Novel Solution of Wheeler-DeWitt Theory

Lukasz Andrzej Glinka

  Open Access OPEN ACCESS  Peer Reviewed PEER-REVIEWED

Novel Solution of Wheeler-DeWitt Theory

Lukasz Andrzej Glinka

B.M. Birla Science Centre, Hyderabad, India


Taking into account the global one-dimensionality conjecture recently proposed by the author, the Cauchy-like analytical wave functional of the Wheeler-DeWitt theory is derived. The crucial point of the integration strategy is canceling of the singular behavior of the effective potential, which is performed through the suitable change of variables introducing the invariant global dimension. In addition, the conjecture is extended onto the wave functionals dependent on both Matter felids as well as the invariant global dimension. Through application of the reduction within the quantum gravity model, the appropriate Dirac equation is obtained and than solved. The case of superposition is also briey discussed.

Cite this article:

  • Glinka, Lukasz Andrzej. "Novel Solution of Wheeler-DeWitt Theory." Applied Mathematics and Physics 2.3 (2014): 73-81.
  • Glinka, L. A. (2014). Novel Solution of Wheeler-DeWitt Theory. Applied Mathematics and Physics, 2(3), 73-81.
  • Glinka, Lukasz Andrzej. "Novel Solution of Wheeler-DeWitt Theory." Applied Mathematics and Physics 2, no. 3 (2014): 73-81.

Import into BibTeX Import into EndNote Import into RefMan Import into RefWorks

1. Introduction

The Wheeler-DeWitt theory, also known as quantum geometrodynamics or quantum General Relativity, is the foundational model of quantum gravity considered in modern theoretical physics, Cf. Ref. [1-38][1]. This model straightforwardly arises from the canonical General Relativity, formulated on the basis of the Arnowitt-Deser-Misner decomposition, well known as the 3 + 1 splitting, of a four-dimensional space-time metric, applied to the Einstein-Hilbert action supplemented by the York-Gibbons-Hawking boundary term. This procedure produces the Hamiltonian action, as well as the primary and secondary constraints satisfying the first-class algebra whose are canonically quantized according to the Dirac method. The quantized Hamiltonian constraint is the quantum evolution known as the Wheeler-DeWitt equation, which is the second order functional differential equation on the abstract configuration space known as the Wheeler superspace and containing all three-dimensional embedded geometries, whose solutions known as wave functionals in general depend on an induced three-metric and Matter fields.

The heart of the matter, however, is the question of integrability of the Wheeler-DeWitt equation, and, for this reason, the possible new physical meaning of the quantum geometrodynamics could arise along with the integration strategy. Since the 1970s, S. W. Hawking and his coauthors [39-63][39] have proposed to solve the Wheeler-DeWitt equation through making use of the formal analogy with the Schrödinger equation of usual quantum mechanics, and applied the Feynman path integral method which, however, generates manifestly non-analytical wave functionals, that is the solutions which do not form the Cauchy surface necessary to the rational analysis of any differential equation. The approach, sometimes called the Hartle-Hawking wave function, is correct from the point of view of quantum field theory, but actually instead of concrete calculations of the path integrals and development of the method beyond the simplest cosmological solutions of the Einstein field equations, the only qualitative ideas which link the Feynman integration with quantum cosmology have been proposed. Since this approach is far from a mathematical consistency, still the question which is neglected in the literature are other possible solutions to the Wheeler-DeWitt equation, including both non-analytical and analytical ones, which could be beyond the solutions defined through the method of path integration. It is worth stressing that, in fact, both any one analytical solution, that is the Cauchy-like wave functional, to quantum geometrodynamics and its possibly interesting physical meaning are still unknown. This point is very unsatisfactory, and, consequently, makes the quantum geometrodynamics a theory produced in the way of a false analogy with the formalism of quantum mechanics.

Nevertheless, the discussion of the qualitative matter of quantum geometrodynamics is not the subject of this paper, whereas we will present here the way to receive the analytical solutions to the Wheeler-DeWitt equation throughout a systematic construction. For this reason, we apply the global one-dimensional conjecture, recently discussed by the author's writings [64, 65, 66], which is immediately rooted in the generic quantum cosmology [67, 68, 69, 70] dedicated to the Einstein-Friedmann Universe. The main result of this conjecture are the wave functional which are dependent on the determinant of the threedimensional metric, which is named the global dimension, and the resulting theory is the Schrödinger quantum mechanics in the one dimension. The crucial point of the integration strategy is application of the suitable change of variables which removes the singular behavior of the effective potential. In result, we receive the concept of invariant global dimension, where the word invariance is related to the invariant integral measure on a spacelike hypersurface, and the presence of Matter fields is included. Finally, we show the way to transform the theory into the suitable Dirac equation, which defined the new strategy for quantum geometrodynamics and is solved to receive the analytical wave functionals.

