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

TL;DR

This paper establishes an upper bound on the radii of photonspheres in regular asymptotically flat compact stars, considering various gravity theories and matter field effects, with implications for understanding ultra-compact objects.

Contribution

It provides a general proof of an upper bound on photonsphere radii for regular compact stars, including cases with matter backreaction and different gravity theories.

Findings

Photonsphere radius has an upper bound related to ADM mass.

Stronger bounds are derived for stars with negative isotropic trace.

Results apply to a broad class of regular ultra-compact stars.

Abstract

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.

Equations

$$d{s}^2$$

$$g(r\to 0)=1+O(r^2)and\chi (0)<\infty .$$

$$g(r\to \infty )=1and\chi (r\to \infty )=0.$$

$$g^{\prime }=-8\pi r\rho +\frac{1-g}{r},$$

$${\chi }^{\prime }=\frac{-4\pi r(\rho +p)}{g},$$

$$m(r)={\int }_0^{r}4\pi r^{\prime 2}\rho (r^{\prime })d{r}^{\prime }.$$

$$g=1-\frac{2m(r)}{r}.$$

$$r^3\rho (r)\to 0asr\to \infty .$$

$$T_{r;\mu }^{\mu }=0.$$

$$p^{\prime }(r)=\frac{1}{2rg}[(3g-1-8\pi r^{2}p)(\rho +p)+2gT-8gp]$$

$$P^{\prime }(r)=\frac{r}{2g}[N(\rho +p)+2gT-4gp],$$

$$\rho ⩾|p|,|p_T|⩾0.$$

$$P(r\to 0)\to 0andrP(r\to \infty )\to 0.$$

$$N(r=0)=2andN(r\to \infty )\to 2.$$

$$V_r=E^{2}and{V}_r^{\prime }=0,$$

$$V_r=(1-e^{2\chi })E^2+g\frac{L^2}{r^2}.$$

$$N(r_{\gamma })=3g(r_{\gamma })-1-8\pi (r_{\gamma }{)}^{2}p=0.$$

$$N^{\prime }(r=r_{\gamma }^{out})⩾0.$$

$$N^{\prime }(r=r_{\gamma })=\frac{2}{r_{\gamma }}[1-8\pi r_{\gamma }^2(\rho +p)].$$

$$8\pi (r_{\gamma }^{out}{)}^2(\rho +p)⩽1$$

$$g(r_{\gamma }^{out})=\frac{1+8\pi (r_{\gamma }^{out}{)}^{2}p}{3}=\frac{1}{3}+\frac{8\pi (r_{\gamma }^{out}{)}^{2}p}{3}⩽\frac{1}{3}+\frac{4\pi (r_{\gamma }^{out}{)}^2(\rho +p)}{3}⩽\frac{1}{2}+\frac{1}{6}=\frac{1}{2}.$$

$$\frac{2m(r_{\gamma }^{out})}{r_{\gamma }^{out}}⩾\frac{1}{2}.$$

$$r_{\gamma }⩽r_{\gamma }^{out}⩽4m(r_{\gamma }^{out})=4{\int }_0^{r_{\gamma }^{out}}4\pi r^{\prime 2}\rho (r^{\prime })d{r}^{\prime }⩽4{\int }_0^{\infty }4\pi r^{\prime 2}\rho (r^{\prime })d{r}^{\prime }=4M.$$

$$T=-\rho +3p<0or\rho >3p.$$

$$g(r_{\gamma }^{out})=\frac{1+8\pi (r_{\gamma }^{out}{)}^{2}p}{3}⩽\frac{1+2\pi (r_{\gamma }^{out}{)}^2(\rho +p)}{3}⩽\frac{1+\frac{1}{4}}{4}=\frac{5}{12}.$$

$$\frac{2m(r_{\gamma }^{out})}{r_{\gamma }^{out}}⩾\frac{7}{12}.$$

$$r_{\gamma }⩽r_{\gamma }^{out}⩽\frac{24}{7}m(r_{\gamma }^{out})⩽\frac{24M}{7}.$$

Full text

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 c1 ; c2 . 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 ad1 ; ad2 ; ad3 ; ad4 ; ad5 ; c3 ; c4 ; c5 ; z1 ; z2 ; z3 .

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 c6 . 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 s1 ; s2 ; s3 ; s4 ; s5 ; s6 . And it was also shown that the photonsphere provide the fast way to circle a black hole s7 ; s8 ; s9 . In addition, it was found that stable photonspheres of compact star can trigger nonlinear instabilities to massless field perturbations us1 ; us2 ; us3 ; us4 ; us5 ; us6 ; us7 . And the characteristic resonances of black holes are related to unstable circular null geodesics ad1 ; r1 ; r2 ; r3 ; r4 ; r5 ; r6 ; r7 ; r8 .

