Derivation of Newton's Law of Gravitation Based on a Fluid Mechanical Singularity Model of Particles
Xiao-Song Wang
We speculate that the universe may be filled with a kind of fluid which may be called aether or tao. Thus, Newton's law of gravitation is derived by methods of hydrodynamics based on a sink flow model of particles.
physics0506062.tex
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\preprint{APS/000-000}

\title{Derivation of Newton's Law of Gravitation Based on
a Fluid Mechanical Singularity Model of Particles}
% Force line breaks with \\

\author{Xiao-Song Wang}
%\altaffiliation[Also at ]{Physics Department, XYZ University.}
\email{wangxs1988@yahoo.com}
%Lines break automatically or can be forced with \\
\affiliation{State Key Laboratory of Nonlinear Mechanics (LNM),
Institute of Mechanics, Chinese Academy of Sciences, Beijing,
100080, China}
\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified

\begin{abstract}
  We speculate that the universe may be filled with a kind of fluid
which may be called aether or tao. Thus, Newton's law of
gravitation is derived by methods of hydrodynamics based on a sink
flow model of particles.
\end{abstract}

\keywords{gravitation; hydrodynamics; source; sink; aether; tao.}

\maketitle \setcounter{section}{0}
\stepcounter{section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\thesection . \ \label{Introduction}Introduction}
%\lowercase{via} \textbackslash\textbackslash}
\thispagestyle{plain}

\newtheorem{definition}{Definition}

\newtheorem{theorem}[definition]{Theorem}

\newtheorem{wcorollary}[definition]{Corollary}

\newtheorem{hypothesis}[definition]{Hypothesis}

  Newton's law of gravitation can be written as
\begin{equation}\label{Newton's law of gravitation}
\mbox{\upshape\bfseries{F}}_{21}=-G\frac{m_1m_2}{r^2}
\hat{\mbox{\upshape\bfseries{r}}}_{21},
\end{equation}
where $m_1$ and $m_2$ are the masses of two particles, $r$ is the
distance between the two particles, $G$ is the gravitational
constant, $\mbox{\upshape\bfseries{F}}_{12}$
 is the force exerted on the particle with mass $m_2$
 by the particle with mass $m_1$,  $\hat{\mbox{\upshape\bfseries{r}}}_{21}$
denotes the unit vector directed outward along the line from the
particle with mass $m_1$ to the particle with mass $m_2$.


  The main purpose of this paper is to derive Newton's law of
gravitation by means of fluid mechanics based on sink flow model
of particles.

  The motive of this paper is to seek a mechanism of gravitational
phenomena. The reasons why new models of gravity are interesting
may be summarized as follows.

   Firstly, there exists some astronomical phenomena that
could not be interpreted by the present theories of gravitation,
for instance, the Titius-Bode law \cite{Nietro1972}. New theories
of gravity may view these problems from new angles.

   Secondly, whether the gravitational constant $G$ depends on
time and space is still unknown, for instance, see
\cite{Gilvarry-Muller1972,Tomaschitz2000,Gaztanaga2002,Copi2004,Khoury-Weltman2004,Nagata2004,Biesiada-Malec2004}.
It is known that the gravitational constant $G$ is a constant in
 Newton's theory of gravitation and in theory of general relativity.

   Thirdly, the mechanism of the action-at-a-distance gravitation remains an
unsolved problem in physics for more than 300 years
\cite{Whittaker1953,Hesse1961,Jaakkola1996}. Although theory of
general relativity is a field theory of gravity\cite{Fock1959},
the concept of field is different from that of continuum mechanics
\cite{Truesdell1966,Fung1977,Eringen1980,Landau-Lifshitz1987}
because of the absence of a continuum in theory of general
relativity. Thus, theory of general relativity can only be
regarded as a phenomenological theory of gravity.

   Fourthly, we do not have a satisfactory quantum theory of gravity
presently
\cite{Carlip2001,Amelino-Camelia2002,Ahluwalia2002,Mitrofanov2003,Christian2005}
. One of the most important challenges in theoretical physics is
to reconcile quantum theory and theory of general relativity
\cite{Carlip2001,Collins2004}. New theories of gravity may open
new ways to solve this problem.

   Fifthly, one of the puzzles in physics is the problem
of dark matter and dark energy, for instance, see
\cite{Ellis2003,Linder2004,Bernardeau2003,Bergstrom2005,Beacom2005,Akerib2006,Feng2006,Fayet2006,Carena2006}.
New theories of gravity may provide new methods to attack this
problem\cite{Linder2004,Bernardeau2003}.

