Upper bounds on the compactness at the innermost light ring of anisotropic horizonless spheres
aa r X i v : . [ g r- q c ] J un Upper bounds on the compactness at the innermost light ring of anisotropichorizonless spheres
Yan Peng ∗ School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
Abstract
In the background of isotropic horizonless spheres, Hod recently provided an analytical proof of abound on the compactness at the innermost light ring with the dominant energy and non-negativetrace conditions. In this work, we extend the discussion of isotropic spheres to anisotropic spheres.With the same dominant energy and non-negative trace conditions, we prove that Hod’s bound alsoholds in the case of anisotropic horizonless spheres.
PACS numbers: 11.25.Tq, 04.70.Bw, 74.20.-z ∗ [email protected] I. INTRODUCTION
According to general relativity, highly curved spacetimes may possess closed light rings (null circulargeodesics), on which massless particles can orbit in a circle [1, 2]. It is well known that closed light ringsare 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 studiedin 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 tothe existence of light rings [14]. In addition, the light ring can be used to describe the distribution of exteriormatter fields outside black holes [15–20]. And it was also proved that the innermost light ring provides thefastest way to circle a central black hole as measured by observers at the infinity [21–23]. Moreover, theexistence 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 bedescribed by the parameter m ( r ) r , where m ( r ) is the gravitational mass within the radius r. In the case of blackholes, the compactness parameter at the innermost light ring is characterized by the lower bound m ( r inγ ) r inγ ) > with r inγ as the innermost light ring radius. However, very differently in the horizonless case, numerical datain [39] suggests that the compactness parameter may satisfy an upper bound m ( r inγ ) r inγ ) for sphericallysymmetric ultra-compact isotropic spheres. Hod has provided compact analytical proofs of the characteristicintriguing bound m ( r inγ ) r inγ ) for the spherically symmetric spatially regular spheres with isotropic tensor( p = p τ ), where p and p τ 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 horizonless spheres. We shallprove the bound m ( r inγ ) r inγ ) in the case of anisotropic sphere with p = p τ . We point out that this bound inthe isotropic case of p = p τ has been proved in [40]. Our main results are included in the last section. II. INVESTIGATIONS ON THE COMPACTNESS AT THE INNERMOST LIGHT RING
We study the closed light ring of spherically symmetric configurations. In the standard Schwarzschildcoordinate, these spacetimes are expressed by line element [40] ds = − e − δ µdt + µ − dr + r ( dθ + sin θdφ ) , (1)where the metric has two functions δ ( r ) and µ ( r ) = 1 − m ( r ) r . For horizonless asymptotically flat spacetimes,the metric functions are characterized by the near origin behavior [40–42] µ ( r →
0) = 1 + O ( r ) and δ (0) < ∞ (2)and the far region behavior [40–42] µ ( r → ∞ ) = 1 and δ ( r → ∞ ) = 0 . (3)We denote the components of the energy-momentum tensor as ρ = − T tt , p = T rr and p τ = T θθ = T φφ , (4)where ρ , p and p τ are respectively the energy density, the radial pressure and the tangential pressure of thehorizonless configurations [17, 43]. For the case of isotropic energy-momentum tensor, there is the relation p = p τ [40]. In this work, our discussion also covers the case of p = p τ . The unknown metric functions aredetermined by the Einstein equations G µν = 8 πT µν . With the energy density and pressures (4), one can expressthe Einstein field equations in the form µ ′ = − πrρ + 1 − µr , (5) δ ′ = − πr ( ρ + p ) µ . (6)Using the Einstein field equations (5) and (6), it has been explicitly proved that the closed light rings arecharacterized by the relation [41] R ( r ) = 3 µ ( r ) − − πr p ( r ) = 0 f or r = r γ , (7)where r γ is the radius of the closed light ring.From equation (2) and the regular condition p (0) < ∞ , the function R ( r ) is characterized by asymptoticalbehaviors R ( r ) = 3 µ ( r ) − − πr p ( r ) → f or r → . (8)We label r inγ as the radius of the innermost closed