e4ψΣ
Σ ,θ Σ . (25)
− 2rΣ(rΣ 2M)sin2θ Ω0 =Ω′0 =Ω′0′ =Ω′0′′ =0
−
Ω˙ =Ω˙′ =0
This last condition (T2) =0, which follows from the 0 0
Darmois conditions, and1thΣerefore is necessary, in order Ω¨0 =Ω¨′0 =Ω¨′0′ =Ω¨′0′′ =0, (28)
to avoid the presence of shells on the boundary surface,
can be obtained at once from a simple inspection of the where (15, 16) have been used, and the subscript 0 indi-
equation (22) in [22]. cates that the quantity is evaluated at the center of the
distribution.
Usingtheconditionsabovein(26)wemaywriteforΛ,
E. The ansatz for the metric functions Ξ and Π
We shall provide a general procedure to choose aˆ, gˆ Λ(r,θ) = Λ (θ)+Λ′(θ)r+F(r,θ)
0 0
and Ω producing physically meaningful models. With
Ξ(r,θ) = Ξ (θ)+Ξ′(θ)r+Ξ′′(θ)r2+G(r,θ)
thisaim,besidesthefulfillmentofthejunctionconditions 0 0 0
(10),weshallrequirethatallphysicalvariablesberegular Π(r,θ) = Π0(θ)+Π′0(θ)r+Π′0′(θ)r2+
withinthefluiddistributionandtheenergydensitytobe + Π′′′(θ)r3+H(r,θ) (29)
0
positive.
To ensure the fulfillment of the junction conditions with F(0,θ) = F′(0,θ) = 0 and G(0,θ) = G′(0,θ) =
(10), we may write without loss of generality, G′′(0,θ) = 0, as well as H(0,θ)= H′(0,θ)= H′′(0,θ) =
H′′′(0,θ) = 0. Then we can finally write for aˆ and gˆ
aˆ = ψˆΣ(θ)+ψˆΣ′ (θ)(r−rΣ)+Λ(r,θ)(r−rΣ)2 (please notice a that there is a misprint in Eq.(43) in
gˆ = Γˆ (θ)+Γˆ′ (θ)(r r )+Ξ(r,θ)(r r )2, [18], there it shouldreadΓ insteadofψ inthe twoterms
Σ Σ − Σ − Σ
Ω = w (θ)+w′ (θ)(r r )+Π(r,θ)(r r )2,(26) withintheroundbracketsforthe metricfunctiong˜(r,θ))
Σ Σ − Σ − Σ
whereΛ(r,θ), Ξ(r,θ) andΠ(r,θ) aresofartwoarbitrary ψˆ′ ψˆ ψˆ′ ψˆ
functions of their arguments. aˆ(r,θ) = r2 Σ +3 Σ +r3 Σ 2 Σ
On the other hand, to guarantee a good behaviour of −rΣ rΣ2 ! −rΣ2 − rΣ3 !
thephysicalvariablesatthecenterofthedistributionwe + (r r )2F(r,θ) (30)
Σ
−
6
gˆ(r,θ) = r4 Γˆ′Σ 3ψˆΣ +r3 Γˆ′Σ +4ψˆΣ w = j(2M +r1+r2)(4M2(1−j2)−(r1−r2)2) ,
rΣ2 − rΣ3 ! −rΣ2 rΣ3 ! (r1+r2)2(1−j2)−4M2(1−j2)+j2(r1−r2)2
(35)
+ (r rΣ)2G(r,θ). (31) where j J =a/M denotes the dimensionless param-
− ≡ M2
eter representing the angular momentum of the source,
and is related to the rotation parameter a of Kerr in
w′ w w′ w its well-known Boyer-Lindquist representation. Also,
Ω(r,θ) = r4 − Σ +5 Σ +r5 Σ 4 Σ
(cid:18) rΣ3 rΣ4 (cid:19) (cid:18)rΣ4 − rΣ5 (cid:19) r1,2 ≡ ρ2+(z±M 1−j2)2,whichintheErez-Rosen
+ (r rΣ)2H(r,θ). (32) coordinqates becomes p
−
2
The metric functions obtained so far, satisfy the junc- r2 = (r M) My(r M) 1 j2 M2j2(1 y2).
