# Deriving the Gibbons-Maldacena-Nunez no-go theorem from the Raychaudhuri
equation

Mir Mehedi Faruk  Department of Physics, McGill University, Montreal, Quebec,
H3A 2T8, Canada.

###### Abstract

In this article, we point out that to solve the null Raychaudhuri equation for
higher dimensional spacetime with accelerating FRW solution in external
directions and static compact internal directions, it is necessary to violate
the Strong Energy condition in higher dimensions. This constraint is well-
known in obtaining accelerating cosmological solutions in string
compactification, first described by Gibbons-Maldacena-Nunez. In deriving this
constraint, we do not make any assumptions regarding the matter content.

††preprint: APS/123-QED

We have compelling evidence that the present universe is dominated by dark
energy and going through a period of accelerated expansion. It is, therefore,
natural to ask ourselves what quantum gravity theories, such as
string/M-theory can tell about accelerating cosmological solutions[1].
However, the construction of accelerating cosmological solutions in string
theory is still an open problem even after almost twenty years of the subject.
The research direction took an interesting turn due to the obstacles such as
the no-go theorems[2, 3] and the proposed swampland conjectures[4, 5, 6]. The
most interesting of them is the well-known no-go first proposed by Gibbons[7,
2]; later on, a more refined version came from Maldacena and Nunez[3] which
took into account the supergravity fluxes and the (Anti) D-Branes. The most
significant point made in this no-go theorem is that to obtain a
$d$-dimensional accelerating cosmological solution by compactifying a $D>d$
dimensional theory, it is necessary to violate the $D$-dimensional strong
energy condition (SEC). Afterward, many other no-go theorems were constructed
in string cosmology paradigm from different viewpoints, including metric-based
constraints[8, 9, 10, 11], worldsheet symmetry[12, 13, 14], energy conditions
[15, 16, 17, 18, 8, 19, 6, 20, 21], supersymmetry[22, 23, 24] string/M theory
[25, 26, 27, 20, 28, 29, 30, 31, 32, 33, 34, 35, 36, 26, 37, 38, 27, 39, 40,
41, 42, 43], spacetime entropy and quantum gravity in dS space [44, 45, 46,
47, 48]. We have made considerable progress in understanding the properties of
gravitational thermodynamics [49, 50, 51, 52, 53]. In these works as well as
in the Hawking-Penrose singularity theorems[54, 55] Raychaudhuri equation
provides a fundamental contribution.

Raychaudhuri equation, a well-known geometric identity is used extensively to
enhance our understanding of various disciplines involving gravity, from
astrophysics[56] to holography[57] and quantum gravity[58, 59]. Recently, in
an interesting article[60] this geometric identity has been used to further
derive new no-go theorems in string compactification. In the no-go theorem,
the article concluded that accelerating backgrounds in string theory can only
solve the Raychaudhuri equation when the null energy condition (NEC) is
violated and/or the internal directions have positive curvature. Besides, the
well studied flux compactification schemes in dS, such as KKLT, are re-visited
in light of this new no-go theorem in the article[60]. The authors point out
that the matter sources or geometries that can potentially evade many of the
previous no-go’s and are considered to be essential ingredients in building
putative dS solutions are unfortunately ruled out by the NEC violation
constraint111Third paragraph of page 2 of ref. [60]. This is bad news for de
Sitter (dS) compactifications; as we know, four-dimensional dS maintains
NEC[62]. The NEC is also well-known to be satisfied by a large set of matter
content, which will further restrict the models with extra dimensions. In this
article, we carefully study the conditions to satisfy the Raychaudhuri
equation for a $D$-dimensional spacetime solution where we have a
$d$-dimensional FRW solution in the external direction and a compact internal
manifold of dimension $n$[63]. There are several arguments[1] whether such
cosmological background with accelerating cosmology can be obtained in string
compactification. The pursuit of constructing four-dimensional accelerating
solutions in string theory has led to claims ranging from multiple
solutions[25, 64, 65] to none at all[6, 66]. But here, we re-examine the
constraints for such backgrounds to satisfy the Raychaudhuri equation. In
particular, we focus if any violation of NEC is really necessary to satisfy
the Raychaudhuri equation. It is well known that NEC also plays a significant
role in establishing the existence of the Big Bang singularity, as well as
proving the second law of thermodynamics for black holes[67]. There are also
some hints that accelerating cosmology should satisfy the NEC to have a UV
completion in string theory[15]. For example, the Virasoro constraint in
string theory coming from worldsheet theory precisely gives rise to the NEC in
the geometry[13]. Because of NEC violation, it is quite difficult to obtain
wormhole solutions, the creation of laboratory universes, and the building of
time machines[13]. We demonstrate here that to solve the Raychaudhuri equation
with four-dimensional accelerating cosmology in external direction
successfully, we rather must violate the SEC in higher dimensions. It is quite
remarkable as this is the precise statement of the GMN no-go theorem. We
briefly discuss the energy conditions, the null Raychaudhuri equation, and the
GMN no-go theorem. Then, in section IV, we work out the details to look for
the constraints to solve the Raychaudhuri equation for the background metric
in (III).

##  I A brief review on Energy conditions

We quickly discuss the Null and Strong energy conditions in this section and
the constraints it provides on the ”scale factor” of the FRW metric[62]. The
NEC simply states that for all the nulllike vector $l^{M}$ we have the
following constraint on Ricci tensor,

$R_{MN}(x)l^{M}l^{N}\geq 0,\quad g_{MN}(x)l^{M}l^{N}=0.$ (1)

Similarly, SEC implies that for any timelike vector $t^{M}$,

$R_{MN}t^{M}t^{N}\geq 0,\quad t^{2}<0.$ (2)

Considering the FRW metric in the physical time co-ordinates,

$\tilde{g}_{\mu\nu}dx^{\mu}dx^{\nu}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j}.$
(3)

As we know, $H=\frac{\dot{a}}{a}$ is the Hubble scale. Accelerating solutions
are identified with

$\displaystyle\ddot{a}/a=\dot{H}+H^{2}>0.$ (4)

For a power law scale factor, $a(t)\propto t^{\gamma}$, we have

SEC $\displaystyle\iff 0<\gamma\leq 1,$ (5) NEC $\displaystyle\iff\gamma\geq
0,$ (6)

If NEC violation is necessary to maintain the Raychaudhuri equation, then it
would rule out many the four-dimensional cosmology such as dS. It is much more
difficult to violate the NEC than it is to violate the SEC. Violating the NEC,
on the other hand, is very difficult, and no known classical energy-momentum
sources or fields are known do so222We express our gratitude to the referee
for assisting us in clarifying this issue..

##  II The null Raychaudhuri equation

In order to establish a singularity theorem, it is necessary to have an
effective means of anticipating the appearance of focal points along
geodesics. Raychaudhuri’s equation offers such a methodology333Interested
readers please look into[78]. Raychaudhuri’s equation reveals that the
occurrence of focal points is quite common, as gravity has a propensity to
focus nearby geodesics. The Raychaudhuri equation for null geodesic
congruences is the following,

$\displaystyle\frac{d\theta}{d\lambda}=-\frac{1}{D-2}\theta^{2}-\sigma^{2}-R_{MN}l^{M}l^{N}$
(7)

Here, $l^{N}$ are the Null vectors. Now, the expansion parameter and the shear
tensor are identified as,

$\displaystyle\theta=\frac{1}{\sqrt{-g_{D}}}\partial_{M}(\sqrt{-g_{D}}l^{M})$
(8)
$\displaystyle\sigma_{MN}=\frac{1}{2}(\nabla_{M}l_{N}+\nabla_{N}l_{M})-\frac{1}{D-2}\hat{h}_{MN}\theta$
(9)

And, $\hat{h}_{MN}$ is the transverse metric, i.e.,

$\displaystyle\hat{h}_{MN}l^{M}=0$ (10)

These are of course transverse to null rays.

##  III Gibbons-Maldacena-Nunez no go theorem

We quickly review the GMN no-go theorem [2, 7, 3] in this section. We will
consider the following warped product $D=d+n$ dimensional manifold as a
solution of higher dimensional quantum gravity theory such as string theory
($D=10$) following[63]. In the external direction, we took the four-
dimensional FRW metric, and the internal direction is given by a (time-
independent) compact $n$-dimensional manifold.

