Upper bounds on the compactness at the innermost light ring of anisotropic horizonless spheres
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Regular Article - Theoretical Physics
Upper bounds on the compactness at the innermost light ring of anisotropic horizonless spheres Yan Penga School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China
Received: 14 July 2020 / Accepted: 14 August 2020 © The Author(s) 2020
Abstract In the background of isotropic horizonless spheres, Hod recently provided an analytical proof of a bound on the compactness at the innermost light ring with the dominant energy condition. In this work, we extend the discussion of isotropic spheres to anisotropic spheres. With the dominant energy and non-negative trace conditions, we prove that Hod’s bound also holds in the case of anisotropic horizonless spheres.
by the lower bound
According to general relativity, highly curved spacetimes may possess closed light rings (null circular geodesics), on which massless particles can orbit in a circle [1,2]. It is well known that closed light rings are usually related to black hole spacetimes. In fact, closed light rings may also exist in the horizonless ultra-compact spacetime. From theoretical and astrophysical aspects, the light rings have been extensively studied in various gravity backgrounds [3–13]. The closed light ring plays an important role in understanding properties of curved spacetimes. For example, the interesting phenomenon of strong gravitational lensing in highly curved spacetimes is closely related to the existence of light rings [14]. In addition, the light ring can be used to describe the distribution of exterior matter fields outside black holes [15–20]. And it was also proved that the innermost light ring provides the fastest way to circle a central black hole as measured by observers at the infinity [21–23]. Moreover, the existence of stable light rings suggests that the central compact stars may suffer from nonlinear instabilities [24–30]. And unstable light rings can be used to determine the characteristic resonances of black holes [31–38]. Recently, the compactness at the innermost closed light ring was investigated. The compactness can be described ) by the parameter m(r r , where m(r ) is the gravitational mass [email protected] (corresponding author)
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m(rγin ) rγin )
1 3
with rγin as the innermost
light ring radius. However, very differently in the horizonless case, numerical data in [39] suggests that the compactness parameter may satisfy an upper bound
m(rγin ) rγin )
1 3
for
spherically symmetric ultra-compact isotropic spheres. Hod has provided compact analytical proofs of the characteristic intriguing bound
1 Introduction
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within the radius r. In the case of black holes, the compactness parameter at the innermost light ring is characterized
m(rγin ) rγin )
1 3
for the spherically symmet-
ric spatially regular spheres with isotropic tensor ( p = pτ ), where p and pτ are interpreted as the radial pressure and the tangential pressure respectively [40]. In the present paper, we study the compactness at the innermost light ring of
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