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Making the Case for Conformal Gravity

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Abstract

We review some recent developments in the conformal gravity theory that has been advanced as a candidate alternative to standard Einstein gravity. As a quantum theory the conformal theory is both renormalizable and unitary, with unitarity being obtained because the theory is a PT symmetric rather than a Hermitian theory. We show that in the theory there can be no a priori classical curvature, with all curvature having to result from quantization. In the conformal theory gravity requires no independent quantization of its own, with it being quantized solely by virtue of its being coupled to a quantized matter source. Moreover, because it is this very coupling that fixes the strength of the gravitational field commutators, the gravity sector zero-point energy density and pressure fluctuations are then able to identically cancel the zero-point fluctuations associated with the matter sector. In addition, we show that when the conformal symmetry is spontaneously broken, the zero-point structure automatically readjusts so as to identically cancel the cosmological constant term that dynamical mass generation induces. We show that the macroscopic classical theory that results from the quantum conformal theory incorporates global physics effects that provide for a detailed accounting of a comprehensive set of 138 galactic rotation curves with no adjustable parameters other than the galactic mass to light ratios, and with the need for no dark matter whatsoever. With these global effects eliminating the need for dark matter, we see that invoking dark matter in galaxies could potentially be nothing more than an attempt to describe global physics effects in purely local galactic terms. Finally, we review some recent work by ’t Hooft in which a connection between conformal gravity and Einstein gravity has been found.

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Notes

  1. Since W μν is constructed as the functional variation with respect to the metric of a gravitational action that is both general coordinate invariant and locally conformal invariant, it kinematically obeys \(W^{\mu\nu}_{\phantom{\mu\nu};\nu}=0\), g μν W μν=0. Likewise a matter energy momentum tensor constructed as the functional variation with respect to the metric of a matter action that is general coordinate and locally conformal invariant will obey \(T_{\mathrm{M}~;\nu}^{\mu\nu}=0\), \(g_{\mu\nu}T_{\mathrm{M}}^{\mu\nu}=0\). While the tracelessness of \(T_{\mathrm{M}}^{\mu\nu}\) forbids the matter fields from having kinematical or mechanical masses, it is important to note [1] that it it does not prevent them from acquiring dynamical masses via spontaneous symmetry breaking. Specifically, as for instance seen explicitly in Sect. 7 below, the scalar order parameter \(\langle S|\bar{\psi}\psi|S\rangle\) that gives a fermionic matter field a mass also carries energy density and momentum that serve to maintain \(g_{\mu\nu}T_{\mathrm{M}}^{\mu\nu}=0\). Moreover, since in flat space it does this by adding a cosmological constant term \(T_{\mathrm{COS}}^{\mu\nu}\) on to the standard kinematic perfect fluid \(T_{\mathrm{KIN}}^{\mu\nu}\) (see e.g. (45) below and [1]), it does not affect energy differences or the conservation of \(T_{\mathrm{KIN}}^{\mu\nu}\). In flat space \(T_{\mathrm{COS}}^{\mu\nu}\) is not observable, and one can use the non-traceless \(T_{\mathrm{KIN}}^{\mu\nu}\) to describe macroscopic systems. It is only in its coupling to gravity that the presence of \(T_{\mathrm{COS}}^{\mu\nu}\) can be felt.

  2. In [10] and also in [11] it is suggested that one treat the functional variation of the gravitational action with respect to the metric as the energy-momentum tensor of gravity. Specifically it was noted that if in Einstein gravity one perturbs the Einstein equations around some background \(g^{(0)}_{\mu\nu}\) according to \(g_{\mu\nu}=g^{(0)}_{\mu\nu }+h_{\mu\nu}\), the first-order term in h μν gives the wave equation obeyed by h μν , while the term that is quadratic in h μν is both covariantly conserved with respect to the background metric and gives the energy density carried by a gravity wave. In Sect. 6 below we provide an equivalent analysis in the conformal case.

  3. The Chandrasekhar mass limit \(M_{\mathrm{CH}}\sim(\hbar c/G)^{3/2}/m_{p}^{2}\) for white dwarfs and the Stefan-Boltzmann constant \(\sigma=2\pi^{5}k_{\mathrm{B}}^{4}/15c^{2}\hbar^{3}\) for black-body radiation both intrinsically depend on ħ. Both of these parameters would expressly have to appear in the macroscopic gravitational equations of motion, and for neither of them could one set ħ to zero.

