General relativistic polytropes with a repulsive
Spherically symmetric equilibrium configurations of perfect fluid obeying a polytropic equation of state are studied in spacetimes with a repulsive cosmological constant. The configurations are specified in terms of three parameters—the polytropic index , the ratio of central pressure and central energy density of matter , and the ratio of energy density of vacuum and central density of matter . The static equilibrium configurations are determined by two coupled first-order nonlinear differential equations that are solved by numerical methods with the exception of polytropes with corresponding to the configurations with uniform distribution of energy density, when the solution is given in terms of elementary functions. The geometry of the polytropes is conveniently represented by embedding diagrams of both the ordinary space geometry and the optical reference geometry reflecting some dynamical properties of the geodesic motion. The polytropes are represented by radial profiles of energy density, pressure, mass, and metric coefficients. For all tested values of , the static equilibrium configurations with fixed parameters , , are allowed only up to a critical value of the cosmological parameter . In the case of , the critical value tends to zero for special values of . The gravitational potential energy and the binding energy of the polytropes are determined and studied by numerical methods. We discuss in detail the polytropes with extension comparable to those of the dark matter halos related to galaxies, i.e., with extension and mass . For such largely extended polytropes the cosmological parameter relating the vacuum energy to the central density has to be larger than . We demonstrate that extension of the static general relativistic polytropic configurations cannot exceed the so called static radius related to their external spacetime, supporting the idea that the static radius represents a natural limit on extension of gravitationally bound configurations in an expanding universe dominated by the vacuum energy.
Also at ]Research Centre of Theoretical Physics and Astrophysics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, CZ-746 01 Opava, Czech Republic Also at ]Research Centre of Computational Physica and Data Processing, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, CZ-746 01 Opava, Czech Republic
Data from cosmological observations indicate that in the framework of the inflationary paradigm Linde (1990), a very small relict repulsive cosmological constant , i.e., vacuum energy, or, generally, a dark energy demonstrating repulsive gravitational effect, has to be invoked in order to explain the dynamics of the recent Universe Krauss and Turner (1995); Ostriker and Steinhardt (1995); Krauss (1998); Bahcall et al. (1999); Caldwell et al. (1998); Armendariz-Picon et al. (2000); Wang et al. (2000). The total energy density of the Universe is very close to the critical energy density corresponding to almost flat universe predicted by the inflationary scenario Spergel et al. (2007). Observations of distant Ia-type supernova explosions indicate that starting at the cosmological redshift expansion of the Universe is accelerated Riess et al. (2004). The cosmological tests demonstrate convincingly that the dark energy represents about of the energy content of the observable universe Spergel et al. (2007); Caldwell and Kamionkowski (2009). These results are confirmed by recent measurements of cosmic microwave background anisotropies obtained by the space satellite observatory PLANCK Adami et al. (2013); Planck Collaboration et al. (2014).
There are strong indications that the dark energy equation of state is very close to those corresponding to the vacuum energy, i.e., to the repulsive cosmological constant Caldwell and Kamionkowski (2009). Therefore, it is important to study the cosmological and astrophysical consequences of the effect of the observed cosmological constant implied by the cosmological tests to be , and the related vacuum energy that is comparable to the critical density of the universe. The presence of a repulsive cosmological constant changes dramatically the asymptotic structure of black-hole, naked singularity, or any compact-body backgrounds as such backgrounds become asymptotically de Sitter spacetimes, not flat spacetimes. In such spacetimes, an event horizon (cosmological horizon) always exists, behind which the geometry is dynamic.
