A stability analysis of the static EKG Boson Stars
AA stability analysis of the static EKG BosonStars.
D.S. Fontanella , A. Cabo , Theoretical Physics Department,Instituto de Cibern´etica, Matem´atica y F´ısica,Calle E, No. 309, Vedado, La Habana, Cuba. [email protected] , [email protected] January 14, 2021
Abstract
The stability of the recently proposed static solutions for bosonstars is analyzed. These solutions of Einstein-Klein-Gordon (EKG)equations arise from considering the interaction of a real scalar fieldwith matter. We assume that the inclusion of the scalar field in addi-tion with matter, allows to justify that stability implies that the totalmass of the solution should grow when the initial condition for thedensity of matter at the origin is also increased. Employing numericalvalues for the static boson star based on a linear relation between thesource and the energy density and between this and the pressure, wefound the relation that linked the the scalar field at the origin withthe matter energy density (MED) in the same point. We also deter-mine the behavior of the total mass (TM) with the matter energydensity in the origin, by also obtaining through this and the weak en-ergy condition, two possible ranges for stable solutions of static bosonstars. a r X i v : . [ g r- q c ] J a n Introduction
The search for stable configurations for compact objects play an importantrole in contemporary Astrophysics. There is a great variety of theoreticallyproposed objects of the most diverse natures [3], [2], [6], [4] which continuebeing studied. Boson stars are ones of the most widely accepted configura-tions assumed for exploring the possibility of the existence of such compactobjects. In the article [1] a static boson star configuration was found solvingthe EKG equations for a real scalar field, with spherical symmetry beingcoupled with matter. These solutions were found numerically by introducinga source being proportional to the energy density of the matter and lookingfor the value of the scalar field at the origin that would determine a Yukawapotential at large distance, for the scalar field. A main result of that work isto show that the inclusion of the interaction of the scalar field with matterallows for the existence of static solutions of the EKG equations, a propertythat these equations lack [2].The present work is composed of the following parts, first we present the sys-tem of equations that were solved in our previous paper and that determinethe boson star solution. In second place, an investigation of the stabilitycriterion is carried out. The relation between matter energy density at theorigin and the scalar field also in the origin, which determine the existenceof the solutions, is evaluated . Finally, we find the TM vs MED in the originplot which combined with the weak energy condition, results in two possibleranges for stable solutions of the configuration.
Consider a metric defined by the following squared interval and coordinates ds = v ( ρ ) dx o − u ( ρ ) − dρ − ρ ( sin θ dϕ + dθ ) , (2.1) x = c t , x = ρ, (2.2) x = ϕ, x ≡ θ, (2.3)and an energy-momentum tensor for a real scalar field in interaction withmatter: T νµ = − δ νµ g αβ Φ ,α Φ ,β + m Φ + 2 J ( ρ ) Φ)+ g αν Φ ,α Φ ,µ + P δ νµ + u ν u µ ( P + e ) . (2.4)2fter making several coordinate changes as detailed in [1], the EKG systemcan be reached: u (cid:48) ( r ) r − − u ( r ) r = −
12 ( u ( r ) φ (cid:48) ( r ) + φ ( r ) + 2 j ( r ) φ ( r )) − e ( r ) , (2.5a) u ( r ) v ( r ) v (cid:48) ( r ) r − − u ( r ) r = −
12 ( − u ( r ) φ (cid:48) ( r ) + φ ( r ) + 2 j ( r ) φ ( r )) + p ( r ) , (2.5b) p (cid:48) ( r ) + ( e ( r ) + p ( r )) v (cid:48) ( r )2 v ( r ) − φ ( r ) j (cid:48) ( r ) = 0 , (2.5c) j ( r ) + φ ( r ) − u ( r ) φ (cid:48)(cid:48) ( r ) = φ (cid:48) ( r )( u ( r ) + 1 r − r ( φ ( r ) j ( r ) φ ( r ) + e ( r ) − p ( r )2 )) , (2.5d)where φ ( r ) is the scalar field, j ( r ) is the source and e ( r ), p ( r ) the energydensity and the matter pressure respectively. Note that we used the Bianchiidentity too for completing the system of the equations. So, in general the setof equations is composed as fallow: The two first equations are the Einsteinone referent to the two first component of the metric v ( r ) and u ( r ). The otherone, (2.5c) is the dynamic equation for the energy, the pressure and the scalarfield, and it substitutes the two Einstein equations that are associated to bothangular directions. This relation results from the Bianchi identity. Finally,(2.5d) is the Klein-Gordon equation for the scalar fields.This system describes the dynamics of the space time for a configurationof scalar field and matter coupled through the source j ( r ) .Let us fix the boundary conditions at a very small value δ = 10 − aroundthe origin in the following way: u ( δ ) = 1 , (2.6) v ( δ ) = 1 , (2.7) φ ( δ ) = φ , (2.8) e ( δ ) = e , (2.9)and assume a linear relationship between the source and the matter energydensity as well as between the pressure and the same matter energy density3s j ( r ) = g e ( r ) ,p ( r ) = n e ( r ) . In the work [1] the solutions were determined by setting a value of the energydensity of the matter at the origin and determining the value of the scalarfield also at the origin in such a way that this field takes the form of a Yukawapotential at large radius. In this way a whole family of regular solutions canbe found.
