Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010

Geodesics: its role in geometrical approaches to gravitation since the appearance of General Relativity Light rays and particles moving under the influence of gravity alone follow space- time geodesics An elegant aspect of the geometrization of the gravitational field is introduced by the so-called geodesical postulate :

In general relativity geodesics are completely determined by the metric properties of space-time This is because general relativity also assumes that space-time geometry is Riemannian But in other metrical theories of gravity, based on non-Riemannian geometry, one distinguishes between metrical geodesics and affine geodesics

Perihelium precession Prediction of gravitational and cosmological phenomena are made by analyzing the behaviour of the light-cone and timelike geodesics Light deflection by the sun Gravitational redshift Also…

Gravitational time delay Black hole physics Cosmological redshift Expansion of the Universe Gravitation lensing Gravitational and cosmological singularities

To further develop these ideas let us consider a kind of interplay between two distinct frameworks: the geometry of Riemann and the geometry of Weyl Almost all information is conveyed by the geodesic lines Thus two distinct theories sharing the same geodesic structure are indistinguishable as far as geodesic-related phenomena are concerned

As we will see, there are circumstances in which one can swift from one to another while keeping some basic geometric structure unchanged. The key notion is the concept of gauge invariance (Weyl) The Weyl geometry

Hermann Weyl 1918

What is Weyl geometry? In Weyl geometry, the manifold is endowed with a global 1-form Riemannian geometry

If we want the elements of the holonomy group to correspond to an isometry, then Consider a closed curve C and two vector fields on C.

Weyl integrable geometry We have a global scalar field defined on the embedding manifold, such that

The interesting fact here is that... Consider now the gauge transformations We can relate the Weyl affine connection with the Riemannian metric connection

...geodesics are invariant under gauge transformations The concept of frames in Weyl geometry The Riemann frame General Relativity is formulated in a Riemann frame, i.e. in which there is no Weyl field

One can look at General Relativity in a non-Riemannian frame (a Weyl frame) Conformally flat spacetimes However… Let us now consider the case of…

As we know, a significant …number of space-times of physical interest predicted by general relativity belong to this class For instance, it is well known that all FRWL cosmological models are conformally flat

Let us consider more generally a certain conformally fl‡at space-time M In the Riemannian context we have no Weyl fi…eld as part of the geometry, and so the components of the affine connection are identical to the Christoffel symbols Suppose now that we make the gauge transformation and with f replacing -. In doing so we go to at a new frame, namely (M;g; )

As we have seen, with respect to geodesics both frames are entirely equivalent Nevertheless, in many aspects the geometries that are defi…ned by them are entirely distinct.

In the Riemann frame the manifold M is endowed with a metric that leads to Riemannian curvature, while in the Weyl frame space-time is flat. Another diference concerns the length of non-null curves or other metric - dependent geometrical quantities since in the two frames we have distinct metric tensors.

Null curves, on the other hand, are mapped into null curves. This implies that the light geometry of a conformally ‡at spacetime is identical to that of Minkowski geometry. Let us now consider a (FLRW) metric for the cases k = +1,-1;, which can be written in the form

In this case the Weyl scalar field will be given by This change of perspective leads, in some cases, to new insights in the description of gravitational phenomena. Gravity in the Weyl frame

In this scenario the gravitational …field is not associated with a tensor, but with a geometrical scalar field living in a Minkowski background. We can get some insight on the amount of physical information carried by the scalar …field by investigating its behaviour in the regime of weak gravity, that is, when we take the Newtonian limit of general relativity.

The Newtonian limit in the Weyl frame In the weak field approximation we take And the Weyl scalar field is considered to be of the same order of.

Then, from the geodesic equation we obtain with Thus Weyl scalar field plays the role of the gravitational potential

And from the Einstein’s equations in the Riemannian frame we get with What is the dynamics of the scalar field? Consider the Einstein-Hilbert action

The duality between the Riemann and the Weyl frames seems to suggest that in the Variation of the action we should consider only variations restricted to the class of conformally flat space-times, that is, Then we have And finally

Weyl frames and scalar gravity Nordstrom theory (1913) Minkowski space-time Gravitation is represented by a scalar field

Einstein-Grossmann early attempt towards a geometrical theory of gravity Conformally flat space-time Einstein-Grossmann theory is may be viewed as a scalar theory in a Weyl flat spacetime.

Weyl frames and quantum gravity Conformal transformation has widely been used in General Relativity as well as in scalar- tensor theories. In fact, there has been a long debate on whether different frames related by conformal transformations have any physical meaning. To our knowledge this debate has, apparently, being restricted to the context of classical physics.

Quantum gravity is widely recognized as one of the most difficult problems of modern theoretical physics. There is currently a vast body of knowledge which includes several different approaches to this area of research. Among the most popular are string theory and loop quantum gravity. There is, however, a feeling among theorists that a final theory of quantum gravity, if there is indeed one, is likely to emerge gradually and will ultimately be a combination of different theoretical frameworks.

As we have seen, when we go to the Weyl frame all information about the gravitational field is encoded in the scalar field, so it seems reasonable that any quantum aspect emerging in the process of quantization, whatever it is, should somehow involve this field. Moreover, one would also expect that the correspondence between the Riemann and Weyl frames would be preserved at the quantum level. If this is true, then it would make sense to carry over the scheme of quantization from the Riemann to the Weyl frame.

Because in the Weyl frame the scalar field is the repository of all physical information it would seem plausible to treat it as genuine physical field. But then we are left with a situation which is typical of the ones considered by quantum field theory in flat space-time. This not so unusual as in perturbative string theory space-time is also treated as an essentially flat background...

Not to mention that Feynmann used to hold the idea that a quantum theory of gravitation should be quantized in Minkowski space-time. At this point many questions arise: What is the meaning of quantizing the Weyl field, anyway? Would the quantization carried out in the Weyl frame imply the quantization of the metric in the Riemann frame?

What would it mean to quantize the metric in the Riemannian frame? Would the theory be renormalizable? Thank you