FRINK Linked Data Fragments server

Matches in SemOpenAlex for { <https://semopenalex.org/work/W4250839063> ?p ?o ?g. }

• W4250839063 endingPage "1388" @default.
• W4250839063 startingPage "1386" @default.
• W4250839063 abstract "The new version of the Motion4D-library now also includes the integration of a Sachs basis and the Jacobi equation to determine gravitational lensing of pointlike sources for arbitrary spacetimes. Program title: Motion4D-library Catalogue identifier: AEEX_v3_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEEX_v3_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 219 441 No. of bytes in distributed program, including test data, etc.: 6 968 223 Distribution format: tar.gz Programming language: C++ Computer: All platforms with a C++ compiler Operating system: Linux, Windows RAM: 61 Mbytes Classification: 1.5 External routines: Gnu Scientic Library (GSL) (http://www.gnu.org/software/gsl/) Catalogue identifier of previous version: AEEX_v2_0 Journal reference of previous version: Comput. Phys. Comm. 181 (2010) 703 Does the new version supersede the previous version?: Yes Nature of problem: Solve geodesic equation, parallel and Fermi-Walker transport in four-dimensional Lorentzian spacetimes. Determine gravitational lensing by integration of Jacobi equation and parallel transport of Sachs basis. Solution method: Integration of ordinary differential equations. Reasons for new version: The main novelty of the current version is the extension to integrate the Jacobi equation and the parallel transport of the Sachs basis along null geodesics. In combination, the change of the cross section of a light bundle and thus the gravitational lensing effect of a spacetime can be determined. Furthermore, we have implemented several new metrics. Summary of revisions: The main novelty of the current version is the integration of the Jacobi equation and the parallel transport of the Sachs basis along null geodesics. The corresponding set of equations read(1)d2xμdλ2=−Γρσμdxρdλdxσdλ,(2)ds1,2μdλ=−Γρσμdxρdλs1,2σ,(3)d2Y1,2μdλ2=−2ΓρσμdxρdλdY1,2σdλ−Γρσ,νμdxρdλdxσdλYν, where (1) is the geodesic equation, (2) represents the parallel transport of the two Sachs basis vectors s1,2, and (3) is the Jacobi equation for the two Jacobi fields Y1,2. The initial directions of the Sachs basis vectors s1,2=(0,s→1,2)=s1,2μ∂μ are defined perpendicular to the initial direction υ→ of the light ray, see also Fig. 1,(4a)s→1=(−sinξ,cosξ,0)T,(4b)s→2=(−cosχcosξ,−cosχsinξ,sinχ)T.Download : Download full-size imageFig. 1. Initial direction υ=(−1,υ→T)=(−1,sinχcosξ,sinχsinξ,cosχ) of a past-directed null goedesic with respect to the local reference frame e(i). Fig. 1. Initial direction υ=(−1,υ→T)=(−1,sinχcosξ,sinχsinξ,cosχ) of a past-directed null goedesic with respect to the local reference frame e(i). A congruence of null geodesics with central null geodesic γ which starts at the observer O with an infinitesimal circular cross section is defined by the above mentioned two Jacobi fields with initial conditions Y1,2μ|λ=0=0 and (dY1,2μ/dλ)|λ=0=s1,2μ. The cross section of this congruence along γ is described by the Jacobian Jij(λ)=gμνYiμsjν|λ. However, to determine the gravitational lensing of a pointlike source S that is connected to the observer via γ, we need the reverse Jacobian JS→O. Fortunately, the reverse Jacobian is just the negative transpose of the original Jacobian JO→S,(5)J:=JS→O=−(JO→S)T. The Jacobian J transforms the circular shape of the congruence into an ellipse whose shape parameters (M±: major/minor axis, ψ: angle of major axis, ε: ellipticity) read(6a)M±=2αsinζ±cosζ±−βsin2ζ±+J112+J212,(6b)ψ=arctan2(J21cosζ++J22sinζ+,J11cosζ++J12sinζ+),(6c)ε=‖M+−M−‖‖M++M−‖ with(7)ζ−=12arctan2αβ,ζ+=ζ−+π2, and the parameters α=J11J12+J21J22, β=J112−J122+J212−J222. The magnification factor is given by(8)μ=λ2M+M−. These shape parameters can be easily visualized in the new version of the GeodesicViewer, see Ref. [1]. A detailed discussion of gravitational lensing can be found, for example, in Schneider et al. [2]. In the following, a list of newly implemented metrics is given: BertottiKasner: see Rindler [3]. BesselGravWaveCart: gravitational Bessel wave from Kramer [4]. DeSitterUniv, DeSitterUnivConf: de Sitter universe in Cartesian and conformal coordinates. Ernst: Black hole in a magnetic universe by Ernst [5]. ExtremeReissnerNordstromDihole: see Chandrasekhar [6]. HalilsoyWave: see Ref. [7]. JaNeWi: Janis–Newman–Winicour metric, see Ref. [8]. MinkowskiConformal: Minkowski metric in conformally rescaled coordinates. PTD_AI, PTD_AII, PTD_AIII, PTD_BI, PTD_BII, PTD_BIII, PTD_C Petrov-Type D – Levi-Civita spacetimes, see Ref. [7]. PainleveGullstrand: Schwarzschild metric in Painlevé–Gullstrand coordinates, see Ref. [9]. PlaneGravWave: Plane gravitational wave, see Ref. [10]. SchwarzschildIsotropic: Schwarzschild metric in isotropic coordinates, see Ref. [11]. SchwarzschildTortoise: Schwarzschild metric in tortoise coordinates, see Ref. [11]. Sultana-Dyer: A