Transcript Einstein e a Teoria da Relatividade

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
An elegant aspect of the
geometrization of the
gravitational field is introduced
by the so-called geodesical
postulate:
Light rays and particles moving under the
influence of gravity alone follow spacetime geodesics
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
Prediction of gravitational and
cosmological phenomena are made by
analyzing the behaviour of the light-cone
and timelike geodesics
Perihelium precession
Light deflection by the sun
Gravitational redshift
Also…
Gravitational time delay
Black hole physics
Gravitation lensing
Gravitational and cosmological
singularities
Cosmological redshift
Expansion of the Universe
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
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
The Weyl geometry
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)
Hermann Weyl
1918
What is Weyl geometry?
Riemannian geometry
In Weyl geometry, the manifold is
endowed with a global 1-form
Consider a closed curve C
and two vector fields on C.
If we want the elements of the
holonomy group to correspond to an
isometry, then
Weyl integrable geometry
We have a global scalar field defined on the
embedding manifold, such that
We can relate the Weyl affine connection
with the Riemannian metric connection
Consider now the gauge transformations
The interesting fact here is that...
...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
However…
One can look at General Relativity in a
non-Riemannian frame (a Weyl frame)
Let us now consider the case of…
Conformally flat spacetimes
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
with
we obtain
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 scalartensor 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