Effect of a second compact object on stable circular orbits
RRUP-19-24
Effect of a second compact object on stable circular orbits
Keisuke Nakashi ∗ and Takahisa Igata † Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan (Dated: November 7, 2019)We investigate how stable circular orbits around a main compact object appear depending on thepresence of a second one by using the Majumudar–Papapetrou dihole spacetime, which consists ofthe two extremal Reissner–Nordstr¨om black holes with different masses. While the parameter rangeof the separation of the two objects is divided due to the appearance of stable circular orbits, thisdivision depends on its mass ratio. We show that the mass ratio range separates into four parts,and we find three critical values as the boundaries.
I. INTRODUCTION
Recent progress in the observation of gravitational waves supports the existence of binary black hole systems. TheLIGO Scientific and Virgo collaborations have already detected gravitational waves ten times from binary black holemergers and once from a binary neutron star merger so far [1–4]. Furthermore, since they have started the thirdobservation run, the number of detections will increase in the future. These results imply that binary black holesystems are a quite common phenomenon in our Universe.A pure binary black hole system is a highly idealized model, around which a third object or matter distributionusually exists in realistic situations. Therefore, one of the next issues is clarifying perturbative interactions with a thirdbody around them. As traditional problems in Newtonian gravity, there are Poincar´e’s three-body problem and theKozai mechanism. In recent years, some problems related to these topics have been considered in the framework of therelativistic three body problem [5, 6], in the context of resonance in a compound extreme mass ratio inspiral/massiveblack hole binary [7], and in gravitational wave emission induced by a third body [8–11]. If a third body itself is thetarget of observation, we can view it as a test body in a fixed background. As a traditional problem in Newtoniangravity, there is Euler’s three-body problem, test particle motion in two fixed centers. The corresponding relativisticsystem is the main topic of this paper.The study of test body motion is significant for the predictions of astrophysical phenomena around a gravitationalsystem such as a binary. In particular, the circular orbit of a test body plays some essential roles in both theory andobservations. In the black hole spacetime, for instance, the bending of light due to strong gravity makes a photonorbit circular near the horizon. If the circular photon orbit is unstable, it relates to the formation of the black holeshadow. On the other hand, the sequence of stable circular massive particle orbits is relevant to accretion disks anda binary system. The innermost stable circular orbit (ISCO) radius is a distinctive one because it is identified asthe inner edge of a standard accretion disk model and a compact binary switches the stage of the evolution from theinspiraling phase to the merging phase there [12, 13].Actual binary black hole systems exist as highly dynamical systems so that one needs to use the numerical methodto analyze the phenomena around such systems; for example, the study of the shadow of a binary black hole systemrequires a fully nonlinear analysis of the numerical relativity [14]. On the other hand, it is also significant to use ananalytical method for a qualitative understanding. To this end, we often employ some axisymmetric and stationary(or static) dihole spacetime as a toy model. There are some exact dihole spacetime solutions of the Einstein equation(or the Einstein–Maxwell equation) such as the Weyl spacetime [15], the Majumdar–Papapetrou spacetime [16–18],the double-Kerr spacetime [19], etc. We can extract the specific features of phenomena around a binary black holesystem by using these dihole spacetimes. Indeed, the eyebrows structure of the binary black hole shadow is reproducedin the (quasi)static dihole spacetime [20–24].The aim of the present paper is to reveal how the marginally stable circular orbit (MSCO) or ISCO of the diholespacetime varies compared to those of the single black hole spacetime. To achieve this, we adopt the Majumdar–Papapetrou (MP) dihole spacetime, which contains two extremal Reissner–Nordstr¨om black holes. The circular orbitand its stabilities in the equal mass MP dihole spacetime have been investigated [25–27]. In our previous paper [25],we clarified the dependence of the positions of MSCOs and ISCOs on the separation parameter in the equal mass MP ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ g r- q c ] N ov dihole spacetime. We found that the range of the dihole separation is divided into five ranges and obtained the fourcritical values as the boundaries. In this paper, we investigate the sequence of the stable circular orbit in the differentmass MP dihole spacetime, which consists of the two different mass extremal Reissner–Nordstr¨om black holes. Oncewe fix the mass scale of one of the two black holes, the system depends on two parameters: the separation and themass ratio. As the result of our analysis, we divide the mass ratio parameter range into four ranges and obtain threecritical values as the boundaries.This paper is organized as follows. In the following section, we introduce the MP dihole spacetime with differentmasses and derive conditions for circular particle orbits on the background. Furthermore, we clarify the stabilityconditions of these orbits in terms of the Hessian of a 2D potential function. In Sec. III, while changing the massratio of the dihole, we analyze the dihole separation dependence of the positions of stable circular orbits. Due tosome qualitative differences of sequences of stable circular orbits, we classify the range of dihole mass ratios into fourparts and determine three critical values of the mass ratio as the boundaries of the range. Section IV is devoted to asummary and discussions. Throughout this paper, we use units in which G = 1 and c = 1. II. CONDITIONS FOR STABLE CIRCULAR ORBITS IN THE MAJUMDAR–PAPAPETROU DIHOLESPACETIME
The metric and the gauge field of the MP dihole spacetime in isotropic coordinates are given by g µν d x µ d x ν = − d t U + U (d ρ + ρ d φ + d z ) , (1) A µ d x µ = U − d t, (2) U ( ρ, z ) = 1 + M + (cid:112) ρ + ( z − a ) + M − (cid:112) ρ + ( z + a ) , (3)where M ± are each black hole mass located at z = ± a ( a ≥ ν := M − M + . (4)Without loss of generality, we assume that the black hole with mass M + is larger than that with mass M − , i.e.,0 ≤ ν ≤ . (5)We use units in which M + = 1 in what follows.The Lagrangian of a particle freely falling in the MP dihole spacetime is given by L = 12 (cid:20) − ˙ t U + U ( ˙ ρ + ρ ˙ φ + ˙ z ) (cid:21) , (6)where the dot denotes derivative with respect to an affine parameter. Since the coordinates t and φ are cyclic, thecanonical momenta conjugate to them are constants of motion: E = ˙ tU , L = ρ U ˙ φ, (7)which are energy and angular momentum, respectively. We normalize the 4-velocity ˙ x µ so that g µν ˙ x µ ˙ x ν = − κ , where κ = 1 for a massive particle and κ = 0 for a massless particle. Rewriting the normalization condition in terms of E and L , we have ˙ ρ + ˙ z + V = E , (8) V ( ρ, z ) = L ρ U + κU . (9)We can view Eq. (8) as an energy equation and V as a 2D effective potential of particle motion in the ρ - z plane. Interms of V , the equations of motion are written as¨ ρ + 2 U z U ˙ z ˙ ρ − U ρ U ˙ z + V ρ , (10)¨ z + 2 U ρ U ˙ z ˙ ρ − U z U ˙ ρ + V z , (11)where V i = ∂ i V and U i = ∂ i U ( i = ρ, z ).We focus on circular orbits with constant ρ and z . Then, the energy equation (8) immediately reduces to V = E . (12)Hence, V must be positive for circular orbits. In addition, we find that constant ( ρ, z ) can be a solution to Eqs. (10)and (11) when its position corresponds to an extremum of V : V ρ = 0 , (13) V z = 0 . (14)We can rewrite the three conditions (12)–(14), respectively, as E = E ( ρ, z ) := V ( ρ, z ; L ) , (15) L = L ( ρ, z ) := − ρ U U ρ U + 2 ρU ρ , (16) U z = a − z [ ρ + ( z − a ) ] / − ν ( a + z )[ ρ + ( z + a ) ] / = 0 . (17)From Eqs. (15) and (16), both values of E and L depend on positions of circular orbits and must be positive. Thepositivity of L leads to that of E as seen from Eq. (9), so that it is sufficient to pay attention only to the positivityof L .Now we solve Eq. (17). If ν = 0, then we obtain the solution z = a . If 0 < ν ≤
