The minimum mass of a spherically symmetric object in Ddimensions, and its implications for the mass hierarchy problem
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Abstract
The existence of both a minimum mass and a minimum density in nature, in the presence of a positive cosmological constant, is one of the most intriguing results in classical general relativity. These results follow rigorously from the Buchdahl inequalities in fourdimensional de Sitter space. In this work, we obtain the generalized Buchdahl inequalities in arbitrary space–time dimensions with \(\Lambda \ne 0\) and consider both the de Sitter and the antide Sitter cases. The dependence on D, the number of space–time dimensions, of the minimum and maximum masses for stable spherical objects is explicitly obtained. The analysis is then extended to the case of dark energy satisfying an arbitrary linear barotropic equation of state. The Jeans instability of barotropic dark energy is also investigated, for arbitrary D, in the framework of a simple Newtonian model with and without viscous dissipation, and we determine the dispersion relation describing the dark energy–matter condensation process, along with estimates of the corresponding Jeans mass (and radius). Finally, the quantum mechanical implications of the mass limits are investigated, and we show that the existence of a minimum mass scale naturally leads to a model in which dark energy is composed of a ‘sea’ of quantum particles, each with an effective mass proportional to \(\Lambda ^{1/4}\).
Keywords
Black Hole Dark Energy Cosmological Constant Minimum Mass Compton Wavelength1 Introduction
The problem of the maximum mass–radius ratio of a stable compact object is one of the most fundamental problems in both general relativity and theoretical astrophysics. In a classic paper, Buchdahl [1] obtained the famous result that the ratio of the total mass M and radius R of a high density stable star cannot exceed the value 4/9, \({GM}/c^2R<4/9\). The two basic physical assumptions used in the derivation of the upper bound for the mass–radius ratio are that the energy density in the star does not increase outwards and that the pressure is isotropic. On the other hand, by applying the principle of causality and Le Châtelier’s principle, in [2] it was shown through numerical integration of the general relativistic hydrostatic equilibrium equation [the Tolman–Oppenheimer–Volkoff (TOV) equation] that the maximum mass of the equilibrium configuration of a dense star cannot exceed \(3.2M_{\odot }\), where \(M_{\odot } \approx 1.981 \times 10^{33}\) g is the solar mass. This numerical value is presently adopted in the astrophysical literature as indicating the mass limit separating black holes from stable stellar type configurations.
Due to its major astrophysical and theoretical importance, the Buchdahl limit has been extensively investigated. The effects of the presence of a cosmological constant on the stellar mass–radius ratio were considered in [3], while limits on M / R for charged spheres were derived in [4]. In [5], it was argued that some of the assumptions used to derive the Buchdahl inequality were very restrictive. For example, neither of them hold for a simple soap bubble. By relaxing these assumptions and considering any static solution of the spherically symmetric Einstein equations for which the energy density \(\rho \ge 0\) and the radial and tangential pressures, \(p\ge 0\) and \(p_T\), satisfy the condition \(p+2p_T \le \Omega \rho c^2 \), \(\Omega >0\), one can obtain the relation \(\sup _{r>0}[2GM(r)/c^2r]\le [(1+2\Omega )^21]/(1+2\Omega )^2\) [5]. These bounds were generalized to the case of charged compact general relativistic objects in [6]. Bounds on M / R for static objects with a positive cosmological constant \(\Lambda >0\) were obtained in [7], where it was shown that the relation \({GM}/c^2R\le 2/9\Lambda R^2/3+(2/9) \sqrt{1+3\Lambda R^2}\) holds if the energy conditions listed above are satisfied. Buchdahl type inequalities, expressed in terms of the mean fluid density of the sphere, in space–times with arbitrary D and \(\Lambda \ne 0\) were also derived in [8, 9, 10], while the case of stable stars in fivedimensional Gauss–Bonnet gravity was considered in [11]. In [8] it was shown that in D dimensions the Buchdahl inequality for the maximum mass–radius ratio can be formulated as \({GM}/R^{D3}\le 2(D2)/(D1)^2\). The standard assumptions used in deriving the Buchdahl inequality were relaxed in [10], where various matter property depending bounds were obtained.
