General theory of relativity predicts the central densities of massive neutron stars (-MANs) to be much larger than the nuclear density. In the absence of energy production, the lifetimes of MANs should be shorter that their low-mass counterparts. Yet neither black holes nor neutron stars, whose masses are between two and five solar masses have ever been observed. Also, it is not clear what happened to the old MANs that were created through the collapse of first generation of stars shortly after the Big Bang. In this article, it is argued that MANs must end as completely invisible objects, whose cores are made of incompressible quark-gluon-superfluids and that their effective masses must have doubled through the injection of dark energy by a universal scalar field at the background of supranuclear density. It turns out that recent glitch observations of pulsars and young neutron star systems and data from particle collisions at the LHC and RHIC are in line with the present scenario.
Unlike luminous stars, whose energies are generated through nuclear fusion, neutron stars emit the rest energy stored in their interiors from old evolutionary epochs. As in the case of luminous normal stars, the total energy emitted by neutron stars is proportional to their masses, implying therefore that massive neutron stars must also be short-living objects.
Similar to the natural selection scenario of primates, most massive astrophysical objects must have disappeared relatively quickly, but only solar-like objects are able to shine for billions of years and to be observable until the present universe: thanks to the parameters characterizing our universe.
Just for illustration: a ten solar masses star has a lifetime 1000 times shorter than that of the Sun. On the other hand, cosmological simulations reveal that the first stars must have been 100 to 10,000 solar masses and that they should have formed from primordial clouds made solely of hydrogen [
However, an evolutionary track, in which the first stars or at least a part of them, may have collapsed to form pulsars and/or neutron stars cannot be statistically excluded. Moreover, if the parameters characterizing our universe do not indeed allow matter-density to grow indefinitely [
Indeed, the following list of arguments are only a few in favor of this scenario:
• Relativistic compact objects with 2 M ⊙ < M < 6 M ⊙ practically do not exist.
• As illustrated in
• The mass range of black holes is practically unlimited as neither lower nor upper bounds are known, whereas NSs enjoy an unusually narrow mass range.
• Isolated neutron stars that are older than one Gyr haven’t been observed yet.
• Modelling the internal structure of NSs requires their central densities to be far beyond the nuclear density: an unknown density regime in which most EOSs become physically inconsistent [
• All EOSs break down when nuclear fluid becomes weakly compressible.
• The glitch phenomena observed in NSs and pulsars (see e.g.
As a consequence, we expect isolated massive NSs to metamorphose into dark objects, whose interior are made of incompressible gluon-quark superfluids and to subsequently disappear from our observation windows.
Normal matter is usually made of self-interacting particles, non-ideal and therefore of dissipative medium. The exemplify these concepts, consider the
flowing water in a river. Particles at the surface communicate with the motionless ones at the ground, generating thereby a velocity profile that varies with the depth, i.e. normal to the direction of motion. If we were to replace the water by honey, the profile of depth-depending velocity would change dramatically. The same applies for other materials, as each material has its own chemical and physical properties that determine the way particles communicate with each other. The collective effect is called friction, which is mathematically represented by anisotropic stress-tensor. The components normal to the direction of motions is called tension with the dynamical viscosity serves as a coefficient, inside which the chemical properties are encapsulated.
The effect of viscosity is generally to speed up and/or slow down the motions of particles in different portions of the domain toward enforcing a uniform motion. But if the flow is subject to external (non-conservative) forces and the viscosity is sufficiently small, then the motion of the particles become random, where the entropy of the system saturates. Such a fluid flow is said to be dissipative and therefore irreversible.
It turns out that when the temperature of the fluid falls below a certain critical value, the effect of viscosity diminishes. In this case, the fluid enters the so-called superfluid phase, where quantum mechanical effects start to emerge on global scales, for example, climbing up the walls of the container or forming discrete number of vortices that rotate coherently with each other.
