Temperature oscillations of a gas moving close to circular geodesic in Reissner-Nordström spacetime
aa r X i v : . [ g r- q c ] D ec Temperature oscillations of a gas moving close tocircular geodesic in Reissner-Nordstr¨om spacetime
Leandro C. Mehret ∗ and Gilberto M. Kremer † Departamento de F´ısica, Universidade Federal do Paran´aCaixa Postal 19044, 81531-980 Curitiba, Brazil
Abstract
The objective of this work is to analyze the temperature oscillationsthat occur in a gas in a circular motion under the action of a Reissner-Nordstr¨om gravitational field, verifying the effect of the charge term ofthe metric on the oscillations. The expression for temperature oscillationsfollows from Tolman’s law written in Fermi normal coordinates for a co-moving observer. The motion of the gas is close to geodesic so the equationof geodesic deviation was used to obtain the expression for temperatureoscillations. Then these oscillations are calculated for some compact stars,quark stars, black holes and white dwarfs, using values of electric chargeand mass from models found in the literature. Comparing the variousmodels analyzed, it is possible to verify that the role of the charge isthe opposite of the mass. While the increase of the mass produces areduction in the frequencies, amplitude and in the ratio between the fre-quencies, the increase of the electric charge produces the inverse effect. Inaddition, it is shown that if the electric charge is proportional to the mass,the ratio between the frequencies does not depend on the mass, but onlyon the proportionality factor between charge and mass. The ratios be-tween the frequencies for all the models analyzed (except for supermassiveblack holes in the extreme limit situations) are close to the 3 / The Reissner-Nordstr¨om metric is a solution of Einstein’s field equations thatcorresponds to the gravitational field produced by an electrically charged mas-sive object (see e.g. the books [1, 2, 3]). Some theoretical models use the ideaof electrically charged astronomical objects to explain certain observable phe-nomena [4, 5, 6, 7], some of them involving compact objects such as neutronstars, quark stars, white dwarfs and black holes. ∗ [email protected] † kremer@fisica.ufpr.br − , + , + , +) for the components of the metric tensoris used.We consider a rarefied gas inside a spacecraft in a circular orbit around acharged massive object where the center of mass of the gas is moving on a cir-cular orbit with geodesic deviation of the Reissner-Nordstr¨om metric. This is arelativistic gas so it can be described using Boltzmann equation and, consideringthe equilibrium, Maxwell-J¨uttner distribution function [10, 8]. From this, Tol-man’s law √− g T = constant can be obtained, establishing a relation betweenthe temperature and the geometry of the spacetime in equilibrium [11, 13]. Thisdescription using a Boltzmann gas at equilibrium is an idealized toy model, withseveral limitations but perhaps can illustrate some features of a real situation.In the present study, we obtained the proper frequencies of the motion byanalyzing the Lagrangian of a test particle of the gas in Reissner-Nordstr¨ommetric. These frequencies were obtained in order to compare them with thefrequencies calculated considering the geodesic deviation approximation for themotion of the gas. The orbital motion of a test particle around compact objects(in particular, the black hole in the galactic center) is subject for several recentpublications [14, 15, 16].Because of the nature of the geodesic motion, it will be convenient to useFermi normal coordinates. These coordinates make the Christoffel symbolsvanish along the geodesic, leaving the metric locally rectangular [20]. The equi-librium condition can be applied to an approximately geodesic motion by theuse of the geodesic deviation equation, since the perturbation terms are linearin distance. The geodesic deviation equation leads to the same frequencies ob-tained in the orbital motion analysis. The methodology applied in the presentstudy was already used to describe other physical situations, as for example tocalculate gravitational perturbation of the hydrogen spectrum [17], the Shirokoveffect for sattelite orbits [18], and gravitationally induced supercurrents relatedto this effect [19].The phenomena of QPOs (quasi-periodic oscillations) are present in manygalactic black holes and neutron star sources in low-mass X-ray binaries [21].The frequencies of some QPOs are in the kHz range, corresponding to orbitalfrequencies next from the central black hole or neutron star. These are knownas HF (high frequency) QPOs. These oscillations often show up in pairs, namedtwin peak HF QPOs [22]. These two peaks correspond to different frequencies:the upper peak is assumed to be the modulation of the azimuthal frequency,whereas the lower one is a precession frequency. Typically, the ratio betweenthese two frequencies is equal to 3 /
