On the Influence of the Mass Definition in the Stability of Axisymmetric Relativistic Thin Disks

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by  Eduar A. Becerra

Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-dddde2016
Physical Sciences
On the Influence of the Mass Definition in the Stability of Axisymmetric
Relativistic Thin Disks
Eduar A. Becerra1,, Fredy L. Dubeibe1,2, Guillermo A. Gonz´alez1,∗
7
1
1EscueladeF´ısica,UniversidadIndustrialdeSantander,Bucaramanga,Colombia
0 2FacultaddeCienciasHumanasydelaEducacio´n,UniversidaddelosLlanos,Villavicencio,Colombia
2

n
a
J Abstract

3
Thestudyonthestabilityofrelativisticdisksisoneofthemostimportantcriteriaforthecharacterization
1
of astrophysically relevant galactic or accretion disks models. In this paper, we perform an analysis of the

stability of static axisymmetric relativistic thin disks, by introducing a first-order perturbation into the
]
energy-momentumtensorofthefluid. Theformalismisappliedtothreeparticularmodelsbuiltwiththeaid
c
q of the displace-cut-reflect (DCR) method, and previously considered in literature (Ujevic and Letelier,
- 2004),butmodifyingthemasscriteria,i.e.,usingtheKomarmassinsteadofthetotalsurfacemass. Under
r thisconditions,itisfoundthatthetotalmassvaluesareindependentoftheparametersoftheDCR-method,
g
which let us choose the boundary condition for the cutoff radius, such that it takes the maximum value
[
that allows an appreciable and well-behaved perturbation on the disk. As a general result, we found that
1 theKomarmassismoreappropriatetodefinethecutoffradius.
v
6 Key words: GeneralRelativity,Relativisticthindisks,Stability.
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6 Influencia de la definici´on de masa en la estabilidad de discos relativistasaxialsim´etricos
3
Resumen
0
.
1 Uno de los criterios m´as importantes para la caracterizaci´on de modelos gal´acticos o discos de acrecio´n
0 astrof´ısicamente relevantes, es el an´alisis de la estabilidad de dichos modelos. En este trabajo, se re-
7 aliza un an´alisis de la estabilidad de discos delgados esta´ticos relativistas con simetr´ıa axial, mediante
1 la introducci´on de una perturbaci´on de primer orden en el tensor de energ´ıa-impulso del fluido. El
: formalismo se aplica a tres modelos construidos con el m´etodo de desplazamiento-corte-reflexi´on (DCR),
v
previamente considerados en la literatura (Ujevic and Letelier, 2004), pero modificando el criterio de
i
X masa, es decir, usando la masa de Komar en lugar de la masa total superficial. Bajo estas condiciones,
se encuentra que los valores de masa total son independientes de los para´metros del m´etodo DCR, lo
r
a que permite elegir la condici´on de frontera para el radio de corte que tome el valor m´aximo y a la vez
permitauna perturbaci´on apreciable ybiencomportada en el disco. Comoresultado general, seencuentra
queparalamayor´ıademodosdeoscilaci´on,lamasadeKomaresmasapropiadaparadefinirelradiodecorte.
Palabras clave: RelatividadGeneral,Discosdelgados Relativistas,Estabilidad.
Introduction Einstein’s theories of gravity. Such models are of astro-
physical interest because they can be used to model accre-
tion disks, galaxies in thermodynamic equilibrium or galaxies
During the last decades, considerable efforts have been made
with black holes centers (Bic´ak, Lynden-Bell and Pichon,
to obtain exact analytical solutions suitable to modeling ax-
1993, Ledvinka, Zofka and Bic´ak, 1998). Moreover, the
isymmetric thin disks, within the framework of Newton’s and
∗Correspondence: G.A.Gonz´alez, [email protected],ReceivedxxxxxXXXX;Accepted xxxxxXXXX.
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E.A.Becerra,F.L.Dubeibe,G.A.Gonza´lez Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-dddde2016
addition of electromagnetic fields in those space-times allows Derivation of First-Order Perturbation Equa-
studying neutron stars formation, white dwarfs, and quasars tions
(Mun˜oz et. al., 2011, Alpar, 2001).
