Conformally flat Conformally flat spacetimes and Weyl spacetimes and Weyl

framesframes

Carlos RomeroCarlos 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 arecompletely 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 precessionPerihelium precession

Prediction of gravitational and Prediction of gravitational and cosmological phenomena are made by cosmological phenomena are made by

analyzing the behaviour of the light-cone analyzing the behaviour of the light-cone and timelike geodesicsand timelike geodesics

Light deflection by the sun

Gravitational redshift

Also…

Gravitational time delayGravitational 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 To further develop these ideas let us consider a kind of interplay between two distinct kind of interplay between two distinct

frameworks: the geometry of Riemann and the frameworks: the geometry of Riemann and the geometry of Weyl geometry of Weyl

Almost all information is conveyed by theAlmost all information is conveyed by the

geodesic lines geodesic lines

Thus two distinct theories sharing the same Thus two distinct theories sharing the same geodesic structure are indistinguishable as far asgeodesic structure are indistinguishable as far as

geodesic-related phenomena are concernedgeodesic-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 gaugeinvariance (Weyl)

The Weyl geometry

Hermann Weyl1918

What is Weyl geometry?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 geometryWeyl 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 connectionwith the Riemannian metric connection

...geodesics are invariant under gauge transformations

The concept of frames in Weyl geometry

The Riemann frame

General Relativity is formulated ina 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 lengthof 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) metricfor the cases k = +1,-1;, which can be written in the form

In this case the Weyl scalar field willbe 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 generalrelativity.

The Newtonian limit in theWeyl frame

In the weak field approximation we take

And the Weyl scalar field is consideredto be of the same order of .

Then, from the geodesic equation

we obtain

with

Thus Weyl scalar field plays the role ofthe 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 theVariation of the action we should consideronly 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 ascalar 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 quantizingthe Weyl field, anyway?

Would the quantization carried out in theWeyl frame imply the quantization of the metric in the Riemann frame?

What would it mean to quantize the metricin the Riemannian frame?

Would the theory be renormalizable?

Thank you