$\displaystyle
ds^{2}=e^{-2A(y^{m})}{g}_{\mu\nu}dx^{\mu}dx^{\nu}+e^{2A(y^{m})}{g}_{mn}(y^{m})dy^{m}dy^{n}.$

The above metric can also be written in the following way using conformal
rescaling of the internal metric[60],

$\displaystyle
ds^{2}=\Omega^{2}(y^{m})[\tilde{g}_{\mu\nu}dx^{\mu}dx^{\nu}+\tilde{h}_{mn}(y^{m})dy^{m}dy^{n}]$

We will refer to the $\Omega$ as the warp factor. We take the compact manifold
has no boundary, and the warp factor is non-singular. Calculating the Ricci
tensor for the $D$-dimensional metric in the external directions following
(36),

$\displaystyle
R^{(D)}_{\mu\nu}=R^{(d)}_{\mu\nu}(\tilde{g})-\tilde{g}_{\mu\nu}[\nabla^{2}(\ln\Omega)+(D-2)(\nabla\ln\Omega)^{2}]$
(13)

The covariant derivative of the above equation is simply along the compact
directions. Using the Einstein equation for the full metric (III), we find out
that,

$\displaystyle
R^{(D)}_{\mu\nu}=T_{\mu\nu}-\frac{\Omega^{2}}{D-2}\tilde{g}_{\mu\nu}T^{M}_{M}$
(14)

Comparing (13), (14) and taking trace over $\tilde{g}$,

$\displaystyle\frac{1}{(D-2)}\frac{\nabla^{2}\Omega^{D-2}}{\Omega^{D-2}}=R^{(d)}+\Omega^{2}(-T^{\mu}_{\mu}+\frac{d}{D-2}T^{M}_{M})$

As a result[3], when we want to have positive curvature spacetime, i.e.
$R^{(d)}>0$ this implies,

$\displaystyle\boxed{(D-d-2)T_{\mu}^{\mu}>dT_{m}^{m}.}$ (16)

Alternatively, a similar statement is when we want to have accelerated
expansion $\frac{\ddot{a}}{a}\geq 0$ in warped compactification, the $D$\-
dimensional Einstein field equations allow a time-independent compactification
to a accelerating solution of dimension $d<D$ if the $D$-dimensional stress
tensor violates the SEC. This can be viewed easily in physical time
coordinates of FRW metric. From eq. (III) we find out,

$\displaystyle
R_{00}^{(D)}=-(d-1)\left(\dot{H}+H^{2}\right)+\frac{\mathrm{\Omega}^{-(D-2)}}{D-2}\nabla^{2}\mathrm{\Omega}^{(D-2)}$

Multiplying both sides with $\Omega^{D-2}$ and integrating over compact space
leads to,

$(d-1)\frac{G_{D}}{G_{d}}\left(\dot{H}+H^{2}\right)=-\int
d^{n}\tilde{y}\sqrt{\tilde{h}}\,\Omega^{(D-2)}R_{00}^{(D)}$ (18)

Here, $G_{D}$ and $G_{d}$ are Newton’s constant in $D$ and $d$ dimensions
respectively. We can clearly see to obtain the accelerating solution we need
to violate (integrated version of) the $D$-dimensional SEC, i.e.

$\displaystyle\int d^{n}\tilde{y}\sqrt{\tilde{h}}\,\Omega^{D-2}R_{00}^{(D)}<0$
(19)

If the higher dimensional SEC was satisfied we would always have
$\frac{\ddot{a}}{a}\leq 0$, i.e. non accelerating cosmology in the four
dimensional external directions. We should also note when we have accelerating
FRW solution it automatically indicates-

$R^{(4)}=3\left(\frac{\ddot{a}}{a}+(\frac{\dot{a}}{a})^{2}\right)>0,$ (20)

positive curvature444Thanks to the referees for pointing it out. As the second
term is clearly non-negative, by nature. We will further prove in the next
section that the accelerating backgrounds in (III) should maintain the
condition (19) to satisfy the Raychaudhuri equation.

##  IV The Raychaudhuri equation and GMN no go theorem

We start to examine with background (III), if this can satisfy the
Raychaudhuri equation following [60]. In other words, we would like to
understand what are the constraints for the background (III) to maintain the
Raychaudhuri equation. We first take the affine null vectors such as,

$\displaystyle N^{M}=\frac{1}{\Omega^{2}}(1,0,0,0,\tilde{n}^{m})$ (21)

Here, $\tilde{n}_{m}$ is an affine unit $n$-dimensional spacelike vector with
respect to metric $\tilde{h}_{mn}$, which means[60]-

$\displaystyle\tilde{h}_{mn}\tilde{n}^{m}\tilde{n}^{n}=1,\,\,\tilde{n}^{m}\tilde{\nabla}_{m}\tilde{n}^{n}=0$
(22)

The expansion parameter associated with $N_{A}$ is,

$\displaystyle\theta=N^{m}\partial_{m}[\ln(\Omega^{D-2}\sqrt{\tilde{h}})]+\frac{3H}{\Omega^{2}}$
(23)

Here, $\sqrt{\tilde{h}}=[det(\tilde{h}_{mn})]^{\frac{1}{2}}$. The shear
tensors have the following non-zero components.

$\displaystyle\sigma_{tt}=-\tilde{h}^{mn}\partial_{n}(\ln\Omega)N_{m}$ (24)
$\displaystyle\sigma_{tm}=\partial_{m}(\ln\Omega)$ (25)
$\displaystyle\sigma_{mn}=-\Gamma^{p}_{mn}N_{p}-\frac{\theta}{D-2}\hat{h}_{mn}$
(26)
$\displaystyle\sigma_{ij}=\Gamma_{ij}^{M}N_{M}-\frac{\theta}{D-2}\hat{h}_{ij}$
(27)

Here, $\Gamma^{A}_{BC}$ are the Christoffel Coefficients of (III). We have
listed them in the appendix[19]. Combining the equations (7), (23) and
(24)-(27), we find out exactly as ref. [60],

$\displaystyle
3(H^{2}+\dot{H}^{2})=\tilde{R}_{mn}^{(n)}\tilde{n}^{m}\tilde{n}^{n}+A_{mn}(\Omega)\tilde{n}^{m}\tilde{n}^{n}$
$\displaystyle-{\Omega^{4}}R_{MN}^{(D)}N^{M}N^{N}$ (28)

Here, $A_{mn}(\Omega)$ is defined as,

$\displaystyle
A_{mn}(\Omega)=(D-2)[\partial_{m}(\ln\Omega)\partial_{n}(\ln\Omega)-\nabla_{m}\partial_{n}(\ln\Omega)].$

Rewriting (28) as,

$\displaystyle
3(H^{2}+\dot{H})=\tilde{R}_{mn}^{(n)}\tilde{n}^{m}\tilde{n}^{n}+A_{mn}(\Omega)\tilde{n}^{m}\tilde{n}^{n}$
$\displaystyle-R_{00}^{(D)}-R_{mn}^{(D)}\tilde{n}^{m}\tilde{n}^{n}$ (29)

As we can see $R_{mn}^{(D)}$ is related to $\tilde{R}_{mn}^{(n)}$ by conformal
symmetry similar to (13) which we can use to further simplify our calculation
(using (36)).

$\displaystyle{R}_{mn}^{(D)}=\tilde{R}_{mn}^{({n})}({\tilde{h}})-\tilde{h}_{mn}\big{(}(D-2)\partial_{p}\ln\Omega\partial^{p}\ln\Omega+\Box\ln\Omega\big{)}$
$\displaystyle+(D-2)\big{(}\partial_{m}(\ln\Omega)\partial_{n}(\ln\Omega)-\nabla_{m}\partial_{n}(\ln\Omega)\big{)},$
(30)

We can clearly see the last term in the above equation (30) exactly cancels
$A_{mn}(\Omega)$ in eq. (29). So we re write eq. (29),

$\displaystyle
3(H^{2}+\dot{H})=-R_{00}^{(D)}+\big{(}(D-2)\partial_{p}\ln\Omega\partial^{p}\ln\Omega+$
$\displaystyle\Box\ln\Omega\big{)}\tilde{h}_{mn}\tilde{n}^{m}\tilde{n}^{n}$
(31)

Simplifying further (31) using the first equation of (22)555We have also used
a known mathematical identity
$(D-2)\partial_{p}\ln\Omega\partial^{p}\ln\Omega+\Box\ln\Omega=\frac{\Box\Omega^{D-2}}{(D-2)\Omega^{D-2}}$,
also appeared in eq.(33) of [3] and multiplying both sides by
$\Omega^{{D-2}}$,

$\displaystyle
3(D-2)(H^{2}+\dot{H})\Omega^{D-2}=-(D-2)R_{00}^{(D)}\Omega^{D-2}+{\Box\Omega^{D-2}}$

First, let us now perform an integral over the compact internal space,

$\displaystyle 3(H^{2}+\dot{H})\frac{G_{D}}{G_{d}}=-\int
d^{n}\tilde{y}\sqrt{\tilde{h}}\,\Omega^{D-2}R_{00}^{(D)}.$ (33)

Throughout, we have used the fact that the integral of the Laplacian of the
warp factor over the compact manifold is zero as the warp factor is non-
singular and the compact manifold has no boundary. Let us first focus on -

A) The dS solution, where we know that $\dot{H}=0$. As a result, the LHS is a
positive definite quantity.

