Existence and stability of circular orbits in general static and spherically symmetric spacetimes
Junji Jia, Jiawei Liu, Xionghui Liu, Zhongyou Mo, Xiankai Pang, Yaoguang Wang, Nan Yang
aa r X i v : . [ g r- q c ] A p r Existence and stability of circular orbits in generalstatic and spherically symmetric spacetimes
Junji Jia , , Jiawei Liu , , Xionghui Liu , , Zhongyou Mo , ,Xiankai Pang , , Yaoguang Wang , , Nan Yang MOE Key Laboratory of Artificial Micro- and Nano-structures, Wuhan University,Wuhan, 430072, China School of Physics and Technology, Wuhan University, Wuhan, 430072, China Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,China Glyn O. Phillips Hydrocolloid Research Centre, Hubei University of Technology,Wuhan 430068, ChinaE-mail: [email protected]
Abstract.
The existence and stability of circular orbits (CO) in static and sphericallysymmetric (SSS) spacetime are important because of their practical and potentialusefulness. In this paper, using the fixed point method, we first prove a necessary andsufficient condition on the metric function for the existence of timelike COs in SSSspacetimes. After analyzing the asymptotic behavior of the metric, we then show thatasymptotic flat SSS spacetime that corresponds to a negative Newtonian potential atlarge r will always allow the existence of CO. The stability of the CO in a generalSSS spacetime is then studied using the Lyapunov exponent method. Two sufficientconditions on the (in)stability of the COs are obtained. For null geodesics, a sufficientcondition on the metric function for the (in)stability of null CO is also obtained. Wethen illustrate one powerful application of these results by showing that an SU(2)Yang-Mills-Einstein SSS spacetime whose metric function is not known, will allowthe existence of timelike COs. We also used our results to assert the existence and(in)stabilities of a number of known SSS metrics. Keywords
Static spacetime; Spherically symmetric spacetime; Circular orbit; Fixedpoint
Submitted to:
Class. Quantum Grav. xistence and stability of circular orbits in general static and spherically symmetric spacetimes
1. Introduction
Static and spherically symmetric (SSS) spacetimes are the best studied spacetimes inthe general theory of relativity. Birkhoff’s theorem guarantees that any sphericallysymmetric solution of the vacuum field equations is static and asymptotically flat and theexterior solution is always given by the Schwarzschild metric [1, 2, 3]. The generalizationof this theorem to include charge states that any spherically symmetric and electricallycharged solution is stationary and asymptotically flat and can be cast into the Reissner-Nordstr¨om (RN) metric form. Moreover, enormous amount of SSS spacetimes withdifferent energy-momentum distributions, representing real physical system to differentextents have been studied in the literature (see Ref. [5, 4] for a list of these solutions).On the other hand, the study of geodesics of timelike or null test objects in thesespacetimes is also very useful and interesting. Verification of general relativity in its earlydays depended exclusively on the using of timelike or null geodesics, including precessionof the perihelion of Mercury and bending of light ray. In studying the geodesics of theSSS spacetimes, we are often concerned with the geodesics that might be of use thatis more important or practical. Among these are the circular orbits (COs) in SSSspacetimes, for a few reasons. First of all, if the CO is stable, they will permit thepossibility for test objects such as satellites or spacecrafts to fall freely on such orbitswith constant radius but without burning extra fuel or worrying about falling into oraway from the central object. Due to this advantage, it is very desirable to know whethera given SSS spacetime permits any CO, and then adjust one’s own orbit to such COs ifthey do exist. Secondly, COs of some special spacetime might be of great importancefor astrophysics and theoretical research of gravity. For example, the inner-most (ormarginally) stable COs play an important role in the accretion disk theory [7, 6, 8]and consequently closely related to the chaotic motion [9] and behavior of gravitationalwaves originated from the central source such as black hole binaries [10, 11]. Study ofthe COs in black hole spacetimes with electric charge and/or scalar field can also beused to test properties of these black holes such as their no-hair theorem and extremality[13, 12, 14, 15, 16] and even the validity of some modified gravity [17, 18, 19, 20].A CO for an SSS spacetime is mathematically equivalent to a fixed point (FP) of theradical geodesics equation. The radical geodesic equation for the SSS metric, as a secondorder differential equation, can always be casted into a first order differential equationsystem. The FP of this system always contain one equation d r/ d τ = 0 where r is theradius of the SSS metric and τ can be chosen as the proper time or the azimuth angularcoordinate. It is clear then a FP always implies a CO. In the opposite way, for a CO,we always have d r ( τ ) / d τ = 0, d r ( τ ) / d τ = 0 and therefore a FP is also guaranteed.Therefore in this paper we will use the term CO and FP indiscriminately. The studyof FPs, as well as the Lyapunov method that will be used to study their stabilities, aremathematically part of the phase space analysis theory. In the literature, there are a fewworks [10, 23, 21, 24, 22] that applied these methods to study COs, with their emphasison various different properties or applications of these orbits. However all these works xistence and stability of circular orbits in general static and spherically symmetric spacetimes a priori . Rather, the criterions we established for the (non-)existence of FPs can evenbe applied to cases that the metric functions are not completely known. Therefore, ourresults given in the form of a few theorems are not only new but vastly applicable to alarge number of metrics. The power of these results is illustrated using the examples inthe Discussion section.
