Quantum phase shift and neutrino oscillations in a stationary, weak gravitational field
Bernard Linet
We develop a new method for determining the gravitationally induced quantum phase shift for a particle propagating in a stationary weak gravitational field, in the framework of the linear approximation. This method is based on the properties of the Synge's world-function. The result is applied to the neutrino oscillations in the field of an isolated, axisymmetric rotating body. The gravitational field is described using the linearized part of the post-Newtonian metric yielded by the standard Nordtvedt-Will formalism. An explicit calculation of the phase shift is performed for neutrinos produced inside a homogeneous nearly spherical rotating body. Our general formulae are valid even for nonradial propagations. The gravitational effects found here involve only the PPN parameter $\gamma$.
gr-qc0206056.tex
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\title{\bf Quantum phase shift and neutrino oscillations in a 
stationary, weak gravitational field}
\author{Bernard Linet} 
\email{linet@celfi.phys.univ-tours.fr}
\affiliation{Laboratoire de Math\'ematiques et Physique Th\'eorique, 
CNRS/UMR 6083,\\
Universit\'e Fran\c{c}ois Rabelais, F-37200 Tours, France}
\author{Pierre Teyssandier} 
\email{Pierre.Teyssandier@obspm.fr}
\affiliation{D\'epartement Syst\`emes de R\'ef\'erence Temps Espace, 
CNRS/UMR 8630, \\
Observatoire de Paris, 61 avenue de l'Observatoire, F-75014 Paris, France}
\date{\today}

\begin{abstract}

We develop a new method for determining the gravitationally induced quantum phase shift for 
a particle propagating in a stationary weak gravitational field, in the framework of the linear 
approximation. This method is based on the properties of the Synge's world-function. The result is applied 
to the neutrino oscillations in the field of an isolated, axisymmetric rotating body. The gravitational field 
is described using the linearized part of the post-Newtonian metric yielded by the standard Nordtvedt-Will 
formalism. An explicit calculation of the phase shift is performed for neutrinos produced inside a 
homogeneous nearly spherical rotating body. Our general formulae are valid even for nonradial propagations. 
The gravitational effects found here involve only the PPN parameter $\gamma$.      
\end{abstract}

\pacs{04.25.-g 95.30.Sf 14.60.Pq}

\maketitle

\section{Introduction}

In the last few years, a series of papers has been devoted to the effects of gravitation on 
the neutrino oscillation phases (see Refs. \cite{Ahluwalia, Fornengo, Bhattacharya, Wudka} and 
references therein). As far as we know, all the proposed derivations of the neutrino oscillation formula 
within the WKB approximation use the explicit determination of the timelike geodesics corresponding to 
freely falling particles. For this reason, even in the simple case of the Schwarzschild metric treated 
in the linear approximation, the calculations become heavy for nonradial propagations and the results 
remain limited to the case where the neutrinos are created outside the matter generating the gravitational field. 
Our purpose in the present paper is to provide a method based on the world-function developed by 
Synge \cite{Synge}, which does not require the determination of the geodesics. This method, 
which has recently proved its worth for electromagnetic signals \cite{Linet2}, can be easily applied to 
neutrinos and more generally to all kinds of particles, as long as we may ignore the spin 
contributions \cite{Piriz}.

We begin by giving a derivation of the quantum mechanical phase valid for any particle freely propagating 
in a given stationary, weak gravitational field. We determine both the travel time and the accumulated phase 
between two points of space-time in the limit of the semi-classical approximation, where the particle is 
assumed to follow its classical path. Next we apply the results to the neutrino oscillation phases. We adopt 
the general assumptions of Bhattacharya {\em et al.} \cite{Bhattacharya} concerning the interference of 
two mass eigenstates observed at the same point of space-time. For applications, we consider the
linear terms in $G$ of the post-Newtonian metric given by the standard Nordtvedt-Will formalism \cite{Will}. 
This kind of approximation enables us to carry out the calculations for neutrinos travelling across the 
gravitational field of an isolated, axisymmetric rotating body. We obtain explicit formulae for a homogeneous, 
nearly spherical rotating body, including the case of nonradial propagations.

