Abstract
We define the rest-frame instant form of tetrad gravity restricted to Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of gauge transformations generated by the 14 first class constraints of the theory, we define and solve the multitemporal equations associated with the rotation and space diffeomorphism constraints, finding how the cotriads and their momenta depend on the corresponding gauge variables. This allows to find a quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal gauges and to find the Dirac observables for superspace in these gauges. The construction of the explicit form of the transformation and of the solution of the rotation and supermomentum constraints is reduced to solve a system of elliptic linear and quasi-linear partial differential equations. We then show that the superhamiltonian constraint becomes the Lichnerowicz equation for the conformal factor of the 3-metric and that the last gauge variable is the momentum conjugated to the conformal factor. The gauge transformations generated by the superhamiltonian constraint perform the transitions among the allowed foliations of spacetime, so that the theory is independent from its 3+1 splittings. In the special 3-orthogonal gauge defined by the vanishing of the conformal factor momentum we determine the final Dirac observables for the gravitational field even if we are not able to solve the Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted to this completely fixed gauge.
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REFERENCES
Lusanna, L., and Russo, S. (2002). “A New Parametrization for Tetrad Gravity” (GR-QC/ 0102074), Gen. Rel. Grav. 34 189.
Lusanna, L., and Russo, S., “Tetrad Gravity: I) A New Formulation,” Firenze Univ. preprint (GR-QC/9807072).
Lusanna, L., and Russo, S., “Tetrad Gravity: II) Dirac's Observables,” Firenze Univ. preprint (GR-QC/9807073).
De Pietri, R., and Lusanna, L., “Tetrad Gravity III: Asymptotic Poincar´e Charges, the Physical Hamiltonian and Void Spacetimes,” Firenze Univ. preprint (GR-QC/9909025).
Arnowitt, R., Deser, S., and Misner, C.W. (1960). Phys. Rev. 117, 1595; (1962) in “Gravitation: an Introduction to Current Research,” ed. Witten, L. (Wiley, New York).
Lusanna, L. (2001). “The Rest-Frame Instant Form of Metric Gravity,” Gen. Rel. Grav. 33, 1579.
Dirac, P. A. M. (1950). Can. J. Math. 2, 129; (1964) Lectures on Quantum Mechanics,” Belfer Graduate School of Science, Monographs Series (Yeshiva University, New York, N.Y.).
Anderson, J. L., and Bergmann, P. G. (1951). Phys. Rev. 83, 1018. Bergmann, P. G., and Goldberg, J. (1955). Phys. Rev. 98, 531.
Lusanna, L., (1990). Phys. Rep. 185, 1; (1991) Riv. Nuovo Cimento 14, n. 3, 1; (1990) J. Math. Phys. 31, 2126; (1990) J. Math. Phys. 31, 428. (1992) Contemp. Math. 132, 531.
Lusanna, L., “Towards a Unified Description of the Four Interactions inTerms of Dirac-Bergmann Observables,” invited contribution to the book “Quantum Field Theory: a 20th Century Profile,” of the Indian National Science Academy, ed. A. N. Mitra, foreward F. J. Dyson (Hindustan Book Agency, New Delhi) (HEP-TH/9907081). “Tetrad Gravity and Dirac's Observables,” talk given at the Conf. “Constraint Dynamics and Quantum Gravity 99,”Villasimius 1999, eds.V. DeAlfaro, J. E. Nelson, M. Cadoni, M. Cavaglia’ and A. T. Filippov, (2000). Nucl. Phys. B (Proc. Suppl.) 88, 301 (GR-QC/9912091). “The Rest-Frame Instant Form of Dynamics and Dirac's Observables,” talk given at the Int. Workshop “Physical Variables in Gauge Theories,” Dubna 1999. “Solving Gauss' Laws and Searching Dirac Observables for the Four Interactions,” talk at the “Second Conf. on Constrained Dynamics and Quantum Gravity,” S. Margherita Ligure 1996, eds. V. De Alfaro, J. E. Nelson, G. Bandelloni, A. Blasi, M. Cavagli`a and A. T. Filippov, (1997). Nucl. Phys. B (Proc.Suppl.) 57, 13 (HEP-TH/9702114). “Unified Description and Canonical Reduction to Dirac's Observables of the Four Interactions,” talk at the Int. Workshop “New non Perturbative Methods and Quantization on the Light Cone', Les Houches School 1997, eds. P. Grang´e, H. C. Pauli, A. Neveu, S. Pinsky and A. Werner (Springer, Berlin, 1998) (HEP-TH/9705154). “The Pseudoclassical Relativistic Quark Model in the Rest-Frame Wigner-Covariant Gauge,” talk at the Euroconference QCD97, ed. S. Narison, Montpellier 1997, (1998) Nucl. Phys. B (Proc. Suppl.) 64, 306.
