In this work we simulate null geodesics for the Bonnor massive dipole metric
by implementing a symbolic-numerical algorithm in Sage and Python. This program
is also capable of visualizing in 3D, in principle, the geodesics for any given
metric. Geodesics are launched from a common point, collectively forming a cone
of light beams, simulating a solid-angle section of a point source in front of
a massive object with a magnetic field. Parallel light beams also were
considered, and their bending due to the curvature of the space-time was
simulated.Comment: 9 pages, 5 figures, Presented at XIX Simposio Internacional de
  M\'etodos Matem\'aticos Aplicados a las Ciencias (19th International
  Symposium of Mathematical Methods Applied to Sciences, XIX-SIMMAC

Bonatti-González, Javier

Cordero-García, Iván

Frutos-Alfaro, Francisco

Oliva-Mercado, Guillermo Andree

Revista de Matemática: Teoría y Aplicaciones

English

arXiv

In this work we simulate null geodesics for the Bonnor massive dipole metric by implementing a symbolic-numerical algorithm in Sage and Python. This program is also capable of visualizing in 3D, in principle, the geodesics for any given metric. Geodesics are launched from a common point, collectively forming a cone of light beams, simulating a solid-angle section of a point source in front of a massive object with a magnetic field. Parallel light beams also were considered, and their bending due to the curvature of the space-time was simulated