The repulsive cosmological constant was discussed mainly in the scope of the cosmological models Misner et al. (1973). Its role in the vacuola models of mass concentrations immersed in the expanding universe has been considered in Stuchlík (1983, 1984); Uzan et al. (2011); Grenon and Lake (2010); Fleury et al. (2013); Firouzjaee and Feghhi (2016). Recently, relevance of the repulsive cosmological constant has been found in the McVittie model McVittie (1933) of mass concentrations immersed in the expanding universe Nolan (1998); Nandra et al. (2012a, b); Kaloper et al. (2010); Lake and Abdelqader (2011); da Silva et al. (2013); Nolan (2014). Significant role of the repulsive cosmological constant has been demonstrated also for astrophysical situations related to active galactic nuclei and their central supermassive black holes Stuchlík (2005). The black hole spacetimes with the term are described in the spherically symmetric case by the vacuum Schwarzschild–(anti-)de Sitter (SdS) geometry Kottler (1918); Stuchlík and Hledík (1999), while the internal, uniform density SdS spacetimes are given in Stuchlík (2000); Böhmer (2004a). In axially symmetric, rotating case, the vacuum spacetime is determined by the Kerr–de Sitter (KdS) geometry Carter (1973). In the spacetimes with the repulsive cosmological term (and the related solutions of the f(R) gravity), motion of photons is treated in a series of papers Stuchlík (1990); Stuchlík and Calvani (1991); Stuchlík and Hledík (2000); Lake (2002); Bakala et al. (2007); Sereno (2008, 2008); Müller (2008); Schücker and Zaimen (2008); Villanueva et al. (2012); Zhao and Tang (2015); Zhao et al. (2016), while motion of test particles was studied in Stuchlík (1983); Stuchlík and Hledík (1999, 2002); Stuchlík and Slaný (2004); Kraniotis (2004, 2005, 2007); Cruz et al. (2005); Kagramanova et al. (2006); Aliev (2007); Chen and Wang (2008); Iorio (2009); Hackmann et al. (2010); Olivares et al. (2011); Chavanis and Harko (2012); Chauvineau and Regimbau (2012); Zou et al. (2014); Sarkar et al. (2014); González et al. (2015); Maciel et al. (2015); Kunst et al. (2015); Zakharov (2015); Arraut (2015a, b); Sporea and Borowiec (2016). Oscillatory motion of current carrying string loops in SdS and KdS spacetimes was treated in Jacobson and Sotiriou (2009); Kološ and Stuchlík (2010); Gu and Cheng (2007); Wang and Cheng (2012); Stuchlík and Kološ (2012a, b, 2014).
The cosmological constant can be relevant in both the geometrically thin Keplerian accretion disks Stuchlík (2005); Stuchlík and Hledík (1999); Stuchlík and Slaný (2004); Müller and Aschenbach (2007); Slaný and Stuchlík (2008) and geometrically thick toroidal accretion disks Stuchlík et al. (2000a); Slaný and Stuchlík (2005); Rezzolla et al. (2003); Aschenbach (2008); Stuchlík et al. (2005); Müller and Aschenbach (2007); Slaný and Stuchlík (2008); Kučáková et al. (2011); Perez et al. (2013); Chakraborty (2015) orbiting supermassive black holes in the central parts of giant galaxies, or in the recently discussed ringed accretion disks Pugliese and Stuchlík (2015, 2016). Spherically symmetric, stationary polytropic accretion in the spacetimes with the repulsive cosmological constant has been studied in Karkowski and Malec (2013); Mach and Malec (2013); Mach et al. (2013); Mach (2015); Ficek (2015).
In spherically symmetric spacetimes Keplerian and toroidal disk structures can be described with high precision by an appropriately chosen Pseudo-Newtonian potential Stuchlík and Kovář (2008); Stuchlík et al. (2009a) that appears to be useful also in studies of motion of interacting galaxies Stuchlík and Schee (2011); Schee et al. (2013); Stuchlík and Schee (2012a). It should be mentioned that the KdS geometry can be relevant also in the case of Kerr superspinars representing an alternate explanation of active galactic nuclei Gimon and Hořava (2004); Boyda et al. (2003); Gimon and Hořava (2009); Stuchlík and Schee (2012b). The superspinars breaking the black hole bound on the spin exhibit a variety of unusual physical phenomena de Felice (1974); Cunningham (1975); de Felice (1978); Stuchlík (1980); Stuchlík et al. (2011); Hioki and Maeda (2009); Stuchlík and Schee (2010, 2012c, 2013).
