Lorentz violation with an invariant minimum speed as foundation of the Gravitational Bose Einstein Condensate of a Dark Energy Star
Claudio Nassif Cruz, Rodrigo Francisco dos Santos, A. C. Amaro de Faria Jr
aa r X i v : . [ phy s i c s . g e n - ph ] O c t Lorentz violation with an invariant minimum speed as foundation of theGravitational Bose Einstein Condensate of a Dark Energy Star **Cl´audio Nassif Cruz, *Rodrigo Francisco dos Santos and A. C. Amaro de Faria Jr. **CPFT: Centro de Pesquisas em F´ısica Te´orica,Rua Rio de Janeiro 1186/s.1304, Lourdes, 30.160-041, Belo Horizonte-MG, Brazil.*UFF: Universidade Federal Fluminense, Instituto de F´ısica,Av. Gal. Milton Tavares de Souza s/n º , Gragoat´a, 24.210-346, Niter´oi-RJ, Brazil.IEAv: Instituto de Estudos Avan¸cados, Rodovia dos Tamoios Km 099, 12220-000, S˜ao Jos´e dos Campos-SP, Brazil.**[email protected], *[email protected], [email protected] We aim to search for the connection between the spacetime with an invariant minimum speedso-called Symmetrical Special Relativity (SSR) with Lorentz violation and the Gravitational BoseEinstein Condensate (GBEC) as the central core of a star of gravitational vacuum (gravastar), whereone normally introduces a cosmological constant for representing an anti-gravity. This usual modelof gravastar with an equation of state (EOS) for vacuum energy inside the core will be generalizedfor many modes of vacuum (dark energy star) in order to circumvent the embarrassment generatedby the horizon singularity as the final stage of a gravitational collapse. In the place of the problem ofa singularity of an event horizon, we introduce a phase transition between gravity and anti-gravitybefore reaching the Schwarzschild (divergent) radius R S for a given coexistence radius R coexistence slightly larger than R S and slightly smaller than the core radius R core of GBEC, where the metricof the repulsive sector (core of GBEC) would diverge for r = R core , so that for such a given radiusof phase coexistence R S < R coexistence < R core , both divergences at R S of Schwarzschild metric(a fine shell of baryonic matter involving the core of GBEC) and at R core of the repulsive core areeliminated, thus preventing the formation of the event horizon. So the causal structure of SSR helpsus to elucidate such puzzle of singularity of event horizon by also providing a quantum interpretationfor GBEC and thus by explaining the origin of a strong anisotropy due to the minimum speed(dark cone) that leads to the phase transition gravity/anti-gravity during the collapse of the star.Furthermore, due to the absence of an event horizon of black hole (BH) where any signal cannotpropagate, the new collapsed structure presents a signal propagation in its region of coexistence ofphases where the coexistence metric does not diverge, thus leading to emission of radiation. PACS numbers: 11.30.Qc,98.80.Qc,04.20.Dw,04.20.Jb,04.70.BwKeywords: quantum black hole, dS-space, dark energy star, vacuum energy, gravitational Bose-Einsteincondensate, minimum speed, Planck length, phase transition, Lorentz violation.
I. INTRODUCTION
The search for understanding the cosmological vacuum has been the issue of hard investigations[1]. Severalmodels have been suggested[2] in order to postulate the existence of a quantum vacuum (e.g: fluid of Zeldovich),but without proposing a physical interpretation for the vacuum until the emergence of a new kind of DeformedSpecial Relativity (DSR) so-called Symmetrical Special Relativity (SSR)[3][4][5][6], which has brought a newinterpretation for the quantum vacuum by means of the concept of an invariant minimum speed V directlyrelated to the miminum length (Planck length L P ), i.e., V ∝ L P [3], which changes the causal structure ofspacetime and geometrizes the quantum phenomena[5][7]. This minimum speed modifies the Minkowski metricof the spacetime by providing the SSR-metric[3], which is a kind of conformal metric as is the de-Sitter (dS)-metric[8]. Such SSR-metric is able to represent the metric of the Gravitational Bose Einstein Condensate(GBEC), which represents the core of a gravastar[10][11][12][13][19][20] [21][22][23]. So we are able to map theSSR-metric into the GBEC-metric in such a way that we can associate the cosmological constant of GBEC withthe minimum speed connected to the vacuum energy density[3]. Thus by linking the star structure equationswith the causal structure of spacetime of SSR, a phase transition appears in place of the event horizon postulatedby Chapline and others[14][15][17][18]. Such a phase transition can be related to this new causal structure ofSSR, where SSR describes perfectly a fluid, which is similar to a relativistic superfluid of type of cosmologicalfluid, being the constituent of GBEC.In the 2nd. section, we make a brief review of the concept of an invariant minimum speed in spacetime andits implications in the existence of the cosmological constant[3], basing on the Dirac’s Large Number Hypothesis(LNH). Thus we present a conformal metric of spacetime due to the presence of the minimum speed and so weshow the cosmological fluid (vacuum energy) or its equation of state (EOS) generated by Symmetrical SpecialRelativity (SSR).In the 3rd. section, we introduce the GBEC metric of a gravastar. We make a study about the well-knownconcept of anisotropy inside the GBEC, which permits the emergence of the phase transition (gravity/anti-gravity), which appears in place of the event horizon during the gravitational collapse. Such phase transitionis an implication of a more general equation of state (EOS) for a real structure of GBEC with many modes ofvibrational states instead of a unique EOS of vacuum ( p = − ρ with w = − p = 0 with w = 0) given in the region of phase coexistence(gravity/anti-gravity).In the section 4, we map the well-known dS-metric that governs GBEC into the SSR-metric, by showing aninteresting similarity between them, as they can be written