2. 2his 3 / kHz range and the ratio of them is compared with the 3 / Consider a relativistic gas of particles with rest mass m in a spacetime withmetric tensor g µν . The mass shell condition g µν p µ p ν = − m c allows us todescribe the state of this gas by the one-particle distribution function f ( ~x, ~p, t )in the phase space spanned by the spacetime coordinates ( x µ ) = ( ct, ~x ) and themomenta ( p µ ) = ( p , ~p ) [8]. The evolution of the distribution function in thephase space is given by the Boltzmann equation [10] p µ ∂f∂x µ − Γ σµν p µ p ν ∂f∂p σ = Q ( f, f ) , (1)where Γ σµν are the Christoffel symbols and Q ( f, f ) the collision operator. Inequilibrium, the distribution function becomes the Maxwell-J¨uttner distributionfunction f (0) = n πkT m cK ( ζ ) exp (cid:18) U τ p τ kT (cid:19) . (2)Here n is the particle number density, T is the temperature, U τ is the four-velocity, k is the Boltzmann constant. K n ( ζ ) denotes modified Bessel functionsof the second kind with ζ = mc /kT , given explicitly by K n ( ζ ) = (cid:18) ζ (cid:19) n Γ(1 / n + 1 / Z ∞ exp( − ζy )( y − n − dy (3)Using the equilibrium distribution function (2) we can calculate the energy-3omentum tensor T µν = c Z p µ p ν f (0) √− g d pp = (cid:18) en + pc (cid:19) U µ U ν + pg µν (4)In the equation (4), g denotes the determinant of the metric tensor, and theenergy per particle e and pressure p are given by e = mc (cid:20) K ( ζ ) K ( ζ ) − ζ (cid:21) , p = nkT. (5)In equilibrium the entropy-flow vector reduces to S µ = − kc Z p µ f (0) ln f (0) √− g d pp = nsU µ (6)where s denotes the entropy per particle s = k (cid:20) ln (cid:18) πkT m cK ( ζ ) n (cid:19) + ζ K ( ζ ) K ( ζ ) − (cid:21) . (7)The chemical potential µ is associated with the Gibbs function per particle µ = e − T s + p/n and it is given by µ = kT (cid:20) n πkT m cK ( ζ ) (cid:21) . (8)Now using (8) the equilibrium distribution function (2) can be written as f (0) = exp (cid:18) µkT − U τ p τ kT (cid:19) . (9)The equilibrium distribution function (2) is obtained from the condition thatat equilibrium the collision operator Q ( f, f ) vanishes (for more details see e.g.[10]). If we insert (9) into the left-hand side of Boltzmann equation (1) results p ν ∂ ν h µkT i − p µ p ν (cid:20) U ν kT (cid:21) ; µ + (cid:20) U µ kT (cid:21) ; ν ! = 0 . (10)The expression (10) is valid for all p µ so ∂ ν h µkT i = 0 , (cid:20) U ν kT (cid:21) ; µ + (cid:20) U µ kT (cid:21) ; ν = 0 . (11)The right-hand expression of the above equation is known as Killing equation,and U ν /kT is a (timelike) Killing vector. This expression can be rewritten as[ U µ T − ] ; ν + [ U ν T − ] ; µ = U µ ; ν + U ν ; µ − T − ( U µ T ,ν + U ν T ,µ ) = 0 (12)4y choosing suitable projections proportional and perpendicular to the four-vector U α we get from (12)˙ T = 0 , ˙ U µ + c T ∇ µ T = 0 . (13)In the above relations we used the definitions ˙ T ≡ U µ ∂ µ T, ˙ U µ ≡ U ν U µ ; ν , ∇ µ T ≡ h νµ T ,ν with h µν = g µν + c − U µ U ν denoting the projector.The interpretation of equations (13) is that at equilibrium a gas must havea stationary temperature and its acceleration must be counterbalanced by aspatial temperature gradient [8]. Note that there are situations where equation(13) is compatible with geodesic motion, as for example the case for equilibriumtori [12], where even if temperature is not uniform within the gas, there maybe one world line on which it reaches a maximum value, and the world linewould be geodesic. However, for a rarefied gas in circular motion around acharged massive object with strong gravitational field, the right-hand expressionof condition (13) is not compatible with a geodesic fluid motion, which wouldrequire ˙ U µ = 0. In section 4.2 we shall return to this point.We can obtain Tolman’s law by considering a fluid in rest with U µ = (cid:16) c/ √− g ,~ (cid:17) . Indeed, by taking into account the existence of a timelike Killingvector amounts to a stationary metric, the acceleration equation reduces to˙ U µ = − c g Γ µ = c g g µν g ,ν . (14)Hence, from the right-hand expression of (13) and from (14) we get c g µν [(ln √− g T ) ,µ ] = 0 , (15)and it follows Tolman’s law √− g T = constant . (16)The equilibrium condition given by the first equation (11) together withTolman’s law (16) implies Klein’s law [13], namely √− g µ = constant. The Reissner-Nordstr¨om metric reads ds = − (cid:18) − M ′ r + Q r (cid:19) ( dx ) + 1 (cid:16) − M ′ r + Q r (cid:17) dr + r ( dθ + sin θdφ ) = − c dτ , (17)where M ′ = GM/c and Q = q G/ (4 πǫ c ). M and q denote the mass and theelectric charge of a massive and charged object, G is the gravitational constant, c ǫ the vacuum permittivity. The coordinate x corresponds to the temporal coordinate, and ( r, θ, φ ) are the spatial coordinatesin the spherical coordinate system. The invariant ds = − c dτ defines theproper time τ .The event horizon for the Reissner-Norstr¨om metric is defined by the ex-pression [3]: ∆ = r − M ′ r + Q . (18)The roots of (18) are r + = M ′ + p M ′ − Q and r − = M ′ − p M ′ − Q . Theseroots will be real and distinct if M ′ > Q . In this work we will adopt as theReissner-Norstr¨om radius R RN = M ′ + p M ′ − Q , i.e., the root r + of (18).This choice was taken in order to avoid the interchange of the spacelike andtimelike behavior of the coordinates