Self-consistent solutions of canonical proper self-gravitating quantum systems
aa r X i v : . [ g r- q c ] A ug Self-consistent solutions of canonical properself-gravitating quantum systems
James Lindesay ∗ Computational Physics LaboratoryHoward University, Washington, D.C. 20059
Abstract
Generic self-gravitating quantum solutions that are not critically dependent onthe specifics of microscopic interactions are presented. The solutions incorporatecurvature effects, are consistent with the universality of gravity, and have ap-propriate correspondence with Newtonian gravitation. The results are consistentwith known experimental results that indicate the maintenance of the quantumcoherence of gravitating systems, as expected through the equivalence principle.
The incorporation of quantum mechanics into gravitational dynamics remainsa perplexing issue in modern physics. In contrast to other interactions like elec-tromagnetism, the trajectory of a gravitating system is independent of the masscoupling to the gravitational field. Thus the gravitation of arbitrary test parti-cles can be described in terms of local geometry only. However, the equations ofgeneral relativity are quite complex and non-linear in the interrelations betweensources and geometry. This makes solutions of even classical systems somewhatcomplicated.The behavior of quantum objects in Minkowski space-time is well describedby standard quantum theory. The various interpretations (Copenhagen, manyworlds, etc.) of the underlying fundamentals of the quantum world must all beconsistent with the standard theory, which has yet to be contradicted by exper-iment. There have been experiments that examined the behaviors of quantumobjects in gravitational fields. Gravitating quantum systems do maintain theircoherence, demonstrating that the structure of the interaction need not breakcoherence in order to localize the system in the field. Moreover, those systemscontinue to gravitate after coherence is broken by detection, as well as themselvesserve as source energy densities. Even highly dynamic gravitational environmentssuch as the big bang can redshift cosmic microwave background radiation without ∗ e-mail address, [email protected] reaking the coherence of the individual quanta. These behaviors are sensibleusing fundamental principles of relativity. Due to the principle of equivalence,the motions of detectors and screens should not break the coherence of inertial(freely falling) systems prior to detection. These basics will be briefly discussedin Section 2Often, a problem considerably simplifies if the parameters are properly chosen.In this treatment, space-like surfaces of simultaneity will be defined by fixed propertime τ , which are generally not coincident with space-like surfaces of simultaneitydefined by coordinate time t . The canonical proper time formulation is particularlyuseful for describing gravitational dynamics using proper time. This formulationwill be discussed in Section 3.Dynamics described using the proper time and convenient spatial coordinateswill provide straightforward solutions for generic self-gravitating quantum sys-tems. The description will be particularly robust with regards to arbitrary micro-scopic interactions that might contribute to generate the inertial masses. In fact,no form of microscopic interaction is mentioned or utilized in the discussion. Theself-gravitating solutions will be developed and discussed in Section 4. Gravitation is an interaction of considerable mathematical subtlety, despiteits familiarity. The geometrodynamics of classical general relativity is most di-rectly expressed using localized geodesics. However, quantum dynamics incor-porate measurement constraints that disallow complete localization of physicalsystems. The subtleties of observed gravitation of quantum systems should offerinsight into the fundamentals of quantum self-gravitation, as will here be exam-ined.