light ring, which corresponds to the smallest positive rootof R ( r ) = 0. In the range [0 , r inγ ], the function R ( r ) satisfies the relation R ( r ) > f or r ∈ [0 , r inγ ] . (9)In particular, at the innermost closed light ring, there is the relation R ( r ) = 3 µ ( r ) − − πr p ( r ) = 0 f or r = r inγ . (10)Substituting equations (5) and (6) into the conservation equation T µr ; µ = 0, one obtains a relation P ′ ( r ) = r g [ R ( ρ + p ) + 2 gT ] , (11)where P ( r ) = r p ( r ), R = 3 µ − − πr p and T = − ρ + p + 2 p τ .In proving the following bound (22), Hod has imposed the conditions of the dominant energy and the non-negative trace of the energy-momentum tensor [40]. In this work, we also impose the same energy condition.The dominant energy condition is ρ > | p | , | p τ | > . (12)And the non-negative trace condition is [39–42] T = − ρ + p + 2 p τ > . (13)In fact, for a polytropic pressure density equation of the form p = p τ = k p ρ , Hod has obtained a bound m ( r inγ ) r inγ ) k p +26( k p +1) , where ρ , p and p τ are interpreted as the energy density, the radial pressure and the tangentialpressure respectively [40]. In the case of k p > or a non-negative trace T = − ρ + p + 2 p τ = − ρ + 3 p > m ( r inγ ) r inγ ) , which is the same as (22). So Hod proved the bound (22) in the case of p = p τ .In the present work, we generalize the discussion to cover the case of p = p τ .Relations (9-13) yield that the function P ( r ) satisfies the inequality P ′ ( r ) > f or r ∈ [0 , r inγ ] . (14)Near the origin, the pressure function P ( r ) has the asymptotical behavior P ( r →
0) = 0 . (15)With relations (14) and (15), one obtains P ( r ) > f or r ∈ [0 , r inγ ] . (16)The relation (16) and P ( r ) = r p ( r ) yield that p ( r ) > f or r ∈ [0 , r inγ ] . (17)In particular, at the innermost light ring, there is the relation p ( r inγ ) > . (18)According to (10) and (18), one finds that3 µ ( r ) − πr p ( r ) > f or r = inγ . (19)The relation (19) yields that µ ( r inγ ) > . (20)The inequality (20) can be expressed as 1 − m ( r inγ ) r inγ > . (21)Then we obtain an upper bound on the compactness at the innermost light ring m ( r inγ ) r inγ . (22) III. CONCLUSIONS
We studied the compactness at the innermost light ring of anisotropic horizonless spheres. We assumedthe dominant energy and non-negative trace conditions. At the innermost light ring, we obtained an upperbound on the compactness expressed as m ( r inγ ) r inγ , where r inγ is the innermost light ring and m ( r inγ ) is thegravitational mass within the sphere of radius r inγ . In fact, Hod firstly proved this bound in the spacetime ofhorizonless spheres with isotropic tensor p = p τ [40]. In the present work, we proved the same bound in thebackground of horizonless spheres with generalized anisotropic tensor covering the case of p = p τ . Acknowledgments
This work was supported by the Shandong Provincial Natural Science Foundation of China under GrantNo. ZR2018QA008. This work was also supported by a grant from Qufu Normal University of China underGrant No. xkjjc201906. [1] J. M. Bardeen, W. H. Press and S. A. Teukolsky, Rotating black holes: Locally nonrotating frames, energyextraction, and scalar synchrotron radiation, Astrophys. J. 178,347(1972).[2] S. Chandrasekhar, The Mathematical Theory of Black Holes, (Oxford University Press, New York, 1983).[3] Goebel, C. J.,Comments on the “vibrations” of a Black Hole,Astrophysical Journal, vol. 172, p.L 95.[4] Teo, E., Spherical Photon Orbits Around a Kerr Black Hole, General Relativity and Gravitation (2003) 35: 1909.[5] Pedro V.P. Cunha, Carlos A.R. Herdeiro, Eugen Radu, Fundamental photon orbits: black hole shadows andspacetime instabilities, Phys. Rev. D 96(2017)no.2,024039.[6] Jai Grover, Alexander Wittig, Black Hole Shadows and Invariant Phase Space Structures, Phys. Rev. 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Hod, Upper bound on the radii of black-hole photonspheres, Phys. Lett. B 727, 345 (2013).[27] P. V. P. Cunha, E. Berti, and C. A. R. Herdeiro, Light-Ring Stability for Ultracompact Objects, Phys. Rev. Lett.119 (2017)251102.[28] S. Hod, On the number of light rings in curved spacetimes of ultra-compact objects, Phys. Lett. B 776, 1 (2018).