Besides the vacuum black-hole (naked-singularity) spacetimes, we have to study the role of a repulsive cosmological constant also in non-vacuum spacetimes representing static mass configurations. Such general relativistic non-vacuum solutions can be interesting, e.g., in connection to the cold dark matter (CDM) halos that are recently widely discussed as an explanation of hidden structure of galaxies enabling correct treatment of the motion in the external parts of galaxies Bosma (1981); Rubin (1982) and are at present assumed usually in the Newtonian approximation Binney and Tremaine (1988); Iorio (2010); Navarro et al. (1997); Stuchlík and Schee (2011); Cremaschini and Stuchlík (2013). There is a variety of candidates for the CDM Weinberg (2008), nevertheless none of these candidates is considered to be confirmed in the present state of knowledge. Therefore, it is important to test the possibility to represent such CDM halos in a relatively simple manner that enables to estimate easily the role of the cosmological constant. We shall discuss the most simple case of spherically symmetric static configurations of perfect fluid with a polytropic equation of state generalizing thus the standard discussion of Tooper Tooper (1964) by introducing the vacuum energy represented by the repulsive cosmological constant. Outside of these polytropic spheres the spacetime is described by the vacuum Schwarzschild–de Sitter geometry.
Choosing the polytropic equation of state means that details of the processes inside the polytropic spheres are not considered, and a simple power law relating the total pressure to the total energy density of matter is assumed. Such an approximation seems to be applicable in the dark matter models that assume weakly interacting particles (see, e.g., Börner (1993); Kolb and Turner (1990); Cremaschini and Stuchlík (2013)). In fact, such a simple assumption enables to obtain basic properties of the non-vacuum configurations governed by the relativistic laws. For example, the equation of state of the ultrarelativistic degenerate Fermi gas is determined by the polytropic equation with the adiabatic index corresponding to the polytropic index , while the non-relativistic degenerate Fermi gas is determined by the polytropic equation of state with , and Shapiro and Teukolsky (1983). It should be noted that a similar case of the adiabatic equation of state can be used in the case of a general ideal gas. This case was appropriately applied to describe the (test) perfect fluid toroidal configurations orbiting black holes Stuchlík et al. (2009a) and can be, in principle, applied for modeling of self-gravitating adiabatic spherically symmetric general relativistic configurations. The special case of polytropes with polytropic index corresponding to the simplest, although rather unphysical and artificial case of spheres with uniform distribution of energy density (but radius dependent distribution of pressure) can be treated as a very useful model—it can serve as a test bed for properties of general relativistic polytropes (GRP hereinafter) because its structure equations can be solved in terms of elementary functions Stuchlík (2000); Stuchlík et al. (2001); Böhmer (2004a); Nilsson and Uggla (2000a); Böhmer and Fodor (2008). For non-zero values of the polytropic index, the structure equations have to be solved by numerical methods.
The Einstein equations with a non-zero cosmological constant lead in the case of spherically symmetric, static equilibrium configurations to generalized Tolman–Oppenheimer–Volkoff (TOV) equation. By using the standard ansatz for the polytropic equation of state, the equations are transferred into dimensionless form of two coupled first-order nonlinear differential equations that are solved by numerical methods under boundary conditions requiring regularity of the solution at the center of the polytrope, and smooth matching of the internal spacetime at the surface of the polytrope to the external Schwarzschild–de Sitter spacetime characterized by the same mass parameter (and the cosmological constant) as the internal spacetime. The configurations are specified in terms of three parameters—the polytropic index , the ratio of central pressure and central energy density of matter , and the ratio of energy density of vacuum and central density of matter . By simultaneously solving the coupled equations, the structure of the polytrope is obtained; it is characterized by the profiles of the energy density, pressure, mass, and two metric coefficients () giving the geometry of the internal spacetime of the polytropic sphere. The spacetime structure can be reflected by the embedding diagrams of the ordinary space and the optical reference geometry reflecting some hidden properties of the spacetime Abramowicz (1990); Stuchlík et al. (2000b). The other relevant characteristics of the polytropes are the gravitational potential energy and the binding energy Tooper (1964).
