Mixed neutron-star-plus-wormhole systems:
We study gravitationally bound, spherically symmetric equilibrium configurations consisting of ordinary (neutron-star) matter and of a phantom/ghost scalar field which provides the nontrivial topology in the system. For such mixed configurations, we show the existence of static, regular, asymptotically flat general relativistic solutions. Based on the energy approach, we discuss the stability as a function of the core density of the neutron matter for various sizes of the wormhole throat.
pacs:04.40.Dg, 04.40.–b, 97.10.Cv
The discovery of the accelerated expansion of the present Universe at the end of the 1990s started a new era in our understanding of the Universe. It became clear that besides visible and dark matter which are gravitationally clustered in galaxies and galaxy clusters there should exist, in addition, a fundamentally different form of energy in the Universe – dark energy, accounting for about three quarters of the energy content of the Universe. This dark energy is not gravitationally clustered, but distributed rather uniformly in space. Since it has a high negative pressure, which is rather unusual from the point of view of ordinary matter, it can provide for the current acceleration of the Universe.
The key property of dark energy lies in its ability to violate various energy conditions. Of greatest interest for cosmology are the following energy conditions: (i) the strong energy condition, ; (ii) the weak/null energy condition (where and are the effective energy density and the pressure of the matter fields filling the Universe). To provide for the accelerated expansion, it is sufficient to violate the strong energy condition. In turn, the failure to satisfy the weak energy condition results in an accelerated expansion of the Universe that is even faster than exponential.
If some form of matter violating these energy conditions does indeed exist in the present Universe, then it may serve not only as a mechanism for providing the current acceleration. It may as well be present in compact astrophysical objects. Indeed, despite the fact that in describing the evolution of the Universe it is assumed that dark energy is homogeneously distributed in the Universe, it is possible to imagine a situation where inhomogeneities may arise due to gravitational instabilities. This could lead to a collapse of lumps of dark energy with the subsequent creation of compact configurations – so-called dark energy stars Mazur:2004ku (); Dymnikova:2004qg (); Lobo:2005uf (); DeBenedictis:2005vp (); DeBenedictis:2008qm (); Gorini:2008zj (); Gorini:2009em (); Dzhunushaliev:2008bq (); Dzhunushaliev:2011ma (); Folomeev:2011aa (); Yazadjiev:2011sd (); Mota:2004pa (); Cai:2005ew (); Debnath:2006iz (); Lee:2009qq (); Folomeev:2011uj ().
For dark energy stars, typically at least the strong energy condition is violated. At the present time, investigations of dark energy stars are done in two main directions: (i) the construction of stationary configurations to demonstrate the existence of such objects and their potential stability Mazur:2004ku (); Dymnikova:2004qg (); Lobo:2005uf (); DeBenedictis:2005vp (); DeBenedictis:2008qm (); Gorini:2008zj (); Gorini:2009em (); Dzhunushaliev:2008bq (); Dzhunushaliev:2011ma (); Folomeev:2011aa (); Yazadjiev:2011sd (); (ii) the study of the possible formation process of such compact objects during gravitational collapse Mota:2004pa (); Cai:2005ew (); Debnath:2006iz (); Lee:2009qq (); Folomeev:2011uj ().
Although the true nature of dark energy is currently unknown, several ways have been suggested to model it. The simplest variant seems to be the assumption that dark energy is nothing else than Einstein’s cosmological constant , which arises as the energy density of the vacuum. However, the well-known “cosmological constant problem” has led to the consideration of models where the dark energy evolves in time. Perhaps one of the most developed directions here are theories with various types of scalar fields.
Scalar fields arise naturally in particle physics including string theory. Using them, a wide variety of scalar-field dark energy models have been proposed (for a review, see, e.g., sahni:2004 (); Copeland:2006wr ()). One of the tools developed for modeling dark energy are the so-called phantom/ghost fields. Their important property is that they violate the weak/null energy conditions. Based on such fields, in Ref. Caldwell:1999ew () a simple cosmological model was suggested, where the phantom dark energy is provided by a ghost scalar field with a negative kinetic term (for the further development of such models, see the review Copeland:2006wr ()).