1,2 − ± − − − −
tionconditions(10)andproducephysicalvariableswhich
h p i (36)
areregularwithinthefluiddistribution. Furthermorethe
As is well known, the relativistic multipole moments
vanishing of gˆ on the axis of symmetry, as required by
(RMM) [26–30], of the Kerr solution, are easily given
theregularityconditions,necessarytoensureelementary
in terms of the rotation parameter j (as well as a). In
flatness in the vicinity of the axis of symmetry, and in
fact, using the FHP method [31] in order to calculate
particularatthecenter(see[23],[24],[25]),isassuredby
the RMM, these are expressedin terms of the expansion
the fact thatΓˆΣ and Γˆ′Σ vanishon the axis of symmetry. coefficients (mk) of the Ernst potential on the axis of
Furthermore,thegoodbehaviourofthefunctionΩonthe symmetry,buttheKerrsolutionistheonlyoneverifying
symmetry axis is fulfilled since wΣ and wΣ′ vanish when that each RMM (Mk) at any order k is just equal to
y = 1. Finally, let us note that the energy-momentum the corresponding coefficient m for such order, which
± k
tensor components (14), (15) do not divergeonthe sym- implies,
metry axis because the first derivative with respect to
the angularvariableθ ofbothΩ′ andΩ′′ vanishes onthe m =M =M(ia)k (37)
k k
symmetry axis since not only w and w′ vanish there,
Σ Σ
buttheirfirstderivativeswithrespecttoθvanishaswell.
Therefore, the massive RMM (even orders) and the ro-
Sofarwehavepresentedthegeneralproceduretobuild
tational RMM (odd orders) can be expressed as follows
up sources for any stationary metric, in what follows,
we shall illustrate the method with the example of Kerr
metric. M =( 1)lM2l+1j2l , M =i( 1)lM2l+2j2l+1.
2l 2l+1
− −
(38)
In particular let us remind that a first conclusion de-
III. PARTICULAR SOLUTION
rived from these RMM (38), is that the rotation of the
object leads to a negative quadrupole massive moment,
A. A source for the exterior Kerr’s solution M
q 2 = j2,i.e. allthe possible sourcesofKerrsolu-
≡ M3 −
The Kerr metric in Weyl coordinates is given by the tion are oblate.
following metric functions Let us next, consider the line element (8) with metric
functions given by (30) and (31)
(r +r )2(1 j2) 4M2(1 j2)+j2(r r )2
1 2 1 2
f = − − − − ,
(r +r +2M)2(1 j2)+j2(r r )2 aˆ(r,θ) = ψˆ s2(3 2s)+r ψˆ′ s2(s 1),
1 2 − 1− 2 Σ − Σ Σ −
(33) gˆ(r,θ) = Γˆ s3(4 3s)+r Γˆ′ s3(s 1),
Σ − Σ Σ −
Ω(r,θ) = w s4(5 4s)+r w′ s4(s 1). (39)
(r +r )2(1 j2) 4M2(1 j2)+j2(r r )2 Σ − Σ Σ −
e2Γ = 1 2 − − − 1− 2 ,
4r1r2(1−j2) with s≡r/rΣ ∈[0,1].
(34) For the Kerr solution we may write:
7
1 τ N +rΣrΣ(2j2 1)
ψˆ ψ ψs = ln 1 2 − ,
Σ ≡ Σ− Σ 2 τ 2N +rΣrΣ(2j2 1) 2(1 j2)(rΣ+rΣ+2)
(cid:26) − 1 2 − − − 1 2 (cid:27)
1 (τ 1)2 y2N +rΣrΣ(2j2 1)
Γˆ Γ Γs = ln − − 1 2 − ,
Σ ≡ Σ− Σ 2 τ(τ 2) 2rΣrΣ(j2 1)
(cid:26) − 1 2 − (cid:27)
(N +rΣrΣ)(2+rΣ+rΣ)
w =Mj 1 2 1 2 , (40)
Σ (j2 1) (j2 1)(N +rΣrΣ)+2(N +j4)
− − 1 2
(cid:2) (cid:3)
with verifytheinequalitye−aˆ(y=1) >egˆ(y=0)−aˆ(y=0) forallval-
ues of s in the range s [0,1], no matter the sign of the
2 parameter j, leading to∈the well known result that the
rΣ = τ 1 y 1 j2 j2(1 y2), (41)
1,2 − ± − − − rotation of the object always generates an oblate source
r
(cid:16) p (cid:17) (q < 0) since l < l . In the figure 1, one example is
z ρ
shown.
