scholarly journals On the discrete Christoffel symbols

2019 ◽  
Vol 34 (30) ◽  
pp. 1950186 ◽  
Author(s):  
V. M. Khatsymovsky
Keyword(s):  
Affine Connection ◽  
Flat Space ◽  
Discrete Analogue ◽  
Connection Form ◽  
Leading Order ◽  
Piecewise Affine ◽  
Order Form ◽  
Christoffel Symbols ◽  
Difference Form ◽  
Torsion Free

The piecewise flat space–time is equipped with a set of edge lengths and vertex coordinates. This defines a piecewise affine coordinate system and a piecewise affine metric in it, the discrete analogue of the unique torsion-free metric-compatible affine connection or of the Levi-Civita connection (or of the standard expression of the Christoffel symbols in terms of metric) mentioned in the literature, and, substituting this into the affine connection form of the Regge action of our previous work, we get a second-order form of the action. This can be expanded over metric variations from simplex to simplex. For a particular periodic simplicial structure and coordinates of the vertices, the leading order over metric variations is found to coincide with a certain finite difference form of the Hilbert–Einstein action.

Leading order of quantum corrections to black hole entropy sum relations

2019 ◽  
Vol 34 (32) ◽  
pp. 1950216
Author(s):  
Tairan Liang ◽  
Wei Xu
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Quantum Physics ◽  
Black Hole Entropy ◽  
Flat Space ◽  
Quantum Corrections ◽  
Leading Order ◽  
Physics Of Black Holes ◽  
Mass Dependent ◽  
Modified Entropy

It has been found recently that the entropy relations of horizons have the universality of black hole mass-independence for many black holes. These universal entropy relations have some geometric and CFT understanding, which may provide further insight into the quantum physics of black holes. In this paper, we present the leading order of black hole entropy sum relations under the quantum corrections. It is found that the modified entropy sum becomes mass-dependent for some black holes in asymptotical (A)dS and flat space–times. We also give an example that the modified entropy sum of regular Bardeen AdS black holes is mass-independent, which may be quantized in the form of the electric charge and the cosmological constant.

scholarly journals WEAK FIELD EXPANSION OF GRAVITY: GRAPHS, MATRICES AND TOPOLOGY

1999 ◽  
Vol 14 (17) ◽  
pp. 2705-2743
Author(s):  
SHOICHI ICHINOSE ◽  
NORIAKI IKEDA
Keyword(s):  
Graphical Representation ◽  
Young Tableau ◽  
Flat Space ◽  
Weak Field ◽  
Main Theme ◽  
Leading Order ◽  
Group Approach ◽  
And Topology ◽  
Diagram Approach ◽  
Scalar Gravity

We present some approaches to the perturbative analysis of the classical and quantum gravity. First we introduce a graphical representation for a global SO (n) tensor (∂)dhαβ, which generally appears in the weak field expansion around the flat space: gμν=δμν+hμν. Making use of this representation, we explain (1) Generating function of graphs (Feynman diagram approach), (2) Adjacency matrix (Matrix approach), (3) Graphical classification in terms of "topology indices" (Topology approach), (4) The Young tableau (Symmetric group approach). We systematically construct the global SO (n) invariants. How to show the independence and completeness of those invariants is the main theme. We explain it taking simple examples of ∂∂h-, and (∂∂h)2-invariants in the text. The results are applied to the analysis of the independence of general invariants and (the leading order of) the Weyl anomalies of scalar-gravity theories in "diverse" dimensions (2,4,6,8,10 dimensions).

scholarly journals Simplicial Palatini action

2018 ◽  
Vol 33 (01) ◽  
pp. 1850004
Author(s):  
V. M. Khatsymovsky
Keyword(s):  
General Relativity ◽  
Probability Distribution ◽  
Path Integral ◽  
Parity Violation ◽  
Equations Of Motion ◽  
Functional Integration ◽  
Diagram Technique ◽  
Connection Form ◽  
Flat Spacetime ◽  
Christoffel Symbols

We consider the piecewise flat spacetime and a simplicial analog of the Palatini form of the general relativity (GR) action where the discrete Christoffel symbols are given on the tetrahedra as variables that are independent of the metric. Excluding these variables with the help of the equations of motion gives exactly the Regge action. This paper continues our previous work. Now, we include the parity violation term and the analog of the Barbero–Immirzi parameter introduced in the orthogonal connection form of GR. We consider the path integral and the functional integration over the connection. The result of the latter (for certain limiting cases of some parameters) is compared with the earlier found result of the functional integration over the connection for the analogous orthogonal connection representation of Regge action. These results, mainly as some measures on the lengths/areas, are discussed for the possibility of the diagram technique where the perturbative diagrams for the Regge action calculated using the measure obtained are finite. This finiteness is due to these measures providing elementary lengths being mostly bounded and separated from zero, just as the finiteness of a theory on a lattice with an analogous probability distribution of spacings.

