Journal of Modern Physics, 2012, 3, 18821890 http://dx.doi.org/10.4236/jmp.2012.312237 Published Online December 2012 (http://www.SciRP.org/journal/jmp) Wavefronts and Light Cones for Kerr Spacetimes Francisco FrutosAlfaro1, Frank Grave2, Thomas Müller2, Daria Adis3 1Department of Physics, University of Costa Rica, San Pedro, Costa Rica 2Institute for Visualization und Interactive Systems, University of Stuttgart, Stuttgart, Germany 3Theoretical Astrophysics, University of Tübingen, Tübingen, Germany Email: frutos@fisica.ucr.ac.cr Received September 26, 2012; revised October 27, 2012; accepted November 5, 2012 ABSTRACT We investigate the light propagation by means of simulations of wavefronts and light cones for Kerr spacetimes. Simu lations of this kind give us a new insight to better understand the light propagation in presence of massive rotating black holes. A relevant result is that wavefronts are backscattered with winding around the black hole. To generate these visualizations, an interactive computer program with a graphical user interface, called JWFront, was written in Java. Keywords: General Relativity; Wavefronts; Lightcones; Simulations 1. Introduction In general relativity and astrophysics, Kerr spacetimes are useful to study, for example, stellar compact objects, like accretion disks in neutron stars. This metric was found by Kerr in 1963 [1], since then this spacetime ap pears in many articles on these topics and it is currently one of the most used metric, because this represents a spacetime of a massive rotating object. Friedrich and Stewart [2,3] based on Arnold’s catas trophe theory [4] developed the theory of wavefronts and singularities (caustics) in general relativity. Recently, Hasse [5] et al., Low [6], and Ehlers and Newman [7] have revived this topic from a mathematical viewpoint. The wavefront propagation, caustic and the light cone structures for a non rotating object, described with Schwarzschild spacetime, was discussed by Perlick [8]. Caustics for the Kerr metric were numerically computed by Rauch and Blandford [9]. Grave studied the gravita tional collapse and wavefronts for this spacetime [10]. More recently, Sereno and De Luca [11] computed these caustics using a Taylor expansion of lightlike geodesics. A numerical treatment on the structure of Kerr caustics was done by Bozza [12]. Qualitative descriptions of wave fronts and caustics for gravitational lensing were pre sented by Blandford and Narayan [13], Schneider et al. [14], and Ohanian and Ruffini [15]. Petters et al. [16,17], and Frittelli and Petters [18] addressed formally this sub  ject. Ellis et al. [19] discussed qualitatively the lig ht con e structure for gravitational lensing. In this work, wavefronts, caustics and light cones for the Kerr spacetime are investigated. The best way to tackle it is through computer simulations. Nowadays, these simulations are becoming relevant in general rela tivity, because they can help understand complex phe nomena. With the new technologies, these simulations can practically be done in real time. Thus, the aim of this work is to provide a new perspective about wavefront propagations in Kerr spacetimes by means of computer simulation. For this purpose, we have designed JWFront [20], an interactive Java program using OpenGL (Open Graphics Library), to visualize wavefronts and light cones for this spacetime. In the next section, the Kerr spacetime and its tetrads will be briefly introduced and discussed. The equatio n of motion, i.e. the geodesic equation and how it is solved, will be discussed in the third section. Definitions for the sake of visualizations about the wavefront, caustic and light cone structures are presented in the fourth section. A succinct discussion about our program JWFront will be given in the fifth section. The last section is devoted to discuss the results of the visualizations for the Kerr spacetime. From these simulations, one can see the evo lution of wavefronts and light cones providing new per spectives for understanding them. 2. Kerr Spacetimes 2.1. Kerr Metric The Kerr metric is an exact solution of the vacuum Ein stein field equation and represents the spacetime of a massive rotating black hole. In this spacetime, the rotat ing body would exhibit an inertial frame dragging (LenseThirring effect), i.e., a particle moving close to it would corotate. This is not because of any force or torque applied on the particle, but rather because of the space C opyright © 2012 SciRes. JMP
