8 research outputs found
Inducing Barbero-Immirzi Connections along SU(2)-reductions of Bundles on Spacetime
We shall present here a general apt technique to induce connections along
bundle reductions which is different from the standard restriction. This
clarifies and generalizes the standard procedure to define Barbero-Immirzi (BI)
connection, though on spacetime. The standard spacial BI connection used in LQG
is then obtained by its spacetime version by standard restriction. The general
prescription to define such a reduced connection is interesting from a
mathematical viewpoint and it allows a general and direct control on
transformation laws of the induced object. Moreover, unlike what happens by
using standard restriction, we shall show that once a bundle reduction is
given, then any connection induces a reduced connection with no constraint on
the original holonomy as it happens when connections are simply restricted.Comment: 6 pages, some comments adde
New Cases of Universality Theorem for Gravitational Theories
The "Universality Theorem" for gravity shows that f(R) theories (in their
metric-affine formulation) in vacuum are dynamically equivalent to vacuum
Einstein equations with suitable cosmological constants. This holds true for a
generic (i.e. except sporadic degenerate cases) analytic function f(R) and
standard gravity without cosmological constant is reproduced if f is the
identity function (i.e. f(R)=R). The theorem is here extended introducing in
dimension 4 a 1-parameter family of invariants R' inspired by the
Barbero-Immirzi formulation of GR (which in the Euclidean sector includes also
selfdual formulation). It will be proven that f(R') theories so defined are
dynamically equivalent to the corresponding metric-affine f(R) theory. In
particular for the function f(R)=R the standard equivalence between GR and
Holst Lagrangian is obtained.Comment: 10 pages, few typos correcte
Gauge Formalism for General Relativity and Fermionic Matter
A new formalism for spinors on curved spaces is developed in the framework of
variational calculus on fibre bundles. The theory has the same structure of a
gauge theory and describes the interaction between the gravitational field and
spinors. An appropriate gauge structure is also given to General Relativity,
replacing the metric field with spin frames. Finally, conserved quantities and
superpotentials are calculated under a general covariant form.Comment: 18 pages, Plain TEX, revision, explicit expression for superpotential
has been adde
On the universality of the Carter and McLenaghan formula
It is shown that the formula of the isometry generators of the spinor
representation given by Carter and McLenaghan is universal in the sense that
this holds for any representation either in local frames or even in natural
ones. The point-dependent spin matrices in natural frames are introduced for
any tensor representation deriving the covariant form of the isometry
generators in these frames.Comment: 7 pages, no figure
Conserved Quantities from the Equations of Motion (with applications to natural and gauge natural theories of gravitation)
We present an alternative field theoretical approach to the definition of
conserved quantities, based directly on the field equations content of a
Lagrangian theory (in the standard framework of the Calculus of Variations in
jet bundles). The contraction of the Euler-Lagrange equations with Lie
derivatives of the dynamical fields allows one to derive a variational
Lagrangian for any given set of Lagrangian equations. A two steps algorithmical
procedure can be thence applied to the variational Lagrangian in order to
produce a general expression for the variation of all quantities which are
(covariantly) conserved along the given dynamics. As a concrete example we test
this new formalism on Einstein's equations: well known and widely accepted
formulae for the variation of the Hamiltonian and the variation of Energy for
General Relativity are recovered. We also consider the Einstein-Cartan
(Sciama-Kibble) theory in tetrad formalism and as a by-product we gain some new
insight on the Kosmann lift in gauge natural theories, which arises when trying
to restore naturality in a gauge natural variational Lagrangian.Comment: Latex file, 31 page
Two-spinor Formulation of First Order Gravity coupled to Dirac Fields
Two-spinor formalism for Einstein Lagrangian is developed. The gravitational
field is regarded as a composite object derived from soldering forms. Our
formalism is geometrically and globally well-defined and may be used in
virtually any 4m-dimensional manifold with arbitrary signature as well as
without any stringent topological requirement on space-time, such as
parallelizability. Interactions and feedbacks between gravity and spinor fields
are considered. As is well known, the Hilbert-Einstein Lagrangian is second
order also when expressed in terms of soldering forms. A covariant splitting is
then analysed leading to a first order Lagrangian which is recognized to play a
fundamental role in the theory of conserved quantities. The splitting and
thence the first order Lagrangian depend on a reference spin connection which
is physically interpreted as setting the zero level for conserved quantities. A
complete and detailed treatment of conserved quantities is then presented.Comment: 16 pages, Plain TE