r and t , that occurs in the region between r + and r − [25]. Other choices are possible but it’s needed to deal with pointswhere the behavior of spacelike and timelike coordinates change.The orbital motion of a test particle with rest mass m in the plane θ = π/ M and electric charge q isdescribed by the Lagrangian L = m (cid:16) − M ′ r + Q r (cid:17) (cid:18) drdτ (cid:19) + r (cid:18) dφdτ (cid:19) − (cid:18) − M ′ r + Q r (cid:19) (cid:18) dx dτ (cid:19) , (19)while the generalized momenta corresponding to the cyclic coordinates x and φ are expressed by p = ∂ L ∂ (cid:0) dx dτ (cid:1) = − m (cid:18) − M ′ r + Q r (cid:19) (cid:18) dx dτ (cid:19) = − Ec , (20) p φ = ∂ L ∂ (cid:16) dφdτ (cid:17) = mr (cid:18) dφdτ (cid:19) = l φ . (21)In the above equations E is the energy of the particle and l φ its angular mo-mentum. In order to keep the dimensions, we will keep the value of c in ourcalculations. The center of the spherical coordinate system are defined as thecenter of the source of the gravitational field.By introducing the dimensionless energy ǫ = E/mc and the dimension-less angular momentum l = l φ /mc and taking into account a circular orbit ofconstant radius, one can find from (17), (20) and (21) the relation: ǫ = (cid:18) − M ′ r + Q r (cid:19) (cid:18) l r (cid:19) . (22)Furthermore, from the equation of motion for the test particle it follows that (cid:18) drdcτ (cid:19) + V = ǫ , (23)6here V is an effective potential defined by V = (cid:18) − M ′ r + Q r (cid:19) (cid:18) l r (cid:19) . (24)From the extreme values of the effective potential (maximum and minimumpoints of the function, where dV /dr = 0) one can get the possible circular orbitsfor the test particle, namely l = r (cid:0) M ′ r − Q (cid:1) r − M ′ r + 2 Q . (25)Now the insertion of (25) into (22) leads to the following expression for thedimensionless energy: ǫ = (cid:18) − M ′ r + Q r (cid:19) (cid:18) r r − M ′ r + 2 Q (cid:19) . (26)Equations (25) and (26) correspond to the values of the angular momentum andenergy of the test particle with rest mass m in orbital motion with θ = π/ dφ/dτ and for the angularmomentum l and integrating the resulting equation yields φ = 1 r s M ′ r − Q r − M ′ r + 2 Q cτ, (27)where the integration constant disappear with a simple redefinition of the vari-able φ . The angular frequency ω φ for the particle motion can be defined as: ω φ ≡ dφdτ = ω N s r ( r − Q /M ′ ) r − M ′ r + 2 Q , ω N = r GMr . (28)In (28), ω N is the Newtonian frequency in the limits r ≫ M ′ and r ≫ Q . Notethat the definition for M ′ = GM/c was used.According to ref. [1] another oscillation frequency can be obtained when theparticle is slightly displaced from the circular motion in the radial direction.This radial frequency ω r is defined in terms of the second derivative of theeffective potential, namely ω r ≡ r c d Vdr = ω N s r − M ′ r + 9 Q − Q /M ′ rr − M ′ r + 2 Q . (29)Here we have used the expression (25) for l .By considering a vanishing electric charge Q = 0 the two frequencies (28)and (29) reduce to the expressions for a Schwarzschild metric [1]: ω φ = ω N r rr − M ′ , ω r = ω N r r − M ′ r − M ′ . (30)7s was pointed out by Wald [1] the difference of the two frequencies impliesin a precession of the particle motion inside the range of the orbital plane. Inthe Reissner-Nordstr¨om metric the precession frequency ω p is given by ω p = ω φ (cid:18) − ω r ω φ (cid:19) = ω φ − s − M ′ /r + 9 Q /r − Q /M ′ r − Q /M ′ r ! . (31)There are other frequencies that can be calculated from the Lagrangian ofthe free particle (for example, frequencies related to the coordinate time x = t such as dφ/dt ). In the present work we calculated ω r and ω φ in order to comparethem with the frequencies obtained in the next section. We follow ref. [8] and consider a rarefied gas inside a spacecraft which is in acircular orbit around an object with mass M and electric charge q . We assumethat the contributions of the spacecraft and the gas to the gravitational fieldcan be neglected and consider that the center of mass of the gas is movingon a circular geodesic of the Reissner-Nordstr¨om metric. The geometry andsymmetries of the problem are explored by using the Fermi normal coordinates(exploring the geodesic motion) and spherical coordinates (exploring the circularorbit). In 1922, Enrico Fermi showed that, given any curve in a Riemannian manifold,it’s possible to introduce coordinates near this curve in such a way that theChristoffel symbols vanish along the curve, leaving the metric locally rectangular[20]. If we consider the curve in question a geodesic, this particularization ofFermi’s idea leads to a coordinate system known as Fermi normal coordinates.In order to describe the local gravitational effects in the vicinity of thegeodesic, a smart choice for an observer at the center of mass is to use Ferminormal coordinates [20], which are comoving and time-orthogonal coordinateswith the center of mass at rest in the origin. The proper time τ of the center onthe geodesic is the time coordinate while the spatial coordinates are orthogonalspace-like geodesics parametrized by the proper distance.We begin by writing the non-vanishing components of the Riemann tensor8or Reissner-Nordstr¨om metric in the case where θ = π/ R ¯0¯1¯0¯1 = − M ′ r + 3 Q r , R ¯2¯3¯2¯3 = 2 M ′ r − Q ,R ¯0¯2¯0¯2 = R ¯0¯3¯0¯3 = M ′ − Q /rr (cid:18) − M ′ r + Q r (cid:19) ,R ¯1¯2¯1¯2 = R ¯1¯3¯1¯3 = − M ′ − Q /rr (cid:18) − M ′ r + Q r (cid:19) − . (32)In the equations (32), the overbar denotes the original Reissner-Nordstr¨omcoordinates according to (17). The numerical index 0 denotes the temporalcoordinate, and the indexes (1 , ,