B) In the case of other accelerating FRW solutions,

$\displaystyle\dot{H}+H^{2}=\frac{\ddot{a}}{a}>0$

and finally, LHS of (33) is also a positive definite quantity as well. In both
cases, we find out that the LHS is positive. The only way the RHS of (33) is a
positive quantity if,

$\int d^{n}\tilde{y}\sqrt{\tilde{h}}\,\Omega^{D-2}R_{00}^{(D)}<0$ (34)

this looks line an (integrated) constraint which indicates towards (averaged)
SEC violation in higher dimensions. But this is the main essence of the GMN
no-go theorem (19), i.e one needs to violation SEC in the higher dimensions to
obtain four dimensional accelerating FRW solution. As we point out here the
Raychaudhuri equation results an integrated (averaged) SEC violation
constraint. This follows entirely following a geometric identity, i.e., the
Raychaudhuri equation, not having any presumption on the matter content of the
theory. In ref. [60], the authors have argued the constraints one obtains to
solve this geometric identity leads to much stronger condition than the
existing no-go theorems such that the NEC must be violated at every point.
Therefore the NEC violation constrain put considerable restriction to
construct any accelerating cosmology. The NEC is the weakest of the energy
conditions in a sense that a violation of the NEC implies a violation of the
other energy conditions, such as weak, dominant, and strong energy conditions.
The minimal coupling of NEC violating matter to Einstein gravity is likely
inconsistent with string theory and black hole thermodynamics[72]. Also, NEC
violating theories often display unsettling characteristics, namely
superluminal propagation[73], unbounded negative Hamiltonians [74]. Therefore,
if NEC violation is essential for the background (III) to satisfy the
Raychaudhuri equation, it could be problematic as NEC violation would cause
the aforementioned problems.

But as we have investigated the problem carefully, we have realized from eq.
(33) that it is important to violate the $D$ dimensional SEC (at least the
integrated version(34)) to satisfy the Raychaudhuri equation when backgrounds
have accelerating FRW solutions in the external directions. This geometric
identity (7) does not impose any condition on the curvature of compact
internal space and/or on the NEC for the background under consideration in
(III). As a consequence violating NEC is not an essential condition for
accelerating geometries. Satisfying this geometric identity for the background
(III) leads to a constrain which indicates the same conclusion as GMN no-go
theorem.

This can be understood from the seminal work of Jacobson. The main result of
the GMN no-go theorem is dependent upon satisfying the Einstein equation. We
basically take the trace over the higher dimensional Einstein equation in the
external direction (see Section III). We do not explicitly perform such a step
to find the relevant constraints to satisfy the Raychaudhuri equation. But as
pointed out by works of Jacobson[49, 50], making the use of null Raychaudhuri
equation, along with the fundamental heat flow equation $\delta Q=TdS$ one can
obtain the Einstein equation as the equation of state. As a result, although
we have not explicitly evoked the Einstein Equation in the derivation of
section IV, it is already implied due to Jacobson[49, 52]. One of the main
reason we reach a different conclusion compared to [60], is when the authors
reach (29) they do no relate $\tilde{R}_{mn}^{(n)}$ with $R_{mn}^{(D)}$.
However, as we know these quantities are related by simple conformal
transformation (30). When we use it to derive the constrain, we find out that
not only the contribution from spacelike parts completely disappears but also
the warp factor contribution can be nicely removed when we perform an integral
over compact space. Consequently, we are left with an integrated constrain
involving only timelike directions of the Ricci tensor, i.e. $R_{00}$. This
allows us to conclude that we need to violate the SEC to satisfy the
Raychaudhuri equation and in the process we re-invent the main essence of GMN
no-go theorem. We have not discussed the status of apparent horizon and
antitrapped surfaces in this article[60]. In future we would like to revisit
these issues on time dependent background, because as pointed out by
Townsend[29, 41] and Steinhardt[11] to evade all the no-go theorems for dS
compactification we might need to go beyond the time independent setup. In
particular we would like to understand how such backgrounds can be realised as
coherent state[75, 76] to bypass the Swampland conjectures[6] and Trans-
Planckian problems[77].

###### Acknowledgements.

We thank F. Rost, L. Aalsma, S. Ahmed, H. Bernardo and K. Dasgupta for
discussions and comments on the draft.

##  Appendix A Christoffel Coefficients

The list Christoffel Coefficients associated with (III) are,

$\displaystyle\Gamma^{\rho}_{\alpha
m}=\partial_{m}(\ln\Omega)\delta^{\rho}_{\alpha}$
$\displaystyle\Gamma_{mn}^{\rho}=0$ $\displaystyle\Gamma^{m}_{\mu r}=0$
$\displaystyle\Gamma_{\mu\nu}^{m}=\tilde{g}_{\mu\nu}\tilde{h}^{mn}\partial_{n}(\ln\Omega)$
$\displaystyle\Gamma^{m}_{nr}=\tilde{\Gamma}^{m}_{nr}(\tilde{h})+\frac{1}{\Omega}(\delta_{r}^{m}\partial_{n}\Omega+\delta_{n}^{m}\partial_{r}\Omega-\tilde{h}_{nr}\partial^{m}\Omega)$
$\displaystyle\Gamma^{\rho}_{\mu\nu}=\tilde{\Gamma}^{\rho}_{\mu\nu}(\tilde{g})$

##  Appendix B Ricci Tensor

In any warped geometry represented with metric

$\displaystyle
ds^{2}=\tilde{h}(z)\tilde{ds}^{2}=\tilde{h}(z)\tilde{G}_{MN}dz^{M}dz^{N}$

The Ricci tensor for the full metric $R(G_{MN})$ is related to Ricci tensor of
the metric $R(\tilde{G}_{MN})$,

$\displaystyle
R(G_{MN})=R(\tilde{G}_{MN})-(D-2)\big{(}\nabla_{M}\nabla_{N}(\frac{\ln\tilde{h}}{2})$
$\displaystyle-\nabla_{M}(\frac{\ln\tilde{h}}{2})\nabla_{N}(\frac{\ln\tilde{h}}{2})\big{)}-\tilde{h}_{MN}\big{(}\nabla_{P}\nabla^{P}(\frac{\ln\tilde{h}}{2})$
$\displaystyle+(D-2)\nabla_{P}(\frac{\ln\tilde{h}}{2})\nabla^{P}(\frac{\ln\tilde{h}}{2})\big{)}$
(36)

Block letter indices such as $M$ can be thought of direction $0,,,,D-1$ for
the metric (III).

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# Gravitational effects on neutrino decoherence in Lense-Thirring metric

Giuseppe Gaetano Luciano111email<EMAIL_ADDRESS>(corresponding
author)${}^{\hskip 0.28453pt1,2}$ and Massimo Blasone222email:
<EMAIL_ADDRESS>0.28453pt1,2}$ 1Dipartimento di Fisica,
Università di Salerno, Via Giovanni Paolo II, 132 I-84084 Fisciano (SA),
Italy.
2INFN, Sezione di Napoli, Gruppo collegato di Salerno, Via Giovanni Paolo II,
132 I-84084 Fisciano (SA), Italy.

###### Abstract

We analyze gravity effects on neutrino wave packet decoherence. As a specific
example, we consider the gravitational field of a spinning spherical body
described by the Lense-Thirring metric. By working in the weak-field limit and
employing Gaussian wave packets, we show that the characteristic coherence
length of neutrino oscillation processes is nontrivially affected, the
corrections being dependent on the mass and angular velocity of the gravity
source. Possible experimental implications are finally discussed.

Neutrino oscillations, decoherence, gravity, Lense-Thirring metric, wave
packets

##  I Introduction

Neutrinos are among the elementary particles in the Standard Model (SM) of
fundamental interactions. In spite of this, their essential nature has not yet
been fully revealed, becoming even more puzzling after Pontecorvo’s pioneering
idea of neutrino mass and mixing Pontec1 ; Pontec4 ; Pontec2  and the
subsequent discovery of flavor oscillations OscExp1 ; OscExp2 ; OscExp3 ;
OscExp4 . Further studies in Quantum Field Theory (QFT) have highlighted the
shortcomings of the standard quantum mechanical (QM) predictions by pointing
out the unitary inequivalence between the Fock spaces for definite flavor
fields and definite mass fields BV95 ; Caplup ; Ji . Phenomenological
implications of this inequivalence have been investigated in a variety of
contexts, ranging from vacuum effects Casimir ; Unruh ; Unruhbis  to particle
decays Protdec1 ; Protdec2 ; Protdec3 ; Protdec4  and apparent violations of
the weak equivalence principle WeP .

