Time-energy uncertainty relation for neutrino oscillations in curved spacetime

We derive the Mandelstam-Tamm time-energy uncertainty relation for neutrino oscillations in a generic stationary curved spacetime. In particular, by resorting to Stodolsky covariant formula of the quantum mechanical phase, we estimate gravity effects on the neutrino energy uncertainty. Deviations from the standard Minkowski result are explicitly evaluated in Schwarzschild, Lense-Thirring and Rindler (uniformly accelerated) geometries. Finally, we discuss how spacetime could affect the characteristic neutrino oscillation length in connection with the recent view of flavor neutrinos as unstable particles.


I. INTRODUCTION
Neutrino mixing and oscillations are among the most active research areas within the framework of particle physics. Since Pontecorvo's original treatment in quantum mechanics (QM) [1], the theoretical bases of these phenomena have been extensively analyzed [2], and a quantum field theoretical (QFT) formalism has been developed [3,4], providing continuous insights toward understanding of novel effects [5][6][7]. In spite of this, such puzzling questions as the dynamical origin of the non-vanishing neutrino masses and mixings in the Standard Model still remain open [8].
Starting from the results of Ref. [9] in the context of unstable particle physics, in Refs. [10] it was argued that the Mandelstam-Tamm time-energy uncertainty relation (TEUR) [11] reduces to the well-known condition for neutrino oscillations described by Pontecorvo states [12]. Such an outcome was recently revisited in Ref. [13], where it was found that TEUR can be rephrased in terms of a flavor-energy uncertainty principle as a consequence of the noncommutativity between the Hamiltonian and the lepton charge operators [3,4]. In that case, flavor-energy uncertainty relations imply that neutrino energy cannot be sharply determined, and the study of its distribution reveals to be important from both theoretical and phenomenological viewpoints. Furthermore, following the quantum field theoretical approach of Refs. [3,4], the aforementioned oscillation condition was interpreted as a fundamental bound on neutrino energy precision: as for unstable particles, only energy distributions are meaningful for flavor neutrinos and the width of the distribution is related to the oscillation length.
The above analysis has been only performed in flat spacetime. The first attempt to accommodate gravity effects into the standard picture of neutrino oscillations was made in Ref. [14]. Further investigation was later carried out in Refs. [15][16][17], showing that gravity-induced corrections are related to the energy redshift. Recently, similar results were derived in accelerated frames in both QM [18,19] and QFT [20]. The question thus arises as to how the framework of neutrino oscillations and, in particular, the condition on the energy uncertainty stemming from TEUR is modified in curved backgrounds.
Based on the outlined scenario, in the present paper we derive the Mandelstam-Tamm version of TEUR for neutrino oscillations in a generic stationary curved spacetime. To this aim, we employ the quantum mechanical covariant formalism for oscillations developed in Ref. [16]. Corrections to the lower bounds on the neutrino energy uncertainty are explicitly evaluated for the cases of Schwarzschild, Lense-Thirring and Rindler (i.e. uniformly accelerated) backgrounds. In particular, we find that the characteristic oscillation length gets non-trivially modified by gaining additional terms which depend on the specific features of the considered geometry. In light of neutrino interpretation as unstable particle, this amounts to state that neutrino lifetime is not an intrinsic quantity. Note that such a result is not entirely uncommon in literature; in Ref. [21], indeed, it was argued that the decay properties of particles are not fundamental, since they depend on the reference frame.
The paper is organized as follows: in Section II we prove that Mandelstam-Tamm TEUR can be derived in a generic stationary curved spacetime. In Section III, after a brief review of the standard treatment of neutrino oscillations within Minkowski framework, we discuss the covariant approach of Ref. [16]. We also show how to recast TEUR in a form suitable for generalization to curved backgrounds. Section IV is devoted to the calculation of the neutrino energy uncertainty in Schwarzschild, Lense-Thirring and Rindler spacetimes. In order to separate metric-induced corrections out, we adopt the weak gravity/acceleration approximation. Finally, conclusions and discussion can be found in Section V.
Throughout the work we shall assume natural units = c = 1 and the conventional timelike signature for Minkowski metric, Furthermore, we shall denote local (general curvilinear) coordinates by indices with (without) hat.

