Neutrino oscillations in accelerated frames

TL;DR

This paper explores how neutrino oscillations are affected by uniform acceleration, introducing a covariant phase definition and analyzing inertial effects, with implications for atmospheric neutrinos and extreme acceleration scenarios.

Contribution

It presents a covariant framework for neutrino oscillations in accelerated frames, extending the standard formalism to include inertial effects and energy redshift.

Findings

Inertial effects modify the Pontecorvo formula for neutrino oscillations.

Energy redshift plays a key role in accelerated frame oscillations.

Preliminary phenomenological analysis suggests observable effects in atmospheric neutrinos.

Abstract

We discuss neutrino oscillations in vacuum from the point of view of a uniformly accelerated observer. A covariant definition of quantum phase is introduced with the aim of generalizing the standard expression of the oscillation amplitude to the accelerating frame. By way of illustration, we address a simplified two-flavor model with relativistic neutrinos, showing that inertial effects on the usual Pontecorvo formula are intimately related to the energy redshift. Phenomenological aspects are preliminarily analyzed in the context of atmospheric neutrinos. Finally, we discuss a gedanken experiment in order to investigate our formalism in regime of extreme acceleration.

Equations

$${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1).$$

$$\left(\begin{array}{c}|{\nu }_e⟩\\ |{\nu }_{\mu }⟩\end{array}\right)=U(\theta )\left(\begin{array}{c}|{\nu }_1⟩\\ |{\nu }_2⟩\end{array}\right),$$

$$U(\theta )=\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right).$$

$$|{\nu }_k(t,\text{x})⟩=\exp [-i{\Phi }_k(t,\text{x})]|{\nu }_k⟩,(k=1,2),$$

$${\Phi }_k=E_{k}t-{\text{p}}_k\cdot \text{x}$$

$$E_k^2=m_k^2+{|{\text{p}}_k|}^2.$$

$${\Phi }_k=E_k(t_B-t_A)-|{\text{p}}_k||{\text{x}}_B-{\text{x}}_A|\approx \frac{m_k^2}{2E_0}{L}_p.$$

$$t_B-t_A\simeq \left|{\text{x}}_B-{\text{x}}_A\right|,$$

$$E_k\simeq E_0+\mathcal{O}\left(\frac{m_k^2}{2E_0}\right),$$

$$P_{{\nu }_e\to {\nu }_{\mu }}$$

$$=$$

$${\Phi }_{12}\simeq \frac{\Delta m_{12}^2}{2E_0}{L}_p\equiv \frac{m_1^2-m_2^2}{2E_0}{L}_p.$$

$${\Phi }_k={\int }_A^B{p}_{\mu }^{(k)}d{x}^{\mu },$$

$$p_{\mu }^{(k)}=m_k{g}_{\mu \nu }\frac{d{x}^{\nu }}{ds}$$

$$d{s}^2=f(\text{a},\text{x}){(dt)}^2-d\text{x}\cdot d\text{x},$$

$$f(\text{a},\text{x})\equiv {\left(1+\text{a}\cdot \text{x}\right)}^2,$$

$$p_t^{(k)}$$

$$p_x^{(k)}$$

$$m_k^2=g^{\mu \nu }{p}_{\mu }^{(k)}{p}_{\nu }^{(k)},$$

$$E_k^{(loc)}(x)={\left|g_{tt}\right|}^{-1/2}{E}_k={f(a,x)}^{-1/2}{E}_k.$$

$${\Phi }_k={\int }_A^B\left[E_k{\left(\frac{dt}{dx}\right)}_0-p_k(x)\right]dx,$$

$$p_k(x)=-{\left(\frac{E_k^2}{f(a,x)}-m_k^2\right)}^{1/2},$$

$${\left(\frac{dt}{dx}\right)}_0=-f(a,x{)}^{-1/2},$$

$${\tilde{\Phi }}_k=-{\int }_A^B{E}_k\left\{1-\sqrt{1-f(a,x)\frac{m_k^2}{E_k^2}}\right\}{f(a,x)}^{-1/2}dx,$$