G. Andree Oliva Mercado

Javier Bonatti González

Iván Cordero García

Francisco Frutos Alfaro

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REVISTA DE MATEMÁTICA: TEORÍA Y APLICACIONES 2015 22(2) : 255–264CIMPA – UCR ISSN: 1409-2433 (PRINT), 2215-3373 (ONLINE)A VISUALIZATION OF NULL GEODESICS FORTHE BONNOR MASSIVE DIPOLEVISUALIZACIÓN DE GEODÉSICAS NULAS PARAEL DIPOLO MASIVO DE BONNORG. ANDREE OLIVA–MERCADO∗ JAVIER BONATTI–GONZÁLEZ†IVÁN CORDERO–GARCÍA‡ FRANCISCO FRUTOS–ALFARO§Received: 25/Feb/2014; Revised: 19/Apr/2015;Accepted: 11/May/2015∗Escuela de Física, Universidad de Costa Rica. San José, Costa Rica. E-Mail:gandreoliva@gmail.com†Misma dirección que/Same address as: G.A. Oliva-Mercado, & Centro de Investigación enCiencias Atómicas, Nucleares y Moleculares (CICANUM). E-Mail: jbonatti2011@gmail.com‡Misma dirección que/Same address as: G.A. Oliva-Mercado. E-Mail: ivancg@msn.com§Centro de Investigaciones Espaciales (CINESPA), Universidad de Costa Rica. San José,Costa Rica. E-Mail: frutos62@gmail.com255256 A. OLIVA – J. BONATTI – I. CORDERO – F. FRUTOSAbstractIn this work we simulate null geodesics for the Bonnor massive dipolemetric by implementing a symbolic-numerical algorithm in Sage and Python.This program is also capable of visualizing in 3D, in principle, the geodesicsfor any given metric. Geodesics are launched from a common point, col-lectively forming a cone of light beams, simulating a solid-angle section ofa point source in front of a massive object with a magnetic field. Parallellight beams also were considered, and their bending due to the curvatureof the space-time was simulated.Keywords: general relativity; geodesics; differential geometry; free software.ResumenEn este trabajo se simulan geodésicas nulas para la métrica de dipolomasivo de Bonnor, implementando un algoritmo simbólico-numérico enSage y Python. Este programa es capaz, en principio, de visualizar en 3Dlas geodésicas de cualquier métrica. Estas geodésicas inician en un puntocomún, formando colectivamente un cono de rayos de luz, simulando unasección de ángulo sólido de una fuente puntual frente a un objeto masivocon campo magnético. Se consideran también rayos de luz paralelos, y sesimuló su cambio de trayectoria debida a la curvatura del espacio-tiempo.Palabras clave: relatividad general; geodésicas; geometría diferencial; softwarelibre.Mathematics Subject Classification: 83C10.1 IntroductionIn General Relativity, Einstein field equations are a set of non-linear partial dif-ferential equations that describe the curvature of space-time due to the presenceof mass-energy-momentum. One of the exact solutions for the Einstein fieldequations in the vacuum was proposed in 1965 by W.B. Bonnor[2]; it describesa massive spherical stationary object with a dipole magnetic field. This solution,expressed using the metric line element ds2 = gµνdxµdxν (1), in which gµν(x)is the metric describing the manifold solution to these equations is:ds2 = −Y2P 2Q3Z(dr2 + Zdθ2)− Z(YP)2sin2 θdϕ2 +(PY)2dt2 (1)1Einstein summation convention applies.Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 22(2): 255–264, July 2015VISUALIZATION OF NULL GEODESICS FOR THE BONNOR MASSIVE... 257whereP = r2 − 2mr − b2cos2θ (2a)Q = (r −m)2 − (b2 +m2)cos2θ (2b)Y = r2 − b2cos2θ (2c)Z = r2 − 2mr − b2. (2d)t, r, θ, ϕ are the standard spherical coordinates. The geometric system of unitsis used along this paper (c = G = 1). The parameters m and b are defined,according to Bonnor, as b being such that if r becomes large, the magnetic fieldis that of a magnetic dipole along θ = 0 of moment mbπ−12 and the gravitationalfield becomes that of a mass 2m at the origin [2].Trajectories on this metric must be found by using the geodesic equations,which can be thought as lines of minimum length (result from δ∫ds = 0):d2xκdλ2= −Γκµνdxµdλdxνdλ(3)subject to initial conditions that determine the kind of geodesics2: time-like(ds2 > 0, trajectories of particles with mass), light-like or null (ds2 = 0, tra-jectories of photons), or space-like (ds2 < 0, trajectories of hypothetical faster-than-light particles). λ is an affine parameter, and Γκµν are the components of theChristoffel symbol.In this paper null geodesics for the Bonnor metric are calculated in orderto simulate the effect of the curvature of the space-time on a point source ofelectromagnetic radiation located at a certain distance from the origin, where themassive object is located. To achieve this purpose, a program in Python and Sagewas developed to calculate analytically the Christoffel symbols from the metricand use them to build the geodesic equations. Once this is done, the equations aresolved numerically using user-specified initial conditions, and these conditionsare then varied to simulate a cone of light beams that represents a solid-anglesection of the source.Other efforts have been made in this topic. For example, Müler and Frauen-diener (2010, [4]) developed Geodesic Viewer, which implements a selection ofmetrics which does not include the one studied here; and unlike this program, alocal frame (tetrads) is required to specify the initial direction of the light beam.The justification for not using a local frame is that our scope is to study the2The sign of ds2 that determines kind of geodesic depends on the signature of the metric. Inthis paper, the signature of the metric is (1,−1,−1,−1)Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 22(2): 255–264, July 2015258 A. OLIVA – J. BONATTI – I. CORDERO – F. FRUTOScollective behavior of geodesics, and high precision knowledge of the local di-rections is not required, as it would be for calculating gravitational lensing forsuch a point source. The Bonnor geodesics were also previously studied in 2Dby Sanabria and Valenzuela [5], for charged test particles, as well as by Kovár̆ etal. [3], for neutral particles, from the