Generalizing optical geometry

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Abstract. Assuming a general timelike congruence of worldlines as a reference frame, we derive a covariant general formalism of inertial forces in General Relativity. Inspired by the works of Abramowicz et. al. (see e.g. Abramowicz and Lasota 1997 Class. Quantum Grav. 14 A23-30), we also study conformal rescalings of spacetime and investigate how these affect the inertial force formalism. While many ways of describing spatial curvature of a trajectory has been discussed in papers prior to this, one particular prescription (which differs from the standard projected curvature when the reference is shearing) appears novel. For the particular case of a hypersurface-forming congruence, using a suitable rescaling of spacetime, we show that a geodesic photon is always following a line that is spatially straight with respect to the new curvature measure. This fact is intimately connected to Fermat's principle, and allows for a certain generalization of the optical geometry as will be further pursued in a companion paper (Jonsson and Westman 2006 Class. Quantum Grav. 23 61). For the particular case when the shear-tensor vanishes, we present the inertial force equation in threedimensional form (using the bold face vector notation), and note how similar it is to its Newtonian counterpart. From the spatial curvature measures that we introduce, we derive corresponding covariant differentiations of a vector defined along a spacetime trajectory. This allows us to connect the formalism of this paper to that of Jantzen et. al. (see e.g. Bini et. al. 1997 Int. J. Mod. Phys. D 6 143-98).

Section: Discussionmentioning

confidence: 95%

“…Thus also relative to the original spacetime geometry must the shear (and rotation) vanish. This result will be used in a companion paper [3] on generalizing the theory of optical geometry.…”

Section: The Projected Curvaturementioning

confidence: 98%

Inertial forces and the foundations of optical geometry

Jonsson

2005

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“…The suffix 'ns' stands for 'New-Straight'. This particular curvature is connected to Fermat's principle, and optical geometry [3,5]. For brevity we let the suffix 's' denote either 'ps', or 'ns'.…”

Section: Including Shear and Expansionmentioning

confidence: 96%

A covariant formalism of spin precession with respect to a reference congruence

Jonsson

2005

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We derive an effectively three-dimensional relativistic spin precession formalism. The formalism is applicable to any spacetime where an arbitrary timelike reference congruence of worldlines is specified. We employ what we call a stopped spin vector which is the spin vector that we would get if we momentarily make a pure boost of the spin vector to stop it relative to the congruence. Starting from the Fermi transport equation for the standard spin vector we derive a corresponding transport equation for the stopped spin vector. Employing a spacetime transport equation for a vector along a worldline, corresponding to spatial parallel transport with respect to the congruence, we can write down a precession formula for a gyroscope relative to the local spatial geometry defined by the congruence. This general approach has already been pursued by Jantzen et. al. (see e.g. Jantzen, Carini and Bini 1997 Ann. Phys. 215 1), but the algebraic form of our respective expressions differ. We are also applying the formalism to a novel type of spatial parallel transport introduced in Jonsson (2006 Class. Quantum Grav. 23 1), as well as verifying the validity of the intuitive approach of a forthcoming paper (Jonsson 2007 Am. Journ. Phys. 75 463) where gyroscope precession is explained entirely as a double Thomas type of effect. We also present the resulting formalism in explicit three-dimensional form (using the boldface vector notation), and give examples of applications.PACS numbers: 04.20.-q, 95.30.Sf

“…In more complicated (less symmetric) backgrounds or for other choices of n α , its Newtonian reading fails. The decomposition, however, remains as generally valid formalism in relativistic dynamics [46,47], as well as different kinds of four-acceleration decomposition can be employed [54][55][56][57].…”

Section: Force Formalismmentioning

confidence: 99%

“…The first force representation benefits from the hypersurface projection of the fluid four-acceleration field, and from its decomposition in the framework of the so-called 'optical reference geometry' that was introduced along with the mapping of test particle motion in the proper conformal hypersurface [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50]. This decomposition enables us to introduce the so-called 'inertial forces' that are represented by the velocity and charge independent gravitational force, the velocity dependent and charge independent centrifugal, Coriolis and Euler forces, whereas the sum of these forces vanishes along geodesics -trajectories of free test particles.…”

Section: Introductionmentioning

confidence: 99%

Charged fluids encircling compact objects: force representations and conformal geometries

Kovář¹,

Kojima²,

Slaný³

et al. 2020

Charged fluids rotating around compact objects can form unique equilibrium structures when ambient large-scale electromagnetic fields combine with strong gravity. Equatorial as well as off-equatorial toroidal structures are among such figures of equilibrium with a direct relevance for astrophysics. To investigate their geometrical shapes and physical properties in the near-horizon regime, where effects of general relativity play a significant role, we commonly employ a scheme based on the energy-momentum conservation written in a standard representation. Here, we develop its interesting alternatives in terms of two covariant force representations, both based on a hypersurface projection of the energy-momentum conservation. In a proper hypersurface, space-like forces can be defined, following from a decomposition of the fluid four-acceleration. Each of the representations provides us with an insight into properties of the fluid flow, being well reflected in related conformal hypersurface geometries; we find behaviour of centrifugal forces directly related to geodesics of these conformal hypersurfaces and their embedding diagrams. We also reveal correspondence between the charged fluid flow world-lines from an ordinary spacetime, and world-lines determined by a charged test particles equation of motion in a conformal spacetime.