\section{Introduction}

The non-linearly coupled Einstein-massive-scalar field equations
have a physically interesting and mathematically elegant structure
that has attracted the attention of researches during the last five
decades (see
\cite{Fra,Lie,Cha,Bek1,Teit,Heu,Bek2,BekMay,Bek20,Hodnmc,Hodrc,Herkr}
and references therein). In particular, this physical system is
known to possess asymptotically flat bound-state solutions in the
form of rotating hairy black holes that support spatially regular
{\it stationary} massive scalar field configurations
\cite{Hodrc,Herkr}.

Intriguingly, the phenomenologically rich Einstein-massive-scalar
system also admits horizonless bound-state solutions known as boson
stars. These physically interesting compact objects are made of
spatially regular {\it stationary} massive scalar fields. The
physical and mathematical properties of these self-gravitating
horizonless matter configurations have been extensively explored by
physicists and mathematicians (see \cite{Fra,Lie} and references
therein). In particular, the intriguing possibility has been
considered in the physics literature that boson stars may contribute
to the elusive dark matter distributions in galaxies \cite{Fra,Lie}.
In addition, compact bound-state scalar configurations (boson stars)
have been considered in the literature as physically exotic
horizonless mimickers of classical black-hole spacetimes
\cite{Fra,Lie}.

The main goal of the present paper is to reveal the intriguing fact
that, as opposed to {\it stationary} matter fields, {\it static}
self-gravitating massive scalar fields with non-minimal coupling to
gravity {\it cannot} form spatially regular horizonless bound-state
matter configurations (boson stars). In this context, it is
interesting to note that the asymptotic analysis of the non-linearly
coupled Einstein-scalar field equations presented in \cite{Hodrs1}
can be used to rule out the existence of static boson stars
(spatially regular matter configurations) made of nonminimally
coupled massive scalar fields in the dimensionless regime $\xi<0$ of
the nonminimal coupling parameter \cite{Notexi}.

In the present paper we shall explore the physical and mathematical
properties of the Einstein-scalar equations in the complementary
physical regime $\xi\geq0$ of the dimensionless nonminimal coupling
parameter. In particular, below we shall present a remarkably
compact no-go theorem for static boson stars which reveals the fact
that the non-linearly coupled Einstein-massive-scalar field
equations do not admit static spatially regular bound-state
solutions made of self-gravitating scalar fields with nonminimal
coupling to gravity.

\section{Description of the system}

We consider a spatially regular asymptotically flat spacetime whose
curvature is sourced by a static self-gravitating massive scalar
field $\psi$ with nonminimal coupling to gravity. The curved
spherically symmetric spacetime is described by the line element
\cite{BekMay,Notecor,Noteunit}
\begin{equation}\label{Eq1}
ds^2=-e^{\nu}dt^2+e^{\lambda}dr^2+r^2(d\theta^2+\sin^2\theta
d\phi^2)\ ,
\end{equation}
where $\nu=\nu(r)$ and $\lambda=\lambda(r)$.

The metric functions describing the spatially regular matter
configurations are characterized by the near-origin functional
behaviors \cite{Hodt1}
\begin{equation}\label{Eq2}
e^{\lambda}(r\to 0)=1+O(r^2)\ \ \ \ ; \ \ \ \ 0<e^{\nu}<\infty\
\end{equation}
with \cite{Notetag}
\begin{equation}\label{Eq3}
\lambda'(r\to0)\to0\ \ \ \ ; \ \ \ \ \nu'(r\to0)\to0\  .
\end{equation}
In addition, the metric functions of the asymptotically flat
spacetime are characterized by the far-region ($r\to\infty$) functional relations
\cite{BekMay}
\begin{equation}\label{Eq4}
\nu(M/r\to0)\sim M/r\ \ \ \ ; \ \ \ \ \lambda(M/r\to0)\sim
M/r\ ,
\end{equation}
where $M$ is the asymptotically measured finite ADM mass of the
spatially regular static matter configuration.

