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.

The basic problem of equivalence of gravitational fields in the framework of torsion theories of gravitation is solved. A coordinate-invariant description of the gravitational field is then presented. Explicit expressions for the dimensions of the group of transformations and its subgroup of isotropy are derived.

In the attempts toward a quantum gravity theory, general relativity faces a serious difficulty since it is non-renormalizable theory. Hořava-Lifshitz gravity offers a framework to circumvent this difficulty, by sacrificing the local Lorentz invariance at ultra-high energy scales in exchange of power-counting renormalizability. The Lorentz symmetry is expected to be recovered at low and medium energy scales. If gravitation is to be described by a Hořava-Lifshitz gravity theory there are a number of issues that ought to be reexamined in its context, including the question as to whether this gravity incorporates a chronology protection, or particularly if it allows Gödel-type solutions with violation of causality. We show that Hořava-Lifshitz gravity only allows hyperbolic Gödeltype space-times whose essential parameters m and ω are in the chronology respecting intervals, excluding therefore any noncausal Gödel-type space-times in the hyperbolic class. There emerges from our results that the famous noncausal Gödel model is not allowed in Hořava-Lifshitz gravity. The question as to whether this quantum gravity theory permits hyperbolic Gödel-type solutions in the chronology preserving interval of the essential parameters is also examined. We show that Hořava-Lifshitz gravity not only excludes the noncausal Gödel universe, but also rules out any hyperbolic Gödel-type solutions for physically well-motivated perfect-fluid matter content.PACS numbers: 95.30. Sf, 98.80.Jk, 04.50.Kd, 95.36.+x

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We reformulate the general theory of relativity in the language of Riemann-Cartan geometry. We start from the assumption that the space-time can be described as a non-Riemannian manifold, which, in addition to the metric field, is endowed with torsion. In this new framework, the gravitational field is represented not only by the metric, but also by the torsion, which is completely determined by a geometric scalar field. We show that in this formulation general relativity has a new kind of invariance, whose invariance group consists of a set of conformal and gauge transformations, called Cartan transformations. These involve both the metric tensor and the torsion vector field, and are similar to the well known Weyl gauge transformations. By making use of the concept of Cartan gauges, we show that, under Cartan transformations, the new formalism leads to different pictures of the same gravitational phenomena. We show that in an arbitrary Cartan gauge general relativity has the form of a scalar-tensor theory. In this approach, the Riemann-Cartan geometry appears as the natural geometrical setting of the general relativity theory when the latter is viewed in an arbitrary Cartan gauge. We illustrate this fact by looking at the one of the classical tests of general relativity theory, namely the gravitational spectral shift. Finally, we extend the concept of space-time symmetry to the more general case of Riemann-Cartan space-times endowed with scalar torsion. As an example, we obtain the conservation laws for auto-parallel motion in a static spherically symmetric vacuum space-time in a Cartan gauge, whose orbits are identical to Schwarzschild orbits in general relativity.Comment: 28 pages. arXiv admin note: text overlap with arXiv:1201.146

A class of Riemann-Cartan Gödel-type space-times are examined in the light of the equivalence problem techniques. The conditions for local space-time homogeneity are derived, generalizing previous works on Riemannian Gödel-type space-times. The equivalence of Riemann-Cartan Gödel-type space-times of this class is studied. It is shown that they admit a five-dimensional group of affine-isometries and are characterized by three essential parameters ℓ, m 2 , ω: identical triads (ℓ, m 2 , ω) correspond to locally equivalent manifolds. The algebraic types of the irreducible parts of the curvature and torsion tensors are also presented. * ja@physto.se † jfonseca@dfjp.ufpb.br ‡ M.A.H.MacCallum@qmw.ac.uk §

We verify the consistency of the Gödel-type solutions within the four-dimensional ChernSimons modified gravity with the non-dynamical Chern-Simons coefficient, for different forms of matter including dust, fluid, scalar field and electromagnetic field, and discuss the related causality issues. We show that, unlike the general relativity, a vacuum solution is possible in our theory. Another essentially new result of our theory having no analogue in the general relativity consists in the existence of the hyperbolic causal solutions for a physically wellmotivated matter.

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 conformallyflat spacetimes. We revisit Einstein and Fokker's interpretation of Nordström 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.