1, we find from Eq. (17) that therange of z is bounded in | z | < a . Solving it for ρ in this range, we obtain the root ρ ( z ) = ( a − z ) / ( a + z ) − ν / ( a + z ) / ( a − z ) ν / ( a + z ) / − ( a − z ) / , z (cid:54) = 1 − ν ν a. (18)When z = a (1 − ν ) / (1 + ν ) holds, then Eq. (17) leads to z = 0, and hence ν = 1. Note that the root ρ is real andpositive in the range − a < z < − − √ ν √ ν a, − ν ν a < z < a. (19)The curve ρ = ρ asymptotically approaches the line z = 1 − ν ν a. (20)In particular, the intersection point of the line with the symmetric axis ρ = 0 corresponds to the center of mass ofthe dihole. On the other hand, the curves terminate on ρ = 0 at z = ± a (i.e., the horizons) and z = − − √ ν √ ν a. (21)Therefore, we can find a circular orbit at a point in the ρ - z plane if it is located on the curve ρ = ρ ( z ) and satisfies E ≥ L ≥ V ij , where V ij = ∂ j ∂ i V ( i, j = ρ, z ). Let h be its determinant, h ( ρ, z ; L ) =det V ij , and k be its trace, k ( ρ, z ; L ) = tr V ij . By using these we define the region D in the ρ - z plane by D = (cid:8) ( ρ, z ) | L > , h > , k > (cid:9) , (22)where h ( ρ, z ) = h ( ρ, z ; L ) | U z =0 , (23) k ( ρ, z ) = k ( ρ, z ; L ) | U z =0 , (24)where the restriction U z = 0 means to eliminate the terms proportional to U z . We can find stable circular orbits onthe curve ρ = ρ ( z ) included in the region D . III. DEPENDENCE OF THE SEQUENCE OF STABLE CIRCULAR ORBITS ON THE MASS RATIO
In this section, focusing on stable circular orbits in the MP dihole spacetime, we analyze the dependence of sequencesof their orbits on the separation a for various values of the mass ratio ν . A. ν = 1 In the beginning, let us recall how sequences of stable circular orbits change as the separation a varies in theequal unit mass MP dihole spacetime (i.e., ν = 1 and M + = M − = 1) [25]. For a > a = 1 . · · · , a sequence ofstable circular orbits exists in the range ρ ∈ ( √ a, ∞ ) on the equidistant symmetric plane z = 0 from each blackhole. Furthermore, it bifurcates at ( ρ, z ) = ( √ a,
0) and extends towards each black hole. As a result, we havethree MSCOs, two of which are the ISCOs. At a = a , the three MSCOs degenerate at ( ρ, z ) = ( √ a , a ≥ a > a ∗ = 0 . · · · , a single sequence of stable circular orbits appears on z = 0 in the range ρ ∈ ( √ a, ∞ ),and this inner boundary corresponds to the ISCO. At a = a ∗ , the single sequence is marginally connected at a pointwhere h has a saddle point. For a ∗ ≥ a > a c = 0 . · · · , we have two sequences of stable circular orbits on z = 0. This phenomenon implies the possibility of double accretion disk formation in this system. In particular, for a ∗ > a > a ∞ = 0 . · · · , the outer sequence exists from infinity to an MSCO and the inner sequence from an MSCOto the ISCO, while for a ∞ ≥ a > a c , the outer boundary of the inner sequence is no longer a marginally stable circularmassive particle orbit but turns into a circular photon orbit; that is, infinitely large energy would be required for thestable circular orbit. At a = a c , the inner sequence just disappears. For a c ≥ a ≥
0, we only have a single sequenceof stable circular orbits from infinity to the ISCO on z = 0.In the following subsections, dividing the range of ν into four parts, we consider the dependence of sequences ofstable circular orbits on the separation a in each range of ν . B. > ν > ν ∞ = 0 . · · · We consider sequences of stable circular orbits for various values of a in the MP dihole spacetime with mass ratio ν (cid:39) ν (cid:54) = 1. We show sequences of stable circular orbits for several values of a in the case ν = 0 . a . Specific numerical values for critical values a , a ∗ , a ∞ , and a c in this subsection are those for ν = 0 . a , we have two sequences of stable circular orbits on both sides of the dihole [see Fig. 1(a)]. Thesequence on the large black hole side exists from infinity to the ISCO near the large black hole. On the other hand,the sequence on the small black hole side is restricted within a finite region. The inner boundary near the small blackhole corresponds to the ISCO, and the outer boundary to an MSCO. As the value of a approaches a critical value a (= 2 . · · · ) from above, the MSCO and the ISCO on the small black hole side approach each other. At a = a ,these merge into one, and then the sequence on the small black hole side just disappears [see Fig. 1(b)]. If a becomessmaller than a , the sequence on the small black hole side no longer exists. We can interpret this disappearanceas a consequence of relativistic effects because the corresponding sequences in Euler’s three-body system, which isgoverned by Newtonian gravity, always exist for arbitrary values of a .In the range a ≥ a > a ∗ (= 0 . · · · ), there exists a sequence of stable circular orbits only on the large black holeside, which appears from infinity to the ISCO [see Fig. 1(c)]. At a = a ∗ , the boundary of the region D touches thecurve ρ = ρ [see Fig. 1(d)]. This