Neutrino oscillations in Kerr-Newman space-time
aa r X i v : . [ g r- q c ] F e b Neutrino oscillations in Kerr-Newman space-time
Jun Ren ∗ and Cheng-Min Zhang (Dated: December 27, 2018)The mass neutrino oscillation in Kerr-Newman(K-N) space-time is studied in theplane θ = θ , and the general equations of oscillation phases are given. The effectof the rotation and electric charge on the phase is presented. Then, we considerthree special cases: (1) The neutrinos travel along the geodesics with the angularmomentum L = aE in the equatorial plane. (2) The neutrinos travel along thegeodesics with L = 0 in the equatorial plane. (3) The neutrinos travel along theradial geodesics at the direction θ = 0. At last, we calculate the proper oscillationlength in the K-N space time. The effect of the gravitational field on the oscillationlength is embodied in the gravitational red shift factor. When the neutrino travelsout of the gravitational field, the blue shift of the oscillation length takes place. Wediscussed the variation of the oscillation length influenced by the gravitational fieldstrength, the rotation a and charge Q .PACS number: 95.30.Sf, 14.60.PqKeywords: neutrino oscillation; Kerr-Newman space-time; oscillation length I. INTRODUCTION
Mass neutrino mixing and oscillations were proposed by Pontecorvo[1], and Mikheyev,Smirnov and Wolfenstein (MSW for short) explored the effect of transformation of oneneutrino flavor into another in a medium with varying density[2, 3]. Recently, the con-sideration of the mass neutrino oscillations has been a hot topic. There have been many ∗ Email: [email protected] theoretical[4–11] and experimental[12–18] studies about the neutrino oscillations. Then, theneutrino oscillations in the flat space time were extended to the cases in the curved space-time[19–26]. Neutrino oscillation experiments were also considered to test the equivalenceprinciple[27]. Calculating the phase along the geodesic line will produce a factor 2 in thehigh energy limit, compared with the value along the null line, which often exists in the flatand Schwarzschild space-time[23, 24, 28–31]. This issue of the factor 2 is due to the differ-ence between the time-like and null geodesics. Furthermore, some alternative mechanismshave been proposed to account for the gravitational effect on the neutrino oscillation[32–34].The inertial effects on neutrino oscillations and neutrino oscillations in non-inertial frameswere also called attention[35, 36]. As a further theoretical exploration, neutrino oscillationsin space-time with both curvature and torsion[37–39] have been studied.In recent years, the researches about the neutrino oscillation have been made newprogress. A further mechanism to generate pulsar kicks, which was based on the spin flavorconversion of neutrinos propagating in a gravitational field, and the neutrino geometricaloptics in gravitational field (in particular in a Lense-Thirring background), have been pro-posed by Lambiase[40, 41]. Some publications were centered on the theoretical study andexperimental measurement of the mixing angle θ [42–44]. And CP violation in neutrino os-cillations were considered by some authors[45–48]. In addition, Cuesta and Lambiase studiedthe neutrino mass spectrum[49]. Akhmedov, Maltoni and Smirnov presented the neutrinooscillograms for different oscillation channels and discussed the effects of non-vanishing 1-2mixing[50].In this paper, we extend the mass neutrino oscillation work from Schwarzschild space-time to Kerr-Newman space-time, since the Kerr-Newman metric is rather important inblack hole physics, where a most generally stationary solution with axial symmetry hasbeen existing[51]. For the reason of simplicity, we confine our treatment in two generationneutrinos (electron and muon). We give the general equations of the oscillation phasesalong the arbitrary null and the time-like geodesics, respectively, in the equal θ plane, θ = θ . The phase along the