The paper is organized in the following way. In the Section 2, the basic facts about quantum geometrodynamics are collected. The Section 3 briey discusses the global one-dimensional conjecture, including the concept of the invariant global dimension. The suitable Dirac equation is obtained in the Section 4, and the new type of analytical wave functionals is constructed in the Section 5. In Section 6, certain consequences are presented, whereas in the Section 7 all results are summarized.

2. Canonical Quantum Gravity

Let us recall the basic facts, for details Cf. Ref. [71, 72, 73, 74]. General Relativity, governed by the Einstein field equations{1}


where is a cosmological constant and is a stress-energy tensor of Matter fields, models space-time as a four-dimensional pseudo-Riemannian manifold equipped with a metric the Riemann-Christoffel curvature tensor the Ricci second fundamental form and the Ricci scalar curvature If is closed and has an induced spacelike boundary with an induced metric the second fundamental form and an extrinsic curvature then the field equations (1) are the equations of motion following from the variational principle applied to the Einstein-Hilbert action with the York-Gibbons-Hawking term [75, 76]


and the stress-energy tensor generated by the variational principle is


where is Matter fields Lagrangian. The appropriate embedding theorems allow to make use of the 3 + 1 splitting [77, 78].


for which the action (2) takes the Hamiltonian form with


where are canonical conjugate momenta, and are [79].


with and holds


where is an intrinsic covariant derivative of are generators of the spatial diffeomorphisms [80].


where and the first-class algebra holds


where are the structure constants of the diffeomorphism group, and all other Lie's brackets vanish. Timepreservation [81, 82, 83, 84] of the primary constraints, that is and leads to the secondary constraints - scalar (Hamiltonian) and vector respectively


where the scalar constraint yields dynamics, while the vector one merely reects diffeoinvariance. Making use of the canonical momentum one obtains the Einstein-Hamilton-Jacobi equation


where is the DeWitt metric on the Wheeler superspace [85, 86, 87, 88]. The Dirac quantization method [81, 82, 83, 84]


applied to the constraint (14), yields the Wheeler-DeWitt equation [80, 89, 90].


whereas other first class constraints merely reect diffeoinvariance


and are not important in this model, called quantum geometrodynamics.

3. Global One-dimensional Conjecture

The global one-dimensionality conjecture [64, 65, 66], establishes the strategy within quantum geometrodynamics which allows to receive analytical solutions. Making use of the Jacobi rule for differentiation of a determinant of a metric one obtains


where is the diffeoinvariant variable which is third order in and is the Levi-Civita density. Consequently, one has the differentiation rule


which applied to the quantum geometrodynamics (17), with making the double contraction of the supermetric with an embedding metric, leads to


and finally the Wheeler-DeWitt equation (17) becomes the usual differential equation


where is the effective potential


The first term in (23) describes contribution due to an embedding geometry only, the second one is mix of the cosmological constant and an embedding geometry, and the third component is due to Matter fields and an embedding geometry. In result, one has to deal with wave functional what agrees with the basic diffeoinvariace (18).

The potential (23) has a manifestly singular behavior which however can be canceled through the appropriate change of variables


where we have introduced the new global dimension called here the invariant dimension, which is a functional of the global dimension and, therefore, also diffeoinvariant. With (24) the equation (22) becomes




In fact, is a kind of the gauge, wherein is generic. Note that the following choice


cancels the singularity in (23), and the equation (27) becomes


with the appropriate normalization condition


where is the invariant product functional measure. Note that both and are the Lebesgue-Stieltjes (Radon) integral measures which can be rewritten as the Riemann measures


what relates the superspace to the space-time.

4. The Dirac equation

Eq. (27) can be derived as the Euler-Lagrange equation of motion by variational principle applied to the action


where partial differentiation was used. Choosing the coordinate system so that the boundary term vanishes


and using the standard definition


one obtains the Lagrangian of the Euclidean field theory


for which the corresponding canonical conjugate momentum is


and, therefore, the choice (36) actually means orthogonal coordinates


for any values of and . Applying (39) in (27), one receives


and combining with (39), the appropriate Dirac equation is obtained


where we have employed the notation


and the -matrices algebra consists only one element - the Pauli matrix


where I is the identity matrix, that in itself obey the algebra


Dimensional reduction of the one component second order theory (27) yields the two component first order one (42) determined by the Eucludean Clifford algebra , Cf. Ref. [91], that is the matrix algebra having a complex two-dimensional representation, which decomposes into a direct sum of two isomorphic central simple algebras or a tensor product


Restricting to yield a two-dimensional spin representations; splits it onto a sum of two one-dimensional Weyl representations.

5. Analytic Wave Functional

The Dirac equation (42) can be rewritten in the Schrödinger form


whose most general solution can be written as


where is an initial data vector with respect to only, is a unitary evolution operator


and is a finite integration area in -space, whereas the volume of full configuration space and the averaged energy are


where is a finite integration region of full configuration space. Explicitly




and, consequently, the received wave functional are


whereas the canonical conjugate momentum is


where and are initial data with respect to . Applying (39) in (56), one obtains


where and and calculating


with using (57), one receives the formula


which compared with (56) leads to the system of equations


The first equation of the system (61) yields the relation


where the last integral arises by the first formula in (51), which after application to the second equation gives simply and, moreover, the volume is -invariant


The probability density can be deduced easily by (55)


and, in the light of the relation (40), one has


Assuming the following separation conditions


where and are functionals of only, while and are constant functionals, and applying the usual normalization, one obtains


where the constants A and B are given by the integrals


assumed to be convergent and _nite. The solution to the equation (67)


joined with (39) and (66) gives the di_erential equation for the initial data


which can be integrated


where C is a constant of integration, and gives the formula


which is equivalent to


Because must be a functional of , one has with a constant , and, moreover, Taking into account (70), one obtains


In the light of the equation (40), however, one of the relations is always true


One sees that in any case has discrete values. By the first relation in (76)


where is an integer, while the second relation in (76) gives


For the first case one has


whereas and for the second one


Finally, the invariant one-dimensional wave functional (55) becomes


in the first case of (76), while for the second one


6. Developments

6.1. General Solutions

The general analytic solutions of the reduced quantum geometrodynamics can be now constructed for any induced metric from the solutions (81) and (82). It can be easily seen that






Making use of (83) in the solutions (81) and (82), one obtains the general solutions according to the global one-dimensionality conjecture




are assumed to be convergent and finite constants. The normalization condition


applied in the solutions (87) and (88), leads to


which yield and, therefore,




The solutions (93) and (94) describe two independent quantum gravity states.

6.2. Superposition

Because, the equations (17) and (31) are linear, the superposition


where are arbitrary constants, could be considered as the most general solution, for which the normalization condition (91) is the constraint




For (98) gives simply


The case is more complicated. Note that the constraint (98) gives


or, equivalently, for




which after mutual adding and making use of (98) gives


and, consequently,


The complex decomposition for and applied in (105) leads to


or, equivalently,


Employing (107) within the constraint (98), one obtains


Because both as squares of absolute values, one obtains the values of the constant in dependence on the integral


where for the condition holds.

7. Summary

We have discussed few consequences of quantum geometrodynamics according to the global one-dimensional conjecture. Employment of the conjecture immediately led us to construction of the analytic solutions, wherein the strategy of integration used the concept of invariant dimension instead of the global dimension introduced to remove the singular behavior of the effective potential. In general, the procedure has used for computations the Lebesgue-Stieltjes, or Radon, one-dimensional integrals, and, therefore, meaningfully simplified considerations of quantum gravity and led to analytical wave functionals. Finally, we have discussed developments of the strategy. The first one was construction of the solutions for any induced metric, which differ from the Feynman path integral solutions, whereas the second one was the question of superposition. Certainly, there are open problems related to the novel wave functionals. The reader interested in advancements is advised to take into account the author's monograph [92].