Shanmugadhasan, S. (1973). J. Math. Phys. 14, 677. Lusanna, L. (1993). Int. J. Mod. Phys. A 8, 4193.
Chaichian, M., Louis Martinez, D., and Lusanna, L. (1994). Ann. Phys. (N.Y.) 232, 40.
Kuchar, K. (1976). J. Math. Phys. 17, 777, 792, 801; (1977) 18, 1589.
Dirac, P. A. M. (1949). Rev. Mod. Phys. 21, 392.
Lusanna, L. (1997). Int. J. Mod. Phys. A 12, 645.
Crater, H., and Lusanna, L. (2001). Ann. Phys. (NY) 289, 87 (HEP-TH/0001046). Alba, D., Crater, H., and Lusanna, L. (2001). Int. J. Mod. Phys. A 16, 3365 (HEP-TH/0103109).
Lusanna, L. (1995). Int. J. Mod. Phys. A 10, 3531 and 3675.
Møller, C. (1949). Ann. Inst. H. Poincar´e 11, 251; (1957) “The Theory of Relativity” (Oxford Univ. Press, Oxford).
Christodoulou, D., and Klainerman, S. (1993) “The Global Nonlinear Stability of the Minkowski Space” (Princeton University Press, Princeton).
Nakahara, M. (1990). “Geometry, Topology and Physics” (IOP, Bristol).
O'Neil, B. (1983). “Semi-Riemannian Geometry” (Academic Press, New York).
Bleecker, D. (1981). “Gauge Theory and Variational Principles” (Addison-Wesley, London).
Schwinger, J. (1963). Phys. Rev. 130, 1253.
Longhi, G., and Lusanna, L. (1986). Phys. Rev. D 34, 3707.
Dirac, P. A. M. (1951). Canad. J. Math. 3, 1.
Regge, T., and Teitelboim, C. (1974). Ann. Phys. (N.Y.) 88, 286.
Alba, D., Lusanna, L., and Pauri, M. (2002). J. Math. Phys. 43, 1677. (HEP-TH/0102087) and “Multipolar Expansions for the Relativistic N Body Problem in the Rest-Frame Instant Form (HEP-TH/0103092).
Lusanna, L., and Materassi, M. (1999). Int. J. Mod. Phys. A 15, 2821 (HEP-TH/9904202).
Hanson, A. J., and Regge, T. (1974). Ann. Phys. (N.Y.) 87, 498. Hanson, A. J., Regge, T., and Teitelboim, C. (1975) “Constrained Hamiltonian Systems,” in Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche, Fisiche e loro Applicazioni, n.22 (Accademia Nazionale dei Lincei, Roma).
Longhi, G., and Materassi, M. (1999). J. Math. Phys. 40, 480 (HEP-TH/9803128); (1999). Int. J. Mod. Phys. A 14, 3397 (HEP-TH/9890024).
Beig, R., and Murchadha, Ó. (1987). Ann. Phys. (N.Y.) 174, 463.
Andersson, L. (1987). J. Geom. Phys. 4, 289.
Landau, L., and Lifschitz, E. (1951) “The Classical Theory of Fields” (Addison-Wesley, Cambridge).
De Witt, B. S. (1967). Phys. Rev. 160, 1113.
