## Final Report Summary - GRAVITYWAVESTARS (Localized objects formed by self-trapped gravitational waves)

It is known that in General Relativity, without additional matter fields, no non-singular time independent solutions exist, implying the absence of localized nonlinear gravitons. Black holes always have singularities, hidden by a horizon. There is, however, a very intriguing object in General Relativity, called geon, whose existence has been conjectured by Wheeler at around 1955. The name “geon” stands for a time-dependent solution of Einstein's vacuum equations created by high frequency gravitational waves, trapped for a very long period in a background geometry created by the waves themselves. The central theme of our project is the study of the formation of geons by gravitational waves. The research group carrying out the reported research at Paris Observatory consisted of the guest researcher Gyula Fodor, the host researcher Philippe Grandclément, and the host researcher’s PhD student Grégoire Martinon. We have also collaborated with Péter Forgács from the Wigner Research Centre in Budapest.

During the two year research period of the project we have developed a new approach to construct geon-type solutions in Einstein's theory of Gravitation based on a combination of analytical and numerical methods. The main challenges have been the following: the mathematical and numerical construction of geons by solving the gravitational field equations, the study of their stability, the computation of their lifetime, and the investigation of their consequences on our Universe. A very important potential application of geons, provided they have good stability properties and sufficiently long lifetimes, is that they can provide a natural component for the mysterious dark matter content of the Universe without invoking unobserved new type of matter fields.

Recently there has been an upsurge of interest in geon-type solutions of the Eintein’s equations in the case when there is a negative cosmological constant. In this case, the spacetime approaches the maximally symmetric anti-de Sitter (AdS) metric. Asymptotically AdS space-times have attracted huge interest in recent years because of the AdS/CFT correspondence. This correspondence connects solutions of Einstein’s equations in asymptotically AdS spacetimes to properties of physically relevant, strongly coupled quantum field theories defined on the outer boundary (representing spatial infinity). The correspondence has been used to study various systems in condensed matter physics and quantum chromodynamics.

As an introductory task for the complicated vacuum gravitational wave problem we have first considered an analogous simpler model, a self-gravitating massless scalar field. In the negative cosmological constant case there are spherically symmetric spatially localized periodically oscillating solutions, which are called AdS breathers in the literature. We have studied these objects, imposing anti-de Sitter asymptotics on space-time. Using a code constructed with the KADATH library, that enables the use of spectral methods, breather solutions were explored in detail. It was found that there are discrete families of solutions indexed by an integer and by their frequency. Using a time evolution code these AdS breathers were found to be stable up to a critical central density. Using a perturbative expansion procedure, small amplitude breathers were worked out for arbitrary spacetime dimensions. Our results indicate that the expansion is likely to have a finite radius of convergence. We have published this work in Physical Review D in 2015.

Our main project during the two year research period was the detailed study of localized vacuum gravitational wave structures in the negative cosmological constant case, which are called AdS geons. We have used two complementary methods to investigate these objects, an analytical expansion procedure, and high precision direct numerical calculations.

To leading linear order, gravitational perturbations of the maximally symmetric anti-de Sitter spacetime are described by the Kodama-Ishibashi formalism. For the 3+1 dimensional spacetimes that we have considered, there are two classes of perturbations, scalar and vector types. In both classes, regular and asymptotically AdS linear perturbation modes are labeled by three integers. The frequency of these perturbative modes is always integer. Because of this, any linear combination of these large number of perturbation modes give solutions, which are perfectly valid periodic configurations to linear order. However, it turns out that only some of these survive to higher order in the expansion procedure, because of the nonlinearities in the system.

The main technical difficulty in our pertubational calculations was the generalization of the already highly complicated Kodama-Ishibashi formalism to higher order perturbations. We had to use an algebraic manipulation system (Maple, Mathematica, or Sage) in order to write down the resulting complicated system of partial differential equations, and had to work out an efficient method for solving them. The second order perturbations can be always calculated, and describe periodicaly oscillating solutions. However, to third order in the expansion, secular terms arise, which are increasing without bound as time passes. Only some exceptional combinations of the linearized solutions can remain time-periodic when nonlinearities are taken into account. Still there remains several families of them. There are two main classes of surviving nonlinear solutions, non-rotating ones, and rotating solutions with a helical symmetry. Non-rotating solutions can be axially symmetric, but there are also non-rotating solutions without any continuous symmetries. Helically symmetric solutions appear to be time independent in a uniformly rotating coordinate system. In each of these classes there is a one-parameter family of solutions, parametrized by an amplitude variable.

Our numerical calculations employ the spectral numerical library KADATH, which has been developed by Philippe Grandclément, the host researcher of the project. The second order expansion solution was used as an initial guess for the iterative process when constructing the numerical solution. Then the amplitude of the solution was increased step-by-step, using smaller amplitude solutions as the initial guess for the numerical iteration. The gauge in which the expansion solution was obtained is unfortunately not suitable for the numerical calculations. Our numerical code uses the maximal slicing condition and spatial harmonic gauge. Since the coordinate transformation between the two applied gauges is quite nontrivial, we use invariant quantities, such as the frequency, mass and angular momentum to compare the results obtained by the two methods, and obtain good agreement even for moderately large amplitudes. The largest amplitude solutions that we were able to calculate require surprisingly high radial and angular resolutions. The calculation of each such configuration requires several hours of computation time on clusters with a few hundred computational cores. Currently we are in the process of writing up our results into a research paper which we intend to publish in Physical Review D.