In Sect. 5, we also consider a new interpretation of the bound on \(l_\mathrm{Pl}^4/\Lambda \), in terms of the Chandrasekhar mass for a condensate of quantum dark energy particles. An intriguing possibility is that dark energy, obeying an equation of state of the form \(\rho _\mathrm{DE}c^2+p_\mathrm{DE}=0\), may condense to form stable selfgravitating objects. This scenario was originally considered in [18, 19, 20, 21, 22, 23, 24]. Dark energy stars or, in a wider sense, objects with negative pressure in their interiors, are interesting alternatives to the standard black hole paradigm. In one implementation of this idea, hypothetical compact general relativistic objects called gravastars (gravitational vacuum stars) have been proposed as an alternative explanation for the astrophysical characteristics usually associated with black holes [25, 26, 27, 28, 29, 30, 31, 32]. The basic physical idea of this scenario is that the quantum vacuum undergoes a phase transition at the moment the event horizon is formed. Therefore, the structure of a gravastar consists of an interior de Sitter condensate obeying the dark energy equation of state, \(\rho c^2=p\). This interior is matched to an exterior consisting of a shell of finite thickness described by the equation of state \(\rho c^2=p\), and the shell is then matched at its vacuum boundary to an exterior Schwarzschild solution.
In this paper, we consider the Buchdahl limit, and the resulting minimum mass and density, for static spherically symmetric compact objects in an arbitrary Ddimensional geometry. From the Einstein field equations in arbitrary dimensions with \(\Lambda \ne 0\) and the hydrostatic equilibrium equation, the generalizations of the Buchdahl limit for arbitrary D, and of the minimum mass allowed for any classical elementary particle, are obtained. In the particular case of \(D=4\), these limits reduce to the corresponding expressions given in [1, 12], respectively. On the other hand, cosmological observations such as those of high redshift supernovae or the cosmic microwave background (CMB) data from the Planck mission [33, 34, 35, 36, 37] suggest that the dark anergy equation of state is linear, with the state parameter lying in the range \( 1 < w =p_\mathrm{DE}/\rho _\mathrm{DE} < 1/3\), where \(p_\mathrm{DE}\) and \(\rho _\mathrm{DE}\) are the thermodynamic pressure and the dark energy density, respectively [38]. Therefore, the possibility that dark energy is not exactly a cosmological constant cannot be rejected a priori. Taking into account this possibility, we obtain the Buchdahl and minimum mass limits in arbitrary space–time dimensions for dark energy obeying a linear barotropic equation of state. The conditions for the collapse of a star embedded in the dark energy fluid are also derived in both the de Sitter and antide Sitter cases.
In addition, an interesting physical possibility is that dark energy may undergo a phase transition leading to a condensation process, which could be either gravitational or of Bose–Einstein type. In this scenario, the condensation of dark energy ‘particles’ could produce compact supermassive objects. We study the condensation process using the classical method of Jeans instability [39], generalized to arbitrary space–time dimensions, and by taking into account the possibility of the presence of viscous dissipative effects in the dark energy fluid. The role of the dissipative processes in the occurrence of the Jeans instability was studied in [40, 41, 42, 43, 44, 45]. As a first step we derive the mass of the dark energy condensate in the framework of the Newtonian dissipationless approximation by assuming that the dark energy fluid condenses, or transforms via a phase transition, into an ideal nonrelativistic fluid. In four dimensions, the corresponding Jeans mass is proportional to \(\Lambda ^{1/2}\) and its numerical value is close to the observationally estimated total mass of the Universe. We also show that the Jeans mass can be represented in a form similar to the Chandrasekhar mass that depends only on the fundamental constants. The effect of the bulk viscosity of the dark energy fluid on the condensation process is also investigated.
Although a selfconsistent and credible theory of quantum gravity has not been found, we use general arguments to investigate the possible quantum mechanical implications of the existence of a minimum mass and minimum density. Firstly, we show that, in four dimensions, the Jeans mass of the dark energy \(M_J\propto (c^2/G)\Lambda ^{1/2}\), having a numerical value of the order of the mass of the Universe, can be obtained from thermodynamic considerations related to the physics of Schwarzschild–de Sitter black holes. We then associate the minimum mass with a temperature which, intriguingly, is very close to the present day temperature of the CMB radiation. Moreover, we show that the existence of a minimum mass bound leads naturally to a hierarchical model in which quantum ‘particles’ above a certain size effectively decohere via interaction with the cosmological constant (or dark energy fluid) and in which dark energy itself is composed of quantum particles with an effective mass proportional to \(\Lambda ^{1/4}\).
This paper is organized as follows. In Sect. 2, the Ddimensional Einstein field equations with a nonzero cosmological constant are introduced and the expressions for the Buchdahl limit and the classical minimum mass are obtained. The more general case of a Ddimensional sphere of matter embedded in a dark energy fluid obeying an arbitrary barotropic equation of state is considered in Sect. 3, and the corresponding limiting masses are derived. The Jeans instability of the dark energy fluid is discussed in Sect. 4, within the framework of a simple Newtonian model. Finally, the quantum mechanical implications of the existence of a classical minimal mass and density are investigated in Sect. 5 and a brief discussion of our main results is presented in Sect. 6.