De Witt, B. S. (1967). Phys. Rev. 162, 1195; (1967). “The Dynamical Theory of Groups and Fields” (Gordon and Breach, New York) and in (1964). “Relativity, Groups and Topology,” Les Houches 1963, eds. DeWitt, C., and DeWitt, B. S. (Gordon and Breach, London); (1984). “The Spacetime Approach to Quantum Field Theory,” in “Relativity, Groups and Topology II,” Les Houches 1983, eds. DeWitt, B. S., and Stora, R. (North-Holland, Amsterdam). De Witt, B. S., and Brehme, R. W. (1960). Ann. Phys. (N.Y.) 9, 220.
Hawking, S. W., and Horowitz, G. T. (1996). Class. Quantum Grav. 13, 1487.
Brill, D. M., and Jang, P. S. (1980). “The Positive Mass Conjecture,” in “General Relativity and Gravitation,” Vol. 1, ed. Held, A. Plenum, New York.
Stephani, H. (1996). “General Relativity” Cambridge Univ. Press, Cambridge.
Trautman, A. (1962). In: “Gravitation, an Introduction to Current Research,” ed. Witten, L. Wiley, New York.
Solov'ev, V. O. (1985). Theor. Math. Phys. 65, 1240; (1988) Sov. J. Part. Nucl. 19, 482.
Bergmann, P. G. (1961). Rev. Mod. Phys. 33, 510.
Marolf, D. (1996). Class Quantum Grav., 13, 1871.
Barbour, J. (1995). “General Relativity as a Perfectly Machian Theory,” in “Mach's Principle: From Newton's Bucket to Quantum Gravity,” eds. Barbour, J. B., and Pfister, H., Einstein's Studies n.6 (Birkh äuser, Boston).
Pauri, M., and Prosperi, M. (1975). J. Math. Phys. 16, 1503.
Choquet-Bruhat, Y., Fischer, A., and Marsden, J. E. (1979). “Maximal Hypersurfaces and Positivity of Mass,” LXVII E. Fermi Summer School of Physics “Isolated Gravitating Systems in General Relativity,” ed. Ehlers, J. (North-Holland, Amsterdam).
Soffel, M. H. (1989). “Relativity in Astrometry, Celestial Mechanics and Geodesy” (Springer, Berlin).
Abbati, M. C., Cirelli, R., Maniá, A., and Michor, P. (1989). J. Geom. Phys. 6, 215.
Schmidt, R. (1987). “Infinite Dimensional Hamiltonian Systems” (Bibliopolis, Napoli.) J. Milnor, (1984). In: “Relativity, Groups and Topology II,” Les Houches 1983, (De Witt, B. S., and Stora, R. Eds.), (Elsevier, Amsterdam).
Bao, D., Isenberg, J., and Yasskin, P. B. (1985). Ann. Phys. (N.Y.) 164, 103.
Helgason, S. (1962). “Differential Geometry and Symmetric Spaces” (Academic Press, New York).
Kobayashi, S., and Nomizu, K. (1963). “Foundations of Differential Geometry,” Vol. I (Interscience, New York, 1963).
Fischer, A. E. “The Theory of Superspace,” (1970). In: “Relativity,” eds. Carmeli, M., Fickler, L., and Witten, L. (Plenum, New York; (1983) Gen. Rel. Grav. 15, 1191; (1986) J. Math. Phys. 27, 718. Rainer, M. (1996). “The Moduli Space of Local Homogeneous 3-Geometries,” talk at the Pacific Conf. on Gravitation and Cosmology, Seoul.
Timothy Swift, S. (1992). J. Math. Phys. 33, 3723; (1993). 34, 3825 and 3841.
Arms, J. M., Marsden, J. E., and Moncrief, V. (1981). Commun. Math. Phys. 78, 455.
Moncrief, V. (1979). J. Math. Phys. 20, 579. Cantor, M. (1981). Bull. Am. Math. Soc. 5, 235.
Cendra, H., Ibort, A., and Marsden, J. (1987). J. Geom. Phys. 4, 183. Balachandran, A. P.,Marmo, G., Skagerstam, B. S., and Stern, A. (1991). “Classical Topology and Quantum States” (World Scientific, Singapore).