geodesic will also produce a factor of 2 in the K-N space-time,Φ( geod ) = 2Φ( null ), in the high energy limit. In our derivation we have not assumed a weakfield approximation.We discuss three spacial cases. Firstly, the oscillation phases along the geodesics with L = aE are considered in the equatorial plane. E is the energy per unit mass of the particle. L and a are the angular momentum per unit mass of the particle and K-N space-time,respectively. The geodesics with L = aE in K-N space-time play the same roles as theradial geodesics in the Schwarzschild and in the Reissner-Nordstrom geometry. In this case,the phases both along the null geodesic and the time-like geodesic are similar in form tothe phases along the radial geodesics (null and time-like) in the Schwarzschild space-time.Secondly, we calculate the oscillation phases along the geodesics with L = 0 in the equatorialplane. This kind of geodesics is also important in K-N space-time. In the Schwarzschildspace-time with non-rotating spherically symmetry, particles with L = 0 can propagate alongthe radial geodesics. But in K-N space-time, because of dragging effect, the coordinate ϕ must change if a particle with L = 0 travels along the geodesics. Thirdly, the phases alongthe radial geodesics at the direction θ = 0 are given. Only at the poles θ = 0 and θ = π ,the ergosphere coincides with the event horizon. At the direction θ = 0, the effects of therotation of the space-time on the oscillation length are found to be more than those in theother directions.At last, we calculate the proper oscillation length in the K-N space time. The oscillationlength is proportional to the local energy (local measurement), E loc = E/ √ g , of theneutrino, where E is a constant along the geodesic. The decrease in the local energy leadsto the decrease in the oscillation length as the neutrino travels out of the gravitational field.So, the blue shift of the oscillation length occurs, which is unlike the case of the gravitationalred shift for light signal. In the equatorial plane in K-N space-time, the rotation haveno contribution to the oscillation length because g has nothing to do with the rotatingparameter a in this plane. The rotation a of the gravitational field shortens the oscillationlength in other equal θ plane, compared with the length in R-N space time. We also givethat the length varies according to θ . And charge Q shortens it too, compared with theKerr space-time case. But, the gravitational field lengthens it, compared with the case inflat space-time.In this paper, we take the neutrino as a spin-less particle to go along the geodesic becausethe spin and the curvature coupling has a little contribution to the geodesic derivation[52].Moreover, the neutrino is a high energy particle, so we do not think the neutrino spin hasmore contribution to the geodesic.The paper is organized as the follows. In Sec.2, we briefly review the standard treatmentof neutrino oscillation in the flat space-time. In Sec.3, we give the general expressionsof the oscillation phases along the null and time-like geodesics in arbitrary equal θ = θ plane. In Sec.4, we discuss the neutrino phase in three special cases. In Sec.5, we discussthe proper oscillation length in K-N space-time. At last, the conclusion and discussion aregiven. Throughout the paper, the units G = c = ~ = 1 and η µν = diag (+1 , − , − , −
1) areused.
II. THE STANDARD TREATMENT OF NEUTRINO OSCILLATION IN FLATSPACE-TIME
