Naked singularities for the Einstein vacuum equations: The exterior solution

Igor Rodnianski, Yakov Shlapentokh-Rothman

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


In this work we initiate the mathematical study of naked singularities for the Einstein vacuum equations in 3 + 1 dimensions by constructing solutions which correspond to the exterior region of a naked singularity. A key element is our introduction of a new type of self-similarity for the Einstein vacuum equations. Connected to this is a new geometric twisting phenomenon which plays the leading role in singularity formation. Prior to this work, the only known examples of naked singularities were the solutions constructed by Christodoulou for the spherically symmetric Einstein-scalar-field system, as well as other solutions explored numerically for either the spherically symmetric Einstein equations coupled to suitable matter models or for the Einstein equations in higher dimensions.

Original languageEnglish (US)
Pages (from-to)231-391
Number of pages161
JournalAnnals of Mathematics
Issue number1
StatePublished - 2023

All Science Journal Classification (ASJC) codes

  • Mathematics (miscellaneous)


  • naked singularity
  • singularity
  • weak cosmic censorship


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