In view of such highly unusual properties of matter for which the weak/null energy conditions are violated, localized objects containing such matter should differ strongly from ordinary stars as well. A striking example of the effects of the presence of such exotic matter is the possibility to allow for traversable wormholes – compact configurations with a nontrivial space-time topology. The term “wormhole” has appeared in the middle of the 1950s in the work of Wheeler Wheeler:1955zz (), who suggested the geometrical model of electric charge in the form of a tunnel connecting two space-time regions and filled by an electric field. The Wheeler wormhole was not traversable. But, this idea has stimulated a great deal of interest in studying models with a nontrivial space-time topology. One of the most significant achievements in this direction is the model of a traversable wormhole suggested in 1988 by Morris and Thorne Thorne:1988 () as a toy model allowing for interstellar travel. Moreover, since recent observational data indicates that matter violating the weak/null energy conditions may indeed exist in the present Universe Star1 (), traversable wormholes have received increasing attention ever since as objects which could really exist in nature.
Providing a nontrivial topology, massless ghost scalar fields were employed in the early pioneering work of Refs. Bronnikov:1973fh (); Ellis:1973yv () (see also earlier work by Bergmann and Leipnik Bergmann:1957 (), where they found a solution for a massless ghost scalar field, but not dealing with the question of the wormhole interpretation of the solution obtained). Further examples of configurations with nontrivial topology were found in Kodama:1978dw (); Kodama:1979 (), based on a ghost scalar field with a Mexican hat potential. Interestingly, it was found that this system has only regular, stable solutions for a topologically nontrivial (wormhole-like) geometry. In kuhfittig (); lxli (); ArmendarizPicon:2002km (); Sushkov:2002ef (); lobo (); sushkov () traversable Lorentzian wormholes were further investigated, refining the conditions on the type of matter fields that would lead to such space-times. A general overview on the subject of Lorentzian wormholes and violations of the various energy conditions can be found in the book of Visser Visser ().
A recent development concerning such wormhole models is based on the expectation, that such topologically nontrivial objects, being traversable, could be bound to ordinary matter satisfying all energy conditions. In arXiv:1102.4454 (), we have suggested such a model of a mixed configuration, consisting of a wormhole (supported by a massless ghost scalar field) filled by a perfect polytropic fluid. We have shown that there exist static, regular solutions describing mixed star-plus-wormhole configurations which possess new physical properties that distinguish them from ordinary stars. A preliminary stability analysis performed only for the external region of the configuration suggested that those solutions could be stable with respect to linear perturbations.
The objective of the present paper is to continue the study of the influence of a wormhole on the structure and the physical properties of compact stars made otherwise from ordinary matter. To provide a nontrivial topology in this model, we here employ a massless ghost scalar field as one of the simplest possibilities allowing for a nontrivial topology. For the ordinary matter filling the wormhole, we choose neutron matter modeled within the polytropic approximation [in the form of Eq. (13)].
The paper is organized as follows: In Sec. II, the general set of equations is derived, describing equilibrium configurations consisting of a massless ghost scalar field and ordinary matter approximated by a polytropic equation of state. Here, the boundary conditions are also discussed. In Sec. III, we discuss the physical properties of the solutions, the radii, the masses, the binding energies and the pressures. We then present the numerically obtained static solutions for two sets of polytropic parameters and consider the issue of their stability within the energy approach. Finally, in Sec. IV we summarize the results obtained and address possible observable effects which may follow from the model considered.
Ii Derivation of the equations for equilibrium configurations
ii.1 Lagrangian and general set of equations
We consider a model of a gravitating massless ghost scalar field in the presence of a perfect fluid. Our starting point is the Lagrangian
Here, is the ghost scalar field, is the Lagrangian of the perfect isotropic fluid (where isotropic means that the radial and the tangential pressure of the fluid agree) which has the form Stanuk1964 (); Stanuk (). This Lagrangian leads to the corresponding energy-momentum tensor
where and are the energy density and the pressure of the fluid, is the four-velocity. For our purposes, it is convenient to choose the static metric in the form
where and are functions of the radial coordinate only, and is the metric on the unit 2-sphere. Also, let us present the metric in Schwarzschild-type coordinates which are frequently used in modeling wormholes
This parametrization of the metric is employed later on to derive the boundary conditions at the wormhole throat at the core of the configuration.
where a “prime” denotes differentiation with respect to .
The field equation for the scalar field is obtained by varying the Lagrangian (1) with respect to ,
Using the metric (3), this equation is integrated to give
where is an integration constant.
