Innermost stable circular orbits of a Kerr-like Metric with ... - arXiv

Innermost stable circular orbits of a Kerr-like Metric with Quadrupole arXiv:1707.08663v1 [gr-qc] 26 Jul 2017

Fabián Chaverri Miranda Francisco Frutos Alfaro Pedro Gómez Ovarez Andree Oliva Mercado School of Physics and Space Research Center of the University of Costa Rica

July 28, 2017 Abstract The innermost stable circular orbit equation of a test particle is obtained for an approximate Kerr-like spacetime with quadrupole moment. We derived the effective potential for the radial coordinate by the Euler-Lagrange method. This equation can be employed to measure the mass quadrupole by observational means, because from this equation a quadratic polynomial for the quadrupole moment can be found. As expected, the limiting cases of this equation are found to be the known cases of Kerr and Schwarzschild.

I.

Introduction

In classical mechanics the orbit of a test particle around a massive object is arbitrary. This is because the effective potential is minimum, for any value of the angular momentum. Nevertheless, in general relativity this is not the case. The effective potential in the Schwarzschild metric has two extrema. When the angular momentum is minimum the two extrema become a single radius which describes the innermost stable circular orbit (ISCO) [15]. Naturally, the rotation of the central body influences the motion around of a particle orbiting it, this is why the orbits around black holes differ between metrics [9]. Another important feature of compact object is its quadrupole moment (q) [12]. It is expected that it would affect the orbits around compact objects. For many exact spacetimes, one cannot find analytical expresions for radii and frequencies of the ISCO. Moreover, Geodesics analysis is cumbersome for such 1

Innermost stable circular orbits of a Kerr-like metric with Quadrupole metrics. It would be useful for studying wave emission and chaotic trajectories of particles around compact objects [1, 10, 13]. Studying the ISCO of black holes is important, because it gives information about the spacetime near the black hole and its background geometry [11, 12]. It is important to recall the no hair conjecture. It states that the geometry outside the black hole horizon (Kerr-Newman metric) is expressed only in terms of three parameters M (mass), a (rotation parameter) and e (charge). Nevertheless, for neutron stars this is not the case. Other parameters, such as deformation (mass quadrupole), and magnetic dipole should be taken into account, but they require the equation of state of the neutron star to completely describe the spacetime near them. However, we are concerned with compact objetcs with three parameters M, a and q [14]. As an example, in a Schwarzschild black hole the radius of the ISCO is r ISCO = 6M. If the electric charge is zero, a Kerr type black hole is obtained from the no hair conjecture [11, 14]. For the Kerr metric the radius of the ISCO depends on the direction of motion of the particle in comparison with the black hole. If the particle moves in the direction of the rotation of the black hole, then the radius becomes smaller r ISCO = M, if the particle moves counter rotation then the radius is bigger than Schwarzschild’s and becomes r ISCO = 9M [15]. In this paper, we are interested in analysing the ISCO of a Kerr-like metric with mass quadrupole. The importance of this metric is that it reduces to the Kerr spacetime and to the Hartle-Thorne (HT) case for certain limits [6, 5]. In principle, it is possible to find an inner solution for the studied metric, by a match with a metric that already matches HT. Moreover, the ISCO equation was found for HT metric [2, 3]. This Kerr-like metric was deduced from the Kerr metric, for this reason it is expected to obtain the known results for Kerr and Schwarzschild metrics for the variables energy, angular momentum and the ISCO radius [8, 7]. We are also interested in the observational applications. Nowadays, there are no direct measurements of the quadrupole moment for compact objects, an analysis of the ISCO structure for this Kerr-like metric could give us a hint to derive q via observational methods. This paper is organized as follows. The Kerr-like metric is introduced in section 2. A detailed calculation of the ISCO equation using the Euler-Lagrange method is in section 3. This method was developed by Chandrasekhar [4]. In section 4, the ISCO equation is compared with the known solutions, Kerr and Schwarzchild black holes by means of a REDUCE program. The summary and discussion of the results are presented in section 5.