II. MANDELSTAM-TAMM TEUR IN STATIONARY SPACETIME
In this Section, we show how to generalize the Heisenberg equation of motion to the case of a generic stationary curved background. For this purpose, let us observe that, for stationary metrics, there exists a global timelike Killing vector field K µ , such that the quantity does not depend on the choice of the spacelike hypersurface Σ if T µν is the (conserved) stress-energy tensor [22]. Since K µ is globally timelike, one can introduce a coordinate t upon which the metric does not depend and with respect to which K µ can be written as K µ = (1, 0, 0, 0). In this case, Eq. (3) can be recast as where we have introduced the normal ordering. Moreover, K satisfies the relation [22] [ where L K is the Lie derivative with respect to K µ . This is exactly the Heisenberg equation (2), since L K O(x) ≡ dO(x)/dt in the present case.
Using the Cauchy-Schwarz inequality, from Eq. (2) it follows that the Mandelstam-Tamm version of TEUR [11] in a generic stationary background can be formulated as with where σ A denotes the standard deviation of the generic operator A, i.e.
In the above expressions, we have used the shorthand notation . . . ≡ ψ| . . . |ψ , where |ψ denotes the state of the system, O(t) represents the "clock observable" that regulates time changes through its dynamics and ∆t is the characteristic time interval that the mean value of O takes to vary by a standard deviation σ O [23].

III. TEUR FOR NEUTRINO OSCILLATIONS
Let us now derive TEUR for neutrino oscillations in a generic stationary curved background. As a first step, we review the standard analysis of flavor oscillations in Minkowski spacetime. Then, we discuss the covariant formalism based on Stodolsky definition of quantum mechanical phase [14]. We restrict our analysis to a simplified two-flavor model. However, the obtained result can be straightforwardly generalized to a more rigorous three-flavor description.

A. Neutrino oscillations in Minkowski spacetime
In the standard treatment of neutrino oscillations, the flavor eigenstates |ν α are written as a superposition of the mass eigenstates |ν k according to [2] where U αk (θ) is the generic element of the Pontecorvo mixing matrix, The spacetime evolution of mass eigenstates is described by 1 where is the quantum-mechanical phase of the k th state with mass m k , energy E k and three-momentum p k , respectively. Mass, energy and momentum are related by the usual mass-shell condition as The oscillation (survival) probability P α→β (P α→α ) that a neutrino produced as |ν α at a spacetime point x 0 is detected as |ν β (|ν α ) at a point x is, therefore, where ϕ 12 ≡ ϕ 1 − ϕ 2 and ϕ k (k = 1, 2) are the phases acquired by the mass eigenstates during their propagation, i.e.
Furthermore, we have defined the projector on the neutrino flavor state |ν β (x) . Now, let us take O(x) = P α (x) and |ψ = |ν α (x 0 ) in Eq. (6). By simple calculations, it is possible to show that [24] 2∆E By observing that the l.h.s. of Eq. (18) reaches its maximum when P α→α = 1/2, we then obtain If we now integrate both sides over time and employ the triangular inequality, the above relation reads [13] ∆E where T ≡ t t0 dt ′ = t − t 0 provides the neutrino time of flight. In order to evaluate the oscillation probability P α→β , let us consider, for simplicity, relativistic neutrinos propagating along a fixed direction (for example, the x-axis). In this case, one can check that the phase Eq. (12) takes the form where we have assumed neutrino mass eigenstates to be energy eigenstates with a common energy E, 2 and is the proper distance travelled by neutrinos. Therefore, from Eqs. (9) and (11) we have By virtue of this relation, the oscillation probability (14) becomes where we have defined the characteristic oscillation length L osc as with ∆m 2 = m 2 2 − m 2 1 . By inserting into Eq. (20), this allows us to derive the following time-energy uncertainty condition for neutrino oscillations: Since the above inequality holds true for any spacetime point, we can set x in such a way to maximize the r.h.s., which implies L p = L osc /2. By means of Eq. (22), this yields Note that, since H is time-independent, the same holds for ∆E. Furthermore, for L osc → ∞, we obtain a vanishing lower bound for ∆E, as one could expect in the absence of flavor oscillations. The same is true for θ = 0: in that case, indeed, neutrino states with definite flavor do coincide with the states with definite mass, which are characterized by a well-defined energy. From the point of view of Ref. [13], this means that neutrinos can be described by standard asymptotic states and not as unstable particles. Therefore, flavor and energy are commuting quantities and no fundamental bounds on energy measurements can be inferred. Inequalities of the form (27) are widely discussed in literature and are usually interpreted as conditions of neutrino oscillation [10,12,13].