   Finally, we do not have a successful unified field theory presently.
Great progress has been made towards an unification of the four
fundamental interactions in the universe in the 20th century.
However, gravitation is still not unified successfully. New
theories of gravity may shed some light on this puzzle.

   To conclude, it seems that new considerations on gravitation is
needed. It is worthy keeping an open mind with respect to all the
theories of gravitaty before the above problems been solved.

   Now let us briefly review the long history of mechanical
interpretations of gravitational phenomena.
   Many philosophers and scientists, such as \cite{Laozi1995}, Thales,
Anaximenes, believed that everything in the universe is made of a
kind of fundamental substance\cite{Whittaker1953}. Descartes was
the first to bring the concept of aether into science by
suggesting that it has mechanical properties\cite{Whittaker1953}.
Descartes interpreted the celestial motions of celestial bodies
based on the hypothesis that the universe is filled by an fluidic
vortex aether.
  Since Newton's law of gravitation was published in
1687\cite{Newton-a}, this action-at-a-distance theory was
criticized by the French Cartesian\cite{Whittaker1953}. Newton
admitted that his law did not touch on the mechanism of
gravitation\cite{Cohen1980}. He tried to obtain a derivation of
his law based on Descartes' scientific research program. At last,
he proved that Descartes' vortex aether hypothesis could not
explain celestial motions properly\cite{Newton-a}. Newton himself
even suggested an explanation of gravity based on the action of an
aetherial medium pervading the space\cite{Newton-b,Cohen1980}.
Euler attempted to explain gravitaty based on some hypotheses of a
fluidic aether\cite{Whittaker1953}.

     In a remarkable paper published in 1905,
Einstein abandoned the concept of aether\cite{Einstein1905}.
However, Einstein's assertion did not cease the explorations of
aether, for instance, see
\cite{Whittaker1953,Vigier1980,Carvalho2003,Arminjon2003,Carvalho-Oliveira2003,Jacobson2004,Davies2005,Levin-Wen2006}.
Einstein changed his view later and introduced his new concept of
ether\cite{Einstein1920,Kostro2000}. I regret to admit that it is
impossible for me to mention all the works related to this field
in history.

Inspired by the above mentioned thoughts and
others\cite{Lagally1922,Landweber-Yih1956,Yih1969,Faber1995,Currie2003},
we show that Newton's law of gravitation is derived based on the
hypothesis that all the particles are made of singularities of a
kind of ideal fluid.

\stepcounter{section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\thesection . \  \label{sink}Forces acting on sources and sinks in ideal fluids}
The purpose of this section is to calculate the forces between
sources and sinks in inviscid incompressible fluids which is
called ideal fluids usually.

Suppose the velocity field $\mbox{\upshape\bfseries{u}}$ of an
ideal fluid is irrotational, then we have
\cite{Lamb1932,Kochin1964,Landau-Lifshitz1987,Faber1995,Currie2003}
\begin{equation}\label{velocity}
\mbox{\upshape\bfseries{u}}=\nabla\phi ,
\end{equation}
where $\phi$ is the velocity potential, $\nabla =
\mbox{\upshape\bfseries{i}}\frac{\partial }{\partial x} +
\mbox{\upshape\bfseries{j}}\frac{\partial }{\partial y} +
\mbox{\upshape\bfseries{k}}\frac{\partial }{\partial z}$ is the
Hamilton operator.

It is known that the equation of mass conservation  of an ideal
fluid becomes Laplace's equation
\cite{Lamb1932,Kochin1964,Landau-Lifshitz1987,Faber1995,Currie2003}
\begin{equation}\label{Laplace's equation}
\nabla^2\phi=0,
\end{equation}
where $\phi$ is velocity potential, $\nabla^2 = \frac{\partial^2
}{\partial x^2} + \frac{\partial^2 }{\partial y^2} +
\frac{\partial^2 }{\partial z^2}$ is the Laplace operator.