2 Mass limits for spherically symmetric stars in D dimensions in the presence of a cosmological constant
In this section, we first introduce the Einstein field equations in arbitrary space–time dimensions with \(\Lambda \ne 0\) and derive the hydrostatic equilibrium equation [the Tolman–Oppenheimer–Volkoff (TOV) equation] for spherically symmetric objects. We then derive the Ddimensional generalizations of the Buchdahl limit and of the minimum mass in general relativity. In estimating critical length and mass scales throughout this paper we use the approximate values \(c \approx 2.998 \times 10^{10} \ \mathrm{cm}\,\mathrm{s}^{1}\), \(G \approx 6.674 \times 10^{8} \;\ \mathrm{cm}^{3}\;\mathrm{g}^{1}\,\mathrm{s}^{2}\), \(h \approx 2\pi \times 1.055 \times 10^{27} \ \mathrm{erg}\;\mathrm{s}\), and \(\Lambda \approx 3 \times 10^{56} \ \mathrm{cm}^{2}\) for the fundamental constants.
2.1 Tolman–Oppenheimer–Volkoff equation in Ddimensional space–time
2.2 Upper and lower bounds for the mass–radius ratio in D space–time dimensions
3 Mass limits for spherically symmetric systems in the presence of dark energy
 (1)
the de Sitter case, with \(\Lambda _{D}>0,\ w<(D2)/(D1)\);
 (2)
the antide Sitter case, with \(\Lambda _{D}<0,\ w>(D2)/(D1)\).
4 Jeans instability of the dark energy fluid in arbitrary space–time dimensions
An interesting physical possibility is that the dark energy fluid, satisfying an equation of state \(p_\mathrm{DE}=w\rho _\mathrm{DE}\) with \(w=w_0=1\), could condense gravitationally to form stellar type stable compact objects, satisfying the same equation of state as the initial medium, but with a different parameter \(w\ne 1\). The condensation process can also be described phenomenologically as the result of a viscous type dissipation process, which triggers the transition between the two fluids. Therefore, to study the dark energy condensation process in the linear Newtonian regime we need to include dissipative effects into the fluid dynamical description of the transition. Hence, we assume that the dark energy fluid has an initial density \(\rho _\mathrm{DE}^{(0)}=\Lambda _Dc^{2}/8\pi G_D\) and pressure \(p_\mathrm{DE}^{(0)}\), which satisfy the equation of state \(\rho _\mathrm{DE}^{(0)}c^{2}+p_\mathrm{DE}^{(0)}=0\). The fluid also has dissipative properties, characterized by the bulk viscosity \(\xi _0\), and the shear or first viscosity \(\eta _0\) [48], describing the internal ‘friction’ of the decaying dark energy. The possibility that dark energy may have some anisotropic stresses, which can be modeled with the help of a viscosity parameter, in addition to the standard sound speed equation of state parameters was considered in [49, 50, 51, 52, 53].
We assume that the dark energy condenses, or experiences a phase transition into a nonrelativistic fluid, which can be characterized by a density \(\rho _\mathrm{DE}\), a pressure \(p_\mathrm{DE}=w\rho _\mathrm{DE}\), a velocity \(\vec {v}\), a gravitational acceleration \(\vec {g}\), a bulk viscosity coefficient \(\xi \), and a shear viscosity coefficient \(\eta \).
4.1 Hydrodynamical and first order perturbation equations
4.2 Condensation of the ideal dark energy fluid
4.2.1 Ideal dark energy jeans mass as a Chandrasekhar mass
4.3 The effect of the bulk viscosity on dark energy condensation
5 Quantum implications of a classical minimum mass density
Nonetheless, it is interesting that this mass scale has previously been proposed as a minimum mass for stable dark matter relics, based on loop quantum gravity calculations [64]. SubPlanck mass loop black hole (LBH) solutions have been shown to exist if quantum gravity effects give rise to a quadratic generalized uncertainty principle (GUP) [65]. In this case, the usual relations between the black hole mass, its horizon and its temperature invert at \(M \sim M_P\), so that \(R_S \propto M^{1}\) and \(T_{H} \propto M\) for \(M < M_P\). Such objects behave like ‘black atoms’, but continue to decay via radiation emission until they reach thermal equilibrium with the CMB photon bath [65] (see also [66] and references therein).