3) denote spatial coordinates.For a circular geodesic in a Reissner-Nordstr¨om field, the Fermi normaltetrads has the same form of the tetrads for a Schwarzschild field [26, 17], andare given by: ( e ¯ α ˆ0 ) = (cid:18) ǫX , , , lr (cid:19) , (33)( e ¯ α ˆ1 ) = (cid:18) − l sin( αφ ) r √ X , √ X cos( αφ ) , , − ǫ sin( αφ ) r √ X (cid:19) , (34)( e ¯ α ˆ2 ) = (cid:18) , , r , (cid:19) , (35)( e ¯ α ˆ3 ) = (cid:18) l cos( αφ ) r √ X , √ X sin( αφ ) , , ǫ cos( αφ ) r √ X (cid:19) , (36)where the following abbreviations were introduced: α = p r − M ′ r + 2 Q r , X = 1 − M ′ r + Q r . (37)Given some initial time, ( e ¯ α ˆ1 ) points to the radial direction and ( e ¯ α ˆ3 ) points tothe tangential direction, while ( e ¯ α ˆ2 ) is always perpendicular to the orbital plane.On the circular geodesics g αβ ∂x α ∂x ˆ µ ∂x β ∂x ˆ ν = η ˆ µ ˆ ν (38)is valid, where η ˆ µ ˆ ν is the Minkowski metric. The tetrads are parallel transportedalong the circular geodesic De ¯ µ ˆ α dτ = 0 , (39)where the operator D/dτ is the absolute derivative with respect to the propertime τ [25].The components of Riemann tensor in Fermi normal coordinates can becalculated by using the expression [20]: R ˆ µ ˆ ν ˆ σ ˆ τ = R ¯ α ¯ β ¯ γ ¯ δ ˆ e ¯ α ˆ µ ˆ e ¯ β ˆ ν ˆ e ¯ γ ˆ σ ˆ e ¯ δ ˆ τ , (40)9here the hat over the indexes refers to coordinates in the Fermi system. Thenon-vanishing components of the Riemann tensor are: R ˆ0ˆ1ˆ0ˆ1 = (cid:0) r Q − M ′ rQ − r M ′ (cid:1) r ( r − M ′ r + 2 Q )+ cos (2 αφ )2 r ( r − M ′ r + 2 Q ) (cid:0) Q − M ′ r (cid:1) (cid:0) r − M ′ r + Q (cid:1) , (41) R ˆ0ˆ2ˆ0ˆ2 = M ′ r − Q r ( r − M ′ r + 2 Q ) , (42) R ˆ0ˆ3ˆ0ˆ3 = (cid:0) r Q − M ′ rQ − r M ′ (cid:1) r ( r − M ′ r + 2 Q ) − cos (2 αφ )2 r ( r − M ′ r + 2 Q ) (cid:0) Q − M ′ r (cid:1) (cid:0) r − M ′ r + Q (cid:1) , (43) R ˆ0ˆ1ˆ0ˆ3 = (cid:0) Q − M ′ r (cid:1) (cid:0) r − M ′ r + Q (cid:1) sin (2 αφ )2 r ( r − M ′ r + 2 Q ) , (44) R ˆ0ˆ1ˆ1ˆ3 = p ( r − M ′ r + Q ) ( M ′ r − Q ) r ( r − M ′ r + 2 Q ) (cid:0) M ′ r − Q (cid:1) cos( αφ ) , (45) R ˆ0ˆ2ˆ1ˆ2 = − p ( r − M ′ r + Q ) ( M ′ r − Q ) r ( r − M ′ r + 2 Q ) (cid:0) M ′ r − Q (cid:1) sin( αφ ) . (46)The following relations also hold: R ˆ0ˆ3ˆ1ˆ3 = − (4 Q − M ′ r )(2 Q − M ′ r ) R ˆ0ˆ2ˆ1ˆ2 , R ˆ0ˆ2ˆ2ˆ3 = − (2 Q − M ′ r )(4 Q − M ′ r ) R ˆ0ˆ1ˆ1ˆ3 , (47) R ˆ1ˆ2ˆ2ˆ3 = − (2 Q − M ′ r )(4 Q − M ′ r ) R ˆ0ˆ1ˆ0ˆ3 , R ˆ1ˆ3ˆ1ˆ3 = (2 Q − r ) r R ˆ0ˆ2ˆ0ˆ2 , (48) R ˆ2ˆ3ˆ2ˆ3 = 2 Q ( r − M ′ r + Q ) cos ( αφ ) r ( r − M ′ r + 2 Q ) − R ˆ0ˆ1ˆ0ˆ1 , (49) R ˆ1ˆ2ˆ1ˆ2 = 2 Q ( r − M ′ r + Q ) sin ( αφ ) r ( r − M ′ r + 2 Q ) − R ˆ0ˆ3ˆ0ˆ3 . (50)The component g ˆ0ˆ0 of the metric tensor up second order in geodesic deviationin the Fermi normal coordinates is given by (see refs. [2, 20]): g ˆ0ˆ0 = − − R ˆ0ˆ n ˆ0 ˆ m x ˆ n x ˆ m . (51)Because of the non-vanishing component R ˆ0ˆ1ˆ0ˆ3 , the second order contribu-tion in g ˆ0ˆ0 is not diagonal. Here we use the same procedure adopted in ref.[8] to obtain a simpler expression by performing a tetrad rotation around thedirection x ˆ2 perpendicular to the orbital plane. The transformation is given by E ¯0 ≡ e ˆ0 , E ¯1 ≡ e ˆ1 cos( αφ ) + e ˆ3 sin( αφ ) , (52) E ¯2 ≡ e ˆ2 , E ¯3 ≡ e ˆ1 sin( αφ ) − e ˆ3 cos( αφ ) , (53)10here ( E ¯ α ˆ1 ) always shows in the radial direction and ( E ¯ α ˆ3 ) always shows in thetangential direction. So that the components of the Riemann tensor in therotated system can be calculated through R abcd = ∂ E ¯ µ ∂ e ˆ a ∂ E ¯ ν ∂ e ˆ b ∂ E ¯ σ ∂ e ˆ c ∂ E ¯ τ ∂ e ˆ d R ˆ µ ˆ ν ˆ σ ˆ τ . (54)In this frame, the relevant Riemann tensor components for the calculationof g take a simpler form: R = 2 Q − M ′ rQ + 3 Q r − M ′ r + 3 M ′ r r ( r − M ′ r + 2 Q ) , (55) R = M ′ r − Q r ( r − M ′ r + 2 Q ) , R = M ′ r − Q r . (56)Using the above expressions for the components of the Riemann tensor in(51) we obtain: g = − M ′ r − M ′ + Q ) r + 6 M ′ Q r − Q r ( r − M ′ r + 2 Q ) ( x ) − M ′ r − Q r ( r − M ′ r + 2 Q ) ( x ) − M ′ r − Q r ( x ) . (57)One important consequence of the choice of the tetrads (52) and (53) is thatthey are no longer parallel transported. From the condition of parallel transportof the tetrads D e ˆ1 /dτ = 0 and D e ˆ3 /dτ = 0 and the relationships (52) and (53)it follows that D E ¯1 dτ = r M ′ − Q /rr E ¯3 , D E ¯3 dτ = − r M ′ − Q /rr E ¯1 . (58)The above equations imply that the tetrad transformations describe a rotationwith frequency ω = r M ′ − Q /rr , (59)with non-vanishing Christoffel symbols on the geodesic and mixed space-timeterms linear in x i in the metric, namely (see ref. [8])Γ = r M ′ − Q /rr Γ = − r M ′ − Q /rr , (60) g = r M ′ − Q /rr x g = − r M ′ − Q /rr