hellfire said:
Thank you for your answers, Garth. I am making an effort to understand this, but I still have no success.
But how does this claim follow? To my understanding this implies a kind of unphysical degree of freedom (a gauge) that leaves physics invariant. However, you wrote that it is not an invariant conformal transformation.
Thank you for that observation and question, sorry about the delay I have not had the time to answer properly until now.
First consider the Brans Dicke theory (BD):
The BD Lagrangian density, in which energy-momentum is conserved, is given by
L^{BD}[g,\phi ]=\frac{\sqrt{-g}}{16\pi }\left( \phi R-\frac{\omega }{\phi }g^{\mu \nu }\nabla _{\mu }\phi \nabla _{\nu }\phi \right) +L_{matter}[g]
where R is the curvature scalar, \omega a coupling constant and L_{matter}[g] is the Lagrangian density for ordinary matter minimally coupled to the scalar field, i.e.
\nabla _{\mu }T_{M\;\nu }^{\;\mu }=0 .
This ensures the rest mass of a particle m(x^{\mu }) , at x^{\mu } , is constant for all x^{\mu },
m(x^{\mu })=m_{0}
BD is a specific case of Jordan's general theory [Jordan, (1959)] and so this representation is known as the Jordan conformal frame (JF). However Dicke in 1962 showed that this Lagrangian can be conformally transformed into a form in which G is a constant and m(x^{\mu}) varies, which is termed the Einstein conformal frame (EF) in the
literature. The conformal dual is given by
L^{BD}[\tilde{g},\widetilde{\phi }]=\frac{\sqrt{-\tilde{g}}}{16\pi G_{N}}\left[ \tilde{R}-\left( \omega +\frac{3}{2}\right) \tilde{g}<br />
^{\mu \nu }\tilde{\nabla }_{\mu }\tilde{\phi }\tilde{\nabla }<br />
_{\nu }\tilde{\phi }\right] +\tilde{L}_{matter}[\tilde{g},\tilde{\phi }]
where \tilde{R} is the curvature scalar in the EF metric \tilde{g}^{\mu \nu }, conformally dual to g^{\mu \nu } according to
g_{\mu \nu }\rightarrow \tilde{g}_{\mu \nu }=\Omega ^{2}g_{\mu \nu } in which \Omega ^{2}=\phi G_{N}
The scalar function \tilde{\phi }=\ln \phi is the BD field in the EF and \tilde{L}_{matter}[\tilde{g},\tilde{\phi }] is the EF Lagrangian density for the ordinary matter, which is now non-minimally coupled to the scalar field, i.e. in the EF \nabla _{\mu }T_{M\;\nu}^{\;\mu } \neq 0.
The principle of Least Action can now be applied to this JF action to obtain the gravitational and scalar field equations and the equivalence principle is guaranteed in this frame.
Conformal duality has also been applied to GR in order to include a scalar field as an additional source of gravity. (See, for example, Quiros' paper
Dual geometries and spacetime singularities) In this case, in contrast to BD, ordinary matter is non-minimally coupled to the scalar field in the JF and it is minimally coupled in the EF. In this case the Lagrangian density in the JF is given by
L^{GR}[g,\phi ]=\frac{\sqrt{-g}}{16\pi }\left( \phi R-\frac \omega \phi<br />
g^{\mu \nu }\nabla _\mu \phi \nabla _\nu \phi \right) +L_{matter}[g,\phi ]
and in the EF
L^{GR}[\tilde{g},\tilde{\phi }]=\frac{\sqrt{-\tilde{g}}}{16\pi G_N}\left[ \tilde{R}-\left( \omega +\frac 32\right) \tilde{g}^{\mu\nu }\tilde{\nabla }_\mu \tilde{\phi }\tilde{\nabla }_\nu \tilde{\phi }\right]+\tilde{L}_{matter}[\tilde{g}].
In this case applying the principle of least action produces the gravitational field equation and the scalar field wave equation in which:
\tilde{\Box }\tilde{\phi }=0
i.e. the scalar field is decoupled from matter and Mach’s principle as understood by BD has been lost.
SCC adapts this conformal gravity action to include the original BD field equation. This is possible if \omega = -\frac32 when the scalar field drops out of the EF action and \phi becomes indeterminate, the principle of the local conservation of matter is used instead to fix \Omega and determine \phi.
Its JF Lagrangian density is therefore, (with \omega general),
L^{SCC}[g,\phi ]=\frac{\sqrt{-g}}{16\pi }\left( \phi R-\frac{\omega }{\phi } <br />
g^{\mu \nu }\nabla _{\mu }\phi \nabla _{\nu }\phi \right) + L_{matter}^{SCC}[g,\phi ]
the conformal dual, by a general transformation
\tilde{g}_{\mu \nu }=\Omega ^{2}g_{\mu \nu } , is
L^{SCC}[\tilde{g},\tilde{\phi }] =\frac{\sqrt{-\tilde{g}}}{16\pi }\left[ \tilde{\phi }\tilde{R}+6\tilde{\phi }\tilde{\Box }\ln \Omega \right] +\tilde{L}_{matter}^{SCC}[\tilde{g},<br />
\tilde{\phi }] [/itex] <br />
-\frac{\sqrt{-\tilde{g}}}{16\pi }\left[ 2\left( 2\omega +3\right) <br />
\frac{\tilde{g}^{\mu \nu }\tilde{\nabla }_{\mu }\Omega \tilde{\nabla }_{\nu }\Omega }{\Omega ^{2}}+4\omega \frac{\tilde{g}^{\mu \nu }<br />
\tilde{\nabla }_{\mu }\Omega \tilde{\nabla }_{\nu }\tilde{\phi }<br />
}{\Omega }+\omega \frac{\tilde{g}^{\mu \nu }\tilde{\nabla }_{\mu }<br />
\tilde{\phi }\tilde{\nabla }_{\nu }\tilde{\phi }}{\tilde{\phi }}\right].<br />
<br />
With m\left( x^{\mu }\right) =\Omega \tilde{m}_{0} where m\left( x^{\mu }\right) is the mass of a fundamental particle in the JF and \tilde{m}_{0} its invariant mass in the EF then SCC has \Omega =\exp \left[ \Phi {N}\left( x^{\mu }\right) \right], and we select the SCC EF by requiring G = \phi^{-1} to be constant, then the Lagrangian density in the EF is given by <br />
L^{SCC}[\tilde{g},\tilde{\phi }]=\frac{\sqrt{-\tilde{g}}}{16\pi G_{N}}\tilde{R}+\tilde{L}_{matter}^{SCC}[\tilde{g}]+\frac{3\sqrt{<br />
-\tilde{g}}}{8\pi G_{N}}\tilde{\square }\tilde{\Phi }_{N}\left(\tilde{x}^{\mu }\right) , <br />
which becomes canonical GR when \tilde{\square }\tilde{\Phi}_{N}\left(\tilde{x}^{\mu }\right) = 0 <i>in vacuo</i>.<br />
<br />
This argument can be found in the 2002 Astrophysics and Space Science paper http://www.kluweronline.com/oasis.htm/5092775 and the eprint <a href="http://arxiv.org/pdf/gr-qc/0212111" target="_blank" class="link link--external" rel="nofollow ugc noopener"> The Principles of Self Creation Cosmology and its Comparison with General Relativity</a>.<br />
<br />
IMHO I think both BD and conformal gravity have not gone far enough in modifying GR; the SCC approach may be wrong, and if so then it is surprising that it produces such a concordant gravitational and cosmological model, but the next and real test will be GP-B!<br />
<br />
Garth