In this paper, $G$ is the Newtonian gravitational constant and $c$ is the speed of light 
in a vacuum. We suppose that space-time is covered by a global quasi Cartesian coordinates 
system $(x^{\mu})$. We put $x^0 = ct$, $t$ being a time coordinate and we use the notation 
$\bx =(x^i)$. The signature adopted for the metric tensor $g_{\mu\nu}$ is $(+---)$.

\section{Phase shift in a weak stationary gravitational field}

\subsection{Phase shift and world-function}  

Let us consider a free particle with proper mass $m$ propagating from a point $x_A$ to a 
point $x_B$. The quantum mechanical phase of this particule accumulated between $x_A$ and 
$x_B$ is given by
\be \lb{2}
\Phi_{AB}=\frac{mc}{\hbar}s_{AB} \, ,   
\ee
where $s_{AB}$ is the path length of the timelike geodesic $\Gamma_{AB}$ joining $x_A$ 
and $x_B$ \cite{Stodolsky}. Denoting by 
$\lambda$ the unique affine parameter along $\Gamma_{AB}$ which satisfies the boundary 
conditions $\lambda_{A} = 0$ and $\lambda_{B} = 1$, we can write
\be \lb{t2a}  
s_{AB}^2 = 2 \, \Omega (x_A, x_B) \, ,
\ee
where $\Omega (x_A, x_B)$ is the so-called world-function, defined as
\be \lb{t2b}
\Omega (x_A, x_B) = \frac{1}{2} \int_{0}^{1} g_{\mu\nu}(x^{\alpha}(\lambda))\frac{dx^{\mu}}{d\lambda}
\frac{dx^{\nu}}{d\lambda} d\lambda \, ,
\ee
the integral being taken over $\Gamma_{AB}$. Consequently, the accumulated phase $\Phi_{AB}$ may be 
written as
\be \lb{2a}
\Phi_{AB}=\frac{mc}{\hbar}\sqrt{2\Omega (x_A, x_B)} \, .
\ee

It follows from Eq. (\ref{2a}) that the knowledge of the world-function corresponding to a given metric 
enables to determine the accumulated phase $\Phi_{AB}$. In addition, the world-function enables to 
perform the calculation of the energy-momentum vector $p^{\alpha} = m c u^{\alpha}$ of the particle 
at $x_A$ and $x_B$ ($u^{\alpha}$ denotes the unit tangent vector $dx^{\alpha}/ds$ along 
$\Gamma_{AB}$). Indeed, it is shown in Ref. \cite{Synge} that the covariant components of $u$ 
at $x_A$ and $x_B$ are given by the relations
\be \lb{t2c}
(u_{\alpha})_A = - \frac{1}{\sqrt{2\Omega (x_A, x_B)}} \frac{\partial \Omega}{\partial x^{\alpha}_A} \, , 
\quad (u_{\alpha})_B = \frac{1}{\sqrt{2\Omega (x_A, x_B)}} \frac{\partial \Omega}{\partial x^{\alpha}_B}\, .
\ee
These relations are very useful, especially in the case where the metric does not depend on one or 
several coordinates, as we shall see in the next subsection. 

\subsection{Phase shift in a stationary space-time}  

Throughout this paper, we restrict our attention to stationary gravitational fields and 
we choose $x^0$ so that $g_{\mu \nu}$ does not depend on $x^0$. The world-function has then the
form $\Omega (x_{B}^{0}-x_{A}^{0},\bx_A,\bx_B)$ and the component $u_0$ of
the tangent vector $u^{\mu}$ to the classical trajectory is a constant of the 
motion, i.e. 
\be \lb{t2d}
u_0 = E/mc^2 \, , \quad E = const.
\ee 
When the particle is passing through a point $\bx$, the local energy as measured by a stationary 
observer staying at $\bx$ is $E_{loc}(\bx) = E/\sqrt{g_{00}(\bx)} $, so the constant $E$ may be 
considered as the asymptotic energy of the particle at spatial infinity. It 
follows from Eqs. (\ref{t2c}) that
\be \lb{n1}
u_0 = \frac{1}{\sqrt{2\Omega (x_{B}^{0}-x_{A}^{0},\bx_A,\bx_B)}} 
\frac{\partial \Omega (x_{B}^{0}-x_{A}^{0},\bx_A,\bx_B)}{\partial (x_{B}^{0}-x_{A}^{0})} = \frac{E}{mc^2} \, .
\ee
Eliminating $2\Omega (x_A, x_B)$ between Eq. (\ref{2a}) and Eq. (\ref{n1}), we obtain
\be \lb{n2}
\Phi_{AB}=\frac{m^2c^3}{\hbar E}\frac{\partial
\Omega (x_{B}^{0}-x_{A}^{0},\bx_A,\bx_B)}{\partial (x_{B}^{0}-x_{A}^{0})} \, .
\ee
Then, eliminating $x_{B}^{0}-x_{A}^{0}$ between Eqs. (\ref{n1}) and (\ref{n2}),
it will be possible -- at least in principle -- to carry out 
the calculation of the accumulated phase $\Phi_{AB}$ as a function of 
$\bx_A$, $\bx_B$ and $E$.

