Innermost stable circular orbits of a Kerr-like Metric with Quadrupole
Fabian Chaverri-Miranda, Francisco Frutos-Alfaro, Pedro Gomez-Ovarez, Andree Oliva-Mercado
aa r X i v : . [ g r- q c ] J u l Innermost stable circular orbits of aKerr-like Metric with Quadrupole F abián C haverri M iranda F rancisco F rutos A lfaro P edro G ómez O varez A ndree O liva M ercado 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 obtainedfor an approximate Kerr-like spacetime with quadrupole moment. We de-rived the effective potential for the radial coordinate by the Euler-Lagrangemethod. This equation can be employed to measure the mass quadrupoleby observational means, because from this equation a quadratic polynomialfor the quadrupole moment can be found. As expected, the limiting cases ofthis equation are found to be the known cases of Kerr and Schwarzschild.
I. I ntroduction
In classical mechanics the orbit of a test particle around a massive object is ar-bitrary. This is because the effective potential is minimum, for any value of theangular momentum. Nevertheless, in general relativity this is not the case. Theeffective potential in the Schwarzschild metric has two extrema. When the an-gular momentum is minimum the two extrema become a single radius whichdescribes the innermost stable circular orbit (ISCO) [15].Naturally, the rotation of the central body influences the motion around ofa particle orbiting it, this is why the orbits around black holes differ betweenmetrics [9]. Another important feature of compact object is its quadrupole mo-ment ( 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 andfrequencies of the ISCO. Moreover, Geodesics analysis is cumbersome for such1nnermost stable circular orbits of a Kerr-like metric with Quadrupolemetrics. It would be useful for studying wave emission and chaotic trajectoriesof particles around compact objects [1, 10, 13].Studying the ISCO of black holes is important, because it gives informationabout the spacetime near the black hole and its background geometry [11, 12]. Itis important to recall the no hair conjecture. It states that the geometry outsidethe black hole horizon (Kerr-Newman metric) is expressed only in terms of threeparameters M (mass), a (rotation parameter) and e (charge). Nevertheless, forneutron stars this is not the case. Other parameters, such as deformation (massquadrupole), and magnetic dipole should be taken into account, but they requirethe equation of state of the neutron star to completely describe the spacetime nearthem. 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 = M . If the electric charge is zero, a Kerr type black hole is obtainedfrom the no hair conjecture [11, 14]. For the Kerr metric the radius of the ISCOdepends on the direction of motion of the particle in comparison with the blackhole. If the particle moves in the direction of the rotation of the black hole, thenthe radius becomes smaller r ISCO = M , if the particle moves counter rotationthen the radius is bigger than Schwarzschild’s and becomes r ISCO = M [15].In this paper, we are interested in analysing the ISCO of a Kerr-like metricwith mass quadrupole. The importance of this metric is that it reduces to theKerr spacetime and to the Hartle-Thorne (HT) case for certain limits [6, 5]. Inprinciple, it is possible to find an inner solution for the studied metric, by amatch with a metric that already matches HT. Moreover, the ISCO equation wasfound for HT metric [2, 3].This Kerr-like metric was deduced from the Kerr metric, for this reason it isexpected to obtain the known results for Kerr and Schwarzschild metrics for thevariables energy, angular momentum and the ISCO radius [8, 7].We are also interested in the observational applications. Nowadays, thereare no direct measurements of the quadrupole