# Analytic study of self-gravitating polytropic spheres with light rings

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## Abstract

Ultra-compact objects describe horizonless solutions of the Einstein field equations which, like black-hole spacetimes, possess null circular geodesics (closed light rings). We study *analytically* the physical properties of spherically symmetric ultra-compact isotropic fluid spheres with a polytropic equation of state. It is shown that these spatially regular horizonless spacetimes are generally characterized by two light rings \(\{r^{\text {inner}}_{\gamma },r^{\text {outer}}_{\gamma }\}\) with the property \(\mathcal{C}(r^{\text {inner}}_{\gamma })\le \mathcal{C}(r^{\text {outer}}_{\gamma })\), where \(\mathcal{C}\equiv m(r)/r\) is the dimensionless compactness parameter of the self-gravitating matter configurations. In particular, we prove that, while black-hole spacetimes are characterized by the lower bound \(\mathcal{C}(r^{\text {inner}}_{\gamma })\ge 1/3\), horizonless ultra-compact objects may be characterized by the opposite dimensionless relation \(\mathcal{C}(r^{\text {inner}}_{\gamma })\le 1/4\). Our results provide a simple analytical explanation for the interesting numerical results that have recently presented by Novotný et al. (Phys Rev D 95:043009, 2017).

## 1 Introduction

Curved spacetimes describing highly compact astrophysical objects may be characterized, according to the Einstein field equations, by null circular geodesics (closed light rings) [1, 2, 3] on which photons and gravitons can orbit the central self-gravitating compact object. These null orbits are interesting from both a theoretical and an astrophysical points of view and their physical properties have been studied extensively by physicists and mathematicians during the last five decades (see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] and references therein).

As demonstrated in [4, 5], the optical appearance of a highly compact collapsing star is determined by the physical properties of its null circular geodesic [4, 5]. Likewise, the intriguing phenomenon of strong gravitational lensing by highly compact objects is related to the presence of light rings in the corresponding curved spacetimes [6]. In addition, as explicitly shown in [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], the discrete quasinormal resonant spectra of compact astrophysical objects are related, in the eikonal limit, to the physical properties (the circulation time and the characteristic instability time scale) of the null circular geodesics that characterize the corresponding curved spacetimes^{1} [23, 24].

^{2}

*horizonless*matter configurations which, like black-hole spacetimes, possess light rings, have attracted much attention in recent years as possible exotic alternatives to the canonical black-hole spacetimes [26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. In particular, in a very interesting work, Novotný, Hladík, and Stuchlík [35] (see also [24]) have recently studied numerically the physical properties of spherically symmetric self-gravitating isotropic fluid spheres with a polytropic pressure-density equation of state of the form [36]

*n*is the polytropic index of the fluid system [36]. It is worth emphasizing that the self-gravitating ultra-compact trapping polytropic spheres were first mentioned in [37].

*two*light rings \(\{r^{\text {in}}_{\gamma },r^{\text {out}}_{\gamma }\}\) (see [26, 33] for related discussions) which are characterized by the compactness inequality

*horizonless*self-gravitating polytropic spheres, violates the lower bound (1) which, as explicitly proved in [25], characterizes the innermost light rings of spherically symmetric black-hole spacetimes.

The main goal of the present paper is to study analytically the physical and mathematical properties of the horizonless ultra-compact polytropic matter configurations. In particular, below we shall provide compact *analytical* proofs for the characteristic intriguing relations (3) and (4) that have recently been observed *numerically* in [35] for the spherically symmetric spatially regular isotropic fluid stars.

## 2 Description of the system

^{3}

*r*functional relations [18, 19, 38]

^{4}

*m*(

*r*) of the matter fields contained within a sphere of radius

*r*is given by the simple integral relation [18, 19]

## 3 Null circular geodesics of spherically symmetric curved spacetimes

In the present section we shall follow the analysis presented in [2, 12, 18, 19] in order to determine the radii of the null circular geodesics (closed light rings) which characterize the spherically symmetric self-gravitating ultra-compact objects. We first note that the energy *E* and the angular momentum *L* provide two conserved physical parameters along the null geodesics of the static spacetime (5) [2, 12, 18, 19].

^{5}

^{6}

## 4 An analytical proof of the characteristic relation \(\mathcal{C}(r^{\text {in}}_{\gamma })\le \mathcal{C}(r^{\text {out}}_{\gamma })\) for horizonless isotropic ultra-compact objects

The physical properties of spherically symmetric self-gravitating isotropic ultra-compact objects have recently been studied numerically in the interesting work of Novotný, Hladík, and Stuchlík [35] (see also [24]). Intriguingly, it has been explicitly shown in [35] that the horizonless curved spacetimes of these spatially regular compact matter configurations generally possess two light rings \(\{r^{\text {in}}_{\gamma },r^{\text {out}}_{\gamma }\}\) (see [26, 33] for related studies) which are characterized by the dimensionless compactness relation (3).

