CIRCULAR ORBITS AROUND SCHWARZSCHILD–AdS SPACETIME

In this paper we discuss parallel transport of vectors and spinors around circular orbits in Schwarzschild–AdS spacetime. We study the global properties of this spacetime via loop variables or holonomy. A set of paths in this background is considered. We demonstrate that for some special radii there appears the so-called quantized band structure of holonomy invariance. This analysis is also extended to parallel transport of a spinor in this spacetime.

Download Full-text

GEOMETRIC TRANSPORT ALONG CIRCULAR ORBITS IN STATIONARY AXISYMMETRIC SPACETIMES

International Journal of Modern Physics D ◽

10.1142/s0218271804005237 ◽

2004 ◽

Vol 13 (09) ◽

pp. 1771-1803 ◽

Cited By ~ 3

Author(s):

DONATO BINI ◽

CHRISTIAN CHERUBINI ◽

GIANLUCA CRUCIANI ◽

ROBERT T. JANTZEN

Keyword(s):

Black Hole ◽

Angular Momentum ◽

Angular Velocity ◽

Electromagnetic Fields ◽

Equatorial Plane ◽

Parallel Transport ◽

Inner Product ◽

Circular Orbits ◽

Kerr Spacetime ◽

Parameter Dependent

Parallel transport along circular orbits in orthogonally transitive stationary axisymmetric spacetimes is described explicitly relative to Lie transport in terms of the electric and magnetic parts of the induced connection. The influence of both the gravito-electromagnetic fields associated with the zero angular momentum observers and of the Frenet–Serret parameters of these orbits as a function of their angular velocity is seen on the behavior of parallel transport through its representation as a parameter-dependent Lorentz transformation between these two inner-product preserving transports which is generated by the induced connection. This extends the analysis of parallel transport in the equatorial plane of the Kerr spacetime to the entire spacetime outside the black hole horizon, and helps give an intuitive picture of how competing "central attraction forces" and centripetal accelerations contribute with gravitomagnetic effects to explain the behavior of the 4-acceleration of circular orbits in that spacetime.

Download Full-text

Curvature

10.1093/oso/9780192895646.003.0015 ◽

2021 ◽

pp. 189-212

Author(s):

Andrew M. Steane

Keyword(s):

Curvature Tensor ◽

Torsion Tensor ◽

Parallel Transport ◽

Lie Derivative ◽

Riemannian Curvature ◽

Riemann Curvature Tensor ◽

Geodesic Deviation ◽

Minkowski Metric ◽

Covariant Derivatives ◽

Global Properties

The mathematics of Riemannian curvature is presented. The Riemann curvature tensor and its role in parallel transport, in the metric, and in geodesic deviation are expounded at length. We begin by defining the curvature tensor and the torsion tensor by relating them to covariant derivatives. Then the local metric is obtained up to second order in terms of Minkowski metric and curvature tensor. Geometric issues such as the closure or non-closure of parallelograms are discussed. Next, the relation between curvature and parallel transport around a loop is derived. Then we proceed to geodesic deviation. The influence of global properties of the manifold on parallel transport is briefly expounded. The Lie derivative is then defined, and isometries of spacetime are discussed. Killing’s equation and properties of Killing vectors are obtained. Finally, the Weyl tensor (conformal tensor) is introduced.

Download Full-text

Circular orbits in cosmic string and Schwarzschild–AdS spacetime with Fermi–Walker transport

The European Physical Journal C ◽

10.1140/epjc/s10052-009-1076-1 ◽

2009 ◽

Vol 63 (1) ◽

pp. 149-155 ◽

Cited By ~ 13

Author(s):

K. Bakke ◽

A. M. de M. Carvalho ◽

C. Furtado

Keyword(s):

Cosmic String ◽

Circular Orbits ◽

Ads Spacetime

Download Full-text

The evolutions of the innermost stable circular orbits in dynamical spacetimes

The European Physical Journal C ◽

10.1140/epjc/s10052-021-09623-6 ◽

2021 ◽

Vol 81 (10) ◽

Author(s):

Yong Song

Keyword(s):

Angular Momentum ◽

Orbital Angular Momentum ◽

Effective Potential ◽

Potential Method ◽

Orbital Energy ◽

Slow Rotation ◽

Schwarzschild Spacetime ◽

Circular Orbits ◽

Ads Spacetime ◽

Stationary Spacetimes

AbstractIn this paper, we studied the evolutions of the innermost stable circular orbits (ISCOs) in dynamical spacetimes. At first, we reviewed the method to obtain the ISCO in Schwarzschild spacetime by varying its conserved orbital angular momentum. Then, we demonstrated this method is equivalent to the effective potential method in general static and stationary spacetimes. Unlike the effective potential method, which depends on the presence of the conserved orbital energy, this method requires the existence of conserved orbital angular momentum in spacetime. So it can be easily generalized to the dynamical spacetimes where there exists conserved orbital angular momentum. From this generalization, we studied the evolutions of the ISCOs in Vaidya spacetime, Vaidya-AdS spacetime and the slow rotation limit of Kerr–Vaidya spacetime. The results given by these examples are all reasonable and can be compared with the evolutions of the photon spheres in dynamical spacetimes.

Download Full-text