in the same form with equivalent conformal factors,where there is a relationship between the GBEC (core) radius and the minimum speed for the core radius.Finally, in the section 5, we investigate deeper what occurs in the region of phase transition, i.e., at thecoexistence radius ( R coexistence ) of the two phases. We aim to obtain the spacetime metric in such a region,where we verify that the metric does not diverge as occurs in the classical case for the Schwarzchild radius R S (no phase transition). We show that there is no divergence in such a region of phase transition due to the factthat the coexistence radius of phases ( R coexistence ) is slightly larger than the Schwarzchild radius by preventingthe singularity of event horizon, i.e., we must find R coexistence > R S . II. A BRIEF REVIEW OF SYMMETRICAL SPECIAL RELATIVITY (SSR): SPACETIMETRANSFORMATIONS WITH A UNIVERSAL MINIMUM SPEED EMERGING FROM DIRAC’SLARGE NUMBER HYPOTHESIS
Let us first show the need of emergence of a universal minimum speed as a new fundamental constant of nature,according to a careful analysis of Dirac’s Large Number Hypothesis (LNH). Such a universal minimum speed V has the same status of the invariance of the speed of light ( c ), however V is given for lower energies related togravity, which is the weakest interaction, whereas c is well-known as being associated with the electromagneticfields. A. An extension of Dirac’s Large Number Hypothesis (LNH): the emergence of a minimum speed as anew constant of nature related to gravity and the cosmological constant
We will also show the relationship between the minimum speed V and the cosmological constant Λ, so thatthe so-called ultra-referential (preferred referential) S V (Fig.2) associated with V represents a kind of Machianbackground field (a vacuum energy) that leads to the cosmological constant Λ. In order to do that, we willstart from Dirac’s LNH by introducing the ratio of the forces of gravitational and electric interaction betweenan electron and a proton, namely: F e F g = e Gm p m e = √ N ∼ , (1)where e = q e / πǫ . N ( ∼ ) is the well-known magic number of Eddginton, which is of the order of thenumber of particles in the universe. m e and m p are the electron and proton masses respectively.The large number of the order of 10 is the well-known Dirac’s large number. Is is interesting to notice thatsuch large number can also be obtained in other ways, as for instance, the ratio F e /F g ∼ r p /R H ∼ , where r p is the proton radius and R H is the Hubble radius. This indicates that such large number also connects lengthscales of the micro-world (proton radius) with the macro-world (universe radius given by the Hubble radius).We know that the orbital speed of the electron in the ground state of the Bohr’s hydrogen atom is given asfollows: v B = v Bohr = αc = e ~ = q e πǫ ~ , (2)where α (= e / ~ c = q e / πǫ ~ c ≈ / v B = v Bohr ( ≈ (1 / c ∼ m/s ) is the velocity of the electron in the fundamental state of the hydrogen atom so-called Bohrvelocity.Now by making an extension of Dirac’s LNH, so that we use the work-energy theorem to implement bothworks to ionize the real hydrogen atom (with Coulombian interaction) and a hypothetical hydrogen atom withonly gravitational interaction between the proton and the electron to be carried from fundamental (Bohr) radius a to infinite, we find the following ratios of work (of applied forces) and kinetic energy, namely: − W F e ( a → ∞ ) − W F g ( a → ∞ ) = W F e ( ∞ → a ) W F g ( ∞ → a ) = q e πǫ R a ∞ r drGm p m e R a ∞ r dr ≡ F e F g = q e πǫ Gm p m e = m e v B m e v G ∼ , (3)where v G is the most fundamental velocity (a too small kinetic energy), since it has origin from the work ofthe gravitational force as being the negative of the same applied force to ionize a hypothetical gravitationalhydrogen atom.From Eq.(3), we get v B v G = q e πǫ Gm p m e ∼ , (4)where v B is the Bohr velocity [Eq.(2)]By substituting Eq.(2) ( v B ) in Eq.(4) of the extended LNH for v G , and after by performing the calculations,we finally find v G = r Gm e m p πǫ q e ~ , (5)where v G ≈ . × − m/s .Eq.(5) shows clearly the fundamental (lowest) speed v G due to its gravitational origin, since it depends on theconstant of gravity G , i.e., v G ∝ G / , such that, if gravity vanishes ( G → v G would be zero; however, as thepresence of gravity cannot be eliminated in anywhere, rest is prevented due to a zero-point energy associatedto the most fundamental vacuum energy of gravitational origin. In this sense, we are led to postulate v G asbeing a new kinematic constant connected to such a vacuum of quantum gravity at very low energies, i.e., itis a unattainable minimum speed associated with a preferred reference frame of background field, which is alsounattainable for any particles, so that the speed of any particles must belong to the interval v G < v ≤ c withina new scenario of Deformed Special Relativity (DSR) so-called Symmetrical Special Relativity (SSR), v G beingthe inferior cut-off of speed related to the vacuum energy.Let us simply use the notation V for representing the universal minimum speed v G , such that we write F e F g = (cid:18) v B v G (cid:19) = (cid:16) v B V (cid:17) ∼ , (6)where V = v G [Eq.(5)] represents the invariant kinematic aspect at lower energies in SSR.It has also been shown that the existence of V in the spacetime of SSR leads to the uncertainty principle[7].As we have already been able to obtain the universal minimum speed V [Eq.(5)] by means of the extendedLNH given in Eq.(4) [or Eq.(6)], we will look for the relationship between V and the cosmological constant Λ.To do this, we should first remember that the ratio of the Hubble radius ( R H ) and the radius of the proton ( r p )is exactly of the order of the square root of the magic number of Eddginton ( √ N ∼ ) with N ∼ . So,let us just write Dirac’s LNH, as follows: F e F g = (cid:16) v B V (cid:17) ∼ R H r p ∼ , (7)where the proton radius r p ∼ − m coincides in being the classical electron radius, which is obtained toexplain the energy of the electron ( m e c ) as originating from the Columbian self-interaction of the electroncharge, i.e., m e c = q e / πǫ r electron , from where one obtains r electron = r classical ∼ r proton = r p .As we have r p ∼ q e / πǫ m e c , so by substituting r p in Eq.