Since the seminal works on exact solutions represent- Following Ujevic and Letelier (2004), and for the sake of
ing static thin disks carried out by Bonnor and Sack- self-consistencyofthecurrentpaper,inwhatfollowswepresent
field (Bonnor and Sackfield, 1968) and Morgan and Mor- thederivation ofthefirst-orderperturbationequations forrel-
gan (Morgan and Morgan, 1969), more realistic mod- ativistic thin disks. Let us start considering that the energy-
els have been proposed (Pichon and Lynden-Bell, 1996, momentum tensor for an isotropic fluid with a discoid shape
Gonz´alez and Letelier, 2000). The superposition of static and without heat flow can bewritten as
and stationary thin disks with black holes at the cen- Tµν =Qµνδ(z), (1)
ter, has been considered by Lemos, Letelier and Semerak
(Lemos and Letelier, 1993, Semer´ak, 2002b, 2004). Vogt where δ denotes theDirac delta function and
andLetelierstudiedtheinclusionofelectromagneticfieldsinto
Qµν =σUµUν +p XµXν +p YµYν,
thin disks made of dust (Vogt and Letelier, 2004a) and of r ϕ
charged perfect fluid (Vogt and Letelier, 2004b). Also in- with σ the surface energy density, p and p the radial and
r ϕ
teresting is the case of thick disks proposed by Gonz´alez and azimuthalpressure,respectively,andUµ,Xµ,andYµ thenon-
Letelier (Gonz´alez and Letelier, 2004), who extended the zerocomponentsoftheorthonormaltetrad. Assumingthatthe
DCR method to include thick disks in their models.∗ first-orderperturbations in theEinstein field equations do not
modify thebackground metric, theperturbedequation for the
Stability is an essential criterion to determine whether or not energy-momentum tensor reads as
a model can be applied to describe an astrophysical system
presentinnature. Ingeneral,therearetwoapproachestostudy (δTµν);µ=0. (2)
the stability of relativistic disks: The first option is based on
Introducing the definition of energy-momentum tensor for a
analyzing the stability of particle orbits along geodesics (see
thin disk (1), in the perturbed equation (2), and integrating
e.g., (Letelier, 2003) and (Vogt and Letelier, 2003)), while
with respect to thecoordinate z, we find
thesecondoptionconsistsinperturbingtheenergy-momentum
tensor (see e.g., (Seguin, 1975)). From a theoretical point of (δQµν);µδ(z)+δQzν[δ(z)],z √gzzdz=0. (3)
view,thelatteroptionismorerigorous,becauseinthiscasethe
Z n o
collectivebehavioroftheparticlesistakenintoaccount. Work- As a consecuence of theDCR method, themetric components
ing on this line, Ujevic and Letelier(2004) investigated the should only depend on r and z (Vogt and Letelier, 2003).
stability of three particular models for relativistic thin disks, Additionally,ifwedefinetheva|lu|eof z atz=0equalszero,
,z
performingafirst-orderperturbationanalysiswithvariableco- theperturbed equation reduces to | |
efficients. However, as a pathological result, the authors find
that the total mass of the disk depends on the parameters of (δQµν) =0. (4)
;µ
the DCR method, such that the boundary conditions are also (cid:12)z=0
(cid:12)
dependenton theseparameters. Takingintoaccountthattheper(cid:12)turbedvectorsmustsatisfythe
(cid:12)
orthonormality condition, and assuming that δXϕ = 0 (since
With the aim to avoid the undesired dependencies between
thefour-velocityandthethermodynamicalvariablesdonotde-
parameters and to observe the possible changes in the sta-
pend on thisquantity),we obtain
bility, in the present paper we redo the calculations made by
Ujevic and Letelier(2004)usingtheKomarmassdefinition δYr =δUt =δXr =δYϕ =0,
instead of the mass definition (along the paper we will call it X Y (5)
total surface mass) introduced in Vogt and Letelier (2003). δXt = rδUr, δYt = ϕδUϕ.
−Ut −Ut
Thenewresultsletustochoosetheboundaryconditionforthe
Duetothefactthatthemetricisstaticandaxisymmetric,and
cutoffradiussuchthatittakesthemaximumvalueallowingan
the lack of z-dependences in Tµν, all the coefficients depend
appreciable and well-behaved perturbation on thedisk.
only on the radial coordinate; therefore, the general perturba-
tion can bechosen as
δξµ(t,r,ϕ)=δξµ(r)ei(kϕ−wt), (6)
∗Fortheinterestedreader,acompletereviewofthestate-of-the-artonrelativisticdiskswasmadebySemer´ak(2002a).
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Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-dddde2016 StabilityofAxisymmetricRelativisticThinDisks
with k the wave number and w the angular frequency of the p B B B +B +B B
C =A ,r 1 2,r 1,r 4 5 3,r+
perturbation. We will focus on the term δξµ(r), which has a 1((cid:18)σ,r(cid:19),r(cid:20) B22 − B2 (cid:21)− B2
significant role for thesystem stability,.
p B B +B
,r 2,r (B +B ) 4,r 5,r +
Therefore,aftersubstitutingtheperturbation(6)intoequation (cid:18)σ,r(cid:19)(cid:20) B22 4 5 − B2 (cid:21)
(4),andreplacingtheconditions(5)intheresultingexpression,
B B B p A C p
the respective perturbedequations for t,r, and ϕ, read as 2B,r22 3 − B12 (cid:18)σ,,rr(cid:19),rr)− C312 (cid:18)σ,,rr(cid:19)−A4
δσUUr,tr(cid:18)2σΓUt t−+ΓUpµrt(cid:19)+δUXrnr(cid:0)σUpt(cid:1)X,rr+ whereA−i,BBAi22,(anBd1C(cid:18)i,σpa,,rrre(cid:19)t,rhe+fa(cid:18)cσtpo,,rrrs(cid:19)m[Bul4ti+plByi5n]g+thBe3p)er,turbed
tr µr −(cid:18)Ut(cid:19),r r − (7) variables in equations (7), (8), and (9), respectively, with the
(cid:0) (cid:1) indexes in numerical order according to their order of appear-
X
Ur (prXr),r+prXr 2Γttr+Γµµr + ance.