[29] S. Hod,Upper bound on the gravitational masses of stable spatially regular charged compact objects, Phys. Rev.D 98, 064014 (2018).[30] Yan Peng, On instabilities of scalar hairy regular compact reflecting stars,JHEP 1810(2018)185.[31] Bahram Mashhoon, Stability of charged rotating black holes in the eikonal approximation, Phys. Rev. D31(1985)no.2,290-293.[32] S. Hod,Universal Bound on Dynamical Relaxation Times and Black-Hole Quasinormal Ringing,Phys. Rev. D75(2007)064013.[33] S. Hod, Black-hole quasinormal resonances: Wave analysis versus a geometric-optics approximation, Phys. Rev.D 80(2009)064004.[34] Sam R. Dolan, The Quasinormal Mode Spectrum of a Kerr Black Hole in the Eikonal Limit, Phys. Rev. D82(2010)104003.[35] Yves Decanini, Antoine Folacci, Bernard Raffaelli, Unstable circular null geodesics of static spherically symmetricblack holes, Regge poles and quasinormal frequencies, Phys. Rev. D 81(2010)104039.[36] Yves Decanini, Antoine Folacci, Bernard Raffaelli, Resonance and absorption spectra of the Schwarzschild blackhole for massive scalar perturbations: a complex angular momentum analysis, Phys. Rev. D 84(2011)084035.[37] S. Hod, Resonance spectrum of near-extremal Kerr black holes in the eikonal limit, Phys. Lett. B 715(2012)348-351.[38] Huan Yang, David A. Nichols, Fan Zhang, Aaron Zimmerman, Zhongyang Zhang, Yanbei Chen, Quasinormal-ez, H. Quevedo, and D. Sudarsky,Black Holes Have No Short Hair, Phys. Rev. Lett. 76, 571(1996).[16] Shahar Hod,A no-short scalar hair theorem for rotating Kerr black holes, Class.Quant.Grav. 33(2016)114001.[17] S. Hod,Hairy Black Holes and Null Circular Geodesics, Phys. Rev. D 84, 124030 (2011).[18] Yun Soo Myung, Taeyoon Moon,Hairy mass bound in the Einstein-Born-Infeld black hole, Phys. Rev. D 86.084047.[19] Yan Peng, Hair mass bound in the black hole with nonzero cosmological constants, Phys. Rev. D 98(2018)104041.[20] Yan Peng, Hair distributions in noncommutative Einstein-Born-Infeld black holes,arXiv:1808.07988.[21] S. Hod, The fastest way to circle a black hole, Physical Review D 84, 104024 (2011).[22] S. Hod, Fermat’s principle in black-hole spacetimes, Int.J.Mod.Phys. D27 (2018) no.14, 1847025.[23] Yan Peng,The shortest orbital period in scalar hairy kerr black holes,arXiv:1901.02601 [gr-qc].[24] J. Keir, Slowly decaying waves on spherically symmetric spacetimes and ultracompact neutron stars, ClassicalQuantum Gravity 33, 135009 (2016).[25] V. Cardoso, A.S. Miranda, E. Berti, H. Witek, and V.T. Zanchin, Geodesic stability, Lyapunov exponents andquasinormal modes, Phys. Rev. D 79, 064016 (2009).[26] S. Hod, Upper bound on the radii of black-hole photonspheres, Phys. Lett. B 727, 345 (2013).[27] P. V. P. Cunha, E. Berti, and C. A. R. Herdeiro, Light-Ring Stability for Ultracompact Objects, Phys. Rev. Lett.119 (2017)251102.[28] S. Hod, On the number of light rings in curved spacetimes of ultra-compact objects, Phys. Lett. B 776, 1 (2018).[29] S. Hod,Upper bound on the gravitational masses of stable spatially regular charged compact objects, Phys. Rev.D 98, 064014 (2018).[30] Yan Peng, On instabilities of scalar hairy regular compact reflecting stars,JHEP 1810(2018)185.[31] Bahram Mashhoon, Stability of charged rotating black holes in the eikonal approximation, Phys. Rev. D31(1985)no.2,290-293.[32] S. Hod,Universal Bound on Dynamical Relaxation Times and Black-Hole Quasinormal Ringing,Phys. Rev. D75(2007)064013.[33] S. Hod, Black-hole quasinormal resonances: Wave analysis versus a geometric-optics approximation, Phys. Rev.D 80(2009)064004.[34] Sam R. Dolan, The Quasinormal Mode Spectrum of a Kerr Black Hole in the Eikonal Limit, Phys. Rev. D82(2010)104003.[35] Yves Decanini, Antoine Folacci, Bernard Raffaelli, Unstable circular null geodesics of static spherically symmetricblack holes, Regge poles and quasinormal frequencies, Phys. Rev. D 81(2010)104039.[36] Yves Decanini, Antoine Folacci, Bernard Raffaelli, Resonance and absorption spectra of the Schwarzschild blackhole for massive scalar perturbations: a complex angular momentum analysis, Phys. Rev. D 84(2011)084035.[37] S. Hod, Resonance spectrum of near-extremal Kerr black holes in the eikonal limit, Phys. Lett. B 715(2012)348-351.[38] Huan Yang, David A. Nichols, Fan Zhang, Aaron Zimmerman, Zhongyang Zhang, Yanbei Chen, Quasinormal-