145 research outputs found
Introductory lectures to loop quantum gravity
We give a standard introduction to loop quantum gravity, from the ADM
variables to spin network states. We include a discussion on quantum geometry
on a fixed graph and its relation to a discrete approximation of general
relativity.Comment: Based on lectures given at the 3eme Ecole de Physique Theorique de
Jijel, Algeria, 26 Sep -- 3 Oct, 2009. 52 pages, many figures. v2 minor
corrections. To be published in the proceeding
Bi-gravity with a single graviton
We analyze a bi-gravity model based on the first order formalism, having as
fundamental variables two tetrads but only one Lorentz connection. We show that
on a large class of backgrounds its linearization agrees with general
relativity. At the non-linear level, additional degrees of freedom appear, and
we reveal the mechanism hiding them around the special backgrounds. We further
argue that they do not contain a massive graviton, nor the Boulware-Deser
ghost. The model thus propagates only one graviton, whereas the nature of the
additional degrees of freedom remains to be investigated. We also present a
foliation-preserving deformation of the model, which keeps all symmetries
except time diffeomorphisms and has three degrees of freedom.Comment: 29 page
A note on the Plebanski action with cosmological constant and an Immirzi parameter
We study the field equations of the Plebanski action for general relativity
when both the cosmological constant and an Immirzi parameter are present. We
show that the Lagrange multiplier, which usually gets identified with the Weyl
curvature, now acquires a trace part. Some consequences of this for a class of
modified gravity theories recently proposed in the literature are briefly
discussed.Comment: 14 pages. v2: factor 2 corrected, updated reference
Null twisted geometries
We define and investigate a quantisation of null hypersurfaces in the context
of loop quantum gravity on a fixed graph. The main tool we use is the
parametrisation of the theory in terms of twistors, which has already proved
useful in discussing the interpretation of spin networks as the quantization of
twisted geometries. The classical formalism can be extended in a natural way to
null hypersurfaces, with the Euclidean polyhedra replaced by null polyhedra
with space-like faces, and SU(2) by the little group ISO(2). The main
difference is that the simplicity constraints present in the formalims are all
first class, and the symplectic reduction selects only the helicity subgroup of
the little group. As a consequence, information on the shapes of the polyhedra
is lost, and the result is a much simpler, abelian geometric picture. It can be
described by an Euclidean singular structure on the 2-dimensional space-like
surface defined by a foliation of space-time by null hypersurfaces. This
geometric structure is naturally decomposed into a conformal metric and scale
factors, forming locally conjugate pairs. Proper action-angle variables on the
gauge-invariant phase space are described by the eigenvectors of the Laplacian
of the dual graph. We also identify the variables of the phase space amenable
to characterize the extrinsic geometry of the foliation. Finally, we quantise
the phase space and its algebra using Dirac's algorithm, obtaining a notion of
spin networks for null hypersurfaces. Such spin networks are labelled by SO(2)
quantum numbers, and are embedded non-trivially in the unitary,
infinite-dimensional irreducible representations of the Lorentz group.Comment: 22 pages, 3 figures. v2: minor corrections, improved presentation in
section 4, references update
On the Relations between Gravity and BF Theories
We review, in the light of recent developments, the existing relations
between gravity and topological BF theories at the classical level. We include
the Plebanski action in both self-dual and non-chiral formulations, their
generalizations, and the MacDowell-Mansouri action.Comment: v3: journal typeset versio
Sachs' free data in real connection variables
We discuss the Hamiltonian dynamics of general relativity with real
connection variables on a null foliation, and use the Newman-Penrose formalism
to shed light on the geometric meaning of the various constraints. We identify
the equivalent of Sachs' constraint-free initial data as projections of
connection components related to null rotations, i.e. the translational part of
the ISO(2) group stabilising the internal null direction soldered to the
hypersurface. A pair of second-class constraints reduces these connection
components to the shear of a null geodesic congruence, thus establishing
equivalence with the second-order formalism, which we show in details at the
level of symplectic potentials. A special feature of the first-order
formulation is that Sachs' propagating equations for the shear, away from the
initial hypersurface, are turned into tertiary constraints; their role is to
preserve the relation between connection and shear under retarded time
evolution. The conversion of wave-like propagating equations into constraints
is possible thanks to an algebraic Bianchi identity; the same one that allows
one to describe the radiative data at future null infinity in terms of a shear
of a (non-geodesic) asymptotic null vector field in the physical spacetime.