δUtϕhik σUt pϕ (cid:0) δσ iwUtU(cid:1)ti(cid:27)=0, Duetothecumbersomeformofequation(10),itmustbesolved
(cid:20) (cid:18) − Ut(cid:19)(cid:21)− numerically. Forthispurpose,weshallimposeDirichletbound-
(cid:0) (cid:1) aryconditions,oneatthecenterofthediskandtheotheroneat
thefinalboundaryofthedomain. However,itshouldbenoted
thatthediskhasaninfiniteradialextension,forthisreason,it
p is necessary to introduce a cutoff on theradial coordinate.
δUr iw r σUt +δσ UtUtΓr +
(cid:20) (cid:18)Ut − (cid:19)(cid:21) tt
δp (XrXr) +XrXr Γr (cid:0)+Γµ (cid:1)+ (8)
r ,r rr µr Thermodynamic variables
δp h(XrXr)+δp Yϕ(cid:0)YϕΓr =0(cid:1)i,
r,r ϕ ϕϕ
In the previous section, we explicitly wrote down the per-
(cid:0) (cid:1)
turbed equations for thin disks, nevertheless, such expressions
aregivenintermsofthethermodynamicvariablesofthefluid,
δUϕ w pϕ σUt +δp (kYϕYϕ)=0, (9) which requires finding explicit formulas for the surface energy
ϕ
(cid:20) (cid:18)Ut − (cid:19)(cid:21) density and the pressures on the disk. To this end, let us rep-
resent thematterdistribution on ahypersurfaceΣ, definedby
the function l(xα) = z, which divides the space-time into two
Finally, from the set of differential equations presented above, regions: M+ on top and M− at the bottom. Therefore, the
and the equation of state of a perfect fluid, δp = δσ(p,r/σ,r), normalvectortothehypersurfaceΣisgivenbyn =l =δz,
α ,α α
thedifferentialequationfortheperturbedenergydensitytakes
and the componentsof themetric tensor must satisfy
the form
g− (r,z)=g+ (r, z), (11)
µν µν −
Aδσ +Bδσ +Cδσ=0, (10)
,rr ,r such that
g− (r,z)= g+ (r, z), (12)
whose coefficient A, B, and C, are given by µν,z − µν,z −
where g+ and g− , should be understood as the metric ten-
αβ αβ
sors for the regions defined by z > 0 (M+) and z < 0 (M−),
A B p
A= 1 1 ,r , respectively.
− B2 (cid:18)σ,r(cid:19)
Bytakingthelimitz 0,thediscontinuitiesinthefirstderiva-
→
tives of themetric tensor take theform
p B B B +B +B B
B=A1(cid:26)(cid:18)σ,,rr(cid:19)(cid:20) 1B222,r − 1,r B24 5(cid:21)− B32− bµν =[gµν,z]=gµ+ν,z(cid:12)z=0− gµ−ν,z(cid:12)z=0=2gµ+ν,z(cid:12)z=0, (13)
2B1 p,r A2B1 p,r , withbµν,thejumpofthe(cid:12)(cid:12)(cid:12)firstderivat(cid:12)(cid:12)(cid:12)ivethrough(cid:12)(cid:12)(cid:12)Σ. Usingthe
B2 (cid:18)σ,r(cid:19),r)− B2 (cid:18)σ,r(cid:19) distributionsmethod(Papapetrou,1968,Poisson,2004),the
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E.A.Becerra,F.L.Dubeibe,G.A.Gonza´lez Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-dddde2016
metric can bewritten as which,inquasi-cylindricalcoordinatesxα=(t,r,ϕ,z)andcon-
g =Θ(l)g+ +Θ( l)g− , (14) sidering thehypersurface Σ definedby the function l(xα)=z,
αβ αβ − αβ with normal vectorn =δz, takes theform
α α
where Θ(l) is the usual Heaviside function. Performing the
1
derivativeof equation (14) we obtain κQµ = bzµδz gµzδzbα +gµzbz gzzbµ
β 2 β− β α β− β (24)
gαβ,γ =Θ(l)gα+β,γ+Θ(−l)gα−β,γ+nγδ(l)[gαβ]. (15) −(cid:8)(bzz−gzzbαα)δβµ .