There is now considerable experimental evidence that quantum coherence ismaintained by the nearly static gravitational field near Earth’s surface. Duringthe early and mid 1970’s, experiments performed by Overhauser, et.al.[1] exam-ined the gravitation of coherent neutrons diffracting from an apparatus whoseorientation could be changed relative to the Earth’s gravitational field. The grav-itating neutrons were seen to maintain spatial coherence, exhibiting a patternconsistent with self-interference through two apertures at different gravitationalpotentials. A more recent experiment measured the small difference between theticks of two interfering quantum clocks[2]. In that experiment, very cold cesiumatoms gravitated vertically across a laser beam that superposed single atoms intostates at differing gravitational potentials. The resultant difference in the phaseemonstrated interference in the quantum oscillations associated with the rela-tivistic energy of the atoms. Thus, gravitating atoms have also been shown tomaintain temporal coherence.The experimental results discussed imply that the quasi-static near-Earthgravitational field does not break the phase coherence of neutrons or atoms asneeded for interference. The experiments also test the principle of equivalence,since motions of the observer do not break the coherence of the inertial (gravitat-ing) particles. The experiments involve both Newton’s gravitational constant G N and Planck’s constant ¯ h in a single equation form. Experiments such as those discussed in the previous section provide illustra-tive examples of the usefulness of the canonical proper time formalism developedin references [3, 4] for describing gravitating systems. The canonical proper timeformulation of relativistic dynamics provides a framework from which one can de-scribe the dynamics of classical and quantum systems using the clocks of those verysystems. The approach presumes that any gravitating quantum system maintainscoherence on surfaces defined by its proper time. The various regions across acoherent state propagate through varying gravitational potentials, with space-likesurfaces of simultaneity defined by fixed proper time τ . The formulation utilizes acanonical transformation on the time variable conjugate to the Hamiltonian thatis used to describe the dynamics, but does not transform other dynamical vari-ables such as momenta or positions. This gives insight into the fundamentals ofan interaction, since the response and back-response of the interacting system isbest parameterized using its proper-scaled dynamics. For quantum systems, Heisenberg’s equations describe the dynamics of anobservable W ( q, p, t ) in terms of the commutator of that observable with theHamiltonian: d ˆ W ( x, p, t ) dt = i ¯ h [ ˆ H, ˆ W ( x, p, t )] + ∂ ˆ W∂t ! x,p . (3.1)In special relativity, the inertial time t is related to the proper time τ through thestandard Lorentz factor γ using dt = γ dτ = HM c dτ. (3.2)The second form in Eqn. 3.2 follows from the relationship between the energyof a system compared to its rest energy. The canonical proper energy form K s defined to generate dynamic changes with regards to the proper time of thatsystem: d ˆ Wdτ = d ˆ Wdt dtdτ ≡ i ¯ h [ ˆ K, ˆ W ] + ∂ ˆ W∂τ ! x,p . (3.3)From this equation, along with Heisenberg’s equation, it then follows that [ K, W ] =
HMc [ H, W ].The canonical proper energy form K is expected to correspond to the Hamil-tonian when the Hamiltonian itself corresponds to the rest energy, K | H = Mc = H = M c . Holding the system mass M fixed during the canonical “boost” from H to K results in the form ˆ K [ H ] = ˆ H M c + M c . (3.4)As an example, a direct substitution of the non-interacting relativistic form H o = q ( pc ) + ( M c ) into this equation yieldsˆ K o = ˆ p M + M c . (3.5)In this case, both temporal parameters are inertial. A few points of interest shouldbe noted: • The form of the equation for K o is that of a non-relativistic free particle,despite the system being completely relativistic; • The momentum in K o is the same canonical momentum of the particle inthe Hamiltonian formulation. This is clearly not a Lorentz transformationof the dynamical parameters of the system; • The sometimes troublesome square root does not appear in the expressionfor K o .Since the positions and momenta of a gravitating particle are typically describedrelative to fiducial observers, rather than the