(7), we obtain (cid:16) v B V (cid:17) ∼ πǫ m e c R H q e ∼ . (8)In a previous work, where SSR theory was introduced[3], it has already been shown that the cosmologicalconstant depends on the Hubble radius ( R H ∼ m ), i.e., Λ = 6 c /R H ∼ − s − [3], where we obtain R H = c r . (9)We know that v B = q e / πǫ ~ . So by substituting Eq.(9) above in Eq.(8) and performing the calculations, wefind v G = V ∼ q e / (4 πǫ c ) / m / e ~ Λ / . (10)Thus we realize that the most fundamental state of vacuum associated with the minimum speed V has directconnection to the cosmological constant Λ (vacuum energy), whose equation of state (EOS) is p ( presuure ) = − ρ ( energy density of vacuum ), thus leading to the anti-gravity. B. The vacuum energy: the minimum speed and the cosmological conatant
We realize that the EOS p = − ρ is the limiting case of EOS associated with the cosmological vacuum(cosmological constant Λ) connected to the minimum speed, i.e., V ∝ Λ / [Eq.(10)].SSR will be able to describe in detail a superfluid, which is very similar to what we see in the literature onGravastar/Dark Energy Star.Our goal is to investigate a more complex structure of GBEC of gravastar, which has many vibrational modesof vacuum, so that the single mode of EOS ( p = − ρ ) for a given cosmological constant Λ should be generalizedfor a variable cosmological parameter (Λ( r )) inside the spherical repulsive core of GBEC with radius r , where r ≤ R core (section 4). C. Spacetime transformations with an invariant minimum speed
A breakdown of Lorentz symmetry for very low energies[3][4][5][6] generated by the presence of a backgroundfield (a vacuum energy related to the cosmological constant) creates a new causal structure in spacetime, wherewe have a mimimum speed V , which is unattainable for all particles, and also a universal dimensionless constant ξ [3], which couples the gravitational field to the electromagnetic one ( c ), as shown in a previous work[3], namely: FIG. 1: The external and internal conical surfaces represent respectively the speed of light c and the unattainableminimum speed V , which is a definitely prohibited boundary for any particle. For a point P in the world line of aparticle, in the interior of the two conical surfaces, we obtain a corresponding internal conical surface, such that we musthave V < v p ≤ c . The 4-interval S is of type time-like. The 4-interval S is a light-like interval (surface of the lightcone). The 4-interval S is of type space-like (elsewhere). The novelty in spacetime of SSR are the 4-intervals S (surfaceof the dark cone) representing an infinitly dilated time-like interval[3], including the 4-intervals S , S and S inside thedark cone for representing a new space-like region[3]. ξ = v G c = Vc = r Gm p m e π q e ~ c , (11)where m p and m e are respectively the mass of the proton and electron. The value of such a minimum speed is v G = V = 4 . × − m/s. We find ξ = 1 . × − [3].Therefore the light cone contains a new region of causality called dark cone [3], so that the velocities of theparticles must belong to the following range: V (dark cone) < v < c (light cone) (Fig.1).The breaking of Lorentz symmetry group destroys the properties of the transformations of Special Relativity(SR) and so generates intriguing kinematics and dynamics for speeds very close to the minimum speed V , i.e.,for v → V , we find new relativistic effects such as the contraction of the improper time and the dilation ofspace[3][9]. In this new scenario, the proper time also suffers relativistic effects such as its own dilation withregard to the improper one when v → V [3][4], namely:∆ τ r − V v = ∆ t r − v c . (12)As the minimum speed V is an invariant quantity as the speed of light c , V does not alter the value of the speed v of any particle[3][4]. Therefore we have called ultra-referential S V [3][4] as being the preferred (background)reference frame in relation to which we have the speeds v of any particle (Fig.2). In view of this, the referenceframe transformations change substantially in the presence of S V , as shown first in the special case (1 + 1) D ,namely: FIG. 2: The reference frame S ′ moves in x -direction with a speed v ( > V ) with respect to the universal background fieldconnected to the unattainable (absolute) ultra-referential S V associated with v G = V . If v G = V → G = 0 inEq.(5)), S V is eliminated (no vacuum energy of quantum gravity) and, thus the galilean (inertial) frame S takes place,recovering the Lorentz transformations. x ′ = Ψ( X − vt + v G t ) = Ψ( X − vt + V t ) (13)and t ′ = Ψ( t − vX/c + v G X/c ) = Ψ( t − vX/c + V X/c ) , (14)where v G = V = p Gm p m e e/ ~ and Ψ = p − v G /v / p − v /c = p − V /v / p − v /c .As the transformations above in Eq.(13) and Eq.(14) are given in (1 + 1) D for the simple case of onedimensional motion as an approximation of the real motion given in (3 + 1) D spacetime of SSR, they justappear in their scalar form, so that we simply consider the scalar V for representing the minimum speed at onespatial dimension, which just represents an ideal case. However, it is important to stress that the real case of(3 + 1) D spacetime has a 3 D -vectorial background field represented by the vector ~V that breaks the Lorentzsymmetry, being invariant at any direction of 3 D -space (Fig.3), in such a way that it reduces to the scalar V at the ideal case of one spatial dimension. The general case (3 + 1) D with the background 3-vector ~V (Fig.3)will be shown soon. This will ensure that rest must be prevented in the real case of (3 + 1) D spacetime of SSR.The transformations shown in Eq.(13) and Eq.(14) are the direct transformations from S V [ X µ = ( ct, X )] to S ′ [ x ′ ν = ( ct ′ , x ′ )], where we have x ′ ν = Λ νµ X µ ( x ′ = Λ X ), so that we write the matrix of transformation forone-dimensional motion in x -direction (Fig.2), as follows:Λ = θγ − θγβ ∗ − θγβ ∗ θγ , (15)or simply Λ = (cid:18) Ψ − Ψ β ∗ − Ψ β ∗ Ψ (cid:19) , (16)such that we recover Λ → L (Lorentz matrix of rotation) for α →