x . (61) A geodesic fluid motion is characterized by the condition that the four-velocityobeys the equation ˙ U µ = 0. The four-velocity of a gas at equilibrium is not11ompatible with the equation of the geodesic motion, since it depends on thegradient of the temperature (see ref. [8, 10]), namely˙ U µ + c T ∇ µ T = 0 . (62)However, that terms that ”perturb” the geodesic behavior are linear in dis-tance, so the equation of geodesic deviation can be applied in our problem [8].The general equation for geodesic deviation for a vector ξ α orthogonal to thegeodesics is D ξ α dτ + R αγµν U γ U ν ξ µ = 0 , (63)where D ξ α /dτ is given by D ξ α dτ = d ξ α dτ + Γ αβγ,ρ ξ β dx ρ dτ dx γ dτ + 2Γ αβγ dξ β dτ dx γ dτ +Γ αβγ Γ βρσ ξ ρ dx σ dτ dx γ dτ + Γ αβγ ξ β d x γ dτ . (64)The last term in (64) can be written asΓ αβγ ξ β d x γ dτ = − Γ αβγ ξ β (cid:20) Γ γκλ U κ U λ + c T ∇ γ T (cid:21) , (65)thanks to the second equilibrium equation (13) which can be rewritten as˙ U γ = DU γ dτ = D x γ dτ = − c T ∇ γ T, d x γ dτ = − Γ γκλ U κ U λ − c T ∇ γ T. (66)By neglecting the second-order contributions of the temperature gradient,approximating the four-vector by ( U µ ) = ( c,~
0) and neglecting the second-ordercontributions to g = − O ( x ), the expression (63) can be written as d x a dτ + 2 c Γ aβ dx β dτ − c Γ aβγ x β Γ γ + c Γ aβ Γ βρ x ρ + c g aβ R β µ x µ = 0 , (67)when one considers only the perturbation terms with linear dependence in thedistance and makes use of the expression R αγµν U γ U ν ξ µ = c g αβ R β µ ξ µ . (68)In (67) the spatial components of ξ a were identified with the components ofthe tetrad system x a .Now by using the expressions for the components of the Riemann tensor (55),(56), Christoffel symbol (60) and metric tensor (61) and replacing a = 1 , , x a : d x dτ + c (cid:20) Q − M ′ rQ + 4 Q r − M ′ r + 6 M ′ r r ( r − M ′ r + 2 Q ) (cid:21) x − c r M ′ − Q r dx dτ = 0 , (69) d x dτ + c ( M ′ r − Q ) r ( r − M ′ r + 2 Q ) x = 0 , (70) d x dτ + 2 c r M ′ r − Q r dx dτ = 0 . (71)From the above system of differential equations we infer that (70) decouplesfrom the two other equations and has a real solution given by x = x sin(Ω τ ) , (72)where Ω denotes the frequency of the harmonic motion of the x componentΩ = Ω N s r ( r − Q /M ′ ) r − M ′ r + 2 Q , Ω N = r GMr . (73)Note that Ω coincides with the orbital frequency ω φ for the test particle in (28).The real solutions of the coupled system of equations (69) and (71) read x = x sin( ωτ ) , (74) x = 2 s ( M ′ r − Q )( r − M ′ r + 2 Q ) M ′ r − M ′ r + 9 Q M ′ r − Q x cos( ωτ ) . (75)Here ω characterizes oscillations in the plane ( x , x ) ω = ω N s r − M ′ r + 9 Q r − Q /M ′ r ( r − M ′ r + 2 Q ) , ω N = r GMr , (76)and it coincides with the radial frequency ω r for the test particle (29). Thisfrequency refers to oscillations in the tangential direction and describes an ellipsein the ( x , x ) plane. However here there is no precession in the orbital planefor a comoving observer.In the limit r ≫ M ′ and r ≫ Q these frequencies coincide, i.e., Ω = Ω N = ω = ω N . By analyzing the oscillation frequencies Ω and ω given by (73) and (76), we caninfer that the roots of the following polynomials: r − M ′ r + 9 Q r − Q /M ′ = 0 , (77) r − M ′ r + 2 Q = 0 , (78)13epresent critical points for the analysis of the frequencies, because these poly-nomials are the denominators of frequency expressions (73) and (76). The poly-nomial in equation (77) has one real root, namely r ω = 2 M ′ + 4 M ′ − Q D / + D / , (79)where D = 8 M ′ − M ′ Q + 2 Q /M ′ + p M ′ Q − Q + 4 Q /M ′ . (80)The polynomial in equation (78) has two real roots: r ω − = 12 (3 M ′ − p M ′ − Q ) , r ω + = 12 (3 M ′ + p M ′ − Q ) (81)It’s important to note that the radii from expressions (79) and (81) arenot related with the event horizons defined by the roots of equation (18), butrepresent limit points of orbit instability. Another important point for thisanalysis is when r = Q /M ′ , since at this point Ω = 0. At this point we cananalyze the following regimes for orbital stability: • r > r ω : in this region, all circular orbits are stable. When r reaches thelimit r → r ω , we have ω → x − x plane, being restricted to the x plane. • r ω + < r < r ω : in this region, there exist unstable circular trajectories.Here, ω becomes imaginary and we have exponential instabilities in the x − x plane. The oscillations grows to infinity when the limit r = r ω + is approached. • r ω − < r < r ω + : in this region there also unstable trajectories. But here,Ω becomes imaginary and the exponential instabilities are in the x plane.When r reaches the limit r → Q /M ′ , we have Ω → x plane, being restricted to the x − x plane.If the charge values Q are small, we can expand the frequencies ω and Ω inseries around Q = 0, yielding ωω N = r r − M ′ r − M ′ + Q (7 r − M ′ )2 r p ( r − M ′ )( r − M ′ ) + O ( Q ) , (82)ΩΩ N = r rr − M ′ − Q ( r − M ′ )2 M ′ p r ( r − M ′ ) + O ( Q ) . (83)In the limit Q → ω reduce to those found in ref. [8]for a Schwarzschild metric. 14 Temperature oscillations