Fromtheaboveequation,andasn(cid:9)otedbyVogt and Letelier
The last term in the right-hand side of equation (15) is singu-
(2003), it can be shown that the non-zero components of the
lar, consequentlytheChristoffel symbolswould not bedefined
surface energy-momentum tensor are
as a distribution. An alternative way to avoid this difficulty,
istoimposethatthemetriciscontinuousonthehypersurface, Qt =σ= 1 gzz br +bϕ , (25)
i.e.,[g ]=g+ g− =0, such that t 16π r ϕ
αβ αβ− αβ (cid:0) (cid:1)
Γαβγ =Θ(l)Γαβ+γ+Θ(−l)Γαβ−γ, (16) Qrr =pr = 161πgzz btt+bϕϕ , (26)
whose derivativereads as 1 (cid:0) (cid:1)
Qϕ =p = gzz bt +br . (27)
Γαβγ,δ =Θ(l)Γαβ+γ,δ+Θ(−l)Γαβ−γ,δ+nδ(l)[Γαβγ]. (17) ϕ ϕ 16π t r
(cid:0) (cid:1)
The corresponding Riemann curvaturetensor is given by
Rα =Θ(l)Rα+ +Θ( l)Rα− +δ(l)Rα , (18) Mass Definition
βγδ βγδ − βγδ βγδ
where Rα± are the tensors defined in M± and
βγδ b
Themass concept in general relativity isnot uniqueand there
Rα =[Γα ]n [Γα ]n , (19)
βγδ βδ γ− βγ δ are several different definitions that are applicable under dif-
1 ferent circumstances. Inthecaseof stationary spacetimes, the
with [Γαβγ]= 2 bbαβnγ +bααnβ−bβγnα . commonly accepted definitions are: the total volumetric (or
(cid:0) (cid:1) surface) mass, Komar, ADM and Bondi-Sach masses. How-
From the last expression, it is clear that Riemann tensor, the
ever, it is a well-known fact that for stationary spacetimes
Ricci tensor, and Ricci scalar on the hypersurfaceare
the ADM and Bondi-Sach masses are exactly alike (Poisson,
1
Rα = bαn n bαn n +b nαn b nαn , 2004). Besides, in the asymptotically flat case, it can be
βγδ 2 δ β γ− γ β δ βγ δ− βδ γ
shown that the ADM and the Komar masses are equivalent
Rbβδ = 21{b(cid:8)αδnβnα−bααnβnδ+bβαnαnδ−bβδnαnα}(cid:9), (Jaramillo and Gourgoulhon, 2009). Then, in the station-
ary, axisymmetric, asymptotically flat, vacuum solutions of
Rb={bµαnαnµ−bααnαnα}. Einstein’s equation, there are only two possibilities to choose
(20) from,thetotalvolumetric(orsurface)massortheKomarmass.
On tbhe other hand, the energy-momentum tensor Tβδ can be Ascommentedintheintroduction,themaindifferencebetween
expressed as thepresentpaperandtheonebyUjevic and Letelier(2004),
T =Θ(l)T+ +Θ( l)T− +δ(l)Q , (21) is that they used the total surface mass, while along this pa-
βδ βδ − βδ βδ per,weusetheKomarmassdefinition,whichisindependentof
whereQ istheenergy-momentumtensorassociatedwiththe
βδ themethodsusedtobuildtheparticularsolutionsofEinstein’s
hypersurface,and T± aretheenergy-momentumtensorsasso-
βδ equation.
ciatedtoM±. Hence,wecanwritetheEinsteinfieldequations
as follows A necessary condition for the definition of mass, is that must
1
R± g R± =κT± not involve any dependence with the specific choice of coor-
βδ− 2 βδ βδ (22) dinates. This property is achieved for stationary and axially
1
Rβδ− 2gβδR=κQβδ. symmetricspacetimes, through theKomar formula Mk,which
reads as
Given that the only source of gravitational field is a thin dis-
b b 1
tribution of matter, theenergy-momentum tensor satisfies M =2 T Tg nαξβ √h d3y, (28)
k αβ− 2 αβ (t)
κQ =1 b n nα bα n n +b nαn b nαn ZΣ(cid:18) (cid:19)
βδ 2{ αδ β − α β δ βα δ− βδ α (23) whereΣisspacelikehypersurface,nα istimelikevectornormal
(b nαnµ bα nαn )g . toΣ, ξβ istimelikeKillingvectorandhisthedeterminantof
− µα − α α βδ} (t)
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Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-dddde2016 StabilityofAxisymmetricRelativisticThinDisks
metric associated to Σ. For a static and diagonal metric, the From (30), (31) and (32), and in accordance with Eq. (12) of
following relations holds Vogt and Letelier(2003), we find
nα =− gαgβtt δβt, ξ(βt) =δtβ, h=gzz grr gϕϕ, Ms =m 1+ 4ma , (35)
| | and (cid:16) (cid:17)
such that thpeKomar mass takes theform m ma m
M =m 1+ 2+ . (36)
Mk = 2π ∞(σ+pr+pϕ) gtt √gzz√grr√gϕϕ drdϕ. rc (cid:16) 4a(cid:17)− 2rc (cid:18) 2rc(cid:19)
| | On the other hand, from the Komar mass definition (29) and