point of view of orbit stability.Sage is an integrated environment consisting of several open-source pack-ages (last count 89) with a common Python-based interface. It is capable ofhandling algebraic expressions (through Maxima and Sympy), as well as numer-ical calculations (through Scipy and Numpy), and many other tasks.Vpython (“visual Python”) is a 3D modeling library for Python, that usesOpenGL for rendering graphics. Its main features are its simplicity, and real-time visualization capabilities [6].2 Equations of motion for the Bonnor metricExpressions for the Bonnor geodesic equations were also calculated by means ofthe variational principle, to contrast them with the results generated with Sage.These expressions are:r̈ = −12∂∂rln(Y 2P 2Q3Z)ṙ2 − ∂∂θln(Y 2P 2Q3Z)θ̇ṙ +Z2∂∂rln(Y 2P 2Q3Z)θ̇2− Q3Z2[ε2P 4∂∂rln(P 2Y 2)+l2Y 4Z sin2 θ∂∂rln(Y 2Z sin2 θP 2)](4)θ̈ = −12∂∂θln(Y 2P 2Q3)θ̇2 − ∂∂rln(Y 2P 2Q3)θ̇ṙ +12Z∂∂θln(Y 2P 2Q3)ṙ2− Q32Y 4[ε2∂∂θln(PY 2)− l2Z sin2 θ∂∂θln(Y 2 sin2 θP 2)](5)where the dots mean derivative with respect to λ, and ε and l are integrationconstants.3 Our program3.1 Analytical parts of the programAs sketched in the figure 1, the program is formed by several modules. Inprinciple, the program is capable of taking any metric written in any coordi-Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 22(2): 255–264, July 2015VISUALIZATION OF NULL GEODESICS FOR THE BONNOR MASSIVE... 259Figure 1: Structure of the program.nate system and plotting geodesics with very little changes. The first module,equations.sage, computes the Christoffel symbols for the given metric an-alytically using Sage, and stores the result as a Python script that can be laterused to do calculations using only Python. This Python script, named by defaultgeodesicEquations.py, contains the right-hand side of the Eq. 3 definedas a function, as well as the metric.3.2 Numerical calculationsThe script geodesicEquations.py is used in the second module,geodesics.py to calculate and visualize the geodesics. The 4th-order Runge-Kutta method is used to solve the set of differential equations, and it is pro-grammed in rk4.py. The numeric calculations may also be handled usingNumpy, a Python library with a variety of optimized functions written for thispurpose; or a specific numerical method can also be integrated to the code thisway.The initial conditions are given to this second module, and a function iscalled to vary them in order to form the light beam cone. Also, these initialconditions are varied in order to get a parallel set of beams. The given initialconditions are: the initial four-vector for the position, and the three spatial initialdirections v[1], v[2], v[3] for the light beam; the other one is calculatedby a function named calculate_v0() that uses the condition that ds2 = 0for null geodesics. Since we are not using local frames, these spatial directionsare really specified respect to a distant observer, assuming an asymptotically flatmetric. Using a local frame and the condition of null geodesics, the spatial initialdirections can be specified using only two angles, as done in [4].Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 22(2): 255–264, July 2015260 A. OLIVA – J. BONATTI – I. CORDERO – F. FRUTOSUsers can give input to the program by creating scripts that call the filesequations.sage and geodesics.py. In this way, multiple metrics andinitial conditions can be tested with no changes to the central code, and morefunctionality can be added in the future.The number of iterations and the step size can be specified, if not, defaultvalues are used. To avoid the effects of singularities on the behavior of geodesics,a stopping point based on distance to the origin is implemented and also can bespecified in the main script.3.3 VisualizationVpython is used not only to plot the data, but also to manipulate the view in3D in real time. Objects type curve() are used for the geodesics, and twosmall spheres mark the positions of both the source and the origin, where themassive object is located. A distinct arrow also points in the direction of +z toindicate the direction of the dipole in the case of the Bonnor metric. While itis true that the code works for any coordinate system, a transformation to theCartesian coordinate system is needed in order to represent positions in R3. Sofar, transformations from spherical and cylindrical coordinates to Cartesian andvice-versa are already implemented.4 Results4.1 Test runsThe program was tested first with Minkowski and Schwarzschild metrics. In Fig.2a a light beam cone is drawn in a Minkowski space-time (in spherical coordi-nates), presented here to compare with the results obtained with other metrics.As it can be seen, each geodesic starts at the same point, marked by a smallred sphere in the figure. As expected, geodesics are straight lines. Results forSchwarzchild metric are shown in Fig. 2b. Fewer geodesics are displayed thistime to better show how they are curved by the space-time. As expected, thelight beams even cross each other when passing very close to the event horizon,and they are just slightly deviated if they pass farther away.Now, parallel beams are simulated for both metrics, results are shown inFig. 3. In both cases, the direction of the beams is the negative x axis, and allthe geodesics remain in the x-y plane. The mass for the Schwarzschild metricis set to 0.5. The main scope for running these test metrics, where geodesicsare known, was to detect possible errors and select more carefully the initialconditions for later runs.Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 22(2): 255–264, July 2015VISUALIZATION OF NULL GEODESICS FOR THE BONNOR MASSIVE... 261Figure 2: Test runs with point sources.Figure 3: Test runs with parallel beams.Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 22(2): 255–264, July 2015262 A. OLIVA – J. BONATTI – I. CORDERO – F. FRUTOSFigure 4: Geodesics for the Bonnor metric with b = 0.001.4.2 Point source with b < 1Now, to the Bonnor metric. In Fig. 4, nine geodesics have been simulated andviewed from different angles, to give a sense of spatial orientation. The source islocated3 at (10, 0, 0), and the direction of the central geodesic is radially inwards,varied then in −0.8 ≤ v[2] ≤ 0.8 and −0.3 ≤ v[3] ≤ 0.3, correspondingto the θ and ϕ directions. 20 iterations with dlam = 0.1 (step size) have beencarried out. The metric parameters are m = 1 and b = 0.001.The presence of the magnetic field makes the photons move in an helical mo-tion around the z axis, that is, around the poles of the magnetic dipole, forminga jet-like structure. This behavior is consistent to the behavior of null geodesicsfor the Melvin metric, that is, locally, the magnetic field is dominant (see [1]).3These position vectors are given in Cartesian coordinates; units are in the geometric system.Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 22(2): 255–264, July 2015VISUALIZATION OF NULL GEODESICS FOR THE BONNOR MASSIVE... 263Figure 5: Geodesics for the Bonnor metric.4.3 The values of b and mVarious values of b and m were tested. The behavior obtained in the precedingsection corresponds to tested values 1 ≤ b ≤ 10−30. With values of b in theinterval 2 ≤ b ≤ 10 at least, the swirling seems to disappear. As predicted byBonnor, with a value of m = 0 the space-time becomes flat and the result issimilar to Fig. 3a. With a value of b = 0, also pointed out by Bonnor, the metricdoes not tend to the Schwarszchild one, but can be interpreted as a monopole ofmass 2m together with higher mass multipoles dependent on m. Parallel beamsof light on both metrics show the difference (compare Fig. 5a and Fig 3b).4.4 Light beams with b ∼ 5For the tested values of b = 2, 5, 10 and m = 1 the results were similar. InFig. 5b, the same x-y view is shown, for comparison with Figs. 5a and 3b. Themagnetic field bends the light in such a way that a big opening is created aroundthe object, and the light seems not to collide with it. Now, in Fig. 5c, variation inthe z axis is tested to see the effect on the distance for the magnetic dipole. Asexpected, the effect decreases with distance, even in the z axis where the dipoleRev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 22(2): 255–264, July 2015264 A. OLIVA – J. BONATTI – I. CORDERO – F. FRUTOSpoints to. It is important to notice that for Figs. 5b and 5c, the curves shown areall in the same plane.A point source located at (15, 0, 0) shows a different story (Fig. 5d). Geodes̄icsare no longer necessarily contained in a plane, and bend creating an "envelope"around the object that is less evident with the distance.5 Physical remarksParallel beams of light simulate distant sources and therefore can be used forgravitational lensing, helping to diagnose which kind of object causes a particu-lar deformation on an image. Neutron stars can have very strong magnetic fields,up to the point of not being ignorable by General Relativity, but also have veryrapid rotation rates, usually dominant. Light that behaves as shown in this paper,then, alerts of the presence of a massive object with a very strong magnetic field,but with negligible rotation. A point source, in the other hand, is also importantto take into account, since several of these massive objects (black holes or neu-tron stars) also have companions, that is, nearby stars that act like point sourcesof radiation.References[1] Bonatti, J.; Semionova, L.; Páez, J. (1985) “Motion in Melvin metric”, Uni-ciencia 2(1): 13–19.[2] Bonnor, W.B. (1965) “An exact solution of the Einstein-Maxwell equationsreferring to a magnetic dipole”, Zeitschrift für Physik 190: 444–445.[3] Kovár̆, J.; Kopác̆ek, O.; Karas, V.; Kojima, Y. (2013) “Regular and chaoticorbits near a massive magnetic dipole”, Classical and Quantum Gravity30(2), 025010.[4] Müller, T.; Frauendiener, J. (2011) “Studying null and time-like geodesicsin the classroom”, Eur. J. Phys. 32 747 doi:10.1088/0143-0807/32/3/011[5] Sanabria V., J.D.; Valenzuela, C.A. (2011) “Motion of charged test particlesin a static spacetime endowed with magnetic field”, Revista Colombiana deFísica 43(1), 4 pages.[6] Scherer, D.; Dubois, P.; Sherwood, B. (2000) “VPython: 3D interactivescientific graphics for students”, Computing in Science and Engineering,Sept./Oct. 2000: 82–88.Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 22(2): 255–264, July 2015

Revista de Matemática Teoría y Aplicaciones

A visualization of null geodesics for the bonnor massive dipole

In this work we simulate null geodesics for the Bonnor massive dipole metric by implementing a symbolic-numerical algorithm in Sage and Python. This program is also capable of visualizing in 3D, in principle, the geodesics for any given metric. Geodesics are launched from a common point, collectively forming a cone of light beams, simulating a solid-angle section of a point source in front of a massive object with a magnetic field. Parallel light beams also were considered, and their bending due to the curvature of the space-time was simulated.En este trabajo se simulan geodésicas nulas para la métrica de dipolo masivo de Bonnor, implementando un algoritmo simbólico-numérico en Sage y Python. Este programa es capaz, en principio, de visualizar en 3D las geodésicas de cualquier métrica. Estas geodésicas inician en un punto común, formando colectivamente un cono de rayos de luz, simulando una sección de ángulo sólido de una fuente puntual frente a un objeto masivo con campo magnético. Se consideran también rayos de luz paralelos, y se simuló su cambio de trayectoria debida a la curvatura del espacio-tiempo

A Visualization of Null Geodesics for the Bonnor Massive Dipole

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