0, which implies Ψ → γ of SR.We have Ψ = θγ and β x ∗ = β ∗ = β (1 − α ), as v = v x for one-dimensional motion in x -direction (Fig.2).We obtain det Λ = (1 − α )(1 − β ) [1 − β (1 − α ) ], where 0 < det Λ <
1. Since V ( S V ) is unattainable ( v > V ),this assures that α = V /v < det Λ = 0 ( > det Λ = ±
1) and so it does not represent a rotation matrix ( det Λ = 1) in SSR[3]Actually the result det Λ ≈ α ≈ v ≈ V emerges from a new relativistic effect of SSR for treatingvery low energies at a ultra-infrared regime (very large wavelengths) too close to the background frame S V , i.e., v ≈ V .The inverse transformations (from S ′ to S V ) are X = Ψ ′ ( x ′ + β ∗ ct ′ ) = Ψ ′ ( x ′ + vt ′ − V t ′ ) , (17) t = Ψ ′ (cid:18) t ′ + β ∗ x ′ c (cid:19) = Ψ ′ (cid:18) t ′ + vx ′ c − V x ′ c (cid:19) . (18)In matrix form, we get the inverse transformation X µ = Λ µν x ′ ν ( X = Λ − x ′ ), so that the inverse matrix isΛ − = (cid:18) Ψ ′ Ψ ′ β ∗ Ψ ′ β ∗ Ψ ′ (cid:19) , (19)where we can show that Ψ ′ =Ψ − / [1 − β (1 − α ) ] = θ − γ − / (1 − β ∗ ), so that we must satisfy Λ − Λ = I .Indeed we have Ψ ′ = Ψ and therefore Λ − = Λ( − v ). This aspect of Λ has an important physical implication.In order to understand this implication, let us first consider the rotation aspect of Lorentz matrix in SR. UnderSR, we have α = 0 ( V = 0), so that Ψ ′ → γ ′ = γ = (1 − β ) − / . This symmetry ( γ ′ = γ , L − = L ( − v ))happens because the Galilean reference frames permit to exchange the speed v (of S ′ ) for − v (of S ) when weare at rest at S ′ . However, in SSR, as there is no rest at S ′ , we cannot change v (of S ′ ) for − v (of S V ) dueto that asymmetry (Ψ ′ = Ψ, Λ − = Λ( − v )), thus leading to Lorentz violation. Due to this fact, S V must becovariant, namely V remains invariant for any change of reference frames in such spacetime. This issue will bewell-understood for the general case (3 + 1) D , where we have an isotropic 3-vectorial background field ~V , thuspreventing rest ( v = 0) for any particles.The (3 + 1) D (Fig.3)[3] transformations in SSR are ~r ′ = θ (cid:20) ~r + ( γ −
1) ( ~r.~v ) v ~v − γ~v (1 − α ) t (cid:21) = θ (cid:20) ~r + ( γ −
1) ( ~r.~v ) v ~v − γ~vt + γ ~V t (cid:21) , (20)where θ = q − V v and θγ = Ψ = q − V v q − v c .And t ′ = θγ (cid:20) t − ~r.~vc (1 − α ) (cid:21) = θγ " t − ~r.~vc + ~r.~Vc , (21)where ~V = ( ~v/v ) V . It is easy to verify that, if we have ~v || ~r ( ≡ X~e x ), we recover Eq.(13) for (1 + 1) D spacetime.So, in the special case (1 + 1) D with ~v = v x = v , we find the following transformations: x ′ = Ψ( x − vt + V t )and t ′ = Ψ( t − vx/c + V x/c ). The inverse transformations for the general case (3 + 1) D and (1 + 1) D weredemonstrated in ref.[3]. Of course, if we make V →
0, we recover the well-known Lorentz transformations.Putting the transformations given in Eq.(20) and Eq.(21) into a matricial form, we obtain the followingmatrix:
FIG. 3: S ′ moves with a 3 D -velocity ~v = ( v x , v y , v z ) in relation to S V . For the special case of 1 D -velocity ~v = ( v x ), werecover the case (1 + 1) D ; however, in this general case of 3 D -velocity ~v , there must be a background vector ~V (minimumvelocity)[3] with the same direction of ~v as shown in this figure. Such a background vector ~V = ( V /v ) ~v is related to thebackground reference frame (ultra-referential) S V , thus leading to a Lorentz violation. The modulus of ~V is invariant atany direction. Λ (4 X = θγ − θγβ x ∗ − θγβ y ∗ − θγβ z ∗ − θγβ x ∗ h θ + θ ( γ − β x β i h θ ( γ − β x β y β i h θ ( γ − β x β z β i − θγβ y ∗ h θ ( γ − β y β x β i h θ + θ ( γ − β y β i h θ ( γ − β y β z β i − θγβ z ∗ h θ ( γ − β z β x β i h θ ( γ − β z β y β i h θ + θ ( γ − β z β i , (22)where we have defined the compact notations namely β x ∗ = β x (1 − α ), β y ∗ = β y (1 − α ) and β z ∗ = β z (1 − α ).Writing the general matrix of transformation Λ (4 X [Eq.(22)] in a compact form (2 × (2 × = θγ − θγ v T (1 − α ) c − θγ v (1 − α ) c h θI + θ ( γ − vv T v i! , (23)where I = I × is the identity matrix and v T = ( v x , v y , v z ) is the transposed of v .If we make α = 0 ( V = 0), which implies θ = 1, the matrix in Eq.(22) (or Eq.(23)) recovers the generalLorentz matrix.The (3 + 1) D (Fig.3) inverse transformations in SSR were also obtained before[3], namely: ~r = θ − ~r ′ + θ − (cid:20)(cid:18) γ − − β ∗ − (cid:19) (cid:18) ~r ′ .~vv (cid:19) + ( γ − ) ∗ − β ∗ t ′ (cid:21) ~v (24)where we have used the simplified notation β ∗ = β (1 − α ). We also have ( γ − ) ∗ = γ − (1 − α ).And t = θ − γ − − β (1 − α ) (cid:20) t ′ + ~r ′ .~vc (1 − α ) (cid:21) = θ − γ − − β (1 − α ) " t ′ + ~r ′ .~vc − ~r ′ .~Vc . (25)In Eq.(24) and Eq.(25), if we make α = 0 (or ~V = 0), we recover the (3 + 1) D Lorentz inverse transformations.From Eq.(24) and Eq.(25), we obtain the general inverse matrix of transformation as follows:Λ − × = θ − γ − − β ∗ θ − γ − β x ∗ − β ∗ θ − γ − β y ∗ − β ∗ θ − γ − β z ∗ − β ∗ θ − γ − β x ∗ − β ∗ h θ − + θ − (cid:16) γ − − β ∗ − (cid:17) β x β i h θ − (cid:16) γ − − β ∗ − (cid:17) β x β y β i h θ − (cid:16) γ − − β ∗ − (cid:17) β x β z β i θ − γ − β y ∗ − β ∗ h θ − (cid:16) γ − − β ∗ − (cid:17) β y β x β i h θ − + θ − (cid:16) γ − − β ∗ − (cid:17) β y β i h θ − (cid:16) γ − − β ∗ − (cid:17) β y β z β i θ − γ − β z ∗ − β ∗ h θ − (cid:16) γ − − β ∗ − (cid:17) β z β x β i h θ − (cid:16) γ − − β ∗ − (cid:17) β z β y β i h θ − + θ − (cid:16) γ − − β ∗ − (cid:17) β z β i , (26)where we have β x ∗ = β x (1 − α ), β y ∗ = β y (1 − α ), β z ∗ = β z (1 − α ) and β ∗ = β (1 − α ) = v (1 − α ) /c = v ∗ /c .Writing the general inverse matrix of transformation Λ − [Eq.