proper coordinates of the gravi-tating particle, this formulation is particularly useful for describing gravitationaldynamics. The equations of motion generated using the canonical proper time formulationinsures that the canonical proper energy is conserved ( dKdτ = 0) if there is noexplicit temporal dependence in the functional form of any interactions. Foreneric proper potential energy forms U ( r ), the canonical proper energy can oftenbe expressed K = p · p m + U ( r ) + mc . (3.6)A potential form consistent with standard gravitation will next be developed. The potential energy is expected to take the form of Newtonian gravitationto lowest order in the gravitational constant G N . However, space-time curvatureeffects are expected to modify the classical result. To construct the relativisticenergy form, the equations of motion resulting from Eqn. 3.6 with U ( r ) = m V ( r )will be examined: dp j dτ = − m ∂ j V ( r ) , dr j dτ = p j m . (3.7)For a constant mass, this form is analogous to the geodesic equation incorporatingspace-time curvature: d x j dτ + Γ jαβ dx α dτ dx β dτ = 0 . (3.8)The present exploration is interested in the behaviors of stationary quantum grav-itating systems. As is the case with electronic distributions in stable atoms, themass distribution should be stationary in a quantum gravitating system. Fora stationary gravitating distribution, assume that dx α dτ = dx dτ δ α (consistent withquantum expectation values). Therefore, substituting the form of the connectionsΓ jαβ for a metric space-time (Riemannian manifold) in the geodesic equation 3.8,the proper interaction form must satisfy d x j dτ = − ∂ j V ( r ) = 12 g jµ g ,µ dx dτ ! . (3.9)This form will be generated for a straightforward static energy density. The space-time metric for a spherically symmetric, static space-time will bechosen to be a generalization of Schwarzschild geometry with non-vanishing localdensities. The metric is given by ds = − − R M ( r ) r ! ( dct ) + dr − R M ( r ) r + r dθ + r sin θ dφ , (3.10)where the Jacobian √− det g ≡ √− g = r sinθ . In this equation, a finite radialmass scale R M ( r ) ≡ G N M ( r ) /c is the length scale of the interior mass-energyontent of the system, with the mixed Einstein tensor given by G = r ∂∂r R M ( r ).For finite mass distributions, the metric takes the form of Minkowski space-timeboth asymptotically ( r >> R M ) as well as wherever the radial mass scale vanishes.The Ricci scalar R = − r ddr ( r dR M ( r ) dr ) (3.11)for such distributions is non-singular as long as the mass density decreases rapidlyenough for small r . Substituting the metric into Eqn. 3.9, one should note that g rr = − g and (cid:16) dx dτ (cid:17) = − c g , giving the equation ∂ r V ( r ) = − c ∂ r ( g ) (3.12)Using the standard condition V ( ∞ ) = 0, the form of the interaction for the propercanonical energy form is therefore given by V ( r ) = − G N M ( r ) r . (3.13)This relativistic form is the same as the usual Newtonian interaction. The gravitational potential energy from Eqn. 3.13 will next be incorporatedin the quantum form of the canonical proper energy equation 3.6. The equationdeveloped will include both the local dynamics of special relativity as well as thecurvature effects of general relativity. Self-consistent solutions of this equationwill be developed in this section.
Consider the stationary gravitation of a mass m due to an interior source massdistribution M ( r ). An invariant probability form measuring the likelihood thatthe particle will be measured by an observer in the space-time interval ∆ ct ∆ V isexpected to take the form P ∆ ct ∆ V = Z ∆ ct ∆ V dct d r √− g | ψ ( ct, r ) | . (4.1)General quantum systems will be temporally dynamic. However, stationary stateprobability densities are not expected to have time dependencies. The wave func-tion that satisfies the stationary state canonical proper energy equation for thisass, and represents the likelihood for measurement within the time interval ∆ ct ,is given by h ˆ p · ˆ p m − G N mM ℓ ( r ) r + mc i ψ ∆ ctnℓℓ z ( r, θ, φ ) = K nℓ ψ ∆ ctnℓℓ z ( r, θ, φ ) ,ψ ∆ ctnℓℓ z ( r, θ, φ ) = √ ∆ ct R nℓ ( r ) Y ℓ z ℓ ( θ, φ ) . (4.2)The proper energy eigenvalues K nℓ are expected to include relativistic velocitiesand temporal curvature effects.The form of the canonically conjugate momentum components in Eqn. 4.2must be consistent with the Heisenberg equations of motion * d ˆ p r dτ + = (cid:28) i ¯ h [ ˆ K, ˆ p r ] (cid:29) = − m ∂ r V ( r ) , (4.3)where V ( r ) was obtained from the geodesic equation 3.12. This implies that thescale factor of the momentum conjugate to r in the proper energy form shouldbe unity, ˆ p r = ¯ hi ∂∂r . Therefore, the spatial curvature effects are evidently alreadyincorporated in the functional form of the potential V ( r ) and the given conjugatemomentum operator.The square of the momentum for the metric form Eqn. 3.10 is thus given byˆ p · ˆ p = − ¯ h ( r ∂∂r r ∂∂r ! − ˆ L ¯ h r ) , (4.4)while in contrast the spatial Laplacian for this metric satisfies ∇ · ∇ = r s − R m ( r ) r ∂∂r r s − R m ( r ) r ∂∂r − ˆ L ¯ h r . (4.5)Parameters analogous to those of Bohr for hydrogenic systems can be developed.The radial scale of the solutions is given by a ≡ ¯ h G N m = λ m L P ! λ m = (cid:18) M P m (cid:19) λ m , (4.6)where the reduced Compton wavelength is given by λ m ≡ ¯ hmc , the Planck length islabeled L P , and the Planck mass is labeled M P . For the present treatment, onlys-wave ℓ = 0 states will be examined. Since probability densities using the metriccoordinates will take the form r R C,ℓ =0 ( r ), It is convenient to introduce a centralreduced radial wavefunction u C ( r/a ) ∝ rR C, ( r ) parameterized by dimensionlessvariable ζ ≡ r/a . The dynamic parameters can also be scaled using the parameter : P ( r/a ) ≡ R r/a u C ( ζ ′ ) dζ ′ , P ( ∞ ) = 1 , R M ( r ) r = 2 (cid:16) mM P (cid:17) ar P ( r/a ) , − ma ¯ h V ( r ) = 2 ar M ( r/a ) m , − ma ¯ h ( K − mc ) = − (cid:16) M P m (cid:17) K − mc m c . (4.7)Using these identifications, Eqn. 4.2 can then be re-written ǫ C u C ( ζ ) = d u C ( ζ ) dζ + ζ ! M ( ζ ) m ! u C ( ζ ) , (4.8)where the dimensionless parameter ǫ C ≡ − (cid:16) M P m (cid:17) K − mc m c quantifies the gravita-tional binding energy of the mass. In order to examine the scale of the spatialcurvature effects, an equation for which the spatial Laplacian (which incorporatesproper radial distances) replaces − ˆ p · ˆ p ¯ h , will also be examined: ǫ ∗ u ∗ ( ζ ) = d u ∗ ( ζ ) dζ + (cid:16) ζ (cid:17) (cid:16) M ∗ ( ζ ) m (cid:17) u ∗ ( ζ ) + (cid:16) mM P (cid:17) (cid:16)h P ′∗ ( ζ ) ζ − P ∗ ( ζ ) ζ i u ∗ ( ζ ) + h P ∗ ( ζ ) ζ − P ′∗ ( ζ ) ζ i du ∗ ( ζ ) dζ − ζ P ∗ ( ζ ) d u ∗ ( ζ ) dζ (cid:17) . (4.9)The terms containing P ∗ ( r ) demonstrate the modifications to the previous form(with mM P →
0) due to factors R M ∗ r .For all solutions, the source mass M ( ζ ) will be presumed to be generated bythe interior self-sourcing of probability density: M ( ζ ) = Z ζ ρ mass ( ζ ′ ) dζ ′ = Z ζ m u C ( ζ ′ ) dζ ′ = m P ( ζ ) . (4.10)The distribution indicates that the differential equations 4.8 and 4.9 will be non-linear. For this distribution, it should be noted that the particle mass scale m appears nowhere in Eqn. 4.8, while it only appears in the spatial scale terms inEqn. 4.9.A mass whose interior probability density provides its local source gravitationalfield will be referred to as a self-gravitating mass. If, in additional, the overallgravitational bound state energy is that of the mass itself, the self-gravitatingmass will be referred to as a self-generating mass. A self-gravitating central massis expected to have non-vanishing probability density at the center. Such a self-gravitating single mass satisfies Eqn. 4.8 with mass distribution given in Eqn.4.10. This form is clearly non-linear, so that initial conditions and eigenvalues arenon-trivially related to the solution.A solution to Eqn. 4.8 for a system that is self-gravitating, but has non-vanishing binding energy eigenvalue is demonstrated in Figure 1. Expressed in .5 1 1.5 2 2.5 3 r (cid:144) a0.20.40.60.8 È u C H r (cid:144) a LÈ Figure 1: Self gravitating quantum mass distribution.the dimensionless form demonstrated, the solution is completely independent ofthe mass of the system. The binding energy eigenvalue for the normalized prob-ability density was obtained by examining the small r behavior of Eqn. 4.8 ina self-consistent manner, yielding a value ǫ C ≃ . | u C ( r/a ) | , while the diagram on the rightis a density plot of the self-gravitating mass density. For the system, the gravityat a given radial coordinate is a field generated by the integrated mass densitywithin that radial coordinate.To obtain a solution to Eqn. 4.9, a mass value must be chosen, since the radialscale factors explicitly appear in the equation. The mass was chosen ( (cid:16) mM P (cid:17) =0 .