[1]  J. R. Klauder (ed.), Magic without magic: John Archibald Wheeler (Freeman, 1972).
In article      
[2]  C. J. Isham, R. Penrose, and D. W. Sciama (eds.), Quantum Gravity. An Oxford symposium (Oxford University Press, 1975).
In article      
[3]  R. Balian and J. Zinn-Justin (eds.), Methods in Field Theory. Les Houches, École D' Été De Physique Théorique. Session XXVIII (North-Holland, 1976).
In article      
[4]  C. J. Isham, R. Penrose, and D. W. Sciama (eds.), Quantum Gravity 2. A second Oxford symposium (Oxford University Press, 1981).
In article      
[5]  S. M. Christensen (ed.), Quantum Theory of Gravity. Essays in honor of the 60th birthday of Bryce S. DeWitt. (Adam Hilger, 1984).
In article      
[6]  R. Penrose and C. J. Isham (eds.), Quantum concepts in space and time (Oxford University Press, 1986).
In article      
[7]  M. A. Markov, V. A. Berezin, and V. P. Frolov (eds.), Quantum Gravity. Proceedings of the Fourth Seminar, May 25-29, 1987, Moscow, USSR (World Scientific, 1988).
In article      
[8]  J. Audretsch and V. de Sabbata (eds.), Quantum mechanics in curved space-time (Plenum Press, 1990).
In article      CrossRef
[9]  A. Ashtekar and J. Stachel (eds.), Conceptual problems of quantum gravity (Birkhauser, 1991).
In article      
[10]  S. Coleman, J. B. Hartle, T. Piran, and S. Weinberg (eds.), Quantum Cosmology and baby Universes (World Scientific, 1991).
In article      CrossRef
[11]  I. L. Buchbinder, S. D. Odintsov, and I. L. Shapiro, Effective Action in Quantum Gravity (Institute of Physics Publishing, 1992).
In article      
[12]  D. J. Gross, T. Piran, and S. Weinberg (eds.), Two Dimensional Quantum Gravity and Random Surfaces (World Scientific, 1992).
In article      
[13]  M. C. Bento, O. Bertolami, J. M. Mourão, and R. F. Picken (eds.), Classical and quantum gravity (World Scientific, 1993).
In article      
[14]  G. W. Gibbons and S. W. Hawking (eds.), Euclidean Quantum Gravity (World Scientific, 1993).
In article      
[15]  J. C. Baez (ed.), Knots and Quantum Gravity (Clarendon Press, 1994).
In article      
[16]  J. Ehlers and H. Friedrich (eds.), Canonical Gravity: From Classical to Quantum (Springer, 1994).
In article      CrossRef
[17]  G. Esposito, Quantum Gravity, Quantum Cosmology and Lorentzian Geometries (Springer, 1994).
In article      CrossRef
[18]  E. Prugovečki, Principles of Quantum General Relativity (World Scien- tific, 1995).
In article      
[19]  R. Gambini and J. Pullin, Loops, Knots, Gauge Theories and Quantum Gravity (Cambridge University Press, 1996).
In article      CrossRef
[20]  G. Esposito, A. Yu. Kamenshchik, and G. Pollifrone, Euclidean Quan- tum Gravity on Manifolds with Boundary (Springer, 1997).
In article      CrossRef
[21]  P. Fré, V. Gorini, G. Magli, and U. Moschella, Classical and Quantum Black Holes (Institute of Physics Publishing, 1999).
In article      CrossRef
[22]  I. G. Avramidi, Heat Kernel and Quantum Gravity (Springer, 2000).
In article      
[23]  J. Kowalski-Glikman (ed.), Towards Quantum Gravity (Springer, 2000).
In article      CrossRef
[24]  C. Callender and N. Huggett (eds.), Physics meets philosophy at the Planck scale. Contemporary theories in quantum gravity. (Cambridge University Press, 2001).
In article      CrossRef
[25]  B. N. Kursunoglu, S. L. Mintz, and A. Perlmutter (eds.), Quantum Grav- ity, Generalized Theory of Gravitation and Superstring Theory-Based Unif ication (Kluwer Academic Press, 2002).
In article      CrossRef
[26]  S. Carlip, Quantum Gravity in 2+1 Dimensions (Cambridge University Press, 2003).
In article      
[27]  G. W. Gibbons, E. P. S. Shellard, and S. J. Rankin (eds.), The Future of Theoretical Physics and Cosmology (Cambridge University Press, 2003).
In article      