Using spherical coordinates$(r,\theta,\varphi)$, a general form of
solution of Laplace's equation (\ref{Laplace's equation}) can be
obtained by sepatation of variables as\cite{Currie2003}
\begin{equation}\label{solution of Laplace's equation}
\phi(r,\theta)=\sum^{\infty}_{l=0}\left(
A_lr^l+\frac{B_l}{r^{l+1}}\right)P_l(\cos \theta),
\end{equation}
where $A_l$ and $B_l$ are arbitrary constants,
 $P_l(x)$ are Legendre's function of the first kind which is defined as
\begin{equation}\label{Legendre's function}
P_l(x)=\frac{1}{2^ll!}\frac{\mathrm{d}^l}{\mathrm{d}x^l}(x^2-1)^l.
\end{equation}

If there exists a velocity field which is continuous and finite at
all points of the space, with the exception of individual isolated
points, then these isolated points are called singularities
usually.



%\theoremstyle{definition}
%\newtheorem{definition}{Definition}
%\newtheorem{definition}{Definition}
\begin{definition}\label{source or sink}
Suppose there exists a singularity  at point $P_0=(x_0,y_0,z_0)$.
 If the velocity field of the singularity at point $P=(x,y,z)$ is
\begin{equation}\label{velocity field of source or sink}
\mbox{\upshape\bfseries{u}}(x,y,z,t)=\frac{Q}{4\pi
r^2}\hat{\mbox{\upshape\bfseries{r}}},
\end{equation}
where $r=\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}$,
$\hat{\mbox{\upshape\bfseries{r}}}$
 denotes the unit vector directed outward along the line
from the singularity to the point $P=(x,y,z)$, then we call this
singularity a source if $Q>0$ or a sink if $Q<0$. $Q$ is called
the strength of the source or sink.
\end{definition}
From (\ref{solution of Laplace's equation}), we see that the
velocity potential $\phi(r,\theta)$ of a source or sink is a
solution of Laplace's equation $\nabla^2\phi=0$. A calculation
shows that the strength $Q$ of a source or sink evaluates the
volume of the fluid leaving or entering a control surface per unit
time.

%\newtheorem{theorem}[definition]{Theorem}

\begin{theorem}\label{force exerted on sources or sinks by fluids}
Suppose (1)there exists an ideal fluid  (2)the ideal fluid is
irrotational and barotropic, (3)the density $\rho$ is homogeneous,
that is
\begin{math}
\partial\rho / \partial x=\partial\rho / \partial y
=\partial\rho / \partial z=\partial\rho / \partial t=0
\end{math}
(4)there are no external body forces exerted on the fluid, (5)the
fluid is unbounded and the velocity of the fluid at the infinity
is approaching to zero. Suppose a source or sink is stationary and
is immersed in the ideal fluid. Then, there is a force
\begin{equation}\label{force on the source in theorem}
\mbox{\upshape\bfseries{F}}_Q= -\rho
 Q\mbox{\upshape\bfseries{u}}_0
\end{equation}
exerted on the source by the fluid, where $\rho$ is the density of
the fluid, $Q$ is the strength of the source or the sink,
$\mbox{\upshape\bfseries{u}}_0$ is the velocity of the fluid at
the location of the source induced by all means other than the
source itself.
\end{theorem}


\mbox{\upshape\upshape\bfseries{Proof.}}
   Only the proof of the case of a source is needed.
Let us select the coordinates that is attached to the static fluid
at the infinity.

   We set the origin of the coordinates at the location of the source.
We surround the source by an arbitrary small spherical surface
$S$. The surface $S$ is centered at the origin of the coordinates
with radius $r$. The outward unit normal to the surface $S$ is
denoted by $\mbox{\upshape\bfseries{n}}$.
   Let $\tau(t)$ denotes the mass system of fluid enclosed in the volume
between the surface $S$ and the source at time $t$.
   Let $\mbox{\upshape\bfseries{F}}_Q$ denotes the hydrodynamic force
exerted on the source by the mass system $\tau$, then a reaction
of this force must act on the the fluid enclosed in the mass
system $\tau$. Let $\mbox{\upshape\bfseries{F}}_S$ denotes the
hydrodynamic force exerted on the mass system $\tau$ due to the
pressure distribution on the surface $S$,
$\mbox{\upshape\bfseries{K}}$ denotes momentum of the mass system
$\tau$.