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 b1 . And the discussion was also extended to the regular ultra-compact star with negative energy-momentum trace b2 . 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 b3 . 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 c2 ; s1 ; s3

[TABLE]

The solutions $\chi(r)$ and $g(r)$ are functions of the radial coordinate r. Regularity of the gravity at the center requires b1 ; b2

[TABLE]

The spacetime at the infinity is asymptotically flat, which is characterized by

[TABLE]

We state that these spherically symmetric stars could be solutions of a perfect fluid coupled to the gravity background. According to Einstein equations $G^{\mu}_{\nu}=8\pi T^{\mu}_{\nu}$, the anisotropic energy momentum tensor is $T^{\mu}_{\nu}=diag\{\frac{1-g-rg^{\prime}}{8\pi r^{2}},~{}\frac{-1+g+rg^{\prime}-2rg\chi^{\prime}}{8\pi r^{2}},~{}-\frac{2g(\chi^{\prime}-r\chi^{\prime 2}+r\chi^{\prime\prime})+g^{\prime}(3r\chi^{\prime}-2)-rg^{\prime\prime}}{16\pi r},~{}-\frac{2g(\chi^{\prime}-r\chi^{\prime 2}+r\chi^{\prime\prime})+g^{\prime}(3r\chi^{\prime}-2)-rg^{\prime\prime}}{16\pi r}\}$. Here we define $T^{t}_{t}=-\rho$, $T^{r}_{r}=p$ and $T^{\varphi}_{\varphi}=T^{\phi}_{\phi}=p_{T}$ as the energy density, the radial pressure and the tangential pressure respectively s3 ; t1 . And the equations of metric solutions can be expressed as

[TABLE]

[TABLE]

The gravitational mass $m(r)$ within a sphere of radius r is given by the integration

[TABLE]

And the metric solution can be put in the form b2

[TABLE]

According to (6), a finite mass configuration is characterized by b1

[TABLE]

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 c2 ; s3 ; us2 to obtain the characteristic equation of the photonsphere in the spherically symmetric compact star background. The conservation equation $T^{\mu}_{\nu;\mu}$ has only one nontrivial component

[TABLE]

Substituting equations (4) and (5) into (9), we arrive at

[TABLE]

with $T=-\rho+p+2p_{T}$ as the trace of the energy momentum tensor.

With the pressure function $P(r)=r^{2}p$, the relation (10) can be transformed into

[TABLE]

where $N=3g-1-8\pi r^{2}p$.

We assume that the matter fields satisfy the dominant energy condition

[TABLE]

According to (8) and (12), the pressure function $P(r)$ has the asymptotical behavior

[TABLE]

With relations (2), (3) and (13), the radial function $N(r)$ satisfies

[TABLE]

In the spherically symmetric spacetime, the photonsphere is characterized by

[TABLE]

where $V_{r}$ is the effective radial potential that governs the null trajectories in the form

[TABLE]

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 $\phi$.

Substituting Einstein equations (4) and (5) into (15) and (16), the photonsphere is determined by the characteristic relation b1

[TABLE]

The roots of (17) correspond to the discrete radii of the null circular geodesics. For the case of a Schwarzschild black hole, there is $g=1-\frac{2M}{r}$, $\chi=0$ and $p=\frac{-1+g+rg^{\prime}-2rg\chi^{\prime}}{8\pi r^{2}}=0$, which yields the familiar $r_{\gamma}=3M$.

We define $r_{\gamma}^{out}$ as the outermost photonsphere of the regular ultra-compact objects. From Eqs. (14) and (17), one deduces that the outermost photonsphere of the spherically symmetric horizonless ultra-compact objects satisfies the relation b1 ; b2 ; us5

[TABLE]

We point out that spatially regular horizonless spacetimes usually possess an even number of photonspheres and the degenerate case of $N^{\prime}(r=r_{\gamma})=0$ may be characterized by odd number of photonspheres us4 ; us5 .

From relations (4), (11) and (17), we get the function b1 ; b2 ; us5

[TABLE]

Putting (19) into (18), we obtain the inequality

[TABLE]

at the outermost photonsphere of the ultra-compact star.

With (12), (17) and (20), we obtains the relations b2 ; b3

[TABLE]

Considering the relations (7) and (21), we have

[TABLE]

So we obtain an upper bound on the radii of photonspheres

[TABLE]

We also consider the matter field configurations with the negative isotropic trace

[TABLE]

According to (17), (20) and (24), we obtain the series of relations b2

[TABLE]

Taking cognizance of relations (7) and (25), we arrive at the inequality

[TABLE]

So a stronger upper bound can be obtained for $T<0$ as

[TABLE]

We mention that the spherically symmetric asymptotically flat black hole photonsphere has an upper bound $r_{\gamma}\leqslant 3M$ b3 . In the black hole, there is a condition $N(r_{H})\leqslant 0$ 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 b3 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) j1 ; j2 ; j3 ; j4 ; j5 ; j6 ; j7 ; j8 ; j9 . It is known that one way to construct hairy compact objects is enclosing the compact objects in a box j10 ; j11 ; j12 ; j13 ; j14 ; j15 ; j16 ; j17 . 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 $r_{\gamma}\leqslant 4M$, where $r_{\gamma}$ 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 $r_{\gamma}\leqslant\frac{24M}{7}$. The analysis in this work can be applied to the general gravity model considering the matter fields’ backreaction on the compact star in various asymptotically flat gravity theories.

Acknowledgements.

We would like to thank the anonymous referee for the constructive suggestions to improve the manuscript. This work was supported by the Shandong Provincial Natural Science Foundation of China under Grant No. ZR2018QA008.