  4. If one did first renormalize each term separately, the associated conformal anomalies would then have to cancel each other identically. Specifically, with the vanishing of \(T^{\mu\nu}_{\mathrm{UNIV}}\) being due to stationarity with respect to the metric, such stationarity equally guarantees the vanishing of the trace \(g_{\mu\nu}T^{\mu\nu}_{\mathrm{UNIV}}\) without any need to impose conformal invariance. Thus even though the vanishing of the individual gravity sector and matter sector traces \(g_{\mu\nu}T^{\mu\nu}_{\mathrm{GRAV}}\) and \(g_{\mu\nu}T^{\mu\nu}_{\mathrm{M}}\) do require conformal invariance, and even though conformal symmetry Ward identities might be violated by renormalization anomalies, the vanishing of \(g_{\mu\nu}T^{\mu\nu}_{\mathrm{UNIV}}\) cannot be affected by the lack of scale invariance of regulator masses. Any anomalies in \(g_{\mu\nu}T^{\mu\nu}_{\mathrm{GRAV}}+g_{\mu\nu}T^{\mu\nu}_{\mathrm{M}}\) must thus all mutually cancel each other identically.

  5. Even though (33) involves terms that are infinite and thus not well-defined, we note the perfect fluid form given in (34) can be established by integrating over the direction of the 3-momentum vector \(\bar{k}\) alone, an integration that is completely finite. A perfect fluid form for (33) can thus be established prior to the subsequent divergent integration over the magnitude of the momentum, with this latter integration not bringing \(\langle \varOmega |T^{\mu\nu }_{\mathrm{M}}|\varOmega \rangle\) to the form of a cosmological constant. Even though (33) is not well-defined, for our purposes here the perfect fluid form given in (34) is a very convenient way of summarizing the infinities in (33) that need to be canceled.

  6. It was noted in [25–27] that a reduction in dynamical dimension of \(\bar{\psi}\psi\bar{\psi}\psi\) from six to four would render the Nambu-Jona-Lasinio four-Fermi interaction theory non-perturbatively renormalizable. In addition it was suggested that one could use such a \(\bar{\psi}\psi\bar{\psi}\psi\) term as vacuum energy counter-term. Now since the QED study given in [25–27] was a purely flat spacetime study, there one could of course, and indeed one ordinarily does, remove the vacuum energy density by normal ordering. However, once one couples the theory to gravity, one can no longer normal order away any contribution to the energy density, and to obtain a finite vacuum energy density one should instead use a four-Fermi counter-term. The d θ =2 condition is thus seen as not only serving to produce dynamical symmetry breaking in flat spacetime, but also as serving to render zero-point fluctuations finite in curved spacetime.

  7. For galaxies the N ∗ γ ∗ term in (68) is never larger than of order the γ 0 term since for galaxies the number of stars N ∗ is never bigger than of order 1011. Hence for galaxies the maximum size associated with the distance in which the right-hand side of (68) vanishes is never larger than of order 100 kiloparsec. For clusters of galaxies N ∗ is of order 1000 times larger as clusters typically contain 1000 galaxies. From (68) the maximum allowed size for clusters is then well into the megaparsec region.

  8. In deriving (70) we note that if set \(g_{\mu\nu}(x)= \omega^{2}(x)\hat{g}_{\mu\nu}(x)\) we obtain \((-g)^{1/2}R^{\alpha}_{\phantom{\alpha}\alpha}(g_{\mu\nu})=(-\hat {g})^{1/2}[\omega^{2}\hat{R}^{\alpha}_{\phantom{\alpha}\alpha}(\hat {g}_{\mu\nu})+6\bar{g}^{\alpha\beta}\omega\hat{\nabla}_{\alpha}\hat {\nabla}_{\beta}\omega]\) where both \(\hat{R}^{\alpha}_{\phantom{\alpha }\alpha}(\hat{g}_{\mu\nu})\) and the \(\hat{\nabla}_{\alpha}\) derivatives are evaluated in a geometry with metric \(\hat{g}_{\mu\nu}(x)\). An integration by parts then yields (70).

  9. While conformal invariance excludes a fundamental Einstein-Hilbert action, various authors have discussed the possibility that one might be induced or derived starting from the conformal gravity action. See e.g. [42–44].

  10. Many of these issues involve the growth of cosmological inhomogeneities and their interplay with the cosmological background as exhibited in (67). A first step toward addressing these issues and in developing conformal cosmological perturbation theory in general has recently been taken in [49]. It will be of interest to ascertain the degree to which conformal cosmological fluctuation theory is aware of the γ 0 and κ scales, especially since κ is a matter fluctuation moment integral.

  11. In passing we note that even if some dark matter candidate particles are found in an accelerator experiment, to establish that such particles contribute to a possible dark matter halo in the Milky Way Galaxy, one would need to determine their local galactic density. For dark matter to not be excluded, the parameter space allowed by dark matter detection in an accelerator experiment would have to not conflict with the parameter space that has already been excluded by the non-detection to date of dark matter in non-accelerator experiments.

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Acknowledgements

This paper is based in part on a presentation made by the author at the International Conference on Two Cosmological Models, Universidad Iberoamericana, Mexico City, November 2010. The author wishes to thank Dr. J. Auping and Dr. A.V. Sandoval for the kind hospitality of the conference.

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    Philip D. Mannheim

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Mannheim, P.D. Making the Case for Conformal Gravity. Found Phys 42, 388–420 (2012). https://doi.org/10.1007/s10701-011-9608-6

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