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Theoretical Derivation of the Cosmological Constant in the Framework of the Hydrodynamic Model of Quantum Gravity: Can the Quantum Vacuum Singularity Be Overcome?

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Theoretical Derivation of the Cosmological Constant in the Framework of the Hydrodynamic Model of Quantum Gravity: Can the Quantum Vacuum Singularity Be Overcome? galaxies Article Theoretical Derivation of the Cosmological Constant in the Framework of the Hydrodynamic Model of Quantum Gravity: Can the Quantum Vacuum Singularity Be Overcome? Piero Chiarelli 1,2 1 National Council of Research of Italy, Area of Pisa, Pisa 56124, Italy 2 Interdepartmental Center “E.Piaggio”, University of Pisa, Pisa 56124, Italy; [email protected]; Tel.: +39-050-315-2359; Fax: +39-050-315-2166 Academic Editor: Emilio Elizalde Received: 9 October 2015; Accepted: 5 April 2016; Published: 16 April 2016 Abstract: In the present work, it is shown that the problem of the cosmological constant (CC) is practically the consequence of the inadequacy of general relativity to take into account the quantum property of the space. The equations show that the cosmological constant naturally emerges in the hydrodynamic formulation of quantum gravity and that it does not appear in the classical limit because the quantum energy-impulse tensor gives an equal contribution with opposite sign. The work shows that a very large local value of the CC comes from the space where the mass of a quasi-punctual particle is present but that it can be as small as measured on cosmological scale. The theory shows that the small dependence of the CC from the mean mass density of the universe is due to the null contribution coming from the empty space. This fact gives some hints for the explanation of the conundrum of the cosmic coincidence by making a high CC value of the initial instant of universe compatible with the very small one of the present era. Keywords: quantum gravity; cosmological constant; dark energy; quantum vacuum energy density; cosmic coincidence conundrum; vacuum catastrophe PACS: 04.60.-m 1. Introduction General relativity is an example of a theory of impressive power and simplicity, but its ambit is purely classical. The CC, on the other hand, is a typical example of a modification originally introduced [1] by hand to help the fit of the physical data [2]. Even if unpleasant, it is the sign that a more general theory (i.e. quantum gravity (QG)) is necessary for a complete understanding. In fact, the quantum version of general relativity is in progress [3–9], and its contribution leads to the emergence of the CC as a correction to the general relativity [10]. Its original role of allowing static homogeneous solutions in general relativity equations in the presence of matter [11] (and then discharged by Einstein himself) turned out to be unnecessary when the expansion of the universe was discovered [12]. Nevertheless, there have been a number of subsequent episodes in which a non-zero CC was considered as an explanation of astronomical observations [13–16]. Meanwhile, particle physicists have realized that the CC can be interpreted as a measure of the energy density of the vacuum [10–18]. This energy density is the sum of a number of contributions, each of magnitude much larger than the upper limits of the CC known today. The question of why the Galaxies 2016, 4, 6; doi:10.3390/galaxies4020006 www.mdpi.com/journal/galaxies Galaxies 2016, 4, 6 2 of 12 observed cosmological vacuum energy is so small compared to that deriving by the scales of particle physics has become a celebrated puzzle. By using the hydrodynamic model of quantum gravity [19], the present work shows that the CC naturally appears in the theory. The outcome shows that the energy-impulse tensor density (QEITD) introduces a quantum contribution that leads to a non-zero CC-value only in the place where the mass of a particle is present, and globally to an observationally accessible CC. The paper is organized as follows: In the first section, the hydrodynamic model of quantum gravity is synthetically resumed, and the emergence of the CC in the gravitational equations is shown. In the second part of the work, the theoretical structure of the CC is analyzed and discussed respect to its large-scale value and time-dependence. 2. The Hydrodynamic Approach to Quantum Gravity The author has shown [19] that the hydrodynamic model of quantum gravity leads to the equation 1 8pG R ´ g R a “ T ` Lg , (1) nm 2 nm a c4 nm nm ` ˘ where Tnm is the covariant quantum energy-impulse tensor (QEITD) and where 8pG m|y|2 L “ ´ (2) c2 g is coupled to the quantum equations (defined in Section 2.2). 