\subsection{Linear weak field approximation} 

For subsequent applications, we suppose that the gravitational field is weak. So, the metric may be written 
in the form
\be\lb{1}
g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} \, ,
\ee
where $\eta_{\mu \nu} = diag(1, -1, -1 ,-1)$ and where the quantities $h_{\mu \nu}$ are 
considered as perturbation terms of the first order in the Newtonian constant $G$. The 
word-function $\Omega (x_A,x_B)$ may be expanded in powers of the gravitational constant \cite{Synge}  
\be\lb{4}
\Omega (x_A, x_B) = \Omega^{(0)}(x_A, x_B) + \Omega^{(1)}(x_A, x_B) + O(h^2) \, ,
\ee
where $\Omega^{(0)}(x_A,x_B)$ is the world-function in Minkowski
space-time 
\be\lb{5}
\Omega^{(0)}(x_A,x_B)=\frac{1}{2}
\left[ (x_{B}^{0}-x_{A}^{0})^2-R_{AB}^{2} \right] \, ,
\ee
with $R_{AB}=\mid \! \bx_B-\bx_A\! \mid$,
and $\Omega^{(1)}(x_A,x_B)$ is given by 
\be\lb{6}
\Omega^{(1)}(x_A, x_B) = \frac{1}{2} 
(x_{B}^{\mu}-x_{A}^{\mu})(x_{B}^{\nu}-x_{A}^{\nu})
\int_{0}^{1}h_{\mu \nu}(x_{(0)}(\lambda )) d\lambda \, ,
\ee
the integration being performed along the path of parametrized equations 
$x_{(0)}(\lambda )=(x_B-x_A)\lambda +x_{A}$. Since space-time is assumed to be stationary, 
Eqs. (\ref{4}), (\ref{5}) and (\ref{6}) yield  
\bea \lb{8}
\Omega (x_A,x_B) & = & \frac{1}{2} (x_{B}^{0}-x_{A}^{0})^2 \left[ 1 + 2 A(\bx_A,\bx_B)\right]
+(x_{B}^{0}-x_{A}^{0}) R_{AB} B(\bx_A,\bx_B)    \nonumber \\ 
&  & \mbox{} - \frac{1}{2} R_{AB}^{2}\left[1 - 2 C(\bx_A,\bx_B) \right] + O(h^2) \, ,
\eea
in which we have set
\bea\lb{9}
& & A(\bx_A,\bx_B) = \frac{1}{2} \int_{0}^{1}h_{00}(\bx_{(0)}(\lambda ))d\lambda ,
\nonumber \\
& & B(\bx_A,\bx_B) = \frac{(x_{B}^{i} - x_{A}^{i})}{R_{AB}} \int_{0}^{1}h_{0i}(\bx_{(0)}
(\lambda ))d\lambda \, , \\
& & C(\bx_A,\bx_B) = \frac{(x_{B}^{i} - x_{A}^{i})(x_{B}^{j} - x_{A}^{j})}{2 R_{AB}^{2}}
\int_{0}^{1}h_{ij}(\bx_{(0)}(\lambda ))d\lambda \, , \nonumber 
\eea
with 
\be \lb{t9a}
\bx_{(0)}(\lambda )=(\bx_B-\bx_A)\lambda +\bx_A \, .
\ee

