Project A1;     (2003 - 2006)

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Analysis of asymptotically flat space-times

    Principal Investigators: Helmut Friedrich

The goal of project A1 is to further develop the theory of asymptotically flat spacetimes as models for the gravitational fields of isolated bodies, Black Holes and the non-linear self-interation of vacuum fields. One has to distinguish between two different regions of spacetime: the interior, in which violent interactions between vacuum and matter fields can arise, and the exterior, in which the gravitational wave signal takes shape. Further idealization leads quite naturally to an asymptotic description of this signal as a radiation field on the hypersurface S that one can think of as being the set of endpoints of outgoing light-like geodesics.

The study of these regions must be understood in the correct context. On the one hand, the processes in the interior determine the asymptotic behaviour of the fields. On the other hand, controlling the asymptotics including the proof of completeness would essentially amount to a confirmation of `weak cosmic censorship' since any singularities that may crop up must be hidden within Black Holes. Breaking the whole problem up into parts is unavoidable and identifying which of these are soluble and vital constitutes the most important and difficult aspect of the task.

The sub-project A1 intends to (i) continue with the analytic studies begun by Friedrich(1) of the region in which space-like and light-like infinities meet, (ii) analyse the solutions of the constraint equations regarding their asymptotic behaviour and physical properties as begun by Dain(2) and (iii) develop numerical codes for the conformal vacuum field equations that could be used both to aid in these studies and to calculate concrete spacetimes in large regions including their asymptotics and radiation fields.

(1) H. Friedrich: Gravitational fields near space-like and null infintiy, J. Geom. Phys. 24 (1998) 83-163
(2) S. Dain, H. Friedrich: Asymptotically Flat Inital Data with Presribed Regularity, Comm. Math. Phys. 222 (2001) 569-609

Researchers


Former Associates
  Sergio Dain   Postdoc, 2003-2006
  Helmut Friedrich   Professor, PI 2003-2006
  Anil Zenginoglu   PhD Student, 2003-2005

Publications

[1] Generalized Korn's inequality and conformal Killing vectors
S. Dain, Calculus of Variations and Partial Differential Equations 25, 535-540 (2006)

[2] Angular momentum-mass inequality for axisymmetric black holes
S. Dain, Phys. Rev. Lett. 96, 101101 (2006)

[3] Elliptic systems
S. Dain, In: J. Frauendiener, D. Guilini, V. Perlick (eds.): Analytical and Numerical Approaches to Mathematical Relativity, Springer, Berlin (2006).

[4] Proof of the (local) angular momentum-mass inequality for axisymmetric black holes
S. Dain, eprint (2006)

[5] On the existence of initial data containing isolated black holes
S. Dain. J. L. Jaramillo, and B. Krishnan, Phys. Rev. D 71, 064003 (2005)

[6] On the non-linearity of subsidiary systems
H. Friedrich, Class. Quantum Grav. 22, L77-L82 (2005)

[7] Is general relativity `essentially understood?'
H. Friedrich, Ann. Phys. (Leipzig) 15, 84-108 (2005)

[8] Hyperboloidal data and evolution
S. Husa, C. Schneemann, T. Vogel, A. Zenginoglu, In: Proc. Spanish Relativity Meeting, Oviedo, Spain (2005)

[9] Trapped surfaces as boundaries for the constraint equations
S. Dain, Class. Quantum Grav. 21, 555-573 (2004)

[10] A new geometric invariant on initial data for Einstein equations
S. Dain, Phys. Rev. Lett. 93, 231101 (2004)

[11] Smoothness at null infinity and the structure of initial data
H. Friedrich, In: P. T. Chrusciel, H. Friedrich (eds.): The Einstein equations and the large scale behaviour of gravitational fields, Birkhaeuser, Basel (2004)

[12] On black holes as inner boundaries for the constraint equations
S. Dain, In: I. Racz (ed.): Relativity Today, Proceedings of the Seventh Hungarian Relativity Workshop, Akademiai Kiadu, Budapest (2004)

[13] Spin-2 fields on Minkowski space near spacelike and null infinity
H. Friedrich, Call. Quantum Grav. 20, 101-117 (2003)