$$\frac{m_k^2}{{\left[E_k^{(loc)}(x_B)\right]}^2}=f(a,x_B)\frac{m_k^2}{E_k^2}\ll 1.$$

$$f(a,x)\frac{m_k^2}{E_k^2}$$

$${\tilde{\Phi }}_k\simeq -{\int }_A^B\frac{m_k^2}{2E_0}{f(a,x)}^{1/2}dx,$$

$${\tilde{\Phi }}_k\simeq \frac{m_k^2}{2E_0}|x_B-x_A|\left(1-{\varphi }_a\right),$$

$${\varphi }_a\equiv \frac{a}{2}\left(x_B+x_A\right).$$

$${\tilde{\Phi }}_{12}\simeq \frac{\Delta m_{12}^2}{2E_0}|x_B-x_A|\left(1-{\varphi }_a\right)=\frac{\Delta m_{12}^2}{2E_0}{L}_p\left(1-{\varphi }_a\right),$$

$${\tilde{\Phi }}_{12}\simeq \frac{\Delta m_{12}^2{L}_p}{2E_0^{(loc)}(x_B){f(a,x_B)}^{1/2}}\left(1-{\varphi }_a\right).$$

Full text

I Introduction

Neutrino oscillations in flat spacetime have been extensively analyzed since Pontecorvo’s pioneering idea of non-degenerate mass-matrix Pontecorvo . Over the years, however, alternative mechanisms have been proposed: among these, worthy of note are the ones suggested by Gasperini Gasperini:1989rt and Liu Liu:1997km , respectively. Although they have both been rejected by experiments, these solutions represent a first attempt to accommodate gravitational effects into the standard picture of neutrino oscillations. A systematic treatment of flavor oscillations in curved spacetime has been discussed by a number of authors in Refs. Aluw ; Ahluwalia:1996ev ; Cardall . In Ref. Cardall , in particular, the authors introduce a simple formalism to demonstrate that gravitational effects are closely related to the redshift of neutrino energy. The framework becomes even richer in astrophysical regimes, where the presence of strong gravitational and magnetic fields (provided that neutrinos possess a non-vanishing magnetic moment) may significantly affect the oscillation probability Aluw ; Lambiase:2004qk . Due to the equivalence principle, similar results are expected to be valid also in accelerated frames. Along this line, a pilot analysis of phenomenological aspects of neutrino oscillations for an accelerating and rotating observer has been performed in Ref. Capozziello:1999ww . Recently, mixing transformations in Rindler (uniformly accelerated) background have been also studied in Quantum Field Theory (QFT) Blasone:2017nbf ; Blasone:2018byx , showing that non-thermal corrections to the Unruh radiation may arise due to the interplay between the Bogoliubov transformation related to the structure of Rindler spacetime and the one hiding in field mixing Blasone:1995zc .

Apart from phenomenological implications, we stress that a deeper understanding of inertial effects on flavor mixing and oscillations may shed some light on a number of intriguing issues at a theoretical level. Recently, indeed, the role of neutrino mixing in the decay of accelerated protons (inverse $\beta$-decay) has been investigated with controversial results Ahluwalia:2016wmf ; Blasone:2018czm ; Cozzella:2018qew . Specifically, in Refs. Matsas:1999jx it was pointed out that the Unruh effect is necessary to maintain the general covariance of QFT when considering the inverse $\beta$-decay rate in the laboratory and comoving frames, respectively. Subsequently, it was noted that neutrino mixing can spoil this agreement Ahluwalia:2016wmf , and further discussion Blasone:2018czm ; Cozzella:2018qew has narrowed down possible causes to the effective nature of asymptotic neutrino states as mass or flavor eigenstates. Cleary, such an ambiguity affects flavor oscillations too. In particular, since the oscillation probability calculated in the ordinary QFT by means of the exact flavor states Blasone:1998hf contains extra-terms with respect to the usual quantum mechanical formula, one expects corrections to arise also for the non-inertial case.

In the present work, a preliminary step along this direction is taken by analyzing the effects of a linear acceleration on the neutrino oscillation formula in the context of Quantum Mechanics (QM). The obtained result should thus be regarded as a benchmark for the field theoretical treatment of the problem, for which work is in progress.