As an application of Newton's second law of motion to the mass
system $\tau$,we have
\begin{equation}\label{Newton's second law of motion}
\frac{\mathrm{D}\mbox{\upshape\bfseries{K}}}{\mathrm{D}t}=-\mbox{\upshape\bfseries{F}}_Q+\mbox{\upshape\bfseries{F}}_S,
\end{equation}
where $\mathrm{D} / \mathrm{D}t$ represents the material
derivative in the lagrangian system. The expressions of the
momentum $\mbox{\upshape\bfseries{K}}$ and the force
$\mbox{\upshape\bfseries{F}}_S$ are

\begin{equation}\label{volume integral and surface integral}
\mbox{\upshape\bfseries{K}}
  =  \int\hspace{-1.5ex}\int\hspace{-1.5ex}\int_\tau
\rho\mbox{\upshape\bfseries{u}}dV, \quad
\mbox{\upshape\bfseries{F}}_S
 =  \int\hspace{-1.95ex}\int_{S} \hspace{-3.35ex}\bigcirc(-p)\mbox{\upshape\bfseries{n}}dS,
\end{equation}
where the first integral is volume integral, the second integral
is surface integral,
 $\mbox{\upshape\bfseries{n}}$ denotes the unit vector directed outward
along the line from the origin of the coordinates to the field
point$(x,y,z)$.

Since the velocity field is irrotational, we have the following
relation
\begin{equation}
\mbox{\upshape\bfseries{u}}=\mbox{\upshape\bfseries{$\nabla$}}\phi,
\end{equation}
where $\phi$ is the velocity potential.

According to Ostrogradsky--Gauss theorem, we have
\begin{equation}\label{Ostrogradsky-Gauss}
\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int_{\tau}
\rho\mbox{\upshape\bfseries{u}}dV
=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int_{\tau}
\rho\mbox{\upshape\bfseries{$\nabla$}}\phi dV
=\int\hspace{-1.95ex}\int_{S} \hspace{-3.35ex}\bigcirc \
\rho\phi\mbox{\upshape\bfseries{n}}dS.
\end{equation}

Note that the mass system $\tau$ does not include the singularity
 at the origin.
 According to Reynolds' transport theorem, we have
\begin{equation}\label{Reynolds' transport theorem}
\frac{\mathrm{D}}{\mathrm{D}t}\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int_\tau\rho\mbox{\upshape\bfseries{u}}dV
=\frac{\partial}{\partial
t}\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int_V
\rho\mbox{\upshape\bfseries{u}}dV +\int\hspace{-1.95ex}\int_{S}
\hspace{-3.35ex}\bigcirc \
\rho\mbox{\upshape\bfseries{u}}(\mbox{\upshape\bfseries{u}}\cdot\mbox{\upshape\bfseries{n}})dS,
\end{equation}
where $V$ is the volume fixed in space which coincide with the
mass system $\tau(t)$ at time $t$, that is $V=\tau(t)$.

Then, using (\ref{Reynolds' transport theorem}) , (\ref{volume
integral and surface integral}) and (\ref{Ostrogradsky-Gauss}), we
have
\begin{equation}\label{material derivative of K}
\frac{\mathrm{D}\mbox{\upshape\bfseries{K}}}{\mathrm{D}t}
=\int\hspace{-1.95ex}\int_{S} \hspace{-3.35ex}\bigcirc \
\rho\frac{\partial\phi}{\partial t}\mbox{\upshape\bfseries{n}}dS
 +\int\hspace{-1.95ex}\int_{S}\hspace{-3.35ex}\bigcirc \ \rho\mbox{\upshape\bfseries{u}}(\mbox{\upshape\bfseries{u}}\cdot\mbox{\upshape\bfseries{n}})dS.
\end{equation}

According to Lagrange--Cauchy integral\cite{Kochin1964}, we have
\begin{equation}\label{Lagrange-Cauchy integral}
\frac{\partial\phi}{\partial t}+
\frac{(\mbox{\upshape\bfseries{$\nabla$}}\phi)^2}{2}+\frac{p}{\rho}=f(t),
\end{equation}
where $f(t)$ is an arbitrary function of time $t$. Since the
velocity $\mbox{\upshape\bfseries{u}}$ of the fluid at the
infinity is approaching to zero,
 and noticing (\ref{solution of Laplace's equation}),
 $\phi(t)$ must be of the following form
\begin{equation}
\phi(r,\theta,t)=\sum^{\infty}_{l=0}
\frac{B_l(t)}{r^{l+1}}P_l(\cos \theta),
\end{equation}
where $B_l(t), l \geq 0$ are functions of time $t$. Thus, we have
the following estimations at the infinity of the velocity field

\begin{equation}\label{estimation}
\phi =O\left( \frac{1}{r}\right), \quad
\frac{\partial\phi}{\partial t} =O\left( \frac{1}{r}\right), \quad
r \rightarrow \infty,
\end{equation}
where $\varphi(x) = O(\psi(x)), x \rightarrow a$ stands for
$\overline{\lim}_{x \rightarrow a} \mid\varphi(x)\mid / \psi(x) =
k, (0 \leq k < +\infty).$

