changed as the simulation progresses.
Refinement and Harmony
In 1999, a young graduate student, Frans Pretorius of the University of British Columbia, started his Ph.D. work on the topic of solving Einstein’s equations in a computer. That year was in the midst of increasing interest in the solution to this problem. Since at the time the discovery of Figure 8: 8:Three right: rectangular, rectangular,circular circularpolar polaroror Figure Three ways ways in in which which space space can can be be pixelated, pixelated, from left to right: Calabrese and collaborators had yet to be elliptical depictedasasaablue bluedot doton onthe thegrid. grid. ellipticalgrids. grids.Each Each pixels pixel isisdepicted made, Pretorius and his advisor, Matthew Similarly, Hahn and Lindquist split random numerical errors, such as rounding Choptuik, decided to pursue the idea of red in a otherwise space green grid may bethe amplified to were a giant red smudge until picture thedot2-dimensional in which errors, being amplified by the Einstein’s increasing the resolution of the space becomes entirely red! black holes were moving into almost 8,000 equations leading the computers to crash. pixilation near the black holes, hoping But since the problem was related to the way in which the grid was constructed, choospixels separated by a distance equal to one They showed that the amplification was this could purge computer crashes. The ing different grids would get rid of the instabilities. What this means is that it was neceshundredth of the radius of the holes. They independent of the resolution used; that technique was already known in simulations sary to slice space in an astute manner to solve the problem. As an example, if the pixels were interested in creating the film showing meant that no matter how small the are all at the same distance from one another, this is a rectangular grid, in which we move of fluids used in problems of aerodynamics the evolution two holes according distances on the gridor and engineering, but its implementation from one pixelof tothe another in space alongtoa line. But onebetween can also points move along a circle Einstein’s equations. This is done in steps in were, any numerical error would grow an ellipse, giving rise to the so-called circular polar and elliptical grids (Fig. 8). After in gravitational physics was still lacking. time like the of frames per second of agroup, digitalphysicists to dominate the whole calculation after a They developed a computer code in the discovery Calabrese and his sought alternative ways in splitting movie. the first frame, at each pixel they equations few steps— digital movie analogy, space to In obtain a realization of Einstein’s free in of the chaos. This finally lead in which the density of pixels across the grid used Einstein’s equations to the calculate the meaning after asimulation few framesfree of numerical continuously changes with time to make less than three years later to first successful computer space curvature. With theiscurvature of the Wea simple can understand this discovery in sure that at each instant the regions where instabilities. The solution more complicated than static rectangular, elliptical fromgrid: the first one can calculate thecontinuously analogy of achanged movie camera. Suppose the curvature of space is more pronounced orspace circular theframe, grid actually needs to be as the simulation what is the force of gravity acting on each that we are trying to shoot a film on a green the resolution is higher than everywhere progresses. hole, which is then used to calculate where screen that does not move. The camera will else. The curvature is bigger near the black holes will move. The holes then move to capture the first frame, which will be mostly holes, so the code continuously changes the 5.the Refinement and Harmony their new positions predicted by Einstein’s green. However, there are always random resolution of the grid keeping it high near In 1999, ainyoung graduate student, thefilming University of British Columbia, equations the second frame, andFrans the Pretorius errors inofthe process; for instance, the holes. This is known as adaptive mesh started his Ph.D. work on the topic of solving Einstein’s equations in a computer. That process of computing the curvature is one pixel in the picture may come out red. refinement. For his thesis work, Pretorius year was in the midst of increasing interest in the solution to this problem. Since at that repeated, which then gives the position of In digital filming and photography this was awarded in 2003 the Metropolis Prize moment the discovery of Calabrese and collaborators was still in the future, Pretorius the holes for the third frame. This then goes is called “noise”. Einstein’s equations are best dissertation in Computational and his advisor, Matthew Choptuik, decided to pursuit the idea of increasing the res- for of the grid keeping it high near the holes on until the full film is created. After three such that every single small red dot in an Physics from the American Physical olution of the space pixelation near the black holes, hoping this could purge computer For his thesis work, Pretorius was award hours of calculation by the IBM computer, otherwise green grid may be amplified to a crashes. The technique was already known in simulations of fluids used in problems of Society. Pretorius went on to the California tation in Computational Physics from the the black holes were a distance apart giant red smudge until the picture aerodynamics and engineering, but itsequal implementation in gravitational physicsbecomes was still Institute of Technology in Pasadena as the California Institute of Technology i to ten times radius. Ita was at this point red!density of pixels across the grid atopost-doctoral fellow to continue this lacking. Theytheir developed computer code in entirely which the that the computer crashed: numerical sinceinstant the problem was related this research. theofdiscovery After the After discovery the chaotic of the c continuously changes with time to make sure thatBut at each the regions where to the research. errors were dominating computation, the way is inhigher which than the grid was constructed, of choice Einstein’sofequation dependent started on curvature of space is morethe pronounced the resolution everywhere else. The nature on the grid, Pretorius to and the numbers control. After different grids wouldthe get rid of the the choice of grid, Pretorius started to study curvature is biggerwere nearout theofblack holes, so thechoosing code continuously changes resolution the issue. the work of Hahn and Lindquist, other instabilities. What this means is that it was how to select the best pixilation. physicists during the 1970’s attempted to necessary to slice space in an astute manner 12 fro solve Einstein’s equations in a computer to solve the problem. As an example, if the lan to no avail. Even as computers improved, pixels are all at the same distance from one det they were also always breaking down when another, this is a rectangular grid, in which faced with Einstein’s theory! we move from one pixel to another in space Ga The National Science Foundation along a line. But one can also move along coo established in 1997 a Grand Challenge grant a circle or an ellipse, giving rise to the solike to support solving Einstein’s equations on called circular polar and elliptical grids (Fig. con a computer. An understanding of why the 8). After the discovery of Calabrese and his tim computers were crashing finally came with group, physicists sought alternative ways pro a groundbreaking work in 2002 by the in splitting space to obtain a realization equ physicists Gioel Calabrese, Jorge Pullin, of Einstein’s equations free of chaos. This Olivier Sarbach, and Manuel Tiglio, then at finally lead in less than three years later to me the Louisiana State University. They proved the first successful computer simulation aliz that Einstein’s equations once written free of numerical instabilities. The solution lem in a computer could be strongly chaotic is more complicated than a simple static tha Figure 9: Adaptive mesh refinement increases Figure 9: Adaptive mesh refinement independing on the choice of the grid used. rectangular, elliptical or circular grid: the the number of pixels near the black holes, where of creases isthe number ofMax pixels near The chaotic behavior means that small grid actually needs to be continuously resolution more critical (© Planck Inst. the for Image courtesy of Leonardo Motta
Image courtesy of Leonardo Motta
black holes, where resolution Gravitational Physics, Potsdam, Germany).is more
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critical (©Max Planck Inst. for GravitaDARTMOUTH UNDERGRADUATE JOURNAL OF SCIENCE tional Physics, Postdam, Germany).
me bec nat