scholarly journals A generalized Weyl structure with arbitrary non-metricity

2019 ◽  
Vol 79 (10) ◽  
Author(s):  
Adria Delhom ◽  
Iarley P. Lobo ◽  
Gonzalo J. Olmo ◽  
Carlos Romero
Keyword(s):  
Equivalence Class ◽  
Conformal Symmetry ◽  
Affine Connection ◽  
The Other ◽  
Metric Tensor ◽  
Form Field ◽  
Torsion Free

Abstract A Weyl structure is usually defined by an equivalence class of pairs $$(\mathbf{g}, {\varvec{\omega }})$$(g,ω) related by Weyl transformations, which preserve the relation $$\nabla \mathbf{g}={\varvec{\omega }}\otimes \mathbf{g}$$∇g=ω⊗g, where $$\mathbf{g}$$g and $${\varvec{\omega }}$$ω denote the metric tensor and a 1-form field. An equivalent way of defining such a structure is as an equivalence class of conformally related metrics with a unique affine connection $$\Gamma _{({\varvec{\omega }})}$$Γ(ω), which is invariant under Weyl transformations. In a standard Weyl structure, this unique connection is assumed to be torsion-free and have vectorial non-metricity. This second view allows us to present two different generalizations of standard Weyl structures. The first one relies on conformal symmetry while allowing for a general non-metricity tensor, and the other comes from extending the symmetry to arbitrary (disformal) transformations of the metric.

Position, momentum, and wave spreading in curved space

2021 ◽  
Vol 36 (08n09) ◽  
pp. 2150065
Author(s):  
Mohammad Khorrami
Keyword(s):  
Quantum Mechanics ◽  
Wave Packet ◽  
Free Particle ◽  
Flat Space ◽  
Quantum Evolution ◽  
Leading Order ◽  
Position Space ◽  
Curved Space ◽  
Minimum Uncertainty ◽  
Geometry Deviation

The effect of the geometry (deviation from the flat space) on the quantum evolution of the momentum and position of a free particle is discussed. It is shown that beginning with a wave-packet of minimum uncertainty (a Gaussian wave), there is a usual increase in the product of the volume uncertainties in the momentum and position space, as seen in the quantum mechanics on a flat spaces. But there is also a contribution from geometry. The leading order of this contribution is calculated.

Curved Space

2017 ◽  
Author(s):  
Nicholas Manton ◽  
Nicholas Mee
Keyword(s):  
Arbitrary Dimension ◽  
Spherical Geometry ◽  
Flat Space ◽  
Geodesic Equation ◽  
Metric Tensor ◽  
Curved Space ◽  
Covariant Derivatives ◽  
Christoffel Symbols ◽  
Geometry Configuration ◽  
Fisheye Lenses

This chapter develops the mathematical technology required to understand general relativity by taking the reader from the traditional flat space geometry of Euclid to the geometry of Riemann that describes general curved spaces of arbitrary dimension. The chapter begins with a comparison of Euclidean geometry and spherical geometry. The concept of the geodesic is introduced. The discovery of hyperbolic geometry is discussed. Gaussian curvature is defined. Tensors are introduced. The metric tensor is defined and simple examples are given. This leads to the use of covariant derivatives, expressed in terms of Christoffel symbols, the Riemann curvature tensor and all machinery of Riemannian geometry, with each step illustrated by simple examples. The geodesic equation and the equation of geodesic deviation are derived. The final section considers some applications of curved geometry: configuration space, mirages and fisheye lenses.

scholarly journals CANONICAL FORMULATION OF A BOSONIC MATTER FIELD IN (1 + 1)-DIMENSIONAL CURVED SPACE

2007 ◽  
Vol 22 (26) ◽  
pp. 4833-4848 ◽  
Author(s):  
R. N. GHALATI ◽  
D. G. C. MCKEON ◽  
T. N. SHERRY
Keyword(s):  
Scalar Field ◽  
Affine Connection ◽  
Matter Field ◽  
Degree Of Freedom ◽  
Curved Space ◽  
First Order ◽  
Order Form ◽  
Canonical Formulation ◽  
Dynamical Degree ◽  
Hilbert Action