implies that two sequences of stable circular orbits are marginally connected at apoint.In the range a ∗ ≥ a > a c (= 0 . · · · ), two sequences of stable circular orbits appear [see Figs. 1(d)–1(g)]. Theouter sequence exists from infinity to the outermost MSCO. On the other hand, the behavior of the inner sequencedivides this range of a into two parts. For a ∗ ≥ a > a ∞ (= 0 . · · · ), the inner sequence exists between an MSCO andthe ISCO. However, at a = a ∞ , the outer MSCO disappears because infinitely large energy and angular momentumwould be required for a massive particle [see Fig. 1(f)]. In other words, a circular photon orbit appears there. For a ∞ ≥ a > a c , the inner sequence appears between the stable circular photon orbit and the ISCO [see Fig. 1(g)]. At a = a c , the stable circular photon orbit and the ISCO merge into one, and then the inner sequence just disappears [seeFig. 1(h)]. In the range a c ≥ a ≥
0, there only exists a sequence of stable circular orbits, which appears from infinityto the ISCO [see Fig. 1(i)].Consequently, we divide the range of a into five parts on the basis of typical behaviors of the sequence of stablecircular orbits and introduce four critical values of a as the boundaries of these ranges as we have done in the case ν = 1. Note that, however, each meaning of critical values is slightly generalized from those of ν = 1. Here, let ussummarize how we define the four critical values:(i) a = a : The sequence of stable circular orbits on the small black hole side disappears.(ii) a = a ∗ : The sequence of stable circular orbits on the large black hole side is divided into two parts.(iii) a = a ∞ : A stable circular photon orbit appears at the outer boundary of the inner sequence of stable circularorbits on the large black hole side.(iv) a = a c : The inner sequence of stable circular orbits on the large black hole side disappears.In the following, according to the difference in the appearance of these critical values, we classify the range of themass ratio ν into four parts. In each range of ν , we discuss the behavior of the sequence of stable circular orbitsdepending on a . Figure 6(a) shows the dependence of the radii of the MSCOs, the ISCOs, and the circular photonorbits on a in the case ν = 0 .
9. In the range a c < a ≤ a ∞ , the radius of the ISCO (red solid line) is smaller than theone of the stable circular photon orbit (orange solid line in the middle of the three). In addition, the discontinuoustransition of the position of the ISCO occurs at a = a c . These phenomena are also seen in the equal mass MP diholespacetime [25]. C. ν = ν ∞ = 0 . · · · If we decrease the value of ν from ν = 1, then at ν = ν ∞ := 4 √
39 = 0 . · · · , (25)the stable circular photon orbit no longer appears for any value of a . In other words, the critical value a ∞ disappearsat ν = ν ∞ . We can interpret that the gravity of the small black hole is not sufficiently strong to make a photon orbitcircular in the region far from the large black hole even if two black holes get close each other. In what follows, weconsider sequences of stable circular orbits in the case ν = ν ∞ .For a > /
2, the behavior of sequences of stable circular orbits is similar as that discussed in the previous subsection.Indeed, we find two critical values a = 2 . · · · and a ∗ = 0 . · · · . We note that, however, qualitative differencesfrom the case in the previous subsection appear at a = 1 /
2. In the limit as a (cid:38) /
2, we find that the MSCO and theISCO at the boundaries of the inner sequence merge into one at ( ρ, z ) = (2 √ / , /
6) = (0 . · · · , . · · · ) (seeFig. 2). Simultaneously, infinitely large energy and angular momentum are required for a massive particle to orbitcircularly here. In other words, here is a stable/unstable circular photon orbit. These behaviors mean that a c and a ∞ are degenerate at a = 1 /
2, that is, a = a c = a ∞ = 1 / a < /