Ii Equations of structure
In terms of the standard Schwarzschild coordinates, the line element of a spherically symmetric, static spacetime is given in the form
with just two unknown functions of the radial coordinate, and . Matter inside the configuration is assumed to be a perfect fluid with being the density of mass-energy in the rest-frame of the fluid and being the isotropic pressure. The stress-energy tensor of the perfect fluid reads
where denotes the four-velocity of the fluid. We consider here the simplest direct relation between the energy density and pressure of the fluid given by the polytropic equation of state
where is the ‘polytropic index’ assumed to be a given constant (not necessarily an integer) and is a constant that has to be determined by the thermal characteristics of a given fluid sphere, by specifying the density and pressure at the center of the polytrope. Since the density is a function of temperature for a given pressure, contains the temperature implicitly. It can be shown that is determined by the total mass, radius, and ratio. (The polytropic equation represents a limiting form of the parametric equations of state for a completely degenerate gas at zero temperature, relevant, e.g., for neutron stars. Then both and are universal physical constants Tooper (1964, 1964).)
In a static configuration, each element of the fluid must remain at rest in the static coordinate system where the spatial components of 4-velocity field , , vanish, leaving the temporal component
the only nonvanishing one. The structure of a relativistic star is determined by the Einstein field equations
and by the law of local energy-momentum conservation
It is convenient to express the equations in terms of the orthonormal tetrad components using the 4-vectors carried by the fluid elements:
Projection of orthogonal to (by the projection tensor ) gives the relevant equation
which is the equation of hydrostatic equilibrium describing the balance between the gravitational force and pressure gradient.
There are two relevant structure equations following from the Einstein equations. These are determined by the and tetrad components of the field equations (the and components give dependent equations). First we shall discuss the component:
This can be transferred into the form
The component of the field equations reads
Using Eq. (4), we obtain the relation
The component of the Einstein equations can be expressed and applied in the form
For integration of the structure equations it is convenient to introduce, following the approach of Tooper (1964), a new variable related to the density radial profile and the central density , by
with being the polytropic index. The boundary condition on reads . The pressure dependence is given by the relation
The conservation law (5) can be expressed in the form
where the parameter is given by the relation
At the edge of the configuration, , there is . Outside the mass configuration with mass parameter related to the mass of the polytrope by , the spacetime is described by the vacuum Schwarzschild–(anti-)de Sitter metric. Solving Eq. (6) and using the boundary condition that the internal and external metric coefficients are smoothly matched at , we obtain
Thus, the internal metric coefficient is determined by the function and the parameter . The function remains to be expressed in terms of , and we need to find the function using the structure equations. First, we rewrite Eq. (6) in the form
Then we can express the component of the Einstein equations and Eq. (5) in the form
Introducing factor giving a characteristic length scale of the polytrope
and factor giving a characteristic mass scale of the polytrope
Eqs (7) can be transformed into dimensionless form by introducing a dimensionless radial coordinate
and dimensionless quantities
where represents a dimensionless mass parameter and represents a dimensionless cosmological constant related to the polytrope. The vacuum energy density is related to the cosmological constant by
The dimensionless form of Eqs (7) determining the polytrope structure then can be written down as
The boundary of the fluid sphere () is represented by the first zero point of , say at :
The solution determines the surface radius of the polytrope and the solution determines its gravitational mass.
In the Newtonian limit (), the structure equations can be transformed to one differential equation of the second order
that is reduced to the Lane–Emden equation, if the cosmological term vanishes ()
The differential equations governing the structure of GRPs have to be solved by numerical methods (even in the Newtonian limit). Only polytropes with the polytropic index , corresponding to configurations having uniform distribution of the energy density but non-uniform pressure profile, allow for solutions of the differential equations in terms of elementary functions.
Iii Properties of the polytropes
The general relativistic polytropic spheres with given polytropic index are determined by the functions and of the dimensionless coordinate and by the length and mass scales, (8) and (9). The functions and are governed by the structure equations, the values of the central energy density , and the parameters and . A concrete polytropic sphere is then given by the first (lowest) solution of the equation that determines all the characteristics of the polytropic configuration and the radial profiles of its energy density, pressure, metric coefficients, or gravitational and binding energy.
Assuming , , , and are given, then mass , radius , and the internal structure of the polytropes can be easily determined. First, the length scale given by Eq. (8) has to be found. By numerical integration of Eqs (11), functions and are found and , where , is determined together with . The radius of the sphere is
and the mass of the sphere is given by
The density, pressure, and mass-distribution profiles are determined by the relations
The temporal and radial metric coefficients can be expressed in the form
(see also (12)).