moment for compact objects, ananalysis of the ISCO structure for this Kerr-like metric could give us a hint toderive q via observational methods.This paper is organized as follows. The Kerr-like metric is introduced insection 2. A detailed calculation of the ISCO equation using the Euler-Lagrangemethod is in section 3. This method was developed by Chandrasekhar [4]. Insection 4, the ISCO equation is compared with the known solutions, Kerr andSchwarzchild black holes by means of a REDUCE program. The summary anddiscussion of the results are presented in section 5. 2nnermost stable circular orbits of a Kerr-like metric with Quadrupole II. T he K err - like M etric This metric describes the spacetime of a massive, rotating, deformed object. Ithas three parameters, the mass of the object, M , the rotation parameter, a andthe quadrupole parameter, q . It is an approximate solution of the Einstein fieldequations and is given by ds = g tt dt + g t φ dtd φ + g rr dr + g θθ d θ + g φφ d φ , (1)where the components of the metric are g tt = − e − ψ ρ h ∆ − a sin θ i g t φ = − Jr ρ sin θ g rr = ρ e χ ∆ (2) g θθ = ρ e χ g φφ = e ψ ρ (cid:20)(cid:16) r + a (cid:17) − a ∆ sin θ (cid:21) sin θ , (3)with J = Ma , ρ = r − a cos θ and ∆ = r − Mr + a . The exponents ψ and χ are given by ψ = qr P + Mqr P (4) χ = qr P + Mqr (cid:16) − + P + P (cid:17) + q r (cid:16) − P − P + P (cid:17) where P = ( θ − ) /2. As limiting cases this spacetime contains the Kerr( q =
0) and the Schwarzschild metrics ( q = a = III. D eriving the
ISCO
The method, we are using, was devised by Chandrasekhar [4]. The Lagrangianis given by L = µ ds d λ = µ ( g tt ˙ t + g t φ ˙ t ˙ φ + g rr ˙ r + g θθ ˙ θ + g φφ ˙ φ ) . (5)The dot over the variables t , r , θ and φ means derivative with respect to λ . Todetermine the ISCO of a test particle in the plane, one sets ˙ θ = θ = π /2.This leaves the Lagrangian as follows L = µ (cid:16) g tt ˙ t + g t φ ˙ t ˙ φ + g rr ˙ r + g φφ ˙ φ (cid:17) , (6)where the components of the metric becomes g tt = e − ψ ′ r h Mr − r i g t φ = − Jrg rr = r e χ ′ ∆ (7) g θθ = r e χ ′ g φφ = e ψ ′ r h r + Mra + r a i with the exponents ψ and χ are reduced to ψ ′ = − qr − Mqr (8) χ ′ = − qr − Mqr − q r The momenta of the three remaining variables are p t = µ (cid:0) g tt ˙ t + g t φ ˙ φ (cid:1) = − Ep r = µ g rr ˙ r (9) p φ = µ (cid:0) g t φ ˙ t + g φφ ˙ φ (cid:1) = L z µ = b = − g tt g φφ + g t φ , E and L z are constants that represent the energyand the angular momentum. ˙ t and ˙ φ are solved˙ t = b (cid:0) Eg φφ + L z g t φ (cid:1) (10)˙ φ = − b (cid:0) L z g tt + Eg t φ (cid:1) .The Hamiltonian is H = (cid:16) − E ˙ t + g rr ˙ r + L z ˙ φ (cid:17) = ε (11) = (cid:20) − b (cid:16) E g φφ + EL z g t φ + L z g tt (cid:17) + g rr ˙ r (cid:21) The effective potential V e f is then V e f = − ε g rr − g rr b ( E g φφ + EL z g t φ + L z g tt ) (12)Using u = r and the equations 8 to 7, then the effective potential in 12 canbe reduced to V e f = − ε (cid:18) − Mu + a u + qu − Mqu + q u (cid:19) + L z (cid:18) u + qu + Mqu + q u (cid:19) − Mu ( L z − Ea ) − E (cid:18) + a u − Mqu + q u (cid:19) (13)The latter expression has to be differentiated twice to find the values of r where the orbit is stable. dV e f du = − ε (cid:18) − M + a u + qu − Mqu + q u (cid:19) + L z (cid:16) u + qu + Mqu + q u (cid:17) − Mu ( L z − Ea ) − E (cid:18) a u − Mqu + q u (cid:19) (14)5nnermost stable circular orbits of a Kerr-like metric with Quadrupole d V e f du = − ε (cid:18) a + qu − Mqu + q u (cid:19) + L z (cid:16) + qu + Mqu + q u (cid:17) − Mu ( L z − Ea ) − E (cid:18) a − Mqu + q u (cid:19) (15)Now, we rewrite these equations using x = aE − L z as follows˜ V e f = V e f [ − qu + ( qu ) = ε (cid:18) − Mu + a u − qu + Mqu + q u (cid:19) + E (cid:18) − + qu + Mqu − q u (cid:19) − Exau + x u (cid:18) − Mu + Mqu + q u (cid:19) =