Finally, we compute the modification to the spin coefficients and the null
congruence in the presence of torsion.Comment: 23 pages + Appendix, 2 figures. v2: Improved text and some amendments
throughout, added more details on the relation between 2+2 foliations and
null tetrads, updated references. Version submitted for peer reviewing. v3:
Few minor amendments, footnote added on a null congruence in the presence of
torsion; matches published versio
Chiral description of ghost-free massive gravity
We propose and study a new first order version of the ghost-free massive
gravity. Instead of metrics or tetrads, it uses a connection together with
Plebanski's chiral 2-forms as fundamental variables, rendering the phase space
structure similar to that of SU(2) gauge theories. The chiral description
simplifies computations of the constraint algebra, and allows us to perform the
complete canonical analysis of the system. In particular, we explicitly compute
the secondary constraint and carry out the stabilization procedure, thus
proving that in general the theory propagates 7 degrees of freedom,
consistently with previous claims. Finally, we point out that the description
in terms of 2-forms opens the door to an infinite class of ghost-free massive
bi-gravity actions. Our results apply directly to Euclidean signature. The
reality conditions to be imposed in the Lorentzian signature appear to be more
complicated than in the usual gravity case and are left as an open issue.Comment: 26 pages; extended discussion of reality conditions, added reference
Spinfoam 2d quantum gravity and discrete bundles
In 4 dimensions, general relativity can be formulated as a constrained
theory; we show that the same is true in 2 dimensions. We describe a spinfoam
quantization of this constrained BF-formulation of 2d riemannian general
relativity, obtained using the Barrett-Crane technique of imposing the
constraint as a restriction on the representations summed over. We obtain the
expected partition function, thus providing support for the viability of the
technique. The result requires the nontrivial topology of the bundle where the
gravitational connection is defined, to be taken into account. For this
purpose, we study the definition of a principal bundle over a simplicial base
space. The model sheds light also on several other features of spinfoam quantum
gravity: the reality of the partition function; the geometrical interpretation
of the Newton constant, and the issue of possible finiteness of the partition
function of quantum general relativity.Comment: 20 pages, 3 figures. Comments added, to appear in Class.Quant.Gra
2-vertex Lorentzian Spin Foam Amplitudes for Dipole Transitions
We compute transition amplitudes between two spin networks with dipole
graphs, using the Lorentzian EPRL model with up to two (non-simplicial)
vertices. We find power-law decreasing amplitudes in the large spin limit,
decreasing faster as the complexity of the foam increases. There are no
oscillations nor asymptotic Regge actions at the order considered, nonetheless
the amplitudes still induce non-trivial correlations. Spin correlations between
the two dipoles appear only when one internal face is present in the foam. We
compute them within a mini-superspace description, finding positive
correlations, decreasing in value with the Immirzi parameter. The paper also
provides an explicit guide to computing Lorentzian amplitudes using the
factorisation property of SL(2,C) Clebsch-Gordan coefficients in terms of SU(2)
ones. We discuss some of the difficulties of non-simplicial foams, and provide
a specific criterion to partially limit the proliferation of diagrams. We
systematically compare the results with the simplified EPRLs model, much faster
to evaluate, to learn evidence on when it provides reliable approximations of
the full amplitudes. Finally, we comment on implications of our results for the
physics of non-simplicial spin foams and their resummation.Comment: 27 pages + appendix, many figures. v2: one more numerical result,
plus minor amendment
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