scholarly journals First-order minisuperspace model for the Faddeev formulation of gravity

2015 ◽  
Vol 30 (32) ◽  
pp. 1550174 ◽  
Author(s):  
V. M. Khatsymovsky
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Field Equations ◽  
Piecewise Constant ◽  
First Order ◽  
Weyl Connection ◽  
Type Form ◽  
Connection Type ◽  
Discrete Theory

Faddeev formulation of general relativity (GR) is considered where the metric is composed of ten vector fields or a ten-dimensional tetrad. Upon partial use of the field equations, this theory results in the usual GR. Earlier we have proposed some minisuperspace model for the Faddeev formulation where the tetrad fields are piecewise constant on the polytopes like 4-simplices or, say, cuboids into which [Formula: see text] can be decomposed. Now we study some representation of this (discrete) theory, an analogue of the Cartan–Weyl connection-type form of the Hilbert–Einstein action in the usual continuum GR.

scholarly journals Spectrum of area in the Faddeev formulation of gravity

2016 ◽  
Vol 31 (19) ◽  
pp. 1650114 ◽  
Author(s):  
V. M. Khatsymovsky
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Hamiltonian Formalism ◽  
Black Hole Entropy ◽  
Field Equations ◽  
Piecewise Constant ◽  
First Order ◽  
Connection Matrices ◽  
Order Representation ◽  
Connection Type

Faddeev formulation of general relativity (GR) is considered where the metric is composed of ten vector fields or a ten-dimensional tetrad. Upon partial use of the field equations, this theory results in the usual general relativity (GR). Earlier, we have proposed first-order representation of the minisuperspace model for the Faddeev formulation where the tetrad fields are piecewise constant on the polytopes like four-simplices or, say, cuboids into which [Formula: see text] can be decomposed, an analogue of the Cartan–Weyl connection-type form of the Hilbert–Einstein action in the usual continuum GR. In the Hamiltonian formalism, the tetrad bilinears are canonically conjugate to the orthogonal connection matrices. We evaluate the spectrum of the elementary areas, functions of the tetrad bilinears. The spectrum is discrete and proportional to the Faddeev analog [Formula: see text] of the Barbero–Immirzi parameter [Formula: see text]. The possibility of the tetrad and metric discontinuities in the Faddeev gravity allows to consider any surface as consisting of a set of virtually independent elementary areas and its spectrum being the sum of the elementary spectra. Requiring consistency of the black hole entropy calculations known in the literature we are able to estimate [Formula: see text].

scholarly journals Some minisuperspace model for the Faddeev formulation of gravity

2014 ◽  
Vol 29 (27) ◽  
pp. 1450141 ◽  
Author(s):  
V. M. Khatsymovsky
Keyword(s):  
General Relativity ◽  
Distinctive Feature ◽  
Vector Fields ◽  
Measure Zero ◽  
Equations Of Motion ◽  
Zero Set ◽  
Constant Vector ◽  
Field Equations ◽  
Piecewise Constant ◽  
The Continuum

We consider Faddeev formulation of General Relativity (GR) in which the metric is composed of ten vector fields or a 4 ×10 tetrad. This formulation reduces to the usual GR upon partial use of the field equations. A distinctive feature of the Faddeev action is its finiteness on the discontinuous fields. This allows to introduce its minisuperspace formulation where the vector fields are constant everywhere on ℝ4 with exception of a measure zero set (the piecewise constant fields). The fields are parametrized by their constant values independently chosen in, e.g. the 4-simplices or, say, parallelepipeds into which ℝ4 can be decomposed. The form of the action for the vector fields of this type is found. We also consider the piecewise constant vector fields approximating the fixed smooth ones. We check that if the regions in which the vector fields are constant are made arbitrarily small, the minisuperspace action and equations of motion tend to the continuum Faddeev ones.

scholarly journals Off-Shell Noether Currents and Potentials for First-Order General Relativity

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 348
Author(s):  
Merced Montesinos ◽  
Diego Gonzalez ◽  
Rodrigo Romero ◽  
Mariano Celada
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Arbitrary Vector ◽  
Four Dimensions ◽  
First Order ◽  
Direct Form ◽  
Definition Of

We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by n-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local SO(n) or SO(n−1,1) transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do not use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local SO(3,1) or SO(4) off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions.

Exact solution of a static charged sphere in general relativity

1969 ◽  
Vol 47 (21) ◽  
pp. 2401-2404 ◽  
Author(s):  
S. J. Wilson
Keyword(s):  
Exact Solution ◽  
General Relativity ◽  
Gravitational Constant ◽  
The Self ◽  
Spherically Symmetric ◽  
Charged Sphere ◽  
Field Equations ◽  
First Order ◽  
Self Energy ◽  
Spherically Symmetric Distribution

An exact solution of the field equations of general relativity is obtained for a static, spherically symmetric distribution of charge and mass which can be matched with the Reissner–Nordström metric at the boundary. The self-energy contributions to the total gravitational mass are computed retaining only the first order terms in the gravitational constant.