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Innermost stable circular orbits of a Kerr-like metric with Quadrupole

II.

The Kerr-like Metric

This metric describes the spacetime of a massive, rotating, deformed object. It has three parameters, the mass of the object, M , the rotation parameter, a and the quadrupole parameter, q. It is an approximate solution of the Einstein field equations and is given by ds2 = gtt dt2 + 2gtφ dtdφ + grr dr2 + gθθ dθ 2 + gφφ dφ2 ,

(1)

where the components of the metric are i e−2ψ h 2 2 ∆ − a sin θ ρ2 2Jr − 2 sin2 θ ρ e2χ ρ2 ∆ 2 2χ ρ e   2 e2ψ  2 2 2 2 − a ∆ sin θ sin2 θ, r +a ρ2

gtt = − gtφ = grr = gθθ = gφφ =

(2)

(3)

with J = Ma, ρ2 = r2 − a2 cos2 θ and ∆ = r2 − 2Mr + a2 . The exponents ψ and χ are given by Mq q P2 + 3 4 P2 3 r r   1 q2  q 1 Mq  2 3 2 χ = 3 P2 + 2 − 6P − 21P + 25P − 1 + 5P + 5P + 2 2 2 2 2 r 3 r4 9 r6

ψ=

(4)

where P2 = (3 cos2 θ − 1)/2. As limiting cases this spacetime contains the Kerr (q = 0) and the Schwarzschild metrics (q = a = 0).

3

Innermost stable circular orbits of a Kerr-like metric with Quadrupole

III.

Deriving the ISCO

The method, we are using, was devised by Chandrasekhar [4]. The Lagrangian is given by µ ds2 2 dλ2 µ = ( gtt t˙2 + 2gtφ t˙φ˙ + grr r˙ 2 + gθθ θ˙ 2 + gφφ φ˙ 2 ). (5) 2 The dot over the variables t, r, θ and φ means derivative with respect to λ. To determine the ISCO of a test particle in the plane, one sets θ˙ = 0 and θ = π/2. This leaves the Lagrangian as follows L=

 µ 2 2 2 ˙ ˙ ˙ ˙ L= gtt t + 2gtφ tφ + grr r˙ + gφφ φ , 2 where the components of the metric becomes

(6)

′

i e−2ψ h gtt = 2 2Mr − r2 r 2J gtφ = − r ′ 2 r e2χ grr = ∆ 2 2χ′ gθθ = r e ′ i e2ψ h gφφ = 2 r4 + 2Mra2 + r2 a2 r with the exponents ψ and χ are reduced to 3 Mq 1q − 3 2r 2 r4 1q 3 Mq 3 q2 − 6 χ′ = − 3 − 2r 4 r4 8r The momenta of the three remaining variables are ψ′ = −


pt = µ gtt t˙ + gtφ φ˙ = − E

pr = µgrr r˙

(7)

(8)

(9)

pφ = µ gtφ t˙ + gφφ φ˙ = Lz


4

Innermost stable circular orbits of a Kerr-like metric with Quadrupole where µ = 1, b2 = − gtt gφφ + g2tφ , E and Lz are constants that represent the energy and the angular momentum. t˙ and φ˙ are solved
1 Egφφ + Lz gtφ 2 b
1 φ˙ = − 2 Lz gtt + Egtφ . b t˙ =

The Hamiltonian is

 1 2 ˙ ˙ − Et + grr r˙ + Lz φ = ε H= 2   1 1  2 2 2 = − 2 E gφφ + 2ELz gtφ + Lz gtt + grr r˙ 2 b

(10)

(11)

The effective potential Ve f is then Ve f = −

1 2ε − (E2 gφφ + 2ELz gtφ + L2z gtt ) 2 grr grr b

(12)