[28]  D. Giulini, C. Kiefer, and C. Lämmerzahl (eds.), Quantum Gravity. From Theory To Experimental Search (Springer, 2003).
In article      
[29]  C. Rovelli, Quantum Gravity (Cambridge University Press, 2004).
In article      CrossRef
[30]  G. Amelino-Camelia and J. Kowalski-Glikman (eds.), Planck Scale Ef- fects in Astrophysics and Cosmology (Springer, 2005).
In article      
[31]  A. Gomberoff and D. Marolf (eds.), Lectures on Quantum Gravity (Springer, 2005).
In article      CrossRef
[32]  D. Rickles, S. French, and J. Saatsi (eds.), The Structural Foundations of Quantum Gravity (Clarendon Press, 2006).
In article      CrossRef
[33]  B. Carr (ed.), Universe of Multiverse? (Cambridge University Press, 2007).
In article      
[34]  B. Fauser, J. Tolksdorf, and E. Zeidler (eds.) Quantum Gravity. Mathe- matical Models and Experimental Bounds (Birkh¨auser, 2007).
In article      CrossRef
[35]  D. Gross, M. Henneaux, and A. Sevrin (eds.), The Quantum Structureof Space and Time (World Scientific, 2007).
In article      
[36]  C. Kiefer, Quantum Gravity (2nd ed., Oxford University Press, 2007).
In article      CrossRef
[37]  T. Thiemann, Modern Canonical Quantum General Relativity (Cambridge University Press, 2007).
In article      CrossRef
[38]  D. Oriti, Approaches to Quantum Gravity. Toward a New Understanding of Space, Time, and Matter (Cambridge University Press, 2009).
In article      CrossRef
[39]  S. W. Hawking, Commun. Math. Phys. 43, 199 (1975).
In article      CrossRef
[40]  ibid. 55, 133 (1977).
In article      
[41]  Phys. Rev. D 13, 191 (1976).
In article      CrossRef
[42]  ibid. 14, 2460 (1976).
In article      
[43]  ibid. 18, 1747 (1978).
In article      
[44]  ibid. 32, 259 (1985).
In article      
[45]  ibid. 37, 904 (1988).
In article      
[46]  Phys. Lett. B 134, 403 (1984).
In article      CrossRef
[47]  Nucl. Phys. B 239, 257 (1984).
In article      CrossRef
[48]  Phys. Scr. T 117, 49 (2005).
In article      
[49]  contributions in: S. W. Hawking and W. Israel (eds.), General Relativity: An Einstein centenary survey, pp. 746-785 (Cambridge Uni- versity Press, 1979).
In article      
[50]  M. Levy and S. Deser (eds.), Recent Developments in Gravitation. Cargese 1978, pp. 145-175 (Plenum Press, 1979).
In article      CrossRef
[51]  B. S. DeWitt and R. Stora (eds.), Relativity, Groups, and Topology II, pp. 333-381 (Elsevier, 1984).
In article      
[52]  H. J. de Vega and N. Sánchez (eds.), Field Theory, Quantum Gravity, and Strings. Proceedings of a Seminar Se- ries Held at DAPHE , Observatoire de Meudon, and LPTHE, Université Pierre et Marie Curie, Paris, Between October 1984 and October 1985, pp. 1-46 (Springer, 1986).
In article      
[53]  J. J. Halliwell, J. Perez–Marcader, and W. H. Zurek (eds.), Physical Origins of Time Assymetry (Cambridge Uni- versity Press, 1992).
In article      
[54]  S. W. Hawking and W. Israel (eds.), Three hundred years of gravitation, pp. 631-652 (Cambridge University Press, 1987).
In article      
[55]  G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 107 (1977).
In article      
[56]  J. B. Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 (1983).
In article      CrossRef
[57]  J. J. Halliwell and S. W. Hawking, Phys. Rev. D 31, 1777 (1985).
In article      CrossRef
[58]  S. W. Hawking and D. Page, Nucl. Phys. B 298, 789 (1988).
In article      CrossRef
[59]  S. W. Hawking, R. Laflamme, and G. W. Lyons, Phys. Rev. D 47, 5342 (1993).
In article      CrossRef
[60]  R. Bousso and S. W. Hawking, SU-ITP-98-26 DAMTP-1998-87.
In article      
[61]  S. W. Hawking and T. Hertog, Phys. Rev. D 73, 123527 (2006).
In article      CrossRef
[62]  J. B. Hartle, S. W. Hawking, and T. Hertog, Phys. Rev. Lett. 100, 201301 (2008).
In article      CrossRef
