Abstract
We derive Einstein’s equations from a linear theory in flat space-time using free-field gauge invariance and universal coupling. The gravitational potential can be either covariant or contravariant and of almost any density weight. We adapt these results to yield universally coupled massive variants of Einstein’s equations, yielding two one-parameter families of distinct theories with spin 2 and spin 0. The Freund-Maheshwari-Schonberg theory is therefore not the unique universally coupled massive generalization of Einstein’s theory, although it is privileged in some respects. The theories we derive are a subset of those found by Ogievetsky and Polubarinov by other means. The question of positive energy, which continues to be discussed, might be addressed numerically in spherical symmetry. We briefly comment on the issue of causality with two observable metrics and the need for gauge freedom and address some criticisms by Padmanabhan of field derivations of Einstein-like equations along the way.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Schulmann, A. J. Kox, M. Janssen, and J. Illy, eds., The Collected Papers of Albert Einstein, Vol. 8, The Berlin Years: Correspondence, 1914–1918, Princeton Univ. Press, Princeton, N. J. (1998); A. Beck and D. Howard, eds., The Collected Papers of Albert Einstein, Vol. 4, The Swiss Years: Writings, 1912–1914, Princeton Univ. Press, Princeton, N. J. (1996); M. Janssen, Ann. Phys. (8), 14, No. S1, 58 (2005).
M. Janssen and J. Renn, “Untying the knot: How Einstein found his way back to field equations discarded in the Zurich notebook,” in: The Genesis of General Relativity: Sources and Interpretations (Boston Stud. Phil. Sci., Vol. 250, J. Renn, ed.), Vol. 2, Einstein’s Zurich Notebook: Commentary and Essays, Springer, New York (2007); http://www.tc.umn.edu/~janss011/pdf%20files/knot.pdf.
M. Fierz and W. Pauli, Proc. Roy. Soc. London, Ser. A, 173, 211 (1939); N. Rosen, Phys. Rev., 57, 147, 150 (1940); A. Papapetrou, Proc. Roy. Irish Acad. Sect. A, 52, 11 (1948); S. N. Gupta, Phys. Rev., 96, 1683 (1954).
R. H. Kraichnan, Phys. Rev., 98, 1118 (1955).
M. Kohler, Z. Phys., 134, 286 (1953).
R. P. Feynman, F. B. Morinigo, W. G. Wagner, B. Hatfield, J. Preskill, and K. S. Thorne, Feynman Lectures on Gravitation, Addison-Wesley, Reading, Mass. (1995).