N τ(τ 2)+(1 y2) j2. (42)
≡− − − −
A straightforward calculation, using (17)–(21) allows
us to find the explicit expressions for the physical vari-
ables, these are displayed in figures (3)–(7). However,
beforeenteringintoadetaileddiscussionofthesefigures,
we shall carry out some calculations with the purpose
of providing some information about the “shape” of the
source. In particular we shall see how it is related with
the rotation parameter j of the source.
For doing so, using our interior metric, we shall calcu-
late the proper length l of the object along the axis z,
z
and the proper equatorial radius lρ: FIG. 1: Functions exp(gˆ(s,y = 0) − aˆ(s,y = 0)) and
exp(−aˆ(s,y = 1)) for values of the rotation parameter j =
rΣ egˆ(y=1)−aˆ(y=1) rΣ egˆ(y=0)−aˆ(y=0) ±0.1 and τ =2.7.
l dz , l dρ,
z ρ
≡ √A ≡ √A
Z0 Z0
(43) Figure2showsthe ellipticity eofthe sourceasafunc-
where ρ,z are the cylindrical coordinates associated to tion of the rotation parameter j, for different values of
the Erez-Rosen coordinates. the parameter τ. As can be seen, the relation between
Obviously in the spherical case (aˆ =gˆ=Ω=0), both e and j for any value of τ shows that the greater is j,
lengths are identical: the greater is e, and therefore the shape of the source is
moreoblate. Ofcourse,forj =0(staticcase)werecover
ls =ls = rΣ dξ =r τ arcsin 2, (44) the sphericity (e = 0). It is also observed from the fig-
z ρ Z0 1−pξ2 Σr2 rτ ure 2 that the deformation of the source with respect to
the spherical case, for any fixed value of j, is smaller for
p 2 larger values of τ (less compact object).
wherethefactthatp= hasbeentakenintoaccount
τr2 Let us now turn back to the physical variables of our
Σ
and where ls, ls denote the lengths corresponding to the model. Figure3exhibitsthebehaviouroftheradialpres-
z ρ
spherical case. sure P g T1 for different values of j.
rr ≡ rr 1
It would be convenient to introduce here the concept In it, we observe the variation of the radial pressure
ofellipticity(e),whichintermsoflz andlρ,isdefinedas with respect to the spherically symmetric case (j = 0).
e 1 lρ. Thetwoextremevaluesofthisparameterare This variation is smaller for angle values close to the
≡ −lz
e=0,whichcorrespondstoasphericalobject,ande=1 equator,asitisapparentfory =0.3. Noticethatthera-
for the limiting case when the source is representedby a dialpressureis positive,withnegativepressuregradient,
disk. Inbetweenofthesetwoextremeswehavee>0for and vanishes on the boundary surface.
a prolate source and e<0 for an oblate one. Figures 4 and 5 depict the behaviour of different en-
In the general (non–spherical case) we must compare ergy momentum components, for a specific choice of the
function e−aˆ(y=1) with egˆ(y=0)−aˆ(y=0), since gˆ(y = 1) parameters q and τ.
±
vanishes along the axis. It can be seen that the sign of Figure 6, shows the verification of the strong energy
both functions is positive, and their relative magnitudes condition ( T0) T1 > 0 (also for a specific choice of
− 0 − 1
8
(a) (b)
FIG. 4: −rΣ2T00 (graphic a), and rΣ2T11 (graphic b), as func-
tions of y=cosθ and s, with j =0.1 and τ =2.7.
FIG.2: Relationbetween theellipticityofthe source eandits
rotation parameter j for different values of τ.
(a) (b)
(a) (b) FIG. 5: (a) rΣ2T33,and (b) rΣ3T12 as functions of y and s.
FIG. 3: Four profiles of rΣ2Prr ≡ rΣ2grrT11, as function of s,
effectappearinginstationaryEinstein–Maxwellsystems.
for y = 0.3 (graphic a), and y = 1 (graphic b) with τ = 2.7,
and different values of j. Indeed,inallstationaryEinstein–Maxwellsystems,there
is a non vanishing component of the Poynting vectorde-
scribing a similar phenomenon [33, 34] (of electromag-
netic nature, in this latter case). Thus, the appearance
the parameters j and τ).
Figure 7, shows the component T3. of such a component, seems to be a distinct physical
0 property of rotating fluids, which has been overlooked
in previous studies of these sources.