Giulini, D. (1995). Helv. Phys. Acta 68, 86.
Lee, J., and Wald, R. M. (1990). J. Math. Phys. 31, 725.
Antonsen, F., and Markopoulou, F. “4D Diffeomorphisms in Canonical Gravity and Abelian Deformations,” Imperial/TP/96-97/26 (GR-QC/9702046).
Teitelboim, C. (1980). “The Hamiltonian Structure of Space-Time,” In: “General Relativity and Gravitation,” ed. Held, A. Vol.I (Plenum, New York).
Kuchar, K. (1993). “Canonical Quantum Gravity” In: “General Relativity and Gravitation” Int.Conf. GR13, Cordoba (Argentina) 1992, eds. Gleiser, R. J., Kozameh, C. N., and Moreschi, O. M. (IOP, Bristol).
Beig, R. (1994). “The Classical Theory of Canonical General Relativity,” in “Canonical Gravity: From Classical to Quantum,” Bad Honnef 1993, eds. Ehlers, J., and Friedrich, H. Lecture Notes Phys. 434 (Springer, Berlin).
Kuchar, K. (1971). Phys. Rev. D 4, 955; (1970). J. Math. Phys. 11, 3322; 13, 768 (1972).
Misner, C. W. (1969). Phys. Rev. Lett. 22, 1071; (1969). Phys. Rev. 186, 1319 and 1328.
York, jr., J. W. (1979). “Kinematics and Dynamics of General Relativity,” in “Sources of Gravitational Radiation,” Battelle-Seattle Workshop 1978, ed. Smarr, L. L. (Cambridge Univ. Press, Cambridge). Qadir, A., and Wheeler, J. A. (1985). “York's Cosmic Time Versus Proper Time,” in “From SU(3) to Gravity,” Y. Ne'eman's festschrift, eds. E. Gotsma and G. Tauber (Cambridge Univ. Press, Cambridge).
Isham, C. J. (1993). “Canonical Quantum Gravity and the Problem of Time,” in “Integrable Systems, Quantum Groups and Quantum Field Theories,” eds. Ibort, L. A. and Rodriguez, M. A. Salamanca 1993 (Kluwer, London); (1991). “Conceptual and Geometrical Problems in Quantum Gravity,” in “Recent Aspects of Quantum Fields,” Schladming 1991, eds. H.Mitter and H.Gausterer (Springer, Berlin); (1994). “Prima Facie Questions in Quantum Gravity” and “Canonical Quantum Gravity and the Question of Time,” in “Canonical Gravity: From Classical to Quantum,” eds. J.Ehlers and H.Friedrich (Springer, Berlin).
Kuchar, K. (1992). “Time and Interpretations of Quantum Gravity,” in Proc. 4th Canadian Conf. on “General Relativity and Relativistic Astrophysics,” eds. Kunstatter, G., Vincent, D., and Williams, J. (World Scientific, Singapore).
Kuchar, K. (1981). “Canonical Methods of Quantization,” in “Quantum Gravity 2,” eds. Isham, C. J., Penrose, R., and Sciama, D. W. (Clarendon Press, Oxford).
Baierlein, R. F., Sharp, D. H., and Wheeler, J. A. (1962). Phys. Rev. 126, 1864.
Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1973). Gravitation (Freeman, New York).
Parentani, R. (1997). “The Notions of Time and Evolution in Quantum Cosmology,” GR-QC/9710130.
Kiefer, C., (1994). “The Semiclassical Approximation to Quantum Gravity” in “Canonical Gravity - from Classical to Quantum,” ed. Ehlers, J., (Springer, Berlin). (1994). “Semiclassical Gravity and the Problem of Time,” in Proc. Cornelius Lanczos Int.Centenary Conf., eds. Chu, M., Flemmons, R., Brown, D., and Ellison, D., (SIAM). (1996). Nucl. Phys. B 475, 339.
Bartnik, R., and Fodor, G., (1993). Phys. Rev. D 48, 3596.
