TY - JOUR
T1 - The conformal transformation's controversy
T2 - What are we missing?
AU - Quiros, Israel
AU - García-Salcedo, Ricardo
AU - Madriz-Aguilar, Jose Edgar
AU - Matos, Tonatiuh
N1 - Funding Information:
Acknowledgments The authors thank Yun-Song Piao for pointing to us reference [91], which might serve as an additional illustration of our discussion on the singularity issue. This work was partly supported by CONACyT México under grants 49865-F and I0101/131/07 C-234/07 of the Instituto Avanzado de Cosmologia (IAC) collaboration (http://www.iac.edu.mx/), by the Department of Physics, DCI, Guanajuato University, Campus León, and by the Department of Mathematics, CUCEI, Guadalajara University.
PY - 2013/2
Y1 - 2013/2
N2 - An alternative interpretation of the conformal transformations of the metric is discussed according to which the latter can be viewed as a mapping among Riemannian and Weyl-integrable spaces. A novel aspect of the conformal transformation's issue is then revealed: these transformations relate complementary geometrical pictures of a same physical reality, so that, the question about which is the physical conformal frame, does not arise. In addition, arguments are given which point out that, unless a clear statement of what is understood by "equivalence of frames" is made, the issue is a semantic one. For definiteness, an intuitively "natural" statement of conformal equivalence is given, which is associated with conformal invariance of the field equations. Under this particular reading, equivalence can take place only if the metric is defined up to a conformal equivalence class. A concrete example of a conformal-invariant theory of gravity is then explored. Since Brans-Dicke theory is not conformally invariant, then the Jordan's and Einstein's frames of the theory are not equivalent. Otherwise, in view of the alternative approach proposed here, these frames represent complementary geometrical descriptions of a same phenomenon. The different points of view existing in the literature are critically scrutinized on the light of the new arguments.
AB - An alternative interpretation of the conformal transformations of the metric is discussed according to which the latter can be viewed as a mapping among Riemannian and Weyl-integrable spaces. A novel aspect of the conformal transformation's issue is then revealed: these transformations relate complementary geometrical pictures of a same physical reality, so that, the question about which is the physical conformal frame, does not arise. In addition, arguments are given which point out that, unless a clear statement of what is understood by "equivalence of frames" is made, the issue is a semantic one. For definiteness, an intuitively "natural" statement of conformal equivalence is given, which is associated with conformal invariance of the field equations. Under this particular reading, equivalence can take place only if the metric is defined up to a conformal equivalence class. A concrete example of a conformal-invariant theory of gravity is then explored. Since Brans-Dicke theory is not conformally invariant, then the Jordan's and Einstein's frames of the theory are not equivalent. Otherwise, in view of the alternative approach proposed here, these frames represent complementary geometrical descriptions of a same phenomenon. The different points of view existing in the literature are critically scrutinized on the light of the new arguments.
KW - Brans-Dicke Theory
KW - Conformal invariance
KW - Conformal transformations
KW - Einstein's frame
KW - Jordan frame
KW - Weyl-integrable geometry
UR - http://www.scopus.com/inward/record.url?scp=84873060072&partnerID=8YFLogxK
U2 - 10.1007/s10714-012-1484-7
DO - 10.1007/s10714-012-1484-7
M3 - Artículo
SN - 0001-7701
VL - 45
SP - 489
EP - 518
JO - General Relativity and Gravitation
JF - General Relativity and Gravitation
IS - 2
ER -