Using u = 1/r and the equations 8 to 7, then the effective potential in 12 can be reduced to  1 5 2 6 4 Ve f = −2ε 1 − 2Mu + a u + qu − Mqu + q u 2 4   11 1 + L2z u2 + 2qu5 + Mqu6 + q2 u8 − 2Mu3 ( Lz − Ea)2 2 4   3 3 2 6 2 2 2 4 − E 1 + a u − Mqu + q u 4 4 

2 2

3

(13)

The latter expression has to be differentiated twice to find the values of r where the orbit is stable.   dVe f 15 2 5 2 2 3 = −2ε − 2M + 2a u + 3qu − 2Mqu + q u du 2   + L2z 2u + 10qu4 + 3Mqu5 + 22q2 u7 − 6Mu2 ( Lz − Ea) 2   9 2 5 2 3 2 − E 2a u − 6Mqu + q u 2

(14) 5

Innermost stable circular orbits of a Kerr-like metric with Quadrupole

  d2 Ve f 75 2 4 2 2 = −2ε 2a + 6qu − 6Mqu + q u 2 du2   + L2z 2 + 40qu3 + 15Mqu4 + 154q2 u6 − 12Mu ( Lz − Ea) 2   45 2 4 2 2 2 − E 2a − 18Mqu + q u 2

(15)

Now, we rewrite these equations using x = aE − Lz as follows V˜e f = Ve f [1 − 2qu3 + (2qu3 )2   13 2 6 2 2 3 4 = ε 2 − 4Mu + 2a u − 2qu + 7Mqu + q u 2   3 19 + E2 − 1 + 2qu3 + Mqu4 − q2 u6 − 2Exau2 2 4   11 2 6 9 4 2 2 + x u 1 − 2Mu + Mqu + q u = 0, 2 4 dVe f dV˜e f = [1 − 5qu3 + (5qu3 )2 ] du du
= ε − 4M + 4a2 u + 6qu2 + 16Mqu3 − 15q2 u5   9 2 2 3 + E qu 6M − qu − 4Exau 2
+ x2 u 2 − 6Mu + 33Mqu4 + 22q2 u6 = 0, d2 Ve f d2 V˜e f = [1 − 20qu3 + (20qu3 )2 ] du2 du2 = ε(4a2 + 12qu − 12Mqu2 − 165q2 u4 )   45 2 2 2 + E qu 18M − qu − 4Exa 2
2 + x 2 − 12Mu + 255Mqu4 + 154q2 u6 = 0,

(16)

(17)

(18)

where the expressions are set to zero, because we are interested in determining the ISCO equation. From 16 and 17, E2 is found E2 = ε(2 − 2Mu − qu3 − 8Mqu4 + 7q2 u6 )   33 2 5 3 2 3 + x u M − 10Mqu − q u . 4

(19) 6

Innermost stable circular orbits of a Kerr-like metric with Quadrupole A fourth order polynomial for x is obtained from 17 and 19

A x4 + 2B x2 + C = 0,

(20)

where

A = u2 (1 − 6Mu + 9M2 u2 − 4Ma2 u3 + 33Mqu4 + 22q2 u6 )  B = −2εu M + (a2 − 6M2 )u    3 5 2 2 2 3 2 5 + Ma u − q u − Mqu + 6q u 2 2
C = ε2 4M2 − 8Ma2 u + 4a4 u2 − 12Mqu2 + 9q2 u4 .

(21)

The solution for x2 is given by

  √ 2 3 2 2ε  √ 3 2 5 a u ± M − qu − 7Mqu + 6q u , x = uZ∓ 2 2

(22)

with

Z± =



33 1 − 3Mu + Mqu4 + 11q2 u6 2



√ ± 2au Mu.