We study a bosonic scalar in (1 + 1)-dimensional curved space that is coupled to a dynamical metric field. This metric, along with the affine connection, also appears in the Einstein–Hilbert action [Formula: see text] when written in first-order form. After applying the Dirac constraint formalism to the Einstein–Hilbert action and the action of the bosonic scalar field separately, we apply it to these actions when they are combined. Only in the latter case does a dynamical degree of freedom emerge.

scholarly journals THE CANONICAL STRUCTURE OF THE FIRST-ORDER EINSTEIN–HILBERT ACTION

2010 ◽  
Vol 25 (17) ◽  
pp. 3453-3480 ◽  
Author(s):  
D. G. C. MCKEON
Keyword(s):  
Phase Space ◽  
Gauge Invariance ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Affine Connection ◽  
Canonical Structure ◽  
First Order ◽  
Order Form ◽  
Solving Equations ◽  
Hilbert Action

The Dirac constraint formalism is used to analyze the first-order form of the Einstein–Hilbert action in d > 2 dimensions. Unlike previous treatments, this is done without eliminating fields at the outset by solving equations of motion that are independent of time derivatives when they correspond to first class constraints. As anticipated by the way in which the affine connection transforms under a diffeomorphism, not only primary and secondary but also tertiary first class constraints arise. These leave d(d-3) degrees of freedom in phase space. The gauge invariance of the action is discussed, with special attention being paid to the gauge generators of Henneaux, Teitelboim and Zanelli and of Castellani.

scholarly journals Affine connection form of Regge calculus

2016 ◽  
Vol 31 (01) ◽  
pp. 1650010 ◽  
Author(s):  
V. M. Khatsymovsky
Keyword(s):  
Equations Of Motion ◽  
Affine Connection ◽  
Connection Form ◽  
Regge Calculus ◽  
Independent Variables ◽  
Connection Matrices ◽  
Arbitrary Change ◽  
Hilbert Action ◽  
General Connection

Regge action is represented analogously to how the Palatini action for general relativity (GR) as some functional of the metric and a general connection as independent variables represents the Einstein–Hilbert action. The piecewise flat (or simplicial) spacetime of Regge calculus is equipped with some world coordinates and some piecewise affine metric which is completely defined by the set of edge lengths and the world coordinates of the vertices. The conjugate variables are the general nondegenerate matrices on the three-simplices which play the role of a general discrete connection. Our previous result on some representation of the Regge calculus action in terms of the local Euclidean (Minkowsky) frame vectors and orthogonal connection matrices as independent variables is somewhat modified for the considered case of the general linear group GL(4, R) of the connection matrices. As a result, we have some action invariant w.r.t. arbitrary change of coordinates of the vertices (and related GL(4, R) transformations in the four-simplices). Excluding GL(4, R) connection from this action via the equations of motion we have exactly the Regge action for the considered spacetime.

scholarly journals Dispersionless integrable hierarchies and GL(2, ℝ) geometry

2019 ◽  
pp. 1-26 ◽  
Author(s):  
Evgeny Ferapontov ◽  
Boris Kruglikov
Keyword(s):  
Integrable Hierarchies ◽  
Affine Connection ◽  
Dimensional Manifold ◽  
Characteristic Variety ◽  
Weyl Geometry ◽  
Totally Geodesic ◽  
Lax Equations ◽  
Natural Generalisation ◽  
Torsion Free ◽  
Two Parameter

AbstractParaconformal or GL(2, ℝ) geometry on an n-dimensional manifold M is defined by a field of rational normal curves of degree n – 1 in the projectivised cotangent bundle ℙT*M. Such geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann (Doubrov–Wilczynski) invariants. In this paper we discuss yet another natural source of GL(2, ℝ) structures, namely dispersionless integrable hierarchies of PDEs such as the dispersionless Kadomtsev–Petviashvili (dKP) hierarchy. In the latter context, GL(2, ℝ) structures coincide with the characteristic variety (principal symbol) of the hierarchy.Dispersionless hierarchies provide explicit examples of particularly interesting classes of involutive GL(2, ℝ) structures studied in the literature. Thus, we obtain torsion-free GL(2, ℝ) structures of Bryant [5] that appeared in the context of exotic holonomy in dimension four, as well as totally geodesic GL(2, ℝ) structures of Krynski [33]. The latter possess a compatible affine connection (with torsion) and a two-parameter family of totally geodesic α-manifolds (coming from the dispersionless Lax equations), which makes them a natural generalisation of the Einstein–Weyl geometry.Our main result states that involutive GL(2, ℝ) structures are governed by a dispersionless integrable system whose general local solution depends on 2n – 4 arbitrary functions of 3 variables. This establishes integrability of the system of Wünschmann conditions.

Sign in / Sign up

Export Citation Format

Share Document