In a standard treatment, the flavor eigenstate | ν α i is a superposition of the mass eigen-states | ν k i , i.e.[21, 22] | ν α i = X k U αk exp [ − i Φ k ] | ν k i , (1)where Φ k = E k t − ~p k · ~x, ( k = 1 , , (2)and the matrix elements U αk comprise the transformation between the flavor and mass bases. E k and ~p k are the energy and momentum of the mass eigenstates | ν k i , and they are differentfor different mass eigenstates. If the neutrino produced at a space-time point A ( t A , ~x A ) anddetected at B ( t B , ~x B ), the expression for the phase in Eq.(2), which is coordinate indepen-dent and suitable for application in a curved space-time, is[21, 53]Φ k = Z BA p ( k ) µ dx µ , (3)where p ( k ) µ = m k g µν dx ν ds , (4)is the canonical conjugate momentum to the coordinate x µ and m k is the rest mass cor-responding to mass eigenstate | ν k i . g µν and s are metric tensor and an affine parameter,respectively.The following assumptions are often applied in the standard treatment[4]: (1) The masseigenstates are taken to be the energy eigenstates, with a common energy; (2) up to O ( m/E ),there is the approximation E ≫ m ; (3) a massless trajectory is assumed, which means thatthe neutrino travels along the null trajectory. In the case of two neutrinos mixing ν e − ν µ ,we can write ν e = cosθν + sinθν , ν µ = − sinθν + cosθν . (5)Here θ is the vacuum mixing angle. The oscillation probability that the neutrino producedas | ν e i is detected as | ν µ i is[54] P ( ν e → ν µ ) = |h ν e | ν µ ( x, t ) i| = sin (2 θ ) sin ( Φ kj , (6)where, Φ kj = Φ k − Φ j , is the phase shift. From the standard treatment of the neutrinooscillation[21–23], the standard result for the phase isΦ k ≃ m k E | ~x B − ~x A | . (7)Here E is the energy for a massless neutrino. So, the phase shift responsible for theoscillation is given by Φ kj ≃ ∆ m kj E | ~x B − ~x A | , (8)where ∆ m kj = m k − m j . III. NEUTRINO OSCILLATION PHASE ALONG THE NULL AND THETIME-LIKE GEODESIC IN THE PLANE θ = θ In this section, we study the neutrino oscillation in equal θ = θ surface. The line elementof K-N space time takes the form ds = g dt + g dr + g dθ + g dϕ + 2 g dtdϕ. (9)The relevant components of the canonical momentum of the k th massive neutrino in Eq.(4)are p ( k ) t = p ( k )0 = m k g ˙ t + m k g ˙ ϕ ; p ( k ) r = m k g ˙ r ; p ( k ) ϕ = m k g ˙ ϕ + m k g ˙ t, (10)where ˙ t = dtds , ˙ r = drds , ˙ ϕ = dϕds . Because the metric tensor components do not depend on thecoordinate t and ϕ , their canonical momenta p ( k ) t and p ( k ) ϕ are constant along the trajectory.In fact, the momentum p ( k )0 conjugate to t is the asymptotic energy of the neutrino at r = ∞ .It is stressed that it is the covariant energy p (not p ) the constant of motion. Otherwise theambiguous definition of the energy will lead to the confusion in understanding the neutrinooscillation.The phase along the null geodesic from point A to point B is given by [21, 23, 53]Φ nullk = Z BA p ( k ) µ dx µ = Z BA ( p ( k )0 dt + p ( k ) ϕ dϕ + p ( k ) r dr )= Z BA ( p ( k )0 dtdr + p ( k ) ϕ dϕdr + p ( k ) r ) dr. (11)We can obtain the following relations which are useful in the calculation g = − g ∆ sin θ , g = − g ∆ sin θ ,g = g ∆ sin θ , g − g g = ∆ sin θ, (12)where ∆ = r − M r + a + Q . Solving the equation (10) for ˙ t and ˙ ϕ , we obtain˙ t = − g E k + g L k ∆ sin θ , ˙ ϕ = g E k + g L k ∆ sin θ , (13)where E k = p ( k )0 m k and L k = − p ( k ) ϕ m k are the energy and angular momentum per unit mass,respectively.In the standard treatment of the neutrino oscillation, the neutrino is usually supposed totravel along the null[4, 21, 22, 54–56] . Following the standard treatment, we will calculatethe phase along the light-ray trajectory from A to B .The lagrangian appropriate to motions in the plane (for which ˙ θ = 0 and θ = a constant= θ ) is[57] L = 12 ( g ˙ t + 2 g ˙ t ˙ ϕ + g ˙ r + g ˙ ϕ ) . (14)The Hamiltonian is given by H = E k ˙ t − L k ˙ ϕ + p ( k ) r m k ˙ r − L . (15)Because of the independence of the Hamiltonian on t , we can deduce that2 H = E k ˙ t − L k ˙ ϕ + p ( k ) r m k ˙ r = δ = constant. (16)Without loss generality, we can set, δ = 1 for time-like geodesics, δ = 0 for null geodesics.Substituting (13) into (16) and setting δ = 0 for null geodesics, we have the radial equation g ˙ r = g E k + 2 g E k L k + g L k sin θ ∆ . (17)We define a new function V ( r ) = g E k + 2 g E k L k + g L k . (18)The different V ( r ) determines the phase of the