No long hair behaviors of ultra-compact objects

We investigate distributions of matter fields outside spherically symmetric ultra-compact objects with null circular orbits in the asymptotically flat background. Based on the dominant energy and the non-negative trace conditions, we prove the existence of effective radius of matter field hairs outside horizonless ultra-compact objects. We also analytically find a no long hair behavior, which states that the effective radius of matter field hairs cannot extend beyond the outermost null circular orbit of horizonless ultra-compact objects.


I. INTRODUCTION
According to the classical general relativity, a characteristic property of highly curved spacetimes is the existence of null circular orbits outside ultra-compact objects, such as black holes and horizonless ultracompact stars [1,2].The null circular orbits provide valuable information on the properties of the spacetime, such as the gravitational lensing, the black hole shadow and the gravitational waves.Hence, the null circular orbits have attracted a lot of attentions from physicists and mathematicians [3]- [22].
The uniqueness theorem states that black holes are completely characterized by only three parameters: mass, electric charge and angular momentum, which are all conserved quantities defined at the infinity associated with matter fields subject to a Gauss law [23][24][25].In contrast, other exterior matter fields not related to a Gauss law are usually called black hole hairs.The null circular orbit divides matter field hairs into two parts: hairs below the orbit and hairs above the orbit.It is supposed that fields below the orbit tending to fall into the horizon and fields above the orbit tending to radiate away to infinity.The authors in [26] have proposed a physical picture that self-interaction between these two parts bind together fields leading to the existence of black hole matter field hairs.So the null circular orbits may be useful in describing the distributions of matter field hairs outside black holes.In accordance with this picture, the authors in [26,27] proposed a no short hair theorem that the effective radius of matter field hairs must extend beyond the innermost null circular orbit of spherically symmetric hairy black holes.In the horizonless ultra-compact star background, such lower bound on the effective radius of hairs also exists on the non-negative trace condition [28].Interestingly, it was also found that over half of the matter field hair gravitational mass is contained above the null circular orbit of spherically symmetric hairy black holes [27,[29][30][31].As a further step of lower bound analysis in [26][27][28], it is meaningful to examine whether there is any upper bound on the effective radius of matter field hairs.
This paper is to study exterior matter field hairs outside ultra-compact objects.The null circular orbits are useful in describing the distributions of hairs.We analytically find a no long hair behavior that the effective radius of matter field hairs must not extend beyond the outermost null circular orbit.

II. INVESTIGATIONS ON EFFECTIVE RADIUS OF MATTER FIELD HAIRS
We consider the gravity system with spherically symmetric ultra-compact objects surrounded by exterior matter field hairs.The curved spacetime is characterized by the line element [26,27] where χ(r) and g(r) are metric functions depending on the radial coordinate r.In the asymptotically flat background, the metric functions satisfy χ(r → ∞) = 0 and g(r → ∞) = 1 at the infinity.
The Einstein equations G µ ν = 8πT µ ν yield the differential equations where ρ, p and p τ are interpreted as the energy density ρ = −T t t , the radial pressure p = T r r and the tangential pressure p τ = T θ θ = T φ φ respectively [27].In this work, we take the dominant energy condition that the pressures are bounded by the non-negative energy density as [26,27] In this paper, we only focus on the region above the outermost null circular orbit and it was proved that exterior null circular orbit outside black hole horizons always exists [32][33][34][35][36].So we can take the condition (4) without contradiction with the fact that there is ρ = −p on black hole horizons [26,27].
We also take the non-negative trace condition [37-40] An example satisfying the relation ( 5) is the gravitating Einstein-Yang-Mills solitons with T = 0 [41].We point out that the non-positive trace condition T 0 is usually imposed in the curved spacetimes [26,27].
The metric function g(r) can be putted in the form [38] where m(r) is the gravitational mass contained within a sphere of radial radius r.
With the equation ( 2), one can further express the mass term by the integral relation In the following, we deduce the null circular orbit equations [27].The energy and angular momentum are conserved on the geodesic trajectories since the metric is independent of t and φ.And the null circular orbit equations can be obtained from the effective potential along with the characteristic relations where E and L are conserved energy and conserved angular momentum respectively.
Substituting equations ( 2) and ( 3) into ( 8) and ( 9), we get the null circular orbit equation where the discrete roots r γ satisfying N (r γ ) = 0 are the radii of the null circular orbits.
From the expression (7), the finiteness of the gravitational mass implies Since p is bounded by ρ, we obtain the asymptotical infinity behavior According to (12), the null circular orbit characteristic equation asymptotically behaves We define r out γ as the outermost null circular orbit radius, which corresponds to the largest positive root of N (r) = 0. Above the outermost null circular orbit, N (r) is non-negative as The conservation equation T µ ν;µ = 0 has only one nontrivial component Substituting equations ( 2) and ( 3) into (15), we obtain the pressure equation where T = −ρ + p + 2p τ is the trace of the energy momentum tensor.
We introduce a new pressure function P (r) = r 4 p, which is proved to be useful in describing the distributions of matter field hairs.Then the equation ( 16) can be expressed as with N = 3g − 1 − 8πr 2 p.
From ( 4), ( 5), ( 14) and ( 17), P (r) is an increasing function above outermost null circular orbits satisfying We take the condition that the energy density ρ approaches zero faster than r −4 which is imposed to make sure that there are no extra conserved charges besides the ADM mass (the energy density of Einstein-Maxwell fields associated with electric charges decreases as the speed of r −4 at the infinity) defined at the infinity associated with matter fields in the spherically symmetric spacetimes [26,27].Also considering the density dominant condition (4), the effective pressure satisfies the asymptotical infinity behavior It was found that the effective pressure |P (r)| must have a local maximum value at some extremum point r 0 [26,27].We can define r m = r 0 as the effective radii of matter fields.According to ( 18) and ( 19), P (r) is an increasing function of r above the outermost null circular orbit and approaches zero at the infinity.So the effective radii r m = r 0 must have an upper bound

III. CONCLUSIONS
We investigated distributions of matter fields outside spherically symmetric ultra-compact objects.We assumed the dominant energy condition and the non-negative trace condition.We defined an effective matter field radius at an extremum point, where the pressure function |P (r)| possesses a local maximum value.Using analytical methods, we obtained an upper bound on the effective matter field radius expressed as r m r out γ with r m as the effective matter field radius and r out γ corresponding to the outermost null circular orbit radius.
So we found a no long hair behavior that the effective matter field radius must not extend beyond the outermost null circular orbit.