Giulini, D., (1999). J. Math. Phys. 40, 2470.
Lusanna, L., (1981). Nuovo Cimento, B 65, 135.
Lichnerowicz, A. (1944). J. Math. Pure Appl. 23, 37. Faures-Bruhat, Y. (1948). C.R. Acad. Sci. Paris 226, 1071; (1956) J. Rat. Mech. Anal. 5, 951; (1962). “The Cauchy Problem” in “Gravitation: An Introduction to Current Research,” ed. Witten, L., (Wiley, New York).
York, jr, J.W. (1971). Phys. Rev. Lett. 26, 1656; 28, 1082 (1972). (1972) J. Math. Phys. 13, 125; 14, 456. (1974) Ann. Ins. H. Poincar´e XXI, 318. O'Murchadha, N., and York, jr, J. W. (1972). J. Math. Phys. 14, 1551 (1974). Phys. Rev. D 10, 428.
Choquet-Bruhat, Y., and York, jr., J. W. (1980). “The Cauchy Problem,” in “General Relativity and Gravitation,” vol.1, ed. Held, A. (Plenum, New York).
Ciufolini, I., and Wheeler, J. A. (1995). “Gravitation and Inertia” (Princeton University Press, Princeton).
Schoen, R., and Yau, S. T. (1979). Phys. Rev. Lett. 43, 1457; (1979). Commun. Math. Phys. 65, 45 and 79, 231 (1980).Witten, E., (1981). Commun. Math. Phys. 80, 381. Brill, D. M. and Jang, P. S. (1980). “The Positive Mass Conjecture,” in “General Relativity and Gravitation,” Vol. 1, ed. A. Held (Plenum, New York). Choquet-Bruhat, Y., (1984). “Positive Energy Theorems,” in “Relativity, Groups and Topology II,” Les Houches XL 1983, eds. DeWitt, B. S., and Stora, R., (North-Holland, Amsterdam). Horowitz, G. T. (1984). “The Positive Energy Theorem and its Extensions,” in “Asymptotic Behaviour of Mass and Spacetime Geometry,” ed. Flaherty, F. J., Lecture Notes Phys. 202 (Springer, Berlin). Perry, M. J. (1984). “The Positive Mass Theories and Black Holes,” in “Asymptotic Behaviour of Mass and Spacetime Geometry,” ed. Flaherty, F. J. Lecture Notes Phys. 202 (Springer, Berlin).
Isenberg, J., and Marsden, J. E. (1984). J. Geom. Phys. 1, 85.
Moncrief, V. (1965). J. Math. Phys. 16, 1556; Arms, J., Fischer, A., and Marsden, J. E., (1975). C.R. Acad. Sci. Paris A 281, 517; Arms, J., (1986). Acta Phys. Pol. B. 17, 499 and Contemp. Math. 81, 99 (1988). Arms, J. M., Gotay, M. J., and Jennings, G., (1990). Adv. Math. 79, 43.
Isenberg, J. (1987). Phys. Rev. Lett. 59, 2389.
Isenberg, J. (1995). Class. Quantum Grav. 12, 2249.
Isenberg, J., and Moncrief, V., (1996). Class. Quantum Grav. 13, 1819.
Bartnik, R., (1988). Commun. Math. Phys. 117, 615 Brill, D (1982). in Proc. Third Marcel Grossman Meeting, ed. Ning, H. (North-Holland, Amsterdam).
Dirac, P. A. M. (1949). Rev. Mod. Phys. 21, 392.
Gaida, R. P., Kluchkovsky, Yu. B. and Tretyak, V. I. (1983). Theor. Math. Phys. 55, 372; (1987). in “Constraint's Theory and Relativistic Dynamics,” eds. Longhi, G., and Lusanna, L. (World Scientific, Singapore, 1987).
Lusanna, L., (1990). (a) Phys. Rep. 185, 1 (b) (1991). Riv. Nuovo Cimento 14, n.3, 1. (c) (1990). J. Math. Phys. 31, 428 and 2126. (d) (1993). Int. J. Mod. Phys. A 8, 4193. (e) (1992). Contemp. Math. 132, 531.
