General Relativity and Gravitation

, Volume 15, Issue 9, pp 875–902 | Cite as

Scalar-tensor theories of gravitation: Foundations and prospects

  • T. Singh
  • L. N. Rai
Research Articles


A generalization of Einstein's theory is discussed in which the gravitation is described by a tensor and a scalar field. The theory is more consistent with Mach's principle and less reliant on absolute properties of space. The modification involves a violation of the “strong principle of equivalence” on which Einstein's theory is based. In the original version of this new theory, the “constant” of gravitationG is varying and particle masses are fixed. Later on another version of the theory was given in whichG is truly a constant and the particle masses vary. The two versions are related by a conformal transformation. The physical and mathematical foundations of this theory have been discussed and the field equations have been derived. The astrophysical and cosmological consequences of the theory have been elaborately reviewed.


Scalar Field Field Equation Differential Geometry Particle Masse Original Version 
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  1. 1.
    Whittaker, E. T. (1951).History of the Theories of Aether and Electricity (Thomas Nelson and Sons, New York).Google Scholar
  2. 2.
    Newton, I. (1686).Principia Mathematica Philosophiae Naturalis (reprinted by University of California Press, Berkeley, California, 1934).Google Scholar
  3. 3.
    Berkeley, G. (1726).The Principles of Human Knowledge, paragraphs 111–117, 1710-DeMotu.Google Scholar
  4. 4.
    Mach, E. (1872).Conservation of Energy, note No. 1 (reprinted by Open Court Publishing Company, Lasalle, Illinois, 1911);The Science of Mechanics, 1883 (reprinted by Open Court Publishing Company, Lasalle, Illinois, 1902), Chap. II, Sec. VI.Google Scholar
  5. 5.
    Komar, A. (1956). Ph.D. thesis, Princeton University, Princeton, New Jersey (unpublished).Google Scholar
  6. 6.
    Brans, C. H. (1961). Ph.D. thesis, Princeton University, Princeton, New Jersey (unpublished).Google Scholar
  7. 7.
    Brans, C. H., and Dicke, R. H. (1961).Phys. Rev.,124, 925.Google Scholar
  8. 8.
    Sciama, D. W. (1953).Mon. Not. R. Astron. Soc.,113, 34.Google Scholar
  9. 9.
    Dicke, R. H. (1959).Am. Scientist,47, 25.Google Scholar
  10. 10.
    Dicke, R. H. (1959).Science,129, 621.Google Scholar
  11. 11.
    Dirac, P. A. M. (1938).Proc. Roy. Soc. London, A165, 199.Google Scholar
  12. 12.
    Wigner, E. P. (1961). Relativity Seminar, Stevens Institute (unpublished).Google Scholar
  13. 13.
    Hoffman, B. (1953).Phys. Rev.,89, 49.Google Scholar
  14. 14.
    Dicke, R. H. (1960).Am. J. Phys.,29, 344.Google Scholar
  15. 15.
    Eotvos, R. (1922).Ann. Phys. (Leipzig),68, 11.Google Scholar
  16. 16.
    Dicke, R. H. (1962).Evidence for Gravitational Theories, Enrico Fermi Course XX, ed. Moller, C. (Academic Press, New York), pp. 1–49.Google Scholar
  17. 17.
    Jordan, P. (1955).Schwerkraft und Weltall (Friedrich Vieweg and Sohn, Braunschweig).Google Scholar
  18. 18.
    Jordan, P. (1959).Z. Phys.,157, 112.Google Scholar
  19. 19.
    Brill, D. R. (1962).Evidence for Gravitational Theories, Enrico Fermi Course XX, ed. Moller, C. (Academic Press, New York), pp. 50–68.Google Scholar
  20. 20.
    Fierz, M. (1956).Helv. Phys. Acta,29, 128.Google Scholar
  21. 21.
    Landau, L. D., and Liftschitz, E. M. (1962).Classical Theory of Fields (Addison-Wesley Publishing Company, Reading, Massachusetts), p. 95.Google Scholar
  22. 22.
    Weinberg, S. (1972).Gravitation and Cosmology (Wiley, New York).Google Scholar
  23. 23.
