Can mixed star-plus-wormhole systems mimic black holes?

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by  Vladimir Dzhunushaliev

Can mixed star-plus-wormhole systems mimic black holes?
Vladimir Dzhunushaliev,1,2,3,4 ∗ Vladimir Folomeev,1,4 † Burkhard Kleihaus,5 ‡ Jutta Kunz5 §
1 Institute for Basic Research, Eurasian National University, Astana 010008, Kazakhstan
2 Department of Theoretical and Nuclear Physics,
Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan
3 Institute of Experimental and Theoretical Physics,
Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan
4Institute of Physicotechnical Problems and Material Science of the NAS of the Kyrgyz Republic,
265 a, Chui Street, Bishkek 720071, Kyrgyz Republic
5Institut fu¨r Physik, Universita¨t Oldenburg, Postfach 2503 D-26111 Oldenburg, Germany
We consider mixed strongly gravitating configurations consisting of a wormhole threaded bytwo
6 typesofordinarymatter. Forsuchsystems,thepossibilityofobtainingstaticsphericallysymmetric
1 solutionsdescribingcompact massive centralobjects enclosed byhigh-redshiftsurfaces (black-hole-
0 likeconfigurations)isstudied. Usingthestandardthinaccretion diskmodel, weexhibitpotentially
2 observabledifferencesallowing todistinguish themixedsystemsfrom ordinaryblackholeswiththe
same masses.
g
u
PACSnumbers: 04.40.Dg,04.40.–b,97.60.Lf,97.10.Gz
A

0
2 I. INTRODUCTION

]
It is now commonly believed that objects described by solutions with an event horizon – black holes (BH) – can
c
q exist in the Universe [1]. In the simplest case it canbe the well-knownexactspherically symmetric solutionfound by
- Schwarzschild in 1916, immediately after the creation of Einstein’s general relativity. However, even 100 years later,
r
g the physical reality of such a mathematical solution is still sometimes questioned, essentially because it is so far not
[ possible to prove unambiguously that astrophysical BH candidates really possess an event horizon [2, 3]. Therefore,

itseemsthatthereisnoa priori reasontoexcludefromconsiderationastrophysicalobjectsthatdonothaveanevent
2
horizon but are able to mimic the main observationalcharacteristics of BHs.
v
4 In this connection,the literature inthe field offersalternativetypes of configurationswhich, fromthe pointofview
2 of a distant observer, would look almost like BHs but would have no horizon and singularity – the so-called black
1 hole mimickers (BHM). Among them are boson stars [4, 5], gravastars [6–8], and wormholes [9–11]. Obviously, the
4 properties of gravitationalfields produced by such objects will depend onthe particular BHM configuration. Then it
0
will manifest itself, for example, in considering the process of accretion of matter onto objects of this kind. Because
.
1 ofthe differences in their externalgeometry,one mightexpect changesboth inthe structure ofaccretiondisks andin
0 their emission spectra.
6
In the present paper we consider a mixed relativistic configuration consisting of a wormhole filled by ordinary
1
matter. Inthiswaywesuggestonemorepossiblewaytomimica“black-hole-like”configuration. (Bythe latter,asin
:
v the case ofother objects of this kind, we mean a compact massive centralobjectenclosed by a high-redshift surface.)
i In our previous works [12–20] we have already studied systems of this kind in various aspects of the problem. In
X
particular,usingtheobtainedequilibriumneutron-star-plus-wormholeconfigurations,someissuesconcerningpossible
r
a astrophysical manifestations associated with the presence of nontrivial topology in the system have been considered.
Namely,inRef.[17]thepassageoflight–radiatedfromthesurfaceofaneutronstar–throughathroatofawormhole
has been studied. It was shown that in this case there is a characteristic distribution of the intensity of the light
which differs from the one obtained when considering the case where radiation does not pass through the throat. In
principle,suchaneffectcouldbeobservedbyinstrumentswithsufficientlyhighresolution. InRef.[19]theinfluenceof
the nontrivialtopology on the structure of the interior magnetic field of mixed systems supported by neutron matter
modeled by isotropic and anisotropic fluids have been demonstrated. Here we would like to continue searching for
otherpotentiallyobservableastrophysicalmanifestationssharedbymixedstar-plus-wormholesystems. Intheprocess,
our purpose will be twofold: (i) we demonstrate the possibility that BHM solutions can be obtained in such mixed
systems; (ii) we reveal potentially observable effects which distinguish such systems from other BHMs considered in
∗ [email protected]
† [email protected]
‡ [email protected]
§ [email protected]
2
the literature.
