Upper bound on the radius of the innermost photonsphere in the regular compact star spacetime

We study properties of the innermost photonsphere in the regular compact star background. We take the traceless energy-momentum tensor and dominant energy conditions. In the regular compact star background, we analytically obtain an upper bound on the radius of the innermost photonsphere as $r_{\gamma}^{in}\leqslant \frac{12}{5}M$, where $r_{\gamma}^{in}$ is the radius of the innermost photonsphere and $M$ is the total ADM mass of the asymptotically flat compact star spacetime.

An interesting question is how close can the photonspheres be to the central compact objects.For a generic black hole spacetime, Hod analytically derived an upper bound on the radius of the outermost photonsphere with non-positive trace of the energy-momentum tensor T 0. The bound is expressed as r out γ 3M , where r out γ is the radius of the outermost photonsphere and M is the total ADM mass of the black hole spacetime [39].In the background of horizonless ultra-compact stars, it was found that the radius of the photonsphere is bounded from above by r out γ 4M for the dominant energy condition and a stronger upper bound r out γ 24 7 M was obtained for negative isotropic trace T < 0, where r out γ is the radii of outermost photonspheres of compact stars and M is the total ADM mass of the compact star spacetime [39].In the traceless energy-momentum tensor case T = 0, it was proved that the innermost photonsphere cannot be arbitrarily close to the central black hole with a lower bound r in γ 6 5 r H , where r in γ is the radius of the innermost photonsphere and r H is the horizon of the black hole [40].As a further step, it is also interesting to study properties of innermost photonspheres of horizonless compact stars.
In this work, we firstly introduce the gravity system of horizonless compact stars.Then we investigate on properties of innermost photonspheres surrounding compact stars.With analytical methods, we obtain an upper bound on the radii of innermost photonspheres of compact stars.We will summarize our main results in the last section.

II. THE GRAVITY SYSTEM IN THE COMPACT STAR SPACETIME
We are interested in spherically symmetric horizonless compact stars with photonspheres.The curved spacetime is characterized by [2,15,17] where µ(r) and δ(r) are radially dependent metric functions.
At the original center, the regular spacetime satisfies [41,42] At the infinity, the asymptotic flatness requires We define ρ, p and p τ as the energy density ρ = −T t t , radial pressure p = T r r and tangential pressure p τ = T θ θ = T φ φ respectively.The equations of motion for metric functions are We take the dominant energy condition We also assume the traceless energy-momentum tensor condition The gravitational mass contained in a sphere with radius r is given by Without generality, the metric function µ(r) can be putted in the form [42] µ According to (8), the finiteness of the gravitational mass requires [41]

III. UPPER BOUNDS ON RADII OF THE INNERMOST PHOTONSPHERES
In this part, we will obtain an upper bound on radii of innermost photonspheres of horizonless compact stars.We get the characteristic equation of photonspheres following analysis in [2,17,25].The independence of the metric (1) on both t and φ leads to the conserved energy E and conserved angular momentum L. The photonspheres of the spherically symmetric star is described by where V r is the effective radial potential Substituting ( 4) and ( 5) into ( 11) and ( 12), we get the photonsphere equation [41] N The roots of characteristic equation (13) satisfying are the discrete radii of the photonspheres.
According to relations (3), ( 6) and ( 10), the radial function N (r) has the boundary behavior We are interested in compact stars with photonspheres.We define r in γ as the innermost photonsphere of the regular ultra-compact star.From equation (15), one deduces that the innermost photonsphere satisfies the relation [28,41,42] The conservation equation T µ ν;µ = 0 has only one nontrivial component Substituting equations ( 4) and ( 5) into (17), we get the radial pressure equation The relation (18) can be transformed into with N = 3µ − 1 − 8πr 2 p.
With the expression (9), we can transform (23) into We further obtain an upper bound for the radius of the innermost photonsphere as [42] r in IV. CONCLUSIONS We studied innermost photonspheres of horizonless compact stars in the asymptotically flat background.
We assumed the traceless energy-momentum tensor and dominant energy conditions for matter fields.With analytical methods, we obtained an upper bound on the radii of innermost photonspheres as r in

γ 12 5
M , where r in γ is the radii of the innermost photonspheres and M is the total ADM mass of the asymptotically flat compact star spacetime.It means that the innermost photonspheres cannot be too far from the original center of the horizonless compact stars.