Neutrino mixing and oscillations are typically analyzed in the plane wave
approximation. However, a more realistic treatment that accounts for neutrinos
being localized particles should involve the use of wave packets (WP), which
introduce decoherence among the mass eigenstates. The first WP approach was
developed in Nussinov , showing the existence of a coherence length beyond
which the interference between massive neutrinos becomes negligible. This
effect arises from the different group velocities of the different mass
states, which leads WPs to spread over macroscopic sizes and separate during
the propagation. Wave packet models of neutrino oscillations were later
developed within the framework of QM Giunti1 ; Giunti2 ; Dolgov  and QFT
QFTWP1 ; QFTWP2 ; QFTWP3 ; QFTWP4 , both in vacuum and matter Matter1 ;
Matter2 ; Matter3  (see GiuntiWP  for a review). In particular, in dense
environments decoherence by WP separation was shown to depend on the model
chosen for the adiabaticity violation of WP evolution Matter2 ; Matter3 .

All of the above investigations have been performed in flat spacetime. Gravity
effects on neutrino decoherence were addressed in Volpe  considering a static
and spherically symmetric field described by the Schwarzschild metric. By
adopting the density matrix formalism Matter3  with Gaussian WPs and
exploiting previous achievements of Stodo ; Cardall ; Fornengo , neutrino
decoherence was quantified by a coherence coordinate distance and a proper
time. As a result, it was shown that these quantities are nontrivially
modified with respect to the flat case, the corrections being in principle
sizable.

The Schwarzschild solution provides a first useful approximation to describe
the spacetime metric around many astronomical objects, including the Earth and
Sun. However, it does not account for the rotation of the source. In this work
we take a further step forward by studying neutrino decoherence in the
gravitational field of a spinning spherical body of constant density and in
the weak-field regime (Lense-Thirring metric) Lense . Following Volpe , we
resort to the density matrix approach and evaluate the coherence length in
terms of the neutrino local energy, which is the energy actually measured by a
inertial observer at rest at finite radius in the gravitational field. In this
sense, our calculation differs from that of Volpe , where the final result is
exhibited as a function of the asymptotic energy of neutrinos. We show that it
is possible to extract a separate gravitational contribution depending on the
mass and angular velocity of the source. Experimental implications of our
finding are preliminarily discussed at the end.

The layout of the paper is as follows: in Sec. II we review the density matrix
approach to describe neutrino WP decoherence in flat spacetime. For this
purpose, we follow Matter3 ; Volpe . The above considerations are then
extended to curved spacetime in Sec. III. As a specific example, we consider
neutrino propagation in the Lense-Thirring metric. Conclusions and outlook are
summarized in Sec. IV. Throughout all the manuscript, we use natural units
$\hbar=c=G=1$ and the metric with the conventional mostly negative signature
$(+1,-1,-1,-1)$.

##  II WP decoherence in flat spacetime: the density matrix approach

In the SM it is a well-established fact that neutrinos interact in weak
eigenstates that are superpositions of mass eigenstates through Pontecorvo
transformation Pontec1 ; Pontec4 ; Pontec2

$|\nu_{\ell}(t,\vec{p})\rangle=\sum_{i=1,2}U^{*}_{\ell\hskip 0.28453pti}\hskip
0.56905pt|\nu_{i}(t,\vec{p})\rangle\,,$ (1)

where $\ell=e,\mu$ ($i=1,2$) denotes the flavor (mass) index and
$U^{*}_{\ell\hskip 0.28453pti}$ is the generic element of Pontecorvo mixing
matrix333We consider a simplified model involving only two generations of
neutrino. The same considerations and results hold in the case of three
flavors.. The time-dependent neutrino state $|\nu_{i}(t,\vec{p})\rangle$ with
three-momentum $\vec{p}$ is solution of the Schrödinger-like equation
$i\frac{d}{dt}|\nu_{i}(t,\vec{p})\rangle=H|\nu_{i}(t,\vec{p})\rangle$, where
$H$ is the Hamiltonian governing the time evolution. In astrophysical
environments the hamiltonian may include different contributions, such as the
vacuum term, the matter- and self-interactions. However, following Volpe , in
our analysis we neglect matter effects and neutrino self-interactions outside
the compact object. We are then left with
$H_{0}|\nu_{i}(t,\vec{p})\rangle=E_{i}(\vec{p})|\nu_{i}(t,\vec{p})\rangle$,
where $E_{i}(\vec{p})=(m_{i}^{2}+|\vec{p}|^{2})^{1/2}$ is the free energy
eigenvalue of the $i$-th mass eigenstate.

In the coordinate space the $i$-th neutrino state is given by the
$3$-dimensional Fourier expansion

$|\nu_{i}(t,\vec{x})\rangle=\frac{1}{(2\pi)^{3}}\int
d^{3}p\,e^{i\vec{p}\cdot\vec{x}}|\nu_{i}(t,\vec{p})\rangle\,.$ (2)

To streamline the notation, henceforth we denote the momentum integration
$(2\pi)^{-3}\int d^{3}p$ by $\int_{\vec{p}}$ .

In the standard treatment of flavor oscillations, the mass eigenstates are
typically described by plane waves. To account for the localization of
neutrinos to a finite region, a formalism based on WPs should be used. This
has been considered in Nussinov ; Giunti1 ; Giunti2 ; Dolgov ; QFTWP1 ; QFTWP2
; QFTWP3 ; QFTWP4 ; Matter1 ; Matter2 ; Matter3 . In the WP approach, the
neutrino flavor state (1) turns out to be a superposition of mass eigenstate
WPs, each centered around the momentum $\vec{p}_{i}$ with distribution
amplitude $f_{\vec{p}_{i}}(\vec{p})$. At the initial time $t_{0}=0$, the
$i$-th WP component satisfies

$|\nu_{i}(0,\vec{p})\rangle=f_{\vec{p}_{i}}(\vec{p})|\nu_{i}\rangle,$ (3)

where $\langle\nu_{j}|\nu_{i}\rangle=\delta_{ij}$ and

$\int_{\vec{p}}|f_{\vec{p}_{i}}(\vec{p})|^{2}=1\,.$ (4)

The explicit form of the WP distribution $f_{\vec{p}_{i}}(\vec{p})$ will be
given below.

Now, by use of Eq. (2), the $i$-th mass state in the coordinate space can be
written as

$|\nu_{i}(t,\vec{x})\rangle=\psi_{i}(t,\vec{x})|\nu_{i}\rangle\,,$ (5)

where the wave function $\psi_{i}(t,\vec{x})$ in the coordinate space is the
Fourier transform of the corresponding momentum-dependent wave function, i.e.

$\psi_{i}(t,\vec{x})=\int_{\vec{p}}e^{i\vec{p}\cdot\vec{x}}\psi_{i}(t,\vec{p})=\int_{\vec{p}}e^{i\vec{p}\cdot\vec{x}}f_{\vec{p}_{i}}(\vec{p})e^{-iE_{i}(\vec{p})t}\,.$
(6)

Therefore, the flavor state takes the form

$|\nu_{\ell}(t,\vec{x})\rangle=\sum_{i=1,2}U^{*}_{\ell\hskip 0.28453pti}\hskip
0.56905pt\psi_{i}(t,\vec{x})|\nu_{i}\rangle\,.$ (7)

Starting from the above premises, let us employ the density matrix formalism
to describe WP decoherence effects. In flat spacetime, the one-body density
matrix for the neutrino state (7) is defined as usual by Matter3

$\rho^{(\ell)}(t,\vec{x})=|\nu_{\ell}(t,\vec{x})\rangle\langle\nu_{\ell}(t,\vec{x})|\,.$
(8)

By using Eqs. (6) and (7), the $jk$-element of this matrix can be easily
computed, yielding

$\rho^{(\ell)}_{jk}(t,\vec{x})=U^{*}_{\ell\hskip 0.28453ptj}U_{\ell\hskip
0.28453ptk}\hskip 0.56905pt\psi_{j}(t,\vec{x})\psi_{k}^{*}(t,\vec{x})\,.$ (9)

Flavor vacuum oscillations occur due to the interference between different
massive neutrinos. Thus, in the WP approach the condition to be satisfied to
detect oscillations at a given point is that the mass eigenstate WPs still
overlap sufficiently to produce interference in that point (see Fig. 1). To
take account of this, one introduces the _coherence length_
$L_{\mathrm{coh}}$, which is defined as the distance travelled by neutrinos
beyond which the WPs corresponding to different propagation eigenstates
composing the produced neutrino flavor state separate by more than the WP size
$\sigma_{x}$. Recalling that the decoherence is generated by the different
group velocities of the different mass eigenstate WPs, the coherence length
can be estimated heuristically as
$L_{\mathrm{coh}}\simeq\sigma_{x}v_{g}(\Delta v_{g})^{-1}$ Matter3 , where
$v_{g}$ is the average group velocity of the WPs and $\Delta v_{g}$ is the
difference between the group velocities of the mass eigenstate WPs. For
relativistic neutrinos we have Matter3 ; Volpe

$L_{\mathrm{coh}}\simeq\frac{2E^{2}}{|\Delta m^{2}_{jk}|}\sigma_{x}\,,$ (10)

where $\Delta m^{2}_{jk}=m^{2}_{j}-m_{k}^{2}$ and $E\simeq|\vec{p}|$ is the
average energy between the interfering mass eigenstates. Clearly, in the plane
wave approximation we have $L_{\mathrm{coh}}\rightarrow\infty$ because of the
infinite spatial extension of plane waves.