    Lord, E. A. (1976).Tensors, Relativity and Cosmology (Tata McGraw-Hill Pub. Col, New Delhi, India), Chap. 16.Google Scholar
  24. 24.
    Misner, C., and Putnam, P. (1959).Phys. Rev.,116, 1045.Google Scholar
  25. 25.
    Smalley, L. L., and Eby, P. B. (1976).Nuovo Cimento,35B, 54.Google Scholar
  26. 26.
    Hill, H. A.,et al. (1974).Phys. Rev. Lett.,33, 1497.Google Scholar
  27. 27.
    Dicke, R. H. (1962).Phys. Rev.,125, 2163.Google Scholar
  28. 28.
    Morganstern, R. E. (1971).Phys. Rev. D,3, 2946.Google Scholar
  29. 29.
    Synge, J. L. (1971).Relativity: The General Theory (North-Holland, Amsterdam), p. 318.Google Scholar
  30. 30.
    Morganstern, R. E. (1970).Phys. Rev. D.,1, 2969.Google Scholar
  31. 31.
    Peters, P. C. (1969).J. Math. Phys.,10, 1029.Google Scholar
  32. 32.
    Wheeler, J. A. (1962).Geometrodynamics (Academic Press, Inc., New York).Google Scholar
  33. 33.
    Ross, D. K. (1972).Phys. Rev. D,5, 284.Google Scholar
  34. 34.
    Sen, D. K., and Dunn, K. A. (1971).J. Math. Phys.,12, 578.Google Scholar
  35. 35.
    Lyra, G. (1951).Math. Z.,54, 52.Google Scholar
  36. 36.
    Sen, D. K. (1968).Fields and/or Particles (Academic Press, London).Google Scholar
  37. 37.
    Sen, D. K., and Vanstone, J. R. (1972).J. Math. Phys.,13, 990.Google Scholar
  38. 38.
    Jeavons, J. S., McIntosh, C. B. G., and Sen, D. K. (1975).J. Math. Phys.,16, 320.Google Scholar
  39. 39.
    Dunn, K. A. (1974).J. Math. Phys.,15, 2229.Google Scholar
  40. 40.
    Dunn, K. A. (1975).Tensor (N.S),29, 214.Google Scholar
  41. 41.
    Ikeda, S. (1976).Lett. Nuovo Cimento,17, 545.Google Scholar
  42. 42.
    Ikeda, S. (1978).Lett. Nuovo Cimento,21, 165.Google Scholar
  43. 43.
    Ikeda, S. (1978).Lett. Nuovo Cimento,22, 37.Google Scholar
  44. 44.
    Harrison, E. R. (1972).Phys. Rev. D,6, 2077.Google Scholar
  45. 45.
    Schwarte, H. M. (1973).Phys. Rev. D,8, 1915.Google Scholar
  46. 46.
    Einstein, A.,et al (1924).Principle of Relativity (Dover, New York), pp. 167–173.Google Scholar
  47. 47.
    Harrison, E. R. (1973).Phys. Rev. D,8, 1916.Google Scholar
  48. 48.
    Anderson, J. L. (1971).Phys. Rev. D,3, 1689.Google Scholar
  49. 49.
    Will, C. M. (1979).General Relativity: An Einstein Centenary Survey, eds. Hawking, S. W., and Israel, W. (Cambridge University Press), pp. 24–89.Google Scholar
  50. 50.
    Will, C. M. (1979).Proc. R. Soc. London,A368, 5.Google Scholar
  51. 51.
    Bergmann, P. G. (1968).Int. J. Theor. Phys.,1, 25.Google Scholar
  52. 52.
    Wagoner, R. V. (1970).Phys. Rev. D,1, 3209.Google Scholar
  53. 53.
    Nordtvedt, K., Jr. (1970).Astrophys. J.,161, 1059.Google Scholar
  54. 54.
    Barker, B. M. (1978).Astrophys. J.,219, 5.Google Scholar
  55. 55.
    Horndeski, G. W., and Lovelock, D. (1972).Tensor (N.S.),24, 79.Google Scholar
  56. 56.