In such fluids the De Broglie wavelengths λ D B , surpasses the mean free path characterizing the collisions between particles, and then each particle starts to coordinate its motion with its neighbours to finally clothe their quantum state: an extraordinary phenomenon in which micro-quantum states start showing up on the macroscopic scales (
In terrestrial fluids superfluidity phases start to show up when the temperature of the fluid becomes approximately one-hundred times smaller than the corresponding Fermi-temperature. In the cores of neutron stars however, although the temperature is of order one hundred million degrees, nuclear fluids are still about ten thousand times lower than the corresponding Fermi-temperature, implying therefore that NS-cores most likely are in quantum superfluid phase.
Astrophysics of Weakly Compressible and Incompressible Fluid FlowsIn an ever expanding universe the ultimate phase of nuclear fluids inside the cores of isolated NSs should have a vanishing entropy, inviscid and incompressible. Such objects are structurally similar to the so-called gravitational Bose-Einstein condensates (GBEC), the supermassive GBECs were found to dynamically unstable [
surrounding media. In fact, computer simulations of rotating superfluids reveal that the enclosed vortices are not static, but oscillate and intersect with each other, eventually turning the fluid turbulent [
Still, we have to determine whether nuclear fluids inside the very central regions of NSs are compressible or incompressible (see [
In the case of NSs, I argue that incompressibility is an inevitable phase of matter once the number density becomes larger than the nuclear one. Among the reasonable arguments that favor this phase are the following:
1) The spatial variation of the coefficient g r r of the Schwarzschild metric on the length scales of atomic nuclei is roughly ( d g r r / d l ≪ 10 − 19 ) implying therefore that gravity-induced stratification is unmeasurably small.
2) The effective potential of the gluon-field inside individual baryons is predicted to increases with radius as r Γ ( ≥ 1 ) . Thus the gluon-quark effective force inside hadrons opposes compression by gravity.
3) Most EOSs used for modeling the very central regions of NS-cores display sound velocities that do not respect causality. However, fluids with V s = O ( c ) cannot be compressed anymore. In fact recent numerical simulations of classical incompressible Navier-Stokes fluid-like flows reveal a blatant inconsistency in the capturing flow configurations, whenever the employed EOSs are set to depend on the local properties of the fluid only [
Thus, using a local description of the pressure for simulating weakly compressible or incompressible fluids is physically inconsistent.
4) Beyond the nuclear density, most sophisticated EOSs tend to converge to the limiting EOS: P L = E (
The classical form of the first law of thermodynamics: d E = T d s − p d V o l is not valid for incompressible nuclear fluids as both d E and d V o l are unrelated and therefore the local pressure P L cannot be calculated from d E / d V o l .
In fact the imposed regularity condition on the pressure at r = 0 manifests the incompressibility character of the fluid and therefore the break down of formula d E / d V o l = − P L = − n 2 ∂ ∂ n ( E / n ) . Moreover, assuming the matter at the center to obey the EOS: P = E , then the TOV equation can be integrated to yield: E ( r ) e V ( r ) = const . , where V stands for the gravitational potential. However, nuclear matter obeying P = E , cannot be compressed and therefore V must be constant, which means a vanishing gravitation-induced stratification. Nevertheless, most NS-models rely on using a non-vanishing density-gradient even at the vicinity of r = 0 . In order to have a NS of a reasonable mass, the central density must be much beyond the nuclear density: a density regime which is experimentally unverifiable and where our theoretical knowledge is severely limited (
In fact, it appears that nuclear fluids with n c r ≥ 3 n 0 , having a constant internal energy density and E k i n = E t h = E m g = 0 should be incompressible gluon-quark superfluids [
Indeed, recent experimental observations from the Relativistic Heavy Ion Collider (RHIC) and the Large Hadronic Collider (LHC) revealed that when heavy ions are accelerated in opposite directions to reach roughly the speed of light and collide, the outcome was not a gas, but rather a nearly frictionless superfluid, even though the effective temperature was extraordinary high [
The brief answer is that when the number density of zero-temperature quantum fluids surpasses the critical density n ≥ n c r , the sub-nuclear particles, such as mesons and gluons start interacting with the scalar field more frequent, thereby enhancing their effective mass and enabling individual baryons to merge together to form a super-baryon (
What is the underlying physical mechanism for generating “dark energy” in NSs?