2, there is only a single sequence of stable circular orbits that appears from infinity to the ISCO,which is the same as that discussed in the previous section. D. ν ∞ > ν > ν ∗ = 0 . · · · We consider sequences of stable circular orbits for various values of a in the case where ν ∞ > ν > ν ∗ = 0 . · · · .We can see typical sequences of stable circular orbits for ν = 0 . a , a ∗ , and a c in this range. Specific numerical values for these critical values in thissubsection are those for ν = 0 . a , a sequence of stable circular orbits appears from infinity to the ISCO on the largeblack hole side, while a sequence appears between an MSCO and the ISCO on the small black hole side [see Fig. 3(a)].When a becomes smaller and smaller, at a = a (= 2 . · · · ), the sequence on the small black hole side disappears [seeFig. 3(b)]. When a becomes smaller and smaller yet, at a = a ∗ (= 0 . · · · ), the sequence on the large black holeside is divided into two parts. In the range a ∗ ≥ a > a c (= 0 . · · · ), there are two sequences, the inner and theouter. As a result, we find three MSCOs as the boundaries of these sequences, and the innermost one corresponds tothe ISCO. At a = a c , the inner sequence disappears. Note that the critical value a ∞ no longer exists in this range of ν . In the range 0 ≤ a < a c , we find a single sequence that appears from infinity to the ISCO. The dependence of theradii of the MSCOs, the ISCOs, and the circular photon orbits on a in the case ν = 0 . a is divided by a , a ∗ , and a c into four parts. The discontinuous transition of the ISCO on thelarge black hole side still occurs at a = a c . For any value of a , stable circular photon orbits do not exist. (b) a = a = 2 . · · · (a) a = 5 (c) a = 1(d) a = a ⇤ = 0 . · · · (e) a = 0 . a = a = 0 . · · · (g) a = 0 . a = a c = 0 . · · · (i) a = 0 . ▲▲ - - - ρ z ▲▲ - - ρ z ▲▲ - - ρ z ▲▲ - - ρ z ▲▲▲ - - ρ z ▲ ▲▲ - - ρ z ▲ ▲▲ - ρ z ▲▲ - - ρ z ▲▲ - ρ z \nu=0.9 FIG. 1: Sequences of stable circular orbits in the Majumdar–Papapetrou dihole spacetime with mass ratio ν = 0 .
9. Theblack solid lines denote the curve ρ = ρ . The shaded regions denote the region D , where h > L >
0, and k >
0. Theboundaries of D are shown by the blue solid lines on which h vanishes and the blue dashed lines on which L diverges. Theblack solid lines in the shaded regions show the positions of stable circular orbits. The green dots indicate the position ofmarginally stable circular orbits, and the red dots indicate the position of the innermost stable circular orbits. The orange dotsshow the positions of stable circular photon orbits, and the orange triangles show those of unstable ones. E. ν = ν ∗ = 0 . · · · We focus on sequences of stable circular orbits for various values of a in the case ν = ν ∗ = 0 . · · · . For large a ,we can see similar behavior of the sequences of stable circular orbits as is shown in the previous subsection. Indeed,we obtain the critical value a = 2 . · · · . We should note that the inner sequence of stable circular orbits appearingat ( ρ, z ) = (2 . · · · , . · · · ) for a = a ∗ = 0 . · · · disappears as soon as it appears [see Fig. 4(d)]. This meansthat the critical values a ∗ and a c are degenerate. Consequently, we have no inner sequence of stable circular orbits \nu=0.76980035891 ▲▲▲ - ρ z a = a c = a = 0 . FIG. 2: Sequences of stable circular orbits in the Majumdar–Papapetrou dihole spacetime with mass ratio ν = ν ∞ = 0 . · · · .The roles of each element in these plots are the same as those in Fig. 1. on the large black hole side. F. ν ∗ > ν > ν = 0 . · · · Let us consider sequences of stable circular orbits for various values of a in the case where 0 < ν < ν ∗ . Observingtypical sequences for ν = 0 . a in this range. The specificnumerical value of a in this subsection is that for ν = 0 . a , we find two sequences of stable circular orbits on both sides of the dihole [see Fig. 5(a)]. Onthe large black hole side, the sequence appears from infinity to the ISCO. On the small black hole side, the sequenceappears from the outer MSCO to the ISCO. If a becomes smaller and reaches a = a (= 1 . · · · ), the sequence onthe small black hole side disappears. Therefore, there still exists the critical value a [see Fig. 5(b)]. In the range a < a , however, any qualitative change of the sequence of stable circular orbits occurs on the large black hole side.The dependence of the radii of the MSCOs, the ISCOs, and the circular photon orbits on a in the case ν = 0 . a is divided by a into two parts. The continuous transition of the ISCOon the large black hole side no longer occurs because there are no separated sequences of stable circular orbits on thelarge black hole side. G. ≤ ν ≤ ν We mention the sequence of stable circular orbits in the range 0 ≤ ν ≤ ν . When the value of ν reaches ν fromabove, the critical value a is equal to zero. This means that the sequence on the small black hole side does not vanishunless the two black holes coalesce into one. If we make the value of ν smaller than ν , any critical values of a do notappear. According to Fig. 6(d), where we set ν = 0 .