Batalin, I. A., and Vilkoviski, G. A. (1984). Nucl. Phys. B 234, 106.
Synge, J. L. (1960). “Relativity: the General Theory” (North-Holland, Amsterdam, 1960).
Hwang, S., (1991). Nucl. Phys. B351, 425.
Dirac, P. A. M. (1962). in “Recent Developments in General Relativity,” Pergamon Press, Oxford, and PWN-Polish Scientific Publishers, Warsaw.
Isham, C. J. and Kuchar, K., (1984). Ann. Phys. (N.Y.) 164, 288 and 316. Kuchar, K., (1986). Found. Phys. 16, 193.
Cartan, E., (1951). “Lecons sur la Geometrie des Espaces de Riemann,” 2nd edn. (Gauthier-Villars, Paris).
Spivak, M., (1970). “Differential Geometry,” vol. 2 (Publish or Perish, Boston).
Chester, C. R. (1971). “Techniques in Partial Differential Equations” (McGraw-Hill Kogakusha, Tokyo).
Sugano, R., Kagraoka, Y., and Kimura, T., (1992). Int. J. Mod. Phys. A 7, 61.
Lifshitz, E. M., and Khalatnikov, I. M. (1963). Advan. Phys. 12, 185; Khalatnikov, I. M. and Lifshitz, E. M. (1970). Phys. Rev. Lett. 24, 76.
Bona, C., Massó, J., Seidel, E., and Walker, P., “Three Dimensional Numerical Relativity with a Hyperbolic Formulation,” GR-QC/9804052.
Choquet-Bruhat, Y., Isenberg, J., and York, J. W. Jr., “Einstein Constraints on Asymptotically Euclidean Manifolds,” GR-QC/9906095. Anderson, A., Choquet-Bruhat, Y., and York, J., Jr., “Einstein's Equations and Equivalent Dynamical Systems,” GR-QC/9907099 and (1997). “Curvature-Based Hyperbolic Systems for General Relativity,” talk at the 8th M. Grossmann eeting (Jerusalem, Israel), GR-QC/9802027. Anderson, A., and York, J. W., Jr. (1999). Phys. Rev. Letters 81, 1154 and 4384. York, J. W., Jr. (1999). Phys. Rev. Letters 82, 1350.
Bergmann, P. G., and Komar, A. B. (1960). Phys. Rev. Lett. 4, 432.
Stewart, J. (1993). “Advanced General Relativity,” (Cambridge Univ. Press, Cambridge).
d'Inverno, R. A., and Stachel, J. (1978). J. Math. Phys. 19, 2447; d'Inverno, R. (1997). “2+2 Formalism and Applications,” in “Relativistic Gravitation and Gravitational Radiation,” Les Houches 1995, eds. Marck, J. A., and Lasota, J. P. (Cambridge Univ. Press, Cambridge). d'Inverno, R., and Smallwood, J. (1980). Phys. Rev. D. 22, 1233; Smallwood, J. (1983). J. Math. Phys. 24, 599; Torre, C. G. (1986). Class. Quantum Grav. 3, 773; Hayward, S. A. (1993). Class. Quantum Grav. 10, 779.
Kuchar, K., (1972). J. Math. Phys. 13, 768.
Smolin, L., “The present moment in quantum cosmology: challenges to the arguments for the elimination of time,” (GR-QC/0104097).
Witten, E. (1981). Commun. Math. Phys. 80, 381.
Sen, A. (1981). J. Math. Phys. 22, 1781; (1982). Phys. Lett. 119B, 89.
Sen, A. (1982). Int. J. Theor. Phys. 21, 1; Sommers, P. (1980). J. Math. Phys. 21, 2567.
Ashtekar, A., (1988). “New Perspectives in Canonical Gravity” (Bibliopolis, Napoli).
Penrose, R., and Rindler, W., (1986). “Spinors and Space-Time” vol.1 and 2. (Cambridge Univ. Press, Cambridge).
