Abstract
We discuss the concepts of Weyl and Riemann frames in the context of metric theories of gravity and state the fact that they are completely equivalent as far as geodesic motion is concerned. We apply this result to conformally flat spacetimes and show that a new picture arises when a Riemannian spacetime is taken by means of geometrical gauge transformations into a Minkowskian flat spacetime. We find out that in the Weyl frame gravity is described by a scalar field. We give some examples of how conformally flat spacetime configurations look when viewed from the standpoint of a Weyl frame. We show that in the non-relativistic and weak field regime the Weyl scalar field may be identified with the Newtonian gravitational potential. We suggest an equation for the scalar field by varying the Einstein-Hilbert action restricted to the class of conformally-flat spacetimes. We revisit Einstein and Fokker’s interpretation of Nordström scalar gravity theory and draw an analogy between this approach and the Weyl gauge formalism. We briefly take a look at two-dimensional gravity as viewed in the Weyl frame and address the question of quantizing a conformally flat spacetime by going to the Weyl frame.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Einstein, A., Infeld, L., Hoffmann, B.: Ann. Math. 39(2), 65 (1938)
Will, C.M.: Theory and Experiment in Gravitation Physics. Cambridge University Press, Cambridge (1993)
Weyl, H.: Sitzungsber. Preuss. Akad. Wiss. 465 (1918)
Weyl, H.: Space, Time, Matter. Dover, New York (1952)
Adler, R., Bazin, M., Schiffer, M.: Introduction to General Relativity. McGraw-Hill, New York (1975). Chap. 15
Pauli, W.: Theory of Relativity. Dover, New York (1981)
O’Raiefeartaigh, L., Straumann, N.: Rev. Mod. Phys. 72, 1 (2000)
Perlick, V.: Class. Quantum Gravity 8, 1369 (1991)
Novello, M., Heintzmann, H.: Phys. Lett. A 98, 10 (1983)
Novello, M., Oliveira, L.A.R., Salim, J.M., Elbas, E.: Int. J. Mod. Phys. D 1, 641–677 (1993)
Salim, J.M., Sautú, S.L.: Class. Quantum Gravity 13, 353 (1996)
de Oliveira, H.P., Salim, J.M., Sautú, S.L.: Class. Quantum Gravity 14, 2833 (1997)
Melnikov, V.: Classical solutions in multidimensional cosmology. In: Novello, M. (ed.) Proceedings of the VIII Brazilian School of Cosmology and Gravitation II (Editions Frontières), pp. 542–560 (1995). ISBN 2-86332-192-7
Bronnikov, K.A., Konstantinov, M.Yu., Melnikov, V.N.: Gravit. Cosmol. 1, 60 (1995)
Arias, O., Cardenas, R., Quiros, I.: Nucl. Phys. B 643, 187 (2002)
Miritzis, J.: Class. Quantum Gravity 21, 3043 (2004)
Miritzis, J.: J. Phys. Conf. Ser. 8, 131 (2005)
Israelit, M.: Found. Phys. 35, 1725 (2005)
Dahia, F., Gomez, G.A.T., Romero, C.: J. Math. Phys. 49, 102501 (2008)
Madriz Aguilar, J.E., Romero, C.: Found. Phys. 39, 1205 (2009)
Ehlers, J., Pirani, F., Schild, A.: General Relativity. Oxford University Press, London (1972). L.O. Raifeartaigh (ed.)
Audretsch, J.: Phys. Rev. D 27, 2872 (1983)
Audretsch, J., Gäller, F., Straumann, N.: Commun. Math. Phys. 95, 41 (1984)
Infeld, L., Schild, A.: Phys. Rev. 68, 250 (1945)
Narlikar, J., Arp, H.: Astrophys. J. 405, 51 (1992)
Ibison, M.: J. Math. Phys. 48, 122501 (2007)
Polyanin, A.D., Zaitsev, V.F.: Handbook of Nonlinear Partial Differential Equations. Chapmann & Hall, London (2004)
Matsuno, I.: J. Math. Phys. 28, 2317 (1987)
Nordström, G.: Phys. Z. 13, 1126 (1912)
Nordström, G.: Ann. Phys. 40, 856 (1913)
Nordström, G.: Ann. Phys. 42, 533 (1913)
Whithrow, J.G., Murdoch, G.E.: Relativistic theories of gravitation. In: Beer, A. (ed.) Vistas in Astronomy, vol. 6, p. 1. Pergamon, Oxford (1965)
Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. Freeman, San Francisco (1973). Chap. 17
Einstein, A., Fokker, A.D.: Ann. Phys. 44, 321 (1914)
Brown, J.D.: Lower Dimensional Gravity. World Scientific, Singapore (1988)
Sikkema, A.E., Mann, R.B.: Class. Quantum Gravity 8, 219 (1991)
Wald, R.M.: General Relativity. Chicago University Press, Chicago (1984)
Faraoni, V., Gunzig, E., Nardone, P.: Fundam. Cosm. Phys. 20, 121 (1999)
Faraoni, V., Nadeau, S.: Phys. Rev. D 75, 023501 (2007)
Faraoni, V.: Cosmology in Scalar-Tensor Gravity. Kluwer Academic, Dordrecht (2004). Chap. 2
Fujii, Y., Maeda, K.: The Scalar-Tensor of Gravitation. Cambridge University Press, Cambridge (2003). Chap. 1
Green, M.B., Schwarz, J., Witten, E.: Superstring Theory, vol. 1. Cambridge University Press, Cambridge (1988)
Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2007)
Kiefer, C.: Quantum Gravity, 2nd edn. Oxford University Press, London (2007)
Feynmann, R.P., Morinigo, F.B., Wagner, W.G.: Feynman Lectures on Gravitation. Addison-Wesley, Reading (1995)
Padmanabhan, T.: In: Iyer, B.R., Kembhavi, A.K., Narlikar, J.V., Vishveshwara, C.V. (eds.) Highlights in Gravitation and Cosmology, Proceedings, Goa, India, pp. 156–165. Cambridge University Press, Cambridge (1987)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Romero, C., Fonseca-Neto, J.B. & Pucheu, M.L. Conformally Flat Spacetimes and Weyl Frames. Found Phys 42, 224–240 (2012). https://doi.org/10.1007/s10701-011-9593-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10701-011-9593-9