The above relation is to be compared with the characteristic oscillation
length in the plane wave formalism, which is

$L_{\mathrm{osc}}\simeq\frac{4\pi E}{|\Delta m_{jk}^{2}|}\,.$ (11)

Thus, it is evident the central rôle of the finite WP-width assumption in
producing decoherence between the different neutrino eigenstate WPs.

It is worth noting that in the presence of three neutrino generations, one
should define a coherence length for each pair of propagation eigenstates. In
that case, complete decoherence occurs when the distance travelled by
neutrinos is higher than all the coherence lengths. As remarked in Matter3 ,
partial decoherence may also be of interest in some physical contexts.

We now aim at deriving the coherence length (10) more rigorously by using the
density matrix approach. As we shall see in the next Section, this formalism
is also well-suited to be extended to curved spacetime. Toward this end, we
need to specify the explicit form of WPs. There exist in the literature
several examples of WPs, such as square or sech WPs. Here, we resort to the
most common Gaussian WPs of momentum width

$\sigma_{p}=\frac{1}{2\sigma_{x}}\,,$ (12)

for which the (normalized) distribution amplitude $f_{\vec{p}_{i}}(\vec{p})$
reads

$f_{\vec{p}_{i}}(\vec{p})={\left(\frac{2\pi}{\sigma_{p}^{2}}\right)}^{3/4}e^{-\frac{\left(\vec{p}-\vec{p}_{i}\right)^{2}}{4\sigma_{p}^{2}}}\,.$
(13)

Notice that the plane wave limit is recovered for $\sigma_{p}\rightarrow 0$
(i.e. for $\sigma_{x}\rightarrow\infty$), which yields
$f_{\vec{p}_{i}}(\vec{p})=[(2\pi)^{3}/\sqrt{V}]\,\delta^{3}\left(\vec{p}-\vec{p}_{i}\right)$,
where $V$ is the normalization volume.

Now, by plugging Eq. (13) into (9), we are led to

$\rho^{(\ell)}_{jk}(t,\vec{x})=N^{(\ell)}_{jk}\int_{\vec{p},\vec{q}}e^{-i\phi_{jk}(t,\vec{x})}\,e^{-\left[\frac{(\vec{p}-\vec{p}_{j})^{2}}{4\sigma_{p}^{2}}\,+\,\frac{(\vec{q}-\vec{q}_{k})^{2}}{4\sigma_{p}^{2}}\right]}\,,$
(14)

where

$\phi_{jk}(t,\vec{x})=[E_{j}(\vec{p})-E_{k}(\vec{q})]t\,-\,(\vec{p}-\vec{q})\cdot\vec{x}$
(15)

is the standard QM phase shift and

$N^{(\ell)}_{jk}\equiv{\left(\frac{2\pi}{\sigma_{p}^{2}}\right)}^{3/2}U^{*}_{\ell
j}\hskip 0.56905ptU_{\ell k}\,.$ (16)

We have assumed equal dispersion $\sigma_{p}$ for the mass eigenstate WPs.

The integrals in Eq. (14) can be computed by expanding the neutrino energy
$E_{j}(\vec{p})$ around the WP central momentum $\vec{p}_{j}$ according to
Volpe

$E_{j}(\vec{p})\simeq E_{j}+(\vec{p}-\vec{p}_{j})\cdot\vec{v}_{j}\,,$ (17)

(and similarly for $E_{k}(\vec{q})$), where $E_{j}\equiv E_{j}(\vec{p}_{j})$
and $\vec{v}_{j}=\frac{\partial E_{j}}{\partial p}|_{\vec{p}=\vec{p}_{j}}$ is
the group velocity of the $j$-th mass eigenstate. The relation (14) then
becomes

$\rho^{(\ell)}_{jk}(t,\vec{x})=N^{(\ell)}_{jk}\int_{\vec{p},\vec{q}}e^{-i(E_{j}-E_{k})t}\,e^{-i[(\vec{p}-\vec{p}_{j})\cdot\vec{v}_{j}-(\vec{q}-\vec{q}_{k})\cdot\vec{v}_{k}]t}\,e^{i(\vec{p}-\vec{q})\cdot\vec{x}}\,e^{-\left[\frac{(\vec{p}-\vec{p}_{j})^{2}}{4\sigma_{p}^{2}}\,+\,\frac{(\vec{q}-\vec{q}_{k})^{2}}{4\sigma_{p}^{2}}\right]}\,.$
(18)

By explicit calculation of the Gaussian integrals, we get

$\rho^{(\ell)}_{jk}(t,\vec{x})=\frac{N^{(\ell)}_{jk}}{(2\sqrt{\pi}\sigma_{x})^{6}}e^{-i(E_{jk}t-\vec{p}_{jk}\cdot\vec{x})}\,e^{-\left[\frac{(\vec{x}-\vec{v}_{j}t)^{2}}{4\sigma_{x}^{2}}+\frac{(\vec{x}-\vec{v}_{k}t)^{2}}{4\sigma_{x}^{2}}\right]}\,,$
(19)

where

$E_{jk}\equiv E_{j}-E_{k}\,,\quad\,\vec{p}_{jk}\equiv\vec{p}_{j}-\vec{q}_{k}.$
(20)

Let us now evaluate the density matrix averaged over time

$\rho^{(\ell)}_{jk}(\vec{x})\equiv\int dt\,\rho^{(\ell)}_{jk}(t,\vec{x})\,,$
(21)

which is the typical quantity of interest in oscillation experiments. Tedious
but straightforward calculations lead to

$\rho^{(\ell)}_{jk}(\vec{x})=A^{(\ell)}_{jk}\,\rho_{jk}^{\mathrm{osc}}(\vec{x})\,\rho_{jk}^{\mathrm{damp}}(\vec{x})\,,$
(22)

where

$\displaystyle A^{(\ell)}_{jk}$ $\displaystyle=$
$\displaystyle\frac{U^{*}_{\ell j}U_{\ell k}}{\sqrt{2}\pi\hskip
1.13809ptv\hskip
1.13809pt\sigma_{x}^{2}}\,e^{-\frac{(E_{jk}\sigma_{x})^{2}}{v^{2}}}\,,$ (23)
$\displaystyle\rho_{jk}^{\mathrm{osc}}(\vec{x})$ $\displaystyle=$
$\displaystyle
e^{i\left(\vec{p}_{jk}-\frac{2E_{jk}\vec{v}_{g}}{v^{2}}\right)\cdot\vec{x}}\,,$
(24) $\displaystyle\rho_{jk}^{\mathrm{damp}}(\vec{x})$ $\displaystyle=$
$\displaystyle
e^{-\frac{(\vec{v}_{j}-\vec{v}_{k})^{2}}{4v^{2}\sigma_{x}^{2}}x^{2}}\,.$ (25)

The first term does not affect oscillations. The second one is the oscillation
term with the extra factor $2E_{jk}\vec{v}_{g}/v^{2}$, where we have set
$\vec{v}_{g}\equiv(\vec{v}_{j}+\vec{v}_{k})/2$ and
$v\equiv(v_{j}^{2}+v_{k}^{2})^{1/2}$. The last factor is the damping term,
which is actually responsible for decoherence. In passing, we remark that the
above averaged density matrix can be alternatively calculated as a function of
time by integrating Eq. (19) over space coordinates. This has been done in
Matter3 , obtaining a similar expression for the decoherence term through the
identification $t\simeq|\vec{x}|$.

From Eq. (25), we can now estimate the coherence length between the mass
eigenstate WPs as the distance at which the density matrix is suppressed by a
factor $e^{-1}$. This gives Volpe

$L_{\mathrm{coh}}=\frac{2\hskip 1.13809ptv\hskip
1.13809pt\sigma_{x}}{|\vec{v}_{j}-\vec{v}_{k}|}\simeq\frac{4\sqrt{2}E^{2}}{|\Delta
m^{2}_{jk}|}\sigma_{x}\,,$ (26)

which agrees with the heuristic estimate in Eq. (10), up to a numerical
factor.

In the next Section we show how the density matrix formalism is modified when
extended to curved spacetime. In particular, we consider the case of Lense-
Thirring metric and evaluate gravity-induced corrections to the coherence
length (26).