Applying (\ref{Lagrange-Cauchy integral}) at the infinity and
using (\ref{estimation}), we have
$\mid\mbox{\upshape\bfseries{u}}\mid\rightarrow0$,
 $\partial\phi / \partial t\rightarrow0$ and $p=p_{\infty}$,
 where $p_{\infty}$ is a constant. Thus,
 $f(t)=p_\infty / \rho$. Therefore, according to (\ref{Lagrange-Cauchy integral}), we have
\begin{equation}\label{the pressure distribution on the surface S}
p=p_\infty-\rho\frac{\partial\phi}{\partial t}-
\frac{\rho(\mbox{\upshape\bfseries{u}}\cdot\mbox{\upshape\bfseries{u}})}{2}.
\end{equation}

Using (\ref{volume integral and surface integral}) and (\ref{the
pressure distribution on the surface S}), we have
\begin{equation}\label{force due to the pressure distribution on the surface S}
\mbox{\upshape\bfseries{F}}_S =\int\hspace{-1.95ex}\int_{S}
\hspace{-3.35ex}\bigcirc \ \rho\frac{\partial\phi}{\partial
t}\mbox{\upshape\bfseries{n}}dS
 +\int\hspace{-1.95ex}\int_{S} \hspace{-3.35ex}\bigcirc \
\frac{\rho(\mbox{\upshape\bfseries{u}}\cdot\mbox{\upshape\bfseries{u}})\mbox{\upshape\bfseries{n}}}{2}dS.
\end{equation}

Using (\ref{Newton's second law of motion}), (\ref{material
derivative of K}), (\ref{force due to the pressure distribution on
the surface S}), we have
\begin{equation}\label{force on the source a}
\mbox{\upshape\bfseries{F}}_Q =\int\hspace{-1.95ex}\int_{S}
\hspace{-3.35ex}\bigcirc \
\left[\frac{1}{2}\rho(\mbox{\upshape\bfseries{u}}\cdot\mbox{\upshape\bfseries{u}})\mbox{\upshape\bfseries{n}}
-\rho\mbox{\upshape\bfseries{u}}(\mbox{\upshape\bfseries{u}}\cdot\mbox{\upshape\bfseries{n}})\right]dS.
\end{equation}

Now let us calculate this velocity $\mbox{\upshape\bfseries{u}}$
in order to obtain $\mbox{\upshape\bfseries{F}}_Q$. Since the
velocity field induced by the source $Q$ is (\ref{velocity field
of source or sink}), then according to the superposition principle
 of velocity field of ideal fluids, the velocity on the surface $S$ is
\begin{equation}\label{velocity on the surface S}
\mbox{\upshape\bfseries{u}} =\frac{Q}{4\pi
r^2}\mbox{\upshape\bfseries{n}}+\mbox{\upshape\bfseries{u}}_0,
\end{equation}
where $\mbox{\upshape\bfseries{n}}$ denotes the unit vector
directed outward along the line from the origin of the coordinates
to the field point$(x,y,z)$. Using (\ref{force on the source a})
and (\ref{velocity on the surface S}), we have
\begin{eqnarray}\label{force on the source b}
\mbox{\upshape\bfseries{F}}_Q
 &=&\rho\int\hspace{-1.95ex}\int_{S} \hspace{-3.35ex}\bigcirc
 \left[-\frac{Q^2}{32\pi^2 r^4}\mbox{\upshape\bfseries{n}}
 +\frac{1}{2}(\mbox{\upshape\bfseries{u}}_0\cdot\mbox{\upshape\bfseries{u}}_0)\mbox{\upshape\bfseries{n}}\right.\nonumber \\
 &&\left.-\frac{Q}{4\pi r^2}\mbox{\upshape\bfseries{u}}_0
 -(\mbox{\upshape\bfseries{u}}_0\cdot\mbox{\upshape\bfseries{n}})\mbox{\upshape\bfseries{u}}_0
 \right]dS.
\end{eqnarray}