B. Neutrino oscillations in curved spacetime
Let us now extend the above formalism to curved backgrounds. To this aim, it comes in handy to cast Eqs. (9) and (23) in a manifestly covariant form. Following Ref. [16], we can then write where Φ is defined as Here, P µ is the generator of spacetime translation of neutrino mass eigenstates and dx µ null /dλ is the null tangent vector to the neutrino worldline parameterized by λ.
It is easy to show that Eq. (28) reduces to Eq. (23) for neutrino propagation in flat spacetime. The advantage, however, is that it allows us to generalize the study of flavor oscillations to curved backgrounds. Indeed, consider the covariant Dirac equation for a doublet ν of spinors of different masses [26], where M = diag[m 1 , m 2 ] is the mass matrix in neutrino mass basis and γ µ are the Dirac matrices satisfying the Clifford algebra {γ µ , γ ν } = 2g µν 1 . The general curvilinear and locally inertial coordinates are connected by the tetrads e μ a , as usual.
The above equation can be simplified by observing that the only non-vanishing contribution of the spin connection where g ≡ |det g µν |, γ 5 = iγ 0 γ 1 γ 2 γ 3 and with ǫâbĉd being the totally antisymmetric tensor of component ǫ0123 = +1. The momentum operator P µ appearing in Eq. (29) can be derived from the generalized mass-shell relation, Neglecting terms of O(A 2 ) and O(AM 2 ), one gets that the quantity P µ dx µ null /dλ for relativistic neutrinos moving along generic null trajectories parameterized by λ reads In deriving Eq. (34), we have required the three-momenta of P µ and dx µ null /dλ to be parallel (i.e. P i ≃ dx i null /dλ) and P 0 = dx 0 null /dλ [16]. By substituting the above expression in Eq. (29), the covariant phase takes the form where dλ can be written in terms of the differential proper distance at constant time dℓ by using the condition of null trajectory, Exploiting the above formalism, we can now recast TEUR in a form suitable for generalization to arbitrary stationary backgrounds. In this perspective, let us rephrase Eq. (20) in terms of the affine parameter λ, where T (λ) ≡ t(λ)−t(λ 0 ) and the integration has been performed over the neutrino null worldline. By use of Eqs. (14), this yields where now ϕ k (k = 1, 2) is more generally defined by the action of the operator Φ in Eq. (35) on the k th neutrino state, i.e.
It should be emphasized that, in the above treatment, ∆E is the energy uncertainty for an observer at rest at infinity. This holds true since E is the eigenvalue of the Hamiltonian H in Eq. (2), which is the time component of the covariant momentum P µ (see footnote 2). In order to recast Eq. (38) in terms of the energy uncertainty ∆E ℓ for a local observer momentarily at rest in the curved background (and also with respect to the oscillation experiment), we must employ vierbein fields, so that Furthermore, the phase-shift ϕ 12 must be expressed in terms of the local energy and proper distance.

IV. TEUR IN CURVED SPACETIME
Let us apply Eq. (38) to estimate the energy uncertainty of neutrinos propagating in non-trivial stationary backgrounds. Specifically, we consider the cases of Schwarzschild, Lense-Thirring and Rindler metrics. In our calculations, we adopt the weak field approximation, so that deviations of the energy uncertainty from the corresponding flat expression (27) can be explicitly exhibited.

