The existence of null circular geodesics outside extremal spherically symmetric asymptotically flat hairy black holes

The existence of null circular geodesics in the background of non-extremal spherically symmetric asymptotically flat black holes has been proved in previous works. An interesting question that remains, however, is whether extremal black holes possess null circular geodesics outside horizons. In the present paper, we focus on the extremal spherically symmetric asymptotically flat hairy black holes. We show the existence of the fastest circular trajectory around an extremal black hole. As the fastest trajectory corresponds to the position of null circular geodesics, we prove that null circular geodesics exist outside extremal spherically symmetric asymptotically flat hairy black holes.


I. INTRODUCTION
It is usually believed that highly curved black hole spacetimes may be generally characterized by the existence of null circular geodesics outside horizons [1,2].The null circular geodesics are the way that massless particles can circle the central black holes.It provides valuable information on the structure of the spacetime geometry, which is closely related to various remarkable properties of black hole spacetimes and has attracted a lot of attentions [3][4][5].
The existence of null circular geodesics can be predicted by analyzing the non-linearly coupled Einsteinmater field equations.In non-extremal hairy black-hole spacetimes, the existence of null circular geodesics has been proved in the asymptotically flat background [40].And null circular geodesics also exist in the stationary axi-symmetric non-extremal black hole spacetimes [41].Along this line, Hod raised a physically interesting question that whether null circular geodesics can also exist outside extremal black holes [42].Hod found that the Einstein-matter field equations seem to fail to provide a proof for the existence of null circular geodesics in extremal black hole spacetimes.So it is interesting to further examine whether null circular geodesics exist outside extremal black holes.
In this work, we firstly introduce the extremal hairy black hole spacetimes.We prove the existence of null circular geodesics by analyzing the existence of the circular trajectory around central extremal hairy black holes.At last, we summarize our main results in the conclusion section.

II. THE EXISTENCE OF NULL CIRCULAR GEODESICS IN EXTREMAL BLACK HOLES
We study physical and mathematical properties of null circular geodesics in the background of extremal hairy black hole spacetimes.The spherically symmetric curved line element of four dimensional asymptotically flat hairy black hole is described by [43,44] ds 2 = −e −2δ µdt 2 + µ −1 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ). ( Here δ and µ are metric solutions only depending on the Schwarzschild areal coordinate r.And angular coordinates are θ ∈ [0, π] and φ ∈ [0, 2π].The black hole horizon r H is obtained from µ(r H ) = 0.At the regular horizon, δ is finite.At the infinity, there is µ(∞) = 1 and δ(∞) = 0 in the asymptotically flat spacetimes.We study null circular geodesics in the equatorial plane with θ = π 2 .
In the following, we follow the analysis of [18] to obtain the null circular geodesic equation.The Lagrangian describing the geodesics is given by where the dot represents differentiation with respect to proper time.
The Lagrangian is independent of coordinates t and φ, leading to two constants of motion labeled as E and L. From the Lagrangian (2), we obtain the generalized momenta expressed as [2] The Hamiltonian of the system is H = p t ṫ + p r ṙ + p φ φ − L, which implies The case of null geodesics corresponds to the value δ = 0.
Equations ( 3) and ( 4) yield relations Substituting Eqs. ( 8) into (7), one finds The null circular geodesic equation ṙ2 = ( ṙ2 ) ′ = 0 can be expressed as The null circular geodesics usually correspond to the circular trajectory with the shortest orbital period [39].In the following, we would like to search for the circular trajectory with the shortest orbital period as measured by asymptotic observers.In order to minimize the orbital period for a given radius r, one should move as close as possible to the speed of light.In this case, the orbital period can be obtained from Eq. ( 1) with ds = dr = dθ = 0 and ∆φ = 2π [39,40]: The circular trajectory with the shortest orbital period is characterized by where the solution is the radius of the fast circular trajectory.
The condition (12) yields the fast circular trajectory equation We find that the null circular geodesic equation ( 10) is identical to the fastest circular trajectory equation (13).So the extreme period circle radius equation and the null circular geodesics equation share the same roots.If we can prove the existence of the fastest circular trajectory, the null circular geodesic exists outside black holes.In fact, there are relations T (r H ) = T (∞) = ∞ according to the formula (11).Thus, the function T (r) must possess a minimum at some finite radius r = r extrem .At the radius, the equations ( 10) and (13) hold, which means r = r extrem corresponds to the position of null circular geodesics.In a word, we prove the existence of null circular geodesics outside extremal black holes.

III. CONCLUSIONS
We investigated on the existence of null circular geodesics outside extremal spherically symmetric asymptotically flat hairy black holes.We firstly obtained the null circular geodesic characteristic equations.Then we got the equation of the fastest circular trajectory that particles can circle the central extremal spherically symmetric asymptotically flat hairy black hole.We showed that the null circular geodesic equation is identical to the fastest circular trajectory equation.So the extreme period circular radius equation and the null circular geodesic equation share the same roots.We found that the extreme period circular radius equation possesses solutions.So we proved that the null circular geodesic exists outside extremal spherically symmetric asymptotically flat hairy black holes.It is natural that our proof also holds for non-extremal spherically symmetric asymptotically flat hairy black holes.