The geometry of the trajectories

In this paper, we use of the geometry of a class of the nature flows to define trajectory manifolds. Trajectory connections as a generalization of the Levi-Civita connections are considered. A method for determining the geometry of the flows created by the integral curves of a vector field is presented. The method contains two steps, the first step is finding the connection by the trajectories of a vector field, and the second step is finding a trajectory metric corresponding to the deduced connection. We show that doing the first step is possible, but for some of the vector fields, the second step may not be possible. In the case of existence of a trajectory manifold a new kind of curvature which we called it “trajectory curvature scalar” appears. We calculate trajectory connections for some vector fields and by an example we show that the trajectory curvature scalar for a trajectory manifold may not be equal to the curvature scalar of it. We find trajectory connection for a vector field close to the Schwarzschild black hole.

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Flows of measures generated by vector fields

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210517000312 ◽

2018 ◽

Vol 148 (4) ◽

pp. 773-818

Author(s):

Emanuele Paolini ◽

Eugene Stepanov

Keyword(s):

Vector Field ◽

Vector Fields ◽

Borel Measure ◽

Weak Sense ◽

Positive Borel Measure ◽

Smooth Structure ◽

Smooth Vector ◽

Integral Curves ◽

The Given ◽

Measurable Vector

The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.

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Geometric Description of the Thermodynamics of the Noncommutative Schwarzschild Black Hole

Advances in High Energy Physics ◽

10.1155/2013/641273 ◽

2013 ◽

Vol 2013 ◽

pp. 1-6

Author(s):

Alexis Larrañaga ◽

Natalia Herrera ◽

Juliana Garcia

Keyword(s):

Heat Capacity ◽

Black Hole ◽

Phase Transitions ◽

Equilibrium States ◽

Geometric Description ◽

Curvature Scalar ◽

Schwarzschild Black Hole ◽

Thermodynamic Interaction

The thermodynamics of the noncommutative Schwarzschild black hole is reformulated within the context of the recently developed formalism of geometrothermodynamics (GTD). Using a thermodynamic metric which is invariant with respect to Legendre transformations, we determine the geometry of the space of equilibrium states and show that phase transitions, which correspond to divergencies of the heat capacity, are represented geometrically as singularities of the curvature scalar. This further indicates that the curvature of the thermodynamic metric is a measure of thermodynamic interaction.

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Physics in Curved Spacetime

Covariant Physics ◽

10.1093/oso/9780198864899.003.0006 ◽

2021 ◽

pp. 211-253

Author(s):

Moataz H. Emam

Keyword(s):

Black Hole ◽

Vector Fields ◽

Killing Vector ◽

Schwarzschild Black Hole ◽

Curved Spacetime ◽

Killing Vector Fields ◽

Event Horizons ◽

Free Particles ◽

Generally Covariant

We discuss mechanics in curved spacetime backgrounds, gravitational time dilation, the motion of free particles, geodesics. We use the Schwarzschild metric as a case study and solve for motion along radial and orbital geodesics. This includes the strange behaviour around the event horizons of a Schwarzschild black hole. Isometries and Killing vector fields are explained and applied. Finally a brief presentation of generally covariant electrodynamics is given.

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Complete lift of vector fields and sprays to T∞M

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887815501133 ◽

2015 ◽

Vol 12 (10) ◽

pp. 1550113 ◽

Cited By ~ 1

Author(s):

Ali Suri ◽

Somaye Rastegarzadeh

Keyword(s):

Vector Field ◽

Differential Equations ◽

Tangent Bundle ◽

Vector Fields ◽

Dimensional Manifold ◽

Closed Curves ◽

Infinite Dimensional ◽

Integral Curves ◽

Complete Lift ◽

Infinite Dimensional Manifold

In this paper for a given Banach, possibly infinite dimensional, manifold M we focus on the geometry of its iterated tangent bundle TrM, r ∈ ℕ ∪ {∞}. First we endow TrM with a canonical atlas using that of M. Then the concepts of vertical and complete lifts for functions and vector fields on TrM are defined which they will play a pivotal role in our next studies i.e. complete lift of (semi)sprays. Afterward we supply T∞M with a generalized Fréchet manifold structure and we will show that any vector field or (semi)spray on M, can be lifted to a vector field or (semi)spray on T∞M. Then, despite of the natural difficulties with non-Banach modeled manifolds, we will discuss about the ordinary differential equations on T∞M including integral curves, flows and geodesics. Finally, as an example, we apply our results to the infinite-dimensional case of manifold of closed curves.