Let us turn to the problem of a gas at equilibrium inside a spacecraft in acircular geodesic motion in a spacetime described by the Reissner-Nordstr¨ommetric. As was pointed out the temperature field obeys Tolman’s law √− g T =constant and if T is a constant equilibrium temperature on the geodesics, wecan approximate the temperature field in the vicinity of the central geodesic byusing (57), yielding TT ≈ M ′ r − M ′ + Q ) r + 6 M ′ Q r − Q r ( r − M ′ r + 2 Q ) ( x ) − M ′ r − Q r ( r − M ′ r + 2 Q ) ( x ) − M ′ r − Q r ( x ) . (84)Now the replacement of the values of x , x and x from (74), (72) and (75)in the temperature profile (84) results: T − T T = ∆( τ ) = M ′ ( x ) r [( A − B ) − ( A + B ) cos(2 ωτ ) − C (cid:18) x x (cid:19) (1 − cos(2Ω τ )) , (85)where the coefficients A , B and C are given by: A = 2 M ′ r − Q + M ′ ) r + 6 M ′ Q r − Q M ′ r ( r − M ′ r + 2 Q ) (86) B = 4( M ′ r − Q ) ( r − M ′ r + 2 Q ) M ′ r ( M ′ r − M ′ r + 9 Q M ′ r − Q ) (87) C = r ( M ′ r − Q ) M ′ ( r − M ′ r + 2 Q ) . (88)The coefficients A , B and C as well as the frequencies Ω and ω , can be deter-mined from knowledge of the mass, electric charge and radius of the gravitationalfield source. The constants x e x are free parameters.Note that ∆( τ ) is a dimensionless and normalized amplitude of the tem-perature oscillations. It was chosen because it can be calculated even if we donot know the equilibrium temperature. In the next subsections this amplitudewill be analyzed for some theoretical models from the literature that considerthe existence of electric charge in some compact objects. The problem con-sists of a gas in circular orbit around a charged massive object that produces astrong gravitational field. To avoid the region between the event horizons of theReissner-Nordstr¨om metric, where occurs the interchange of the spacelike andtimelike components of the metric tensor [25], circular orbits with r = 5 R RN ,where R RN is the Reissner-Nordstr¨om radius, are considered. The choice ofthis multiple of the Reissner-Nordstr¨om radius is arbitrary but also takes intoaccount the orbit stability. We also take x = x = 1.15able 1: Charged compact stars. M ( M ⊙ ) r = 5 R RN ( km ) q ( × C ) ω ( kHz ) Ω( kHz ) Ω /ω .
430 21 .
030 0 .
259 3 .
413 5 .
399 1 . .
765 24 .
275 1 .
517 3 .
027 4 .
826 1 . .
728 33 .
769 3 .
434 2 .
250 3 .
657 1 . .
248 59 .
560 7 .
576 1 .
306 2 .
173 1 . .
150 132 .
240 18 .
314 0 .
594 1 .
002 1 . M ( M ⊙ ) r = 5 R RN ( km ) q ( × C ) ω ( kHz ) Ω( kHz ) Ω /ω .
02 29 .
791 0 2 .
407 3 .
806 1 . .
07 29 .
923 0 .
989 2 .
414 3 .
824 1 . .
15 30 .
362 1 .
486 2 .
398 3 .
810 1 . .
25 30 .
824 1 .
982 2 .
387 3 .
808 1 . τ ) and the proper time τ , using different mass and electric chargeconfigurations. For each case the ratio between the frequencies Ω /ω is alsocalculated. In 2003 Ray et al [4] presented a model to describe the effect of electric chargeon compact stars, assuming that the charge distribution is proportional to themass density. This model consider a polytropic equation of state for chargedstars. Based on the mass and electric charge values for compact stars providedfrom ref. [4] the table 1 was elaborated, containing, in addition to mass andelectric charge, the orbit radius r = 5 R RN and the ratio between frequenciesΩ /ω .In 2009 Negreiros et al [9] presented a model for electrically charged quarkstars. These stars are formed from a compression process similar to that formingneutron stars, but much more intense, where the neutrons decay to the quarksthat constitute them. Based on the values provided by ref. [9], the table 2 waselaborated, containing values of mass, electric charge, orbit radius r = 5 R RN and ratio between the frequencies Ω /ω for quark stars.Using the values provided by the tables 1 and 2, graphs relating the nor-malized amplitude of the temperature oscillations and proper time are plottedand shown in the figures 1 and 2. Comparing the plots shown in these figureswe can notice that the increase on the mass appears to affect the amplitude oftemperature oscillations more than the increase on electric charge. It occursdue to the factor M ′ / r in the expression (85). The radius is a multiple of theReissner-Nordstr¨om radius, so it is also a function of the mass and the chargeaccording with the positive root of the expression (18). In our analysis we con-sider always M > Q to get real roots, so the factor M ′ / r is O ( M ′− ). This16 τ -1e-09-8e-10-6e-10-4e-10-2e-1002e-10 ∆ (τ) Figure 1: Normalized amplitude of the temperature oscillations of a gas incircular motion around electrically charged compact stars with mass 1 . M ⊙ (straight line), 1 . M ⊙ (dashed line) and 12 . M ⊙ (dot-dashed line). Theproper time τ is expressed in seconds ( s ). τ -5e-10-4e-10-3e-10-2e-10-1e-1001e-10 ∆ (τ) Figure 2: Normalized amplitude of the temperature oscillations of a gas in circu-lar motion around electrically charged quark stars with mass 2 . M ⊙ (straightline) and 2 . M ⊙ (dashed line). The proper time τ is expressed in seconds ( s ).17able 3: Charged black hole at the galactic center. M ( M ⊙ ) r = 5 R RN ( km ) p q ω ( mHz ) Ω( mHz ) Ω /ω . × . × − . .
471 0 .
764 1 . . × . × .
131 1 .