(26)] in a compact form (2 × − × = θ − γ − − β ∗ θ − γ − v T ∗ c (1 − β ∗ ) θ − γ − v ∗ c (1 − β ∗ ) h θ − I + θ − ( γ − − β ∗ − vv T v i , (27)where v T ∗ = v T (1 − α ), v ∗ = v (1 − α ) and β ∗ = β (1 − α ).We can compare the inverse matrix in Eq.(27) with the matrix in Eq.(23) and verify that Λ − = Λ T , in asimilar way as made before for the particular case (1 + 1) D (one dimensional motion).If we make α = 0 ( ~V = 0), which implies θ = 1, the inverse matrix in Eq.(26) (or Eq.(27)) recovers thegeneral inverse matrix of Lorentz.Although we associate the minimum speed V with the ultra-referential S V , this is inaccessible for any particle.Thus, the effect of such new causal structure of spacetime (SSR) generates a symmetric mass-energy effect towhat happens close to the speed of light c , i.e., it was shown that E = m c Ψ, so that E → v → V [3][4].It was also shown that the minimum speed V is associated with the cosmological constant, which is equivalentto a fluid (vacuum energy) with negative pressure[3][4]: just reminding that we have shown that V ∝ Λ / inEq.(10).The metric of such symmetrical spacetime of SSR is a deformed Minkowsky metric with a global multiplicativefunction Θ, i.e., a conformal factor Θ[8], being similar to a dS-metric, namely: dS = Θ g µν dx µ dx ν , (28)where Θ = θ − = 1 / (1 − V /v )[3][4][8].We can say that SSR geometrizes the quantum phenomena as investigated before (the UncertaintyPrinciple)[5][7] in order to allow us to associate quantities belonging to the microscopic world with a newgeometric structure that originates from Lorentz symmetry breaking. Thus SSR may be a candidate to tryto solve the problems associated with the gravitational collapse, which is a phenomenon that mixes inevitablyquantum properties with the geometric structure of spacetime. III. THE GBEC AND ITS ANISOTROPY
The core of a Gravastar/Dark Energy Star is described as being composed of an exotic material calledGravitational Bose Einstein Condensate (GBEC)[12][19][20]. This is a relativistic superfluid and this region0connects with a shell of ordinary matter (baryonic matter) described by the Schwarzschild metric (Fig.4). Sucha connection would take place by means of a phase transition in spacetime[15][17][18] that occurs near theSchwarzschild radius. Thus, by following the works of CFV and MM[12][13][16][19][20][ ? ], we write the metricof a gravastar[19][20], namely: dS = − f ( r ) c dt + dr h ( r ) + r d Ω , (29)where d Ω is the well-known solid angle. The metric functions f ( r ) and h ( r ) are given for dS-sector (GBEC),namely: f GBEC ( r ) = A (cid:18) − r R core (cid:19) (30)and h GBEC ( r ) = (cid:18) − r R core (cid:19) , (31)where R c = R core (core radius) and A is a given constant or even a certain function, which is obtained dependingon the boundary conditions. In our investigation, as we will see there is a similarity between GBEC-metric (akind of dS-metric) and SSR-metric, we will make a mapping between them in the next section.The constant vacuum energy density ρ inside the simple model of GBEC with a single positive cosmologicalconstant Λ (dS-space) as shown in Fig.4 is ρ = Λ c πG , (32)where Λ is the cosmological constant whose the unique vacuum state is represented by the well-known EOS ofvacuum energy, namely: p = wρ = − ρ, (33)where w = − p is the pressure and ρ is the vacuum energy density.On the other hand, the baryonic region with ultra-relativistic plasma in shell (Fig.4) is described by thefollowing EOS: p = wρ = ρ, (34)with w = +1 by representing the attractive matter of the ultra-relativistic plasma in shell.Here it is important to call attention to the fact that we are proposing a general model of gravastar, i.e., adark energy star so that we have a more general EOS that encompasses several vibrational degrees of vacuumin order to explain the anisotropy inside GBEC, in such a way that the pressure becomes practically zero atthe core radius, which is equivalent to v ≈ V in a general EOS, so that dpdρ = w ( v ) = − Ω = − v c , whereΩ = β = v/c [14][15]. Thus we obtain the following general EOS inside GBEC of our model of dark energy star,namely: p = − w ( v ) ρ = − v c ρ. (35)In the general EOS given by Eq.(35), we can see that there is a correspondence of the well-known EOSfor the fundamental vacuum energy of Λ, i.e., p = − ρ with v = c , which represents the maximum repulsive1pressure p inside GBEC at r = 0 or exactly the center of the spherical repulsive core of the dark energy star,where the so-called anisotropy is ∆ = 0 (subsection A). In other words, we can say that we find p = − ρ associated to the maximum repulsive parameter Λ( r = 0) = Λ in the center of the core. But, when weapproaches to r = R core , Eq.(35) shows us that the repulsive pressure is practically null at the surface of thecore, i.e., p = − ( V /c ) ρ = − ξ ρ ≈ v = V . This important result already indicates firstly that thereis a direct relationship between v = V and r = R core in the sense that both r = R core and v = V leadto the divergenges of GBEC-metric [Eq.(29)] and SSR-metric in Eq.(28) (a kind of dS-(conformal) metric[8])respectively. Therefore we are led to think that there is a direct mapping between Eq.(28) and Eq.(29), wherethe coefficient A [Eq.(30)] should be adjusted for consistency between both metrics whose boundary conditions( V and R core ) are equivalent. This issue will be treated in the next section (section 4).Furthermore, the decreasing of the repulsive pressure p to quasi-zero [Eq.(35)] when the radius r approachesto R core , so that the parameter Λ( r = R core ) ≈
0, allows us to understand the emergence of the phase transitionfrom anti-gravity to gravity as there should be a coexistence region close to R core , i.e., R coexistence < R core ,where GBEC-metric [Eq.(29)] does not diverge with p = 0 (Λ coexistence = 0).Therefore, the emergence of such null pressure at r = R coexistence is the unique way to permit the changeof its signal to positive ( p >