different trajectory. From (17), we get˙ r = √− Vρsinθ , (19)where ρ = r + a cos θ . So, the equations governing t and ϕ are dtdr = − ρ ( g E k + g L k )∆ sinθ √− V , dϕdr = ρ ( g E k + g L k )∆ sinθ √− V . (20)The mass-shell condition is[21] m k = g µν p ( k ) µ p ( k ) ν = p ( k ) µ p ( k ) µ = p ( k )0 p ( k )0 + p ( k ) ϕ p ( k ) ϕ + p ( k ) r p ( k ) r . (21)Substituting p ( k )0 = g p ( k )0 + g p ( k ) ϕ , p ( k ) ϕ = g p ( k ) ϕ + g p ( k )0 and (10) into the equation ofthe mass-sell condition(21), we obtain p ( k ) r = m k √− V − sin θ ∆ ρsinθ . (22)In the process of calculation, the relations (12) are used. Applying the relativistic condition p k ≫ m k , we have the relation p ( k ) r ≃ m k ρsinθ ( √− V − sin θ ∆2 √− V ) . (23)Adopting (20) and p ( k ) r , the phase along the null geodesics (11) is approximated byΦ nullk ≃ Z BA m k ρsinθ dr √− V . (24)The phase (24) is a general result. The different function V ( r ) corresponds to the differentmotion and determines the different phase consequently. If a = 0, we can obtain the oscil-lation phase in the Reissner-Nordstorm space-time; if Q = 0, the Kerr space-time case isgiven. If a = 0 , Q = 0, the function V ( r ) reduces to V ( r ) = − r sin θ E k . (25)The phase (24) becomes toΦ nullk = Z BA m k p ( k )0 dr = m k p ( k )0 ( r B − r A ) . (26)This is just the phase in Schwarzschild space-time[21, 22].The velocity of an extremely relativistic neutrino is nearly the speed of light. In thestandard treatment, the neutrino is supposed to travel along the null line[4, 21, 22, 54–56].Despite of this, the propagation difference between a massive neutrino and a photon can haveimportant consequences and this tiny derivation becomes important for the understandingof the neutrino oscillation. Therefore, for more general situations, we start to calculate thephase along the time-like geodesic. The factor of 2 will be obtained, when compared thetime-like geodesic phase with the null geodesic phase in the high energy limit. The classicalorbit is defined to a plane[21, 23], θ = θ , dθ = 0. The phase along the time-like geodesicis[19, 23, 28, 53] Φ geodk = Z BA p ( k ) µ dx µ = Z BA ( p ( k )0 dtdr + p ( k ) ϕ dϕdr + p ( k ) r ) dr. (27)For time-like geodesic, δ = 1, equation (16) becomes E k ˙ t − L k ˙ ϕ + p ( k ) r m k ˙ r = 1 , (28)while the equations for ˙ t and ˙ ϕ (13) are the same for time-like geodesics[57]. Substituting ˙ t and ˙ ϕ , we have dsdr = 1˙ r = √− g q − − V ∆ sin θ . (29)So, we obtain the equations for dt/dr and dϕ/dr for time-like geodesics dtdr = − √− g ∆ sin θ ( g E k + g L k ) q − − V ∆ sin θ , dϕdr = √− g ∆ sin θ ( g L k + g E k ) q − − V ∆ sin θ . (30)According to mass shell condition, p ( k ) r is given by p ( k ) r = − m k √− g s − − V ∆ sin θ . (31)Thus, the phase along the time like geodesic isΦ geodk = Z BA m k √− g dr q − − V ∆ sin θ . (32)If the high energy limit is taken into account, Eq. (32) reduces toΦ geodk ≃ Z BA m k ρ sin θ dr √− V = 2Φ nullk . (33)It is often noted that the factor 2 of the neutrino phase calculations exists in the flat space-time[29, 30] and in the Schwarzschild space-time[23, 24, 28], which is believed to be thedifference between the null geodesic and the time like geodesic. The neutrino phase inducedby the null condition, as in the standard treatment, comes from the 4-momentum p µ definedalong the time-like geodesic, and the equation (17) governing ˙ r to the null geodesic. If the4-momentum defined along the null geodesic was instead used to compute the null phase,we would obtain zero because of the null condition. When we calculate the phase along thetime-like geodesic, ˙ r in (28) is defined to the time-like geodesic. It is the difference producingthe factor 2. It can be proved that the neutrino phase along the null is the half of the valuealong the time like geodesic in the high energy limit in a general curved space-time(seeAPPENDIX A in literature[23]). IV. THREE SPECIAL CASESA. Oscillation phases