For the sake of brevity, we shall henceforth omit the symbol $O(h^2)$ standing for the 
post-post-Minkowskian terms. Differentiating the r.h.s. of Eq. (\ref{8}) with respect to 
$x_{B}^{0}-x_{A}^{0}$ and taking Eq. (\ref{n2}) into account, we obtain for $\Phi_{AB}$ 
\be \lb{n3}
\Phi_{AB}=\frac{m^2c^3}{\hbar E}\left[ (x_{B}^{0}-x_{A}^{0})
\left( 1 + 2 A(\bx_A,\bx_B)\right) + R_{AB} B(\bx_A,\bx_B) \right] \, .
\ee
Now, substituting for $\Omega$ from its expression (\ref{8}) into Eq. (\ref{n1}) and 
solving the equation thus obtained for $x_{B}^{0}-x_{A}^{0}$, we find   
\bea \lb{11}
& & x_{B}^{0}-x_{A}^{0} = R_{AB} \left\{ \frac{1}
{\sqrt{ \displaystyle 1-\frac{m^2c^4}{E^2}}}
\left[ 1- \left(1 - \frac{ m^2c^4}{E^2 - m^2 c^4} \right)
A(\bx_A,\bx_B) - C(\bx_A,\bx_B) \right] - B(\bx_A,\bx_B) \right\} \, .
\eea
Finally, inserting (\ref{11}) into (\ref{n3}), we get an expression for $\Phi_{AB}$ as follows 
\be \lb{n4}
\Phi_{AB}=\frac{R_{AB}}{\hbar}\frac{m^2c^3}
{\sqrt{ E^2 -m^2c^4}} \left[ 1+  \frac{E^2}{E^2 - m^2c^4} A(\bx_A,\bx_B) 
- C(\bx_A,\bx_B)\right] \, .
\ee

The formulae (\ref{11}) and (\ref{n4}) giving respectively the time delay and the 
accumulated phase between $x_A$ and $x_B$ are valid whatever the speed of the particle.

\section{Neutrino oscillations in the field of an isolated, axisymmetric rotating body} 

\subsection{General formulae}

Let us apply the previous results to the neutrino oscillation phases. We suppose that 
the neutrinos are produced at fixed local energy $E_{loc}$ in a weak flavour eigenstate which is 
a linear superposition of two mass eigenstates $m_1$ and $m_2$, with $m_2 < m_1$. Moreover, we 
suppose that the source of neutrinos is located at a fixed point $\bx_A$ and that the interference 
is observed at the same final space-time point $(x_{B}^{0}, \bx_B)$. As a consequence, the eigenstates 
of masses $m_1$ and $m_2$ are respectively emitted at the times $(x_{A}^{0})_1$ and $(x_{A}^{0})_2$, with 
$(x_{A}^{0})_1 < (x_{A}^{0})_2$. It is generally considered \cite{Bhattacharya} that the phase difference 
$\triangle \Phi =\Phi_2-\Phi_1$ between the two mass eigenstates at the point $x_B$ is given by
\be \lb{n5}
\triangle \Phi =\frac{E}{\hbar c}\left[ (x_{A}^{0})_2 - (x_{A}^{0})_1 \right]
+(\Phi_{AB})_2 - (\Phi_{AB})_1 \, ,
\ee
which can be rewritten in the form
\be \lb{n5a}
\triangle \Phi =\frac{E}{\hbar c}\left[ x_{B}^{0} - (x_{A}^{0})_{1}  
-  \left( x_{B}^{0} - (x_{A}^{0})_{2} \right) \right] 
+ (\Phi_{AB})_{2} - (\Phi_{AB})_{1} \, .
\ee

Using (\ref{11}) and (\ref{n4}), it is easily seen that the phase difference $\triangle \Phi$ 
is given by
\be \lb{n5b}
\triangle \Phi = - \frac{R_{AB}}{\hbar}
\frac{(m_{1}^{2} - m_{2}^{2})c^3}{\sqrt{E^2 - m_{1}^{2}c^4} \, + \sqrt{E^2 - m_{2}^{2}c^4}}
\left[1 + \frac{E^2}{\sqrt{E^2 - m_{1}^{2}c^4} \; \sqrt{E^2 - m_{2}^{2}c^4}} A(\bx_A, \bx_B)
- C(\bx_A, \bx_B) \right] \, .
\ee

We note that the quantity $B(\bx_A,\bx_B)$ does not appear in expression (\ref{n5b}). 
According to (\ref{9}), this means that the components $g_{0i}$ of the metric do not contribute to 
the phase difference $\triangle \Phi$ in the limit of the weak-field, linearized approximation.