2. Metric and geodesic equations
In this section we set up our notation for the metric and derive the basic geodesicequations associated with it. We also derive explicitly the FP autonomous equationsystem from the radial geodesic equation. The most general form of the SSS spacetimemetric can be chosen as the followingd s = f ( r )d t − g ( r )d r − r dΩ d − , (1)where ( t, r, θ, φ ) are the coordinates, dΩ d − is the solid angle element of the d − d is the total dimension of the spacetime. Using this metric,the geodesic equations can be routinely derived. Due to the static and sphericalsymmetry of the metric, we can always set in the geodesic equations θ ( τ ) = π/
2, wherewe use τ to denote proper time for timelike geodesics and the affine parameter for nullgeodesics. Moreover, there always exist two first integrals whose constants are identifiedwith the energy E and angular momentum L of the test object per unit massd φ ( τ )d τ = Lr , d t ( τ )d τ = Ef ( r ) . (2)The equation for r ( τ ) then is (cid:20) d r ( τ )d τ (cid:21) = g ( r ) (cid:20) E f ( r ) − L r − δ n (cid:21) ≡ V ( r ) , (3) xistence and stability of circular orbits in general static and spherically symmetric spacetimes δ n = 1 , V ( r ) is defined as theentire right hand side of Eq. (3). Taking derivative of Eq. (3) with respect to τ again,we obtain d r ( τ )d τ = V ′ ( r )2 , (4)where ′ here and henthforce denotes the derivative with respect to r .Eqs. (3) and (4) combined can be thought as an autonomous system of two firstorder differential equations for x ( τ ) = r ( τ ) and y ( τ ) ≡ d r ( τ ) / d τ . Now the FP for thissystem is at a radius, denoted by x = x ∗ , that satisfyingd x ( τ )d τ = y ( τ ) = 0 , (5)d y ( τ )d τ = V ′ ( x ∗ )2 = 0 (6)at some instantaneous time and all times after. Noting (3), this is equivalent to requirethat at x ∗ = r ∗ V ( r ∗ ) = 0 , and V ′ ( r ∗ ) = 0 . (7)For the timelike geodesics, using (3) again these become r ∗ − f ( r ∗ ) r ∗ f ′ ( r ∗ ) − f ( r ∗ ) ! E L ! = r ∗ f ( r ∗ )0 ! . (8)Treating E and L as unknowns, this is a simple linear system. It is easy to check thatwhen and only when 2 f ( r ∗ ) − r ∗ f ′ ( r ∗ ) = 0 (9)the rank of the augmented matrix is larger than that of the coefficient matrix and thenthere exist no solution. Otherwise, these equations can be transformed to E = 2 f ( r ∗ )2 f ( r ∗ ) − r ∗ f ′ ( r ∗ ) , (10) L = r ∗ f ′ ( r ∗ )2 f ( r ∗ ) − r ∗ f ′ ( r ∗ ) . (11)These equations are in agreement with Ref. [21]. The corresponding equations in thenull geodesic case will be consider in section 6.In this paper, we assume that the metric function f ( r ) is positive in the range of r in which we seek the FP, because otherwise according to the metric (1) the r coordinateshould have to be interpreted as time, not radius and a FP in time is not of our interest.We also assume that in this range of r the spacetime is not singular because otherwisewe cannot study the existence of FP on singular radius anyways. And then we knowthat there always exist coordinate transforms to make f ( r ) continuous if it was not inthe first place. Therefore throughout this paper we assume that f ( r ) is already madecontinuous. xistence and stability of circular orbits in general static and spherically symmetric spacetimes
3. Existence of the Fixed Points
For a given f ( r ), if in the space spanned by energy E and angular momentum L thereexist a non-empty set S for whose element ( E, L ) the solution to Eq. (8) does existat some radius r = r ∗ , then we say for that f ( r ) the FPs can exist (for that ( E, L )at r = r ∗ ). If for a given f ( r ), the set S is empty, then we say that there exist no FPfor the spacetime described by f ( r ). Apparently, if FPs exist for some ( E, L ) thenthe two equations in Eq. (8) will both have solutions simultaneously. If either of (10)or (11) is not satisfied by any r, then the FPs do not exist for that (
E, L ). Underthese definitions, it is clear then if f (0) >