Not all of the Einstein field equations are independent because of the law of conservation of energy and momentum . Taking the component of this equation gives
Thus, we have five unknown functions: , and . Keeping in mind that and are related by an equation of state, there are only four unknown functions. For these functions, there are four equations: the two Einstein equations (5) and (6), the scalar-field equation (8), and the equation of hydrostatic equilibrium (9). Using the energy-momentum tensor (2), the right-hand sides of the Eqs. (5) and (6) take the form
ii.2 Equation of state
To model the neutron matter filling the wormhole, it is necessary to choose an appropriate equation of state. The choice of the equation of state plays a crucial role in modeling the neutron-star structure. A large number of neutron matter equations of state have been suggested, providing different sets of densities and pressures for the neutron stars. These include a parametric dependence as in the pioneering work of Oppenheimer and Volkoff Oppen1939 (), a polynomial dependence as in Cameron1959 (), an equation of state obtained from field theoretical considerations as in DAl1985 (), or a unified equation of state (see, e.g., Ref. Haensel:2004nu () where the analytical representations of unified equations of state of neutron-star matter are given).
Here, we employ a simplified variant for the equation of state, where a more or less realistic neutron matter equation of state is approximated in the form of a polytropic equation of state. In particular, we employ the following parametric relation between the pressure and the energy density of the fluid,
where is the baryon number density, is some characteristic value of , is the baryon mass, and and are parameters whose values depend on the properties of the neutron matter.
For our purpose it is convenient to rewrite the above equation of state in the form
with the constant , the polytropic index , and denotes the rest-mass density of the neutron fluid.
Setting and , we consider below configurations with two sets of values for the parameters and :
Introducing the new variable Zeld ()
where is the density of the neutron fluid at the wormhole throat (or the center of the star in the case without a wormhole) we may rewrite the pressure and the energy density, Eq. (13), in the form
Making use of this expression, we obtain for the internal region with from Eq. (12)
where is a constant, related to the pressure of the fluid at the wormhole throat (or at the center of the star in the case without a wormhole). This equation may be integrated to give in the internal region with the metric function in terms of ,
and is the value of at the throat where . The integration constant is fixed by requiring that the space-time is asymptotically flat, i.e., at infinity.
ii.3 Internal set of equations
We first consider the set of equations in the internal region, where . Here, the system is characterized by three unknown functions: , and . These three functions are determined by the three Eqs. (5), (6), and (8), and also by the relation (17). It is convenient to rewrite these equations by introducing the new function ,
The function can be interpreted as the total mass within the areal radius . Thus, for , where denotes the outer boundary of the fluid where (for more details on this, see Sec. III), we obtain the total mass of the configuration within the boundary of the star of radius . The limit then yields the total mass of the configuration.
With this function , Eq. (5) yields
For the spherically symmetric case without a wormhole (corresponding in our case to the absence of the scalar field), we need to require the boundary condition in order to guarantee regularity at the origin Tooper:1964 (). This corresponds to the fact that there is no mass associated with the origin, . For a wormhole, on the other hand, there exists a finite minimal value of the radius, the throat , and this is associated with a finite value of the mass function at the throat, .
Now, we introduce dimensionless variables
where we have used expression (8) and introduced the dimensionless constant
ii.4 External set of equations
Let us next turn to the external part () of the solutions outside the fluid. Asymptotic flatness of the external solutions requires that the metric function tends to one at infinity. This in turn determines the value of the integration constant at the throat.
To find the external solutions, we start again from the Einstein equations (5) and (6), and the scalar-field equation (8), taking into account that in the external region there is no ordinary matter, i.e., . This leads to the following system of equations,
These can be rewritten in terms of the dimensionless variables , and as follows,
This system still contains the parameter as a trace of the influence of the fluid on the external solution due to the definition of the dimensionless quantities (20).
ii.5 Expansion at the throat
Finally, we need to consider the appropriate boundary conditions at the wormhole throat corresponding to the core of the configurations arXiv:1102.4454 ().
Let us here briefly comment on the nomenclature used. In contrast to the case of ordinary stars having a center at the point , there exists a finite minimal value of the radial coordinate corresponding to the radius of the throat in the case of the star-plus-wormhole systems considered here. Moreover, because of the presence of ordinary matter, the configurations differ also from simple wormholes. To take this into account, we employ the term “core” to describe the “throat” area of such star-plus-wormhole systems.
In terms of the dimensionless variables introduced above, denotes the coordinate at the core of the configuration. Here, the dimensionless density of the fluid is characterized by
This condition corresponds to the fact that at the wormhole throat the density of the fluid is .