*analytical*techniques in order to provide a compact proof for the intriguing property \(\mathcal{C}(r^{\text {in}}_{\gamma })<\mathcal{C}(r^{\text {out}}_{\gamma })\) [see Eqs. (3) and (16)] which characterizes the horizonless isotropic ultra-compact objects. We first point out that, taking cognizance of Eqs. (6), (7), (15), and (19), one finds the simple asymptotic relations

^{7}the function \(\mathcal{R}(r)\) is characterized by the inequality

*P*(

*r*) is a monotonically decreasing function between the two light rings of the horizonless compact object:

## 5 Upper bound on the compactness of the inner light ring of isotropic ultra-compact objects

The characteristic compactness parameter \(\mathcal{C}(r)\equiv m(r)/r\) of the self-gravitating ultra-compact objects can be computed using the numerical procedure described in [35]. Intriguingly, as demonstrated explicitly in [35], the spatially regular horizonless ultra-compact objects may be characterized by inner light rings whose dimensionless compactness parameter \(\mathcal{C}(r^{\text {in}}_{\gamma })\) is well below the lower bound (1) which, as explicitly proved in [25], characterizes the innermost null circular geodesics (light rings) of spherically symmetric asymptotically flat black-hole spacetimes.

*n*, the numerically computed dimensionless compactness parameter \(\mathcal{C}^{\text {numerical}}(r^{\text {in}}_{\gamma })\) of the isotropic ultra-compact objects [35]

^{8}[41]. One finds that \(\mathcal{C}(r^{\text {in}}_{\gamma };n)\) is a monotonically decreasing function of the dimensionless polytropic index

*n*. Interestingly, we find that the numerical results presented in Table 1 are described extremely well by the simple asymptotic formula (see Table 1)

Ultra-compact polytropic fluid spheres with light rings. We present, for various values of the polytropic index *n*, the *numerically* computed [35, 41]\(^{8}\) dimensionless compactness parameter \(\mathcal{C}^{\text {numerical}}(r^{\text {in}}_{\gamma };n)\) of the isotropic matter configurations. We also present the corresponding values of the dimensionless compactness parameter \(\mathcal{C}^{\text {analytical}}(r^{\text {in}}_{\gamma };n)\) as calculated directly from the simple analytical fit (28). One finds a remarkably good agreement between the numerical results [35] and the analytical formula (28). In particular, one deduces from (28) the characteristic asymptotic value \(\mathcal{C}(r^{\text {in}}_{\gamma })\rightarrow 0.2149<1/4\) for \(n\gg 1\)

\(\ \text {Polytropic}\ \) | \(\mathcal{C}^{\text {numerical}}(r^{\text {in}}_{\gamma })\) | \(\mathcal{C}^{\text {analytical}}(r^{\text {in}}_{\gamma })\) |
---|---|---|

\(\ \ \text {index}\ \ n\ \ \) | \(\text {Ref.}\) [35] | \(\text {Eq.}\) (28) |

2.2 | 0.2906 | 0.2877 |

2.4 | 0.2824 | 0.2817 |

2.6 | 0.2767 | 0.2765 |

2.8 | 0.2723 | 0.2721 |

3.0 | 0.2683 | 0.2683 |

3.2 | 0.2649 | 0.2650 |

3.4 | 0.2620 | 0.2620 |

3.6 | 0.2594 | 0.2594 |

3.8 | 0.2570 | 0.2571 |

4.0 | 0.2549 | 0.2550 |

*analytical*explanation for the

*numerically*inferred asymptotic behavior (29) of the dimensionless compactness parameter. In particular, we shall now derive an upper bound on the compactness parameter \(\mathcal{C}(r^{\text {in}}_{\gamma };n)\) of the isotropic ultra-compact objects in the \(n\gg 1\)

^{9}[37] limit of the polytropic index, which corresponds to the limiting pressure-density relation [see Eq. (2)]

^{10}, or equivalently [see Eq. (31)]

^{11}\(^{,}\)

^{12}

*analytically*derived upper bound (35) on the dimensionless compactness parameter is consistent with the asymptotic behavior (29) which stems from the

*numerical*studies [35] of the self-gravitating ultra-compact isotropic fluid configurations. In particular, in this section we have provided an explicit

*analytical*proof to the

*numerically*observed intriguing fact that horizonless ultra-compact objects can violate the lower bound (1) which characterizes spherically symmetric black-hole spacetimes.

^{13}

## 6 Summary

Horizonless spacetimes describing self-gravitating ultra-compact matter configurations with closed light rings (null circular geodesics) have recently attracted much attention as possible spatially regular exotic alternatives to canonical black-hole spacetimes (see [26, 27, 28, 29, 30, 31, 32, 33, 34, 35] and references therein).

In particular, the physical properties of horizonless ultra-compact isotropic fluid spheres with a polytropic equation of state have recently been studied numerically in the physically important work of Novotný, Hladík, and Stuchlík [35]. Interestingly, it has been explicitly shown numerically in [35] that these spherically symmetric spatially regular ultra-compact polytropic matter configurations generally posses *two* closed light rings (see also [26, 33] for related discussions).