0, then there always exists a timelike FPat r ∗ = 0 , E = p f (0) , L = 0. This FP in most physically important cases is notrelavent, either due to the singularity of the r = 0 point, or due to the presence ofmatter at r = 0 which prevents the test particle from doing geodesic motion, or it isjust a trivial FP such as in the case of Minkovski spacetime. Therefore in the followingsections when the FPs of timelike geodesics are discussed, FP at this point is excludedand we concentrate on non-trivial ones. Now let us study Eq. (10) and (11) separatelyfirst and then combine the results to obtain our main conclusion presented as TheoremA. Keeping in mind that E is real, it is clear that as long as the denominator 2 f − rf ′ of Eq. (10) is positive for the given f ( r ) and some r , then there always exist E suchthat Eq. (10) is satisfied. The contrapositive statement is that, no E will satisfy Eq.(10) if and only if for all r the denominator 2 f − rf ′ will remain negative. In this case,and noticing another case (9), we can assume that for all r ,2 f − rf ′ = − δ ( r ) (12)where δ ( r ) is a semi-positive but otherwise arbitrary function. Due to its arbitrarinessand for the sake of a shorter expression for the solution to (12) , we can also change thefunction δ ( r ) to r d κ ( r ) / d r without losing any generality as long as κ ( r ) is monotonicallyincreasing. Then the Eq. (12) becomesd f ( r )d r − fr = r d κ ( r )d r (13)whose solution is readily found to be f ( r ) = ( κ ( r ) + c κ ) r , (14)where c κ is an arbitrary constant. Since the only requirement for κ ( r ) is that it isincreasing, then we can always redefine κ ( r ) such that κ ( r ) + c κ → κ ( r ). Finally, we get f ( r ) = κ ( r ) r . (15)From this and the requirement that f ( r ) >
0, we establish the statement that: if themetric function f ( r ) is expressed as (15) for any positive and monotonically increasing xistence and stability of circular orbits in general static and spherically symmetric spacetimes κ ( r ), then Eq. (10) will have no solution at any r . Some remarks should beemphasized here. First, it is obvious that the opposite is also true: if Eq. (10) is notsatisfied by any r , then f ( r ) must be expressible as (15). Secondly, the form (15) isvalid also locally: in the neighborhood of any r such that the denominator 2 f − rf ′ isnegative, f ( r ) should be expressible as (15), for some locally increasing κ ( r ). Lastly,the contrary is also true: if the κ ( r ) in (15) is locally positive and increasing, then Eq.(10) is not satisfied in the corresponding neighborhood.Now Eq. (11), for a given f ( r ), will not be satisfied by any r if and only if oneof the following two conditions are satisfied: (a) for the r such that the denominator2 f − rf ′ <
0, the factor in the numerator f ′ >
0; (b) for the r such that the denominator2 f − rf ′ > f ′ < f − rf ′ < r such that f ′ >
0. Using (15), one finds f ′ = κ ′ ( r ) r + 2 κ ( r ) r. (16)Since κ ( r ) is an increasing function and positive, we see that f ′ > f ′ < f − rf ′ > f >
0, the only requirement therefore is f ′ < . (17)Combining the above two cases for Eqs. (10) and (11), we know that the FPs ofthe metric will not exist if and only if either in the entire range of r , Eq. (15) is satisfiedso that Eq. (10) is broken, or in some range of r Eq. (15) is satisfied and in the rest ofthe range Eq. (17) is satisfied so that Eq. (11) is broken. In the following we furtherprove a theorem that will remove the possibility that Eq. (15) and (17) are piece-wiselysatisfied:(Theorem A) The FPs for the given metric (1) do not exist if and only if either onlyEq. (15) is satisfied in the entire range of r , or only Eq. (17) is satisfied in the entirerange of r , but not because Eq. (15) and (17) are piece-wisely satisfied.The proof is indeed simple. Let us assume that Eq. (15) is satisfied in some range of r ∈ ( a, b ) and Eq. (17) is satisfied in the range of ( b, c ). Then in ( a, b ) and ( b, c ) wehave respectively f ′ > f ′ <
0, and consequently at r = b , f ′ ( b ) = 0. Now let usshow that the above assumptions are impossible by considering how f ( r ) changes in asmall neighborhood r ∈ ( b − χ, b ]. In this neighborhood according to (15), the function f ( r ) /r = κ ( r ) should have a semi-positive derivative. Evaluate the following derivative xistence and stability of circular orbits in general static and spherically symmetric spacetimes r = b − ǫ ∈ ( b − χ, b ] κ ′ ( r ) | r = b − ǫ = (cid:18) f ( r ) r (cid:19) ′ (cid:12)(cid:12)(cid:12)(cid:12) r = b − ǫ = (cid:20) f ′ r − fr (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) r = b − ǫ (18)= f ′ ( b ) b + (cid:20) f ′′ ( b ) b − f ′ ( b ) b (cid:21) ( − ǫ ) − f ( b ) b = + (cid:20) − f ′ ( b ) b + 6 f ( b ) b (cid:21) ( − ǫ ) + O ( ǫ ) (19)= − f ( b ) b − (cid:20) f ′′ ( b ) b + 6 f ( b ) b (cid:21) ǫ + O ( ǫ ) (20) < , (21)where from (18) to (19) the Taylor expansion to the first order of ǫ is used and in (20) wesubstituted f ′ ( b ) = 0. Clearly to the leading (zeroth) order of ǫ , κ ′ ( r ) | r = b − ǫ <
0, whichconflicts with the requirement that κ ′ ( r ) ≥