The contribution of quarks to the baryon mass is approximately 2%, whereas the energy required to deconfine them is roughly equal or even larger than the energy needed for the creation of the whole baryon, which is roughly equal to 0.94 GeV.
However, as gluons are virtual particles that are generated by vacuum fluctuations that popping into existence and disappearing, then the symmetry between creation and annulation must be perfectly tuned, as otherwise protons would not survive a life time of the order 1052 the light crossing time through a baryon. Thus the numerous complicated interactions between the sub-nuclear particles embedded in the quark-gluon cloud are perfectly organized and fine-tuned, thereby giving rise to a complete suppression of energy loss via dissipation. This implies that quark-gluon plasma most likely acquires a one single quantum state. In this case, the entropy d S = k B log Ω must vanish, where Ω is the number of all possible microscopic states. This is in line with experimental data of the RHIC and the LHC, which showed that slamming heavy ions against each other with almost the speed of light produced almost a frictionless fluid with very low entropy.
In fact, the dynamics of particles of a plasma generally rely on different particle-mediators of various communication speeds, such as the speed of light, speed of sound ( V s ≪ c ), transport velocity, speed of viscous interactions and so on. Here, the ordered/liminar motion of particles will be affected by direct and indirect collisions so that remote fluid parcels would subsequently turn turbulent, where entropy generation is enhanced. This agrees well with numerical simulations, which show that high Reynolds number flows may turn turbulent, whenever the internal interactions between fluid parcels are mediated with different communication speeds [
As the nuclear fluid in the very central region of NSs has density beyond the nuclear one, the only degree of freedom left, where exotic energy could be still created would be through merging baryons and generating new communication channels between the quarks. This however requires enhancing compression by external forces, e.g. enhancing the curvature of the embedding spacetime. Similar to the recently explored pentaquarks, new communication channels must be generated between the quarks to ensure stability of the internal structure of the newly born super-baryon. The energy required for constructing the channels comes mainly from quark-antiquark interactions with the field, which we term here as dark energy.
When the first baryons merge with its neighbors at r = 0 and form a new super-baryon, the number of communication's channels, i.e. the flux tubes through which the strong force is communicated between the quarks, increases non-linearly with the number of participating quarks. For instance, for a given number of baryons, say n, each which is made of three quarks flavor, the number of communication channels scales as n ( n − 1 ) ~ n 2 , so that the number of bonds between the quarks increases as n ( n − 1 ) / 2 (
energy. The excess of dark energy goes to increase the surface tension as well as the volume energy of the bag enclosing the freely moving quarks. Hence the total work , d W t o t per unit volume d V o l needed to increase the volume of the super-baryon reads:
d W t o t d V o l = − P + n ⋅ d T (1)
where P , d T denote the local pressure and the surface tension perpendicular to the normal vector n generated by the enclosed quarks d N , respectively. Taking into account that the concerned super-baryon is spherically symmetric and composed of incompressible fluids ( P , d N / d V o l = const . ), we obtain that n ⋅ d T = α Φ r 2 and therefore the dark energy density would have the form:
E Φ = α Φ r 2 + β Φ (2)
α Φ , β Φ are constants. In fact this is similar to the static quark-antiquark potential inside individual baryons, which can be described as a superposition of Coulomb-like term, i.e., ( const . / r ) and a term, whose action increases with radius.
As the concerned quantum fluid is incompressible, in hydrostatic equilibrium and in a superfluid phase, the scalar field, which is the source of dark energy, can be safely considered as a spatially and temporally constant. This implies that the EOS of dark energy is P Φ = − E Φ , i.e. extremely stiff, non-local and behaves like r 2 . As the mass of the super-baryon continues to grow via merger with the surrounding baryons as well as through the injection of dark energy, the negative pressure must increase as r 2 , to finally attain a global maximum at its surface of the object.