    Horndeski, G. W. (1974),Int. J. Theor. Phys.,10, 363.Google Scholar
  57. 57.
    Lindstrom, U. (1976).Nuovo Cimento,32B, 298.Google Scholar
  58. 58.
    Lindstrom, U. (1976).Nuovo Cimento,35B, 130.Google Scholar
  59. 59.
    Cohn, J. (1974).Gen. Rel. Grav.,6, 143.Google Scholar
  60. 60.
    Smalley, L. L. (1974).Phys. Rev. D,9, 1635.Google Scholar
  61. 61.
    Smalley, L. L. (1975).Phys. Rev. D,12, 376.Google Scholar
  62. 62.
    Canuto, V., Adams, P. J., Hsieh, S. H., and Tsiang, E. (1977).Phys. Rev. D,16, 1643.Google Scholar
  63. 63.
    Canuto, V., Hsieh, S. H., and Adams, P. J. (1977).Phys. Rev. Lett.,39, 429.Google Scholar
  64. 64.
    Canuto, V. M. (1978).Riv. Nuovo Cimento, Ser. 3,1, 1–42.Google Scholar
  65. 65.
    Canuto, V. M. (1979). Invited review paper presented at the Second Marcel Grossman Meeting, Trieste, Italy, (preprint).Google Scholar
  66. 66.
    Dirac, P. A. M. (1937).Nature,139, 323.Google Scholar
  67. 67.
    Dirac, P. A. M. (1973).Proc. R. Soc. London,A333, 403.Google Scholar
  68. 68.
    Callan, C. G., Coleman, S., and Jackiw, R. (1970),Ann. Phys. (N. Y.,59, 42.Google Scholar
  69. 69.
    Petrosian, V. (1975).Int. Astron. Union Symp.,63, 31.Google Scholar
  70. 70.
    Fujii, Y. (1974).Phys. Rev. D,9, 874.Google Scholar
  71. 71.
    Endo, M., and Fukui, T. (1977).Gen. Rel. Grav.,8, 833.Google Scholar
  72. 72.
    Salmona, A. (1967).Phys. Rev.,154, 1218.Google Scholar
  73. 73.
    Hillebrandt, W., and Heintzmann, H. (1974).Gen. Rel. Grav.,5, 663.Google Scholar
  74. 74.
    Bruckman, W. F., and Kazes, E. (1977).Phys. Rev. D,16, 261.Google Scholar
  75. 75.
    Misner, C. W., and Zapolsky, H. S. (1964).Phys. Rev. Lett.,12, 635.Google Scholar
  76. 76.
    Harrison, B. K. (1965).Phys. Rev.,137, B 1644.Google Scholar
  77. 77.
    Harrison, B. K., Thorne, K. S., Wakano, M., and Wheeler, J. A. (1965).Gravitation Theory and Gravitational Collapse (Univ. of Chicago Press, Chicago), Chap. 5.Google Scholar
  78. 78.
    Nariai, H. (1972).Prog. Theor. Phys.,47, 118.Google Scholar
  79. 79.
    Thorne, K. S., and Dykla, J. J. (1971).Astrophys. J.,166, L35.Google Scholar
  80. 80.
    Matsuda, T. (1972).Prog. Theor. Phys.,47, 738.Google Scholar
  81. 81.
    Brans, C. H. (1962).Phys. Rev.,125, 2194.Google Scholar
  82. 82.
    Nariai, H. (1969).Prog. Theor. Phys.,42, 742.Google Scholar
  83. 83.
    Nariai, H. (1970).Prog. Theor. Phys.,43, 334.Google Scholar
  84. 84.
    Nariai, H. (1972).Prog. Theor. Phys.,47, 832.Google Scholar
  85. 85.
    Misner, C. W., and Sharp, D. H. (1964).Phys. Rev.,136, B571.Google Scholar
  86. 86.
    Matsuda, T., and Nariai, H. (1973).Prog. Theor. Phys.,49, 1195.Google Scholar
  87. 87.
    Banerjee, A., and Bhattacharya, D. (1979).Phys. Rev. D,19, 3176.Google Scholar
  88. 88.