0. Similarly to this case, if Eq. (15) and (17)are satisfied respectively in r ∈ ( b, c ) and ( a, b ), the same calculation can also be carriedthrough except the neighborhood we will consider in this case becomes r ∈ [ b, b + χ ) and − ǫ → ǫ . We will still have a confliction on the monotonicity of κ ( r ). Therefore only oneof (15) and (17) can be satisfied on the whole range of r but not both in a piece-wiseway.This theorem greatly simplifies the scenarios that we need to consider when findingmetrics that do not allow FPs and we will refer to it as the timelike FP non-existencetheorem (TFPNET).In the following, we give two simple examples satisfying respectively Eq. (15) and(17) in the entire range of r and therefore do not permit any FPs, and one more examplethat satisfies (15) in some range of r and (17) in some other range and neither of themin the rest where FPs lies. The first example is given simply by κ ( r ) = r so that f ( r ) = r · r = r . The resulting Eq. (10) and (11) become E = − r , L = − r . (22)Both the two equations permit no solution and therefore no FPs. The second exampleis given by f ( r ) = 1 /r and the Eq. (10) and (11) become E = 23 r , L = − r . (23)The second equation eliminates the existence of FPs. The third example is a simplecombination of the above two: f ( r ) = r r · r . (24)At very small and large r , this resembles the first and second examples respectively andtherefore we expect that the FPs will not exist there. However, in the middle range, as xistence and stability of circular orbits in general static and spherically symmetric spacetimes E = 2 r r − , L = − ( r − r r − . (25)Clearly, these two equations will have solution when r ∈ (3 − / , / ) for some ( E, L ).
4. Existence of fixed points for asymptotically flat spacetime
Even though the previous results shows that there exist metric that has no FPs forany (
E, L ), the required f ( r ) in these cases are non-trivial. Usually people are moreinterested in asymptotically flat spacetimes due to their physical relevance. Thereforein this section we discuss under the asymptotic flatness requirement about what willhappen to the TFPNET theorem we obtained in last section.For the result (15) since κ ( r ) is positive and increasing, apparently f ( r ) will divergeat r → ∞ and therefore do not correspond to any asymptotically flat spacetime. Thecase (17) however do permit asymptotically flat spacetime. An example satisfying (17)would be f ( r ) = c + 2 ar β , (26)where c > , a > , β >
0. It is known [25] that a necessary and sufficient condition fora spacetime to be asymptotically flat is that its metric takes the above form with β ≥ r → ∞ . On the other hand, the Eqs. (10) and (11) corresponding to (26) become E = r − β ( r β + 2 a ) cr β + a (2 + β ) , L = − aβr cr β + a (2 + β ) , (27)which clearly exclude the existence of any FPs. Therefore, enforcing the asymptoticflatness condition alone would not make sure that all SSS spacetimes permit FPs.On the other hand, it is well known in perturbation theory of general relativity thatfor an SSS spacetime, at the region of r → ∞ an effective Newtonian potential φ ( r ) canalways be assigned to the g component of the metric by [26]1 + 2 φ ( r ) = lim r →∞ g ( r ) = lim r →∞ f ( r ) . (28)The potential corresponding to example (26) would be φ ( r ) = ar β . (29)For a >
0, this is a positive gravity potential that can only be generated by negative massand it would produce a repelling force to normal test objects, resulting naturally no CO.Inspired by this observation, we can further exclude metric like this by imposing thatthe corresponding Newtonian potential at large r is negative and obtain the followingtheorem:(Theorem B) Any static, spherically symmetric and asymptotically flat spacetimecorresponding at large radius to a negative Newtonian potential would allow theexistence of FP in the space of ( E, L, r ). xistence and stability of circular orbits in general static and spherically symmetric spacetimes
5. Stability of the fixed points
For asymptotically flat, SSS spacetime with negative Newtonian potential at largeradius, then the above theorem B clearly assert that there will exist FP in the spacespanned by (