Equation (96) may have profound implications, since it predicts the existence of a phenomenologically significant length scale \(R_{\Lambda }\) which, we may conjecture, demarcates the boundary between quantum mechanical and classical behavior. This may have implications for the study of gravitational decoherence. Surprisingly, despite the common interpretation of the cosmological constant as a ‘gravitational’ phenomenon, relatively little work had been done on this topic in the context of models with \(\Lambda \ne 0\) (see [67, 68, 69, 70, 71] and references therein).

\(M_{\Lambda } \le M \lesssim M_P\) correspond to elementary particles, whose behavior is manifestly quantum mechanical;

\(M_P \lesssim M \lesssim M'_{\Lambda }\) behave quantum mechanically but have the potential to form black holes (which continue to behave like quantum ‘particles’) if compressed below their Buchdahl limit;

\(M'_{\Lambda } \lesssim M \lesssim M'_W\) behave classically and have the potential to form black holes (which continue to behave like classical particles) if compressed below their Buchdahl radius.
Hence, by combining standard Compton type arguments, which imply a minimum radius for a quantum object with a given mass, with classical Buchdahl type bounds for \(\Lambda >0\), which imply a minimum mass density, we are led naturally to a picture in which dark energy is composed of a sea of quantum particles with effective mass \(M_{\Lambda } \propto \Lambda ^{1/4}\). The associated mass density \(\rho _{\Lambda }\) is given by \(M_{\Lambda }\) divided by the volume occupied by each particle due to its Compton wavelength. In this picture, the dark energy condensation picture discussed in Sect. 4 occurs due to fluctuations which lower the effective mass within a localized region, leading to local overdensities. This is equivalent to a local softening of the effective equation of state.
Since the geometric relations are transitive among \(M_{i1},M_{i},M_{i+1}\), and \(M_{i2},M_{i},M_{i+2}\) triplets, Fig. 1 also implies that \(M'_{W} \simeq M'^{2}_{\Lambda }/M_{P}\) which is straightforward to verify directly. We note that these geometric relations originate from the following three physical constraints: (1) the size of a classical object is larger than or equal to its Compton wavelength, (2) both a minimum mass and minimum density exist, in general relativity, in space–times with a positive cosmological constant, \(\Lambda > 0\), and (3) the maximum mass of a gravitationally stable classical compact object also exists in this scenario.
Remarkably, these mass scales also admit other physical interpretations. As stated above, there exist both sound theoretical reasons and empirical evidence to support the claim that \(M'_W\) should be interpreted as the maximum possible mass of a de Sitter universe. In addition, it is straightforward to verify that the Chandrasekhar mass for a condensate of particles of mass \(M_{\Lambda }\) is also equal, up to numerical coefficients of order unity, to \(M'_W\). Since we require \(M \lesssim M'_W\), this is another way of saying that our (known) universe cannot collapse and must expand forever, as implied by observations suggesting a phase of accelerated expansion beginning at the current epoch [33, 34, 35, 36, 37].
Furthermore, if we assume that dark matter particles, with mass \(M_{\Lambda }\), formed from the condensation of dark energy, this implies that the initial dark matter temperature, at the epoch of formation, is equal to the (constant) temperature of the dark energy fluid in its original phase. Even if this is not the case, by envisaging dark energy as a ‘sea’ of quantum mechanical particles, the onset of accelerated expansion then coincides with the point at which the temperature of the thermal bath of photons drops below the temperature associated with the effective mass of the dark energy particles, \(M_{\Lambda }\), for the first time. It would be interesting to further investigate this in the context of thermodynamic interpretations of gravity (see, for example, [72, 73, 74] and references therein). In particular, one may hope to obtain a ‘thermodynamic interpretation’ of the coincidence problem, posed by the onset of accelerated expansion at the present time.
We also note that the expression for the minimum radius, \(R \ge R_\mathrm{min} = (R_P^2R_W)^{1/3} \approx 10^{15}\) m, which is of the same order of magnitude as the classical electron radius \(r_e\), and which was obtained previously, by two different methods, in [15, 17], may also be obtained a third way by requiring the density of the Universe to be less than the Planck density, \((3/4\pi )M'_W/R^3 \le (3/4\pi )M_P/R_P^3 =: \rho _P\). In this context, \(R_\mathrm{min} = (R_P^2R_W)^{1/3} \approx r_e\) may be interpreted as the minimum classical radius to which the known universe may be compressed before it exceeds the Planck density.
Finally, before concluding this section, we note that, using Eq. (44), we may obtain a further generalization of Eq. (83) which is valid in arbitrary dimensions. An analysis similar to that given above should then yield results valid for any value of \(D \ge 2\). However, in this case a subtlety may arise, since it has recently been proposed in [75] that, for space–times with compact dimensions, the Compton wavelength changes on scales \(R_C < R_E\), where \(R_E\) is the length scale of the compactification. We therefore leave a full analysis of the higherdimensional case, including both spherically symmetric and nonspherically symmetric space–times, to a later publication.