Z0 Z0
p (29) using (31),(32) and (33), we get
The Komar mass definition differs from thetotal surface mass Mk =m, (37)
in the fact that the former one considers all the contributions and
totheenergy-momentumtensor,whilethesurfacemass, given ma
M =m . (38)
by rc − rc
2π ∞ Hence,theresultingexpressionforthecutoffradiuswhencon-
Ms = σ√gzz√grr√gϕϕ drdϕ, (30) sidering Eq. (30) is
Z0 Z0
takes only intoaccount thesurface energy density. 2a2+a 4a2+m(1 n)(4a+m)
r∗ = − , (39)
c (1 n)(4a+m)
p −
while using Eq. (29), we get
Case 1: Isotropic Schwarzschild Thin Disk
a
r = (40)
c 1 n
The isotropic Schwarzschild thin disk metric in quasi- −
InFig. 1weshowacomparisonbetweenthecutoffradiir and
c
cylindrical Weyl-Papapetrou coordinates was obtained by
r∗, using different values of the parameters m and a. Taking
Vogt and Letelier(2005),and can bewritten as c
intoaccountthatvaluesofnsmallerthan0.9arenotusednor
ds2= 2R−m 2dt2+ 1+ m 4(dr2+r2dϕ2+dz2) (31) physically appropriate, we plot the difference rc −rc∗ in the
− 2R+m 2R range n [0.9,1].
(cid:18) (cid:19) (cid:16) (cid:17) ∈
where m is a positive constant and R2 =r2+(z +a)2.
| | 8
Thus, the corresponding expressions for the pressure compo-
nentsandsurfaceenergydensity(seee.g.,Vogt and Letelier m=0.1,a=0.2
6
(2003)), are obtained by using equations (25-27), m=0.2,a=0.3
16maR2 m=0.3,a=0.4
σ= π(2R0+m0)5, (32) *-rc 4 m=0.4,a=0.5
rc m=0.5,a=0.6
8m2aR2
p= 0 , (33)
π(2R +m)5(2R m) 2
0 0
−
with R =R(r,z=0) and p=p =p .
0 r ϕ
0
Let us define a general cutoff radius as the radial distance at
0.90 0.92 0.94 0.96 0.98 1.00
whichthematterwithinthethindiskformeduptosuchradius
n
corresponds to the n% of the total matter of the infinite disk,
FIGURE 1. Comparison between the cutoff radii r and r∗
i.e. c c
using different values of theparameters m and a.
nM =M , (34)
T rc
with M the total mass of the infinite disk and M the mass In all that follows, except where especially noted, we define
T rc
up to the cutoff radius. As noted in the previous section, the cutoff radius r using the 95% of the total mass of infinite
c
the total mass value may depend on theconsidered definition, disk,†and theremaining5% of matterisdistributedalong the
such that the cutoff radius could depend also on this choice. plane z = 0 from r to infinity. Moreover, we assume that
c
†Thispercentage correspondstotheoptimalvaluetogetanon-negligibleperturbationonthedisk.
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E.A.Becerra,F.L.Dubeibe,G.A.Gonza´lez Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-dddde2016
at r = 0, the perturbation equals 10% of the unperturbed en- Fig. 3it can beseen thattheoscillation amplitudeisnot only
ergydensityvalueandatr=r theperturbationvanishes,i.e. smaller for Eq. (29) (upper panel) than for Eq. (30) (upper
c
δσ(r=r )=0. panel),butalso disappearsinapproximatelythemiddleofthe
c
disk. It means that for the set of parameters here considered,
Bysettingm=0.5anda=0.6,whicharethesetofparameters the density and pressure in the disk are stable independently
that best fit the expected energy density profile in a realistic of themass definition used.
model, we find r 12 and r∗ 10. In Fig. 2 we present
c ≈ c ≈
the numerical solution to the differential equation (10) using 1.´10-4
theKomarmassdefinition(upperpanel) andthetotalsurface w=1,k=0
mass (lower panel). In this figure, we show the perturbed en- 5.´10-5 w=1,k=1
ergydensityprofilefordifferentvaluesofw. Itcanbeseenthat
w=1,k=2
theparameterwisproportionaltothefrequencyofoscillations
withinthedisk,i.e. thenumberofring-likestructuresincreases ~∆p 0.
withincreasingw. Also,itisworthnotingthattheoscillations
quicklydecay to zero regardless of theoscillation modes, how- -5.´10-5
ever the oscillation amplitude is smaller for Eq. (29) (upper
panel) than for Eq. (30) (upperpanel).
-1.´10-4
0 2 4 6 8 10 12
6.´10-4 r
1.´10-4
w=1,k=0
w=1,k=0
3.´10-4 w=2,k=0
w=3,k=0 5.´10-5 w=1,k=1
w=1,k=2
~Σ 0.