In doing so, we first construct static spherically symmetric solutions whose nontrivial topology is provided by the
presence of a ghost scalar field. This type of fields are now widely used in modeling the accelerated expansion of the
present Universe [21]. With the opposite sign in front of its kinetic energy term, such a field violates the null energy
condition that may lead to the appearance of a nontrivial wormholelike topology. The aim of the present work is to
study possible observational differences between mixed systems and ordinary BHs with the same masses.
The important observational manifestations of BHs are the effects associated with a process of accretion of sur-
roundingmatterontoaBH.Forthinaccretiondisks,theenergyreleasedinsuchaprocessis 6%to42%(depending
∼
onthe spinof the black hole) ofthe rest mass ofthe accreting matter, andthis energy may be convertedinto observ-
ableradiation[1]. CalculationsoftheaccretionflowontoaBHandtheemittedradiationpatternare,ingeneral,very
difficult. But since our purpose here is just to reveal the differences between the accretion onto BHs and our mixed
systems but not a more or less realistic modeling of the accretion process in itself, we restrict ourselves to the con-
sideration of a relatively simple model. Namely, we will consider a steady-state accretion process for a geometrically
thin and optically thick accretion disc orbiting the mixed configurations. To reveal the differences, we will compare
our results with those obtained for BHs with the same masses.
The paper is organized as follows. In Sec. II we present the general-relativistic equations for the mixed systems
underconsiderationanddescribetwoparticularchoicesoftheequationofstateforordinarymatter. Inordertoobtain
black-hole-like solutions, in Sec. III we numerically solve these equations with different choices for the parameters of
the systems. To demonstratethe observationaldifferences betweenthe obtainedmixed systems andordinaryBHs, in
Sec.IVweconsiderthe processofthin-diskaccretionontosuchconfigurationsandcomparethe energyfluxesemitted
from the disk’s surface. Finally, in Sec. V we discuss and summarize the obtained results.
II. STATEMENT OF THE PROBLEM
We willconsideramixedsystemcontainingtwotypesoffluid: (i)anordinaryfluidsatisfyingallenergyconditions,
and (ii) an exotic fluid violating the null energy condition. For the ordinary fluid, one can take in principle any form
of fluid. For example, it could be ordinary matter that stars are made of, including neutron stars [12–14, 17, 19, 20],
or dark matter, but it could also comprise electromagnetic fields [22, 23], chiral fields [15], Yang-Mills fields [16], or
complex scalar fields [18].
As regards the exotic fluid, we will consider a situation where its presence gives rise to a nontrivial wormholelike
spacetime topology of the system. Modeling of such a fluid can be done in many ways, both within the frameworks
of general relativity and when considering modified theories of gravity.
A. General equations
To demonstrate a possibility of obtaining black-hole-like solutions for the aforementioned mixed systems, let us
consider a situation where:
(i)Ordinarymatterismodeledbyanisotropicperfectfluid,i.e.,byafluidwithequalradialandtangentialpressures.
Its energy-momentum tensor is
Tk =(ε+p)u uk δkp , (1)
i(fl) i − i
where ε, p, and ui are the energy density, the pressure, and the four-velocity of the fluid, respectively.
(ii)Exoticmatterisdescribedbyoneghostscalarfieldϕ,i.e.,byafieldwiththeoppositesigninfrontofitskinetic
energy term, with the following energy-momentum tensor:
1
Tk = ∂ ϕ∂kϕ δk ∂ ϕ∂µϕ V(ϕ) , (2)
i(sf) − i − i −2 µ −
(cid:20) (cid:21)
where V(ϕ) is the potential energy.
Anecessaryconditionforprovidinganontrivialwormholetopologyinthe systemisthe violationofthe nullenergy
condition, T nink 0, where T =Tk +Tk andni is any nullvector. Inour case this implies thatthe following
ik ≥ ik i(fl) i(sf)
inequalities are satisfied (at least in some region of spacetime): T0 T1 <0,T1 >0.