We can apply the formula (\ref{n5b}) when the stationary field is treated in the slow-motion, 
post-Newtonian limit of metric theories of gravity, provided only the metric perturbations 
of order $1/c^2$ and $1/c^3$ are taken into account. We use the Nordtvedt-Will PPN formalism 
and we suppose that the coordinates $(x^{\mu})$ are a standard post-Newtonian gauge \cite{Will}. 
Since the potentials $h_{0i}$ do not influence the phase oscillations, the only relevant 
potentials are then 
\be \lb{14}
h_{00}=-\frac{2}{c^2}U+O(4) \, , \quad  h_{ij}=-\frac{2 \gamma}{c^2} U\delta_{ij}+O(4) \, ,
\ee
where $U$ is the gravitational potential. From Eqs. (\ref{9}), 
we deduce 
\be \lb{15}
A(\bx_A,\bx_B) = -\frac{1}{c^2}\int_{0}^{1}U(\bx_{(0)}(\lambda ))d\lambda , 
\quad C(\bx_A,\bx_B) = -\frac{\gamma}{c^2} 
\int_{0}^{1}U(\bx_{(0)}(\lambda ))d\lambda \, ,
\ee
where $\bx_{(0)}(\lambda )$ is given by (\ref{t9a}). Substituting for $A(\bx_A, \bx_B)$ and 
$C(\bx_A, \bx_B)$ from Eqs. (\ref{15}) into Eq. (\ref{n5b}), we obtain 
the general formula for the phase difference 
\be \lb{15a}
\triangle \Phi = - \frac{R_{AB}}{\hbar}
\frac{(m_{1}^{2} - m_{2}^{2})c^3}{\sqrt{E^2 - m_{1}^{2}c^4} \, + \sqrt{E^2 - m_{2}^{2}c^4}}
\left\{ 1 + \frac{1}{c^2} \left[ \gamma - \frac{E^2}{\sqrt{E^2 - m_{1}^{2}c^4} \; 
\sqrt{E^2 - m_{2}^{2}c^4}}\right] \int_{0}^{1} U( \bx_{(0)}(\lambda)) d\lambda \right\} \, .
\ee

This formula may be applied to neutrinos propagating in the field of an isolated, axisymmetric 
rotating body. It is immediately seen that the angular momentum of the rotating body has no influence 
on the neutrino oscillations in the weak-field limit and that there exists
no coupling with the velocity $\bv_r$ of the center of mass of the body relative to the universe 
rest frame. This last feature is natural since $\gamma$ is the only PPN parameter involved in the 
expression of $\triangle \Phi$.

Specializing to the case where $m_1c^2 \ll E$ and $m_2c^2 \ll E$ and expanding the 
formula (\ref{15a}) in series of powers of $m_1c^2/E$ and of $m_2c^2/E$, we find  
\be \lb{16}
\triangle \Phi = - \frac{R_{AB}}{2\hbar}\frac{(m_{1}^{2}-m_{2}^{2})c^3}{E} \left\{ 1 +
\frac{\gamma -1}{c^2}\int_{0}^{1}U(\bx_{(0)}(\lambda ))d\lambda
+ \frac{(m_{1}^{2} + m_{2}^{2}) c^4}{4E^2} \left[ 1 +
\frac{\gamma -3}{c^2}\int_{0}^{1}U(\bx_{(0)}(\lambda ))d\lambda \right] +\cdots \right\} \, .
\ee

This development shows that in general relativity, there is no contribution of the  Newtonian 
potential $U$ up to the order $m_{1}^{2}-m_{2}^{2}$ since $\gamma =1$. This conclusion holds 
equally well for a propagation inside the body as for a propagation outside the body. Thus we 
significantly extend a result previously obtained for the exterior Schwarzschild 
metric \cite{Fornengo, Bhattacharya}.

