4 research outputs found
Conformal invariance: from Weyl to SO(2,d)
The present work deals with two different but subtilely related kinds of
conformal mappings: Weyl rescaling in dimensional spaces and SO(2,d)
transformations. We express how the difference between the two can be
compensated by diffeomorphic transformations. This is well known in the
framework of String Theory but in the particular case of spaces. Indeed,
the Polyakov formalism describes world-sheets in terms of two-dimensional
conformal field theory. On the other hand, B. Zumino had shown that a classical
four-dimensional Weyl-invariant field theory restricted to live in Minkowski
space leads to an SO(2,4)-invariant field theory. We extend Zumino's result to
relate Weyl and SO(2,d) symmetries in arbitrary conformally flat spaces (CFS).
This allows us to assert that a classical -invariant field does not
distinguish, at least locally, between two different -dimensional CFSs.Comment: 5 pages, no figures. There are slight modifications to match with the
published versio
Conformally flat spacetimes and Weyl frames
We discuss the concepts of Weyl and Riemann frames in the context of metric
theories of gravity and state the fact that they are completely equivalent as
far as geodesic motion is concerned. We apply this result to conformally flat
spacetimes and show that a new picture arises when a Riemannian spacetime is
taken by means of geometrical gauge transformations into a Minkowskian flat
spacetime. We find out that in the Weyl frame gravity is described by a scalar
field. We give some examples of how conformally flat spacetime configurations
look when viewed from the standpoint of a Weyl frame. We show that in the
non-relativistic and weak field regime the Weyl scalar field may be identified
with the Newtonian gravitational potential. We suggest an equation for the
scalar field by varying the Einstein-Hilbert action restricted to the class of
conformally-flat spacetimes. We revisit Einstein and Fokker's interpretation of
Nordstr\"om scalar gravity theory and draw an analogy between this approach and
the Weyl gauge formalism. We briefly take a look at two-dimensional gravity as
viewed in the Weyl frame and address the question of quantizing a conformally
flat spacetime by going to the Weyl frame.Comment: LATEX - 18 page
General Relativity and Weyl Geometry
We show that the general theory of relativity can 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.Comment: LATEX, 22 page