black hole in the Einstein–de Sitter universe by Sultana and Dyer [12]. TaubNUT: see Ref. [13]. To study the behavior of geodesics, it is often useful to determine an effective potential like in classical mechanics. For several metrics, we followed the Euler–Lagrangian approach as described by Rindler [10] and implemented an effective potential for a specific situation. As an example, consider the Lagrangian L=−αt˙2+α−1r˙2+r2φ˙2 for timelike geodesics in the ϑ=π/2 hypersurface in the Schwarzschild spacetime with α=1−2m/r. The Euler–Lagrangian equations lead to the energy balance equation r˙2+V(r)=k2 with the effective potential V(r)=(r−2m)(r2+h2)/r3 and the constants of motion k=αt˙ and h=r2φ˙. The constants of motion for a timelike geodesic that starts at (r=10m,φ=0) with initial direction ξ=π/4 with respect to the black hole direction and with initial velocity β=0.7 read k≈1.252 and h≈6.931. Then, from the energy balance equation we immediately obtain the radius of closest approach rmin≈5.927. Beside a standard Runge–Kutta fourth-order integrator and the integrators of the Gnu Scientific Library (GSL), we also implemented a standard Bulirsch–Stoer integrator. Running time: The test runs provided with the distribution require only a few seconds to run. References: T. Müller, New version announcement to the GeodesicViewer, http://cpc.cs.qub.ac.uk/summaries/AEFP_v2_0.html. P. Schneider, J. Ehlers, E. E. Falco, Gravitational Lenses, Springer, 1992. W. Rindler, Phys. Lett. A 245 (1998) 363. D. Kramer, Ann. Phys. 9 (2000) 331. F.J. Ernst, J. Math. Phys. 17 (1976) 54. S. Chandrasekhar, Proc. R. Soc. Lond. A 421 (1989) 227. H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt, Exact Solutions of the Einstein Field Equations, Cambridge University Press, 2009. A.I. Janis, E.T. Newman, J. Winicour, Phys. Rev. Lett. 20 (1968) 878. K. Martel, E. Poisson, Am. J. Phys. 69 (2001) 476. W. Rindler, Relativity – Special, General, and Cosmology, Oxford University Press, Oxford, 2007. C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, W.H. Freeman, 1973. J. Sultana, C.C. Dyer, Gen. Relativ. Gravit. 37 (2005) 1349. D. Bini, C. Cherubini, Robert T. Jantzen, Class. Quantum Grav. 19 (2002) 5481. T. Muller, F. Grave, arXiv:0904.4184 [gr-qc]." @default.
• W4250839063 created "2022-05-12" @default.
• W4250839063 creator A5027936127 @default.
• W4250839063 date "2011-06-01" @default.
• W4250839063 modified "2023-09-23" @default.
• W4250839063 title "Motion4D-library extended" @default.
• W4250839063 doi "https://doi.org/10.1016/j.cpc.2011.02.009" @default.
• W4250839063 hasPublicationYear "2011" @default.
• W4250839063 type Work @default.
• W4250839063 citedByCount "1" @default.
• W4250839063 countsByYear W42508390632014 @default.
• W4250839063 crossrefType "journal-article" @default.
• W4250839063 hasAuthorship W4250839063A5027936127 @default.
• W4250839063 hasConcept C11413529 @default.
• W4250839063 hasConcept C121332964 @default.
• W4250839063 hasConcept C12426560 @default.
• W4250839063 hasConcept C136119220 @default.
• W4250839063 hasConcept C154504017 @default.
• W4250839063 hasConcept C169590947 @default.
• W4250839063 hasConcept C199360897 @default.
• W4250839063 hasConcept C202444582 @default.
• W4250839063 hasConcept C2524010 @default.
• W4250839063 hasConcept C33923547 @default.
• W4250839063 hasConcept C41008148 @default.
• W4250839063 hasConcept C43364308 @default.
• W4250839063 hasConceptScore W4250839063C11413529 @default.
• W4250839063 hasConceptScore W4250839063C121332964 @default.
• W4250839063 hasConceptScore W4250839063C12426560 @default.
• W4250839063 hasConceptScore W4250839063C136119220 @default.
• W4250839063 hasConceptScore W4250839063C154504017 @default.
• W4250839063 hasConceptScore W4250839063C169590947 @default.
• W4250839063 hasConceptScore W4250839063C199360897 @default.
• W4250839063 hasConceptScore W4250839063C202444582 @default.
• W4250839063 hasConceptScore W4250839063C2524010 @default.
• W4250839063 hasConceptScore W4250839063C33923547 @default.
• W4250839063 hasConceptScore W4250839063C41008148 @default.
• W4250839063 hasConceptScore W4250839063C43364308 @default.
• W4250839063 hasIssue "6" @default.
• W4250839063 hasLocation W42508390631 @default.
• W4250839063 hasOpenAccess W4250839063 @default.
• W4250839063 hasPrimaryLocation W42508390631 @default.
• W4250839063 hasRelatedWork W1497385637 @default.
• W4250839063 hasRelatedWork W1867780321 @default.
• W4250839063 hasRelatedWork W1972375568 @default.
• W4250839063 hasRelatedWork W2026090113 @default.
• W4250839063 hasRelatedWork W2059228205 @default.
• W4250839063 hasRelatedWork W2368465938 @default.
• W4250839063 hasRelatedWork W2375354128 @default.
• W4250839063 hasRelatedWork W2384572477 @default.
• W4250839063 hasRelatedWork W4232190566 @default.
• W4250839063 hasRelatedWork W931781699 @default.
• W4250839063 hasVolume "182" @default.
• W4250839063 isParatext "false" @default.
• W4250839063 isRetracted "false" @default.
• W4250839063 workType "article" @default.