∆
~p 0.
∆
-3.´10-4
-5.´10-5
-6.´10-4
0 2 4 6 8 10 12 -1.´10-4
0 2 4 6 8 10
r
6.´10-4 r
w=1,k=0 FIGURE 3. Perturbed pressure profiles δp˜= √gzzδp for the
isotropic Schwarzschild thindisk using Eq. (29) (upperpanel)
3.´10-4 w=2,k=0 and Eq. (30) (lower panel).
w=3,k=0
Afullanalysisofthesystemrequiresalsoacomparisonbetween
~Σ 0.
∆ the amplitude of the perturbed velocities with the escape ve-
locity of the constituents in the disk. In Fig. 4 we show the
-3.´10-4 profiles for the perturbed radial δUr velocities using Eq. (29)
(upperpanel)andEq. (30)(lowerpanel),settingtherespective
-6.´10-4 cutoff radii for the 95.776% of the total mass of infinite disk‡.
0 2 4 6 8 10 Both velocities exhibit an increase in the frequency of oscilla-
r tionswithincreasingw;moreover,weseethattheenvelopesof
FIGURE 2. Perturbed energy density profiles δσ˜ = √gzzδσ theoscillating functions increase at theexternalradial bound-
for different values of w using Eq. (29) (upperpanel) and Eq. ary, however, the perturbed radial velocity grows faster in the
(30) (lower panel) in theisotropic Schwarzschild thin disk. lower panel than in theupperpanel.
Ontheotherhand,in Fig. 3we presenttheprofile of theper- Concerning the escape velocity v , it is well known that it
e
turbed pressure. When fixing w and varying k, the amplitude should belarger than theperturbed radial velocity, otherwise,
forδphasanoscillatorybehaviorsimilartotheoneofδσ. From themodel will nothaveanyastrophysical validity. Theescape
‡Wesetthispercentage becauseitgivesplacetounstabilities.
6
Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-dddde2016 StabilityofAxisymmetricRelativisticThinDisks
velocityrequiredtoovercometheforcegeneratedbyagravita- ratify that,underfirstorderperturbationsoftheform (6)and
tional potential Φ can be calculated as v =√2Φ, which from using the Komar mass definition, the isotropic Schwarzschild
e
theweak-fieldapproximationisgivenas2Φ=g 1. So,from thin disk has a stable behavior for a large set of parameters
tt
−
(31), we find such that it can be used to describe astrophysical models.
2√2mR 2.´10-4
v = . (41)
e 2R+m w=1,k=0.05
1.´10-4 w=2,k=0.05
For the set of parameters a = 0.6 and m = 0.5, the resulting
w=3,k=0.05
escape velocities for Eq. (29) and Eq. (30) are v = 0.26 and
e
ve = 0.28, respectively. From Fig. 4 it is clear that using Eq. ~j∆U 0.
(29)theperturbedradialvelocityisalwayslessthantheescape
velocity (reddashedline),suchthatallparticles remaininside -1.´10-4
the disk. Conversely, using Eq. (30) the perturbed radial ve-
locitycanbegreaterthantheescapevelocity(reddashedline),
-2.´10-4
meaning that theparticles may escape of thedisk. 0 2 4 6 8 10 12
r
4.´10-1 2.´10-4
w=1,k=0.05
2.´10-1 1.´10-4 w=2,k=0.05
w=3,k=0.05
~r∆U 0. ~j∆U 0.
w=1,k=0
-2.´10-1 w=2,k=0 -1.´10-4
w=3,k=0
-4.´10-10 2 4 6 8 10 12 14 -2.´10-40 2 4 6 8 10
r r
4.´10-1 FIGURE 5. Profiles of the perturbed azimuthal velocities for
different oscillation modes in Schwarzschild isotropic thin disk
2.´10-1 using Eq. (29) (upperpanel) and Eq. (30) (lower panel).