F. FRUTOSALFARO ET AL. 1883 time curvature associated with this black hole. This re gion is called the ergosphere. At large distances this spacetime is flat (asymptotically flat). In BoyerLinquist coordinates the metric has the following form [21,22]: 2 22 22 22 22 2 22 00 031122 ddsind sin dd dddd sta ra at gt gtgr g 222 22 33 dd d d, r g 22 2222 , cos , S rRra ra (1) where R S is the Schwarzschild radius in geometrical units (c = G = 1), M is the mass of the black hole, aJM (angular momentum per unit mass, ), and μν aM are the metric components, which can be read off easily from (1). The Kerr spacetime contains the Minkowski flat metric, the Schwarzschild metric, and the Lense Thirring spacetime. If , i.e. neglecting the sec ond order in powers of a, one gets the LenseThirring metric, which represents the metric of a massive slow rotating body. We get the Schwarzschild metric if 20a 0a , which represents the metric of a massive nonrotating body. 2.2. Local Frames: Tetrad Formalism The tetrad formalism is very useful in general relativity. It defines a mathematical element called tetrad or vier bein, which is used to connect the curved coordinate sys tems with the local flat Lorentz coordinates. These tet rads must fulfill the equation μν η , αβ αβ μ ν ee α (2) where e μν ηis a chosen vierbein element, stands for the Minkowski metric (diag(–1, 1, 1, 1)). For the Kerr spacetime, there are at least two possibili ties to choose these tetrads. The first one is called the lo cally static frame (LSF). In this frame, the observer is static. This kind of observer cannot be located in the er gosphere, because they would move with superluminal velocity in this region to counteract the LenseThirring effect. The local tetrads for this static observer have fol lowing components: 0 00 1 11 1 1 eg eg 2 s tt Rr rr 2 22 03 00 32 200 3303 0000 3303 2 2 11 sin 1, sin SS S eg gg etgg g ggg g Rar Rr t Rr (3) where ,, andtr are understood as unit vector directions. The second one is called locally nonrotatin g frame [23] (LNRF), in which the observer is stationary. An observer in this kind of frames could be in the ergosphere. The local tetrads for this stationary observer have following components: 33 03 022 0300 33330300 33 1 11 2 22 3 33 1 11 1, sin S gg et ggg gg gg Rar t err g eg eg 2 222 2 –sinra a (4) where . These tetrad defini tions are useful to find the trajectories of light rays mov ing in a Kerr spacetime. 3. The Geodesic Equation and Its Solution 3.1. Geodesic Equation In general relativity, the trajectory of particles or light rays can be determined by the geodesic equation. Gener ally, this equation can only be solved using numerical methods. This equation has the following form [21,22]: 2 2 ddd 0, dd d xxx ,,0,1,2,3,and (5) where is an affine parameter, a parameter such that ddx has constant magnitude (affine parametrization). The components , called the Christoffel symbols, are given by . 2 gg g g xxx These symbols for the Kerr metric can be computed by means of symbolic programs. A program using the free Copyright © 2012 SciRes. JMP
F. FRUTOSALFARO ET AL. 1884 symbolic software REDUCE [24] was written to obtain them. In the Appendix, the nonnull Christoffel symbols are listed. Introducing these Christoffel symbols into Equation (5), one has four ordinary second order differ ential equations given by: 200 01 02 2 00 13 23 2 211 00 03 2 11 12 22 33 2 222 00 03 2 ddd d 2dd dd ddd 20 dd d dd 2 dd d dd d dd d dd 2 ddd tt r r rt r tt 2 1 11 22 1 , ddd dd d 20, d d t r 2 2 11 22 2 dd dd d0 d . r 22 12 22 33 233 01 02 2 33 13 23 dd d 2 dd d ddd d 2dd dd ddd 20 dd d r tr r ， (6) For light rays, there is also another equation they have to fulfill, the null g eodesic equation (lightlike geodesics): 2 2 00 03 22 11 22 ddd ddd dd 2 dd dd ddd sxx g tt gg r gg 2 33 d d d 0.g ,,, (7) The fourdimensional trajectories of light rays can be found by solving the Equations (6) with the constraint Equation (7). Now, we need initial conditions in order to solve these equation s numerically. 