2017/08/31 by Pedro V. P. Cunha, Emanuele Berti, Carlos A. R. Herdeiro +1 · 378 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Classical mechanics #Cosmology and Gravitation Theories #Einstein field equations #Energy condition #General relativity #Geodesic #Gravitation #Light cone #Mathematical analysis #Mathematical physics #Mathematics #Null (SQL) #Physics #Pulsars and Gravitational Waves Research #Quantum mechanics #Saddle point #Spacetime #astro-ph.HE #gr-qc #hep-th
paper · pdf · doi:10.1103/physrevlett.119.251102
published in Physical Review Letters 119(25), 251102 (American Physical Society) · 5 pages, 1 figure; v2: minor changes, references added, includes published DOI
openalex publication_date 2017/12/18 · arxiv created 2017/12/19 · arxiv updated 2017/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We prove the following theorem: axisymmetric, stationary solutions of the Einstein field equations formed from classical gravitational collapse of matter obeying the null energy condition, that are everywhere smooth and ultracompact (i.e., they have a light ring) must have at least two light rings, and one of them is stable. It has been argued that stable light rings generally lead to nonlinear spacetime instabilities. Our result implies that smooth, physically and dynamically reasonable ultracompact objects are not viable as observational alternatives to black holes whenever these instabilities occur on astrophysically short time scales. The proof of the theorem has two parts: (i) We show that light rings always come in pairs, one being a saddle point and the other a local extremum of an effective potential. This result follows from a topological argument based on the Brouwer degree of a continuous map, with no assumptions on the spacetime dynamics, and, hence, it is applicable to any metric gravity theory where photons follow null geodesics. (ii) Assuming Einstein's equations, we show that the extremum is a local minimum of the potential (i.e., a stable light ring) if the energy-momentum tensor satisfies the null energy condition.