General relativity and Weyl geometry

Abstract

We show that the general theory of relativity may be formulated in the language of Weyl geometry. We develop the concept of Weyl frames and point out that the new mathematical formalism may lead to different pictures of the same gravitational phenomena. We show that in an arbitrary Weyl frame general relativity, which takes the form of a scalar-tensor gravitational theory, is invariant with respect to Weyl tranformations. A kew point in the development of the formalism is to build an action that is manifestly invariant with respect to Weyl transformations. When this action is expressed in terms of Riemannian geometry we find that the theory has some similarities with Brans-Dicke gravitational theory. In this scenario, the gravitational field is not described by the metric tensor only, but by a combination of both the metric and a geometrical scalar field. We illustrate this point by, firstly, discussing the Newtonian limit in an arbitrary frame, and, secondly, by examining how distinct geometrical and physical pictures of the same phenomena may arise in different frames. To give an example, we discuss the gravitational spectral shift as viewed in a general Weyl frame. We further explore the analogy of general relativity with scalar-tensor theories and show how a known Brans-Dicke vacuum solution may appear as a solution of general relativity theory when reinterpreted in a particular Weyl frame. Finally, we show that the so-called WIST gravity theories are mathematically equivalent to Brans-Dicke theory when viewed in a particular frame.

Invariant quantities in the scalar-tensor theories of gravitation

“…Mathematically this result is the well known transformation rule for Christoffel symbols under the conformal transformation [3,23], or the definition corresponding to Weyl-integrable geometry [13,25]. But here the point is simply thatΓ λ µν remains invariant under the transformations (2) and (3).…”

“…For example, quintessence in general relativity has A = 1, α = 0, thus I 1 ≡ 1 which holds in any frame and parametrization, i.e. in "veiled" [24] or "Weyled" [25] general relativity. We say that the scalar field is nonminimally coupled if I ′ 1 ≡ 0.…”

Some remarks about non-minimally coupled scalar field models

“…Now, defining the total energy density and the total pressure as ρ T = ρ m + ρ DE and p T = p m + p DE , respectively, then the total energy density is indeed conserved in the sense ‡ For a discussion about the regularity of the conformal transformation, or the equivalence issue of the two frames, see for example [173,174,175,176,177,178,179,180,181,182,183,184] and references therein.…”

Acceleration in Weyl Integrable Spacetime

“…eqs (30) and (31) in [28]). More general Lagrangians have also been suggested, especially in the context of the Palatini formalism, for example L = e −φ R in [29]; L = R + βR 2 − 2Λ gravity for FRW models was investigated in [30]; see also [31] for a formulation of a theory invariant with respect to Weyl transformations and [32] for the inclusion of torsion. There is however an alternative view, namely that the pair (Q, g) which defines the Weyl spacetime also enters into the gravitational theory and therefore, the field Q must be contained in the Lagrangian independently from g. In the case of integrable Weyl geometry, i.e.…”