Light ring stability in ultra-compact objects (1708.04211v2)

Abstract: 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.

Light Ring Stability in Ultra-Compact Objects

The paper by Cunha, Berti, and Herdeiro offers a rigorous theoretical exploration into the stability of light rings in ultra-compact objects (UCOs), a subject pertinent in astrophysical and gravitational research. Ultra-compact objects are theoretical configurations that possess circular photon orbits, known as light rings (LRs), but lack event horizons, distinguishing them from black holes (BHs).

The authors establish that any axisymmetric, stationary UCO formed from classical gravitational collapse of matter satisfying the null energy condition will necessarily possess at least two LRs, with one of these being stable. This key finding stems from topological arguments leveraging the Brouwer degree of a continuous map. Such reasoning is independent of specific dynamical assumptions, thus extending its applicability across various spacetime metrics where photons follow null geodesics.

Theoretical Implications

The presence of a stable LR in UCOs is noteworthy because it implies the potential for accumulating electromagnetic or gravitational radiation, which might not decay rapidly, possibly leading to spacetime instabilities. The exploration of such instabilities is crucial as they could challenge the long-standing assertion that UCOs can serve as BH mimickers in gravitational wave (GW) observations. The authors mention that if these instabilities occur on astrophysically short time scales, UCOs cannot present viable observational alternatives to BHs.

Numerical Results and Claims

The theorem, complemented by topological analysis, serves as a foundation to assert that LRs within UCOs always appear in pairs—an assertion the authors support quantitatively without dependence on assumptions about spacetime dynamics. They offer a decisive argument that the extremum among the pair constitutes a local minimum, thereby indicating a stable configuration under the conditions specified by Einstein's equations.

Speculation on Future Developments in AI

Considering advancements in AI and computational astrophysics, future research could leverage machine learning techniques to simulate gravitational collapse scenarios and assess the formation and stability of LRs in theoretical UCO models. Additionally, AI-enhanced data analysis might facilitate the scrutiny of GW signatures to discern subtle distinctions between BHs and horizonless UCOs.

Conclusion

The paper contributes significant insights into the discussion on UCOs as potential BH alternatives, reinforcing the understanding that the existence of stable LRs inherently challenges the plausibility of UCOs as stable astrophysical entities. This investigation advances the contemplation of the astrophysical viability of UCO scenarios within the framework of general relativity, decisively suggesting that smooth, stable UCOs are unlikely under current theoretical assumptions.