along the geodesics with L k = aE k in the equatorial plane It is very important that the geodesic is described in the equatorial plane θ = π/ L k = aE k play the same roles as the radial geodesics inthe Schwarzschild and in the Reissner-Nordstrom geometry. In this case, for null geodesic˙ t , ˙ ϕ and ˙ r reduce to ˙ t = r + a ∆ E k ; ˙ ϕ = a ∆ E k ; ˙ r = ± E k . (34)These equations in fact define the shear-free null-congruences which we use for constructinga null basis for a description of the K-N space-time in a Newman-Penrose formalism[57].The function V ( r ) for null geodesic becomes to, V ( r ) = − r E k . (35)So, the phase along the null isΦ nullk ≃ Z BA m k ρsinθ dr √− V = Z BA m k E k dr = m k p ( k )0 ( r B − r A ) , (36)0which appears the same form as that of the Schwarzschild space-time radial oscillation case.We now turn to a consideration of the time-like geodesic case. The equations for ˙ t, ˙ ϕ arethe same as for the null geodesics, while ˙ r becomes to˙ r = r E k + 1 g . (37)Substituting L k = aE k into (32), we obtain the phase along the time-like geodesicΦ geodk = Z BA m k dr [( p ( k )0 m k ) + g ] / . (38)Compared with the phase along the radial time-like geodesic in the Schwarzschild space-time[23], Φ geodk ( Sch ) = Z BA m k dr r ( p ( k )0 m k ) − g = Z BA m k dr r ( p ( k )0 m k ) + g , (39)we find that the oscillation phase with L k = aE k in K-N space-time has the similar formas the phase along the radial in Schwarzschild space-time. Substituting g = − r ∆ intoequation (38), we have Φ geodk = Z BA m k dr q b + Mr − a + Q r , (40)where b = ( p ( k )0 m k ) −
1. Equation (40)can be integrated directly to giveΦ geodk = m k b q br B + 2 M r B − a − Q − m k b q br A + 2 M r A − a − Q − M m k b / ln br B + M + p b ( br B + 2 M r B − a − Q ) br A + M + p b ( br A + 2 M r A − a − Q ) . (41)Eq.(41) shows the effects of rotation a on the oscillation phase.If a = 0, we can obtain ˙ t, ˙ ϕ, ˙ r along the radial null-geodesics in the equatorial plane inReissner-Nordstrom space-time˙ t = r r − M r + Q , ˙ ϕ = 0 , ˙ r = ± E. (42)Therefore, the phases along the radial null and time-like geodesic in Reissner-Nordstromspace-time are given by, respectivelyΦ nullk ( RN ) = m k p ( k )0 ( r B − r A ) , Φ geodk ( RN ) = Z BA m k dr q b + Mr − Q r . (43)1Letting a = 0 in (41), the integral of equation (43) is given. B. Oscillation phases along the geodesics with L = 0 in the equatorial plane The geodesics with L k = 0 is another important class of geodesics in K-N space-time. Ifthe coordinate t and ϕ has a relation dϕ/dt = − g /g , the canonical momentum p ( k ) ϕ in(10) vanishes. The corresponding ˙ t , ˙ ϕ and ˙ r for null geodesic are˙ t = − g ∆ E k ; ˙ ϕ = g ∆ E k ; ˙ r = √− g r E k . (44)And ˙ r for time-like geodesics is ˙ r = s g E k / ∆ g . (45)Substituting L k = 0 into (24) and (32), the phases along the null and time-like geodesic aregiven by, respectivelyΦ nullk = Z BA m k p ( k )0 rdr √− g = Z BA m k p ( k )0 p − g f g dr, (46)Φ geodk = Z BA p − g f g m k dr r ( p ( k )0 m k ) − f g . (47)where f g = g − g /g . It is difficult to integrate (46) and (47) directly. We can work outthem by expanding as a when a is a small quantity. C. Oscillation phase along the radial geodesic at θ = 0 Unlike in the Schwarzschild and in the Reissner-Nordstrom space-time, the event horizondoes not coincide with the ergosphere where g vanishes in K-N space-time. This is animportant feature which distinguishes the K-N space-time from the others. The ergospherethat is a stationary limit surface coincides with the event horizon only at the poles θ = 0and θ = π . The phase along the null geodesic in the direction θ = 0 can be written asΦ nullk = Z BA m k ρ sin θ dr √− V = Z BA m k ρ sin θ dr q r + a + a ρ (2 M r − Q ) sin θ sin θ . (48)2Substituting θ = 0, the equation (48) becomesΦ nullk = Z BA m k p ( k )0 dr = m k p ( k )0 ( r B − r A ) . (49)By similar calculation, the phase along the time-like geodesics at θ = 0 is given byΦ geodk = Z BA m k dr ( b + Mr − Q r + a ) / , (50)where b = ( p ( k )0 m k ) − V. PROPER OSCILLATION LENGTH