The paper is structured as follows: in Section II, we briefly review the standard treatment of neutrino oscillations in flat spacetime using the plane-wave formalism. Section III is devoted to a heuristic derivation of the oscillation probability for a uniformly accelerated observer. The same result is recovered in Section IV by solving the Dirac equation in accelerated frames. The obtained expression is critically compared with the one in Ref. Capozziello:1999ww , where corrections are calculated in a more geometric framework. As possible applications, in Section V we discuss how Earth’s gravity affects the oscillation probability of atmospheric neutrinos. In addition, we propose a gedanken experiment in which an ideal detector is used for testing inertial effects in proximity of high-density astrophysical objects. Section VI contains conclusions and an outlook at future developments of the present work.

Throughout the paper, we shall use natural units $\hslash=c=1$ and the flat Minkowski metric with the conventional timelike signature

[TABLE]

II Neutrino oscillations in flat spacetime

We start by reviewing the standard theory of neutrino oscillations in Minkowski spacetime. For the sake of simplicity, we focus on a model with only two flavor generations (for a more rigorous three-flavor description, we remand the reader to Ref. Maki:1962mu ).

In the conventional matrix notation, indicating by $|\nu_{\alpha}\rangle$ ($\alpha=e,\mu$) and $|\nu_{k}\rangle$ ($k=1,2$) neutrino flavor and mass eigenstates, respectively, the following relation holds KimPev

[TABLE]

where $\theta$ is the mixing angle and $U(\theta)$ is the Pontecorvo unitary matrix

[TABLE]

In what follows, we describe the propagation of the mass eigenstates by plane-waves, i.e.

[TABLE]

where

[TABLE]

is the quantum-mechanical phase of the $k^{th}$ neutrino state, with $E_{k}$ and $\textbf{p}_{k}$ being its energy and momentum, respectively. Mass, energy and momentum are related by the mass-shell condition

[TABLE]

In the relativistic approximation, labelling with $A\hskip 0.28453pt(t_{A},\textbf{x}_{A})$ and $B\hskip 0.28453pt(t_{B},\textbf{x}_{B})$ the spacetime points in which neutrinos are produced and detected, respectively, the phase acquired by the $k^{th}$ eigenstate after propagating over the distance $L_{p}\,\equiv\,\left|\textbf{x}_{B}-\textbf{x}_{A}\right|$ reads555In order for the interference pattern not to be destroyed, we remark that neutrinos must be produced coherently and measured at the same spacetime point.

[TABLE]

Notice that, in the second step of Eq. (7), we have exploited the relativistic condition $m_{k}\ll E_{k}$, so that

[TABLE]

and the first order expansion for $E_{k}$

[TABLE]

with $E_{0}$ being the energy for a massless neutrino. The last equation amounts to require that mass eigenstates are also energy eigenstates with a common energy $E_{0}$.

Let us now consider an electron neutrino $|\nu_{e}\rangle$ emitted via weak interaction at the point $A\hskip 0.28453pt(t_{A},\textbf{x}_{A})$. Using Eq. (2), the probability that it is revealed as muon neutrino $|\nu_{\mu}\rangle$ at the point $B\hskip 0.28453pt(t_{B},\textbf{x}_{B})$ is given by

[TABLE]

where, according to Eq. (7), the phase-shift $\Phi_{12}\equiv\Phi_{1}-\Phi_{2}$ takes the form

[TABLE]

It should be noted that, in the case where at least one of the states $|\nu_{k}\rangle$ is non-relativistic, a wave packet approach is required instead of the above plane-wave formalism Giunti:1991ca . For our purposes, however, such an analysis would show that the approximations Eqs. (8) and (9) are adequate, leading to the formula Eq. (II) for the oscillation probability. We also stress that this equation represents the quantum-mechanical limit of a more general formula derived within the QFT framework. For a detailed analysis of this, see Refs. Blasone:1995zc .

