The conformal transformation’s controversy: what are we missing?

Abstract

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.

Appendices

Appendix A: General relativity gauge

General relativity can be obtained from (35) in a particular gauge when ${\bar{\varphi }}=\varphi _0$. In fact, after this choice, the solution of the equation in the RHS of Eq. (15) will be

$$\begin{aligned} \varphi -\varphi _0=2\ln \Omega \,\Rightarrow \,\Omega ^2=e^{\varphi -\varphi _0}. \end{aligned}$$

It can be shown that, under (7) with the latter choice of the conformal factor, the affine connection of a Weyl-integrable space maps to the Christoffel symbols of the conformal metric: $\Gamma ^\alpha _{\beta \gamma }\rightarrow \{^{\bar{\alpha }}_{\beta \gamma }\}$, etc. In other words, WI spaces transform under (7)—with $\Omega ^2=e^{\varphi -\varphi _0}$—into Riemannian spaces. Besides, the action (35) transforms into the Einstein-Hilbert action,

$$\begin{aligned} S_{EH}=\frac{1}{16\pi G_{eff}}\int d^4x\sqrt{-{\bar{g}}}\,{\bar{R}}, \end{aligned}$$

where $G_{eff}=e^{-\varphi _0}$ is the (rescaled) effective gravitational coupling. Hence GR is a particular gauge in the conformal equivalence class $\mathcal{C }$ defined in Eq. (16). Going into this particular gauge means that conformal invariance becomes, automatically, into a broken symmetry.

Appendix B: Brans–Dicke theory with matter

Although for sake of simplicity we have considered in this paper only the vacuum sector of Brans–Dicke theory, several qualitative aspects of introducing matter into the theory can be discussed here. We will see that the results of the present work are not modified by the addition of ordinary matter into BD theory. We have seen, in particular, that considering the motion of test particles in vacuum BD theory already reveals several interesting features. For instance, assuming one adheres to the first viewpoint in Sect. 3.1, the test particles which follow geodesics of the JF metric do not fallow geodesics of the conformal EF metric. Hence, if one decides to interpret the results of a given analysis in the EF of BD theory, since deviations from geodesic motion are interpreted by the EF observers as due to the existence of additional interactions of non-gravitational origin between the test particle and the scalar field $\varphi$, one will be faced with confronting the predictions of the theory with “five-force” experiments. The above analysis may be corroborated by adding matter into Brans–Dicke theory. The matter part of the JFBD gravity is assumed to be given by [1], $S_m=\int d^4x\sqrt{-g}\mathcal{L }_m(\psi _i,g_{\mu \nu })$, where $\mathcal{L }_m$ is the Lagrangian density of matter, and $\psi _i$ represent the matter fields. The following continuity equation is obeyed in the JFBD theory:

$$\begin{aligned} \nabla ^\nu T^{(m)}_{\nu \mu }=0,\,T^{(m)}_{\mu \nu }=\frac{2}{\sqrt{-g}}\frac{\partial }{\partial g^{\mu \nu }}({\sqrt{-g}\,\mathcal{L }}_m), \end{aligned}$$

(42)

where $T^{(m)}_{\mu \nu }$—the matter stress-energy tensor. Under (7) with $\Omega ^2=e^\varphi$—in the sense implied by the first viewpoint in Sect. 3.1 (Eq. (24))—the JF matter action above is transformed into the EF one [2, 90], ${\bar{S}}_m=\int d^4x\sqrt{-{\bar{g}}}\,e^{-2\varphi }\bar{\mathcal{L }}_m({\bar{\psi }}_i,e^{-\varphi }{\bar{g}}_{\mu \nu })$, while (42) is mapped into

$$\begin{aligned} {\bar{\nabla }}^\nu {\bar{T}}^{(m)}_{\nu \mu }=-\frac{1}{2}{\bar{\partial }}_\mu \varphi \,{\bar{T}}_{(m)}, \end{aligned}$$

(43)

where we have considered that, under (7), ${\bar{T}}^{(m)}_{\mu \nu }=\Omega ^{-2}T^{(m)}_{\mu \nu }$, and ${\bar{T}}_{(m)}={\bar{g}}^{\mu \nu }{\bar{T}}^{(m)}_{\mu \nu }$ is the trace of the EF stress-energy tensor. Only for traceless (massless) matter fields the continuity Eq. (42) is not transformed by (7). Physically Eq. (43) means that in terms of the EFBD metric there is an additional (non-gravitational) interaction between matter and the scalar field (the five-force) so that these fields exchange energy-momentum. If one invokes instead the point of view of Sect. 3.2, $\gamma _{RW}:(\mathcal{M },g_{\mu \nu })\mapsto (\mathcal{M },{\bar{g}}_{\mu \nu },\varphi )$, then the JF continuity Eq. (42) is mapped into

$$\begin{aligned} {\bar{\nabla }}^\nu _{(w)}{\bar{T}}^{(m)}_{\nu \mu }= -{\bar{\partial }}^\nu \varphi \,{\bar{T}}^{(m)}_{\nu \mu }, \end{aligned}$$

(44)

which is just the continuity equation in WI spaces. Although the term in the RHS of (44) might seem alien, it expresses the fact that the units of measure of the stresses and energy are running units. In no case the RHS of (44) can be interpreted as a source term. To show this notice that Eq. (44) is the trace of the most general equation, ${\bar{\nabla }}^{(w)}_\alpha {\bar{T}}^{(m)}_{\nu \mu }= -{\bar{\partial }}_\alpha \varphi \,{\bar{T}}^{(m)}_{\nu \mu }$, which is, in turn, the equivalent of the metricity condition (27)—with $\Omega ^2=e^\varphi$—in the matter sector of the EFBD theory. The opposite sign in the RHS of the above equation in respect to the sign of the RHS of Eq. (27) is a consequence of the fact that, increasing extent of the units of length and time is correlated with the contrary effect on the units of stresses and energy.