    Johnson, M. (1972).Lett. Nuovo Cimento,4, 323.Google Scholar
  89. 89.
    Hawking, S. W. (1972).Commun. Math. Phys.,25, 167.Google Scholar
  90. 90.
    Morganstern, R. E. (1971).Phys. Rev. D,4, 278.Google Scholar
  91. 91.
    Morganstern, R. E. (1971).Phys. Rev. D,4, 946.Google Scholar
  92. 92.
    Morganstern, R. E. (1971).Phys. Rev. D,4, 282.Google Scholar
  93. 93.
    Morganstern, R. E. (1971).Nature,232, 109.Google Scholar
  94. 94.
    Miyazaki, A. (1978).Phys. Rev. D,7, 1570.Google Scholar
  95. 95.
    Miyazaki, A. (1978).Prog. Theor. Phys.,60, 321.Google Scholar
  96. 96.
    Milne, E. A., and McCrea, W. H. (1934).Quart. J. Math. Oxford Ser.,5, 73.Google Scholar
  97. 97.
    Miyazaki, A. (1978).Phys. Rev. Lett.,40, 725.Google Scholar
  98. 98.
    Miyazaki, A. (1979).Phys. Rev. D,19, 2861.Google Scholar
  99. 99.
    Morganstern, R. E. (1974).Astrophys. J.,191, 39.Google Scholar
  100. 100.
    Nariai, H. (1972).Prog. Theor. Phys.,47, 1824.Google Scholar
  101. 101.
    Nariai, H. (1972).Prog. Theor. Phys.,48, 703.Google Scholar
  102. 102.
    Banerjee, S. (1974).Phys. Rev. D,9, 877.Google Scholar
  103. 103.
    Misner, C. W. (1969),Phys. Rev.,186, 1319.Google Scholar
  104. 104.
    Dehnen, H., and Obregon, O. (1971).Astrophys. Space Sci.,14, 154.Google Scholar
  105. 105.
    Dehnen, H., and Obregon, O. (1972).Astrophys. Space Sci.,15, 326.Google Scholar
  106. 106.
    Luke, S. K., and Szamosi, G. (1972).Astron. Astrophys.,20, 397.Google Scholar
  107. 107.
    Burman, R. (1972).Lett. Nuovo Cimento,4, 643.Google Scholar
  108. 108.
    Burman, R. (1972).Lett. Nuovo Cimento,4, 645.Google Scholar
  109. 109.
    Tinsley, B. M. (1972).Astrophys. J. Lett.,174, L119.Google Scholar
  110. 110.
    Aragone, C., and Restuccia, A. (1972).Lett. Nuovo Cimento,4, 962.Google Scholar
  111. 111.
    Ruban, V. A., and Finkelstein, A. M. (1975).Gen. Rel. Grav.,6, 601.Google Scholar
  112. 112.
    Belinski, V. A., and Khalatnikov, I. M. (1973).Sov. Phys. JETP,36, 591.Google Scholar
  113. 113.
    Nariai, H. (1969).Prog. Theor. Phys.,42, 544.Google Scholar
  114. 114.
    Raychaudhuri, A. K. (1980).Theoretical Cosmology (Oxford University Press), p. 157.Google Scholar
  115. 115.
    Narlikar, J. V., and Kembhavi, A. K. (1980).Fund. Cosmic Phys.,6, 1.Google Scholar
  116. 116.
    Kaufmann III, W. J. (1968).J. Math. Phys.,9, 1053.Google Scholar
  117. 117.
    Bondi, H., Vander Burg, M. G. J., and Metzner, A. W. K. (1962).Proc. R. Soc. London,A269, 21.Google Scholar
  118. 118.
    Vaidya, P. C. (1953).Nature,171, 260.Google Scholar
  119. 119.
    Morganstern, R. E. (1967).Phys. Rev.,163, 1357.Google Scholar
  120. 120.
    Will, C. M. (1977).Astrophys. J.,214, 826.Google Scholar
  121. 121.
    Davis, W. F. (1979). Ph.D. thesis, Massachusetts Institute of Technology (U.S.A.) (unpublished).Google Scholar
  122. 122.