0, (16) d ˜ V e f du = dV e f du [ − qu + ( qu ) ]= ε (cid:0) − M + a u + qu + Mqu − q u (cid:1) + E qu (cid:18) M − qu (cid:19) − Exau + x u (cid:0) − Mu + Mqu + q u (cid:1) =
0, (17) d ˜ V e f du = d V e f du [ − qu + ( qu ) ]= ε ( a + qu − Mqu − q u )+ E qu (cid:18) M − qu (cid:19) − Exa + x (cid:0) − Mu + Mqu + q u (cid:1) =
0, (18)where the expressions are set to zero, because we are interested in determiningthe ISCO equation.From 16 and 17, E is found E = ε ( − Mu − qu − Mqu + q u )+ x u (cid:18) M − Mqu − q u (cid:19) . (19)6nnermost stable circular orbits of a Kerr-like metric with QuadrupoleA fourth order polynomial for x is obtained from 17 and 19 A x + B x + C =
0, (20)where A = u ( − Mu + M u − Ma u + Mqu + q u ) B = − ε u (cid:18) M + ( a − M ) u + (cid:18) Ma u − q (cid:19) u − Mqu + q u (cid:19) C = ε (cid:0) M − Ma u + a u − Mqu + q u (cid:1) . (21)The solution for x is given by x = ε uZ ∓ (cid:20) (cid:16) a √ u ± √ M (cid:17) − qu − Mqu + q u (cid:21) , (22)with Z ± = (cid:18) − Mu + Mqu + q u (cid:19) ± au √ Mu . (23)Inserting 22 in 19 one finds E E = ε Z ∓ (cid:20) ( − Mu ) (cid:16) − Mu ± au √ Mu (cid:17) (24) + (cid:18) a M − q (cid:19) u + Mqu + q u (cid:21) Substituting 24 and 22 in 14, L z is determined L z = ε u A (cid:20) M − M u + (cid:18) Ma − q (cid:19) u + (cid:18) M a − Mq (cid:19) u + M (cid:16) a − M a (cid:17) u + (cid:16) M a + q (cid:17) u ± Mau √ Mu (cid:0) a u − Ma u + a u − Mu + (cid:1)(cid:21) , (25)7nnermost stable circular orbits of a Kerr-like metric with Quadrupolewhere A = u Z + Z − .Finally, substituting 24, 25 and 22 in 15 and changing u = r , the ISCOequation is found P = Mr − M r + (cid:18) M − Ma + q (cid:19) r − (cid:18) M a − Mq (cid:19) r − q ± Mar √ Mr ∆ =
0. (26)
IV. C omparing with known metrics
Now, we proceeded to analyze the limiting cases for the possible r values. Forthis analysis we also used a REDUCE program which finds the solutions forequations such as the one obtained in 26.The first limiting case is when q = a = r = M ). For Schwarzschild, the relation found isthe following P Sch = Mr − M r + M r = Mr ( r − M ) ( r − M ) = r = M and ε = E = − Mu √ − Mu = r
89 (28) L z = s Mu ( − Mu )= √ M (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. TheISCO equation for the Kerr metric found by Chandrasekhar and Pradhan [4, 11]is r − Mr ∓ a √ Mr − a =
0, (30)8nnermost stable circular orbits of a Kerr-like metric with QuadrupoleSquaring the last expression, one gets r − Mr + ( M − a ) r − Ma r + a =
0. (31)The simplification of 26 is P Kerr = Mr − M r + (cid:16) M − Ma (cid:17) r − M a r ± Mar √ Mr ∆ =
0. (32)From last expression, after squaring, we get ( r − Mr + ( M − a ) r − Ma r + a ) × ( r − Mr + M r − Ma ) = E = s Z ∓ (cid:18) − Mu ∓ au √ Mu (cid:19) , (34) x = − a √ u ± √ M √ uZ ∓ , L z = ∓ s MuZ ∓ (cid:16) a u + ± au √ Mu (cid:17) . (35)These values for E and L z are exactly the same as the ones determined byChandrasekhar [4] which validates the original equations 22, 19 and 25. V. D iscussion
We have successfully found the ISCO equation for a Kerr-like metric with quadru-pole, 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 andangular momentum with the mass, rotation parameter, and quadrupole parame-ter will be discussed in a future article.An important feature of the ISCO equation, is that it is quadratic in q , there-fore it is easily solvable given the values of a , M and r ISCO . This means that,9nnermost stable circular orbits of a Kerr-like metric with Quadrupolegiven this model, indirect measurement of the quadrupole parameter of a com-pact 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 ex-tending the ISCO equation for the inclusion of charged compact objects, will bedone in future papers. R eferenceseferences