0 − 1 1
For the mixed system under consideration, it is convenient to use polar Gaussian coordinates. The metric then
reads
ds2 =eν(dx0)2 dr2 R2dΩ2, (3)
− −
3
where ν and R are functions of the radial coordinate r only, dΩ2 is the metric on the unit two-sphere, and the time
coordinate x0 =ct. Then the corresponding components of the energy-momentum tensor take the form
1
T0 =ε ϕ′2+V, (4)
0 − 2
1
T1 = p+ ϕ′2+V, (5)
1 − 2
1
T2 =T3 = p ϕ′2+V. (6)
2 3 − − 2
We will consider here the simplest situation when ordinary matter is located in the central region of the system.
The center is describedby the value of the radialcoordinate r=0,which correspondsto a throatoranequator. The
term center thus refers to the extremal surface, located symmetrically between the two asymptotically flat regions.
We also assume that the matter has a maximum density at the center. (For cases of a shifted maximum density, see
Refs. [19, 20].) Also, without loss of generality, we can set the value of the scalar field at the center to ϕ(0)=0, but
we note that ϕ′(0)=0. Then the potential of the scalar field can be expanded in the neighborhood of the center as
6
1
ϕ ϕ r+ ϕ r3, (7)
1 3
≈ 6
where ϕ is the derivative at the center, the square of which corresponds to the “kinetic” energy of the scalar field.
1
In further calculations we will use this kinetic energy to introduce dimensionless variables.
Takinginto accountthe components ofthe fluid energy-momentumtensor(4)-(6), the (0), (1), and(2) components
0 1 2
of the Einstein equations with the metric (3) take the form
R′′ R′ 2 1 8πG
2 + + = T0, (8)
−" R (cid:18)R(cid:19) # R2 c4 0
R′ R′ 1 8πG
+ν′ + = T1, (9)
−R R R2 c4 1
(cid:18) (cid:19)
R′′ 1R′ 1 1 8πG
+ ν′+ ν′′+ ν′2 = T2, (10)
R 2 R 2 4 − c4 2
where the prime denotes differentiation with respect to r.
The general equation for the scalar field ϕ is
1 ∂ ∂ϕ dV
√ ggik = . (11)
√ g∂xi − ∂xk dϕ
− (cid:18) (cid:19)
Using the metric (3), this equation gives
1 R′ dV
ϕ′′+ ν′+2 ϕ′ = . (12)
2 R −dϕ
(cid:18) (cid:19)
Because of the conservationof energy and momentum, Tk =0, not all of the Einstein field equations are indepen-
i;k
dent. Taking the i=1 component of the conservation equations gives
dT1 1 R′ 1
1 + T1 T0 ν′+2 T1 T2+T3 =0. (13)
dr 2 1 − 0 R 1 − 2 2 3
(cid:20) (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1)
Taking into account the components (4)-(6), and also Eq. (12), we obtain from Eq. (13)
dp 1 dν
= (ε+p) . (14)
dr −2 dr
Keeping in mind that the pressureand the energy density of ordinarymatter are relatedby some equation ofstate
(EOS), we have four unknown functions – R, ν, p, and ϕ – for which there are five equations, (8)-(10), (12), and
(14), only four of which are independent. We will consider below two types of EOS, used both in describing compact
astrophysical objects and in modeling dark matter. In both cases, for simplicity, we assume that the scalar field is
massless and has no self-interaction, i.e., that V =0.
4
B. Polytropic EOS
Consider first the case of ordinary (neutron) matter modeled by a polytropic EOS. Such an EOS, being, on one
hand, relatively simple, reflects adequately the general properties of more realistic EOSs describing matter at small
and high densities and pressures. This EOS can be taken in the following form [24]:
p=Kρ1+1/n, ε=ρ c2+np, (15)
b b
with the constant K = kc2(n(ch)m )1−γ, and the polytropic index n = 1/(γ 1), and where ρ = n m denotes the
b b − b b b
rest-mass density of the neutron fluid. Here n is the baryonnumber density, n(ch) is a characteristicvalue of n , m
b b b b
is the baryon mass, and k and γ are parameters whose values depend on the properties of the neutron matter.
The literatureinthe fieldoffersavarietyofvaluesforthe parametersenteringthis EOS.Thisallowsthe possibility
of getting both weakly and strongly relativistic objects [24]. For simplicity, here we take only one set of parameters
for the neutron fluid. Namely, we choose m =1.66 10−24g, n(ch) =0.1fm−3, k =0.1, and γ =2 [25]. We employ
b × b
these values for the parameters in the numerical calculations of Sec. III.