    Gurevich, L. E., and Dynkin, S. D. (1973).Sov. Phys. JETP,36, 195.Google Scholar
  123. 123.
    Lun, A. W. C., and McIntosh, C. B. G. (1973).Gen. Rel. Grav.,4, 475.Google Scholar
  124. 124.
    Ray, D. (1977).J. Math. Phys.,18, 245.Google Scholar
  125. 125.
    Bandyopadhyay, N. (1978).J. Math. Phys.,19, 1423.Google Scholar
  126. 126.
    Van Stockum, W. J. (1937).Proc. R. Soc. Edinburgh,57, 135.Google Scholar
  127. 127.
    Bandyopadhyay, N. (1979).J. Math. Phys.,20, 1494.Google Scholar
  128. 128.
    Maitra, S. C. (1966).J. Math. Phys.,7, 1025.Google Scholar
  129. 129.
    Majumdar, S. D. (1947).Phys. Rev.,72, 390.Google Scholar
  130. 130.
    Das, A. (1962).Proc. R. Soc. London,A267, 1.Google Scholar
  131. 131.
    De, U. K., and Raychaudhuri, A. K. (1968).Proc. R. Soc. London,A303, 97.Google Scholar
  132. 132.
    Nayak, B. K. (1975).Austr. J. Phys.,28, 585.Google Scholar
  133. 133.
    Tiwari, R. N., and Nayak, B. K. (1976).Phys. Rev. D,14, 2502.Google Scholar
  134. 134.
    Raychaudhuri, A. K., and Bandyopadhyay, N. (1978).Prog. Theor. Phys.,59, 414.Google Scholar
  135. 135.
    Tiwari, R. N., and Nayak, B. K. (1978).Phys. Rev. D,18, 1805.Google Scholar
  136. 136.
    Bonnor, W. B. (1965).Mon. Not. R. Astron. Soc.,129, 443.Google Scholar
  137. 137.
    Tiwari, R. N., and Nayak, B. K. (1978).Phys. Rev. D,18, 2752.Google Scholar
  138. 138.
    Raychaudhuri, A. K., and Bandyopadhyay, N. (1978).Phys. Rev. D,18, 2756.Google Scholar
  139. 139.
    Som, M. M., and Raychaudhuri, A. K. (1968).Proc. R. Soc. London, A304, 81.Google Scholar
  140. 140.
    Tiwari, R. N., and Nayak, B. K. (1976).Phys. Rev. D,14, 395.Google Scholar
  141. 141.
    Bruckman, W. F., and Kazes, E. (1977).Phys. Rev. D,16, 269.Google Scholar
  142. 142.
    Mahanta, M. N., and Reddy, D. R. K. (1974).J. Math. Phys.,15, 1235.Google Scholar
  143. 143.
    Raychaudhuri, A. K. (1955).Phys. Rev.,98, 1123.Google Scholar
  144. 144.
    Janis, A. I., Robinson, D. C., and Winicour, J. (1969).Phys. Rev.,186, 1729.Google Scholar
  145. 145.
    Buchdahl, H. A. (1972).Int. J. Theor. Phys.,6, 407.Google Scholar
  146. 146.
    Buchdahl, H. A. (1973).Int. J. Theor. Phys.,7, 287.Google Scholar
  147. 147.
    147. Tolman, R. C. (1934).Relativity, Thermodynamics and Cosmology (Oxford Univ. Press), p. 92.Google Scholar
  148. 148.
    Buchdahl, H. A. (1973).Gen. Rel. Grav.,4, 319.Google Scholar
  149. 149.
    Singh, T. (1975).J. Math. Phys.,16, 2109.Google Scholar
  150. 150.
    Singh, T. (1976).Indian J. Pure Appl. Math.,7, 482.Google Scholar
  151. 151.
    Singh, T. (1977).Indian J. Pure Appl. Math.,8, 1205.Google Scholar
  152. 152.
    Singh, T. (1979–80).J. Sci. Res. (B.H.U.) India (in press).Google Scholar
  153. 153.
    Goswami, G. K. (1978).J. Math. Phys.,19, 442.Google Scholar
  154. 154.
