conformally flat spacetimes and weyl frames carlos romero cargèse - 11 mai 2010
TRANSCRIPT
![Page 1: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/1.jpg)
Conformally flat Conformally flat spacetimes and Weyl spacetimes and Weyl
framesframes
Carlos RomeroCarlos Romero
Cargèse - 11 Mai 2010
![Page 2: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/2.jpg)
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:
![Page 3: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/3.jpg)
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
![Page 4: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/4.jpg)
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…
![Page 5: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/5.jpg)
Gravitational time delayGravitational time delay
Black hole physics
Cosmological redshift
Expansion of the Universe
Gravitation lensing
Gravitational and cosmological singularities
![Page 6: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/6.jpg)
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
![Page 7: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/7.jpg)
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
![Page 8: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/8.jpg)
Hermann Weyl1918
![Page 9: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/9.jpg)
What is Weyl geometry?What is Weyl geometry?
In Weyl geometry, the manifold is endowed with a global 1-form
Riemannian geometry
![Page 10: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/10.jpg)
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.
![Page 11: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/11.jpg)
Weyl integrable geometryWeyl integrable geometry
We have a global scalar field defined on the embedding manifold, such that
![Page 12: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/12.jpg)
The interesting fact here is that...
Consider now the gauge transformations
We can relate the Weyl affine connectionwith the Riemannian metric connection
![Page 13: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/13.jpg)
...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
![Page 14: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/14.jpg)
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…
![Page 15: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/15.jpg)
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
![Page 16: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/16.jpg)
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; )
![Page 17: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/17.jpg)
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.
![Page 18: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/18.jpg)
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.
![Page 19: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/19.jpg)
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
![Page 20: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/20.jpg)
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
![Page 21: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/21.jpg)
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.
![Page 22: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/22.jpg)
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 .
![Page 23: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/23.jpg)
Then, from the geodesic equation
we obtain
with
Thus Weyl scalar field plays the role ofthe gravitational potential
![Page 24: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/24.jpg)
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
![Page 25: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/25.jpg)
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
![Page 26: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/26.jpg)
Weyl frames and scalar gravity
Nordstrom theory (1913)
Minkowski space-time
Gravitation is represented by ascalar field
![Page 27: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/27.jpg)
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.
![Page 28: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/28.jpg)
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.
![Page 29: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/29.jpg)
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.
![Page 30: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/30.jpg)
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.
![Page 31: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/31.jpg)
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...
![Page 32: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/32.jpg)
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?
![Page 33: Conformally flat spacetimes and Weyl frames Carlos Romero Cargèse - 11 Mai 2010](https://reader035.vdocuments.us/reader035/viewer/2022062718/56649e695503460f94b65e43/html5/thumbnails/33.jpg)
What would it mean to quantize the metricin the Riemannian frame?
Would the theory be renormalizable?
Thank you