A. Schwarzschild metric
It is well known that, in isotropic coordinates, the (linearized) weak field Schwarzschild metric reads where the gravitational potential φ is defined as with G being the gravitational constant and M the mass of the source of gravity. The only non-vanishing tetrad components for this metric are Using Eqs. (32) and (41), one can easily prove that A µ = 0, due to the spherical symmetry of the metric. From Eq. (34), it thus follows that the neutrino phase-shift ϕ 12 along a null geodesic in Schwarzschild spacetime is simply given by or equivalently where E ≡ P 0 = g 00 dt/dλ and we have employed Eq. (36) to relate dλ and dℓ. Without loss of generality, we can consider the case of neutrino radial propagation (for example, along the x-axis). From Eq. (41) we then have so that the phase-shift ϕ 12 takes the form [16] where we have exploited the fact that E is constant along the null trajectory. Therefore, Eq. (38) reads As explained above, in order to make a comparison with the corresponding flat expression (27), we need to rephrase Eq. (48) in terms of quantities measured by an observer at rest in the curved background. Then, by defining the proper distance corresponding to the coordinate difference x − x 0 as and employing Eq. (40) to express the asymptotic energy E in terms of the local one E ℓ as we obtain This can be recast in a form similar to Eq. (26), by introducing the following definition of proper oscillation length L osc [17]: where we have neglected higher-order terms in φ. Moreover, we stress that the gravitational potential is to be regarded as φ(x) = −GM/x. Let us now observe that the maximization of the r.h.s. of Eq. (52) yields L p = L osc /2, which allows us to relate the time interval T to the proper distance L p travelled by neutrinos as in Minkowski analysis. Indeed, by exploiting the condition of neutrino null trajectory ds 2 = 0, we have which leads to By inserting into Eq. (52), we finally get where we have defined an effective oscillation length L osc ef f depending on the mass of the source of gravity according to with L osc given in Eq. (53). In this way, TEUR for neutrino oscillations in Schwarzschild spacetime acquires the same form as in the Minkowski case, Eq. (27). Note that L osc ef f is increased with respect to the corresponding expression in flat spacetime due to the gravitational field [17], thus leading to a more stringent lower bound on the local energy uncertainty. Clearly, for M → 0, L osc ef f (M ) reduces to the oscillation length L osc in Eq. (25), so that Eq. (56) recovers the standard energy condition in flat background. We also note that, for L osc ef f (M ) → ∞, we have ∆E ℓ ≥ 0, in accordance with the discussion at the end of Section III A.
Some comments are in order here: first, the effective oscillation length (57) and the ensuing condition on neutrino energy (56) depend explicitly on the details of the metric, as one could expect. Such a result becomes even more interesting from the point of view of QFT analysis [13]. In that context, indeed, neutrinos are viewed as unstable particles, and Eqs. (27) and (56) are interpreted as a bound on the width of energy distribution. As a consequence, the quantity L osc ef f /(2 sin 2 (2θ)) plays the rôle of neutrino lifetime, which thus turns out to be intimately related to the geometric properties of the spacetime in which neutrinos propagate. Moreover, for neutrino travelling parallel to the x-axis with impact parameter y = b, the condition (57) takes the form The above relation will be useful for comparison with the next analysis in Lense-Thirring metric.

B. Lense-Thirring metric
The weak field solution for the gravitational field generated by a rotating spherical body on the equatorial plane is [27] where Ω ≡ 4R 2 ω/5, φ is the gravitational potential defined as in Eq. (42), and ω is the angular velocity of the source of gravity, which we assume to be oriented along the z axis. This is the metric usually employed to describe gravitomagnetic frame-dragging effect [28]. In this framework, the non-vanishing tetrad components are Unlike Schwarzschild case, A µ has a non-trivial expression, owing to the presence of off-diagonal contributions in the metric. Specifically, from Eq. (32) it follows that Furthermore, since we are considering neutrinos propagating in the equatorial plane, we can set z = const = 0 and dx µ null /dλ = (dx 0 null /dλ, dx 1 null /dλ, dx 2 null /dλ, 0). Hence, by using Eq. (29), one can show that the phase-shift ϕ 12 along a null geodesic in the Lense-Thirring metric has the same form as in Eq. (44), since the contribution due to the potential A µ in Eq. (34) vanishes. The remarkable difference, however, lies in the relation between dℓ and dλ. Indeed, by resorting to Eq. (36), we have Let us restrict our discussion to relativistic neutrinos propagating along the x-axis with impact parameter y = b > R. In this case, Eq. (62) reduces to The above relation can be expressed in terms of quantities measured by a local observer by employing the definitions (40) and (49) of local energy and proper distance, respectively. In particular, we have which lead to Note that Eq. (67) can be still cast in the form provided that the oscillation length is now defined as Once again, the r.h.s. of Eq. (68) is maximized for L p = L osc /2. Thus, by exploiting such a condition and performing calculations similar to the ones in Schwarzschild background, one can show that the time interval T for a null trajectory is related to the proper distance L p travelled by neutrinos as follows: Substitution of Eq. (70) into Eq. (68) entails where the effective oscillation length L osc ef f now depends on both the mass and the angular velocity of the source according to Clearly, for Ω → 0, the corresponding Schwarzschild formulas (56) and (58) are recovered, as expected for a spherically symmetric non-rotating source. Furthermore, we stress that, as in the previous analysis, the effective oscillation length is increased with respect to the flat case.