To carry out numerical calculations, it is convenient to rewrite the above equations in terms of dimensionless
variables. This can be done as follows:
r R √8πG c2
ξ = , Σ= , φ(ξ)= ϕ(r) with L= , (16)
L L c2 √8πGϕ
1
whereListhecharacteristicsizeofthesystem. Inturn,forthefluiddensityonecanusethenewreparametrization[26],
ρ =ρ θn , (17)
b bc
where ρ is the density of the neutron fluid at the center of the configuration. Then Eqs. (8)-(10), (12), and (14)
bc
take the following dimensionless form:
Σ′′ Σ′ 2 1
2 + + =T˜0, (18)
−" Σ (cid:18)Σ(cid:19) # Σ2 0
Σ′ Σ′ 1
+ν′ + =T˜1, (19)
−Σ Σ Σ2 1
(cid:18) (cid:19)
Σ′′ 1Σ′ 1 1
+ ν′+ ν′′+ ν′2 = T˜2, (20)
Σ 2 Σ 2 4 − 2
eνc−ν
φ′2 = , (21)
(Σ/Σ )4
c
1
σ(n+1)θ′+ [1+σ(n+1)θ]ν′ =0. (22)
2
Here the dimensionless right-hand sides of the Einstein equations are:
1 1 1
T˜0 =B(1+σnθ)θn φ′2, T˜1 = Bσθn+1+ φ′2, T˜2 = Bσθn+1 φ′2, (23)
0 − 2 1 − 2 2 − − 2
where B = (ρ c2)/ϕ2 is the dimensionless ratio of the fluid energy density to that of the scalar field at the center;
bc 1
Σ and ν are the values of the corresponding functions at the center [see Eq. (26)]; σ = Kρ1/n/c2 = p /(ρ c2) is
c c bc c bc
a constant, related to the pressure p of the fluid at the center. The values of the fluid parameters appearing here
c
are taken from the above text [see after Eq. (15)]. Eq. (12) has been integrated to give the expression (21) with the
integration constant chosen so as to provide φ′ =1 at the center.
Eq. (22) may be integrated to give in the internal region with θ =0 the metric function eν in terms of θ,
6
2
1+σ(n+1)
eν =eνc , (24)
1+σ(n+1)θ
(cid:20) (cid:21)
and eνc is the value of eν at the center where θ = 1. The integration constant νc is fixed by requiring that the
space-time is asymptotically flat, i.e., eν =1 at infinity.
5
1. Boundary conditions
We here consider neutron-star-plus-wormhole configurations that are asymptotically flat and symmetric under
ξ ξ. ThemetricfunctionΣ(ξ)maybeconsideredasadimensionlesscircumferentialradialcoordinate. Asymptotic
→−
flatness requires that Σ(ξ) ξ for large ξ . Because of the assumed symmetry of the configurations, the center of
the configurations at ξ = 0→sh|o|uld corresp|o|nd to an extremum of Σ(ξ), i.e., Σ′(0) = 0. If Σ(ξ) has a minimum at
ξ = 0, then ξ = 0 corresponds to the throat of the wormhole. If, on the other hand, Σ(ξ) has a local maximum at
ξ = 0, then ξ = 0 corresponds to an equator. In that case, the wormhole will have a double throat surrounding a
belly (see, e.g., Refs. [15–17]).
Expanding the metric function Σ in the neighborhood of the center
Σ Σ +1/2Σ ξ2
c 2
≈
and using Eqs. (18) and (19), we find the relations
1 Σ
c
Σ = , Σ = 1 B[1+σ(n+1)] . (25)
c 2
1/2 Bσ 2 −
− n o
Thus the sign of the expansion coeffipcient Σ determines whether the configurations possess a single throat at the
2
center or an equator surrounded by a double throat.
Equations(18)-(22) aresolvedfor givenparametersof the fluid σ, n, andB, subjectto the boundaryconditions at
the center of the configurationξ =0,
Σ(0)=Σ , Σ′(0)=0, ν(0)=ν , φ(0)=0, φ′(0)=1, (26)
c c
and also θ(0)=1. Note here that, using (16), we can express the dimensional value of the derivative ϕ as follows:
1
c4 1
ϕ2 = . (27)
1 8πGL2
Thus the dimensional “kinetic” energy of the scalar field depends only on the value of the characteristic length L,
which can be chosen arbitrarily subject to some physically reasonable assumptions. Substituting this ϕ2 into the
1
expression for B [see after Eq. (23)], we find
B =8πGρ (L/c)2. (28)
bc
It is seen from the above expressions for ϕ2 and B that by fixing L, one automatically determines the value of ϕ2.