\subsection{Neutrinos in the field of a nearly spherically symmetric rotating body} 

According to (\ref{15a}) and (\ref{16}), an explicit calculation of the phase difference $\triangle \Phi$ 
requires only to perform the integration of the Newtonian potential $U$ along the line defined 
by (\ref{t9a}). As an example, we may consider the gravitational field generated by an isolated, 
nearly spherically symmetric rotating body. For the sake of brevity, we 
neglect the multipole structure of the gravitational field, which means that the body is characterized 
by its mass $M$ and by an external radius $r_0$. We shall examine two cases. 

{\em The geodesic path joining $\bx_{A}$ and $\bx_{B}$ is entirely outside the body}. In this case, we have 
$U(r)= GM/r$, which yields \cite{Buchdahl, Linet2}
\be \lb{19}
\int_{0}^{1}U(\bx_{(0)}(\lambda ))d\lambda =\frac{GM}{R_{AB}}\ln
\frac{r_A+r_B+R_{AB}}{r_A+r_B-R_{AB}} \, ,
\ee
where $r_A = \mid \! \bx_A \! \mid $ and $r_B = \mid \! \bx_B \! \mid$.

Substituting (\ref{19}) into (\ref{16}) yields the explicit expression of the phase difference 
for nonradial geodesics outside the body. For radial geodesics, the formula (\ref{19}) reduces to
\be \lb{20}
\int_{0}^{1}U(\bx_{(0)}))d\lambda =\frac{GM}{r_B-r_A}\ln \frac{r_B}{r_A} \, .
\ee
Inserting (\ref{20}) into (\ref{16}) and putting 
$\gamma = 1$, we recover up to the order $m_{1}^{4}-m_{2}^{4}$ 
a result previously given in Ref. \cite{Papini} for the 
radial geodesics in the Schwarzschild metric (this effect 
was also calculated by the authors of Ref. \cite{Bhattacharya}, but the terms of order 
$m_{1}^{4}-m_{2}^{4}$ in their Eq. (13) must be divided by 2).
More generally, substituting $(1+\omega )/(2+\omega)$ for  $\gamma$ into (\ref{16}), we get 
the expression of $\triangle \Phi$ within the frame of the Bergmann-Wagoner tensor-scalar 
theories of gravity (see, e.g., \cite{Will} and references therein). Thus we extend a result 
previously found \cite{Capozziello} at the order $m_{1}^{2}-m_{2}^{2}$ for radial geodesics. 