3.2. Initial Conditions The initial spacetime event 00000 txyz for all geodesics of the bundle defining the wavefront (see be low) has to be given in order to solve numerically Equa tions (6) with the constraint Equation (7). For each geo desic of the bundle, the fourvelocity at the initial point, 0 ddd ,,, , ddd xyz 0 dd d d xt determines the direction for each geodesic and they are calculated as follows: the tridimensional (3D) initial vec tor, 00 dddd ,, , d ddd xxyz for each geodesic in local flat spacetime is given input. Using the null geodesic condition for this local metric, the initial time derivative 0 ddt is determined. Now, we have all components in local flat spacetime 0 ddx . Finally, the fourvelocity in nonflat space time is determined by transforming from the local flat spacetime to the nonflat spacetime using th e tetrads e for the Kerr metric: 00 dd dd xx e With the purpose of simulating wavefronts and light cones in mind, one has to choose between the two kinds of observers (see Section 2). Now, we have all elements to numerically solve the four ordinary equations with these initial conditions. For this goal, a fourth or der Ru ng eKut t a procedure is used. 4. Wavefronts, Caustics and Light Cones 4.1. Wavefronts Formally speaking, the wavefronts are defined as follows: A wavefront is generated by a bundle of light rays or thogonal to a spacelike 2surface in a fourdimensional Lorentzian manifold [5]. To simulate it, the wavefront is defined as the surface A generated by all points of the null geodesic bundle at a given time: 000000 is a null geodesic with. ,,, , i i i t At ttxyztt Qualitatively speaking, the wavefronts that spread out in all directions from the source are spherical at the very beginning and if they are approaching a deflector, they get distorted and their sheets develop generally singulari ties: cusp ridges, self intersections and caustics. In gravitational lens theory, it is considered that light ray deflection occurs only at the place where the deflec tor is located (thin lens approximation). This approxima tion is very useful in many calculations, specially, if we are dealing with strong lensing. Under this consideration, wavefronts propagate spherically without any perturba tion until the deflector, then wavefronts are distorted by the deflector. The general case is completely different, because wavefronts get already perturbed before they approach the deflector and can wind around the black hole (see Figure 1). An observer which is behind the deflector will see different sheets of the same wavefront coing from different directions. Then, the observer will m Copyright © 2012 SciRes. JMP
F. FRUTOSALFARO ET AL. Copyright © 2012 SciRes. JMP 1885 Figure 1. Differences in the evolution of wavefront in presence of a black hole (top) and a gravitational lens (bottom). think that there are multiple images of the same source. 4.2. Caustics A caustic of a wavefront is formally defined as the set of all points where the wavefront fails to be an (immersed) submanifo ld [5]. Roughly speaking, a caustic is the envelope of refl ec ted or refracted light rays by a curved surface or object. A caustic can be a point, a line or a surface. For instance, for the Schwarzschild black hole the caustic is a line along of the line of sight, and for the point mass lens or nonrotating black hole the caus tic is a point in the line of sight. Interesting caustic shapes can be found in gravita tional lens theory, for example, for some elliptical lens models, it is common to find diamond shape caustics. Another important point to mention about caustics is that if an observer would be on a caustic, he would detect a high light intensity (mathematically speaking, it would be infinity). 4.3. Light Cones The light cone is defined as the surface generated by all points ,,,txyz, that fulfill the geodesic equation with the null geodesic condition for a fixed starting event ,,, 00000 txyz . To visualize the light cones, one has to suppress one space dimension, using for instance the coordinates ,,, ,,txy txz ,,tyz or . Light cones can also be used to visualize caustic structures [13], because
F. FRUTOSALFARO ET AL. 1886 time slices or cuts in the light cones represent the devel opment of the wavefront. The same differences that ap peared in the structures of wavefronts are also expected in light cones. In the present work, we will mainly concentrate on visualizations of wavefronts and light cones. For more mathematical details about wavefron ts, caustics and light cones, the interested reader may consult the references at the end. Details of the simulations will be shown in the sixth section. 