    Papapetrou, A. (1947).Proc. R. Irish, Acad.,A51, 191.Google Scholar
  155. 155.
    Papapetrou, A. (1953).Ann. Phys. (Leipzig),12, 309.Google Scholar
  156. 156.
    Ehlers, J. (1957). thesis, Hamburg.Google Scholar
  157. 157.
    Bonnor, W. B. (1961).Z. Phys.,151, 439.Google Scholar
  158. 158.
    McIntosh, C. B. G. (1974).Commun. Math. Phys.,37, 335.Google Scholar
  159. 159.
    Sneddon, G. E., and McIntosh, C. B. G. (1974).Austr. J. Phys.,27, 411.Google Scholar
  160. 160.
    Geroch, R. (1971).J. Math. Phys.,12, 918.Google Scholar
  161. 161.
    Nayak, B. K., and Tiwari, R. N. (1977).J. Math. Phys.,18, 289.Google Scholar
  162. 162.
    Reddy, D. R. K. (1973).J. Phys. A,6, 1867.Google Scholar
  163. 163.
    Reddy, D. R. K. (1977).J. Phys. A,10, 185.Google Scholar
  164. 164.
    Krori, K. D., and Nandy, D. (1977).J. Phys. A,10, 993.Google Scholar
  165. 165.
    Singh, T., and Rai, L. N. (1979).Indian J. Pure Appl. Math.,10, 1432.Google Scholar
  166. 166.
    Luke, S. K., and Szamosi, G. (1972).Phys. Rev. D,6, 3359.Google Scholar
  167. 167.
    Mahanta, M. N., and Reddy, D. R. K. (1972).J. Math. Phys.,13, 768.Google Scholar
  168. 168.
    Buchdahl, H. A. (1972).Nuovo Cimento,12B, 269.Google Scholar
  169. 169.
    Hart, H. B. (1972).Phys. Rev. D,5, 1256.Google Scholar
  170. 170.
    Hart, H. B. (1975).Phys. Rev. D,11, 960.Google Scholar
  171. 171.
    Cocke, W. J., and Cohen, J. M. (1968).J. Math. Phys.,9, 971.Google Scholar
  172. 172.
    O'Hanlon, J., and Tupper, B. O. J. (1973).Nuovo Cimento,14B, 190.Google Scholar
  173. 173.
    Tupper, B. O. J. (1975).Canad. Math. Bull.,18, 151.Google Scholar
  174. 174.
    Smalley, L. L. (1977).Scientific Application of Lunar Laser Ranging, ed. Mulholland, J. D. (D. Reidel Publ. Co., Holland), pp. 91–102.Google Scholar
  175. 175.
    Smalley, L. L. (1978).Found. Phys.,8, 59.Google Scholar
  176. 176.
    Smalley, L. L. (1980). Reinterpretation of the New Test of the Equivalence Principle from Lunar Laser Ranging (preprint).Google Scholar
  177. 177.
    Bertotti, B. (1979).Recent Developments in Gravitation, eds. Levy, M., and Deser, S. (Plenum Press, New York), p. 3.Google Scholar
  178. 178.
    Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1973).Gravitation (W. H. Freeman and Co., San Francisco), Chap. 39.Google Scholar
  179. 179.
    Ni, W. T. (1972).Astrophys. J.,176, 769.Google Scholar
  180. 180.
    Will, C. M. (1974).Experimental Gravitation, ed. Bertotti, B. (Academic Press, New York), p. 1.Google Scholar
  181. 181.
    O'Connell, R. F. (1968).Phys. Rev. Lett.,20, 69.Google Scholar
  182. 182.
    Morganstern, R. E. (1971).Phys. Rev. D,3, 616.Google Scholar
  183. 183.
    Gurevich, L. E., and Dynkin, S. D. (1976).Sov. Astron.,19, 319.Google Scholar

Copyright information

© Plenum Publishing Corporation 1983

Authors and Affiliations

  • T. Singh
    • 1
  • L. N. Rai
    • 1
  1. 1.Applied Mathematics Section, Institute of TechnologyBanaras Hindu UniversityVaranasiIndia

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