The self-gravitating scalar field is characterized by the action
\cite{BekMay,NoteSEH}
\begin{equation}\label{Eq5}
S=S_{EH}-{1\over2}\int\big(\partial_{\alpha}\psi\partial^{\alpha}\psi+\mu^2\psi^2+\xi
R\psi^2\big)\sqrt{-g}d^4x\ ,
\end{equation}
where $\mu$ \cite{Notemu} is the mass of the matter field and the
dimensionless nonminimal coupling parameter $\xi$ quantifies the
coupling between the scalar curvature $R(r)$ [see Eq. (\ref{Eq22})
below] of the spatially regular spacetime and the massive scalar
field $\psi$. For later purposes we note that asymptotically flat
spacetimes are characterized by the far-region functional behavior
\cite{BekMay}
\begin{equation}\label{Eq6}
R(M/r\to0)\to 0\
\end{equation}
of the scalar curvature. In addition, spatially regular nonminimally
coupled scalar matter configurations are characterized by
asymptotically finite radial eigenfunctions. In particular, as
discussed in \cite{BekMay}, the bounded functional relation
\begin{equation}\label{Eq7}
-\infty<8\pi\xi\psi^2<1\ \ \ \ \text{for}\ \ \ \ M/r\ll1\  .
\end{equation}
for the scalar eigenfunction guarantees that the corresponding
physically acceptable curved spacetimes are characterized by finite
and positive values of the asymptotic effective gravitational
constant $G_{\text{eff}}=G(1-8\pi G\xi\psi^2)$ \cite{BekMay}.

The spatial and temporal functional behaviors of the nonminimally
coupled massive scalar fields with the action (\ref{Eq5}) are
governed by the compact partial differential equation \cite{BekMay}
\begin{equation}\label{Eq8}
\partial_{\alpha}\partial^{\alpha}\psi-(\mu^2+\xi R)\psi=0\  .
\end{equation}
In particular, for spherically symmetric static matter
configurations, one obtains from (\ref{Eq1}) and (\ref{Eq8}) the
characteristic ordinary differential equation
\begin{equation}\label{Eq9}
\psi{''}+{1\over2}\big({{4}\over{r}}+\nu{'}-\lambda{'}\big)\psi{'}-(\mu^2+\xi
R) e^{\lambda}\psi=0\  ,
\end{equation}
which determines the spatial behavior of the scalar eigenfunction in
the curved spacetime (\ref{Eq1}).

The components of the energy-momentum tensor $T^{\alpha}_{\beta}$,
which characterizes the nonminimally coupled static scalar fields
with the action (\ref{Eq5}), are given by the functional expressions
\cite{BekMay}
\begin{equation}\label{Eq10}
T^{t}_{t}=e^{-\lambda}{{\xi(4/r-\lambda{'})\psi\psi{'}+(2\xi-1/2)(\psi{'})^2+2\xi\psi\psi{''}}
\over{1-8\pi\xi\psi^2}}-{{\mu^2\psi^2}\over{2(1-8\pi\xi\psi^2)}}\ ,
\end{equation}
\begin{equation}\label{Eq11}
T^{r}_{r}=e^{-\lambda}{{\xi(4/r+\nu{'})\psi\psi{'}+(\psi{'})^2/2}
\over{1-8\pi\xi\psi^2}}-{{\mu^2\psi^2}\over{2(1-8\pi\xi\psi^2)}}\ ,
\end{equation}
and
\begin{equation}\label{Eq12}
T^{\theta}_{\theta}=T^{\phi}_{\phi}=e^{-\lambda}{{\xi(2/r+\nu{'}-\lambda{'})\psi\psi{'}+(2\xi-1/2)(\psi{'})^2+2\xi\psi\psi{''}}
\over{1-8\pi\xi\psi^2}}-{{\mu^2\psi^2}\over{2(1-8\pi\xi\psi^2)}}\  .
\end{equation}
As explicitly proved in \cite{BekMay}, physically acceptable spacetimes are
characterized by finite components of the energy-momentum tensor:
\begin{equation}\label{Eq13}
\{|T^{t}_{t}|,|T^{r}_{r}|,|T^{\theta}_{\theta}|,|T^{\phi}_{\phi}|\}<\infty\  .
\end{equation}