6 Conclusions and discussions
The existence of dark energy, as proved by a plethora of astrophysical and cosmological observations [33, 34, 35, 36, 37], has fundamentally modified the landscape of theoretical physics. If dark energy, represented by a cosmological constant, is one of the major components of the Universe, it is a reasonable assumption to include it among the fundamental constants of nature [62]. Hence, we can extend the set of fundamental constants, which can be taken as the speed of light c, the gravitational constant G, Planck’s constant h, and the cosmological constant \(\Lambda \). Therefore, the mere existence of the cosmological constant, or, at least, of some form of dark energy, may imply drastic modifications or extensions of the basic laws of physics.
If the set of fundamental constants is enlarged, it follows that there are two different masses that can be constructed from c, G, h, and \(\Lambda \) [62]. The first Wesson mass, \(M_W=\left( h /c\right) \sqrt{\Lambda /3}\approx 10^{66}\) g, may be relevant at the quantum scale, while the second Wesson mass, \(M'_W=(c^2/G)\sqrt{3/\Lambda }\approx 10^{56}\) g, has the same order of magnitude value as the observable mass of the Universe.
From a theoretical point of view, \(M'_W\) can be obtained as the upper bound on the mass of a gravitationally stable spherically symmetric object in fourdimensional general relativity in the presence of a positive cosmological constant, \(\Lambda > 0\), and as the Jeans mass of a gravitationally unstable dark energy condensate, respectively. Alternatively, the Smarr formula for uncharged nonrotating black holes suggests that the mass of the Universe, \(M_U \approx M'_{W}\), follows from thermodynamic properties of the de Sitter horizon.
In addition, we investigated the quantum mechanical implications of the existence of both a classical minimum mass and a minimum density. By combining simple Compton type arguments for the minimum radius of a quantum mechanical object, \(R_C\), with the classical minimum radius, \(R_\mathrm{min}\) (and assuming \(R_C \ge R_\mathrm{min}\)) we obtained a new minimum mass scale, \(M_{\Lambda } \sim \sqrt{M_PM_W} \sim 10^{35}\) g, where \(M_P\) denotes the (nonreduced) Planck mass, which depends on all four ‘universal’ constants, G, c, h, and \(\Lambda \), simultaneously. Interestingly, the temperature associated with this mass scale is of the order of the present day CMB temperature. The associated length scale \(R_{\Lambda } \sim \sqrt{R_PR_W} \sim 10^{3}\) cm, where \(R_P\) denotes the (nonreduced) Planck length, represents the maximum possible Compton wavelength of a quantum mechanical object, suggesting an absolute maximum decoherence length associated with ‘gravitational’ decoherence through the interaction of the system with omnipresent dark energy.
Another interpretation of the dimensionless quantity \(R_{P}^2 \Lambda \) is the ratio between Planck area and the area of the cosmic horizon. Holographically, this is the number of quantum gravity bits present on the boundary. The fact that the total number of bits on the boundary is equal to the total number of quanta in the bulk space indicates that holography is at work in the entire universe.
Finally, we note that the existence of mass/density bounds in the asymptotically AdS case, when the dark energy satisfies the conditions \(w>(D2)/(D1), \ \Lambda _{D}<0\), has interesting implications from the viewpoint of holographic duality. The maximum mass bound for a given radius guarantees that any object with larger mass will inevitably collapse to form a black hole. Holographically, the maximum mass would correspond to the maximum temperature (identified with Hawking temperature of the maximum mass black hole) of the dual gauge matter before the inevitable deconfinement phase transition occurs [47, 76]. On the other hand, the existence of a minimum mass and a minimum density determines the conditions under which a gravitationally stable, static object can be formed in the presence of dark energy. If the average density of the object is too small, it will not be able to support itself gravitationally under the outward pressure. At present, it is unclear what the gauge theory dual of this minimum mass/density should be, since the boundary space is always asymptotically AdS regardless of the mass and size of the static star located at the center. We leave this interesting question for a future publication.
Notes
Acknowledgments
We are grateful to the anonymous referee for comments and suggestions that helped us to significantly improve our manuscript. We would like to thank Taum Wuthicharn for pointing out relation (105). P.B. and K.C. are supported in part by the Thailand Research Fund (TRF), Commission on Higher Education (CHE) and Chulalongkorn University under Grant RSA5780002. M.L. is supported by a Naresuan University Research Fund individual research grant.
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