[63]  Phys. Rev. D 77, 123537 (2008).
In article      CrossRef
[64]  L. A. Glinka, Grav. Cosmol. 16 (1), pp. 7-15, (2010).
In article      CrossRef
[65]  Concepts Phys. 6, pp. 19-41 (2009).
In article      
[66]  New Adv. Phys. 2, pp. 1-62 (2008).
In article      
[67]  L. A. Glinka, Grav. Cosmol. 15 (4), pp. 317-322 (2009).
In article      CrossRef
[68]  AIP Conf. Proc. 1018, pp. 94-99 (2008).
In article      CrossRef
[69]  in E. Ivanov and S. Fedoruk (eds.), Supersym- metries and Quantum Symmetries: Proc. of International Workshop, Dubna, Russia, July 30 - Aug. 4, 2007 (JINR Dubna, 2008), pp. 406-411.
In article      
[70]  SIGMA 3, pp. 087-100 (2007).
In article      
[71]  Ch. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation (Freeman, 1973).
In article      
[72]  R. M. Wald, General Relativity (University of Chicago, 1984).
In article      CrossRef
[73]  S. Carroll, Spacetime and Geometry. An Introduction to General Relativity (Addison-Weseley, 2004).
In article      
[74]  E. Poisson, A relativist's toolkit. The mathematics of black-hole mechan-ics (Cambridge University Press, 2004).
In article      CrossRef
[75]  J. W. York, Phys. Rev. Lett. 28, 1082 (1972).
In article      CrossRef
[76]  G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977).
In article      CrossRef
[77]  R. Arnowitt, S. Deser and C. W. Misner, in L. Witten (ed.) Gravitation: An Introduction to Current Research, pp. 227–264 (John Wiley and Sons, 1962).
In article      
[78]  B. DeWitt, The Global Approach to Quantum F ield Theory (Clarendon Press, 2003).
In article      
[79]  A. Hanson, T. Regge, and C. Teitelboim, Constrained Hamiltonian Sys- tems (Accademia Nazionale dei Lincei, 1976).
In article      
[80]  B. S. DeWitt, Phys. Rev. 160, 1113 (1967).
In article      CrossRef
[81]  P. A. M. Dirac, Lectures on Quantum Mechanics (Belfer Graduate School of Science, Yeshiva University, 1964).
In article      
[82]  Phys. Rev. 114, 924 (1959).
In article      CrossRef
[83]  Proc. Roy. Soc. (London) A 246, 326 (1958).
In article      
[84]  Can. J. Math. 2, 129 (1950).
In article      CrossRef
[85]  A. E. Fischer, in M. Carmeli, S.I. Fickler, and L. Witten (eds.) Rela- tivity. Proceedings of the Relativity Conference in the Midwest held at Cincinnati, Ohio, June 2-6, 1969, pp. 303–359 (Plenum Press, 1970).
In article      
[86]  Gen. Rel. Grav. 15, 1191 (1983).
In article      CrossRef
[87]  J. Math. Phys. 27, 718 (1986).
In article      CrossRef
[88]  B. S. DeWitt, in M. Carmeli, S.I. Fickler, and L. Witten (eds.) Rela- tivity. Proceedings of the Relativity Conference in the Midwest held at Cincinnati, Ohio, June 2-6, 1969, pp. 359-374 (Plenum Press, 1970).
In article      
[89]  J. A. Wheeler, Geometrodynamics (Academic Press, 1962). in C. DeWitt and B. DeWitt (eds.) Relativity, Groups, and Topology. Lectures Delivered at Les Houches During the 1963 Session of the Summer School of Theoretical Physics, pp. 317-501 (Gordon and Breach Science Publishers, 1964).
In article      
[90]  Einsteins Vision (Springer 1968). in C. M. DeWitt and J. A. Wheeler (eds.) Battelle Rencontres 1967 Lectures in Mathematics and Physics, pp. 242-308 (W. A. Benjamin, 1968).
In article      
[91]  V. V. Fernández, A. M. Moya, and W. A. Rodrigues Jr, Adv. Appl. Clifford Alg. 11, 1 (2001).
In article      CrossRef
[92]  L.A. Glinka, AEthereal Multiverse: A New Unifying Theoretical Approach to Cosmology, Particle Physics, and Quantum Gravity (Cambridge International Science Publishing, 2012).
In article      


1In this paper we use the units in units8πG/3=1, c=1,ћ=1.

  • CiteULikeCiteULike
  • MendeleyMendeley
  • StumbleUponStumbleUpon
  • Add to DeliciousDelicious
  • FacebookFacebook
  • TwitterTwitter
  • LinkedInLinkedIn