The foregoing discussion applies to an observer at rest or moving inertially with respect to the oscillation experiment. Nevertheless, due to gravity, any stationary laboratory on Earth experiences a linear acceleration (in the present analysis, we do not take care of rotational effects. A thorough discussion of this subject can be found in Ref. Capozziello:1999ww ). To show how acceleration affects flavor oscillations, let us then recast the quantum mechanical phase Eq. (7) into a covariant form, according to Stodolsky:1978ks

[TABLE]

where

[TABLE]

is the canonical four-momentum conjugated to the coordinates $x^{\mu}$ and $ds$, $g_{\mu\nu}$ are the line element and the metric tensor, respectively. The integration in Eq. (12) has to be performed along the light-ray trajectory linking the spacetime points $A$ and $B$. For $g_{\mu\nu}$ corresponding to the flat metric Eq. (1), it is easy to show that Eqs. (12) and (13) reproduce the standard result Eq. (7), as it should be.

III Inertial effects on neutrino oscillations: a heuristic treatment

We now turn to the discussion of neutrino oscillations for a uniformly accelerated observer. In order to apply the covariant formalism above described, let us recall that the line element in an accelerated frame can be written as (we neglect the effects of the spacetime curvature) Misner

[TABLE]

where

[TABLE]

with a being the proper three-acceleration and $x^{\mu}=(t,\textbf{x})$ the Fermi coordinates for an accelerated observer Misner ; lrf , whose range of validity is limited by the requirement $|\textbf{x}|\ll{|\textbf{a}|}^{-1}$. This occurs because the above reference frame is conceived to describe a neighborhood of the observer’s world line as long as the previous condition holds. However, the confinement on the spatial region does not affect the relevance of our considerations, since typical oscillation lengths of neutrino experiments allow us to deal with even considerable values of a. For instance, for acceleration of the order of Earth’s gravity, the metric is valid within a range of one light-year.

Without loss of generality, we can restrict our analysis to $1+1$ dimensions, assuming the acceleration to be antiparallel to the direction of neutrino propagation (see Fig 1). According to Eq. (13), the components of the neutrino canonical momentum $p_{\mu}^{(k)}$ are

[TABLE]

They are related to each other and to the mass $m_{k}$ by the generalized mass-shell condition

[TABLE]

with $g_{\mu\nu}$ given in Eq. (14). Since the metric does not depend on the coordinate $t$, the timelike momentum component $p_{t}^{(k)}$ is conserved along the geodesic trajectory of the $k^{th}$ neutrino eigenstate. We define such a constant to be $p_{t}^{(k)}\equiv E_{k}$. It represents the energy measured by an observer at rest at the origin. Due to the metric Eq. (14), however, it differs from the energy at any other spacetime point. The local energy, defined as the energy measured by an observer at rest at the generic position $x$, is related to $E_{k}$ by Misner

[TABLE]

Next, by considering relativistic neutrinos and using Eq. (12), the phase of the $k^{th}$ neutrino eigenstate reads

[TABLE]

where the momentum $p_{k}(x)\equiv-p_{x}{(k)}$ is obtained from the generalized mass-shell condition Eq. (18) as

[TABLE]

and the light-ray differential $\left(dt/dx\right)_{0}$ is given by

[TABLE]

where the minus sign in Eqs. (21) and (22) is due to the fact that neutrinos propagation is antiparallel to the $x$-axis. By inserting Eqs. (21) and (22) into Eq. (20), we get

[TABLE]

where the tilde has been introduced to distinguish the above expression of the phase from the standard one in Eq. (7).

Now, since detecting non-relativistic neutrinos is an extremely hard task, it is reasonable to require that

[TABLE]

This amounts to restrict our analysis to neutrinos that are relativistic at the detector position $B\hskip 0.28453pt(t_{B},x_{B})$, and thus along all their path $(x_{B}\leq x\leq x_{A})$, for we have

[TABLE]

Equations (24) and (III) allow us to approximate the covariant phase Eq. (23) as follows

[TABLE]

where, as in the absence of acceleration, we have used the first-order approximation $E_{k}\,\simeq\,E_{0}\,+\,\mathcal{O}\left(\frac{m^{2}_{k}}{2E_{0}}\right)$, with $E_{0}$ being the energy at the origin for a massless particle. Since this energy is constant along the light-ray trajectory between $A$ and $B$, the integration in Eq. (26) can be readily performed, obtaining

[TABLE]

where we have introduced the short-hand notation

[TABLE]

Thus, the phase-shift responsible for the oscillation takes the form

[TABLE]

where we have used the definition of proper distance at constant time $dl\,\equiv\,\sqrt{-g_{xx}}\hskip 1.42262ptdx\,=\,dx$.