2.1. The QEITD Derived by the Hydrodynamic Quantum Model In order to obtain the covariant QEITD, it is enough to calculate its contravariant form in the flat Minkowski space-time. In this case, the quantum hydrodynamic equations of motion (for scalar uncharged particles), as shown by Guvenis [20], read BS BS m mn pq,tq pq,tq 2 BmB y 2 2 g m n ´ } ´ m c “ 0 (3) Bq Bq |y|ˇ ˇ ˇ ˇ and B BS BJm |y|2 “ “ 0, (4) Bq Bqm Bq m ˆ ˙ m where i} By By˚ J “ py ˚ ´ y q (5) m 2m Bqm Bqm is the 4-current, where } y S “ lnr s, (6) 2i y˚ and where the connection between the standard quantum notations and the hydrodynamic ones is given by the relation iS y “ |y|expr s. (7) } Equations (3) and (4), describing the (quantum) motion of the mass density m “ m|y|2 (generalized to the non-Euclidean space in Section 2.2), are coupled with Equation (1) through the QEITD [19] n . BL n 2 . BL n Tm “ qm . ´ Ldm “ |y| qm . ´ Ldm , (8) Bqn ˜ Bqn ¸ Galaxies 2016, 4, 6 3 of 12 where L “|y|2L (9) is the Lagrangian density, and L is the hydrodynamic Lagrangian function that defines the quantum hydrodynamic equations of motion as a function of the couple of real variables |y| and S [20–23] where the momentum reads E BS p “ p , ´p q “ ´ . (10) m c i Bqm The quantum hydrodynamic equation of motion (3) that, in the Lagrangian form, reads [19] BL pm “ ´ . m . (11) Bq and . BL p “ ´ , (12) m Bqm where dS BS BS . m L “ “ ` qi “ ´pmq (13) dt Bt Bqi is coupled to the current conservation Equation (4) where the 4-current Jm Equation (6), after simple manipulation, reads [19] pm . J “ pcr, ´J q “ ´|y|2 “ rq , (14) m i m m where |y|2 BS r “ ´ . (15) mc2 Bt Moreover, by using Equations (14) and (15) it follows that r . 1 BS . pm “ ´m q “ ´ q , (16) |y|2 m c2 Bt m and, hence, that ´1 . m 2 BS m ´1 m L “ ´pmq “ c Bt pm p “ r Jm p ´1 Bln´r y s¯ Blnr y s Blnr y s (17) i} 2 y˚ y˚ y˚ “ ´ c m 2 Bt Bq Bqmi ˆ ˙ and ´1 n 2 2 BS n a n Tm “ |y| c Bt pm p ´ pa p dm ´1 p pn V “ | |2 ´2 4¯ BS` m ´ ´˘ qu n y m c Bt m2c2 1 mc2 dm ´1 V “ | |2 2 ´ 1 ¯BS ´ n´´ ´ ¯ qu ¯ n (18) m y c mc2 Bt umu 1 mc2 dm 2 Blnr y s Blnr y s ´ } ¯ y´ ´1 ´ } ¯ y¯˚ y˚ Blnr s m “ m|y|2c2 2im2c2 y˚ 2mc Bq Bqn Bt V ¨ ´ `¯ ´ qu n ˛ ˆ ˙ 1 mc2 dm g . ˝ ´ ¯ ‚ where um “ c qm and where 2 BS BS E Vqu “ p pm “ ´ p2 “ m2c2 1 ´ , (19) Bqm Bq m c2 mc2 m ˆ ˙ ˆ ˙ m }2 BmB |y| has been used in [19], where the quantum potential reads Vqu “ ´ m |y| . As shown by Bialiniki-Birula et al. and by the author himself [19–23], it is noteworthy to observe that, due to the biunique relation between the quantum hydrodynamic approach (with the quantization bond) and Galaxies 2016, 4, 6 4 of 12 the standard quantum equations [21–24], Equations (3) and (4) are equivalent to the Klein-Gordon equation (KGE) [19–23] 2 2 mn m c g BnBmy “ ´ y, (20) }2 and, hence, the QEITD (18) makes the quantum gravity Equation (1) independent by the hydrodynamic approach used to derive it. 2.2. The Quantum Gravity System of Equations Now we have all the equations of the system that define the quantum gravitational evolution. The Einstein Equation (1) is coupled to the KGE that (for scalar uncharged particles) in a non-Euclidean space-time reads 2 2 m 1 m mn m c B y;m “ ? B ´g pg Bnyq “ ´ y, (21) ´g }2 a a with the covariant form of the QEITD (8) Tmn “ Tm gan that includes the quantum contribution, given by the quantum potential, that reads }2 1 V “ ´ ? Bm ´g pgmnB |y|q . (22) qu m |y| ´g n a 3. The Cosmological Constant In this section, we derive the cosmological constant L by requiring that the classical Einstein equation and the minimum action principle (MAP) are recovered in the classical limit. By considering, for the sake of simplicity, states of matter (or antimatter), without mixed superposition of them, we obtain the identity [19] 1 BS E Vqu ´ “ “ ˘mg 1 ´ , (23) c2 Bt c2 mc2 ˆ ˙ c where, according to the Dirac interpretation, the negative energy states apply to the antimatter. By using Equation (23), the QEITD (18) reads ´1 n 2 2 BS n a n Tm “ |y| c Bt pm p ´ pa p dm ´1 ´ 2 ¯ Vqu “ | |2 c 1 B`S u un ´ m2c2 ˘ 1 BS ´ n y g2 c2 Bt m c2 Bt 1 mc2 dm (24) ˆ ˙ ´2 2 ¯ V ´ ¯ ´ ¯ “ ´ p˘q m|y˘| c ´ qu n ´ n g 1 mc2 umu dm , ˆ ˙ b ` ˘ 2 2 where |y`| and |y´| are the matter and antimatter particle density, respectively, so that we can explicitly write the hydrodynamic motion equation [19], which reads n Vqu dum d Vqu B Vqu g BTm 1 ´ 2 “ ´um 1 ´ 2 ` m 1 ´ 2 “ ˘ 2 n , (25) c mc ds ds ˜c mc ¸ Bq ˜c mc ¸ mc Bq n n Tm where the quantum energy-impulse tensor Tm “ [19] reads |y|2 n 2 n Tm mc Vqu n n Tm “ 2 “ p˘q ´ 1 ´ 2 umu ´ dm .
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