E, L, r ). For the spacetimes considered in this section, we assume that thisis the case and denote the FP radius as r ∗ . Then one important question following is thestability of the FPs. Here we address this issue using the Lyapunov exponent method byfollowing closely Ref. [21]. In the simple SSS spacetime case, the Lyapunov exponentsare proportional by a positive factor to the eigenvalues of the coefficient matrix of thelinear perturbation equation of the autonomous system of Eqs. (5) and (6).First of all, notice that Eqs. (10) and (11) can be considered as algebraicalconstraints for variables appearing in them but not differential equations of f ( r ) asa function of r , simply because that these two equations are only satisfied at the FP r ∗ but not other radius values. One sees then for any given f ( r ), among the threeinputs { E, L, r } into these two equations, generally once any one of them is fixed, theother two will be solvable from these two constraints, if the solution does exist. For thesimplicity of the argument, let us choose E as the free variable, i.e., once a proper E that allows the existence of the FP is chosen, these two constraints will force L and r tobe at some particular values L ∗ and r ∗ respectively. Alternatively, Eqs. (10) and (11)can also be thought as relations that f ( r ∗ ) and f ′ ( r ∗ ) satisfy in terms of E, L ∗ and r ∗ .In the following, we will use both these two interpretations.For the metric (1), the Lyapunov exponents have been computed in Ref. [21] andhere we directly quote as λ = ± s V ′′ ( r )d t ( τ ) / d τ . (30)Substituting Eqs. (2) and (3) and using Eqs. (10) and (11), one finds for the timelikecase λ = ± s − g f (cid:20) f · f ′ r ∗ − f ′ ) + f f ′′ (cid:21) ≡ ± p g · h ( r ∗ , f, f ′ , f ′′ ) / f, f ′ , f ′′ are evaluated at r ∗ and h is defined as h ( r ∗ , f, f ′ , f ′′ ) = − f ′′ + 2( f ′ ) f − f ′ r ∗ . (32)It is known that if λ is imaginary (or real) then the FP is stable (or unstable). In orderto obtain some useful conditions regarding the stability of the FP, we can study theconditions under which either h ( r ∗ ) < h ( r ∗ ) > . (34) xistence and stability of circular orbits in general static and spherically symmetric spacetimes g > r , the FP will be stable (or unstable) if and only if h ( r ∗ ) < h ( r ∗ ) >
0) holds true. Unfortunately, the sign of h cannot be completely fixed byjust using the FP constraints (10) and (11) due to the presence of f ′′ term, which doesnot appear in Eqs. (10) and (11) at all. Indeed, solving f and f ′ from (10) and (11)and substituting into (32), one finds h = − f ′′ ( r ∗ ) + 2( L ∗ − r ∗ ) E L ∗ ( L ∗ + r ∗ ) , (35)from which f ′′ cannot be removed.Even though we cannot find the necessary and sufficient condition on the stabilityor instability of the FP, if we think of h to be positive or negative not only for theFP radius r = r ∗ but for all r , then Eqs. (33) or (34) can be treated as differentialequations. This way we are able to get the following sufficient condition:(Theorem C) If the differential equation − f ′′ + 2( f ′ ) f − f ′ r ≡ σ ( r ) < >
0) (36)for all r , then the FP, if exists, would be stable (or unstable).In what follows we analyze only the case for σ ( r ) < σ ( r ) > f ( r ) and its solution will always make the Lyapunov exponent imaginary andconsequently the FP stable since σ ( r ) < r ∗ . It is also clearthat if σ ( r ) is positive for all r , then the opposite happens: the FP will always beunstable.Eq. (36) though looks simple, has proven to be difficult to get an analytical solution.This is mainly because of the presence of σ ( r ) as an arbitrary and non-homogenous term.This arbitrariness of σ ( r ) however can also be taken advantage of. Without losing anygenerality, we can transform σ ( r ) to terms proportional to the second or third term onthe left hand side of Eq. (36) so that the equation becomes homogenous and can besolved exactly. Each of the solved explicit form of f ( r ) will be a sufficient condition forthe (in)stability of the FP. For the first transform from σ ( r ) to a term proportional to( f ′ ) /f , noticing that at the FP, f ′ ( r ∗ ) > f ( r ∗ ) >
0, we can rewrite σ ( r ) so that (36)becomes − f ′′ + 2( f ′ ) f − f ′ r = r ( f ′ ) ζ ′ f ( r ) , (37)where if ζ ′ ( r ) <
0, the FP will be stable. Notice that in this transform, we made surethe last term has the same sign as the σ ( r ) term. This equation can be easily solved toobtain the solution f ( r ) = c exp (cid:18)Z r [2 r ( ζ ( r ) − c ) + 1] d r (cid:19) . (38)Since ζ ( r ) is only required to be monotonic, then the c can be absorbed into ζ ( r ). Weget f ( r ) = c exp (cid:18)Z r [2 r ζ ( r ) + 1] d r (cid:19) . (39) xistence and stability of circular orbits in general static and spherically symmetric spacetimes − f ′′ + 2( f ′ ) f − f ′ r = − f ′ d ln [ r χ ′ ( r )]d r , (40)and we require that r χ ′ ( r ) is a positive function with a positive derivative but otherwisearbitrary function. This will grantee that σ ( r ) will be negative for all radius andtherefore the FP is stable. Moreover, the equation also becomes solvable and we obtain f ( r ) = 1 c χ ( r ) + c , (41)where c = 0 and c are constants. If f ( r ) can be written as (39) in which ζ ′ ( r ) <