{\em The point of emission $\bx_{A}$ is located inside 
the massive body and the point of reception $\bx_{B}$ is located outside}. In order to simplify 
the evaluation of the integrals in the r.h.s. of (\ref{16}), it is useful to introduce the new 
affine parameter $\overline{\lambda}$ defined by 
\be \lb{t20a}
\overline{\lambda} = \lambda + \frac{r_{B}^2 - r_{A}^2 - R_{AB}^2}{2 R_{AB}^2} \, ,
\ee
the range of $\overline{\lambda}$ being $\overline{\lambda}_{A} \leq \overline{\lambda} 
\leq \overline{\lambda}_{B}$, with
\be \lb{t20b}
\overline{\lambda}_{A} = \frac{r_{B}^2 - r_{A}^2 - R_{AB}^2}{2 R_{AB}^2} \, , \quad 
\overline{\lambda}_{B} = \frac{r_{B}^2 - r_{A}^2 + R_{AB}^2}{2 R_{AB}^2} \, .
\ee
It is easily seen that the value $\overline{\lambda}_{0}$ corresponding to the intersection 
of the segment connecting $\bx_A$ and $\bx_B$ with the sphere $r= r_{0}$ is given by 
\be 
\overline{\lambda}_{0} = \frac{\sqrt{r_0^2 - r_p^2}}{R_{AB}} \, \nonumber
\ee
where $r_p$ is the impact parameter determined by
\be \lb{t20c}
r_{p}^{2} = \frac{r_{A}^{2}r_{B}^{2} - (\bx_A \cdot \bx_B)^{2}}{R_{AB}^{2}} \equiv 
\frac{1}{2}\left( r_{A}^{2}+r_{B}^{2}\right) -\frac{1}{4}R_{AB}^{2}
-\frac{(r_{B}^{2}-r_{A}^{2})^2}{4R_{AB}^{2}} \, .
\ee
We have  
\be
\int_{0}^{1} U(\bx_{(0)}(\lambda)) d\lambda = 
\int_{\overline{\lambda}_{A}}^{\overline{\lambda}_{0}} U_{i}(\bx_{(0)}(\overline{\lambda})) 
d\overline{\lambda} + 
\int_{\overline{\lambda}_{0}}^{\overline{\lambda}_{B}} \frac{GM}{\mid \! 
\bx_{(0)}(\overline{\lambda}) \! \mid} 
d\overline{\lambda} 
\nonumber \, ,
\ee
where $U_{i}$ is the potential inside the body. This relation holds for any mass density law 
compatible with the spherical symmetry. For the sake of simplicity, let us assume that the body 
is homogeneous. Then, $U_{i}$ is given by
\be
U_{i}(r) = \frac{GM}{2 r_0} \left(3 - \frac{r^2}{r_0^2} \right)\, , \quad 0 \leq r <r_0 \, ,  \nonumber 
\ee 
and a straightforward calculation yields  
\bea \lb{21}
& & \int_{0}^{1}U(\bx_{(0)}(\lambda ))d\lambda =\frac{GM}{R_{AB}}
\left[ \frac{\sqrt{r_{0}^{2}-r_{p}^{2}} - \epsilon \sqrt{r_{A}^{2}-r_{p}^{2}}}{r_0}
\left( \frac{4}{3}-\frac{r_{A}^{2}+r_{p}^{2} + \epsilon \sqrt{r_{0}^{2}-r_{p}^{2}}
\sqrt{r_{A}^{2}-r_{p}^{2}}}{6r_{0}^{2}}\right) \right. \nonumber \\
& & \qquad \qquad \qquad \qquad \qquad \qquad \quad \left. +\ln 
\left( \frac{\displaystyle r_B+\frac{R_{AB}}{2}+\frac{r_{B}^{2}-r_{A}^{2}}
{2R_{AB}}}{r_0+\sqrt{r_{0}^{2}-r_{p}^{2}}}\right) \right] \, ,
\eea
where $\epsilon = 1 $ if $R_{AB}^{2}<r_{B}^{2}-r_{p}^{2}$ and $\epsilon = -1$ if 
$R_{AB}^{2}>r_{B}^{2}-r_{p}^{2}$ (if $ R_{AB}^{2} = r_{B}^{2}-r_{p}^{2}$, then 
$r_{A}^{2}-r_{p}^{2} =0$ and all terms containing $\epsilon$ as a factor vanish).

\section{Conclusion} 

It is clear that the world-function constitutes a very powerful tool for finding the quantum phase 
shift for particles freely propagating in a stationary, weak gravitational field. The method 
works equally well for propagations inside matter as for propagations outside matter, provided 
the spin effects may be neglected.

Using this method, we have obtained a set of formulae giving the gravitationally induced neutrino 
oscillations in the field of an isolated, axisymmetric rotating body.  The Newtonian potential 
is the only gravitational quantity occuring in these formulae, which implies 
that the neutrino oscillations are not influenced by the internal angular momentum of the source 
of the gravitational field. 

Owing to the simplicity of the method, it has been possible to perform explicit 
calculations for neutrinos propagating in the field of a homogeneous, nearly 
spherically symmetric rotating body. Let us emphasize that these calculations
are straightforward even for neutrinos emitted inside the body and travelling
along nonradial geodesics.



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%%CITATION = PHRVA,D57,4724;%%
%%CITATION = PHRVA,D56,1895;%%
%%CITATION = PHRVA,D59,067301;%%
%%CITATION = PHRVA,D64,065009;%%
%%CITATION = GR-QC 0204009;%%
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%%CITATION = PHRVA,D55,7960;%%
%%CITATION = GRGVA,11,391;%%
%%CITATION = GR-QC 0110056.%%
%%CITATION = AUJPA,32,241;%%
%%CITATION = MPLAE,A14,2193;%%
\end{thebibliography}