0) when r > R coexistence , where the attractive matter (baryonic plasma in shell)prevails as being gravity represented by the Schwarzchild metric. Actually, here it should be emphasized thatsuch change of a negative pressure to a positive pressure at R coexistence with p = 0 is due to the variation ofanisotropy ∆ inside GBEC as we will investigate in the subsection A. The variation of the anisotropy ∆( r ) isconsistent with the general EOS given in Eq.(35), but the existence of anisotropy is not consistent with thesingle EOS p = − ρ for a constant cosmological constant Λ > r = 0). So, if GBEC were really governed only by the EOS p = − ρ , the phase transition would not occur.This is the failure of the usual model of gravastar (Fig.4).Fig.6 shows the region of phase transition ( R core ) with a certain approximation, in spite of this figure givenin the literature[12][13] is not able to show clearly the small difference between R coexistence where p must beexactly zero, and R core at which Cattoen[12] consider that the pressure vanishes, since Cattoen does not makea clear distinction between R core and R coexistence . However, thanks to the unattainable minimum speed V ofSSR, the little difference between both radius will be elucidated in the section 5, thus preventing the divergenceof GBEC and Schwarzchild metric at R coexistence , so that the singularily of event horizon is eliminated. A. The relationship of DEC and WEC with the anisotropy and compactness
The study of fluid with negative pressure in stars was regulated in refs.[12][13][14][16] by following theBuchdahl-Bondi relation. We have the following conditions:1) The NEC (The Null Energy Condition) ρ + p i ≥ . (36)2) The DEC (The Dominant Energy Condition) | p i | ≤ ρ. (37)The imposition of such conditions[12][13][14][16] for the compactness of the star with mass m results in arange of values that the compactness should obey in such a way that the horizon event is not formed. Thus, forthe existence of a phase transition with the appearance of a repulsive core (GBEC), the values of compactnessmust conform to the following ranges in the respective layers close to the Schwazschild radius defined by thevalue CFV[12], by respecting NEC and DEC conditions of Buchdahl-Bondi, namely:89 < Gmc r < , ∆ ≥ Gmc r − Gmc r > , (38)2 FIG. 4: In the interior of a usual model of gravastar, we find a repulsive core (GBEC) associated with a single positivecosmological constant Λ > p = − ρ ( w = −
1) for a unique Λ > R coexistence and thus the phase transition. where ∆ is the magnitude of the so-called anisotropy, namely ∆ = p t − p r ρ , where p t is the tangencial pressureand p r is the radial pressure inside the repulsive core with radius R core .The coupling between the compactness and anisotropy characterizes the need to prevent the formation of theevent horizon.Usually, in the literature about nuclear Astrophysics, the anisotropy is related to the presence of an electro-magnetic field, but here the situation is different since the anisotropic term is introduced, so that we can obtaina repulsive effect, which is capable of preventing the formation of the event horizon. This is why the connectionbetween anisotropy and compactness is essential, which means that the anisotropy arises to respect the valuesabove, being in accordance with the compactness that must be Gmc r <
1. This means that r > R S = 2 Gm/c ,i.e., a radius which is slightly larger than R S , by preventing the emergence of the event horizon. IV. MAPPING BETWEEN DS-METRIC THAT GOVERNS GBEC AND SSR METRIC
In this section, we map the geometric (metric) structure of SSR [Eq.(28)][3][4][5][6] into the geometry (metric)of spacetime of GBEC [Eq.(29)], as there is a similarity between them.We already know the dS-metric[19][20]. So let us now rewrite Eq.(29) with azimuthal symmetry as therepulsive core is here considered to be a perfect sphere, where we only have Λ = Λ( r ), which does not dependon the angles θ and φ , so that we simply neglect the term of solid angle in the metric, namely:3 dS = − f ( r ) c dt + dr h ( r ) , (39)where we already know the dS-metric functions of GBEC, i.e., f dS ( r ) = A (cid:18) − r R core (cid:19) , h dS ( r ) = (cid:18) − r R core (cid:19) . (40)Now we should rewrite the metric of SSR[3][4][5][6] by considering the effect of the deformed light cone with c ′ << c [21][ ? ] and also the deformed dark cone with V ′ >> V inside the collapsed star, being close to thehypothetical event horizon, i.e., we find that both cones approach to each other close to the coexistence regionof phase transition: see Fig.5 for the classical case of a black hole, where there is only the deformed lightcone. So there is no dark cone in Fig.5, i.e., there is just a drastic decreasing of c close to R S , so that we find c ′ ( << c ) → r = R S , which does not occur in the non-classical case, where the lightcone cannot become completely closed ( c ′ >