One of the basic characteristics of the polytropes is the mass-radius (-) relation. Using Eq. (10), we obtain
and the - relation can be expressed by the formula
is the gravitational radius of the polytropic configuration determined by its total gravitational mass . The quantity determines compactness of the sphere, i.e., effectiveness of the gravitational binding, and it can be represented by the gravitational redshift of radiation emitted from the surface of the polytropic sphere Hladík and Stuchlík (2011).
The external vacuum Schwarzschild–de Sitter spacetime, with the same mass parameter and the cosmological constant as those characterizing the internal spacetime of the polytrope, has the metric coefficients
There are two pseudosingularities of the external vacuum geometry that give two length scales related to the polytropic spheres. The first one is determined by the radius of the black hole horizon
and the second one is given by the cosmological horizon
In astrophysically realistic situations, even for the most massive black holes in the central part of giant galaxies, such as the one observed in the quasar TON 618 with the mass Ziolkowski (2008), or for whole giant galaxies containing an extended CDM halo and having mass up to , the black hole horizon and the cosmological horizon radii are given with very high precision by the simplified formulae
The horizons (black-hole and cosmological) thus give two characteristic length scales of the SdS spacetimes. Clearly, the radius corresponding to the black hole horizon is located inside the polytropic spheres, while the cosmological horizon is located outside the polytropic sphere, usually at extremely large distance from the polytrope for the observationally given value of the relict cosmological constant.
The Schwarzschild–de Sitter geometry can be characterized by a dimensionless parameter Stuchlík and Hledík (1999)
Considering the observationally given repulsive cosmological constant , the cosmological parameter takes extremely small values for astrophysically relevant objects such as the stellar mass black holes and galactic center black holes, and even for the largest compact objects of the universe, i.e., the central supermassive black holes in the active galactic nuclei or for the related giant galaxies Stuchlík et al. (2000a); Stuchlík (2005). However, we can introduce a third characteristic length scale determining the boundary of the gravitationally bound system, where cosmic repulsive effects start to be decisive. This is the so called static radius Stuchlík and Hledík (1999); Stuchlík and Slaný (2004); Stuchlík and Schee (2011, 2012a); Arraut (2013, 2014) defined as
At the static radius, the gravitational attraction of the central mass source is just balanced by the cosmic repulsion and behind the static radius the cosmic repulsive acceleration prevails Stuchlík (2005).
It is relevant and instructive to relate the three characteristic length scales of the external vacuum spacetime to the length scale of the general relativistic polytrope and its radius . In the case of polytropes with very large central density, related to the central densities of neutron stars, quark stars, or other very compact objects, the polytrope length scale is comparable to the scale of the black hole horizon, while with decreasing central density the polytrope length scale increases in comparison to the black hole horizon scale. In the case of extremely low central densities related to extremely extended polytropes that could represent, e.g., the CDM halos, their length scale is comparable to the static radius of the external spacetime. We shall see that the static radius cannot be exceeded by the polytrope extension. For observationally given cosmological constant, the length scale (extension) of all astrophysically relevant polytropes is much lower than the length scale of the cosmological horizon.
Iv Gravitational energy and binding energy of the polytropic spheres
Properties of the GRPs are well characterized by their gravitational potential energy and binding energy. The later reflects amount of the microscopic kinetic energy bounded in the relativistic polytropes. Both the (negative) gravitational potential energy and the binding energy are related to the total energy given by the mass parameter of the polytropes, and are expressed in terms of the parameters characterizing the polytropes that can be determined numerically. In the case of the polytropes, the binding energy must be just negatively valued gravitational potential energy, because the polytropic configurations with uniform distribution of energy density have to be considered as incompressible.
iv.1 Gravitational potential energy
Due to the equivalence of matter and energy, the total energy of the mass configuration, including the internal energy and gravitational potential energy, is given by the gravitational mass generating the external gravitational field:
The proper energy is defined as the integral of the energy density over the proper volume of the fluid sphere
with given by (12). The gravitational potential energy is thus given by
Since , there is and —the gravitational potential energy is always negative. Following the basical work of Tooper Tooper (1964), we can consider the negatively valued gravitational potential energy, , as the gravitational binding energy, i.e., the energy representing the work that has to be applied to the system in order to disperse the matter against the gravitational forces. The intensity of the gravitational binding of the polytropic spheres can be represented by the ratio
The proper energy of a relativistic polytrope consists of the rest energy of gas, the kinetic energy of microscopic motion of the gas, and the radiation energy. The simple polytropic law relates the total energy density and the total pressure which consists of gas pressure related to the kinetic energy of the microscopic motion, and the radiation pressure. Therefore, we have to determine the gas density of the polytropic matter.