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Geodesic Vector Fields on a Riemannian Manifold

Mathematics ◽

10.3390/math8010137 ◽

2020 ◽

Vol 8 (1) ◽

pp. 137 ◽

Cited By ~ 2

Author(s):

Sharief Deshmukh ◽

Patrik Peska ◽

Nasser Bin Turki

Keyword(s):

Riemannian Manifold ◽

Vector Field ◽

Vector Fields ◽

Euclidean Spaces ◽

Smooth Vector ◽

Smooth Vector Field ◽

Geodesic Vector ◽

Integral Curves

A unit geodesic vector field on a Riemannian manifold is a vector field whose integral curves are geodesics, or in other worlds have zero acceleration. A geodesic vector field on a Riemannian manifold is a smooth vector field with acceleration of each of its integral curves is proportional to velocity. In this paper, we show that the presence of a geodesic vector field on a Riemannian manifold influences its geometry. We find characterizations of n-spheres as well as Euclidean spaces using geodesic vector fields.

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Characterization of integral curves of a linear vector field in Lorentz 3-space

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887816500730 ◽

2016 ◽

Vol 13 (06) ◽

pp. 1650073

Author(s):

Tunahan Turhan ◽

Nihat Ayyıldız

Keyword(s):

Vector Field ◽

Vector Fields ◽

Symmetric Matrix ◽

Order System ◽

Linear Vector ◽

Integral Curves ◽

The Matrix ◽

Vector Formula

We propose a detail study of integral curves or flow lines of a linear vector field in Lorentz [Formula: see text]-space. We construct the matrix [Formula: see text] depending on the causal characters of the vector [Formula: see text] by analyzing the non-zero solutions of the equation [Formula: see text], [Formula: see text] in such a space, where [Formula: see text] is the skew-symmetric matrix corresponding to the linear map [Formula: see text]. Considering the structure of a linear vector field, we obtain the linear first-order system of differential equations. The solutions of this system of equations give rise to integral curves of linear vector fields from which we provide a classification of such curves.

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Vector Fields

10.1093/oso/9780198805021.003.0001 ◽

2018 ◽

Author(s):

S. G. Rajeev

Keyword(s):

Fixed Point ◽

Vector Field ◽

Scalar Field ◽

Vector Fields ◽

Integral Curve ◽

Jacobi Matrix ◽

Partial Differential Operator ◽

Material Derivative ◽

First Order ◽

Integral Curves

The velocity of a fluid at each point of space-time is a vector field (or flow). It is best to think of it in terms of the effect of fluid flow on some scalar field. A vector field is thus a first order partial differential operator, called the material derivative in fluid mechanics. The path of a speck of dust carried along (advected) by the fluid is the integral curve of the velocity field. Even simple vector fields can have quite complicated integral curves: a manifestation of chaos. Of special interest are incompressible (with zero divergence) and irrotational (with zero curl) flows. A fixed point of a vector field is a point at which it vanishes. The derivative of a vector field at a fixed point is a matrix (the Jacobi matrix) whose spectrum is independent of the choice of coordinates.

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The Schwarzschild black hole

10.1093/oso/9780198786399.003.0047 ◽

2018 ◽

Author(s):

Nathalie Deruelle ◽

Jean-Philippe Uzan

Keyword(s):

Black Hole ◽

Gravitational Field ◽

Equilibrium Configuration ◽

Original Form ◽

Schwarzschild Black Hole ◽

Schwarzschild Metric ◽

Schwarzschild Geometry

This chapter discusses the Schwarzschild black hole. It demonstrates how, by a judicious change of coordinates, it is possible to eliminate the singularity of the Schwarzschild metric and reveal a spacetime that is much larger, like that of a black hole. At the end of its thermonuclear evolution, a star collapses and, if it is sufficiently massive, does not become stabilized in a new equilibrium configuration. The Schwarzschild geometry must therefore represent the gravitational field of such an object up to r = 0. This being said, the Schwarzschild metric in its original form is singular, not only at r = 0 where the curvature diverges, but also at r = 2m, a surface which is crossed by geodesics.

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