Since the radius $r$ can be arbitrarily small, the velocity
$\mbox{\upshape\bfseries{u}}_0$ can be treated as a constant in
the integral of (\ref{force on the source b}). Thus, (\ref{force
on the source b}) turns out to be
\begin{equation}\label{force on the source c}
\mbox{\upshape\bfseries{F}}_Q = -\rho\int\hspace{-1.95ex}\int_{S}
\hspace{-3.35ex}\bigcirc \ \frac{Q}{4\pi
r^2}\mbox{\upshape\bfseries{u}}_0 dS.
\end{equation}
Since again $\mbox{\upshape\bfseries{u}}_0$ can be treated as a
constant, (\ref{force on the source c}) turns out to be
(\ref{force on the
source in theorem}).This completes the proof.               $\Box$%\begin{flushright}$\Box$\end{flushright}


%\newremark{note}{Note}
%\newremark{remark}{Remarks}
%\begin{remark}
\mbox{\upshape\upshape\bfseries{Remark.}}
Lagally\cite{Lagally1922},
 Landweber and Yih\cite{Landweber-Yih1956,Yih1969}, Faber\cite{Faber1995} and
Currie\cite{Currie2003} obtained the same result of Theorem
\ref{force exerted on sources or sinks by fluids} for the special
case where the velocity field is steady.
%\end{remark}

  Theorem \ref{force exerted on sources or sinks by fluids} only considers the situation
that the sources or sinks are at rest. Now let us consider the
case that the sources or sinks are moving in the fluid.

\begin{theorem}\label{force exerted on
moving sources or sinks by fluids} Suppose the presuppositions
(1),(2),(3),(4) and (5)in Theorem \ref{force exerted on sources or
sinks by fluids} are valid and a source or a sink is moving in the
fluid with a velocity $\mbox{\upshape\bfseries{v}}_s$, then there
is a force
\begin{equation}\label{force on moving source in theorem}
\mbox{\upshape\bfseries{F}}_Q= -\rho
 Q(\mbox{\upshape\bfseries{u}}_f-\mbox{\upshape\bfseries{v}}_s)
\end{equation}
is exerted on the source by the fluid, where $\rho$ is the density
of the fluid, $Q$ is the strength of the source or the sink,
$\mbox{\upshape\bfseries{u}}_f$ is the velocity of the fluid at
the location of the source induced by all means other than the
source itself.
\end{theorem}

%\begin{proof}
\mbox{\upshape\upshape\bfseries{Proof.}}
 The velocity of the fluid relative to the source at the
location of the source is
$\mbox{\upshape\bfseries{u}}_f-\mbox{\upshape\bfseries{v}}_s$. Let
us select the coordinates that is attached to the source and set
the origin of the coordinates at the location of the source. Then
(\ref{force on moving source in theorem}) can be arrived following
the same procedures in the proof of Theorem \ref{force exerted on
sources or sinks by fluids}.     $\Box$
%\end{proof}


Applying Theorem \ref{force exerted on moving sources or sinks by
fluids} to the situation
 that a source or sink is exposed to the velocity field of another source or sink, we have

%\newtheorem{wcorollary}[definition]{Corollary}

\begin{wcorollary}\label{singularity-singularity forces}
Suppose the presuppositions (1),(2),(3),(4) and (5) in Theorem
\ref{force exerted on sources or sinks by fluids} are valid and a
source or a sink with strength $Q_2$ is exposed to the velocity
field of another source or sink with strength $Q_1$, then the
force $\mbox{\upshape\bfseries{F}}_{21}$
 exerted on the singularity with strength $Q_2$
 by the velocity field of the singularity with strength $Q_1$ is
\begin{equation}\label{force exerted on singularity Q_2 by singularity Q_1}
\mbox{\upshape\bfseries{F}}_{21}=-\rho Q_2\frac{Q_1}{4\pi
r^2}\hat{\mbox{\upshape\bfseries{r}}}_{21} + \rho
Q_2\mbox{\upshape\bfseries{v}}_2,
\end{equation}
where $\hat{\mbox{\upshape\bfseries{r}}}_{21}$ denotes the unit
vector directed outward along the line from the singularity with
strength $Q_1$ to the singularity with strength $Q_2$, $r$ is the
distance between the two singularities,
$\mbox{\upshape\bfseries{v}}_2$ is the velocity of the source with
strength $Q_2$.
\end{wcorollary}