We remark that Eq. (26) does not match with the corresponding result Eq. (25) of Ref. Capozziello:1999ww . In that case, indeed, the correction to the neutrino phase-shift depends logarithmically on the acceleration. We suspect that such a discrepancy arises because of an incorrect derivation of the final expression of the phase-shift in Ref. Capozziello:1999ww from the corresponding formula in Ref. Cardall .

Now, in order to compare Eq. (29) with the standard result Eq. (11), let us rewrite $E_{0}$ in terms of the neutrino local energy at the detector position $B\hskip 0.28453pt(t_{B},\textbf{x}_{B})$. Using Eq. (19), it follows that

[TABLE]

By virtue of the condition on the range of validity of the adopted metric (namely $a\,x\ll 1$), Eq. (30) can be further manipulated, thus giving

[TABLE]

where we have neglected higher order terms in the acceleration. The first term on the r.h.s. is the only surviving contribution for vanishing acceleration. As expected, it corresponds to the standard oscillation phase in Eq. (11). The remaining term provides the correction induced by a uniform, linear acceleration on the neutrino oscillation phase.

IV Inertial effects on neutrino oscillations: a geometric treatment

In the previous section, we have derived inertial effects on neutrino oscillations in a simple heuristic way. Using a more geometric treatment, we now want to prove that the same result can be obtained by solving the Dirac equation in an accelerated frame. As a first step, let us write down the covariant Dirac equation in curved spacetime Weinberg

[TABLE]

where $M$ is the neutrino mass matrix and $\psi$ is a column vector of spinors of different neutrino masses666In this section, greek (latin) indices refer to general curvilinear (locally inertial) coordinates.. The vierbein fields $e^{\mu}_{a}$ connect the general curvilinear and locally inertial sets of coordinates. The spinorial connection $\Gamma_{\mu}$ is accordingly defined by

[TABLE]

Using the relation

[TABLE]

we find that the only non-vanishing contribution of the spin connection is

[TABLE]

where we have denoted by $\epsilon^{abcd}$ the totally antisymmetric tensor with $\epsilon^{0123}\,=\,+1$ and

[TABLE]

with $g\,\equiv\,\mathrm{det}\,g_{\mu\nu}$. Following Ref. Cardall , the generalized flavor neutrino state can now be written as

[TABLE]

where $U_{{\alpha k}}\hskip 0.28453pt(\theta)$ is the generic element of the Pontecorvo matrix in Eq. (3). The neutrino oscillation phase is given by

[TABLE]

where $P_{\mu}$ is the four-momentum operator that generates the spacetime translation of the mass eigenstates and $\frac{dx^{\mu}_{null}}{d\lambda}$ is the null tangent vector to the neutrino worldline $x^{\mu}$, parameterized by $\lambda$. For diagonal metrics, denoting with $dl$ the differential proper distance at constant time, we have

[TABLE]

The momentum operator $P_{\mu}$ can be derived from the generalized mass-shell condition

[TABLE]

As in Sec. III, by requiring neutrino mass eigenstates to be energy eigenstates with a common energy $E_{0}$ and assuming the spatial components of $P^{\mu}$ and $\frac{dx^{\mu}_{null}}{d\lambda}$ to be antiparallel, in the relativistic approximation one has Cardall

[TABLE]

where we have neglected terms of $\mathcal{O}(A_{G}^{2})$ and $\mathcal{O}{(A_{G}M^{2})}$.