V. I. Ogievetsky and I. V. Polubarinov, Ann. Phys., 35, 167 (1965).
S. Weinberg, Phys. Rev., 138, B988 (1965).
S. Deser, Gen. Relativity Gravitation, 1, 9 (1970); arXiv:gr-qc/0411023v2 (2004).
L. P. Grishchuk, A. N. Petrov, and A. D. Popova, Comm. Math. Phys., 94, 379 (1984).
A. A. Logunov and M. A. Mestvirishvili, Theor. Math. Phys., 86, 1 (1991).
J. B. Pitts and W. C. Schieve, Gen. Relativity Gravitation, 33, 1319 (2001); arXiv: gr-qc/0101058v2 (2001).
J. B. Pitts and W. C. Schieve, Found. Phys., 34, 211 (2004); arXiv:gr-qc/0406102v1 (2004).
N. Boulanger and M. Esole, Class. Q. Grav., 19, 2107 (2002); arXiv:gr-qc/0110072v2 (2001).
S. Deser, Class. Q. Grav., 4, L99 (1987).
N. Boulanger, T. Damour, L. Gualtieri, and M. Henneaux, Nucl. Phys. B, 597, 127 (2001); arXiv:hep-th/0007220v2 (2000).
S. V. Babak and L. P. Grishchuk, Phys. Rev. D, 61, 024038 (2000); arXiv:gr-qc/9907027v2 (1999).
N. Pinto-Neto, and P. I. Trajtenberg, Braz. J. Phys., 30, No. 1, 181 (2000).
T. Padmanabhan, “From gravitons to gravity: Myths and reality,” arXiv:gr-qc/0409089v1 (2004).
M. Visser, Gen. Relativity Gravitation, 30, 1717 (1998); arXiv:gr-qc/9705051v2 (1997).
S. V. Babak and L. P. Grishchuk, Internat. J. Mod. Phys. D, 12, 1905 (2003); arXiv: gr-qc/0209006v2 (2002).
I. V. Tyutin and E. S. Fradkin, Sov. J. Nucl. Phys., 15, 331 (1972).
P. G. O. Freund, A. Maheshwari, and E. Schonberg, Astrophys. J., 157, 857 (1969).
D. G. Boulware and S. Deser, Phys. Rev. D, 6, 3368 (1972).
W. Israel, Differential Forms in General Relativity (2nd ed.), Dublin Inst. Adv. Studies, Dublin (1979).
P. van Nieuwenhuizen, Nucl. Phys. B, 60, 478 (1973).
P. Havas, “Energy-momentum tensors in special and general relativity,” in: Developments in General Relativity, Astrophysics, and Quantum Theory (F. I. Cooperstock, L. P. Horwitz, and J. Rosen, eds.), IOP, Bristol (1990), p. 131.
E. Kretschmann, Ann. Phys. (8), 53, 575 (1917).
J. L. Anderson, Principles of Relativity Physics, Acad. Press, New York (1967).
M. Friedman, Foundations of Space-Time Theories: Relativistic Physics and Philosophy of Science, Princeton Univ. Press, Princeton, N. J. (1983).
J. D. Norton, Erkenntnis, 42, 223 (1995).
C. Misner, K. Thorne, and J. A. Wheeler, Gravitation, Freeman, San Francisco (1973).
J. B. Pitts, Stud. Hist. Philos. Sci. B. Stud. Hist. Philos. Modern Phys., 37, 347 (2006); arXiv:gr-qc/0506102v4 (2005).
D. Giulini, “Some remarks on the notions of general covariance and background independence,” in: Approaches to Fundamental Physics: An Assessment of Current Theoretical Ideas (Lect. Notes Phys., Vol. 721, E. Seiler and I.-O. Stamatescue, eds.), Springer, New York (2007); arXiv:gr-qc/0603087 (2006).
P. Bergmann, “Topics in the theory of general relativity,” in: Lectures in Theoretical Physics, Benjamin, New York (1957); R. U. Sexl, Fortschr. Phys., 15, 269 (1967); J. Stachel, “The meaning of general covariance: The hole story,” in: Philosophical Problems of the Internal and External Worlds (J. Earman, A. I. Janis, G. J. Massey, and N. Rescher, eds.), Univ. Pittsburgh Press, Pittsburgh (1993), p. 129.
M. J. Gotay and J. E. Marsden, “Stress-energy-momentum tensors and the Belinfante-Rosenfeld formula,” in: Mathematical Aspects of Classical Field Theory (Contemp. Math., Vol. 132, M. J. Gotay, J. E. Marsden, and V. Moncrief, eds.), Amer. Math. Soc., Providence, R. I. (1992), p. 367; http://www.math.hawaii.edu/~gotay/SEMTensors.pdf.
V. I. Ogievetskii and I. V. Polubarinov, Sov. Phys. JETP, 21, 1093 (1965); R. F. Bilyalov, Russ. Math., 46, No. 11, 6 (2002).
L. Rosenfeld, Acad. Roy. Belgique, Cl. Sci. Mém., 18, No. 6, 1 (1940).
N. Rosen, “Flat space and variational principle,” in: Perspectives in Geometry and Relativity (B. Hoffmann, ed.), Indiana Univ. Press, Bloomington (1966), p. 325; Gen. Relativity Gravitation, 4, 435 (1973); R. D. Sorkin, Modern Phys. Lett. A, 17, 695 (2002); http://philsci-archive.pitt.edu/archive/00000565/.