0) due to the internal (repulsive) dark cone (Fig.1) that preventsits closing, avoiding the event horizon, since V ′ >> V . We will investigate the behavior of both c ′ and V ′ insection 5.Let us now write a given SSR-metric for representing a spherical repulsive core (with azimuthal symmetry)inside which the dark cone opens ( V ′ > V ) when r goes to R core , namely: dS = − c dt (cid:0) − V ′ v (cid:1) + dr (cid:0) − V ′ v (cid:1) , (41)from where V changes to V ′ > V (for a hypothetical internal observer inside the core), but c mantains fixedfor him (her) as it just changes to c ′ < c for another exernal observer out of the repulsive core given by matterduring the collapse of the baryonic sector, from where V already mantains fixed for such external observer.In sum, for an internal observer inside the repulsive core, V ′ > V (the dark cone opens) and the light coneremains fixed ( c ), while for an external observer in the baryonic sector (matter), c ′ < c (the light cone closes)and the dark cone remains fixed ( V ), as both observers cannot see clearly beyond the region of phase transition(quasi event horizon). A. Mapping of the components rr and tt between both metrics By substituting both Eqs.(40) in the metric of Eq.(39), we get GBEC-metric, as follows: dS = − A (cid:18) − r R core (cid:19) c dt + dr (cid:16) − r R core (cid:17) . (42)By comparing both metrics of SSR [Eq.(41)] and GBEC [Eq.(42)] with respect to their rr terms and alsotheir tt terms, we obtain two equivalences, namely:1 (cid:0) − V ′ v (cid:1) ≡ (cid:16) − r R core (cid:17) , (43)obtained for their rr terms.And 1 (cid:0) − V ′ v (cid:1) ≡ A (cid:18) − r R core (cid:19) , (44)4 FIG. 5: This figure shows the classical (causal) structure of black hole (BH) with event horizon[21]. This structure ismodified with the introduction of the causal structure of SSR[3][22]. We see that the speed of light changes close to theevent horizon when the gravitational field is extremely high. Thus, we expect that the speed of light, the minimum speedand the cosmological constant also acquire specific values ( c ′ , V ′ and Λ ′ ) during a non-classical gravitational collapse.The formation of a gravastar due to a phase transition[12] that leads to the emergence of GBEC is connected to a newstructure of spacetime having a positive cosmological constant (a highly repulsive core). obtained for their tt terms.First of all, from the equivalence in Eq.(43), we get V ′ v = r R core . (45)By introducing Eq.(45) into Eq.(44), we finally find A = 1 (cid:0) − V ′ v (cid:1) ≡ (cid:16) − r R core (cid:17) . (46)So, now by substituting A above in Eq.(42) of the GBEC-metric, we just verify that Eq.(41) for SSR-metricinside the core is indeed equivalent to Eq.(42). So, in doing this, we obtain dS = − c dt (cid:16) − r R core (cid:17) + dr (cid:16) − r R core (cid:17) ≡ − c dt (cid:0) − V ′ v (cid:1) + dr (cid:0) − V ′ v (cid:1) . (47)5 FIG. 6: Usual graph given in the literature by Cattoen[12][13] for representing the radial pressure versus radius, showingthe region inside which there is a radius of phase transition, i.e., such a radius is in somewhere in the interval r g < r < r max to be determined. The dominant anisotropy occurs for p ( R core ) ≈ R S < R coexistence < R core , where theradial pressure p vanishes, so that the singularity of event horizon is prevented. According to this figure, the pressuredecreases from its maximal value at r max to zero at R core (the transition region for Cattoen) due to the fact that gravity isreduced abruptally in this interval, where the baryonic matter is crushed into a Quark Gluon Plasma (QGP). We believethat the origin of such a rapid decrease of gravity comes from the vacuum anisotropy that already begins to govern thecollapse for r < r max until reaching the phase transition (gravity/anti-gravity) with a null pressure at r = R core in thefigure (actually at r = R coexistence < R core ), where the anisotropy reaches its maximum value. V. REGION OF COEXISTENCE BETWEEN PHASES
Rewritting the Schwarzschild metric, we have dS = − c ′ dt + dr (cid:0) − Gmc r (cid:1) + r d Ω = − c (cid:18) − Gmc r (cid:19) dt + dr (cid:0) − Gmc r (cid:1) + r ( dθ + sin θ dφ ) , (48)from where we obtain an effective speed of light c ′ = c p − Gm/c r < c . This assumption will be betterjustified soon when we consider the light cone in the region of phase transition or close to the event horizon inthe case of classical collapse (Fig.5)[21].We expect that, in the region of coexistence between the two phases (gravity or baryonic phase/anti-gravityor GBEC), i.e., for a given R S < R coexistence < R core , the light cone becomes almost closed, so that the speedof light is reduced to c ′ = c ( R coex. ) = c p − Gm/c R coex. . Thus, thanks to the minimum speed, i.e., thepresence of the dark cone (Fig.1)[3], we intend to show more clearly that the event horizon is almost formed at r = R coex. , which is slightly larger than the Schwarzschild radius ( R S = 2 Gm/c ).In order to obtain the coexistence radius R coex. , we have to admit that the minimum speed increases close6to the region of phase transition in such a way that V ′ ( >> V ) approaches to the speed of light c ′ ( << c ) thatdecreases, so that the light cone becomes almost closed, but not exactly closed at the event horizon as occursin the classical gravitational collapse (Fig.5)[21], by preventing the singularity of event horizon.To know how the minimum speed increases, we have to take into account the concept of reciprocalvelocity[5][7], where we have seen that the minimum speed works like a kind of “inverse” (reciprocal) speed( v rec ) of the speed of light, i.e., v rec = cV /v = v /v [5][7] with Ψ( v ) = Ψ( √ cV = 1, such that we find v rec = V for v = c . Thus, according to this relation for v rec , we can get the effective minimum speed V ′ in the regionof phase transition, namely V ′ ( R coex. ) = cV /c ( R coex. ) = cV /c p − Gm/c R coex. = V / p − Gm/c R coex. ,where V is the universal minimum speed[3].As V ′ approaches to c ′ close to the region of phase transition, both speeds become equal at R coex. , i.e., V ′ ( R coex. ) = c ′ ( R coex. ) so that we write c ′ = c r − Gmc R coex. = V ′ = V q − Gmc R coex. , (49)from where we obtain R coex. = 2 Gmc (1 − ξ ) = 2 Gmc (cid:0) − Vc (cid:1) , (50)where ξ (= V /c ) is the universal dimensionless constant of fine adjustment[3]. And it was already shown thatthere is a direct relationship between the minimum speed V and the minimum length of quantum gravity(Planck length L P ), i.e., V ∝ L P (= p G ~ /c )[3].Eq.(50) shows the expected result by indicating that the event horizon is not formed, since we can see thatthe radius R coex. is in fact slightly larger than the Schwarzschild radius ( R S = 2 Gm/c ) due to the universalminimum speed V ∼ − m/s that has origin in a quantum gravity as V ∝ L P . Thus a quantum gravity isresponsible for preventing the singularity of event horizon and so it is also responsible for the existence of thevacuum energy/dark energy.We find R S /R coex. = (1 − ξ ) <