5. JWFront An interactive frontend or GUI (graphical user interface) to visualize wavefronts and light cones in general relativ ity, called JWFront, was written in Java [20]. Basically, on this GUI, the user have to enter the initial position values and choose the values for mass and angular mo mentum per unit mass (M and a). Later, the user can choose what to see. Among the applications, the user can get from our program, are: Wavefront animations in 2D and 3D, Light cone visualizations. The light cones are visualized using the coordinate systems ,,txy or ,,tzx . All data obtained from solving the equations is processed in the program by means of Java and OpenGL subroutines in order to simu  late wavefronts and light cones. Moreover, this Java program can be easily modified to simulate wavefronts and light cones for other spacetime. The user just has to provide the Christoffel symbols into the program. The interested reader may send us a message request ing for the program or for more information about it. 6. Simulation with JWFront Now, let us discuss some examples of the simulations obtained by JWFront (see Figures 24). Figures 24 are visualizations for the Kerr spaceti me with M = 1 and a = 0.9. Figure 2. Twodimensional wavefront sequence for the Kerr metric (M = 1, a = 0.9). The sequenc e begins on the top left frame. The wavefront is moving from the right to the left in the xy plane. Copyright © 2012 SciRes. JMP
F. FRUTOSALFARO ET AL. 1887 Figure 3. Threedimensional wavefront sequence for the Kerr metric ( M = 1, a = 0.9). The sequence begins on the top left frame. The 2D visualization of a wavefront (light pulse) moving from right to left is shown in Figure 2. In these frames, the inner horizon is displayed as a small filled circle, the ergoregion as a bigger circle. Because of the rapidly rotation of the black hole, the wavefront is not symmetric in this plane. The black hole rotates counter clockwise, and so that the upper part of the wavefront reaches the y axis earlier than the lower part. Further more, because the wavefront infinitely winds around the black hole from left to right and right to left, the ob server will not see a continuously visible Ein stein ring as in the case of a nonrotating black hole. An observer located in the intersection point of the wavefront with itself can see the initial light pulse coming from two direction in this plane. In Figure 3, the 3D visualizations of a wavefront are shown. In this Figure, the wavefront consists of 1/8 of a sphere defined by the initial local directions. Certain steps of the wavefront motion are included in every frame. We can see that the wavefront, starting from be low the z axis, reaches positive z values, because of the above winding effect. As explained with the last figures, every point of the spacetime (excluding those inside of the black hole) is reached by this wavefront. The visu alizations of the light cones are shown in Figure 4 ((t, x, y) coordinates). The structures observed in these frames are similar to the corresponding structur es of Figure 2. 7. Conclusion The simulations produced by JWFront helps understand the light propagation in strong gravitational fields with rotation, such as in Kerr spacetimes. An interesting fea ture of wavefronts propagation appeared: the wavefronts are backscattered and wind around the black hole. Thus, an observer on the line of sight with the deflecto r and the source would see multiple images, and if the black hole does not rotate, the observer would see at least one Ein stein ring, if he or she is aligned with the black hole. For Schwarzschild metric this winding effect is symmetric whereas for the Kerr one it is not, this is due to black hole rotation. JWFront can also displayed the visualiza Copyright © 2012 SciRes. JMP
F. FRUTOSALFARO ET AL. 1888 Figure 4. Light cone evolution for the Kerr metric (M = 1, a = 0.9). The se quence begins on the top left frame. The light cone evolves from the initial point on the xyt space. tions of light cones in th ese spacetimes. The results of the wavefront visualizations showed that the same structures can also be seen with light cone simulations as expected. 8. Acknowledgements F. FrutosAlfaro would like to thank Dr. rer.nat. Antonio Banichevich and Ph.D. HerberthMorales for fruitful dis cussions. REFERENCES [1] R. P. Kerr, “Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics,” Physical Re view Letters, Vol. 11, No. 5, 1963, p. 237. doi:10.1103/PhysRevLett.11.237 [2] H. Friedrich and J. M. Stewart, “Characteristic Initial Data and Wavefront Singularities in General Relativity,” Proceedings of the Royal Society London, Series A, Vol. 385, No. 1789, 1983, pp. 345371. [3] J. M. Stewart, “Advanced General Relativity,” Cambridge Monographs on Mathematical Physics, Cambridge Uni versity, Cambridge, 1993. [4] V. I. Arnold, “Singularities of Caustics and Wavefronts,” Kluwer, Amsterdam, 1990. doi:10.1007/9789401133302 [5] W. Hasse, M. Kriele and V. Perlick, “Caustics of Wave front s in General Rel ativity,” Classical and Quantum Grav Copyright © 2012 SciRes. JMP