On the other hand, astronomical observations reveal that the compactness parameter of most NSs and pulsars are equal or even larger than half. This however imposes a constrain on the EOS of dark energy in NSs: a non-local negative pressure of the form P Φ = − α E Φ , with α < 1 should be excluded, as otherwise these objects would collapse into solar-mass BHs, whose existence is not supported by observations.
As we have mentioned already, gluon-quark superfluid inside baryons must be incompressible. It is however not clear how spatially separated superfluid parcels could be brought together to merge without violating the incompressibility character of the fluid? One could envisage an instantaneous crossover phase transition in which the compressible nuclear fluid consisting of individual baryons and having chemical potential μ ~ n turns into incompressible gluon-quark-superfluid with μ ~ const . (
E = a 0 n 2 → darkenergy E = a q s f × n (3)
where a , a q s f are constant coefficients. Note that the injection of dark energy is necessary for boosting the energy of mesons and to subsequently convert them into gluons needed for forming the new flux tubes between the quarks inside the super-baryon.
In order to insure that the dark energy goes to solely enable a smooth crossover phase transition, we require that there must be a critical number density n c r , where the Gibbs function vanishes. The combined energy density (i.e., the density of internal energy of baryons and that of dark energy E ϕ ) per particle should be larger than or equal to the energy required to de-confine the quarks inside individual baryons:
f ( n ) = E b + E ϕ n − 0.939 GeV ≥ 0 (4)
Using the scalings [ ρ ] = 10 15 g / cm 3 ( ≐ 0.597 / fm 3 ) , chemical potential (energy per particle) [ μ ] = 1 GeV , we then obtain [ a 0 ] = 1.674 GeV ⋅ fm 3 and [ a ϕ ] = 5.97 × 10 − 39 GeV / fm 5 and [ b ϕ ] = 0.597 GeV / fm 3 .
In this case the Gibbs function in non-dimensional units reads:
f ( n ) = a 0 n + b ϕ n − 0.939 (5)
The function f ( n ) may have several minima, depending on the values of a 0 and b ϕ . However, for a crossover phase transition to occur, both f ( n ) and ∂ f ( n ) / ∂ n must vanish, which occurs at n = 0.81 for the most reasonable values: a 0 = 1 and b 0 = 0.37 (
This constant density would characterize the whole baryonic matter in the entire stellar-size SB. Following QCD analysis, such a density yields an approximate coupling constant α s ≈ 0.199 , which still much higher than the experimentally verified value of 0.122 (see
In the present case the potential of vacuum energy at r = 0 is V ϕ = b ϕ , and therefore there is a non-local pressure P N L = − E ϕ = − V ϕ = − b ϕ . For determining the value of a ϕ , we need to study the ultimate global structure of the object.
We note that the radius of the super-baryon behaves like a transition front that propagates outwards through the ultra-weakly compressible nuclear fluid of the NS, leaving the matter behind its front in an incompressible gluon-quark-superfluid phase. When the front reaches the surface of the entire object, which is expected
to occur on the scale of O ( 10 8 ) yrs , then the object becomes a SuSu-objects and disappears from our observational windows, as its radius would be indistinguishable from the corresponding event horizon. Equivalently, we require the following equation to be fulfilled:
R ∗ q = R S = 2 G g c 2 ( M N S + M ϕ ) (6)
where R ∗ q , R S , M N S and M ϕ denote the radius of the object, Schwarzschild radius, Mass of the original NS and the mass-enhancement due to dark energy, respectively. As the number density inside the SuSu-object is constant and equal to 3 × n 0 and as the vacuum energy density obeys the relation E ϕ = α ϕ r 2 + β ϕ , then:
M N S = 4 π ∫ 0 R E b r 2 d r = 4 π 3 ρ c r R 3 M ϕ = 4 π ∫ 0 R E ϕ r 2 d r = 4 π 5 α ϕ R 5 + 4 π 3 β ϕ R 3 (7)
Let us nondimensionlize Equation (6) using the following scaling values:
[ R ] = 10 6 cm , [ β ϕ ] = [ ρ ] = 10 15 g / cc , [ a ϕ ] = [ ρ / R 2 ] ,
M ˜ = [ M ] = 4 π 3 [ ρ ] [ R ] 3 = 2.1 M ⊙
We then obtain the following equivalent form to Equation (6):
M ϕ M N S = 3 5 ( R q ⋆ 2 ρ c r ) α ϕ + β ϕ ρ c r (8)
The term α ϕ r 2 in Equation (2) is the source of the non-local vacuum pressure, which yields the incompressibility character of the gluon-quark superfluid in self-gravitating systems.