0, oras (41) in which r χ ′ ( r ) is positive and increasing, then clearly the FPs will always bestable. On the contrary, if ζ ( r ) in (39) is increasing or r χ ′ ( r ) is decreasing, then thecorresponding σ ( r ) in (36) will be positive and the FPs will be unstable.It is worth emphasizing that the above two f ( r )’s are just some sufficient conditionsobtained using very special form of σ ( r ). It is very possible that there exist other σ ( r ) and consequently f ( r ) which will make the FP stable or unstable too. Moreover,Eq. (36) itself is also the consequence of a quite strong requirement that at all radius h ( r ∗ , f, f ′ , f ′′ ) has a fixed sign. It is not hard to come up metrics that will only make h negative (or positive) locally, i.e., near the FP but not entire r , which then can makethe FP stable (or unstable).
6. Fixed points of null geodesics
For null geodesics, using Eq. (7) the existence of FPs would require [21] EL = ± s f ( r ∗ ) r ∗ , (42)2 f ( r ∗ ) = r ∗ f ′ ( r ∗ ) . (43)An inspection of these two equations and (10) and (11) indicate that the null case isthe limit of 2 f ( r ∗ ) → r ∗ f ′ ( r ∗ ) for the timelike case.Clearly, there always exist ( E, L, r ) satisfies Eq. (42). For Eq. (43), the existenceof FP suggest that there exist some r ∗ that will satisfy this equation. Therefore a non-existence condition of FP would be that there exist no r ∗ satisfying Eq. (43). In otherwords, 2 f ( r ) − rf ′ ( r ) = − δ ( r ) < r where δ ( r ) is a positive but otherwise arbitrary function or2 f ( r ) − rf ′ ( r ) = ρ ( r ) > r where ρ ( r ) is an positive but otherwise arbitrary function. Eq. (44) is similarto Eq. (12) and its solution is f ( r ) = µ ( r ) r (46) xistence and stability of circular orbits in general static and spherically symmetric spacetimes µ ( r ) is a positive function with a positive derivative. Eq. (45) differs from Eq.(44) only by a sign and therefore the solution can be similarly solved as f ( r ) = ξ ( r ) r (47)where ξ ( r ) is a positive function with a negative derivative. Therefore we have thefollowing theorem:(Theorem D) The FP for null geodesics does not exist if and only if f ( r ) takes the formof (46) or (47).Comparing with the TFPNET, Eq. (17) implies that its f ( r ) /r , the functioncorresponding ξ ( r ) in (47), will always have a semi-negative derivative. This meansthat for the ξ ( r )’s that have a strict negative derivative everywhere, there will existneither timelike nor null FP. While for the ξ ( r ) that might has zero derivative at someradius, there is still no timelike FP but there can exist null FP. An example can beconstructed as κ ( r ) = a + ( r − a ) , f ( r ) = κ ( r ) r = [ a + ( r − a ) ] r , ( a >
0) (48)which has κ ′ ( r ) = 3( r − a ) ≥ r = a , 2 f − rf ′ = 0 and therefore there exist a null FP. On theother hand, one can also give examples of metrics that allows timelike FP but no nullFPs, such as ξ ( r ) = (cid:18) e − r + 1 r (cid:19) / , (49)which leads to f ( r ) = ξ ( r ) r = (1 + r e − r + r ) / . (50)Here ξ ′ ( r ) < f ( r ) violates (17). It is straightforward to verify that for this f ( r ) which satisfies Eq. (43), Eqs. (42) and (43) allowno physical solution for ( E, L, r ) while Eq. (10) and (11) still allow real solution.When asymptotic flatness condition is imposed on the spacetime, in general itwill not guarantee the existence of FP of null geodesics. Indeed, it is not hard toverify that the metric (26) will not allow FP for null geodesics. Furthermore, when theasymptotically flat metric is required to correspond to a negative Newtonian potentialat large r , then clearly condition (46) will not be satisfied. There however still existmetrics that satisfies condition (47), as long as ξ ( r ) is chosen not to decrease too fast sothat when multiplied by r , the resultant f ( r ) can still satisfies the desired asymptoticform. An example is given by ξ ( r ) = 1 r ( r + 2 M ) , f ( r ) = ξ ( r ) r = 1 − Mr + 2 M , (51)where
M > r and still satisfies Eq. (47), andtherefore it has no null FP. This point is clearly different from timelike case. xistence and stability of circular orbits in general static and spherically symmetric spacetimes λ = ± s g ( r ∗ )2 r ∗ [2 f ( r ∗ ) − r ∗ f ′′ ( r ∗ )] . (52)Again, the presence of the f ′′ term implies that in the general case the stability of the FPcannot be determined definitely without actually found the FP. We can at most obtainsome sufficient conditions by forcing the term under the square root to be negative (orpositive) for the entire range of r :2 f ( r ) − r f ′′ ( r ) = − µ ′ ( r ) , (53)where µ ( r ) is an arbitrary but with its derivative sign fixed. Unlike in the case of FP oftimelike geodesics, here the Eq. (53) can be solved exactly to produce f ( r ) = (cid:18)Z µ ( r ) r d r + c (cid:19) r . (54)When µ ′ ( r ) > µ ′ ( r ) <
