Cunha Pedro V P, Berti Emanuele, Herdeiro Carlos A R
Departamento de Física da Universidade de Aveiro and CIDMA, Campus de Santiago, 3810-183 Aveiro, Portugal.
CENTRA, Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal.
Phys Rev Lett. 2017 Dec 22;119(25):251102. doi: 10.1103/PhysRevLett.119.251102. Epub 2017 Dec 18.
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.
由服从零能量条件的物质的经典引力坍缩形成的爱因斯坦场方程的轴对称、静态解，若处处光滑且超紧致（即它们有一个光环），则必定至少有两个光环，且其中一个是稳定的。有人认为，稳定的光环通常会导致非线性时空不稳定性。我们的结果表明，每当这些不稳定性在天体物理学的短时间尺度上发生时，光滑的、物理上和动力学上合理的超紧致物体作为黑洞的观测替代物是不可行的。该定理的证明有两个部分：(i) 我们表明光环总是成对出现，一个是鞍点，另一个是有效势的局部极值。这个结果源于基于连续映射的布劳威尔度的拓扑论证，对时空动力学没有任何假设，因此，它适用于任何光子沿零测地线传播的度规引力理论。(ii) 假设爱因斯坦方程，我们表明如果能量 - 动量张量满足零能量条件，则该极值是势的局部最小值（即一个稳定的光环）。