01, for a >
0, both the large and the small black holes have thesequence of the stable circular orbits; i.e., two MSCOs—one of these is also the ISCO—always appear on the smallblack hole.
IV. SUMMARY AND DISCUSSIONS
We have investigated the sequence of stable circular orbits around the symmetric axis in the Majumdar–Papapetrou(MP) dihole spacetime with different masses. Once we fix the mass of the large black hole to 1, the MP dihole spacetimeis characterized by two parameters: the separation a and the mass ratio ν .When ν (cid:39) ν (cid:54) = 1, the sequence of the stable circular orbits changes as a varies in common with the caseof ν = 1, but we have generalized the definitions of the critical values of a to be valid for the case of the differentmass MP dihole from those in the equal mass MP dihole spacetime [25]. When the value of a is relatively large,the sequence of stable circular orbits on the large black hole side exists from infinity to the ISCO while that on thesmall black hole side is restricted to a finite range. At a = a , the sequence on the small black hole side disappears.This phenomenon occurs due to the relativistic effect of the appearance of the ISCOs; that is, since the radius of the \nu=0.7 (a) a = 5 (b) a = a = 2 . · · · (c) a = 1(d) a = a ⇤ = 0 . · · · (e) a = 0 . a = a c = 0 . · · · (g) a = 0 . ▲▲ - ρ z ▲▲ - - - ρ z ▲▲ - - ρ z ▲▲ - - ρ z ▲▲ - - ρ z ▲▲ - - ρ z ▲▲ - - ρ z FIG. 3: Sequence of stable circular orbits in the Majumdar–Papapetrou dihole spacetime with mass ratio ν = 0 .
7. The rolesof each element in these plots are the same as those in Fig. 1. outer MSCO on the small black hole side decreases faster than the one of the ISCO as a decreases, the positions ofthe MSCO and the ISCO coincide with each other at a = a , and then the sequence on the small black hole sidedisappears [see the green and red dashed lines in Figs. 6(a)–6(c)]. For a < a , the sequence of stable circular orbitsappears only on the large black hole side. When a = a ∗ , the sequence on the large black hole side is marginallyconnected at a point. In the range a c < a < a ∗ , two sequences of the stable circular orbits appear on the large blackhole side. The outer boundary of the inner sequence is an MSCO in a ∞ < a < a ∗ while a stable circular photon orbitin a c < a ≤ a ∞ . Finally, for 0 ≤ a ≤ a c , since the inner sequence vanishes, we have a single connected sequence frominfinity to the ISCO on the large black hole side.We have also revealed the dependence of the sequence of stable circular orbits on ν . Figure 7 shows the relationbetween ν and the critical values of a . For ν > ν ∞ = 0 . · · · , the sequences of the stable circular orbits arequalitatively the same as these of the case ν (cid:39)
1. At ν = ν ∞ , the two critical values a ∞ and a c merge with each \nu=0.530607412404 ▲▲ - - ρ z a = a ⇤ = a c = 0 . · · · FIG. 4: Sequences of stable circular orbits in the Majumdar–Papapetrou dihole spacetime with mass ratio ν = ν ∗ = 0 . · · · .The roles of each element in this plot are the same as those in Fig. 1. (a) a = 5 (c) a = 1(b) a = a = 1 . · · · \nu=0.3 ▲▲ - ρ z ▲▲ - - - ρ z ▲▲ - - ρ z FIG. 5: Sequence of stable circular orbits in the Majumdar–Papapetrou dihole spacetime with mass ratio ν = 0 .