0) function, then the FP determined by the above f ( r ) willbe always stable (or unstable).
7. Discussion
In this section, we first discuss two applications of our results obtained in previoussections, and then discuss possible future works.The first application, which is also what initially motivated us from considering theFP problem for SSS spacetimes, is to use these results to study the existence of FP fora metric whose explicit formula is not completely known. In general relativity, even forthe simplest SSS case there are many models whose field equations cannot be solvedanalytically due to the high non-linearity of the Einstein equations. Some of these modelmetrics even lack a known numerical solution. Examples of these include the famous“colored black hole” for the SU(2) Yang-Mills-Einstein model [27], whose solution inthe entire outer region of the black hole is only known numerically. What happens tothese models’ field equations is that we often only know from physical/mathematicalarguments some limited features such as boundary or asymptotic behaviors of the metricfunctions and other fields, but not their analytical solutions. On the other hand, theexistence of FP of these models might be of theoretical or practical interests. We nowillustrate with the colored black hole example that our results provide an opportunityto assert the (non-)existence of FP of the metric without having to know its analyticalformula.For the SU(2) Yang-Mills-Einstein SSS model considered in Ref. [27], if we wereusing a metric of the formd s = f ( r )d t − (cid:18) − m ( r ) r (cid:19) − d r − r dΩ , (55) xistence and stability of circular orbits in general static and spherically symmetric spacetimes f ( r ) becomes f ′ ( r ) = 4 w ′ ( r ) f ( r ) r + 2 f ( r ) m ( r ) − m ′ ( r ) rr [ r − m ( r )] , (56)while the field equations for the mass function m ( r ) and gauge field w ( r ) is still givenby Eqs. (10a) and (12) of the same paper. It is also known from asymptotic analysisthat for the mass and gauge functions respectively m ( r ) → M, and | w ( r ) | → − c/r as r → ∞ , (57)where 0 < M < ∞ and c >
0. For these equations, the author then used the shootingmethod to find their numerical solution. Up to this point, the analytical solution of thismetric is still not known yet. Now in our work however we show that without knowingthe solutions of Eq. (56) but with only the equation itself and the asymptotics (57),using theorems obtained previously we can prove the existence of FP for this metric.First of all, we see that the first term on the right hand side of Eq. (56) is positivedefinite. For the second term, at large r its sign is determined by the numerator, whichwe set to the following m ( r ) − m ′ ( r ) r = − p ′ ( r ) r . (58)Solving this, we obtain m ( r ) = p ( r ) r. (59)Now due to the boundness (57) of m ( r ), clearly p ( r ) → M/r + O ( r − b ) where b > r → ∞ . Then we see that p ′ ( r ) → − M/r + O ( r − b − ) < r → ∞ . Therefore f ( r )is an increasing function at large enough r and will not satisfy Eq. (17). Further, weshow that f ( r ) will not increase as fast as Eq. (15). Substituting Eq. (15) and (59) into(56) one finds κ ′ ( r ) r + 2 κ ( r ) r = 2 κ ( r ) r − p ( r )] w ′ ( r ) − rp ′ ( r )1 − p ( r ) . (60)Now because of the asymptotics of p ( r ) and w ( r ) in (57), one see that the fraction partof the right hand side is of order 1 /r at large r and there exist no κ ( r ) that can satisfythe above equation to the leading order for all large r . Now since neither of Eq. (15)and (17) are satisfied for the entire r , according to Theorem A metric (55) in the SU(2)Yang-Mills-Einstein SSS model will allow the existence of timelike FP.The second application of the theorems in this paper is that we can use them tostudy the existence and stability of the FPs of some general SSS metrics. A total of8 metrics from Ref. [4], whose f ( r )’s are given in the first column of Table 1, areexamined and their allowance of the timelike and null FPs are listed in the secondand fourth columns respectively. These spacetimes are chosen mainly because of theirmathematical simplicity. For metrics that do allow FPs, the columns three and fiveshows their stabilities obtained by using the sufficient conditions in section 5 and 6, xistence and stability of circular orbits in general static and spherically symmetric spacetimes Table 1.