Let us now apply Eqs. (36) and (38) to the particular case of a uniformly accelerated frame. In the same fashion as the previous heuristic analysis, we restrict to $1+1$ dimensions; with reference to the metric tensor Eq. (14), the only non-trivial component of the vierbein fields is thus given by

[TABLE]

where ${f(a,x)}$ is defined as in Eq. (15). Inserting Eq. (42) into Eq. (36), one directly obtains $A^{\mu}_{G}\,=\,0$, yielding

[TABLE]

By use of Eqs. (39) and (43), the phase in Eq. (38) then becomes

[TABLE]

where we have exploited Eq. (19) and the definition of proper distance introduced above. After the mass operator in Eq. (37) has acted on $|\nu_{k}\rangle$, we obtain

[TABLE]

that is exactly the same expression derived in Eq. (26).

V Applications

In this section, we analyze some illustrative physical applications of our result. We begin by discussing the phenomenological implications of Eq. (31) in the framework of atmospheric neutrinos. In this case, mimicking the metric of a stationary observer on Earth with the one in Eq. (14) and exploiting the equivalence principle, we can estimate the corrections induced by gravity to the probability of neutrino oscillations (we stress again that we are not concerned with effects of Earth’s rotation). Then, we present a gedanken experiment in which these corrections are evaluated in more exotic regimes. We remark that, in both cases, the condition $a\hskip 0.56905ptx\ll 1$ is satisfied.

V.1 Earth’s gravity effects on atmospheric neutrinos

In the context of the atmospheric neutrino problem, it is known that flavor oscillations can be faithfully analyzed using a simplified two-generations model, since they largely occur between muonic and tauonic flavors ($\nu_{\mu}\leftrightarrow\nu_{\tau}$)777This happens because the mixing angle $\theta_{13}$ is much smaller than the others, and two of the neutrino mass states are very close in mass compared to the third ($\Delta m^{2}_{21}\ll\Delta m^{2}_{32}\approx\Delta m^{2}_{31}$ in the normal mass hierarchy) Tanabashi ..

Atmospheric neutrinos are produced in hadronic showers resulting from the interaction of cosmic rays with nuclei in the atmosphere. Typical flight distances in experiments involving these neutrinos range from $10^{2}$ km (for neutrinos downward-going from an interaction above the detector) to more than $10^{4}$ km (for neutrinos upward-going from collisions on the other side of the Earth). We restrict to the first case, where no background matter effect occurs.

Consider a detector comoving with the Earth: by restoring proper units in Eqs. (II) and (31), a straightforward calculation then leads to

[TABLE]

where we have indicated with $P$ ($\widetilde{P}$) the oscillation probability as measured by the inertial (accelerated) observer. To numerically evaluate Eq. (46), we have set a neutrino mean flight path $L_{p}\sim 10^{2}\,{\mathrm{km}}$, an acceleration of the order of Earth’s gravity, $a\sim 10\,{\mathrm{m/s}}^{2}$, $\Delta m_{\mathrm{atm}}^{2}\sim 10^{-3}\,{\mathrm{eV}}^{2}$, $E_{0}\sim 1\,\mathrm{GeV}$ and maximal mixing $\theta_{atm}\approx\pi/4$ Tanabashi .

The obtained correction is far below the uncertainty on the current best-fit value of the oscillation probability $P_{\nu_{\mu}\rightarrow\nu_{\tau}}$, thus preventing any possibility of detecting gravitational effects on atmospheric neutrino oscillations at present. Future experiments, however, may give new insights in this direction.

V.2 Neutrino oscillations in extreme acceleration regimes: a gedanken experiment

We now propose a gedanken experiment in order to test our formalism in astrophysical regimes. In this framework, it is reasonable to expect a larger contribution of gravitational effects on the oscillation probability, due to the extremely high accelerations that might be reached in this case.

As proof of this, let us consider an ideal accelerated detector in proximity of a high-density object; by way of illustration, we focus on the case of Sirius B, the nearest (known) white dwarf to the Earth. It is known that the gravity on the surface of this star is of the order of $10^{6}\,\mathrm{m/s^{2}}$ wd . For such an acceleration, using Eqs. (II) and (31), we obtain

[TABLE]

where we have set $L_{p}\sim 10^{4}\,{\mathrm{km}}$ and, as for the previous case, $\Delta m^{2}\sim 10^{-3}\,{\mathrm{eV}}^{2}$, $E_{0}\sim 1\,\mathrm{GeV}$ and maximal mixing $\theta\approx\pi/4$. It is worth observing that the chosen value for the oscillation length still allows us to work with the metric of Eq. (14) for the given acceleration.