1, that is to say R coex. > R S by preventing the event horizon. But, if we make ξ = 0 ( V = 0), we recover the classical case of singularity at the Schwarzschild radius (no phase transition),thus leading to the black hole (BH).In view of this quantum gravity effect given by the miminum speed connected to the cosmological constantΛ[3], we realize that the metric in Eq.(48) cannot diverge for the baryonic phase of the star, since its minimumradius (of matter) is now R coex. > R S , so that the divergence of the Schwarzschild metric (Eq.(48)) is prevented.Therefore, the divergence of the metric at R S is replaced by a too high value, being still finite. In order toobtain such a finite result, we just substitute Eq.(50) into Eq.(48), and so we find the metric in the region ofcoexistence of phases for r = R coex. , namely: dS coex. = − ξc dt + 1 ξ dr + r d Ω = − v dt + cV dr + r d Ω , (51)where 1 /ξ = c/V ∼ is a too large pure number, and v = √ cV represents a universal speed that providesthe transition from gravity to anti-gravity in the cosmological scenario[3][8]. So it is interesting to note thatsuch a speed v also plays the role of an order parameter obtained just in the region of phase transition of anon-classical gravitational collapse. This connection between the cosmological scenario with anti-gravity[8] andthe phase of a repulsive core inside a gravastar by means of the same universal order parameter of transitiongiven by v seems to be a holographic aspect of spacetime. We will explore deeply this issue elsewhere.Dirac has already called attention to the importance of the well-known Large Number Hypothesis (LNH) evenbefore the obtaining of ξ [3]. So a given infinite greatness that appears in Physics could be removed by a morefundamental principle. In view of this, it is interesting to notice that the metric in Eq.(51) shows us that thetiny pure number ξ in the denominator of the spatial term dr prevents its singularity and thus also prevents an7interval dS of pure space-like as occurs at the event horizon of a BH (Fig.5), because the light cone does notbecome completely closed in the region of phase transition given by the metric in Eq.(51), since the temporalterm of the metric above does not vanish ( c ′ ( R coex. ) = √ ξc = v ), i.e., ξc dt = V cdt = 0 as v = √ cV = 0,which is exactly the order parameter of transition that indicates the begining of a new phase of anti-gravity for r < R coex. .It is also interesting to note that a signal could be transmitted with speed c ′ = v in the region of phasetransition ( R coex. ), which does not occur at the event horizon of BH, where the light cone is completely closedso that c ′ = 0 (Fig.5) for r = R S (no signal).In any way, it is important to realize that such collapsed structure (dark energy star with a thin shell ofbaryonic matter) can emit radiation, since the temporal term of the metric in the phase transition region[Eq.(51)] is non zero, i.e., − ξc dt , which indicates that there is no event horizon.We sill realize that the Buchdahl-Bondi relation for preventing the event horizon is now better justified bythe constant ξ = V /c , since we find89 < GmR c c = (1 − ξ ) < , ∆( R c ) = 14 (1 − ξ ) ξ ≈ ξ >> , (52)where ξ ∼ − [3] and thus the anisotropy ∆ = p t − p r ρ is in fact so large at R coex. , i.e., ∆( R coex. ) >> p t is the tangencial pressure and p r is the radial pressure.In the isotropic case, we find p = p r = p t , however the works of CFV and Debenedicts et al[12][13][14][16] hasdemonstrated the relevance of the tangencial (transverse) pressure for a gravastar in spite of they were not ableto provide a satisfactory explanation for the anisotropy ∆, which is now well-understood by the new structureof GBEC with infinite vibrational modes of vacuum [Eq.(35)]. VI. CONCLUSIONS AND REMARKS
This work establishes the connection between the spacetime with an invariant minimum speed, i.e., theso-called Symmetrical Special Relativity (SSR) with Lorentz violation and the Gravitational Bose EinsteinCondensate (GBEC) as the central core of a star of gravitational vacuum (gravastar/dark energy star). So itwas introduced a new causal structure of spacetime that reveals the existence of a vacuum inside the core withvarious vibrational modes, which naturally explain the well-known anisotropy that leads to the phase transition(gravity/anti-gravity) at the coexistence radius ( R S < R coex. < R core ).The model eliminates the formation of a singularity of event horizon in an simple way and leads to the emissionof radiation by means of a phase transition between gravity and antigravity before reaching the Schwarzschildradius ( R S < R coex. ).This fundamental mechanism for eliminating the singularity of the event horizon can open a window and newinteresting perspectives on the study of preventing of physical (central) singularities of black holes replaced byblack hole mimickers like the present model of dark energy star.The information paradox and other related issues will be also investigated within this new causal structureof spacetime with an invariant minimum speed. Acknowledgements
The first author of this long research on SSR since 1988 dedicates his present work to the memory of AlbertEinstein and Stephen Hawking who searched for understanding the true quantum black hole, whose foundationsare provided by the spacetime with an invariant minimum speed as the kinematic origin of an anti-gravity givenby the dark energy within a new quantum gravity scenario. In sum, SSR shows that classical black holes arenot formed during a gravitational collapse. Thus, Einstein was right. [1] F. R. Klinkhamera and G. E. Volovik, Physics Letters A 347, 8 (2005).8