C. Rindler metric
Finally, let us discuss how the condition (27) gets modified for an observer moving with eternal uniform linear acceleration. First, we remark that, from the point of view of such an observer, the line element can be written in terms of the Rindler Fermi coordinates as [19,29] where we have assumed the acceleration a to be parallel to the x-axis. Note that Eq. (73) only holds in the approximation x ≪ a −1 [19,29].
In the above setting, one can show that the only non-trivial tetrad component is and A µ = 0. Thus, by resorting to Eqs. (34) and (36), the neutrino phase-shift takes the form where, in the second step, we have used the relations Note that, unlike the two previous examples, the "asymptotic" energy E is now to be regarded as the energy measured by an observer at rest at the origin of the coordinate system (where the metric Eq. (73) reduces to the flat Minkowski one), rather than the energy measured at infinity [19]. From Eq. (38), we have where the oscillation length has been now defined as By using this equation and observing that, for null trajectories, the time interval T is related to the proper distance L p by we finally obtain the following bound on the local energy uncertainty: where and we have imposed the usual condition L p = L osc /2. Differently from Schwarzschild and Lense-Thirring geometries, however, for a given finite acceleration a, we cannot consider arbitrarily large effective oscillation lengths, since this would violate the approximation ax ≪ 1 in which the metric (73) is valid.

V. CONCLUDING REMARKS
In this work, we have derived the time-energy uncertainty relation for neutrino oscillations in the Mandelstam-Tamm form in the case of a generic stationary background. In particular, it has been shown that the standard oscillation condition calculated in Refs. [10,12,13] is non-trivially modified in the presence of gravity/acceleration. By means of Stodolsky covariant definition of the quantum mechanical phase, we have evaluated the oscillation probability formula in curved spacetime, and then we have estimated lower bounds on the energy uncertainty ∆E. Specifically, this has been done in Schwarzschild, Lense-Thirring and Rindler geometries. Deviations from the corresponding result in Minkowski spacetime have been explicitly calculated in the weak field approximation, emphasizing how the flat outcome is recovered in the limit of vanishing gravitational field/acceleration.
The aforementioned corrections to the energy uncertainty condition have been expressed by defining an effective oscillation length depending on the features of the metric itself for each treated background. This becomes particularly interesting if we regard such a result as the ultrarelativistic approximation of some more fundamental relation which has to be derived in the context of Quantum Field Theory. As argued in Ref. [13], indeed, TEUR for neutrino oscillations in QFT can be recast in the form of a flavor-energy uncertainty principle; moreover, a field theoretical approach to the phenomenon of flavor transitions leads to a novel interpretation of the condition in Eq. (27) (and similarly for the relations obtained in curved spacetimes): flavor neutrinos can be formally viewed as unstable particles and the quantity (L osc ef f )/(2 sin 2 (2θ)) can be interpreted as the neutrino lifetime. The derived conditions of neutrino oscillations thus provide estimations of the energy distribution width, which cannot be eliminated in experiments and characterizes the definition of neutrino flavor states. As a result of our analysis, this definition turns out to be closely related to the details of the spacetime in which neutrinos propagate. We remark that a similar achievement was obtained in Ref. [21], where it was shown that the decay properties of particles are less fundamental than commonly thought.
Apart from its relevance in understanding of neutrino flavor oscillations in curved spacetime, we remark that the above discussion fits in the longstanding controversy about the real physical nature of asymptotic neutrino states in QFT [3,4]. In this sense, it should be emphasized that, whilst in the QM treatment mixing relations between flavor and mass neutrinos act as pure rotations on massive states, in QFT they emerge from the interplay between rotations and Bogoliubov transformations at level of ladder operators [3]. The non-commutativity between these two transformations turns out to play a key rôle, being responsible for the unitary inequivalence of mass and flavor representations and their related vacuum structures. Considerations based upon the conservation of the total lepton number in the production/detection vertex of a weak decay process involving neutrinos [13] seem to indicate that flavor states provide the most appropriate description of the behavior of asymptotic neutrinos. In that case, flavor-energy uncertainty relations imply that neutrino energy cannot be exactly measured, and the analysis of its distribution may offer valuable informations at both theoretical and phenomenological levels. Similar arguments in favor of flavor states have been given in Refs. [30] on the basis of the General Covariance of QFT in the context of the decay of accelerated protons.
As a further development, we emphasize that the present analysis may open new perspectives in the study of entanglement in curved spacetime. Indeed, quantum information properties related to neutrino oscillations were extensively explored [31]. In particular, in Refs. [6,32] linear entropies were employed to quantify dynamical entanglement, showing that they are proportional to the uncertainty of the number operator on neutrino flavor states. Such an achievement was later generalized to QFT [33]. It is thus evident that the study of uncertainty relations for neutrino oscillations is closely related to the quantum information properties of this phenomenon, as it was already pointed out in Ref. [13]. Neutrino entanglement in curved spacetime is already under investigation.
More details about the intimate nature of neutrino mixing and oscillations can only be understood by developing a full-fledged field theoretical treatment of these phenomena. Progress along this direction has been recently made in Refs. [34], where these issues have been addressed in connection with the non-trivial structure of vacuum in QFT [35].