Choquet-Bruhat, Y., and Christodoulou, D., (1981). Acta Math. 146, 129; Reula, O., (1982). J. Math. Phys. 23, 810; Reula, O., and Todd, K., (1984). J. Math. Phys. 25, 1004; Parker, T., and Taub, C. H. (1982). Commun. Math. Phys. 84, 223; Parker, T. (1985). Commun. Math. Phys. 100, 471; Biz´on, P., and Malec, E., (1986). Class. Quantum Grav. 3, L123.
Frauendiener, J., (1991). Class. Quantum Grav. 8, 1881.
Frauendiener, J., (1989). Class. Quantum Grav. 6, L237.
Einstein, A. (1916). Sitzungsber Preuss. Akad. Wiss. Phys. Math. Kl. 42, 1111.
Møller, C., (1961). Ann. Phys. (N.Y.) 12, 118; in Proc. Int. School of Physics Fermi, E., (1962). Course XX (Academic Press, New York).
Pirani, F. A. E. (1962). “Gauss' Theorem and Gravitational Energy,” in “Les Theories Relativistes de la Gravitation,” Proc. Int. Conf. at Royaumont 1959, eds. Lichnerowicz, A., and Tonnelat, M. A. CNRS, Paris.
Goldberg, J. N. (1988). Phys. Rev. D 37, 2116.
Rosen, N. (1940). Phys. Rev. 57, 147; (1963) Ann. Phys. (N.Y.) 22, 1; Found. Phys. (1986) 15, 998); in “From SU(3) to Gravity,” Ne'eman's, Y., festschrift, eds. Gotsman, E., and Tauber, G., (Cambridge Univ. Press, Cambridge); (1986) in “Topological Properties and Global Structure of Space-Time,” eds. Bergmann, P. G., and de Sabbata, V. (Plenum, New York).
Petrov, A. Z. (1969). “Einstein Spaces” (Pergamon, Oxford.
Robinson, D. C. (1989). Class. Quantum Grav. 6, L121.
Bailey, I., and Israel, W. (1980). Ann. Phys. (N.Y.) 130, 188.
Dixon, W. G. (1967). J. Math. Phys. 8, 1591. (1979). “Extended Objects in General Relativity: their Description and Motion,” in “Isolated Gravitating Systems in General Relativity,” ed. J. Ehlers (North-Holland, Amsterdam).
Kovalevvsky, J., Mueller, I. I., and Kolaczek, B. (eds.) (1989). “Reference Frames in Astronomy and Geophysics��� (Kluwer, Dordrecht).
Dixon, W. G. (1973). Gen. Rel. Grav. 4, 199.
DeWitt, B. S., and Brehme, R. W. (1960). Ann. Phys. (N.Y.) 9, 220.
Norton, J. D. (1989). “What was Einstein's Principle of Equivalence?,” in “Einstein and the History of General Relativity: Einstein Studies,” Vol. 1, eds. Howard, D., and Stachel, J. (Birkhäuser, Boston). (1993) Rep. Prog. Phys. 56, 791 (1993).
Abramowicz, M. A. (1993). “Inertial Forces in General Relativity,” in “The Renaissance of General Relativity and Cosmology,” eds. G. Ellis, A. Lanza and J. Miller (Cambridge Univ. Press, Cambridge, 1993). Sonego, S., and Massar,M. (1996). Class. Quantum Grav. 13, 139. De Felice, F. (1991). Mon. Not. R. Astron. Soc. 252, 197.
Pauri, M., and Vallisneri, M. (1999). “Classical Roots of the Unruh and Hawking Effects,” Found. Phys. 29, 1499 (GR-QC/9903052).
Lusanna, L., and Nowak-Szczepaniak, D. (2000). Int. J. Mod. Phys. A 15, 4943 (HEP-TH/0003095).
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De Pietri, R., Lusanna, L., Martucci, L. et al. Review: Dirac's Observables for the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge. General Relativity and Gravitation 34, 877–1033 (2002). https://doi.org/10.1023/A:1016369931750
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DOI: https://doi.org/10.1023/A:1016369931750