Annales Henri Poincaré

, Volume 20, Issue 3, pp 813–857 | Cite as

On the Asymptotic Behavior of Static Perfect Fluids

  • Lars Andersson
  • Annegret Y. BurtscherEmail author
Open Access


Static spherically symmetric solutions to the Einstein–Euler equations with prescribed central densities are known to exist, be unique and smooth for reasonable equations of state. Some criteria are also available to decide whether solutions have finite extent (stars with a vacuum exterior) or infinite extent. In the latter case, the matter extends globally with the density approaching zero at infinity. The asymptotic behavior largely depends on the equation of state of the fluid and is still poorly understood. While a few such unbounded solutions are known to be asymptotically flat with finite ADM mass, the vast majority are not. We provide a full geometric description of the asymptotic behavior of static spherically symmetric perfect fluid solutions with linear equations of state and polytropic-type equations of state with index \(n>5\). In order to capture the asymptotic behavior, we introduce a notion of scaled quasi-asymptotic flatness, which encompasses the notion of asymptotic conicality. In particular, these spacetimes are asymptotically simple.



The authors would like to thank Patryk Mach for discussions on the Lane–Emden equation and Naresh Dadhich for pointing out the conformal asymptotically flat geometry. We would also like to thank Claes Uggla for correspondence regarding their dynamical systems approach, Piotr Chruściel for information on the asymptotically flat situation, Anna Sakovich for pointing out references related to notions of mass and Shadi Tahvildar-Zadeh for general discussions on the equations of state. A.B. acknowledges financial support during an extended research stay at the Riemann Center for Geometry and Physics at the Leibniz University Hannover, during which this project was initiated. The major part of the project was carried out at the University of Bonn and supported by the Sofja Kovalevskaja award of the Humboldt Foundation endowed by the German Federal Ministry of Education and Research (held by Roland Donninger). The authors also gratefully acknowledge financial support in connection with a research stay at the Erwin Schrödinger Institute in Vienna during the “Geometry and Relativity" program in 2017, where the project was finalized.


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Authors and Affiliations

  1. 1.Albert Einstein Institute (Max Planck Institute for Gravitational Physics)PotsdamGermany
  2. 2.Department of MathematicsRutgers, The State University of New JerseyPiscatawayUSA
  3. 3.Department of MathematicsRadboud UniversityNijmegenThe Netherlands

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