Now we will impose the restrictions placed by the weak energy condition andthe fact that the total mass of the system must increase when the value ofthe energy density is also increased at the origin [5]. These criteria will beexamined here for the entire family of solutions, determining those that meetthem. These conditions will be easily checked if we start from consideringthe formula for the total mass M ( e ) = (cid:90) dr π r e t ( r ) , (2.10) e t ( r ) = 12 ( u ( r ) φ (cid:48) ( r ) + φ ( r ) + 2 j ( r ) φ ( r )) + e ( r ) . As stated in [1] the value of the initial field at the origin is a function ofthe matter energy density at the origin. In other words, depending on thedensity of matter that the object in question has, it will need a specific valueof scalar field at the origin in order to show a Yukawa like behaviour of thescalar field in the faraway regions. This relationship was determined usingMathematica software version 11.1 and the results are shown in the table 1.The function φ ( e ) defined by the table is shown in the figure 1.Note how that, if the MED at the origin increases the scalar field needful toobtain a plausible solution increases too. Doing a fit for the set of points in(table1[1-2]) we obtain that the functional relation between scalar field andthe MED can be expressed as φ [ e ] = 1 . √ e + 0 . − . . ϕ o ( e o ) Figure 1: Relation between matter energy density and the scalar field at theorigin ( φ ) needful for obtaining plausible solutions. The relation was foundfixing g = 0 . n = 0 . Firstly, after solving the (2.5a)-(2.5d) system of equations and fixing the val-ues of the MED at the origin we find the value of the scalar field at the origin,such that the field behaves like a Yukawa potential for large distances. Af-terwards, by integrating the total energy from r = δ to r = 10, at which thetotal energy density is approximately close to zero, we find the total mass vsMED curve as shown in figure 2.Note that the total mass of the system initially increases with the corre-sponding increase in the MED at the origin, until it reaches a maximum of17 . .
3. From this point it begins to decrease until it reachesa minimum of 16 . e = 7 , where it starts to increase again, this timewith a lower slope. Then, the slope remains positive for the intervals e < . e >
7. Let us analyze the energetic conditions in such ranges. Calcu-lating the total energy density e t for few values of e , we obtained the plotsshown in the figure 3. As it can be observed, for certain values of the en-ergy density of matter at the origin, the total energy begins to have negative5able 1: The scalar field and total mass values corresponding to the fixedmatter energy density at the origin. e φ M | r =10 . The value of the total energy density at which the curves begin tohave a negative region was determined graphically to be e = 8 .
15 in figure4. Also, there are regions where, despite the fact that the derivative of thetotal mass with respect to the energy is positive, the physical system is notmeeting the weak energy condition. Hence, for the system (2.5a)-(2.5d) thestability regions e < . < e < .
15 result, which are shown as theshaded areas in figure 2. For a better illustration, the graphs were only taken for certain values of e . .30.50.80.9 11.1 1.3 22.38 4 5 6 7 7.5 8 8.59 10 M ( e o ) r = Figure 2: Plot of the total mass vs matter energy density at the origin forthe values g = 0 . n = . e values over 8 .
15 the system isunstable because the total energy density starts to have a negative region. e = e = e = e =
50 1 2 3 4 5 - - e t ( r ) r Figure 3: Behavior of the total energy density with the radial distance fromthe origin for different values of e . 7 = e = e = - - - e t ( r ) r Figure 4: Behavior of the total energy density with the radial distance fromthe origin around the value e = 8 . We study the stability of the static solutions of the EKG equations for a realscalar field with spherical symmetry and including matter. This analysis wascarried out under two simple criteria: The weak energy condition and theneed for growth of the mass with the increase in the energy density at theorigin [5]. Under these criteria and for a specific configuration described indetail in [1], two stability zones were determined, as well as the relationshipbetween the scalar field and the matter energy density at the origin. Theanalysis will be extended for other values of the pressure and energy rela-tionships as well as for the coupling between the scalar field and matter. Inparticular, the plans also include studying polytropic relationships.
Acknowledgments
The authors acknowledge the partial support of the Office of External Activi-ties of ICTP (OEA), through the Network on Quantum Mechanics, Particlesand Fields (Net-09). 8 eferenceseferences