1 1
But the value of B can still change depending on the value of the fluid density ρ at the center. Therefore one can
bc
consider B as a parameter describing the ratio of the fluid energy density at the center to the energy density of the
scalar field at the center.
Asidefromgivingtheboundaryconditionsatthecenter,itisimportantforustokeeptrackofthebehaviourofthe
system on the other boundary – the surface of the fluid. Like the characteristics of the central region, the properties
of the fluid’s boundary (in particular, magnitudes of the surface red shift) will also be determined by the parameters
of the system. For more discussion of this issue, see Sec. III.
C. Completely degenerate Fermi gas
One more simple type of EOSused in the literature to model compactobjects (white dwarfs and neutronstars [1],
dark matter stars [27]) is an EOS describing an ideal completely degenerate Fermi gas at zero temperature. Its
equation of state can be obtained by using usual expressions for the energy density and pressure [1]:
1 kF c5m4 1 c5m4
ε= k2 m2c4+k2c2dk = f z 1+z2(1+2z2) sinh−1(z) fχ , (29)
π2~3 f ~3 8π2 − ≡ ~3 1
1 Z0 kF q k4c2 c5m4 1 h p i c5m4
p= dk = f z 1+z2(2/3z2 1)+sinh−1(z) fχ , (30)
3π2~3 Z0 m2fc4+k2c2 ~3 8π2 h p − i≡ ~3 2
q
where χ ,χ are the dimensionless energy density and pressure, respectively. Here m is the fermion mass, k is
1 2 f F
the Fermi momentum, z = k /(m c) is the relativity parameter. Eqs. (29) and (30) yield a parametric dependence
F f
p=p(ε).
6
Intwolimiting cases,this EOScanbe representedinsimplepower-lawforms: (i)inthe nonrelativisticcase,z 1,
≪
we get the polytropic law, χ χ5/3, and (ii) in the ultrarelativistic case, z 1, we have χ =χ /3.
2 ∝ 1 ≫ 2 1
Using this EOS and the dimensionless variables (16), we get for the right-hand sides of Eqs. (18)-(20)
1 1 1
T˜0 =B χ φ′2, T˜1 = B χ + φ′2, T˜2 = B χ φ′2, (31)
0 1 1− 2 1 − 1 2 2 2 − 1 2− 2
where B = c5m4f 8πGL2m4fc. In turn, instead of Eq. (22), we have
1 ~3ϕ21 ≡ ~3
dχ 1 χ +χ dν
1 1 2
= . (32)
dξ −2dχ /dχ dξ
2 1
1. Boundary conditions
As before, we choose the boundary conditions in the form of (26). Then, taking into account the expansion in the
vicinity of the center,
1 1
χ χ + χ ξ2, χ χ + χ ξ2, (33)
1 1c 12 2 2c 22
≈ 2 ≈ 2
where χ ,χ are the values of the dimensionless energy density and pressure at the center, one can obtain the
1c 2c
following expressions for the expansion coefficients of the metric function Σ:
1 Σ
c
Σ = , Σ = [1 B (χ +χ )]. (34)
c 2 1 1c 2c
1/2 B χ 2 −
1 2c
−
Again, the sign of Σ determines whepther the system is a single- or double-throat one.
2
As in the case of the polytropic matter, the dimensional “kinetic” energy of the scalar field (27) is determined
completely by the characteristic size of the system. For the fermionic gas under consideration, it is natural to use as
a characteristicsizethe Landauradiusderivedinconsideringcompactconfigurationsconsistingofanultrarelativistic
degenerate Fermi gas within the framework of Newtonian gravity (see, e.g., Ref. [27]):
~M
Pl
R = , (35)
L c m2
f
where M is the Planck mass.
Pl
In considering our mixed systems with the fermionic fluid, it is then natural to choose L =αR , where α is some
L
free scale parameter. Using this L in the expression for B [see after Eq. (31)], we get B = 8πα2. By choosing
1 1
different values of α, we can change the contribution in the right-hand sides of the Einstein equations (31) coming
from the fermionic matter.
