According to Einstein this curvature is the reason for gravity. It predicts that all objects which are subject only to gravity move on straight lines. But a straight line through curved spacetime may look like a curve for us. Hence gravity is described not as a force, but rather as curvature of spacetime. For example, if a heavy object like a star causes sufficient spacetime curvature around itself, any smaller object moving on a straight line through this curved spacetime, looks like moving on a curved orbit for us. Hence gravity is an effect caused purely by the curvature of spacetime.

The waveform shown here is in principle in the frequency region audible to human beings. In the example here the amplitude has been artificially increased so that it is loud enough for you to hear, click on the graph to listen to the waveform! This waveform was computed using the so called Post-Newtonian theory which approximates General Relativity for the case of slow moving particles. The reason why we use this approximation is that computations with it are much easier than when we use full General Relativity. Yet near the end of the graph the two objects may move quite fast so that the Post-Newtonian approximation starts to break down. Notice that the calculation of the waveform was stopped when the two objects started to merge. The final plunge and merger of the two objects in principle emits the strongest gravitational waves. Yet at this point the objects are moving so fast that Post-Newtonian theory is no longer valid and we have to use General Relativity in order to do our calculations. Unfortunately General Relativity is so complicated that nobody has manged to do this analytically so far. The only way out seems to put the equations of General Relativity on a computer and to try to simulate them there. This however, can be very difficult as well, due to numerical instabilities.

For example, I am working on constructing initial data for binary black holes. Such binaries are believed to spiral toward each other on quasi-circular orbits. I have constructed initial data for binary black holes based on Post-Newtonian data, which are astrophysically realistic as long as the black holes are well separated. Below are pictures of such initial data for two black holes in a Post-Newtonian circular orbit.

Both pictures show the so called conformal factor as seen from different angles. This conformal factor is closely related to the spacetime curvature. The two spikes are the black holes. Since the curvature at each black hole center is infinite each spike should in principle be infinitely long. However due to limited resolution the spikes are cut off at some finite value.

Together with Bernd Bruegmann and Pablo Laguna I have also investigated how to find coordinate systems which corotate with the two orbiting black holes. Such corotating coordinate systems have the advantage that the rapid circling motion of the two black holes is transformed away so that one has to simulate only the slower drift of the holes toward each other. It is hoped that then the numerical simulations will be more accurate and stable. In the formulation we are using coordinates are fixed by choosing a lapse function and a shift vector. Our objective is to find a lapse and shift, which yield approximately corotating coordinates on the initial data slice. As a first step we have applied this idea to puncture initial data, which are similar but much simpler than the Post-Newtonian based initial data shown above. In addition, I am investigating the properties of different formulations of the Einstein equations in numerical applications. The aim is to find out which of the formulations is numerically more stable.