## Final Activity Report Summary - EXOB (Excision and outer boundary conditions for the Einstein equations)

According to Albert Einstein's theory of General Relativity (1916), gravitational waves are ripples in the fabric of spacetime generated whenever two massive objects (such as neutron stars or black holes) orbit each another and collide. A number of ground-based facilities, such as GEO600 and LIGO, aimed at detecting and measuring gravitational radiation have been constructed, and a space detector, LISA, is scheduled for launch in 2015. Gravitational wave detectors will offer us an entirely new way of studying the Universe, especially its most dramatic events. Whereas most of the electromagnetic radiation emitted by the collision and coalescence of two black holes or neutron stars will be absorbed by the surrounding matter and therefore will not be able to reach us, gravitational waves can travel through matter without being absorbed. These waves therefore have the capability to reveal things about the cosmos that we otherwise could not possibly know.

These gravitational waves are very weak. Their effect will be to alternately stretch and shrink distances by roughly a factor of 10^(-21). If two test masses were placed one kilometre apart, gravity waves would change their separation by one millionth of a millionth of a millionth of a metre. To be able to detect and interpret gravitational waves we need a prior knowledge of the possible kinds of signals we expect to see. The only viable way to gain a complete knowledge of the gravitational wave signals is to carry out numerical simulations. This is one of the goals of Numerical Relativity. However, this has proven to be much more challenging than expected and there are many issues which remain unsolved. It has become clear that just using more and more computational resources is not the answer and that a better mathematical understanding of the structure of the continuum and discrete initial-boundary value problem is needed.

The primary goal of this research project was to investigate analytical and numerical tools for the treatment of boundary conditions for Einstein's equations. Two types of boundaries appear in numerical relativity: excision of black holes (inner boundary) and outer boundary. The first type of boundaries allows to eliminate the singularity of the black hole from the computational domain, while the second type is introduced because of finite computational resources.

As a result of this project we now better understand how to handle moving black holes and how to implement smooth boundaries using overlapping grids, i.e. by combining a Cartesian main grid with overlapping grids adapted to the boundaries. Another important results of this project are:

1. The realisation that, using formulations that contain second spatial derivatives, can give rise to difficulties which are not present in fully first order formulations. In some cases, techniques to overcome these problems were developed.

2. An improved understanding of how to treat boundaries when second order in space formulations are used.

3. The introduction of new constraint damping mechanisms. One is based on adding spatial derivatives of the constraints to the main system, thus obtaining a mixed hyperbolic-parabolic system. The other one is based on adding suitable lower order terms without affecting the consistency of the system. This method is now being used very effectively by researchers around the world.

4. The use of asymptotically null slices as a tool to push outer boundaries as far away as possible. Preliminary tests have shown that this technique leads to higher accuracy at lower computational cost.

These gravitational waves are very weak. Their effect will be to alternately stretch and shrink distances by roughly a factor of 10^(-21). If two test masses were placed one kilometre apart, gravity waves would change their separation by one millionth of a millionth of a millionth of a metre. To be able to detect and interpret gravitational waves we need a prior knowledge of the possible kinds of signals we expect to see. The only viable way to gain a complete knowledge of the gravitational wave signals is to carry out numerical simulations. This is one of the goals of Numerical Relativity. However, this has proven to be much more challenging than expected and there are many issues which remain unsolved. It has become clear that just using more and more computational resources is not the answer and that a better mathematical understanding of the structure of the continuum and discrete initial-boundary value problem is needed.

The primary goal of this research project was to investigate analytical and numerical tools for the treatment of boundary conditions for Einstein's equations. Two types of boundaries appear in numerical relativity: excision of black holes (inner boundary) and outer boundary. The first type of boundaries allows to eliminate the singularity of the black hole from the computational domain, while the second type is introduced because of finite computational resources.

As a result of this project we now better understand how to handle moving black holes and how to implement smooth boundaries using overlapping grids, i.e. by combining a Cartesian main grid with overlapping grids adapted to the boundaries. Another important results of this project are:

1. The realisation that, using formulations that contain second spatial derivatives, can give rise to difficulties which are not present in fully first order formulations. In some cases, techniques to overcome these problems were developed.

2. An improved understanding of how to treat boundaries when second order in space formulations are used.

3. The introduction of new constraint damping mechanisms. One is based on adding spatial derivatives of the constraints to the main system, thus obtaining a mixed hyperbolic-parabolic system. The other one is based on adding suitable lower order terms without affecting the consistency of the system. This method is now being used very effectively by researchers around the world.

4. The use of asymptotically null slices as a tool to push outer boundaries as far away as possible. Preliminary tests have shown that this technique leads to higher accuracy at lower computational cost.