\section{The no-go theorem for the static nonminimally coupled
massive scalar field configurations (boson stars) in the regime
$\xi\geq0$}

As emphasized above, the asymptotic ($r\gg M$) analysis of the
Einstein-scalar field equations presented in \cite{Hodrs1,Notegib}
rules out the existence of static boson stars made of
self-gravitating massive scalar fields in the dimensionless regime
$\xi<0$ of the nonminimal coupling parameter. In the present section
we shall complete the no-go theorem for the static sector of
horizonless boson stars. In particular, we shall explicitly prove
that nonminimally coupled massive scalar fields with a dimensionless
field-curvature coupling parameter in the physical regime $\xi\geq0$
cannot form spatially regular asymptotically flat static matter
configurations.

To this end, we first note that the radial scalar equation
(\ref{Eq9}) takes the simple asymptotic form [see Eqs. (\ref{Eq4}) and
(\ref{Eq6})] \cite{Hodrs1}
\begin{equation}\label{Eq14}
\psi{''}+{{2}\over{r}}\psi{'}-\mu^2\psi=0\
\end{equation}
in the $M/r\ll1$ region. The physically acceptable (normalizable)
radial solution of (\ref{Eq14}) which respects the bounded asymptotic
behavior (\ref{Eq7}) of physically acceptable spacetimes \cite{BekMay} is given by
\begin{equation}\label{Eq15}
\psi(r)=A\cdot{{e^{-\mu r}}\over{\mu r}}\ \ \ \ \text{for}\ \ \ \
M/r\ll1\ ,
\end{equation}
where $A$ is a normalization constant.

We shall now prove that the eigenfunction $\psi(r)$, which
characterizes the spatial behavior of the spherically symmetric
static scalar field configurations, has a non-monotonic dependence on the areal
coordinate $r$. In particular, if the radial scalar eigenfunction is
characterized by the near-origin relation $\psi(r\to0)\to0$, then
its asymptotic behavior [see Eq. (\ref{Eq15})]
\begin{equation}\label{Eq16}
\psi(M/r\to0)\to0\
\end{equation}
would immediately imply that it must possess (at least) one extremum
point $r_{\text{peak}}\in(0,\infty)$ in the curved spacetime (\ref{Eq1}). It therefore remains to be
proved that nonminimally coupled massive scalar fields which are
characterized by the near-origin behavior
\begin{equation}\label{Eq17}
\psi(r\to0)\neq0\
\end{equation}
must also posses an extremum point.

For later purposes, we note that one finds from the energy-momentum
components (\ref{Eq10}) and (\ref{Eq12}) the simple functional relation
\begin{equation}\label{Eq18}
T^{t}_{t}-T^{\phi}_{\phi}=e^{-\lambda}{{\xi(2/r-\nu{'})\psi\psi{'}}
\over{1-8\pi\xi\psi^2}}\  .
\end{equation}
In particular, taking cognizance of Eqs. (\ref{Eq2}), (\ref{Eq3}), (\ref{Eq13}), and (\ref{Eq18}),
one deduces the characteristic near-origin functional behavior
\begin{equation}\label{Eq19}
|{{1}\over{r}}\psi\psi{'}|<\infty\ \ \ \ \text{for}\ \ \ \ r\to0
\end{equation}
for spatially regular scalar matter configurations.
Likewise, taking cognizance of Eqs. (\ref{Eq2}), (\ref{Eq3}), (\ref{Eq13}), (\ref{Eq17}), (\ref{Eq19}),
and the relation [see Eqs. (\ref{Eq10}) and (\ref{Eq11})]
\begin{equation}\label{Eq20}
T^{t}_{t}-T^{r}_{r}=e^{-\lambda}{{(2\xi-1)(\psi{'})^2+2\xi\psi\psi{''}-\xi(\nu+\lambda){'}\psi\psi{'}}
\over{1-8\pi\xi\psi^2}}\  ,
\end{equation}
one finds the bounded near-origin functional behavior
\begin{equation}\label{Eq21}
|\psi{''}|<\infty\ \ \ \ \text{for}\ \ \ \ r\to0\
\end{equation}
for spatially regular static field configurations.