*analytical*techniques in order to explore the physical and mathematical properties of the ultra-compact polytropic stars. In particular, it has been explicitly proved that the two light rings of these horizonless matter configurations are characterized by the relation [see Eqs. (16) and (27)]

Finally, it is interesting to emphasize the fact that the analytical results derived in the present paper provide a simple *analytical* explanation for the interesting *numerical* results that have recently presented by Novotný, Hladík, and Stuchlík [35] for the physical properties of the self-gravitating ultra-compact polytropic spheres.

## Footnotes

- 1.
- 2.
We shall use natural units in which \(G=c=1\).

- 3.
Here \((t,r,\theta ,\phi )\) are the familiar Schwarzschild spacetime coordinates.

- 4.
Here a prime \('\) denotes a spatial derivative with respect to the radial coordinate

*r*. - 5.
Here a dot \(^{.}\) denotes a derivative with respect to an affine parameter.

- 6.
- 7.
It is important to note that, as explicitly shown in [33], there are special horizonless matter configurations with degenerate light rings, which are characterized by the relations \(\mathcal{R}(r=r_{\gamma })=\mathcal{R}'(r=r_{\gamma })=0\), that may violate the inequality (24). We shall henceforth consider in this section generic [26, 33] self-gravitating ultra-compact objects which respect the relation (24).

- 8.
The numerical results presented in Table 1 correspond to self-gravitating isotropic fluid spheres which are characterized by the limiting pressure-to-density ratio \(p_{\text {c}}/\rho _{\text {c}}=n/(n+1)\) at their centers [35]. These matter configurations saturate the upper bound \(v_{\text {s}}\le 1\) on the speed of sound which is imposed by causality requirements [41].

- 9.
- 10.
It is important to emphasize that in the present section we consider generic spatially regular field configurations with light rings [that is, we consider the case of matter configurations with non-degenerate light rings as well as the special case of matter configurations with degenerate light rings for which \(\mathcal{R}'(r=r_{\gamma })=0\)].

- 11.
It is worth noting that the upper bound (34) on the physically allowed values of the dimensionless compactness parameter \(\mathcal{C}(r^{\text {in}}_{\gamma };n\gg 1)\) is a monotonically decreasing function of the physical parameter \(k_{\text {p}}\). For example, substituting the opposite limit \(k_{\text {p}}\rightarrow 0^+\) into (34), one obtains the upper bound \(\mathcal{C}(r^{\text {in}}_{\gamma };n\gg 1,k_{\text {p}}\rightarrow 0^+)<1/3\). The numerical results presented in Table 1 correspond to horizonless ultra-compact isotropic fluid spheres which are characterized by the limiting central ratio \(p_{\text {c}}/\rho _{\text {c}}=n/(n+1)\) which is imposed by causality requirements [35, 41]. In particular, in the \(n\gg 1\) regime one finds the limiting behavior \(p_{\text {c}}/\rho _{\text {c}}\rightarrow 1^-\) for the physically acceptable matter configurations, which corresponds to the limiting value \(k_{\text {p}}\rightarrow 1^-\) [see Eq. (30)].

- 12.
- 13.
It is worth emphasizing that the dimensionless lower bound (1) was derived for spherically symmetric hairy black-hole spacetimes with matter fields which are characterized by an energy-momentum tensor with a negative trace [25, 39]. Interestingly, for the isotropic matter fields that we consider in the present paper, this lower bound can be extended to the regime of hairy black-hole spacetimes with generic values of the energy-momentum trace. To see this, we recall that spherically symmetric black-hole spacetimes are characterized by the relation \(\mathcal{R}(r_{\text {H}}\le r\le r^{\text {in}}_{\gamma })\le 0\) [25], where \(r_{\text {H}}\) is the radius of the outer black-hole horizon. Substituting this inequality into (13) and taking cognizance of (11), one finds the characteristic inequality \(P'(r_{\text {H}}\le r\le r^{\text {in}}_{\gamma })\le 0\). This inequality, together with the fact that black-hole spacetimes with regular horizons are characterized by the relation \(p(r_{\text {H}})=-\rho (r_{\text {H}})\le 0\) [38], yield the important inequality \(P(r=r^{\text {in}}_{\gamma })\le 0\). We therefore conclude that hairy black-hole spacetimes with isotropic matter fields and

*generic*values of the energy-momentum trace are characterized by \(\mu (r=r^{\text {in}}_{\gamma })\le 1/3\) [see Eqs. (14) and (15)], or equivalently \(\mathcal{C}(r^{\text {in}}_{\gamma })\ge 1/3\) [see Eqs. (16) and (18)].

## Notes

### Acknowledgements

This research is supported by the Carmel Science Foundation. I would like to thank Yael Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B. Tea for stimulating discussions.

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