To this end, supercomputers are often employed to study black holes, gravitational waves, neutron stars and many other phenomena described by Albert Einstein's theory of general relativity.
In this line, much attention is paid to the gauge conditions, coordinates, and various formulations of the Einstein equations and the effect they have on the ability to produce accurate numerical solutions.
[2] It, like his earlier theory of special relativity, described space and time as a unified spacetime subject to what are now known as the Einstein field equations.
At the time that ADM published their original paper, computer technology would not have supported numerical solution to their equations on any problem of any substantial size.
Applying symmetry reduced the computational and memory requirements associated with the problem, allowing the researchers to obtain results on the supercomputers available at the time.
[11] In their paper they point out that In the years that followed, not only did computers become more powerful, but also various research groups developed alternate techniques to improve the efficiency of the calculations.
With respect to black hole simulations specifically, two techniques were devised to avoid problems associated with the existence of physical singularities in the solutions to the equations: (1) Excision, and (2) the "puncture" method.
In the excision technique, which was first proposed in the late 1990s,[12] a portion of a spacetime inside of the event horizon surrounding the singularity of a black hole is simply not evolved.
In 2005, a group of researchers demonstrated for the first time the ability to allow punctures to move through the coordinate system, thus eliminating some of the earlier problems with the method.
The Lazarus project (1998–2005) was developed as a post-Grand Challenge technique to extract astrophysical results from short lived full numerical simulations of binary black holes.
It combined approximation techniques before (post-Newtonian trajectories) and after (perturbations of single black holes) with full numerical simulations attempting to solve Einstein's field equations.
[25] All previous attempts to numerically integrate in supercomputers the Hilbert-Einstein equations describing the gravitational field around binary black holes led to software failure before a single orbit was completed.
Mesh refinement first appears in the numerical relativity literature in the 1980s, through the work of Choptuik in his studies of critical collapse of scalar fields.
[35] The technique has now become a standard tool in numerical relativity and has been used to study the merger of black holes and other compact objects in addition to the propagation of gravitational radiation generated by such astronomical events.
[36][37] In the 21st century, hundreds of research papers have been published leading to a wide spectrum of mathematical relativity, gravitational wave, and astrophysical results for the orbiting black hole problem.