01, or m ≃ . M P ) such that some spatial curvature effects would be apparentin the calculations. For this mass, the binding energy eigenvalue ǫ ∗ was found to belarger than ǫ C by 0.38%, while the central density remained essentially unchanged.A solution to Eqn. 4.8 for which the gravitational potential results in vanishingnet binding energy can also be found. Figure 2 demonstrates a self-generatingsolution to this equation with ǫ C = 0. The diagram on the left again demonstratesthe probability density | u C ( r/a ) | , while the diagram on the right is a density plotof the self-gravitating mass density consistent with vanishing overall gravitationalbinding energy. The scales of the diagrams have been chosen to be consistent withthe prior self-gravitating mass. The self-generating mass density is seen to be moreconcentrated at the center relative to the self-gravitating mass density, developinga greater integrated gravitational potential energy, with a commensurate changein integrated kinetic energy. The central density | u ∗ (0) | solving Eqn. 4.9 (with m ≃ . M P ) is modified from that solving Eqn. 4.8 by an increase of about3.8%.The solutions demonstrated in Figures 1 and 2 are independent of mass. How- .5 1 1.5 2 2.5 3 r (cid:144) a0.511.522.53 È u C H r (cid:144) a LÈ Figure 2: Self generating quantum mass distribution.ever, as previously mentioned, the radial mass scale used to generate the Einsteintensor is given by R M ( r ) = 2 (cid:16) mM P (cid:17) a P ( r/a ). The crucial factor 1 − R M ( r ) r in themetric is demonstrated in Figure 3. If this factor changes sign, space-like behav- (cid:144) a0.20.40.60.81 1 - R M H r L €€€€€€€€€€€€€€€€ r Figure 3: Metric factor − g and g − rr .iors become time-like (and vice versa), and a trapped region for which outgoingphotons must propagate towards decreasing radial parameter r will be present.As long as the mass is not chosen to be too large, there is no trapped region. Alarger mass lowers the y-intercept of this curve. For the self-generating mass, themaximum value the mass can take without introducing a trapped region is givenby about 0.63 M P . Masses smaller than this will not generate a black hole.For completeness, the non-vanishing components of the Einstein tensor G ctct = G rr and G θθ = G φφ are demonstrated in Figure 4. The tensor satisfies the vacuumsolution G µβ = 0 in the exterior region ra > .
64. All solutions presented have .1 0.2 0.3 0.4 0.5 0.6 r (cid:144) a100200300400500600700G = G rr (cid:144) a-0.4-0.3-0.2-0.1G ΘΘ Figure 4: Einstein tensor components for self-generating mass.non-vanishing densities of finite extent.