Comparing the form of the metric (4) with the form of the metric (3) used for the derivation of the Eqs. (21) and (22), and taking into account expression (18), together with the dimensionless variables (20), we find
where we have introduced the dimensionless function . From this expression, taking into account the relation from (31), we obtain the boundary condition for the function at the throat
This boundary condition implies that, as , the following expression vanishes,
This in turn means that the coefficient in front of the derivative of the function in Eq. (22) goes to zero, which leads to a singularity in a generic solution.
To obtain solutions, which are regular at the throat we consider a Taylor series expansion of the functions and in the neighborhood of the point . At a point close to the throat the expansion for reads to first order
Substituting this into Eq. (21), we obtain for the following expression,
Regularity of the solutions of Eq. (22) is achieved, when we assume that, together with (34), the right-hand side of (22) goes simultaneously to zero. Proceeding from this requirement, we obtain the following expression for :
Substituting this into (36), we finally obtain
The Taylor series expansion for the function at at the point reads to first order
where we obtain for the coefficient ,
Nonsymmetric wormholes (with respect to the two asymptotically flat space-times) could also be obtained, but these would satisfy a different set of boundary conditions.
Thus, a static equilibrium solution is obtained as follows: We start the numerical integration at the point , solving numerically the system of equations (21) and (22) subject to the boundary conditions (35) with (33) and (38) for , and (39) with (40) for . We then proceed with the integration until we reach the point , where the function becomes zero arXiv:1102.4454 (). Here, the energy density associated with the fluid vanishes. The surface bounded by the radius represents the boundary of the neutron matter. Note, that the energy density associated with the scalar field is, in general, still finite at this boundary.
Now, the external part of the solution is sought, starting from the surface of the fluid at with the boundary conditions and as determined from the internal part of the solution. Requiring asymptotical flatness of the space-time finally allows to determine the value of the integration constant from (17) by requiring to be equal to unity at infinity. (The values of for the examples shown in Fig. 1 are given in the caption.) Thus, the complete solution for the configuration under consideration is derived by matching of the internal fluid solutions given by Eqs. (21)-(23) with the external solutions obtained from the system (27)-(29).
Iii Static solutions
In this section, we discuss the numerical solutions of the above sets of internal and external equations and their physical properties.
The radial coordinate describes the areal radius of a sphere with area . The throat radius corresponds to the areal radius , and the neutron matter of the star is contained within the areal radius (denoted in the tables). The gravitational radius of the system corresponds to the areal radius , where is the total mass. In dimensionless coordinates, the areal radius is given in terms of the coordinate . Thus, denotes the throat radius and the radius of the neutron fluid.
Another physically relevant radial coordinate is given by the coordinate associated with the proper radius, which gives the distance from the throat. It is defined as follows,
Then, the proper radius of the fluid is obtained in dimensional variables as .
iii.2 Mass contributions
The system under consideration consists of two parts: the internal region and the external region . Correspondingly, the energy density of the system is given by the internal energy density, obtained from the expressions (10), (13), (14), (17), and (20)
and the external energy density, ,
The total energy density of the configuration is
with , where the quantity
fixes the scale of the mass.
The asymptotic value corresponds to the total mass of the configuration, while in dimensionless units . For later reference, we now subdivide this expression for the total mass into four dimensionless components according to
with the mass at the throat
the mass of the fluid
the internal part of the mass of the scalar field
and the external part of the mass of the scalar field
iii.3 Binding energy
We start from the total energy of the system given by
In order to derive a physically motivated expression for the binding energy of the system, let us consider the relativistic continuity equation
with the baryon number density of the neutron fluid and the four-velocity . It follows that the neutron particle number is given by
as discussed, e.g., in Ref. Zeld (). An analogous definition holds for a boson star composed of massive bosons. Indeed, in order to disperse the particles to infinity, one has to supply precisely this amount of energy to the system.
For the combined star-plus-wormhole system, on the other hand, we still have to consider how to deal with the massless ghost scalar field that supports the wormhole. Thus, let us first address an isolated wormhole made from a massless ghost scalar field without any neutron matter. With the above set of boundary conditions, such a wormhole has zero energy (respectively, zero mass) Visser (). This agrees with the energy of flat space in the absence of a wormhole.
Let us therefore consider the binding energy of the star-plus-wormhole system as the difference between the following contributions: The energy of free particles dispersed to infinity with a (therefore) vanishing ghost scalar field and the energy of the combined star-plus-wormhole system. Alternatively, we could consider the difference between the energy of free particles dispersed to infinity together with the energy of an isolated wormhole and the energy of the combined star-plus-wormhole system. Both cases yield the same result, namely the above expression (51), obtained without a wormhole.
|Without a wormhole|
|Without a wormhole|