##  III Gravity effects on WP decoherence: the Lense-Thirring metric example

Gravitational effects on neutrino oscillations have been extensively analyzed
in the literature by using several approaches, e.g. the plane wave method
Fornengo ; Konno  or geometric formalisms Cardall ; Zhang  and in different
metrics, such as Schwarzschild Fornengo ; Burgard , Kerr Wudka ; Konno ,
Friedmann-Robertson-Walker Visinelli  and Lense-Thirring Lambiase  metrics,
among others. Recently, nontrivial results have been obtained in extended
theories of gravity ETG , quantum gravity scenarios Mavromatos ; Nicolini  and
in QFT on Rindler background Unruh .

The first systematic WP treatment of decoherence in neutrino oscillations has
been developed in Volpe  by relying on the Stodolsky’s covariant
generalization of the quantum mechanical phase shift in Eq. (15) Stodo . By
way of illustration, calculations have been explicitly performed for
Schwarzschild geometry.

In curved spacetime a neutrino flavor state produced at the point
$P(t,\vec{x}_{P})$ is described by
$|\nu_{\ell}(P)\rangle=\sum_{i=1,2}U^{*}_{\ell i}|\nu_{i}(P)\rangle$. During
the propagation to the detection point $D(t_{D},\vec{x}_{D})$, the $i$-th mass
eigenstate evolves according to

$|\nu_{i}(P,D)\rangle=e^{-i\Phi_{i}(P,D)}|\nu_{i}(P)\rangle\,,$ (27)

where the QM phase $\Phi_{i}(P,D)$ in its covariant form is given by Stodo

$\Phi_{i}(P,D)=\int_{P}^{D}p^{(i)}_{\mu}dx^{\mu}\,.$ (28)

Here $p^{(i)}_{\mu}$ ($\mu=0,1,2,3$) is the canonical four-momentum conjugated
to the coordinate $x^{\mu}$,

$p^{(i)}_{\mu}=m_{i}g_{\mu\nu}\frac{dx^{\nu}}{ds}\,,$ (29)

satisfying the generalized mass-shell relation

$p^{(i)}_{\mu}p^{(i)\hskip 0.56905pt\mu}=m_{i}^{2}\,,$ (30)

where $g_{\mu\nu}$ is the metric tensor and $ds$ the line element along the
trajectory described by the $i$-th neutrino state. In the approximation of
relativistic neutrinos, it is reasonable to assume that this trajectory is
close to a null-geodesic. Clearly, the detection point $D$ is such that the
mass eigenstate WPs can still interfere in it (see Fig. 1).

Figure 1: Pictorial representation of a neutrino propagating in the
gravitational field of a spinning spherical body from the production point
$\mathrm{P}$ to the detection point $\mathrm{D}$, where the WPs can still
interfere. We are assuming that each eigenstate WP follows a trajectory close
to null-geodesics. The (online) colored widths represent the distribution of
the trajectories due to the WP finite extension.

The relation (27) along with the definition (28) of the QM phase generalize
Eq. (2) to an arbitrary curved space. For $g_{\mu\nu}$ equal to Minkowski
tensor, it is easy to check that the standard evolution in flat spacetime is
recovered.

###  III.1 The Lense-Thirring metric example

The Lense-Thirring metric describes the gravitational field around a spinning
spherical source of constant density. Let us denote by $M$ and $R$ the mass
and radius of the central source, respectively. By assuming the neutrino
propagation to be confined to the equatorial plane ($z=0$), the line element
in the (linearized) weak-field limit can be written in cartesian coordinates
as

$\mathrm{d}s^{2}=(1+2\varphi)\mathrm{d}t^{2}-\frac{\varphi\hskip
0.56905pt\Omega}{r^{2}}(x\,\mathrm{d}y-y\,\mathrm{d}x)\,\mathrm{d}t-(1-2\varphi)(\mathrm{d}x^{2}+\mathrm{d}y^{2})\,,$
(31)

where

$\Omega\equiv\frac{4R^{2}\omega}{5}\,,$ (32)

and $\omega$ is the angular velocity of the source, supposed to be constant
and oriented along the $z$ axis. The gravitational potential
$\varphi\equiv\varphi(r)$ is defined by

$\varphi(r)=-\frac{M}{r}\equiv-\frac{M}{\sqrt{x^{2}+y^{2}}}\,,$ (33)

where, for simplicity, we have denoted the radial distance in the equatorial
plane by $r$.

Equation (31) provides the metric typically employed to describe
gravitomagnetic frame-dragging effects Ruggi ; Ciufo . Moreover, in TEUR  it
has been used to compute gravity corrections to the Mandelstam-Tamm time-
energy uncertainty relation for oscillations.

In the above setting, it is easy to show that the only nontrivial components
of the four-momentum $p_{\mu}^{(i)}$ are

$\displaystyle p_{t}^{(i)}$ $\displaystyle=$ $\displaystyle
m_{i}\left[(1+2\varphi)\frac{dt}{ds}+\frac{\varphi\hskip 0.56905pt\Omega\hskip
0.56905pty}{2r^{2}}\frac{dx}{ds}-\frac{\varphi\hskip 0.56905pt\Omega\hskip
0.56905ptx}{2r^{2}}\frac{dy}{ds}\right],$ (34) $\displaystyle p_{x}^{(i)}$
$\displaystyle=$ $\displaystyle m_{i}\left[\frac{\varphi\hskip
0.56905pt\Omega\hskip
0.56905pty}{2r^{2}}\frac{dt}{ds}-(1-2\varphi)\frac{dx}{ds}\right],$ (35)
$\displaystyle p_{y}^{(i)}$ $\displaystyle=$ $\displaystyle
m_{i}\left[-\frac{\varphi\hskip 0.56905pt\Omega\hskip
0.56905ptx}{2r^{2}}\frac{dt}{ds}-(1-2\varphi)\frac{dy}{ds}\right],$ (36)

where we have neglected higher-order terms in the potential $\varphi$.

Following Ruggi ; TEUR , we can now consider a first approximation in which
the neutrinos propagate along a direction parallel to the $x$-axis with impact
parameter $y=b>R$ (Fig. 1). This implies that $dy=0$ in Eqs. (34) and (36).
Furthermore, since the metric (31) does not depend on $t$, the component
$p_{t}^{(i)}\equiv E_{i}(\vec{p})$ is a constant of motion, corresponding to
the neutrino energy as measured by an inertial observer at rest at infinity
(asymptotic energy). The local energy, which is the quantity actually measured
by an observer at rest at finite radius in the gravitational field Fornengo ;
Cardall ; Petruz , is related to $E_{i}(\vec{p})$ trough the transformation
law that connects the local Lorentz frame $\\{x^{\hat{a}}\\}$ to the general
frame $\\{x^{\mu}\\}$

$x^{\mu}=e^{\mu}_{\hat{a}}x^{\hat{a}},\qquad
g_{\mu\nu}e^{\mu}_{\hat{a}}e^{\nu}_{\hat{b}}=\eta_{\hat{a}\hat{b}}\,,$ (37)

where $e^{\mu}_{\hat{a}}$ are the vierbein fields and we have used the
standard convention of denoting general coordinates and local Lorentz frame
indexes by Greek and hatted Latin letters, respectively. With reference to the
metric (31), the relevant tetrad components are TEUR

$e^{0}_{\hat{0}}=1-\varphi,\quad\,\,e^{1}_{\hat{0}}=\frac{\varphi\hskip
0.56905pt\Omega\hskip
0.56905pty}{r^{2}},\quad\,\,e^{2}_{\hat{0}}=-\frac{\varphi\hskip
0.56905pt\Omega\hskip
0.56905ptx}{r^{2}},\quad\,\,e^{i}_{\hat{j}}=(1+\varphi)\delta^{i}_{j}\,.$ (38)

From Eq. (37), it is a matter of calculation to show that the local and
asymptotic energies are related by TEUR

$E_{L}=\left[1-\varphi\left(1+\frac{\Omega\hskip
0.56905pty}{r^{2}}\right)\right]E(\vec{p})\,,$ (39)

where the subscript $L$ stands for “local” and we have omitted for simplicity
the space- and momentum-dependence of $E_{L}$.

A comment is now in order: in Volpe  all the quantities are expressed in terms
of the asymptotic energy $E_{i}(\vec{p})$. To make the comparison with the
formulas of Volpe  easier, in what follows we retain the dependence on
$E_{i}(\vec{p})$ and implement the substitution (39) only at the end.