Classical gravitating systems are expected to satisfy various energy conditionseverywhere. These conditions assert that any observer should locally measuregravitational fields generated by time-like or light-like sources, regardless of theirmotion. This is consistent with an expectation that no energy source can propa-gate at a speed greater than that of light. However, quantum systems do exhibitspace-like coherent behaviors. Space-like coherence allows the evaporation of blackholes, thereby locally violating energy conditions. Also, systems with significantbinding might violate these conditions. It is therefore of interest to examine theenergy conditions of these self-gravitating systems.The null and weak energy conditions assert that the form I null/weak ≡ − u µnull/weak T µβ u βnull/weak ≤ T µβ represents components of the energy-momentum tensor sourcingthe gravitational field in Einstein’s equation. The dominant energy condition di-rectly develops the form of the 4-momentum of the gravitational source as seenby the observer with four velocity ~u observer , given by p µsource ≡ − T µβ u βobserver . Thisfour-momentum is expected to be time-like or light-like, i.e., I DEobserver ≡ ~p source · ~p source ≤ , where the dot product is defined by the metric of the geometry. For all of the self-gravitating and the self-gravitating, self-generating solutions given, the null andweak energy conditions are satisfied everywhere for all types of motions. Likewise,the dominant energy condition for fiducial (static) observers and arbitrary radialmotions is also satisfied everywhere for all solutions. However, the dominantenergy condition for rapid pure azimuthal motions of the observer was found toe violated only in the region just inside of the surface in each solution, likely dueto coherence and gravitational binding from the interior mass distribution. Rapidmotions were found to be those motions exceeding the condition r u θ > R ′ M ( r ) q ( r R ′′ M ( r )) − (2 R ′ M ( r )) . (4.11)Exterior to the region of coherence, as well as proximal to the center, all energyconditions were found to be satisfied. A further exposition of energy conditions,as well as a more detailed development of the general formulation based on theequivalence principle, including co-gravitating masses and cluster decomposability,will be found in reference [6]. Self-gravitating quantum solutions, consistent with the equivalence principle,have been found using coherence parameterized by the local proper time of thegravitating system. The solutions required no specific form for micro-physicalinteractions, consistent with the universal nature of gravitation. The approachconsiders space-time as an emergent construct of quantum measurement, withcurvatures generated by Einstein’s equation in the form G µν = − πG N c h ˆ T µν i . Thedynamics developed is consistent with the measurement constraints of standardquantum theory.The quantum stationary solutions developed incorporate curvature effects. Forweak curvatures and slow motions, the solutions exhibit both quantum and clas-sical Newtonian correspondence through proper time Heisenberg equations of mo-tion. The formulation, being generally representation independent, demonstratesthat the exhibition of quantum coherent behavior for gravitating systems need notrequire second quantization of the gravitation field itself. The solutions satisfy sen-sible conditions of physicality on the energy densities sourcing the gravitationalfields, including non-singular behavior everywhere and non-negative mass densi-ties R ′ M ( r ) ≥ Acknowledgments
The author gratefully acknowledges useful past discussions with Tepper Gill,Tehani Finch, and Lenny Susskind. eferences [1] R. Colella, A.W. Overhauser, and S.A. Werner, “Observation of Gravitation-ally Induces Quantum Interference”,
Phys.Rev.Lett. , 1472-1474 (1975).See also A.W.Overhauser and R.Colella, Phys.Rev.Lett . , 1237 (1974).[2] H. Muller, A. Peters, and S. Chu, “A precision measurement of the gravi-tational redshift by the interference of matter waves”. Nature , 926-929(2010), doi:10:1038/nature08776.[3] T. Gill and J. Lindesay, “Canonical Proper Time Formulation of RelativisticParticle Dynamics”,
International Journal of Theoretical Physics , 2087-2098 (1993).[4] J. Lindesay and T. Gill, “Canonical Proper Time Formulation for PhysicalSystems”, Foundations of Physics (1), 169-182 (2004).[5] L. Susskind and J. Lindesay , An Introduction to Black Holes, Informa-tion, and the String Theory Revolution: The Holographic Universe ;World Scientific: Singapore 2005.[6] J. Lindesay,