Recalling that numerical and theoretical studies of the internal structure of NSs predict a compactness parameter α s ( ≐ R s / R N S ) ≥ 1 / 2 , which, in combination with the requirement that the object should turn invisible at the end of its cosmological life time, we conclude that its final total mass M t o t ≤ 2 × M N S . As it is shown in
To summarize the parameter determination procedure:
1) Let the isolated NS has the mass M N S .
2) The baryonic fluid at the verge of phase transition obeys the EOS E b = a 0 n 2 , whereas the EOS of the gluon-quark superfluid is E b = const . and for the dark energy E ϕ = α ϕ r 2 + β ϕ .
3) From the minimization requirement of the Gibbs function we obtained the coefficient a 0 and β ϕ , where the latter was set to equalize the bag constant in terms of the MIT-description of quarks in QCD. Here we use B 1 / 4 = 220 MeV , which is equivalent to 0.37 in the here-used non-dimensional units.
4) The coefficient α ϕ has been determined by requiring that the radius of the original NS coincides with the corresponding Schwarzschild radius after the object has entirely metamorphosed into a stellar-size SB. In this case, the baryonic matter amounts just to one half of the total mass-energy of the SB.
Baryon matter in QCD is made of gluon-quark plasmas [
15 × ( 0.938 / 3 ) GeV / channel ≈ 4.6 GeV
which is only slightly higher than the value revealed from pentaquark.
However, due to the strong confinement effect, quarks and gluons exist exclusively inside baryons and never in free space.
In the here-presented model, the density of matter at the very central region of massive NSs is beyond the nuclear density, and therefore mergers of baryons to form super-baryons cannot be excluded.
As more baryons are dissolved and join the super-baryon core, its volume and mass will increase to finally reach the surface of the entire object on the cosmological time scale.
Similar to gluon-quark plasmas inside individual baryons, the ocean of the incompressible gluon-quark superfluid inside the object would be shielded from the outside world by a repulsive quantum membrane, whose strength is proportional to the number of the enclosed quarks (
We conjecture that this membrane, which would be located outside the horizon, would be sufficiently strong to prohibit quantum tunnelling of particles both from inside and outside the wall, except for gravitons. If this is indeed the case, then there must be a length scale Λ r m , so that when the separation length, d, between two arbitrary SB-objects becomes comparable to Λ r m , the objects, being fermions, must experience repulsive forces similar to those operating between individual nucleons in atomic nuclei.
Hydrodynamically, the generation of the dark energy inside the cores of massive NSs can be modelled by introducing a scalar field, which, together with the baryonic energy, may be used to solve the TOV-equation inside these general relativistic objects [
The author thanks Johanna Stachel and Friedel Thielemann for the useful discussions on various aspects of neutron stars and quark physics.
Hujeirat, A.A. (2018) On the Ultimate Fate of Massive Neutron Stars in an Ever Expanding Universe. Journal of Modern Physics, 9, 51-69. https://doi.org/10.4236/jmp.2018.91004