The f ( r ) component of the metrics, their existence of timelike FP (Yes:Y,No:N) and FP stabilities (Stable: S, unstable: U) are shown from column 1 to 3. Theexistence of null FP (Yes:Y, No:N) and FP stabilities (Stable: S, unstable: U) areshown in column 4 and 5. The asymptotic flatness of the metrics (Flat:F, non-flat: N)and sign of corresponding Newtonian potential at infinity (positive: +, negative: − )are shown in column 6. See Ref. [4] for other components of the metrics except theRN metric whose g ( r ) = f ( r ) − . a, b, n are real constants except otherwise stated. f ( r ) T. Exist. T. Stab. N. Exist. N. Stab. Asympt.1 + a/r Y ( a <
0) Y ( a <
0) U F, − N ( a >
0) N ( a >
0) F, +1 + a/r + 9 b / (32 r ) Y ( a <
0) Y ( a < −| b | ) F, − ( b = 0) N ( −| b | < a <
0) F, − N ( a >
0) N ( a >
0) F, + ar b , ( a >
0) Y (0 < b <
2) S N (0 < b <
2) NN ( b < , b >
2) N ( b < , b >
2) N b + a r /b Y S N N( a + br n ) , ( a >
0) Y ( b > , n >
1) Y ( b > , n >
1) S NY ( b > , S N ( b > , N0 < n <
1) 0 < n < b < , n >
0) U ( n >
1) Y ( b < , n >
0) S ( n >
1) NN ( b > , n <
0) N ( b > , n <
0) F, +Y ( b < , n <
0) U ( n < −
2) Y ( b < , n <
0) F, − a (5 + br ) (2 − br ) Y ( a >
0) Y ( a >
0) S N( ab <
0) N ( a <
0) N ( a <
0) N r ( a + b ln r ) Y Y S N a exp( br ) Y ( b >
0) S Y ( b >
0) S N( a >
0) N ( b <
0) N ( b <
0) N when they are applicable. In column six we list for the metrics allowing FPs whetherthey are asymptotically flat and if yes the sign of the corresponding Newtonian potentialat large r . Apparently, as dictated by Theorem B, all asymptotically flat spacetime withnegative effective Newtonian potential at large r allows timelike FP. In Table 1, we alsolisted the most famous Schwarzschild and RN metrics, their FP existence and stabilities.These properties for these two metrics have been extensively studied elsewhere [28] andit is seen that our results agrees very well with them. The difference in our approach isthat after using the theorems in this paper, the calculations needed to obtain the sameconclusion for these two spacetimes, and also for other metrics in Table 1 which havenever been studied before, become very elementary. This clears shows the power of ourresults.Regarding ways to extend the current work, there exist at least a few. The firstis to notice that Eqs. (10) and (11) and all the analysis followed do not rely on thedimension of the spacetime [29]. Therefore all conclusions given by the Theorems inthis paper also apply to SSS spacetime of higher dimension. Since these spacetimes andthe geodesics in them are of their own importance [30], our work can shed light on the xistence and stability of circular orbits in general static and spherically symmetric spacetimes r , but theirdependence on r and their asymptotics should be enough. Since enormous difficultiesusually lie in the solution process from these dependence to the analytical form of themetric, our results can be an important tools for determine the existence and stabilitiesof COs for these metrics. The last extension is to relax the SSS requirement but tostudy using a similar strategy the existence and stability of FPs in equatorial planeof axially symmetric spacetime. Since a general form of an axially symmetric metric,which is the only starting point that our analysis needs, is already known in the Weyl-Lewis-Papapetrou coordinates [31, 32], we expect that an analogous approach shouldalso be applicable here. Research in these directions are now pursued. Acknowledgments
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