789 1 . . × . × .
443 5 .
462 2 . . M ⊙ and 1 . M ⊙ are closer from each other than the curvefor 12 . M ⊙ , because the increasing in mass and charge for the later is muchlarger than for the former ones. In 2011 Bin-Nun [5] presented a model for describing electrically charged massiveobjects. This model considers the charge term in the Reissner-Nordstr¨om metricin terms of a proportion of the mass ( Q ′ = Q = 4 p q M , with p q as a freeparameter). According to this model, the quantity Q ′ = Q of the Reissner-Nordstr¨om metric can be interpreted as a free parameter rather than a physicalquantity such as electric charge. This interpretation considers the charge termas a result of a tidal gravitational effect [27], known as TRN ( Tidal Reissner-Nordstr¨om ). According to the TRN, the parameter p q can assume negativevalues.A modification to this model was made in 2014 by Zakharov [6], expressingthe free parameter p q as p q = Q /M without the 1 / M BH =(4 , ± , × M ⊙ and another located in the elliptic galaxy M
87 with mass M M = (6 , ± , × M ⊙ . These values were obtained by observationalmeasurements [28, 29].The charge parameter p q assumes the values 0 or 1, where p q = 0 representsthe absence of electric charge (Schwarzschild) and p q = 1 represents the casewhere Q = M , a situation known as ERN ( Extremal Reissner-Nordstr¨om ).The negative value of the charge parameter p q represents the TRN configuration.Based on the article by Zakharov [6] the tables 3 and 4 were elaborated,containing the mass, the different values of the charge parameter p q , the orbitradius r = 5 R RN and the ratio between frequencies Ω /ω for the two black holesconsidered in the article.Using the values provided by tables 3 and 4, graphs similar to the one18able 4: Charged black hole at M87. M ( M ⊙ ) r = 5 R RN ( km ) p q ω ( µHz ) Ω( µHz ) Ω /ω . × . × − . .
326 0 .
529 1 . . × . × .
784 1 .
240 1 . . × . × .
694 3 .
788 2 . τ -1e-11-5e-120 ∆ (τ) Figure 3: Normalized amplitude of the temperature oscillations of a gas incircular motion around a electrically charged black hole at the galactic centerwith charge parameter p q = 0 (dashed line) and p q = 1 (straight line). Theamplitude ∆( τ ) is multiplied by a 10 scale factor. The proper time τ isexpressed in seconds ( s ). τ -1e-12-5e-130 ∆ (τ) Figure 4: Normalized amplitude of the temperature oscillations of a gas incircular motion around a electrically charged black hole at the galactic centerwith charge parameter p q = 0 (straight line) and p q = − . τ ) is multiplied by a 10 scale factor. The proper time τ isexpressed in seconds ( s ). 19hown in the figure 3 have been plotted. The plot in 3 relates the amplitudeof temperature oscillations and the proper time for the situations where p q = 0(Schwarzschild case) and p q = 1 (ERN case). The plot in 4 relates the ampli-tude of temperature oscillations and the proper time for the cases where p q = 0(Schwarzschild case) and p q = − . p q = − . M ′ / r in the expression (85), where the radius is a multiple of theReissner-Nordstr¨om radius. The negative value of the parameter p q causes thesquare root in expression (18) always to have real roots, even if M < Q . So thefactor M ′ / r is O ( M ′− ), with the charge contributing to increase the massand subsequently decreasing the amplitude of the temperature oscillations. Wecan also infer that the time scale in 3 and 4 is much larger than the time scalefor the other previous models. It is related with the values of the masses of theblack holes, which are much larger than the other compact objects.By analyzing the tables 3 and 4 we can realize that the values of the ratioΩ /ω for the two black holes are the same for all values of p q , even for differentmasses. This suggests that for this model, the ratio Ω /ω does not depend onthe mass, but only on the parameter p q . In fact, we will show that for a modelthat considers the electric charge proportional to mass by a factor p q , the ratioΩ /ω depends only on p q . To proof this assertion, let us build the ratio Ω /ω from (73) and (76) by considering Q = p q M ′ as in Bin-Nun’s model, namelyΩ ω = s r − r p q M ′ r − M ′ r + 9 p q M ′ r − p q M ′ . (89)Now by taking the orbit radius as a multiple of the Reissner-Nordstr¨om radius,we have: r = N (cid:2) M ′ (1 + p − p q ) (cid:3) = N M ′ h, (90)where N is a real positive number and h = 1 + p − p q . Replacing this in (89)we obtain: Ω ω = s N h − N h p q N h − N h + 9 N p q h − p q . (91)Therefore we conclude that Ω /ω is only function of p q . This model presented in 2014 by Liu et al [7] suggests the possibility that electri-cally charged white dwarf stars are responsible for the formation of supernovae.These stars would have masses above the Chandrasekhar limit of 1 , M ⊙ , sothey would be unstable white dwarfs that would continue to collapse. In thismodel the polytropic equation of state was considered for exact solutions, and20able 5: Charged white dwarfs. M ( M ⊙ ) r = 5 R RN ( km ) q ( × C ) ω ( kHz ) Ω( kHz ) Ω /ω . .
622 2 .
35 5 .
899 10 .
559 1 . . .
967 2 .
35 4 .
955 8 .
460 1 . . .
816 2 .
35 3 .
502 5 .
719 1 . .
954 24 .
678 2 .
35 3 .
060 4 .
953 1 . . .
906 2 .
35 2 .
248 3 .