3. The rolesof each element in these plots are the same as those in Fig. 1. other, so that the parameter range of a is divided into four parts. At ν = ν ∗ = 0 . · · · , the critical values a ∗ and a c coincide with each other. For ν < ν ∗ , the sequence on the large black hole side does not separate into two parts.The remaining critical value a also disappears when ν = ν = 0 . · · · . When we make the value of ν smallerthan ν , the sequences of stable circular orbits on both sides do not vanish until the two black holes merge into one[see Fig. 6 (d)].The phenomena we have revealed are not caused by electric charges of the black holes, but by the presence of twoblack holes. Hence, in our Universe, we can observe such phenomena occurring around a compact object accompaniedby a second compact object. The sequence of stable circular orbits is not on a flat plane because of the existence of asecond compact object. Therefore, we may observe a deformed accretion disk that indicates the existence of anothergravitational source. Furthermore, for any value of ν , the radius of the ISCO on the large black hole side tends to bemore inner than the case of a single black hole. This suggests that high energy X-rays can be detected compared tothe single black hole case because the effective temperature of the standard disk is higher as the radius of the ISCO issmaller [28, 29]. In the ranges ν ∗ < ν ≤ a c < a < a ∗ , we may observe double accretion disks. Since a relativisticeffect causes the inner disk, we can use it in the testing of gravitational theories. The observation of the accretiondisk of a main black hole with the second companion object (e.g., blazar OJ287) may help us to explore the effectof the second compact object [30]. The presence of the stable circular photon orbit in the ranges ν ∞ < ν ≤ a c < a ≤ a ∞ is a characteristic property of the dihole spacetime and is associated with distinctive phenomenologicalfeatures, such as the chaotic behavior of the null geodesics. We can observe qualitatively different chaotic features inthe dihole shadow [24].Our findings indicate that the existence of the second compact object can affect gravitational wave emission froma test particle orbiting a main supermassive black hole because the inspiral phase tends to be longer than the case ofa single black hole because of the shift of the ISCO radius. In the context of the quasinormal mode, the frequency0 a c ρ a ∞ a * a ρ a c a * a a ρ (a) ν = 0.9 (b) ν = 0.7 (c) ν = 0.3 ρ ρ (d) ν = 0.01 FIG. 6: Dependence of the radii of MSCOs and circular photon orbits on the separation parameter a in the MP diholespacetime with mass ratio: (a) ν = 0 .
9, (b) ν = 0 .
7, (c) ν = 0 .
3, and (d) ν = 0 .
01. The green and red solid lines show theradii of MSCOs on the large black hole side, and the green and red dashed lines show those on the small black hole side. Inparticular, the red lines indicate each ISCO. The orange solid lines show the radii of circular photon orbits on the large blackhole side, and the orange dashed lines show those on the small black hole side. The dashed green and red lines merge at a = a and then the sequence of the stable circular orbits on the small black hole side disappears. In cases (a) and (b), the green solidlines emerge at a = a ∗ , and the outer exist in the range a c < a < a ∗ . The inner in case (a) exists in the range a ∞ < a < a ∗ while the one in case (b) exists in the range a c < a < a ∗ . In case (a), the stable circular photon orbits appear on the largeblack hole side in a c < a < a ∞ whereas they do not in the other cases. In case (d), the sequence on the small black hole sidealways exists because there is no critical value of a . is known to correspond to the orbital frequency of the unstable circular photon orbit. Since our results show thatthe orbital frequency can be comparable to that of a circular massive particle orbit, we can expect that the resonantexcitation of the quasinormal mode occurs [31].The MP dihole spacetime we have used in the background is static, but a realistic binary system is a dynamicalsystem. Therefore, we should take into account dynamical features in future work. Acknowledgments
The authors thank T. Harada, M. Kimura, T. Kobayashi, Y. Koga, and Y. Mizuno for their fruitful discussions anduseful comments. This work was supported by a Grant-in-Aid for Early-Career Scientists from the Japan Society forthe Promotion of Science (JSPS KAKENHI Grant No. JP19K14715) (T.I.) and the Rikkyo University Special Fundfor Research (K.N.). K.N. also thanks the Yukawa Institute for Theoretical Physics at Kyoto University, where this1 ν * ν ∞ a a * a ∞ a c ν νa FIG. 7: Dependence of critical values of the separation a on the mass ratio ν . The blue, orange, red, and green lines show a = a , a ∗ , a ∞ , and a c , respectively. The parameter range of ν is divided into four parts, and the appearance of the criticalvalues changes drastically at the boundaries, ν = ν ∞ , ν ∗ , and ν . At ν = ν ∞ , the critical values a ∞ and a c degenerate eachother. At ν = ν ∗ , the critical values a ∗ and a c coincide with each other. 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