~r∆U 0. Case 2: Chazy-Curzon Thin Disk
w=1,k=0
-2.´10-1 w=2,k=0 TheChazy-Curzon thindisk metricin Weylcoordinates is de-
w=3,k=0 scribed as (Bic´ak, Lynden-Bell and Katz, 1993)
-4.´10-1
0 2 4 6 8 10 ds2 = e2Φdt2+e−2Φr2dϕ2+e2(Λ−Φ)(dr2+dz2), (42)
−
r
where themetric functions Φ y Λ are given by
FIGURE 4. Profiles of the perturbed radial velocities δU˜r for
different oscillation modes in Schwarzschild isotropic thin disk m m2r2
Φ= , Λ= , (43)
using Eq. (29) (upperpanel) and Eq. (30) (lower panel). The −R − 2R4
red dashed line denotes the value of the respective escape ve- with R2 =r2+(z +a)2. From (42) and (43), and equations
locities. | |
(25),(26),and(27),weobtaintheexpressionsforthepressure
and thesurface energydensity of theChazy-Curzon thindisk,
In Fig. 5 we show the profiles of the perturbed azimuthal ve-
locities. Inbothcasesthevelocitiesarestableandsomeorders ma mr2
σ= 1 e2(Φ0−Λ0), (44)
of magnitude smaller than the escape velocity. These results 2πR3 − R3
0 (cid:20) 0 (cid:21)
7
E.A.Becerra,F.L.Dubeibe,G.A.Gonza´lez Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-dddde2016
m2a r2 Case 3: Zipoy-Voorhees Thin Disk
p = e2(Φ0−Λ0), p =0, (45)
ϕ 2πR4R2 r
0 0
where R0, Λ0 and Φ0 are thefunctions evaluated at z=0. The Zipoy-Voorhees thin disk metric in Weyl coordinates has
theform (Ujevic and Letelier, 2004)
1.5´10-1 ds2= e2Φdt2+e−2Φr2dϕ2+e2(Λ−Φ)(dr2+dz2), (46)
k=0,w=0.7 −
with metric functions Φ and Λ, given by
k=0,w=1
7.5´10-2
m R + z +a
k=0,w=1.5 Φ= ln a | | ,
b a R + z +b
− (cid:20) b | | (cid:21)
~Σ 0.
∆ (47)
-7.5´10-2 1.´101000 Λ= (b2ma2)2 ln (Ra+R4bR)2R−(b−a)2 ,
-1.´10100 − (cid:20) a b (cid:21)
0 2 4 6 8 where R2 = r2+(z +a)2, R2 = r2+(z +b)2 and b > a.
-1.5´10-1 a | | b | |
0.0 0.2 0.4 0.6 0.8 1.0 Hence, by applying the same procedure used for the previous
r disksmodels,weobtainexpressionsforthesurfaceenergyden-
1.5´10-1 sity and theazimuthal pressure,
w=1,k=0
a b
7.5´10-2 ww==11,,kk==01..64 σ= m2e−2(Λ0−Φ0) 2(Ra0+Rb0)(cid:18)Ra0 + Rb0(cid:19)
− 2π(b a)2 (Ra0+Rb0)2 (b a)2
− − − (48)
~∆Σ 0. 
a b  me−2(Λ0−Φ0) 1 1
1.´10100 + + ,
-7.5´10-2 0 −(cid:18)Ra20 Rb20(cid:19)(cid:21) 2π(b−a) (cid:18)Ra0 − Rb0(cid:19)
-1.´10100 a b
-1.5´10-10.0 0.2 0.4 r 0 0.26 4 60.88 1.0 pϕ =m22πe−(b2(−Λ0a−)Φ20) 2(R(Ra0a0++RRb0b)0)(cid:18)2R−a(0b+−Ra)b20(cid:19)− (49)


FIGURE6. Profiles of theperturbed energy density δσ˜ in the a b 
Chazy-Curzon thin disk using Eq. (29), for different values of + ,
R2 R2
wandk. Theparametershavebeensetasa=0.4andm=0.5. (cid:18) a0 b0(cid:19)(cid:21)
The insets show that for finite values of r, the perturbations with R , R , Λ and Φ the respective functions evaluated
a0 b0 0 0
tend to infinity. at z = 0. Just like in the Chazy-Curzon model, the thin disk
describedbytheZipoy-Voorheesmetrichasnoradialpressure.
Substituting equations (43), (44) and (45), into the Komar
Replacing (47), (48) and (49) into equation (29), we can cal-
massequation(29),weobtain,accordingly,M =m. Apartic-
k culate the Komar mass for the Zipoy-Voorhees thin disk as
ularsetofvaluesthatsatisfy theenergyconditionsarea=0.4
M =m.
and m=0.5, in which case r 8. In Fig. 6 we show the nu- T
c
≈
mericalsolutiontothedifferentialequation(10)withboundary To be consistent with the previous models, the calculations
conditions at r =0 and r =rc = 8. As can be noted, regard- were performed for a particular set of values that satisfy the
less of the values of w (upper panel) and k (lower panel), the energy conditions, m = 0.5, a = 1 and b = 2.15, such that
profiles of the perturbed energy density grow rapidly to infin- the cutoff radius takes the value r 31. In the same way as
c
ity. To rule out that the above effect is due to the chosen cases 1 and 2, in Fig. 7 we show the≈numerical solution to the
values of a, we perform an analysis varying δσ in the inter- differential equation (10) with boundary conditions at r = 0
val 0 m/a 1.3, observing the same tendency. The same and r=rc=31.