F. FRUTOSALFARO ET AL. 1889 ity, Vol. 13, No. 5, 1996, pp. 11611182. doi:10.1088/02649381/13/5/027 [6] R. Low, “Stable Singularities of Wavefronts in General Relativity,” Journal of Mathematical Physics, Vol. 36, No . 8, 1998, pp. 33323335. doi:10.1063/1.532257 [7] J. Ehlers and E. Newman, “The Theory of Caustics and Wavefront Singularities with Physical Applications,” Jour nal of Mathematical Physics, Vol. 41, 2000, pp. 3344 3378, arXiv:grqc/9906065. doi:10.1063/1.533316 [8] V. Perlick, “Gravitati onal Lensing from a Spacetime Per spective,” Living Reviews in Relativity, Vol. 7, No. 9, 2004, arXiv:1010.3416. [9] K. P. Rauch and R. Blandford, “Optical Caustics in a Kerr Spacetime and the Origin of Rapid XRay Variabil ity in Active Galactic Nuclei,” The Astrophysical Journal, Vol. 421, No. 1, 1994, pp. 4668. doi:10.1086/173625 [10] F. Grave, “Visualization of Gravitational Collapse and Wavefronts in the General Relativity Theory, Diplomar beit,” Master Thesis, Eberhard Karls Universität Tübin gen, Tübingen, 2004. [11] M. Sereno and F. De Luca, “Primary Caustics and Critical Points behind a Kerr Black Hole,” Physical Review D, Vol. 78, No. 2, 2008, Article ID: 023008, arXiv: astro ph/0710.5923. [12] V. Bozza, “Optical Caustics of Kerr Spacetime: The Full Structure,” Physical Review D, Vol. 78, No. 6, 2008, Ar ticle ID: 063014, arXiv:grqc/0806.4102. [13] R. D. Blandford and R. Narayan, “Cosmological Applica tions of Gravitational Lensing,” Annual Review of As tronomy and Astrophysics, Vol. 30, 1992, pp. 311358. doi:10.1146/annurev.aa.30.090192.001523 [14] P. Schneider, J. Ehlers and E. E. Falco, “Gravitational Lenses,” Springer, Berlin, 1992. doi:10.1007/9781461227564 [15] H. Ohanian and R. Ruffini, “Gravitation and Spacetime,” W. W. Norton & Company, New York, 1994. [16] A. O. Petters, “Arnold’s Singularity Theory and Gravita tional Lensing,” Journal of Mathematical Physics, Vol. 34, No. 8, 1993, pp. 35553581. doi:10.1063/1.530045 [17] A. O. Petters, H. Levine and J. Wambsganss, “Singularity The o r y a n d Gra vi t a t i o nal L ensin g,” S p ring e r , B e r l in, 2001. doi:10.1007/9781461201458 [18] S. Frittelli and A. O. Petters. “Wave Fronts, Caustic Sheets, and Caustic Surfing in Gravitational Lensing,” Journal of Mathematical Physics, Vol. 43, No. 11, 2002, pp. 55785611, arXiv:astroph/0208135. doi:10.1063/1.1511790 [19] G. F. R. Ellis, B. A. Bassett and P. K. S. Dunsby, “Lens ing and Caustic Effects on Cosmological Distances,” Classical and Quantum Gravity, Vol. 15, No. 8, 1998, pp. 23452361, arXiv:grqc/9801092v1. doi:10.1088/02649381/15/8/015 [20] F. Grave, F. FrutosAlfaro, T. Müller and D. Adis, “JWFront in the Eleventh Marcel Grossmann Meeting: On Recent Developments in Theoretical and Experimental General Relativity,” In: H. Kleinert and R. T. Jantzen, Eds., Gravitation and Relativistic Field Theories, Proceedings of the MG11 Meeting, World Scientific Publishing Com pany, London, 2008. [21] R. d’Inverno, “Introducing Einstein’s Relativity,” Oxford University Press, Oxford, 1992. [22] C. W. Misner, K. S. Thorne and J. A. Wheeler, “Gravita tion,” Freeman, San Francisco, 1973. [23] J. M. Bardeen, W. H. Press and S. A. Teukolsky, “Rotat ing Black Holes: Locally Nonrotating Frames, Energy Extraction, and Scalar Synchrotron Radiation,” The As trophysical Journal, Vol. 178, 1972, pp. 347369. doi:10.1086/151796 [24] A. C. Hearn, “REDUCE (User’s and Contributed Pack ages Manual),” KonradZuseZentrum für Information stechnik, Berlin, 1999. [25] O. Semerák, “Spinning Test Particles in a Kerr FieldI,” Monthly Notices of Royal Astronomical Society, Vol. 308, No. 3, 1999, pp. 863875. doi:10.1046/j.13658711.1999.02754.x Copyright © 2012 SciRes. JMP
F. FRUTOSALFARO ET AL. 1890 Appendix: Christoffel Symbols The nonzero Christoffel symbols for the Kerr metric are given by 222 2r 02 01 4 2 s Rra 0 02 2 4sin cos aJr 2 022 13 4 sin 2 Jra 2222 rra 2 03 cos sin r 23 4 2aJ 122 62 Rr 00 2 s 1222 2sin Jr 03 6 12 1 2 s Rrr 11 2 2 1 12 2sin c aos 1 22 2 r 2 14222 33 6 sin 2sinraJr 2 00 6 2sin cos aJr 222 03 6 2sin cos Jr ra 2 2 11 2sin cos a 2 12 2 r 21 22 12 2 2422 33 6 sin cos s Rr ra 322 01 42 Jr 3 02 4 2cos sin Jr 322222 13 4 1sin 2 s rRraJr 342 23 4 cos 2sin sin aJr These Christoffel symbols coincide with the ones ob tained by Smerák [25]. Copyright © 2012 SciRes. JMP
