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)