(23)

Inserting 22 in 19 one finds E2    √ 2ε E = (1 − 2Mu) 1 − 2Mu ± 2au Mu Z∓    1 25 29 2 6 2 3 4 + a M − q u + Mqu + q u 2 2 2 2

(24)

Substituting 24 and 22 in 14, L2z is determined L2z

     2εu 3 5 2 2 2 2 2 = M − 3M u + 2Ma − q u + 6M a − Mq u3 A 2 2     + M a4 − 12M2 a2 u4 + 5M2 a4 + 6q2 u5  √
4 4 2 3 2 2 ± 2Mau Mu a u − 2Ma u + 4a u − 6Mu + 3 ,

(25) 7

Innermost stable circular orbits of a Kerr-like metric with Quadrupole where A = u2 Z+ Z− . Finally, substituting 24, 25 and 22 in 15 and changing u = 1/r, the ISCO equation is found 

 1 P = Mr − 9M r + 3 6M − Ma + q r3 2   √ 33 29 − 7M2 a2 − Mq r2 − q2 ± 6Mar Mr∆ = 0. 2 2 5

IV.

2 4

3

2

(26)

Comparing with known metrics

Now, we proceeded to analyze the limiting cases for the possible r values. For this analysis we also used a REDUCE program which finds the solutions for equations such as the one obtained in 26. The first limiting case is when q = a = 0 which reduces to the known Schwarzschild metric case (r = 6M). For Schwarzschild, the relation found is the following

PSch = Mr5 − 9M2 r4 + 18M3 r3 = Mr3 (r − 6M) (r − 3M) = 0

(27)

Clearly, the Schwarzschild case is contained in 27. It is also found the values of the energy and angular momentum, reducing 22 and 19 and using r = 6M and ε = 1/2. 1 − 2Mu E= √ 1 − 3Mu r 8 = 9 s M Lz = u (1 − 3Mu) √ = 2 3M

(28)

(29)

As seen in 28 and 29 both values are the ones found by Chandrasekhar [4]. The other important case is Kerr, reducing 26 to Kerr means that q = 0. The ISCO equation for the Kerr metric found by Chandrasekhar and Pradhan [4, 11] is

√ r2 − 6Mr ∓ 8a Mr − 3a2 = 0,

(30) 8

Innermost stable circular orbits of a Kerr-like metric with Quadrupole Squaring the last expression, one gets r4 − 12Mr3 + 6(6M2 − a2 )r2 − 28Ma2 r + 9a4 = 0.

(31)

The simplification of 26 is 5

2 4



3

PKerr = Mr − 9M r + 3 6M − Ma

2



√ r3 − 7M2 a2 r2 ± 6Mar Mr∆ = 0. (32)

From last expression, after squaring, we get

(r4 − 12Mr3 + 6(6M2 − a2 )r2 − 28Ma2 r + 9a4 )×

(r3 − 6Mr2 + 9M2 r − 4Ma2 ) = 0

(33)

Obviously equation 30 is contained in 32, giving us the solutions for the Kerr case. The energy and angular momentum are s

  √ 1 1 − 2Mu ∓ au Mu , E= Z∓ √ √ a u± M , x=− √ uZ∓ s  √ M  2 2 Lz = ∓ a u + 1 ± 2au Mu . uZ∓

(34)

(35)

These values for E and Lz are exactly the same as the ones determined by Chandrasekhar [4] which validates the original equations 22, 19 and 25.

V. Discussion We have successfully found the ISCO equation for a Kerr-like metric with quadrupole, that reduces to the known equations for Kerr (q = 0) and Schwarzchild (q = a = 0) metrics. Graphical analysis concerning the dependence of the ISCO radius, energy and angular momentum with the mass, rotation parameter, and quadrupole parameter will be discussed in a future article. An important feature of the ISCO equation, is that it is quadratic in q, therefore it is easily solvable given the values of a, M and r ISCO . This means that, 9

Innermost stable circular orbits of a Kerr-like metric with Quadrupole given this model, indirect measurement of the quadrupole parameter of a compact object can be completely done from observational data regarding the mass, rotational parameter and ISCO radius of the object. Further applications, including analysis of observational data, as well as extending the ISCO equation for the inclusion of charged compact objects, will be done in future papers.

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