R. M. Wald, General Relativity, Univ. Chicago Press, Chicago (1984).
E. R. Huggins, “Quantum mechanics of the interaction of gravity with electrons: Theory of a spin-two field coupled to energy,” Doctoral dissertation, Calif. Inst. Tech., Pasadena, Calif. (1962).
A. N. Petrov and J. Katz, Proc. Roy. Soc. London A, 458, 319 (2002); arXiv:gr-qc/9911025v3 (1999).
M. G. Hare, Can. J. Phys., 51, 431 (1973); A. S. Goldhaber and M. M. Nieto, Phys. Rev. D, 9, 1119 (1974); L. S. Finn and P. J. Sutton, Phys. Rev. D, 65, 044022 (2002); arXiv:gr-qc/0109049v2 (2001).
S. Deser, “Bimetric gravity revisited,” in: Developments in General Relativity, Astrophysics, and Quantum Theory (F. I. Cooperstock, L. P. Horwitz, and J. Rosen, eds.), IOP, Bristol (1990), p. 77.
V. I. Zakharov, JETP Letters, 12, 312 (1970); H. van Dam and M. Veltman, Nucl. Phys. B, 22, 397 (1970); Gen. Relativity Gravitation, 3, 215 (1972); M. Carrera and D. Giulini, “Classical analysis of the van Dam-Veltman discontinuity,” arXiv:gr-qc/0107058v2 (2001).
Yu. M. Loskutov, Theor. Math. Phys., 107, 686 (1996).
K. Sundermeyer, Constrained Dynamics: With Applications to Yang-Mills Theory, General Relativity, Classical Spin, Dual String Model (Lect. Notes Phys., Vol. 169), Springer, Berlin (1982).
J. B. Pitts, J. Phys. Conf. Ser., 33, 279 (2006); arXiv:hep-th/0601185v2 (2006).
A. Anderson and J. W. York Jr., Phys. Rev. Lett., 81, 1154 (1998); arXiv:gr-qc/9807041v1 (1998).
S. S. Gershtein, A. A. Logunov, and M. A. Mestvirishvili, “On one fundamental property of gravitational field in the field theory,” arXiv:gr-qc/0412122v5 (2004).
G. Velo and D. Zwanziger, Phys. Rev., 186, 1337 (1969).
I. Schmelzer, “General ether theorie and graviton mass,” arXiv:gr-qc/9811073v1 (1998).
Yu. V. Chugreev, Theor. Math. Phys., 138, 293 (2004).
N. Arkani-Hamed, H. Georgi, and M. D. Schwartz, Ann. Phys., 305, 96 (2003); arXiv:hep-th/0210184v2 (2002).
R. Delbourgo and A. Salam, Lett. Nuovo Cimento, 12, 297 (1975); S. Hamamoto, Progr. Theoret. Phys., 95, 441 (1996); 97, 141 (1997); M. J. Du., J. T. Liu, and H. Sati, Phys. Lett. B, 516, 156 (2001); arXiv:hep-th/0105008v1 (2001); F. A. Dilkes, M. J. Du., J. T. Liu, and H. Sati, Phys. Rev. Lett., 87, 041301 (2001); arXiv:hep-th/0102093v3 (2001).
I. A. Batalin, and I. V. Tyutin, Internat. J. Mod. Phys. A, 6, 3255 (1991).
A. S. Vytheeswaran, Internat. J. Mod. Phys. A, 13, 765 (1998); arXiv:hep-th/9701050v2 (1997).
Author information
Authors and Affiliations
Additional information
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 311–336, May, 2007.
Rights and permissions
About this article
Cite this article
Pitts, J.B., Schieve, W.C. Universally coupled massive gravity. Theor Math Phys 151, 700–717 (2007). https://doi.org/10.1007/s11232-007-0055-7
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11232-007-0055-7