# Gravity, Black Holes and Gravitational Waves

Spacetime is the combination of the 3-dimensional space we live in plus time. According to Albert Einstein's theory of General Relativity spacetime can be visualized as a rubber sheet which gets deformed by any object which has mass or energy. This deformation is called curvature of spacetime. All the stars, planets, particles and things in the universe cause spacetime to curve. The amount of curvature caused by any one object is proportional to its mass and energy. Below is an example of the spacetime curvature caused by a spherical object. 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.

### Black Holes

If enough mass is concentrated in a small enough region of spacetime, the spacetime curvature can become infinite. The pull of gravity in this case becomes so strong that nothing, not even light, can escape this region. Such objects are called black holes and the boundary which marks the region out of which nothing can escape anymore is called an event horizon. Things which move through the event horizon can never return! Black holes can form for example when very massive stars collapse under the influence of their own gravity. Black holes are indeed black since they only swallow but never reflect light. They however influence their surroundings by the curvature they produce. So smaller objects can orbit around black holes just like the planets around the sun, except that the orbits are more complicated. The orbits are in general not closed and if the black hole rotates it can even drag other objects with it. Three examples of Orbits around Black Holes can be found here.

### Gravitational Waves

If an object moves through spacetime (e.g. if one star or black hole orbits around another star) the time-varying curvature of spacetime caused by this object can create little ripples in spacetime, which then propagate through spacetime, just like ripples on a pond or waves on a rubber sheet. These ripples are called gravitational waves. Such gravitational waves can in principle propagate through the whole universe and will shake all objects in their path. The amplitude of these waves however is usually very small, so that they are very hard to detect. The strongest waves are produced by some of the most violent crashes in the universe, such as the merger of two black holes. Several detectors are currently operating that can detect and measure gravitational waves. One of the most promising sources of gravitational waves are binary systems of compact objects, such as two neutron stars or two black holes orbiting around each other. Such binaries loose energy due to the emission of gravitational waves and thus the orbital radius shrinks. Hence the two objects spiral toward each other emitting gravitational waves of increasing frequency and amplitude. Below is a graph of how part of the waveform for the inspiral of two compact objects might look like:

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.

### My Research

Currently my research is mainly focused on numerical relativity, i.e. simulating General Relativity on a computer. One of the goals in this field is to numerically solve the Einstein equations to simulate the merger of two black holes or two neutron stars. For such simulations three ingredients are crucial: (i) We need to start the simulation with astrophysically realistic initial data, (ii) the equations used in the computer simulation have to be written in such a form that they are numerically stable long enough to simulate the entire black hole merger. (iii) especially for neutron stars we have to include enough physics (e.g. a realistic matter equation of state or magnetic fields) to produce results that model real neutron star mergers. I am interested in all of the of these aspects.
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.