With our assumptions, the mass-shell relation (30) takes the form

$(1+2\varphi)\left(\frac{dt}{ds}\right)^{2}-(1-2\varphi)\left(\frac{dx}{ds}\right)^{2}+\frac{\varphi\hskip
0.56905pt\Omega\hskip 0.56905ptb}{r^{2}}\frac{dt}{ds}\frac{dx}{ds}=1\,,$ (40)

where $r$ must now be intended as $r(x,y=b)$. From Eq. (34) along with
$p_{t}^{(i)}=E_{i}(\vec{p})$, we derive

$\frac{dt}{ds}=\frac{E_{i}(\vec{p})}{m_{i}}\left(1-2\varphi\right)-\frac{\varphi\hskip
0.56905pt\Omega\hskip 0.56905ptb}{2r^{2}}\frac{dx}{ds}\,,$ (41)

which can be replaced into (40) to give

$\frac{dx}{ds}=\pm\frac{E_{i}(\vec{p})}{m_{i}}\left[1-\frac{m^{2}_{i}}{2E_{i}^{2}(\vec{p})}(1+2\varphi)\right].$
(42)

Here we have exploited the condition of relativistic neutrinos at infinity,
which ensures that they are even more relativistic for $r<\infty$, according
to Eq. (39). Without loss of generality, we assume that neutrinos propagate
toward increasing values of $x$ as $s$ increases, so that the solution with
the positive sign must be considered.

Let us now consider the covariant phase in Eq. (28). Combining Eqs. (34)-(36)
and (41),(42), one gets for the integral argument

$p_{\mu}^{(i)}dx^{\mu}=E_{i}(\vec{p})\left\\{dt-\left[(1-2\varphi)-\frac{\varphi\hskip
0.56905pt\Omega\hskip
0.56905ptb}{r^{2}}-\frac{m_{i}^{2}}{2E_{i}^{2}(\vec{p})}\right]dx\right\\}.$
(43)

The covariant phase then reads

$\Phi_{i}(P,D)=E_{i}(\vec{p})\left(t_{PD}-{b}_{PD}-c^{(\Omega)}_{PD}\right)+\frac{m_{i}^{2}}{2E_{i}(\vec{p})}x_{PD}\,,$
(44)

where, for brevity, we have defined $t_{PD}\equiv t_{D}-t_{P}$, $x_{PD}\equiv
x_{D}-x_{P}$ and

$\displaystyle{b}_{PD}$ $\displaystyle\equiv$ $\displaystyle
x_{PD}+2M\log\left(\frac{x_{D}+r_{D}}{x_{p}+r_{P}}\right),$ (45)
$\displaystyle c^{(\Omega)}_{PD}$ $\displaystyle\equiv$
$\displaystyle\frac{\Omega
M}{b}\left(\frac{x_{D}}{r_{D}}-\frac{x_{P}}{r_{P}}\right),$ (46)

with

$r_{D}\equiv r(x=x_{D},b)=\sqrt{x_{D}^{2}+b^{2}},\quad\,\,\,r_{P}\equiv
r(x=x_{P},b)=\sqrt{x_{P}^{2}+b^{2}}.$ (47)

Clearly, for a non-rotating source, $c^{(\Omega)}_{PD}=0$. In this case, Eq.
(44) reproduces the linearized result of Volpe  for the Schwarzschild metric,
up to a global sign due to the different signature convention adopted for the
metric.

From Eq. (44), the phase difference $\Phi_{jk}=\Phi_{j}-\Phi_{k}$ of the mass
eigenstate WPs reads

$\Phi_{jk}(P,D)=\left[E_{j}(\vec{p})-E_{k}(\vec{q})\right]\left(t_{PD}-{b}_{PD}-c^{(\Omega)}_{PD}\right)+\left(\frac{m_{j}^{2}}{2E_{j}(\vec{p})}-\frac{m_{k}^{2}}{2E_{k}(\vec{q})}\right)x_{PD}\,.$
(48)

By use of the first-order expansion (17), this becomes

$\displaystyle\Phi_{jk}(P,D)$ $\displaystyle=$ $\displaystyle
E_{jk}\left(t_{PD}-{b}_{PD}-c^{(\Omega)}_{PD}\right)+\left(\frac{m_{j}^{2}}{2E_{j}}-\frac{m_{k}^{2}}{2E_{k}}\right)x_{PD}$
$\displaystyle+\,\,\vec{v}_{j}\cdot(\vec{p}-\vec{p}_{j})(t_{PD}-{\lambda}_{j})-\vec{v}_{k}\cdot(\vec{q}-\vec{q}_{k})(t_{PD}-{\lambda}_{k})\,,$

where we have used Eq. (20) and introduced the shorthand notation

${\lambda}_{i}(P,D)\equiv
b_{PD}+c^{(\Omega)}_{PD}+\frac{m_{i}^{2}}{2E_{i}^{2}}x_{PD}\,.$ (50)

Let us now evaluate the one-body density matrix describing the neutrino mass
eigenstates as (non-covariant) Gaussian WPs. This is given by the
generalization of Eq. (14) with the QM phase being given by Eq. (III.1), i.e.

$\displaystyle\widetilde{\rho}^{(\ell)}_{jk}(P,D)$ $\displaystyle=$
$\displaystyle
N^{(\ell)}_{jk}\int_{\vec{p},\vec{q}}e^{-i\Phi_{jk}(P,D)}\,e^{-\left[\frac{(\vec{p}-\vec{p}_{j})^{2}}{4\sigma_{p}^{2}}\,+\,\frac{(\vec{q}-\vec{q}_{k})^{2}}{4\sigma_{p}^{2}}\right]}\,,$
$\displaystyle=$
$\displaystyle\frac{N^{(\ell)}_{jk}}{(2\sqrt{\pi}\sigma_{x})^{6}}\,e^{-iE_{jk}(t_{PD}-b_{PD}-c^{(\Omega)}_{PD})}\,e^{i\left(\frac{m^{2}_{k}}{2E_{k}}-\frac{m_{j}^{2}}{2E_{j}}\right)x_{PD}}\,e^{-\sigma_{p}^{2}\left[v_{k}^{2}\left(t_{PD}-{\lambda}_{k}\right)^{2}+v_{j}^{2}\left(t_{PD}-{\lambda}_{j}\right)^{2}\right]},$

where we have used the tilde to distinguish the density matrix in curved
spacetime from the corresponding flat expression. The normalization
$N^{(\ell)}_{jk}$ is defined in Eq. (16).

As in Sec. II, the averaged density matrix is obtained by integrating
$\widetilde{\rho}^{(\ell)}_{jk}$ over the coordinate time. Also in this case,
the resulting expression can be factorized into the product of three terms as
follows

$\widetilde{\rho}^{(\ell)}_{jk}(x_{P},x_{D})=A^{(\ell)}_{jk}\,\widetilde{\rho}_{jk}^{\mathrm{\,\,osc}}(x_{P},x_{D})\,\widetilde{\rho}_{jk}^{\mathrm{\,\,damp}}(x_{P},x_{D})\,,$
(52)

to compare with the corresponding flat result (22). The amplitude
$A^{(\ell)}_{jk}$ is independent of the travelled distance and exhibits the
same expression as in Minkowski spacetime (see Eq. (23)). The second term,
which is responsible for flavor oscillations, is given by

$\widetilde{\rho}_{jk}^{\mathrm{\,\,osc}}(x_{P},x_{D})=e^{i\left(\frac{m_{k}^{2}}{2E_{k}}-\frac{m_{j}^{2}}{2E_{j}}\right)x_{PD}}\,e^{-i\frac{E_{jk}}{v^{2}}\left(v_{j}^{2}\frac{m_{j}^{2}}{2E^{2}_{j}}+v_{k}^{2}\frac{m_{k}^{2}}{2E^{2}_{k}}\right)x_{PD}}\,.$
(53)

Finally, the damping term reads

$\widetilde{\rho}_{jk}^{\mathrm{\,\,damp}}(x_{P},x_{D})=e^{-\frac{(v_{j}v_{k}x_{PD})^{2}}{4v^{2}\sigma_{x}^{2}}\left(\frac{m_{k}^{2}}{2E^{2}_{k}}-\frac{m_{j}^{2}}{2E^{2}_{j}}\right)^{2}}\,,$
(54)

which for relativistic neutrinos becomes

$\widetilde{\rho}_{jk}^{\mathrm{\,\,damp}}(x_{P},x_{D})\simeq
e^{-\frac{(m_{k}^{2}-m_{j}^{2})^{2}\hskip
0.56905pt}{32E^{4}\sigma_{x}^{2}}x_{PD}^{2}}.$ (55)

Here $E$ is the average energy between the mass eigenstates, as defined below
Eq. (10).

We notice that Eq. (55) is formally the same as the damping term found in
Volpe  in Schwarzschild metric. As argued in Petruz , to make the dependence
on $M$ and $\Omega$ explicit, $\widetilde{\rho}_{jk}^{\mathrm{\,\,damp}}$
should be recast in terms of the local (rather than asymptotic) energy (39).
By implementing the transformation (39) into Eq. (55), we are led to

$\widetilde{\rho}_{jk}^{\mathrm{\,\,damp}}(x_{P},x_{D})\simeq
e^{-\frac{(m_{k}^{2}-m_{j}^{2})^{2}\hskip
0.56905ptx_{PD}^{2}}{32\,E_{L}^{4}\sigma_{x}^{2}}\,\left[1+\frac{4M}{r_{D}}\left(1+\frac{\Omega\hskip
0.56905ptb}{r_{D}^{2}}\right)\right]},$ (56)

where the local energy must be considered as being evaluated at the detection
point $D$.