596 1 . τ -4e-09-3e-09-2e-09-1e-090 ∆ (τ) Figure 5: Normalized amplitude of the temperature oscillations of a gas incircular motion around a electrically charged white dwarf with mass ≈ . M ⊙ (straight line) and ≈ . M ⊙ (dashed line). The proper time τ is expressed inseconds ( s ).a general equation of state (based on the equation of state of the free electron)was considered for numerical solutions. For the present analysis the electriccharge was considered with a constant value, with variable values of the mass.In this way we can see the role of the mass in the temperature oscillations.Based on the values provided by the article of Liu et al [7], the table 5 waselaborated, containing, in addition to mass and electric charge, the value ofthe orbit radius r = 5 R RN and the ratio between frequencies Ω /ω of the whitedwarf stars analyzed in the article.Using the values provided by the table 5 graphs similar to the one shownin the figure 5 have been plotted. The plot in 5 relates the amplitude of tem-perature oscillations and proper time for the mass values of ≈ , M ⊙ and ≈ , M ⊙ . In this work the normalized amplitudes of temperature oscillations of a gas in acircular geodesic motion in Reissner-Nordstr¨om space-time was determined by21ollowing the same methodology of ref. [8] where the Schwarzschild metric wasused. The proper frequencies derived from the equations of the geodesic devia-tion in Reissner-Nordstr¨om metric were the same found by using the equationsof motion. The expressions obtained for the temperature profile calculated fromTolman’s law and the behavior of the oscillations were also consistent with theresults of ref.[8].After these calculations the obtained expression were used to calculate theamplitudes of temperature oscillations for some theoretical models for chargedcompact objects: compact stars, quark stars, black holes and white dwarfs.These oscillations presented frequencies of the same range ( kHz ) as the fre-quencies found in QPO phenomena (with exception of the supermassive blackholes, but the frequencies scale with the inverse of the mass and for this reasonare in the µHz range [24]), so we calculate the ratio between the frequencies andcompare with the 3 / / /ω ratio is closer to 3 / /ω doesnot depend on mass but only on the proportionality factor between charge andmass. We can also observe that the ratio Ω /ω increases with the electric charge.For the situation with charge parameter p q = 1, this value is much larger than3 /
2, but this is an extreme limiting case.The white dwarf model allowed to analyze the variation of mass with con-stant electric charge. Comparing the analysis of this model with that of theblack holes, it was possible to verify that the role of the charge term is the op-posite of the mass term, that is, while the increase of mass produces a reductionin the frequencies, amplitude and in the ratio between frequencies, the increaseof the electric charge produces an inverse effect. This behavior reflects the factthat the mass term and the electric charge term have opposite signs in the ex-pression of the Reissner-Nordstr¨om metric. In this way, the main objective ofthe work was reached, which was the determination of the role of the chargeterm in the behavior of temperature oscillations of a gas in geodesic motion inthe presence of a Reissner-Nordstr¨om metric.Although the QPOs and the problem treated in the present work are not thesame problem, they have some features in common. The study of a gas in circu-22ar geodesic motion around a charged compact object based in the Lagrangianof a free particle motion reveals the existence of two proper frequencies: ω asso-ciated with the radial frequency ω r and Ω associated with the orbital frequency ω φ . The existence of these two frequencies and the relativistic strong field regimesuggests that these two problems can share some common behaviors. For thesereasons we also calculate the ratio between the frequencies Ω /ω in our analysis.As said in the introdution, this is an idealized toy model with several limitationsbut some results are compatible with the values found for QPOs. Acknowledgments
L. C. M. has been supported by CAPES (Coordena¸c˜ao de Aperfei¸coamento dePessoal de N´ıvel Superior), Brazil and G. M. K by CNPq (Conselho Nacional deDesenvolvimento Cient´ıfico e Tecnol´ogico), Brazil. The authors thank ProfessorWinfried Zimdahl for useful discussions.
References [1] R. M. Wald,
General Relativity , (The University of Chicago Press, Chicago,1984).[2] C. Misner, K. Thorne, and J. Wheeler,
Gravitation , (W.H. Freedman andCompany, San Francisco, 1973).[3] S. Chandrasekhar,
The Mathematical Theory of Black Holes , (Oxford Uni-versity Press, Oxford, 2009).[4] S. Ray et al.,
Phys. Rev. D , (2003) 84004.[5] A. Bin-Nun, Class. Quantum Grav. , (2011) 114003.[6] A. Zakharov, Phys. Rev. D , (2014) 62007.[7] H. Liu et al., Phys. Rev. D , (2014) 104043.[8] W. Zimdahl and G. M. Kremer, Phys. Rev. D , (2015) 24003.[9] R. Negreiros et al., Physical Review D , (2009) 83006.[10] C. Cercignani and G. M. Kremer, The Relativistic Boltzmann Equation:Theory and Applications , (Birkh¨auser Verlag, Berlin, 2002).[11] R. Tolman and P. Ehrenfest,
Phys. Rev. , (1930) 1791.[12] J. Kov´aˇr et al., Phys. Rev. D , (2011) 84002.[13] O. Klein, Rev. Mod. Phys. , (1949) 531.[14] A. Hees et al., Phys. Rev. Lett. , (2017) 211101.2315] A. Broderick et al., Astrophys. J. , vol. (2014) 7.[16] F. Zhang, Y. Lu and Q. Yu,
Astrophys. J. , (2015) 27.[17] L. Parker and L. Pimentel, Phys. Rev. D , (1982) 3180.[18] M. Shirokov, Gen. Relativ. Gravit. , (1973) 131.[19] W. Zimdahl, Exp. Tech. Phys. , (1985) 403.[20] F. Manasse and C. Misner, J. Math. Phys. , (1963) 735.[21] G. T¨or¨ok et al., Astron. Astrophys. , (2005) 1.[22] C.German´a, Mon. Not. R. Astron. Soc. , (2013) L1.[23] W. Klu´zniak and M. Abramowicz, Astron. Astrophys. , (2001) L19.[24] M. Abramowicz et al., Astrophys. J. , (2004) L63.[25] R. D’Inverno, Introducing Einstein’s Relativity , (Oxford University Press,Oxford, 1998).[26] P. Collas and D. Klein,
Gen. Rel. Grav. , (2007) 737.[27] N. Dadhich et al., Phys. Lett. B , (2000) 1.[28] A. Ghez et al., Astrophys. J. , (2008) 1044.[29] S. Gillessen et al., Astrophys. J. , (2009) L114.[30] M. Abramowicz et al., Class. Quantum Grav. , (2006) 1689.[31] M. Bursa et al., Astrophys. J. ,617