First-order field equations in general relativity

1991 ◽  
Vol 106 (3) ◽  
pp. 273-289 ◽  
Author(s):  
M. Mattes ◽  
M. Sorg
Keyword(s):  
General Relativity ◽  
Field Equations ◽  
First Order

scholarly journals PRE-MULTISYMPLECTIC CONSTRAINT ALGORITHM FOR FIELD THEORIES

2005 ◽  
Vol 02 (05) ◽  
pp. 839-871 ◽  
Author(s):  
MANUEL DE LEÓN ◽  
JESÚS MARÍN-SOLANO ◽  
JUAN CARLOS MARRERO ◽  
MIGUEL C. MUÑOZ-LECANDA ◽  
NARCISO ROMÁN-ROY
Keyword(s):  
Fiber Bundle ◽  
General Framework ◽  
Vector Fields ◽  
Classical Field ◽  
Field Theories ◽  
Second Step ◽  
Field Equations ◽  
Jet Bundle ◽  
First Order ◽  
Classical Field Theories

We present a geometric algorithm for obtaining consistent solutions to systems of partial differential equations, mainly arising from singular covariant first-order classical field theories. This algorithm gives an intrinsic description of all the constraint submanifolds. The field equations are stated geometrically, either representing their solutions by integrable connections or, what is equivalent, by certain kinds of integrable m-vector fields. First, we consider the problem of finding connections or multivector fields solutions to the field equations in a general framework: a pre-multisymplectic fiber bundle (which will be identified with the first-order jet bundle and the multi-momentum bundle when Lagrangian and Hamiltonian field theories are considered). Then, the problem is stated and solved in a linear context, and a pointwise application of the results leads to the algorithm for the general case. In a second step, the integrability of the solutions is also studied. Finally, the method is applied to Lagrangian and Hamiltonian field theories and, for the former, the problem of finding holonomic solutions is also analyzed.

scholarly journals Einstein gravity as a gauge theory for the conformal group

2021 ◽  
Author(s):  
Yannick Herfray ◽  
Carlos Scarinci
Keyword(s):  
General Relativity ◽  
Gauge Theory ◽  
Principal Bundle ◽  
A Priori ◽  
Conformal Group ◽  
Standard Theory ◽  
Field Equations ◽  
First Order ◽  
Torsion Freeness ◽  
Soldering Form

Abstract General Relativity in dimension $n = p + q$ can be formulated as a gauge theory for the conformal group $\SO\left(p+1,q+1\right)$, along with an additional field reducing the structure group down to the Poincaré group $\ISO\left(p,q\right)$. In this paper, we propose a new variational principle for Einstein geometry which realizes this fact. Importantly, as opposed to previous treatments in the literature, our action functional gives first order field equations and does not require supplementary constraints on gauge fields, such as torsion-freeness. Our approach is based on the ``first order formulation'' of conformal tractor geometry. Accordingly, it provides a straightforward variational derivation of the tractor version of the Einstein equation. To achieve this, we review the standard theory of tractor geometry with a gauge theory perspective, defining the tractor bundle a priori in terms of an abstract principal bundle and providing an identification with the standard conformal tractor bundle via a dynamical soldering form. This can also be seen as a generalization of the so called Cartan-Palatini formulation of General Relativity in which the ``internal'' orthogonal group $\SO\left(p,q\right)$ is extended to an appropriate parabolic subgroup $P\subset\SO\left(p+1,q+1\right)$ of the conformal group.

scholarly journals Curvature tensor for the spacetime of general relativity

2017 ◽  
Vol 14 (05) ◽  
pp. 1750078 ◽  
Author(s):  
Zafar Ahsan ◽  
Musavvir Ali
Keyword(s):  
General Relativity ◽  
Perfect Fluid ◽  
Curvature Tensor ◽  
Vector Fields ◽  
Killing Vector ◽  
Conformal Killing ◽  
Field Equations ◽  
Text Structures ◽  
Conformal Killing Vector Fields ◽  
Perfect Fluid Spacetime

In the differential geometry of certain [Formula: see text]-structures, the role of [Formula: see text]-curvature tensor is very well known. A detailed study of this tensor has been made on the spacetime of general relativity. The spacetimes satisfying Einstein field equations with vanishing [Formula: see text]-tensor have been considered and the existence of Killing and conformal Killing vector fields has been established. Perfect fluid spacetimes with vanishing [Formula: see text]-tensor have also been considered. The divergence of [Formula: see text]-tensor is studied in detail and it is seen, among other results, that a perfect fluid spacetime with conserved [Formula: see text]-tensor represents either an Einstein space or a Friedmann-Robertson-Walker cosmological model.

General Relativity

2019 ◽  
Author(s):  
Steven Carlip
Keyword(s):  
General Relativity ◽  
High Energy Physics ◽  
Hamiltonian Formulation ◽  
Experimental Tests ◽  
High Energy ◽  
Gravitational Energy ◽  
Field Equations ◽  
Physics First ◽  
Condensed Matter Theory ◽  
Introductory Textbooks

This work is a short textbook on general relativity and gravitation, aimed at readers with a broad range of interests in physics, from cosmology to gravitational radiation to high energy physics to condensed matter theory. It is an introductory text, but it has also been written as a jumping-off point for readers who plan to study more specialized topics. As a textbook, it is designed to be usable in a one-quarter course (about 25 hours of instruction), and should be suitable for both graduate students and advanced undergraduates. The pedagogical approach is “physics first”: readers move very quickly to the calculation of observational predictions, and only return to the mathematical foundations after the physics is established. The book is mathematically correct—even nonspecialists need to know some differential geometry to be able to read papers—but informal. In addition to the “standard” topics covered by most introductory textbooks, it contains short introductions to more advanced topics: for instance, why field equations are second order, how to treat gravitational energy, what is required for a Hamiltonian formulation of general relativity. A concluding chapter discusses directions for further study, from mathematical relativity to experimental tests to quantum gravity.

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