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-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, Queen's 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.: 219441 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 [GRAPHICS] The initial directions of the Sachs basis vectors s(1.2) = (0, (s) over right arrow (1,) (2)) are defined perpendicular to the initial direction (upsilon) over right arrow of the light ray, see also Fig. 1, (S) over right arrow (1) = (- sin xi, 0)(T.) (S) over right arrow (2) (-cos chi cos xi, - cos chi sin xi, sin chi)(T). A congruence of null geodesics with central null geodesic gamma which starts at the observer O with an infinitesimal circular cross section is defined by the above mentioned two Jacobi fields with initial conditions Yl(1.2)vertical bar,lambda=0 = 0 and (dY(1.2)/d lambda)vertical bar lambda=0 = s1.2. The cross section of this congruence along y is described by the Jacobian J(ij)(lambda) = gijY(t)sj vertical bar lambda. However, to determine the gravitational lensing of a pointlike source S that is connected to the observer via gamma, we need the reverse Jacobian JS -> O. Fortunately, the reverse Jacobian is just the negative transpose of the original Jacobian J(O -> s), J:J(S -> O) = - (J(O -> S))(T). The Jacobian J transforms the circular shape of the congruence into an ellipse whose shape parameters (M-+/-: major/minor axis, psi: angle of major axis, epsilon: ellipticity) read M-+/- = root 2 alpha sin zeta(+/-) cos zeta(+/-) - beta sin(2) zeta(+/-) + J(11)(2) + J(21)(2) (6a) psi = arctan 2(J(21) cos zeta(+) + J(22) sin zeta(+). J(11) cos zeta(+) + J(12) sin zeta(+)). (6b) epsilon = parallel to M+ - M-parallel to (6c) parallel to M+ - M-parallel to with zeta(-) = 1/2 arctan 2 alpha/beta. zeta(+) = zeta(-) + pi/2, (7) and the parameters alpha = J(11)J(12) + J(21)J(22), beta = J(11)(2) - J(12)(2) + J(21)(2) - J(22)(2). The magnification factor is given by mu = lambda(2)/M+M-. (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: 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 L = -alpha(t) over dot(2) + alpha(-1)(r) over dot(2) + r(2)(phi) over dot(2) for timelike geodesics in the v = pi/2 hypersurface in the Schwarzschild spacetime with alpha = 1 - 2m/r. The Euler-Lagrangian equations lead to the energy balance equation (r) over dot(2) + V(r) = k(2) with the effective potential V(r) = (r - 2m)(r(2) + h(2))/r(3) and the constants of motion k = alpha(t) over dot and h = r(2)(phi) over dot The constants of motion for a timelike geodesic that starts at (r = 10m, phi = 0) with initial direction xi = pi/4 with respect to the black hole direction and with initial velocity beta = 0.7 read k approximate to 1.252 and h approximate to 6.931. Then, from the energy balance equation we immediately obtain the radius of closest approach r(min) approximate to 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. (C) 2011 Elsevier B.V. All rights reserved.
sachs basis jacobi equation general relativity timelike and lightlike geodesics