Upper bound on the radii of regular ultra-compact star photonspheres

We investigate the photonsphere in the background of regular asymptotically flat compact stars. The analysis includes the general hairy compact star considering the matter fields' backreaction on the metric in various gravity theories. We prove that the photonsphere of the compact star has an upper bound expressed in terms of the ADM mass of the spacetime. In the case of negative isotropic trace, a stronger upper bound can be obtained.


I. INTRODUCTION
General relativity predicts that null bound geodesics may exist outside compact objects, such as black holes and horizonless compact stars [1,2]. The null geodesics can provide valuable information about the structure and geometry of the curved spacetime. In particular, the circular null geodesics on which photon can orbit the central compact object are known as photonspheres. The photonspheres are of great importance from the astrophysical and theoretical aspects [3][4][5][6][7][8][9][10][11][12][13].
On the side astrophysics, it was found that photonspheres play an important role in the optical appearance of a compact star to external observers in the asymptotic region. For example, the strong gravitational lensing phenomenon by black holes is mainly due to the existence of null circular geodesics [14]. On the other side of theoretical studies, the photonsphere is useful in determining the effective length of the hair above the hairy black hole horizon [15][16][17][18][19][20]. And it was also shown that the photonsphere provide the fast way to circle a black hole [21][22][23]. In addition, it was found that stable photonspheres of compact star can trigger nonlinear instabilities to massless field perturbations [24][25][26][27][28][29][30]. And the characteristic resonances of black holes are related to unstable circular null geodesics [3,[31][32][33][34][35][36][37][38].
The horizonless star with photonspheres is usually called regular ultra-compact star. Bounds on the compactness and photonsphere radii were studied. In the case of positive energy-momentum trace, the lower bound on the compactness parameter of horizonless ultra-compact star was studied in [39]. And the discussion was also extended to the regular ultra-compact star with negative energy-momentum trace [40]. In the black hole background, it was shown that the photonsphere radius has an upper bound expressed in terms of the total ADM mass of the spacetime [41]. Along this line, we try to examine whether there are similar bounds on photonspheres of horizonless ultra-compact stars. This paper is planed as follows. In section II, we introduce the regular compact star with photonspheres in the asymptotically flat gravity. In section III, we analytically obtain upper bounds on the radius of regular compact star photonsphere. And the last section is devoted to our main results.

II. THE GRAVITY MODEL OF REGULAR COMPACT STARS
We consider static spherically symmetric horizonless ultra-compact star which possesses null circular geodesics. In Schwarzschild coordinates, the compact star geometry are described by the line element [2,15,17] The solutions χ(r) and g(r) are functions of the radial coordinate r. Regularity of the gravity at the center requires [39,40] g(r → 0) = 1 + O(r 2 ) and The spacetime at the infinity is asymptotically flat, which is characterized by We state that these spherically symmetric stars could be solutions of a perfect fluid coupled to the gravity background. According to Einstein equations G µ ν = 8πT µ ν , the anisotropic energy momentum tensor is T µ ν = Here we define T t t = −ρ, T r r = p and T ϕ ϕ = T φ φ = p T as the energy density, the radial pressure and the tangential pressure respectively [17,42]. And the equations of metric solutions can be expressed as The gravitational mass m(r) within a sphere of radius r is given by the integration And the metric solution can be put in the form [40] g = 1 − 2m(r) r .
According to (6), a finite mass configuration is characterized by [39]

III. UPPER BOUNDS ON RADII OF COMPACT STAR PHOTONSPHERES
In this part, we prove a generic upper bound on the photonsphere of compact stars. We firstly follow the analysis in [2,17,25] to obtain the characteristic equation of the photonsphere in the spherically symmetric compact star background. The conservation equation T µ ν;µ has only one nontrivial component Substituting equations (4) and (5) into (9), we arrive at with T = −ρ + p + 2p T as the trace of the energy momentum tensor.
With the pressure function P (r) = r 2 p, the relation (10) can be transformed into where N = 3g − 1 − 8πr 2 p.
We assume that the matter fields satisfy the dominant energy condition ρ |p|, |p T | 0.
With relations (2), (3) and (13), the radial function N (r) satisfies In the spherically symmetric spacetime, the photonsphere is characterized by where V r is the effective radial potential that governs the null trajectories in the form Here E is the conserved energy and L is the conserved angular momentum in accordance with the independence of the metric (1) on both t and φ.
We point out that spatially regular horizonless spacetimes usually possess an even number of photonspheres and the degenerate case of N ′ (r = r γ ) = 0 may be characterized by odd number of photonspheres [27,28].
So a stronger upper bound can be obtained for T < 0 as We mention that the spherically symmetric asymptotically flat black hole photonsphere has an upper bound r γ 3M [41]. In the black hole, there is a condition N (r H ) 0 at the horizon r H , which play an important role in the analysis. In this horizonless compact star, we have no such relation and instead there is N (0) = 2 > 0 at the center.For this reason, we cannot simply follow the analysis of black hole photonsphere in [41] to obtain the bound on the regular star photonsphere. We believe it is interesting to further search for stronger upper bounds on the compact star photonsphere and examine whether there is regular star photonsphere, which can saturates the bound (23) and (27) [43][44][45][46][47][48][49][50][51]. It is known that one way to construct hairy compact objects is enclosing the compact objects in a box [52][53][54][55][56][57][58][59]. So it is also very interesting to examine the photonsphere radius bound in the confined gravity.

IV. CONCLUSIONS
We studied photonspheres in the background of horizonless asymptotically flat ultra-compact stars. We showed that the radius of the compact star photonsphere is bounded from above by r γ 4M , where r γ is the radius of the photonsphere and M is the total ADM mass of the spacetime. In the case of negative isotropic trace, we obtained a stronger upper bound in the form r γ