Classical Weyl transverse gravity

The European Physical Journal C, May 2017

We study various classical aspects of the Weyl transverse (WTDiff) gravity in a general space-time dimension. First of all, we clarify a classical equivalence among three kinds of gravitational theories, those are, the conformally invariant scalar tensor gravity, Einstein’s general relativity and the WTDiff gravity via the gauge-fixing procedure. Secondly, we show that in the WTDiff gravity the cosmological constant is a mere integration constant as in unimodular gravity, but it does not receive any radiative corrections unlike the unimodular gravity. A key point in this proof is to construct a covariantly conserved energy-momentum tensor, which is achieved on the basis of this equivalence relation. Thirdly, we demonstrate that the Noether current for the Weyl transformation is identically vanishing, thereby implying that the Weyl symmetry existing in both the conformally invariant scalar tensor gravity and the WTDiff gravity is a “fake” symmetry. We find it possible to extend this proof to all matter fields, i.e. the Weyl-invariant scalar, vector and spinor fields. Fourthly, it is explicitly shown that in the WTDiff gravity the Schwarzschild black hole metric and a charged black hole one are classical solutions to the equations of motion only when they are expressed in the Cartesian coordinate system. Finally, we consider the Friedmann–Lemaitre–Robertson–Walker (FLRW) cosmology and provide some exact solutions.

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Classical Weyl transverse gravity

Eur. Phys. J. C Classical Weyl transverse gravity Ichiro Oda 0 0 Department of Physics, Faculty of Science, University of the Ryukyus , Nishihara, Okinawa 903-0213 , Japan We study various classical aspects of the Weyl transverse (WTDiff) gravity in a general space-time dimension. First of all, we clarify a classical equivalence among three kinds of gravitational theories, those are, the conformally invariant scalar tensor gravity, Einstein's general relativity and the WTDiff gravity via the gauge-fixing procedure. Secondly, we show that in the WTDiff gravity the cosmological constant is a mere integration constant as in unimodular gravity, but it does not receive any radiative corrections unlike the unimodular gravity. A key point in this proof is to construct a covariantly conserved energy-momentum tensor, which is achieved on the basis of this equivalence relation. Thirdly, we demonstrate that the Noether current for the Weyl transformation is identically vanishing, thereby implying that the Weyl symmetry existing in both the conformally invariant scalar tensor gravity and the WTDiff gravity is a “fake” symmetry. We find it possible to extend this proof to all matter fields, i.e. the Weyl-invariant scalar, vector and spinor fields. Fourthly, it is explicitly shown that in the WTDiff gravity the Schwarzschild black hole metric and a charged black hole one are classical solutions to the equations of motion only when they are expressed in the Cartesian coordinate system. Finally, we consider the Friedmann-Lemaitre-RobertsonWalker (FLRW) cosmology and provide some exact solutions. 1 Introduction The physical importance of Weyl (local conformal) symmetry has not been clearly established in quantum gravity thus far. It is usually believed that if the energy scale under consideration goes up to the Planck mass scale, all elementary particles, which are either massive or massless at the low Dedicated to the memory of Mario Tonin. energy scale, could be regarded as almost massless particles where the Weyl symmetry would become a gauge symmetry and play an important role. However, it is true that a concrete implementation of the Weyl symmetry as a plausible gauge symmetry in quantum gravity encounters a lot of difficulties. For instance, if one requires an exact Weyl symmetry to be realized in gravitational theories at the classical level, only two candidate theories are deserved to be studied though they possess some defects in their own right. The one theory is the conformal gravity, for which the action is described in terms of the square term of the conformal tensor. The conformal gravity belongs to a class of higher derivative gravities so that it suffers from a serious problem, i.e. violation of the unitarity because of the emergence of massive ghosts, although it has an attractive feature as a renormalizable theory [1, 2]. The other plausible candidate as a gravitational theory with the Weyl symmetry, which we consider in this article intensively, is the conformally invariant scalar–tensor gravity [3, 4]. In this theory, a (ghost-like) scalar field is introduced in such a way that it couples to the scalar curvature in a conformally invariant manner. Even if this theory is a unitary theory owing to the presence of only second-order derivative terms, it suffers from a sort of triviality problem in the sense that when we take a suitable gauge condition for the Weyl symmetry (we take the scalar field to be a constant), the action of the conformally invariant scalar–tensor gravity reduces to the Einstein–Hilbert action of Einstein’s general relativity. It is therefore unclear to make use of the conformally invariant scalar–tensor gravity as an alternative theory of general relativity. Of course, the conformally invariant scalar–tensor gravity is not a renormalizable theory like general relativity. One reason why we would like to consider a gravitational theory with the Weyl symmetry stems from the cosmological constant problem [5], which is one of the most difficult problems in modern theoretical physics. The Weyl symmetry forbids the appearance of operators of dimension zero such as the cosmological constant in the action so it is expected that the Weyl symmetry might play an important role in the cosmological constant problem [6]. In this respect, a difficulty is that the Weyl symmetry is broken by quantum effects and its violation emerges as a trace anomaly of the energymomentum tensor [7,8]. Thus, the idea such that one utilizes the Weyl symmetry as a resolution of the cosmological constant problem makes no sense at the quantum level even if it is an intriguing idea at the classical level. Here a naive but natural question arises: Is the Weyl symmetry always violated by radiative corrections? We think that it is not always so. What kind of the Weyl symmetry is not broken? In a pioneering work by Englert et al. [9], it has been clarified that the conformally invariant scalar–tensor gravity coupled to (...truncated)


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Ichiro Oda. Classical Weyl transverse gravity, The European Physical Journal C, 2017, pp. 284, Volume 77, Issue 5, DOI: 10.1140/epjc/s10052-017-4843-4