Substituting the Ricci scalar curvature \cite{Noteric}
\begin{equation}\label{Eq22}
R=-{{8\pi}\over{1-8\pi\xi\psi^2}}\Big\{e^{-\lambda}\Big[\xi\big({{12}\over{r}}+3\nu{'}-3\lambda{'}\big)\psi\psi{'}+
6\xi\psi\psi{''}+(6\xi-1)(\psi{'})^2\Big]-2\mu^2\psi^2\Big\}\
\end{equation}
which characterizes the spherically symmetric self-gravitating massive scalar fields
into Eq. (\ref{Eq9}), one obtains the ordinary differential equation
\begin{equation}\label{Eq23}
{\cal F}\cdot\psi{''}
+\psi{'}\cdot\Big[{1\over2}\big({{4}\over{r}}+\nu{'}-\lambda{'}\big){\cal F}
+8\pi\xi(6\xi-1)\psi\psi{'}\Big]
-\mu^2e^{\lambda}\big(1+8\pi\xi\psi^2\big)\psi=0\  ,
\end{equation}
which determines the radial behavior of the nonminimally coupled scalar matter configurations, where
\begin{equation}\label{Eq24}
{\cal F}(r;\xi)\equiv 1+8\pi\xi(6\xi-1)\psi^2\  .
\end{equation}
For later purposes we note that, taking cognizance of Eq. (\ref{Eq16}),
one finds that ${\cal F}(r;\xi)$ is characterized by the simple
asymptotic behavior
\begin{equation}\label{Eq25}
{\cal F}(M/r\to0)\to1\  .
\end{equation}

The radially-dependent function ${\cal F}(r;\xi)$ is obviously
positive definite for nonminimally coupled massive scalar fields
in the dimensionless regimes $\xi\geq 1/6$ and $\xi\leq0$ [see Eq. (\ref{Eq24})].
We shall now prove that this property also characterizes
the function ${\cal F}(r)$ in the complementary dimensionless regime $0<\xi<1/6$ of the
nonminimal coupling parameter.
Let us assume that the function ${\cal F}(r)$ has a root at some
point $r_0\in(0,\infty)$, in which case one finds from (\ref{Eq23}) the functional relation
\cite{Notepno}
\begin{equation}\label{Eq26}
8\pi\xi(6\xi-1)(\psi{'})^2=\mu^2e^{\lambda}(1+8\pi\xi\psi^2)\ \ \ \
\text{at}\ \ \ \ r=r_0\  .
\end{equation}
Taking cognizance of the fact that, in the dimensionless
regime $0<\xi<1/6$, the l.h.s of (\ref{Eq26}) is non-positive
whereas the r.h.s of (\ref{Eq26}) is positive definite, one deduces that
the equality sign in (\ref{Eq23}) cannot be respected at the assumed
root $r=r_0$ of ${\cal F}(r)$.
We therefore learn that the radial function ${\cal F}(r)$ has no roots.
In particular, using the characteristic asymptotic relation (\ref{Eq25}),
one finds that the radially-dependent function ${\cal F}(r;\xi)$ is
characterized by the simple inequality
\begin{equation}\label{Eq27}
{\cal F}(r)>0\ \ \ \ \text{for}\ \ \ \ r\in(0,\infty)\
\end{equation}
in the entire curved spacetime.