In analogy with the flat-spacetime case, one can now define the proper
coherence length as the distance at which the density matrix is suppressed by
the factor $e^{-1}$. We then obtain

$\widetilde{L}^{PD}_{\mathrm{coh}}\simeq\frac{4\sqrt{2}\hskip
0.56905ptE_{L}^{2}\hskip 0.56905pt\sigma_{x}}{|\Delta
m_{jk}^{2}|}\left(1-2\frac{M}{r_{D}}-2\frac{M\hskip 0.56905pt\Omega\hskip
0.56905ptb}{r_{D}^{3}}\right).$ (57)

We remark that in the $M\rightarrow 0$ limit, Eq. (26) is straightforwardly
recovered, since $E_{L}\rightarrow E$. This is indeed the expected outcome in
the absence of gravitational field. Therefore, respect to the flat case, the
coherence length in the linearized Lense-Thirring metric turns out to be
decreased by acquiring some nontrivial corrections. Specifically, the second
term in the brackets is a constant factor that only depends on the source mass
and the detection point. The presence of this correction could be somehow
expected, since the spacetime metric is no longer translation-invariant. The
third term is the genuinely Lense-Thirring imprint, as it carries the
information about the rotational velocity $\Omega$ of the central object and
the impact parameter $b$ of neutrino’s trajectory.

It would be interesting to estimate the gravity corrections in Eq. (57). For
this purpose, we suppose to consider neutrinos of typical energy $E_{L}\simeq
0.5\,\mathrm{MeV}$ and mass difference $\Delta m_{jk}^{2}\simeq
10^{-5}\,\mathrm{eV}^{2}$. For the WP width, we take $\sigma_{x}\simeq
10^{-12}\,\mathrm{cm}$ as in Matter2 ; Matter3 ; Volpe . By assuming the mass
of the Sun $M_{\odot}\simeq 10^{30}\,\mathrm{Kg}$ and its rotational frequency
$\omega^{-1}\simeq 27\,\mathrm{d}$ as sampling values for the gravity source
and setting $b\gtrsim R_{\odot}\simeq 10^{8}\,\mathrm{m}$, $r_{D}\simeq
10^{11}\,\mathrm{m}$, we obtain the following estimate for the relative
difference between the coherence length (57) and the corresponding flat-
spacetime expression

$\eta\equiv\frac{|\widetilde{L}^{PD}_{\mathrm{coh}}-\widetilde{L}^{PD}_{\mathrm{coh}}(M=0)|}{\widetilde{L}^{PD}_{\mathrm{coh}}(M=0)}\simeq
10^{-8}\,,$ (58)

thus showing the negligible impact of Sun’s gravity on decoherence effects, as
predictable for very weak fields. However, for neutrinos in astrophysical
environments, such as supernova neutrinos or neutrinos propagating in the
field of massive compact objects, these effects are expected to become
significant. For instance, in Fig. 2 the relative difference (58) is plotted
as a function of $M$ and for distances $r_{D}$ such that the weak-field
approximation is still valid. This shows that the contribution of gravity
corrections becomes increasingly relevant as one approaches strong-gravity
regimes. Of course, a more rigorous analysis of decoherence effects in this
scenario would require us to go beyond the linearized approximation. This
study is under active investigation and will be presented elsewhere In prepa .

Figure 2: Plot of the relative correction $\eta$ versus the source mass $M$
(in solar mass units). The logarithmic scale is used on the horizontal axis.

Finally, in order to compare Eq. (56) with the result of Petruz , we need to
express the coordinate distance $x_{PD}$ in terms of the proper distance
$L_{PD}$ between the production and detection points. Under our assumptions,
it is possible to show that these two quantities are related by TEUR

$L_{PD}=\int_{P}^{D}\sqrt{-g_{xx}}dx=x_{PD}+M\log\left(\frac{x_{D}+r_{D}}{x_{P}+r_{P}}\right),$
(59)

where $r_{D}$ and $r_{P}$ are defined in Eq. (47). By plugging into Eq. (56),
the averaged density matrix takes the form

$\widetilde{\rho}_{jk}^{\mathrm{\,\,damp}}(x_{P},x_{D})\simeq
e^{-\frac{(m_{k}^{2}-m_{j}^{2})^{2}\hskip
0.56905ptL_{PD}^{2}}{32\,E_{L}^{4}\sigma_{x}^{2}}\,\left[1+\frac{4M}{r_{D}}\left(1+\frac{\Omega\hskip
0.56905ptb}{r_{D}^{2}}\right)-\frac{2M}{L_{PD}}\log\left(\frac{x_{D}+r_{D}}{x_{P}+r_{P}}\right)\right]},$
(60)

which gives rise to the modified coherence length

$\widetilde{L}^{PD}_{\mathrm{coh}}\simeq\frac{4\sqrt{2}\hskip
0.56905ptE_{L}^{2}\hskip 0.56905pt\sigma_{x}}{|\Delta
m_{jk}^{2}|}\left[1-2\frac{M}{r_{D}}-2\frac{M\hskip 0.56905pt\Omega\hskip
0.56905ptb}{r_{D}^{3}}\right]+M\log\left(\frac{x_{D}+r_{D}}{x_{P}+r_{P}}\right)\,.$
(61)

For vanishing spinning velocity $\Omega=0$, it is easy to check that the above
formula reproduces the coherence length obtained in Petruz  for the case of a
non-rotating Schwarzschild source444Strictly speaking, the second term in the
square brackets is found with opposite sign with respect to Petruz , due to a
possible mistake therein.. By comparison with Eq. (57), it follows that
$\widetilde{L}^{PD}_{\mathrm{coh}}$ acquires a further correction with the
same logarithmic behavior as in Schwarzschild spacetime Petruz . We plan to
further investigate the physical meaning of Eq. (61) in order to understand
how to express the quantities in terms of proper distance in a more consistent
way.

##  IV Discussion and Conclusions

The influence of gravity on neutrino decoherence has been investigated within
the framework of the density matrix with Gaussian WPs. As a specific
background, we have considered the gravitational field around a spinning
spherical body described by Lense-Thirring metric. By working in the weak-
field, we have derived the effective coherence length for relativistic
neutrinos, showing that it is nontrivially modified with respect to the flat
case. A rough estimation of gravity corrections has highlighted that they are
below the sensitivity of current experiments for neutrinos propagating in the
gravitational field of the Sun. However, significant deviations from the
standard result are expected in strong-field regimes, e.g. in the case of
neutron stars formed from a core-collapse supernova Matter2 ; Volpe  or in the
presence of supermassive black holes. This is in line with Volpe .

We remark that our study relies on some preliminary assumptions. For instance,
we have neglected matter and neutrino self-interactions outside the central
compact object. As claimed in Volpe , these effects can be embedded by using a
similar procedure involving the use of the matter eigenstate basis instead of
the mass one. This investigation turns out to be necessary to explore whether
WP decoherence suppresses flavor oscillations and its impact on the supernova
dynamics and $r$-process nucleosynthesis. Furthermore, we have considered
rectilinear propagation of mass eigenstate WPs. In a more general treatment,
bending effects should also be contemplated. To give a rough estimation of the
correction we have neglected, let us think of neutrinos as nearly massless
particles. In this case, we can refer to Ruggi ; Cohen , where it has been
shown that the magnitude of the bending angle $\delta\phi$ for a light ray in
Lense-Thirring metric and in the equatorial plane turns out to be

$\delta\phi\simeq\frac{M}{b}\left(1-\frac{\vec{J}\cdot\vec{n}}{M\hskip
0.56905ptb}\right),$ (62)

where $\vec{J}$ is the angular momentum of the source and $\vec{n}$ is a unit
vector in the direction of the angular momentum of light about the center of
the source-body. The term outside the parenthesis provides the pure
Schwarzschild correction. For instance, in the case of the Sun, the relative
correction to the pure mass term is of order of $10^{-6}$, which in principle
justifies our assumption.

Apart from their intrinsic relevance, let us emphasize that decoherence
effects on neutrino oscillations are also studied to constrain quantum gravity
models Mavromatos ; Nicolini ; Test . In particular, in Mavromatos  quantum
gravitational analogues of the MSW effect and of foam models endowed with
stochastic fluctuations of the background are presented as possible
alternative sources of decoherence in neutrino oscillations. A similar
analysis has been recently developed in Stutt , where the influence of quantum
gravity on neutrino propagation and decoherence has been investigated with a
focus on the case of neutrino interactions with virtual black holes produced
by spacetime fluctuations. The study of the above aspects is quite demanding
and will be object of future works In prepa .

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