We have carried out a systematic search for the range
IV. DISCUSSION
ofvalues ofτ andj for whichour models exhibit accept-
able physical properties. We have focused on the fulfill-
By extending the general procedure developed in [18] ment of Positive Energy Density (P.E.D.) ( T0 > 0),
to the stationary case, we have been able to build up a Strong EnergyCondition (S.E.C.) (( T0) T−i >0 0)and
− 0 − i
physicallymeaningfulsourcefortheKerrmetric,satisfy- PositiveRadialPressure(P.R.P.)(P =g T1 >0).The
rr 11 1
ing the matching conditions on the boundary surface of resultsofthissearchareshowninTableI.Itappearsev-
the matter distribution. In spite of the fact that a per- ident that physically meaningful sources exist for a wide
fect fluid sourcefor the Kerrmetric might exist [32], our range of values of the parameters.
source is necessarily anisotropic in the pressure. Ontheotherhand,thechosenrangeofvaluesofthepa-
Particular attention deserves the presence of a non– rameters, incorporates values considered in the existing
vanishing T3 component of the energy–momentum ten- literature, to describe realistic models of rotating neu-
0
sor. Indeed, definingas usualanenergy–momentumflux tron stars and white dwarfs. Let us elaborate on this
vector as: Fν = VµT (where Vµ denotes the four issue with some detail.
νµ
−
velocity of the fluid), it appears that, in the equatorial Therotationofthesourceisdeterminedbytheparam-
planeofoursystem,energyflowsroundincirclesaround eters a,M of the exterior solution. Indeed, the rotation
the symmetry axis. This result is a reminiscence of an parameterj =J/M2 =a/M standsforthedimensionless
9
angular momentum of the rotating source. So, if we re-
strict ourselves to a sub-extreme Kerr solution (a<M),
then j <1.
In [35] numerical models of rotating neutron stars
are constructed for different EOSs. For each EOS the
star’s angular momentum, ranges from J = 0 to the
Keplerian limit J = J , and the dependence of the
max
quadrupole moment on the rotation parameter j, is es-
tablished. Within the interval1.0 to 1.8 solar masses for
the mass M, the parameter j is in the interval between
0.1 and 0.8.
FIG. 6: rΣ2[(−T00)−T11] as function of y and s, with j =0.1
and τ =2.7. In [36] upper limits on the parameter j are found
forrepresentativeEOSs;uniformly rotatingneutronstar
modelswithmaximummassforvariousequationsofstate
are studied, and the parameter j does not exceed the
value 0.7.
In [37] realistic equations of state for rapidly rotating
neutron stars are explored, including a wider range of
values for j.
Finally, we would like to conclude with the following
comment. Insomestaticsourcesitmayoccurthatl <l
z ρ
does not imply that the source is oblate (q < 0) (see
[18, 38] for a discussion on this issue). However, as we
FIG.7: rΣ3T03asfunctionofyands,withj =0.1andτ =2.7. have seen, this is not the case of our source.
V. ACKNOWLEDGMENTS No. FIS2015-65140-P (MINECO/FEDER), and the
Consejer´ıade Educaci´on of the Junta de Castilla y Le´on
ThisworkwaspartiallysupportedbytheSpanishMin- undertheResearchProjectGrupodeExcelenciaGR234.
isterio de Ciencia e Innovacio´n under Research Projects
[1] R.P. Kerr, Phys. Rev. Lett. 11, 237 (1963). [11] S.DrakeandR.Turolla, Classical Quantum Gravity 14,
[2] W. C. Hernandez,Phys. Rev. 159, 1070 (1967). 1883 (1997).
[3] J. Cohen, J. Math. Phys. 8, 1477 (1967). [12] T. Papakostas, J.Phys: Conf. Ser. 8, 13 (2005).
[4] H.Wahlquist, Phys. Rev. 172, 1291 (1968). [13] S. Viaggiu, Int. J. Modern. Phys.D, 15,1441 (2006).
[5] W. C. Hernndez,Phys. Rev. 167, 1180 (1968). [14] S. Viaggiu, Int. J. Modern. Phys.D, 19,1783 (2010).
[6] W. Israel, Phys. Rev. D 2, 641 (1970). [15] E. Kyriakopoulos, Int. J. Modern. Phys.D, 22,1350051
[7] A.Krasinski, Ann.Phys.112, 22 (1978). (2013).