As predicted, inertial effects may not be completely negligible in this case. However, it is worth saying that experiments like the one above considered are far from being viable nowadays. Indeed, it would be technically cumbersome to build a detector capable of withstanding the mechanical stress arising in those regimes without breaking; on the other hand, even if it were possible, then the problem would arise of how to send and retrieve a probe from the surface of such remote sources (Sirius B, for example, lies at a distance of $8.60$ light-years away from the Sun).

Notwithstanding these technical difficulties, some of the implications of the result Eq. (47) in the physics of neutrino oscillations will be discussed in the next section.

VI Discussion and conclusion

We have analyzed neutrino flavor oscillations from the point of view of a uniformly accelerated observer. Corrections to the standard result have been derived by use of Stodolsky covariant definition of neutrino quantum phase. Relying on phenomenological considerations, we have restricted our discussion to relativistic neutrinos, so that a plane-wave treatment could be applied. In order to realize how acceleration affects the usual Pontecorvo formula, the formalism of neutrino oscillations in curved spacetime has been used. Within such a framework, we have found that inertial effects are intimately related to the redshift of neutrino energy, according to Ref. Cardall . Furthermore, it has been pointed out that a separate “acceleration phase” can be extracted from the standard result only for small accelerations.

As a possible application of our analysis, we have calculated the correction induced by Earth’s gravity on the oscillation probability of atmospheric neutrinos. In that case, simulating the metric of an observer comoving with the Earth with the one in Eq. (14), we have found that the contribution to the neutrino phase-shift is negligible, thus leading to effects which are currently unmeasurable. It is clear that the origin of this outcome can be traced back to the difficulty of detecting gravitational effects on oscillations in the weak-field regime, as it is near to the Earth. On the other hand, in astrophysical regimes (e.g. outside a black hole or in proximity of pulsars) we expect these corrections to be far more relevant (as also suggested by the analysis carried out for a white dwarf), resulting in a possible modification of the oscillation probability induced by gravity (see also Ref. Miller:2013wta for a quasi-classical treatment of neutrino oscillations in the gravitational field of a heavy astrophysical object). If confirmed, such an effect could be exploited for investigating the gravity-induced interactions that neutrinos may have experienced during their travel throughout the Universe, and thus the mass distribution of the Universe itself. Oscillations of neutrinos from supernovae and active galactic nuclei may be valuable to search traces of space-time quantum foam KlapdorKleingrothaus:2000fr . Non-trivial implications may also arise in the context of supernova nucleosynthesis, and, in particular, in the production of heavy elements in neutrino-driven winds from proto-neutron stars and neutrino-induced nucleosynthesis in outer shells of supernovae Wu . A further interesting scenario to explore is the rôle of neutrino oscillations in the generation of the rotational pulsar velocity in the presence of intense magnetic fields Kusenko . These aspects, however, will be investigated in future publications.

Aside from phenomenological aspects, we emphasize that investigating inertial effects on neutrino mixing and oscillations may be useful for clarifying a variety of controversial problems at a theoretical level. Recently, for instance, some concerns have been raised regarding the agreement between the decay rates of accelerated protons in the inertial and comoving frames when neutrino mixing is taken into account Ahluwalia:2016wmf ; Blasone:2018czm ; Cozzella:2018qew . Besides, the evolution of neutrinos in a background matter moving with a linear acceleration has been analyzed in Ref. Dvornikov . Relevant processes in non-inertial frames are studied also in condensed matter physics. In Ref. Basu:2013kt , in particular, an enhancement of the spin current for a linearly accelerating semiconductor system has been predicted.

Finally, we remark that our whole analysis has been performed in the context of Quantum Mechanics. Along the line of Refs. Blasone:2017nbf ; Blasone:2018byx , it naturally arises the question of how the oscillation probability for an accelerated observer would appear within the framework of Quantum Field Theory. Work is already in progress along this line prepa .