iv.2 Adiabatic processes and speed of sound
In the relativistic polytropes, the special case of adiabatic processes implies a unique relation between the gas density and the total mass density , or between and Tooper (1964). The assumption of an adiabatic process is consistent with the absence of heat terms in the energy-momentum tensor. For an adiabatic process the relativistic generalization of the first law of thermodynamics takes the form
where is the change in the energy density due to a change in the specific volume. Since
we arrive at
and using the variable , we find equation
Because the internal energy density is small being compared to the rest energy density near the boundary of the polytropic sphere, we obtain the profile of the rest mass density in the form
In the nonrelativistic limit (), the gas density and the total density are nearly equal.
The standard relativistic (Landau–Lifshitz) formula for the phase velocity of sound in an adiabatic process Landau and Lifshitz (1987)
yields the phase sound speed at the center of the polytrope to be given by
For a given there is a maximum value of the parameter that guarantees :
For the nonrelativistic Fermi gas we have , while for the ultrarelativistic Fermi gas we have ; for the case of there is . However, these limits hold for the phase sound velocity, not the group velocity, so they should not be taken too literally Tooper (1964).
iv.3 Binding energy
The proper mass and the total rest energy of gas in a polytropic sphere are determined by the relation
The energy represents the sum of the rest masses of the elementary particles in the polytrope in units of energy, and gives number of nucleons in the polytrope multiplied by the nucleon rest mass. The proper rest energy of the gas in the polytropic configuration is given by integration over the proper volume and is determined by the relation
with given by (12). The binding energy of the gas of the polytropic sphere is then given by the formula
Considering an ‘initial’ state where the particles are widely dispersed and the system has zero internal energy, and assuming conservation of the number of nucleons, the binding energy represents the difference in energy between the ‘initial’ state and the ‘final’ state in which the particles with given internal energy are bounded by gravitational forces.
We can consider the quantity giving the difference of the proper energy and the proper rest energy , describing the internal ‘kinetic’ energy of the polytropic sphere (more precisely of particles constituting the polytrope)
Polytropic fluid spheres can be characterized by relating the gravitational potential energy, the binding energy, and the kinetic energy, to the total energy, introducing the following parameters. The internal energy parameter
binding energy parameter
and the kinetic energy parameter
Clearly, the parameters are not independent. They are related by
It is not apparent, if the binding energy is positive, or negative. The gas density is smaller than the total density , but the radial metric coefficient is in general greater than unity. Recall that in the Newtonian limit (with ), we obtain in the first approximation
and the binding energy is determined by the well known formula Shapiro and Teukolsky (1983)
The Newtonian limit demonstrates immediately that the binding energy can be positive or negative, in dependence on the polytropic index . Since the gravitational energy is always negative, we can conclude that in this limit the binding energy is positive (negative) for (). In the fully GRPs, the situations is clearly more complex. The fully general relativistic polytropic spheres are characterized by the most important quantity relating the binding energy and the gravitational potential energy through the formula
that enables to find easily the regions of positively valued binding energy since the gravitational energy is again always negative.
V Embeddings of the ordinary and optical space
We concentrate our attention into visualization of the structure of the internal spacetime of the GRPs, considering both the ordinary and the optical geometry of the spacetime.
The curvature of the internal spacetime of the polytropes can conveniently be represented by the standard embedding of 2D, appropriately chosen, spacelike surfaces of the ordinary 3-space of the geometry (here, these are sections of the central planes) into 3D Euclidean space Misner et al. (1973).