Taking cognizance of Eqs. (\ref{Eq2}), (\ref{Eq3}), (\ref{Eq19}), (\ref{Eq21}),
and (\ref{Eq23}), one obtains the near-origin ($r\to0$) radial equation
\begin{equation}\label{Eq28}
\psi{''}+{{2}\over{r}}\psi{'}
-\mu^2{\cal F}^{-1}\big(1+8\pi\xi\psi^2\big)\psi=0\  ,
\end{equation}
whose physically acceptable solution \cite{Notepac} is characterized
by the small-$r$ spatial behavior \cite{Notefoh,Noteaq0}
\begin{equation}\label{Eq29}
\psi(r\to0)=B\Big[1+{1\over6}\mu^2{{1+8\pi\xi
B^2}\over{1+8\pi\xi(6\xi-1)B^2}}\cdot r^2\Big]+O(r^3)\  ,
\end{equation}
where $B$ is a constant. Interestingly, and most importantly for our
analysis, one learns from Eqs. (\ref{Eq24}), (\ref{Eq27}), and
(\ref{Eq29}) that spatially regular nonminimally coupled massive
scalar fields with $\xi\geq0$ are characterized by the near-origin
functional relations
\begin{equation}\label{Eq30}
\psi\psi'(r\to0)=0\ \ \ \ ; \ \ \ \ \psi\psi''(r\to0)>0\ .
\end{equation}

The asymptotic large-$r$ and small-$r$ functional behaviors
(\ref{Eq16}) and (\ref{Eq30}), when combined together, reveal the important fact that the
characteristic radial eigenfunction $\psi(r)$ of the spatially
regular nonminimally coupled massive scalar fields must possess, in
accord with our previous assertion, (at least) one extremum point
$r_{\text{peak}}\in(0,\infty)$ in the curved spacetime (\ref{Eq1}). In particular, at this extremum
point the scalar eigenfunction is characterized by the functional
relations
\begin{equation}\label{Eq31}
\{\psi\neq0\ \ \ ; \ \ \ \psi{'}=0\ \ \ ; \ \ \
\psi\cdot\psi{''}<0\}\ \ \ \ \text{for}\ \ \ \ r=r_{\text{peak}}\  .
\end{equation}

Substituting (\ref{Eq31}) into the radial differential equation
(\ref{Eq23}), one finds
\begin{equation}\label{Eq32}
{\cal F}\cdot\psi\psi{''}=\mu^2e^{\lambda}(1+8\pi\xi\psi^2)\psi^2\ \
\ \ \text{at}\ \ \ \ r=r_{\text{peak}}\  .
\end{equation}
Interestingly, taking cognizance of the analytically derived
relations (\ref{Eq27}) and (\ref{Eq31}) one learns that, in the
dimensionless physical regime $\xi\geq0$, the l.h.s of (\ref{Eq32})
is negative definite whereas the r.h.s of (\ref{Eq32}) is positive
definite. Thus, for spatially regular static scalar configurations,
the radial differential equation (\ref{Eq23}) cannot be satisfied at
the characteristic extremum point (\ref{Eq31}).

We therefore conclude that static spatially regular massive scalar
fields whose nonminimal coupling parameter lies in the
dimensionless physical regime $\xi\geq0$ cannot form spherically symmetric asymptotically
flat horizonless matter configurations (boson stars).

\section{Summary}

Boson stars represent self-gravitating horizonless compact
objects which are made of massive scalar fields. The
physical and mathematical properties of these spatially
regular bound-state matter configurations have attracted
the attention of physicists and mathematicians during the last three decades
(see \cite{Fra,Lie} and references therein).

In the present paper we have presented a novel non-existence theorem for static
boson stars. In particular, the static sector of the non-linearly coupled
Einstein-matter field equations has been studied analytically for spatially
regular self-gravitating massive scalar fields with nonminimal coupling to
gravity. Interestingly, our compact no-go theorem explicitly rules out the
existence of asymptotically flat horizonless spacetimes that contain
spatially regular boson stars which are made of nonminimally coupled
\cite{Notexip} massive scalar fields.

\bigskip
\noindent
{\bf ACKNOWLEDGMENTS}
\bigskip

This research is supported by the Carmel Science Foundation. I would
like to thank Yael Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B.
Tea for helpful discussions.