III. NUMERICAL RESULTS
A. Procedure of finding solutions
Whensolvingtheobtainedequationsnumerically,weproceedasinRef.[17],wherethesolutionsearchprocedureis
describedindetail. Theprocedure,briefly,isasfollows. Wesolvethesystemofequations(18)-(21)and(22),(23)(for
the polytropic fluid) or (31), (32) (for the Fermi gas) with the correspondingboundary conditions (26) together with
(25) [for the polytropic fluid] or (33) and(34) [for the Fermi gas]. In doing so,the configurationsunder consideration
can be subdivided into two regions: (i) the internal one, where both the scalarfield and the fluid are present; (ii) the
external one, where only the scalar field is present. Here the solutions are obtained by using Eqs. (18)-(21), in which
θ,χ ,χ are set to zero.
1 2
The internal solutions must be matched with the external ones at the boundary of the fluid, ξ = ξ , by equating
b
the corresponding values of the functions φ, Σ, ν and their derivatives. The boundary of the fluid ξ is defined by
b
p(ξ ) = 0. The value of the integration constant ν at the center is determined proceeding from the requirement of
b c
asymptotic flatness of the external solutions.
7
As pointed out in Ref. [17], there exists a critical value of B, B , at which Σ [see Eq. (25)]. Beyond
crit c
→ ∞
this critical value, physically reasonable solutions no longer exist. A similar situation takes place for the Fermi gas,
where some critical value of the coefficient B = Bcrit is also involved, see Eq. (34). In the present paper we will be
1 1
interested in solutions correspondingto the values of B and B close to the critical ones. Aside from this, as one can
1
seefrom(25), asB B (thatcorrespondstoBσ 1/2)the expansioncoefficientΣ [(1 n)/2 B]. Thenfor
crit 2
→ → ∼ − −
the polytropic index n 1, which is often used in the literature in modeling relativistic objects, Σ will be certainly
2
≥
negative. That is, if for small values of B there is a single throat located at the center of the configuration, then, as
B increases, the center of the configuration no longer represents a throat but instead corresponds to an equator. On
eachsideoftheequatoraminimalareasurface(athroat)islocated. Inthiscasetheresultingconfigurationsrepresent
double-throat systems, where the space between the throats can be completely or partially filled by the fluid. The
latter situation is exactly the one that we consider below.
A similar situation is also found in the case of the Fermi gas.
B. Total mass of the system
For the spherically symmetric metric (3), the mass m(r) of a volume enclosed by a sphere with circumferential
radius R , corresponding to the center of the configuration, and another sphere with circumferential radius R > R
c c
can be defined as follows:
c2 4π r c2 4π r dR
m(r)= R + T0R2dR R + T0R2 dr′, (36)
2G c c2 0 ≡ 2G c c2 0 dr′
ZRc Z0
wherewerefertothefirsttermasthemassassociatedwiththecenter,M ,whilethemassassociatedwiththethroat,
c
M , is obtained by integrating up to the throat radius R = R(r ). As pointed out in Ref. [17], despite the fact
th th th
that the size of the equator at the center R diverges as B B , the size of the throat R , and correspondingly
c crit th
→
the mass associated with the throat, remain finite. This is because the divergence of the positive mass at the center
M is exactly canceled by the mass associated with the mass of the ordinary fluid, which is negative in the case of
c
double-throat systems. This comes about because the derivative dR/dr is negative in the region where the ordinary
fluid is located, and therefore the mass integral associated with this fluid gives a negative contribution to the total
mass.
In the dimensionless variables the expression (36) takes the form
ξ dΣ
m(ξ)=M Σ + T˜0Σ2 dξ′ , (37)
∗ c 0 dξ′
( Z0 )
where T˜0 is taken from (23) (for the polytropic fluid) or from (31) (for the Fermi gas). The coefficient M in front of
0 ∗
the curly brackets has the dimension of mass
c3 B 1 ~3c3B
Mpoly = or MFermi = 1
∗ 2 s8πG3ρbc ∗ 2s8πG3m4f
for the polytropic and Fermi fluids, respectively. Note that the total mass M is then obtained by taking the upper
limit of the integralto infinity, since the energy density of the scalar field becomes equal to zero only asymptotically,
as Σ .