The 3D optical reference geometry Abramowicz et al. (1988), associated with the spacetime under consideration, enables introduction of a natural ‘Newtonian’ concept of gravitational and inertial forces and reflects some hidden properties of the test particle motion Abramowicz (1990); Abramowicz et al. (1993); Stuchlík et al. (2000b); Kovář and Stuchlík (2007). (In accord with the spirit of general relativity, alternative approaches to the concept of inertial forces are possible, e.g., the ‘special relativistic’ one Semerák (1995).) Properties of the inertial forces can be reflected by the embedding diagrams of appropriately chosen 2D sections of the optical geometry, as reviewed, e.g., in Hledík (2002); Stuchlík et al. (2000b). The embedding diagrams of the polytropes with the uniform distribution of the energy density were presented in Stuchlík et al. (2001), here they are constructed for typical GRPs with . Note that it can be directly shown by using the optical reference geometry that extremely compact configurations, allowing the existence of bound null geodesics, can exist Stuchlík et al. (2001, 2012). Such extremely compact relativistic polytropes have a turning point of the embedding diagram of the optical geometry as shown in Stuchlík et al. (2000b). However, as we show later, such configurations can have compactness parameter .
We embed the equatorial plane of the ordinary space geometry and optical reference geometry into the 3D Euclidean space with the line element
The embedding is a rotationally symmetric surface with the line element (2D):
v.1 Ordinary space
Its equatorial plane has the line element
with being the solution of the TOV for the GRP. We have to identify and . Clearly, , and . The embedding formula then takes the form
different signs give isometric surfaces. We take ‘+’ sign. Using Eq. (17), we arrive at the dimensionless embedding formula, if we introduce
in the form
This must be integrated numerically using computer code for . Clearly, the embedding is well defined in whole the range of allowed , as there.
v.2 Optical space (optical reference geometry)
Its equatorial plane has the line element
that has to be identified with . Now, the azimuthal coordinates still can be identified (), however the radial coordinates are related via
and the embedding formula is given by
It is convenient to cast the embedding formula into a parametric form . Then
the turning points of the embedding diagrams are given by the condition
The reality condition, determining the limits of embeddability, reads
For the GRPs, the metric coefficients of the optical geometry are given by the formulae
Introducing a dimensionless coordinate by
we can write
The condition for the turning points of the embedding diagrams thus reads
The embedding formula takes the form
This has to be solved numerically, together with the condition on the limits of embeddability given in the form
Vi Configurations of uniform density
There is a special class of GRPs of the index where the structure equations can be integrated in terms of elementary functions. We shall discuss these polytropes in detail because they can give an intuitive insight into the role of the cosmological constant and can serve as a test bed for the general case of polytropes with .
The polytropes correspond to the special class of the internal Schwarzschild–(anti-)de Sitter spacetimes Stuchlík (2000) where the distribution of density is uniform although the pressure grows monotonically from its zero value on the surface of the configuration to a maximum value at its center. Recall that in the configurations with , it is not necessary to use the unrealistic notion of an incompressible fluid—one can consider fluids with pressure growing as radius decreases, being ‘hand tailored’ Misner et al. (1973). Assuming , Eq. (11b) can be integrated to give
while the Eq. (11a) takes the form
and can be integrated directly after separation of variables. Using the boundary condition , we obtain
This solution determines the dependence of pressure on the radial coordinate, since for there is . The dependence is given in units of the energy density since . From the condition , we find the radius of the configuration to be determined by
We illustrate behavior of the function in Fig. 1. Clearly, the parameters and have to be restricted by the condition
However, the polytropic configurations should behave regularly for all allowed values of the relativistic parameter , but always diverges for low enough, if . Therefore, it is natural to put the restriction of
We can express the pressure profile in terms of , , instead of , and , obtaining thus the form of expression of the polytrope as being discussed in Stuchlík (2000). Introducing a new parameter , having dimension of length, by the relation
we find the relation of , , and to be given by
The presented results for the polytropes are relevant for both the positive and negative values of the cosmological parameter describing thus also effects of the attractive cosmological constant when .
The radial metric coefficient is given by the relation
and the total mass reads
The temporal metric coefficient is determined by the relations
The special case of the attractive cosmological constant corresponding to has to be treated separately as . In such a case, Eq. (22) reduces to
which leads, after integration with the boundary condition , to the formula
The boundary of the configuration is at
The central pressure and the pressure profile can be expressed in terms of the radial coordinates , in the form
The metric coefficients have a special form, too. The radial component corresponds to the flat sections