The propagation of a neutrino is over its proper distance , but dr in (24) is only acoordinate. The proper distance can be written as[58] dl = r ( g µ g ν g − g µν ) dx µ dx ν . (51)In K-N space-time, we have dl = s − g dr + ( g g − g ) dϕ . (52)Substituting dϕdr , we obtain dr = √− g V √ ∆ E k sinθ dl. (53)In order to discuss conveniently, we adopt the differential form of (24) d Φ nullk = m k ρsinθ dr √− V . (54)Substituting (53), we have d Φ nullk = m k p ( k )0 √ g dl. (55)It is assumed that the mass eigenstates are taken to be the energy eigenstates, with a com-mon energy in the standard treatment. The equal energy assumption is considered to becorrect by some authors[29, 31, 59] and studied carefully in papers[24, 60, 61]. In addi-tion, it is adopted widely in many literatures, for example[21–23, 62]. p will represent thecommon energy of different mass eigenstates. In fact, the condition of equal momentum3is also adopted to study the neutrino oscillation. In the flat space-time, both conditions(the equal energy and the equal momentum) present practically the same neutrino oscilla-tion results[24]. There are conditions of time translation invariance and space translationinvariance in the flat space-time. So, energy conservation and momentum conservation ofa free particle are right. In the curved (stationary) space-time, the energy of a particle isconserved along the geodesic due to the existence of a time-like killing vector field. However,the canonical conjugate momentum to r , p r is not conserved because ( ∂∂r ) a is not killing inthe curved (stationary) space-time. Consequently, it is very difficult to study neutrino oscil-lation if the condition of equal momentum is adopted in curved space-time. In this section,our discussion is on the base of the results in the standard treatment which the phase iscalculated along the null. Then, the phase shift which determines the oscillation is d Φ nullkj = d Φ nullk − d Φ nullj = ∆ m k p √ g dl, (56)where ∆ m kj = m k − m j . The equation (56) can be rewritten as dld ( Φ nullkj π ) = 4 πp ∆ m kj √ g = 4 πp loc ∆ m kj . (57)The term πp ∆ m kj √ g in (57) can be interpreted as oscillation length L OSC (which is definedby the proper distance as the phase shift Φ nullkj changing 2 π ) measured by the observer atrest at a position r , and p loc = p / √ g is the local energy. As r → ∞ , p loc approachesto the energy p measured by the observer at infinity. πp ∆ m kj is the oscillation length in theflat space-time. Equation (57) is universal significance in curved space-time. In fact, √ g is the gravitational red shift factor which shows the effect of the gravitational field on theoscillation length. Consider two static observers O (the radial coordinate r ) and O ′ (theradial coordinate r ′ ). The oscillation length measured by O and by O ′ is, respectively L OSC ( r ) = 4 πp ∆ m kj p g ( r ) , L OSC ( r ′ ) = 4 πp ∆ m kj p g ( r ′ ) . (58)We can obtain the relation L OSC ( r ′ ) L OSC ( r ) = p g ( r ) p g ( r ′ ) . (59)If r ′ > r , we have L OSC ( r ′ ) < L OSC ( r ) and blue shift occurs. Physically, the oscillation lengthis proportional to the local energy of the neutrino. When the neutrino travels out of the4gravitational field, the local energy decreases. Consequently, the neutrino oscillation lengthdecreases and blue shift takes place. From equation (57), the oscillation length increasesin the gravitation field because of 0 < g < g = 1 − M/r , wehave L OSC ( Sch ) = 4 πp ∆ m kj p − M/r . (60)In order to study the influence of Charge on the neutrino oscillation, we consider theoscillation length in the Reissner-Nordstrom space-time L OSC ( RN ) = 4 πp ∆ m kj √ g = 4 πp ∆ m kj q − Mr + Q r . (61)Compared with the case in the Schwarzschild space-time, the oscillation length decreasesdue to the influence of charge Q .The metric