→∞
For the case of a massless scalar field consideredhere, it is useful to write downanother, more elegant definition of
the total mass via the Komar integral. The latter, in general, is defined as [28]
2 1
M = T g T naξbdV,
K c2 ab− 2 ab
ZΣ(cid:18) (cid:19)
where na is a normal to Σ and ξb is a timelike Killing vector. Using the above dimensionless variables, we find for
the polytropic fluid
ξb
Mpoly =MpolyB eν/2Σ2[1+σ(n+3)θ]θndξ (38)
K ∗
Z0
and for the Fermi fluid
ξb
MFermi =MFermiB eν/2Σ2(χ +3χ )dξ. (39)
K ∗ 1 1 2
Z0
8
Note that here the integration is performed only in the range from 0 to ξ , where there is a nonzero contribution
b
associated with the fluids.
It is seen from the expressions (38) and (39) that in order to ensure the finiteness of the total mass of the system
it is necessary that as B →Bcrit or B1 →B1crit (when Σc →∞) the metric function eνc →0 simultaneously (keeping
in mind that the functions θ,χ ,χ remain always finite). The numerical solutions presented below indicate that in
1 2
the vicinity of B and Bcrit this is indeed the case.
crit 1
C. The choice of the density and pressure at the center
Tocarryoutnumericalcalculations,itisnecessarytoassignthecorrespondingvaluesofthedensityandpressureat
thecenter. InthevicinityofB andBcrit,theycanbe foundfromtheconditionthatthe radicandinthe expression
crit 1
forΣ isapproximatelyequaltozero. Then, takingintoaccountEq.(28),forthe polytropicEOSwecanobtainfrom
c
Eq. (25):
c4 n/(n+1)
ρ = (1 δ) , (40)
bc 16πGL2K −
(cid:20) (cid:21)
where δ 1 is some constant. In the limit δ 0, the density ρ at the center goes to its critical value.
bc
≪ →
In the same way, for the Fermi gas, using the condition that the radicand in the expression for Σ from (34) is
c
approximately equal to zero, one can find the value of the pressure at the center in the vicinity of Bcrit:
1
1 δ 1 δ
χ = − − . (41)
2c 2B ≡ 16πα2
1
Here we used the expression for B obtained earlier [see after Eq. (35)].
1
Then,using the obtainedexpressionsinthe boundaryconditions(25),(26), (33), and(34),wesolvedthe equations
numerically according to the procedure described in Sec. IIIA.
D. Results of calculations
Examples of the obtained solutions are presented in Figs. 1-4. Fig. 1 shows the typical distributions of the total
and fluid energy densities for the mixed systems under consideration. The calculations indicate that when the scale
parameter α & 1/2 the total energy density has a characteristic well in the neighborhood of the throat located at
ξ = ξth. This is because the “kinetic” energy of the scalar field, which behaves here as eνc−νth(Σc/Σth)4 [see
∼
Eq. (21)], exhibits fast growth due to the large value of Σ , on one hand, and to the small value of Σ , on the other
c th
hand. This is accompanied by a simultaneous fast decrease (modulus) of the metric function ν, which, starting from
large values on the surface of the fluid, undergoes a sharp decrease as the throat is approached (see Fig. 2). The
ultimate resultis that, for instance,at α=1 in the vicinity ofξ =ξ the dimensionless totalenergy density is of the
th
order of 4 105 (at δ =10−15).
| × |
Fig. 2 shows the typical behavior of the metric functions Σ and ν in the external regionof the mixed systems. For
the configurationsunder considerationwith B B andB Bcrit, the throats alwaysreside outside the fluid (for
≈ crit 1 ≈ 1
systemswiththethroatslocatedwithinafluid,seeRef.[17]). Thenumericalcalculationsindicatethatasαdecreases,
the throat shifts further away from the surface of the fluid (in units of relative radii).
Figs.3and4showthe totalmassesandthe valuesofthemetric functionν onthe surfaceofthe fluid, νsurf =ν(ξ )
poly b
and νsurf = ν(x ), as functions of the parameter δ. As δ 0, the mass of the polytropic configurations increases,
Fermi b →
tending to some finite value. This growth of the mass is accompanied by a simultaneous increase of the modulus of
νsurf. Thus, from the point of view of a distant observer,the surface of the fluid will look like a high-redshift surface,
i.e., in this sense the system will be similar to a BH. In turn, for any δ used here the radiiofthe fluids remainalmost
unchanged (for the numerical values, see in captions of Figs. 3 and 4), but the gravitational radius of the systems,
r =2.95(M/M )km, grows as δ 0. Note that for the configurations considered here the radius of the polytropic
g ⊙
→
fluid is always less than the gravitationalradius of the system as a whole.