\stepcounter{section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\thesection . \  \label{gravitation}Derivation of inverse-square-law of gravitation}
Since quantum theory shows that vacuum is not empty and has
physical effects, e.g., the Casimir
effect\cite{Lamoreaux2005,Intravaia-Lambrecht2005,Guo-Zhao2004,Davies2005},
it is valuable to probe vacuum by introducing the following
hypotheses

%\newtheorem{hypothesis}[definition]{Hypothesis}
\begin{hypothesis}\label{tao}
Suppose the universe is filled by an ideal fluid named tao; the
ideal fluid fulfill the conditions (2),(3),(4),(5) in Theorem
\ref{force exerted on sources or sinks by fluids}.
\end{hypothesis}
This fluid may be named tao after \cite{Laozi1995} in order to
distinguish with Cartesian aether. Following Einstein, Infeld and
Hoffmann, who introduced the idea that particles may be looked as
singularities in fields
\cite{Einstein-Infeld-Hoffman1938,Einstein-Infeld1940}, and
noticing (\ref{force exerted on singularity Q_2 by singularity
Q_1}), it is nature to introduce the following
\begin{hypothesis}\label{monad}
All the microscopic particles were made up of a kind of elementary
sinks of tao. These elementary sinks were created
simultaneously.The initial masses and the strengthes of the
elementary sinks are the same.
\end{hypothesis}
We may call these elementary sinks as monads.

Suppose a particle with mass $m$ is composed of $N$ monads. Then,
according to Hypothesis \ref{monad}, we have
\begin{eqnarray}
&& m_0(t) =m_0(0)+\rho q_0 t, \label{the mass of a monad is increasing 1} \\
&& Q  = -N q_0, \quad m(t) = N m_0(t) = - \frac{Q}{q_0}m_0(t), \label{the mass of a monad is increasing 2} \\
 && \frac{dm_0}{dt} =\rho q_0, \quad \frac{dm}{dt}  = -\rho Q,\label{the mass of a monad is increasing 3}
\end{eqnarray}
where $m_0(t)$ is the mass of monad at time $t$, $-q_0( q_0 > 0)$
is the strength of a monad, $m(t)$ is the mass of a particle at
time $t$,
 $Q$ is the strength of the particle,
 $N$ is the number of monads that make up the particle,
 $\rho$ is the density of the tao, $t\geqslant 0$.

From (\ref{the mass of a monad is increasing 3}), we see that the
mass $m_0$ of a monad is increasing since $q_0$
 evaluates the volume of the tao fluid entering the monad per unit time.
From (\ref{the mass of a monad is increasing 3}), we also see that
the mass of a monad or a particle is increasing linearly.

 Based on
Hypothesis \ref{tao} and Hypothesis \ref{monad}, the motion of a
particle is determined by
\begin{theorem}\label{motion of a particle}
The equation of motion of a particle is
\begin{equation}\label{the equation of motion of a particle}
m(t)\frac{\mathrm{d}\mbox{\upshape\bfseries{v}}}{\mathrm{d}t}
=\frac{\rho q_{0}}{m_{0}(t)} m(t) \mbox{\upshape\bfseries{u}}
-\frac{ \rho q_{0}}{m_{0}(t)} m(t) \mbox{\upshape\bfseries{v}}
+\mbox{\upshape\bfseries{F}},
\end{equation}
where $m_0(t)$ is the mass of monad at time $t$, $-q_0$ is the
strength of a monad, $m(t)$ is the mass of a particle at time $t$,
$\mbox{\upshape\bfseries{v}}$ is the velocity of the particle,
$\mbox{\upshape\bfseries{u}}$ is the velocity of the tao at the
location of the particle induced by all means other than the
particle itself, $\mbox{\upshape\bfseries{F}}$ denotes other
forces.
\end{theorem}

\mbox{\upshape\upshape\bfseries{Proof.}}
%\begin{proof}
 Applying Newton's second law and
Theorem \ref{force exerted on moving sources or sinks by fluids}
to this particle, we have
$m\mathrm{d}\mbox{\upshape\bfseries{v}}/\mathrm{d}t = -\rho
Q(\mbox{\upshape\bfseries{u}}-\mbox{\upshape\bfseries{v}})
+\mbox{\upshape\bfseries{F}}$. Noticing (\ref{the mass of a monad
is increasing 2}), we get (\ref{the equation of motion of a
particle}).  $\Box$

%\end{proof}

(\ref{the equation of motion of a particle}) shows that there
exists an universal damping force
\begin{equation}\label{universal damping force}
\mbox{\upshape\bfseries{F}}_d = -\frac{\rho
q_0}{m_0}m\mbox{\upshape\bfseries{v}}
\end{equation}
 exerted on each particle.