component g in the K-N space-time is g = 1 − M r − Q ρ , (62)where ρ = r + a cos θ . In the equatorial plane, there is, g = 1 − M/r + Q /r , whichis the same as g in the Reissner-Nordstrom space-time. Thus, it is concluded that theneutrino oscillation length along the geodesics in the equatorial plane in the K-N space-timeis identical to that in the Reissner-Nordstrom space-time and the rotating parameter a doesnot work in this plane. Therefore, we have to select other plane θ = θ = π/ θ = θ , the oscillation lengthcan be written as L OSC ( K − N ) = 4 πp ∆ m kj q − Mr − Q r + a cos θ (63)It is obvious that the oscillation length decreases too because of the rotation of the gravi-tational field compared with that in R-N space-time. Letting Q = 0 in (63), the oscillationlength in Kerr space-time is given by L OSC ( Kerr ) = 4 πp ∆ m kj q − Mrr + a cos θ . (64)5Comparing with (63), the charge Q shortens the oscillation length. We can obtain that theoscillation length varies with θ by ddθ L OSC ( K − N ) = 4 πp ∆ m kj ( g ) / M r − Q ( r + a cos θ ) a sin θ cos θ . (65)In K-N space-time, we conclude that the oscillation length increases with θ within 0 < θ <π/
2, and it becomes maximum in the equatorial plane. Then, it decreases with θ within π/ < θ < π . At the direction θ = 0 and θ = π , the oscillation length occurs minimum, L OSC ( K − N ) = 4 πp ∆ m kj q − Mr − Q r + a . (66)In summary, the gravitational field lengthens oscillation length; both the rotation a andthe charge Q shorten the oscillation length. VI. CONCLUSION AND DISCUSSION
In this paper, we have given the phase of mass neutrino propagating along the null andthe time like geodesic in the gravitational field of a rotating symmetric and charged object,which is described by Kerr-Newman metric. Most astrophysical bodies in universe haverotation and charge generally. Thus the work about the neutrino oscillation in the K-Nspace time is important and meaningful for the black hole astrophysics. We work out thegeneral formula of oscillation phase on the equal θ = θ plane with the generality. Thephase along the geodesic is the double of that along the null in the high energy limit, whichis the same in the cases in flat and Schwarzschild space-time. By setting θ = π/
2, thephases in the equatorial plane are given. As a = 0 or Q = 0, we obtain the phases in theR-N space-time or in the Kerr space-time. Moreover, we study three special cases in K-Nspace-time: geodesics with L = aE ; geodesics with L = 0; radial geodesics at θ = 0. Amongthem, the geodesics with L = aE have the same importance as the radial geodesics in theSchwarzschild and in the R-N geometry. The phases obtained are very similar in form tothe cases along the radial geodesics in the Schwarzschild and in the R-N space-time.In Sec.5, the proper oscillation length in the K-N space time is studied in detail. Wefind that oscillation length in curved space-time is proportional to the local energy, which is6regraded as the neutrino ”climbs out of the gravitational potential well”. So, the blue shiftoccurs. Then, the effects of rotation and charge of the space-time on the oscillation lengthare given. Because of the correction of the gravitation field, the oscillation length increases,compared with the flat space time case. However, both the rotation a and the charge Q shorten the oscillation length. It is noted, the rotation has null contribution to the lengthin the equatorial plane in K-N space-time, because red shift factor is independence of a inthis plane. Finally, we remark that our result exists generality, which can be exploited tostudy the neutrino oscillation near the rotating compact star, neutron star or black hole. [1] Pontecorvo B 1957 Mesonium and antimesonium Zh. Eksp. Theor. Fiz. Il Nuovo Cimento C Phys. Rev. D Rev. Mod. Phys. Progress inParticle and Nuclear Physics Phys. Lett. B Phys. Rev. D Phys. Rev. D Phys. Rev. D Phys. Rev.D
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