≤ ≤
procedure was performed using the total surface mass defini-
tion,obtainingthesamebehavior. Theseresultsshowthatthe It can be seen from Fig. 7 that thereexist instabilities for the
energy density, and hence the pressure and velocities, exhibit Zipoy-Voorhees thin disk strongly amplified before they reach
instabilities for this model and are independent of the mass 10%ofitscutoffradius,andregardlessoftheangularfrecuency
definition used in thecalculations. w(upperpanel)orwavenumberk(lowerpanel)theperturbed
8
Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-dddde2016 StabilityofAxisymmetricRelativisticThinDisks
energy density tends to infinity. The results for the Zipoy- the disks are made of a perfect fluid, we found the differen-
Voorhees thin disk when using the total surface mass are not tial equation for the perturbed energy density and the non-
shownbecausetheyexhibitpracticallythesamebehaviorthan zerocomponentsofthesurfaceenergy-momentumtensor. The
using the Komar mass. The independence of the results with differential equation was numerically solved, defining a cutoff
the particular chosen values of a and b, was also analyzed ob- radius r , such that thematter up to r is approximately 95%
c c
servingthesametendency. Giventheinstabilityinthesurface ofthemassoftheinfinitediskandtheremaining5%ofmatter
energy density,the other thermodynamic variables of the thin is distributed from r to infinity. Moreover, we assumed that
c
disk also are unstable. at the center of the disk the perturbation equals 10% of the
unperturbed energy density, and at the external cutoff radius
theperturbation vanishes.
4.´10-2
k=0,w=0.13
Once we derived the expressions for the pressure components,
2.´10-2 k=0,w=0.15 the surface energy density, and the metric functions for each
k=0,w=0.17 particular case, we calculated the Komar mass. As a general
result,wefindthatthemassparameterineachoneofthemet-
~Σ 0.
∆ rics equals the Komar mass for the disk, while by using the
1.´10100
totalsurface massdefinition,asisthecaseof thereviewed pa-
-2.´10-2 0. per, the total mass depends on the parameters of the DCR
-1.´10100
0 5 10 15 20 method. TheuseoftheKomardefinitionletsussetthephysi-
-4.´10-2 calparametersthatbestfittheexpectedenergydensityprofile
0.0 0.5 1.0 1.5 2.0 2.5 3.0
in arealistic model andsimultaneously satisfy theenergy con-
r
ditions. With this result, the cutoff radius only depends on
1.´103
1.´10100 the thermodynamic variables and the free parameters of each
metric.
0.
5.´102
-1.´101000 5 10 15 As the main finding, we found that the cutoff radius is larger
for theKomar mass definition than for thetotal surface mass.
~Σ 0.
∆ This result let us to increase the number of parameters that
k=0.01,w=0.17 give place to stable Schwarzschild thin disk models. On the
-5.´102 k=0.05,w=0.17 other hand, the Chazy-Curzon and Zipoy-Voorhees thin disk
modelsarenotstableunderfirst-orderperturbations,because
k=0.3,w=0.17
thethermodynamicvariablesandfluidvelocitiestendtoinfin-
-1.´103
0 1 2 3 4 5 ity for finite values of the radial coordinate. Such instabilities
r are a consequence of the lack of radial pressure and are not
related tothedefinition ofmass. Wealso haveshown that the
FIGURE 7. Profiles of the perturbed energy density δσ˜ for
infinite radial extension of the disk can be the reason for the
different values of w (upper panel) and k (lower panel) in the
instability,ashypothesizedbyUjevicet al.,nevertheless,some
Zipoy-Voorhees thin disk. The parameters have been set as
instabilitiescanbeartificiallyintroducedintothemodeldueto
a=1, b=2.15 and m=0.5. The inset shows thetendency of
theuseofanon-appropriatemassdefinitioninthecalculations.
the curvesfor larger scales.
Acknowledgments. We would like to thank the anonymous
referee for useful comments and remarks, which improved the
Concluding Remarks presentation of the paper. This research was partially sup-
ported by VIE-UIS under grant numbers 1822 and 1785, and
COLCIENCIASundergrantnumber8840. EABwould liketo
In this paper, we have reviewed the results obtained by
thankCOLCIENCIASfortheirsupportthrough theprogram:
Ujevic and Letelier(2004) on thestability of threeparticu-
J´ovenesInvestigadoreseInnovadores2014. FLDacknowledges
larmodels,usinganalternativemassdefinition,i.e.,theKomar
financial support from the Universidad de los Llanos provided
mass. This formal concept of mass can be defined in any sta-
undergrant Commission: Postdoctoral Fellowship Scheme.
tionaryspacetime,soitisapplicabletoallthreeparticularthin
disk models under consideration: the isotropic Schwarzschild,
Chazy-Curzon, and Zipoy-Voorhees metrics. Assuming that
9
E.A.Becerra,F.L.Dubeibe,G.A.Gonza´lez Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-dddde2016
Conflict of interest. The authors declare that they have no Semer´ak, O. (2004). Exact power-law discs around static
conflict of interest. black holes. Class. Quantum Grav. 21, 2203.
Semer´ak,O.(2002).FollowingthePragueInspiration,toCel-
ebratethe60thBirthdayofJiriBic´ak.WorldScientific,Sin-
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10