[8] S.HaggagandJ.Marek,IlNuov.Cim.B62,273(1981). [16] M. Azreg-Ainou,Eur. Phys. J.C,74, 2865 (2014).
[9] L. Herrera and J. Jimenez, J. Math. Phys. 23, 2339, [17] E. Kyriakopoulos, arXiv: 1602.04067v1.
(1982). [18] J.L.Hern´andez-Pastora,L.HerreraandJ.Martin,Clas-
[10] S.Haggag, Il Nuov. Cim. B 105, 365 (1990). sical Quantum Gravity 33, 235005, (2016).
10
TABLE I: Fulfillment (F) or violation (V) of different criteria for good physical behaviour: Positive Energy Density (P.E.D.)
(−T00 > 0), Strong Energy Condition (S.E.C.) ((−T00)−Tii > 0) and Positive Radial Pressure (P.R.P.) (Prr = g11T11 > 0).
The symbol ∗ over F means that although the criterion is fulfilled, nevertheless ∂rPrr changes its sign in theinterval s∈[0,1]
within the source. This table corresponds to an oblate source (dueto the rotation) of the Kerr solution for different values of
j and a sequenceof different valuesof theparameter τ.
P.E.D. / S.E.C. / P.R.P.
τ(cid:31)j 0.9 0.8 0.7 0.5 0.3 0.1 0.05 0.03
2.67 V V V V V V V V F∗ V V F F V F FV F F V F F F F
2.7 V V V V V V V V F∗ V V F F V F F F F FF F F F F
2.8 V V V V V F∗ V V F∗ V V F F V F F F F FF F F F F
2.9 V V V V V F∗ V V F∗ FV F F F F F F F FF F F F F
3 V V F∗ V V F∗ V V F∗ FV F F F F F F F FF F F F F
3.1 V V F∗ V V F∗ V V F∗ FV F F F F F F F FF F F F F
3.5 V V F∗ F V F∗ F FF∗ F F F F F F F F F FF F F F F
4 F F F∗ FF F∗ FF F F F F F F F F F F FF F F F F
4.5 F F F∗ FF F FF F F F F F F F F F F FF F F F F
5 F F F FF F FF F F F F F F F F F F FF F F F F
[19] G.ErezandN.Rosen,Bull. Res. Council Israel,8F,47, [29] R. O. Hansen,J. Math. Phys., 15, 46 (1974).
(1959). [30] K. S. Thorne, Rev. Mod. Phys., 52, 299 (1980).
[20] H.Quevedo,Fortschr. Phys. 38 733 (1990). [31] G. Fodor, C. Hoenselaers and Z. Perj`es, J. Math. Phys.
[21] G. Darmois, M´emorial des Sciences Math´ematiques 30, 2252 (1989).
(Gauthier-Villars, Paris, 1927) Fasc. 25. [32] W. Roos, Gen. Relativ. Gravit. 7, 431 (1976).
[22] L.Herrera, A. DiPrisco, J. Iba´n˜ez and J. Ospino, Phys. [33] W. B. Bonnor, Phys. Lett. A 158, 23 (1991).
Rev. D 87, 024014, (2013). [34] L. Herrera, G. A. Gonz´alez, L. A. Pach´on and J. A.
[23] H. Stephani, D. Kramer, M. MacCallum, C. Honselaers, Rueda, Class. Quantum Grav. 23, 2395 (2006).
and E. Herlt, Exact Solutions to Einsteins Field Equa- [35] W.G.LaarakkersandE.Poisson,Astrophys. J.512,282
tions (Cambridge University Press, Cambridge, Eng- (1999).
land), (2003), 2nd Ed. [36] J.L. Friedman, J.R. Ipser and L. Parker, Astrophys. J.
[24] J. Carot, Classical Quantum Gravity 17, 2675 (2000). 304, 115 (1986).
[25] G. T. Carlson, Jr. and J. L. Safko, Ann. Phys. (N.Y.) [37] G.B.Cook,S.L.ShapiroandS.A.Teukolsky,Astrophys.
128, 131 (1980). J. 424, 823 (1994).
[26] R.Geroch, J. Math. Phys. 11, 2580 (1970). [38] W. B. Bonnor, Gen. Relativ. Gravit. 45, 1403 (2013).
[27] R.Geroch, J. Math. Phys. 11, 1955 (1970).
[28] R.Geroch, J. Math. Phys., 12, 918 (1971).