We study neutrino spin oscillations in black hole backgrounds. In the case of a charged black hole, the maximum frequency of oscillations is a monotonically increasing function of the charge. For a rotating black hole, the maximum frequency decreases with increasing the angular momentum. In both cases, the frequency of spin oscillations decreases as the distance from the black hole grows. As a phenomenological application of our results, we study simple bipolar neutrino system which is an interesting example of collective neutrino oscillations. We show that the precession frequency of the flavor pendulum as a function of the neutrino number density will be higher for a charged/non-rotating black hole compared with a neutral/rotating black hole respectively.
Neutrino physics is an active area of research with important implications for particle physics, cosmology and astrophysics. Cosmological implications of neutrinos include lepto/ baryogenesis and possible connections to the dark sector of the universe [1]. The phenomenon of neutrino oscillations can explain solar and atmospheric neutrino problems. It also provides the first experimental evidence for physics beyond the standard model since it requires nonzero mass for neutrinos.
The effects of gravitational fields on the neutrino oscillations have been studied in the literature. In [2], neutrino flavor oscillations in the Kerr-Newman space-time has been studied. In [3], the authors investigated the effect of a Quantum Gravity-induced minimal length on neutrino oscillations. Neutrino optics in gravitational fields has been studied in [4]. Interaction of neutrinos with an external field provides one of the factors required for a transition between helicity states. In [5], neutrino spin oscillations has been studied in Schwarzschild background which describes the gravitational field of an uncharged and non-rotating black hole. In this paper, we study neutrino spin oscillations in Reissner-Nordstrom (RN) and Kerr backgrounds which describe the gravitational fields of a charged non-rotating and a rotating black hole respectively.
RN metric describes the gravitational field of a charged non-rotating black hole.
(1) (2) Here and M are the charge and the mass of the black hole respectively and we have used the natural units ℏ
. The components of vierbein four velocity are [5]:
(
where
is the four velocity of a particle in its geodesic path, which is related to through . The four velocity of a particle in the relevant metric is related to the world velocity of the particle through where and is the proper time. The nonzero vierbein vectors are : (5) To study the spin evolution of a particle in a gravitational field, we calculate which is the analogue of the electromagnetic field tensor. It is defined as follows: (6) where are the covariant derivatives of vierbein vectors given by : (7) After using Eqs.( 5) and (7) to calculate the covariant derivatives of vierbein vectors, we have: (8) We recall that an antisymmetric tensor in four dimensions can be stated in terms of two three dimensional vectors (such as electric and magnetic fields) :
It is worth to mention that the electric and magnetic fields in (9), are not (real)electric and magnetic fields but they are only their analogues.
Using Eqs.( 4), ( 5), ( 6), (8), and (9), we have the following forms for the electric and magnetic fields: (10) Since the gravitational field around a charged non-rotating black hole is symmetric, we can consider neutrino motion in the equatorial plane (hence ). The four-velocity in the vierbein frame is not constant in general case but it can be shown that for circular orbits which means that the velocity four vector of neutrino is constant with respect to the vierbein frame. We assume that the motion is in a circular orbit with constant radius ( ). It is important to note that not all the orbits with arbitrary radius are stable. Here we concentrate on the stable orbits. The geodesic equation of a particle in a gravitational field is described by [6]: (11) where variable parameterizes the particle’s world line. From Eqs. (1), (2), and (11), we can calculate the values of and :
(
Neutrino spin precession is given by the expression 𝛺 , where vector is defined as follows [7]:
After substituting (3), ( 10), ( 12), ( 13), ( 14) and in 𝛺 , we obtain the only nonzero component of frequency :
In Fig.
(1) we show 𝛺 versus for different values of 𝛼 . For 𝛼 our results reduce to that in the case of the Schwarzschild metric [5]. We stop at 𝛼 since the RN metric with > does not describe a black hole due to the existence of a naked singularity at =0. It is seen that the maximum frequency of oscillations increases with 𝛼. It is also seen that 𝛺 has its largest value for . 𝛼 curve is in agreement with the result obtained for the Schwarzschild metric [5]. We also see that curves for different values of 𝛼 coincide at large . This indicates that the effect of the black hole charge diminishes at long distances. In Fig. (2) we show the neutrino transition probability 𝛺 .
Kerr metric [8] describes the geometry of space-time in the vicinity of a rotating balck hole. The Kerr metric is usually written in Boyer-Lindquist coordinates, which is not diagonal. One can transform the Kerr metric in to the standard form (diagonal form) with an appropriate change of variable [9]. (16) where : (17) and :
(18)
The vierbein vectors and their inverses are as follows:
The nonzero components of the covariant derivatives of vierbein vectors can be found using Eqs. ( 7) and ( 19) :
(20) After using Eqs. ( 4), ( 6), ( 9), ( 19) and ( 20) we arrive at the following expressions for the analogues of the electric and magnetic fields :
The trajectories of particles
This content is AI-processed based on open access ArXiv data.