A similar situation also takes place for the system with the fermionic fluid. The only difference is in the behavior
of the mass of the configuration for large α, when as δ 0 the mass does not grow, but decreases (see the left panel
of Fig. 4). There, as in the case of the polytropic fluid,→the modulus of νsurf increases for any α (see the right panel
Fermi
of Fig. 4). In turn, the radius of the Fermi fluid is always (for any δ) larger than the gravitational radius for α = 1
and less than it for α 1/2.
≤
9
FIG.1: ThetotalenergydensityT˜0 fromEq.(23)(forthesystemwiththepolytropicfluid)andfromEq.(31)(forthesystem
0
with the fermionic fluid) (both in units of ϕ2) are shown as functions of the relative radius ξ/ξ . The inset shows the fluid
1 b
energy densities B(1+σnθ)θn (for the polytropic fluid) and B1χ1 (for the fermionic fluid). The numbers near the curves
correspond to the values of the scale parameter α for the Fermi systems. Since the solutions are symmetric with respect to
ξ=0, thegraphs are shown only for ξ>0. The thin vertical line corresponds to theboundary of thefluids. For all plots, the
parameter δ is taken as 10−15. Asymptotically,as ξ→±∞,the total energy goes to zero for all systems.
FIG. 2: External metric functions Σ and ν for the systems of Fig. 1 are shown as functions of the relative radius ξ/ξ . The
b
minimaofthefunctionΣcorrespondtothelocation ofthethroat. Thebolddotsdenotetheminimumradiusforastableorbit
ξso (see below in Sec. IV).Asymptotically, as ξ →±∞,the spacetime is flat with Σ→|ξ| and eν →1 from below.
10
-20
120
T
h
e
-30 s
u
M 100 rfa
M/ M ce
s re
as 80 -40ds
otal m hift fu
T 60 n
-50ctio
n

40
-60
20
-15 -13 -11 -9 -7 -5
10 10 10 10 10 10
FIG. 3: Total mass and the surface value of the redshift function νsurf for the system with the polytropic fluid are shown as
poly
functions of theparameter δ. For all systems, the radius of thefluid is R ≈10.55km.
poly

-20
1
10 =0.1 n -30
o
M/M ncti =0.1
mass 100 hift fu-40
al ds-50
Tot 10-1 =1 ce re =1
a-60
urf
-2 e s
10 h-70
T
-80
-3
1010-15 10-13 10-11 10-9 10-7 10-5 10-15 10-13 10-11 10-9 10-7 10-5
FIG. 4: Total mass and the surface value of the redshift function νsurf for the systems with the Fermi fluid are shown as
Fermi
functions of the parameter δ. For all systems, the fermion mass is taken to be m = 1GeV. The parameter α takes the
f
values 1/10,1/5,1/3,1/2,1, from top to bottom. The radii of the fluid are: for α = 1 – RFermi ≈ 2.08km; for α = 1/2 –
RFermi ≈1.44km; for α=1/3 – RFermi ≈1.14km; for α=1/5 – RFermi ≈0.84km; for α=1/10 – RFermi ≈0.55km.
The numerical values of the masses and sizes of the systems with the Fermi fluid shown in Fig. 4 are given for
m = 1GeV. For other m the values of the total masses and sizes are derived from those of Fig. 4 by multiplying
f f
them by the factor (1GeV/m )2, where m is taken in GeV. There, the value of νsurf will remain the same for any
f f Fermi
m . Correspondingly, if we assume for definiteness that m lies in the range 1 eV.m .102 GeV (such values are
f f f
used, for instance,in modeling darkmatter [27]), the totalmassesand sizesof the configurationsunder consideration
will lie in a very wide range. For example, for m equal, say, to 1MeV, the system with α=1/3 and δ =10−15 has
f
the total mass M 5 106M , and the radius of the surface of the fluid is R 106km. A configuration with such
⊙
∼ × ∼
characteristics,possessing a high-redshift surface, might mimic BHs at the center of galaxies [29].
IV. THIN ACCRETION DISK
In this section we consider the process of accretionof test particles onto our configurations. The purpose is to find
out what are the differences between the mixed systems under consideration and BHs as regards the observational