During the two year research period of the project we have developed a new approach to construct geon-type solutions in Einstein's theory of Gravitation based on a combination of analytical and numerical methods. The main challenges have been the following: the mathematical and numerical construction of geons by solving the gravitational field equations, the study of their stability, the computation of their lifetime, and the investigation of their consequences on our Universe. A very important potential application of geons, provided they have good stability properties and sufficiently long lifetimes, is that they can provide a natural component for the mysterious dark matter content of the Universe without invoking unobserved new type of matter fields.

Recently there has been an upsurge of interest in geon-type solutions of the Eintein’s equations in the case when there is a negative cosmological constant. In this case, the spacetime approaches the maximally symmetric anti-de Sitter (AdS) metric. Asymptotically AdS space-times have attracted huge interest in recent years because of the AdS/CFT correspondence. This correspondence connects solutions of Einstein’s equations in asymptotically AdS spacetimes to properties of physically relevant, strongly coupled quantum field theories defined on the outer boundary (representing spatial infinity). The correspondence has been used to study various systems in condensed matter physics and quantum chromodynamics.

As an introductory task for the complicated vacuum gravitational wave problem we have first considered an analogous simpler model, a self-gravitating massless scalar field. In the negative cosmological constant case there are spherically symmetric spatially localized periodically oscillating solutions, which are called AdS breathers in the literature. We have studied these objects, imposing anti-de Sitter asymptotics on space-time. Using a code constructed with the KADATH library, that enables the use of spectral methods, breather solutions were explored in detail. It was found that there are discrete families of solutions indexed by an integer and by their frequency. Using a time evolution code these AdS breathers were found to be stable up to a critical central density. Using a perturbative expansion procedure, small amplitude breathers were worked out for arbitrary spacetime dimensions. Our results indicate that the expansion is likely to have a finite radius of convergence. We have published this work in Physical Review D in 2015.

Our main project during the two year research period was the detailed study of localized vacuum gravitational wave structures in the negative cosmological constant case, which are called AdS geons. We have used two complementary methods to investigate these objects, an analytical expansion procedure, and high precision direct numerical calculations.

To leading linear order, gravitational perturbations of the maximally symmetric anti-de Sitter spacetime are described by the Kodama-Ishibashi formalism. For the 3+1 dimensional spacetimes that we have considered, there are two classes of perturbations, scalar and vector types. In both classes, regular and asymptotically AdS linear perturbation modes are labeled by three integers. The frequency of these perturbative modes is always integer. Because of this, any linear combination of these large number of perturbation modes give solutions, which are perfectly valid periodic configurations to linear order. However, it turns out that only some of these survive to higher order in the expansion procedure, because of the nonlinearities in the system.

The main technical difficulty in our pertubational calculations was the generalization of the already highly complicated Kodama-Ishibashi formalism to higher order perturbations. We had to use an algebraic manipulation system (Maple, Mathematica, or Sage) in order to write down the resulting complicated system of partial differential equations, and had to work out an efficient method for solving them. The second order perturbations can be always calculated, and describe periodicaly oscillating solutions. However, to third order in the expansion, secular terms arise, which are increasing without bound as time passes. Only some exceptional combinations of the linearized solutions can remain time-periodic when nonlinearities are taken into account. Still there remains several families of them. There are two main classes of surviving nonlinear solutions, non-rotating ones, and rotating solutions with a helical symmetry. Non-rotating solutions can be axially symmetric, but there are also non-rotating solutions without any continuous symmetries. Helically symmetric solutions appear to be time independent in a uniformly rotating coordinate system. In each of these classes there is a one-parameter family of solutions, parametrized by an amplitude variable.

Our numerical calculations employ the spectral numerical library KADATH, which has been developed by Philippe Grandclément, the host researcher of the project. The second order expansion solution was used as an initial guess for the iterative process when constructing the numerical solution. Then the amplitude of the solution was increased step-by-step, using smaller amplitude solutions as the initial guess for the numerical iteration. The gauge in which the expansion solution was obtained is unfortunately not suitable for the numerical calculations. Our numerical code uses the maximal slicing condition and spatial harmonic gauge. Since the coordinate transformation between the two applied gauges is quite nontrivial, we use invariant quantities, such as the frequency, mass and angular momentum to compare the results obtained by the two methods, and obtain good agreement even for moderately large amplitudes. The largest amplitude solutions that we were able to calculate require surprisingly high radial and angular resolutions. The calculation of each such configuration requires several hours of computation time on clusters with a few hundred computational cores. Currently we are in the process of writing up our results into a research paper which we intend to publish in Physical Review D.