Motion4D-library extended

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.

New version program summary

Program title: Motion4D-libraryCatalogue identifier: AEEX_v3_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEX_v3_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 219 441No. of bytes in distributed program, including test data, etc.: 6 968 223Distribution format: tar.gzProgramming language: C++Computer: All platforms with a C++ compilerOperating system: Linux, WindowsRAM: 61 MbytesClassification: 1.5External routines: Gnu Scientic Library (GSL) (http://www.gnu.org/software/gsl/)Catalogue identifier of previous version: AEEX_v2_0Journal reference of previous version: Comput. Phys. Comm. 181 (2010) 703Does the new version supersede the previous version?: YesNature 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)(2)(3) 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 are defined perpendicular to the initial direction of the light ray, see also Fig. 1,(4a)(4b)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 and . The cross section of this congruence along γ is described by the Jacobian . 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=−T(JO→S). 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)(6b)ψ=arctan2(J₂₁cosζ₊+J₂₂sinζ₊,J₁₁cosζ₊+J₁₂sinζ₊),(6c) with(7) and the parameters α=J₁₁J₁₂+J₂₁J₂₂, . The magnification factor is given by(8) 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].

The Christoffel symbols and the natural local tetrads of these new metrics are given in the Catalogue of Spacetimes, Ref. 14].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 for timelike geodesics in the ?=π/2 hypersurface in the Schwarzschild spacetime with α=1−2m/r. The Euler–Lagrangian equations lead to the energy balance equation with the effective potential V(r)=(r−2m)(r²+h²)/r³ and the constants of motion and . 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:

• 1] 

  T. Müller, New version announcement to the GeodesicViewer, http://cpc.cs.qub.ac.uk/summaries/AEFP_v2_0.html.

• 2] 

  P. Schneider, J. Ehlers, E. E. Falco, Gravitational Lenses, Springer, 1992.

• 3] 

  W. Rindler, Phys. Lett. A 245 (1998) 363.

• 4] 

  D. Kramer, Ann. Phys. 9 (2000) 331.

• 5] 

  F.J. Ernst, J. Math. Phys. 17 (1976) 54.

• 6] 

  S. Chandrasekhar, Proc. R. Soc. Lond. A 421 (1989) 227.

• 7] 

  H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt, Exact Solutions of the Einstein Field Equations, Cambridge University Press, 2009.

• 8] 

  A.I. Janis, E.T. Newman, J. Winicour, Phys. Rev. Lett. 20 (1968) 878.

• 9] 

  K. Martel, E. Poisson, Am. J. Phys. 69 (2001) 476.

• 10] 

  W. Rindler, Relativity – Special, General, and Cosmology, Oxford University Press, Oxford, 2007.

• 11] 

  C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, W.H. Freeman, 1973.

• 12] 

  J. Sultana, C.C. Dyer, Gen. Relativ. Gravit. 37 (2005) 1349.

• 13] 

  D. Bini, C. Cherubini, Robert T. Jantzen, Class. Quantum Grav. 19 (2002) 5481.

• 14] 

  T. Muller, F. Grave, arXiv:0904.4184 gr-qc].