Now let us consider a system consists of two particles. Based on
Hypothesis \ref{monad}, applying Theorem \ref{motion of a
particle} to this system, we have

\begin{wcorollary}\label{motion of a system consists of two particles}
Suppose there is a system consists of two particles and there are
no other forces exerted on the particles, then the equations of
motion of this system are
\begin{eqnarray}\label{equations of motion of a system consists of two particles}
&& m_1 \frac{d\mbox{\upshape\bfseries{v}}_1}{dt}
  =  -\frac{\rho q_0}{m_0} m_1 \mbox{\upshape\bfseries{v}}_1
 -\frac{\rho q^2_0}{4\pi m^2_0} \frac{m_1m_2}{r^2} \hat{\mbox{\upshape\bfseries{r}}}_{12} \label{equations of motion of a system consists of two particles 1}\\
&&m_2 \frac{d\mbox{\upshape\bfseries{v}}_2}{dt}
  =  -\frac{\rho q_0}{m_0}m_2\mbox{\upshape\bfseries{v}}_2
 -\frac{\rho q^2_0}{4\pi
 m^2_0}\frac{m_1m_2}{r^2}\hat{\mbox{\upshape\bfseries{r}}}_{21},\label{equations of motion of a system consists of two particles 2}
\end{eqnarray}

where $m_i, i=1,2$ is the mass of the particles,
$\mbox{\upshape\bfseries{v}}_i, i=1,2$ is the velocity of the
particles, $m_0$ is the mass of a monad,
 $-q_0$ is the strength of a monad,
 $\rho$ is the density of the tao,
$\hat{\mbox{\upshape\bfseries{r}}}_{12}$ denotes the unit vector
directed outward along the line from the particle with mass
$m_2(t)$ to the particle with mass $m_1(t)$,
$\hat{\mbox{\upshape\bfseries{r}}}_{21}$ denotes the unit vector
directed outward along the line from the particle with mass
$m_1(t)$ to the particle with mass $m_2(t)$.
\end{wcorollary}

 Ignoring the damping forces in (\ref{equations of motion of a system consists of two particles 2}), we have
\begin{wcorollary}\label{Inverse-Square-Law of Gravitation}
Suppose
(1)$\mbox{\upshape\bfseries{v}}_i\ll\mbox{\upshape\bfseries{u}}_i,
i=1,2$, where $\mbox{\upshape\bfseries{v}}_i$ is the velocity of
the particle with mass $m_i$, $\mbox{\upshape\bfseries{u}}_i$ is
the velocity of the tao at the location of the particle with mass
$m_i$ induced by the other particle,
 (2)there are no other forces exerted on the particles,
then the force $\mbox{\upshape\bfseries{F}}_{21}(t)$
 exerted on the particle with mass $m_2(t)$
 by the velocity field of tao induced by the particle with mass $m_1(t)$ is
 \begin{equation}\label{the inverse-square-law of gravitation}
\mbox{\upshape\bfseries{F}}_{21}(t)=-G(t)\frac{m_1(t)m_2(t)}{r^2}
\hat{\mbox{\upshape\bfseries{r}}}_{21},
\end{equation}
where $G=\rho q^2_0/(4\pi m^2_0(t))$,
$\hat{\mbox{\upshape\bfseries{r}}}_{21}$ denotes the unit vector
directed outward along the line from the particle with mass
$m_1(t)$ to the particle with mass $m_2(t)$, $r$ is the distance
between the two particles.
\end{wcorollary}
Corollary \ref{Inverse-Square-Law of Gravitation} is coincide with
Newton's inverse-square-law of gravitation (\ref{Newton's law of
gravitation}) except for two differences. The first difference is
that $m_i, i=1,2$ are constants in Newton's law (\ref{Newton's law
of gravitation}) while in Corollary \ref{Inverse-Square-Law of
Gravitation} they are functions of time $t$. The second difference
is that $G$ is a constant in Newton's theory while here $G$ is a
function of time $t$.

Let us now introduce a hypothesis that $G$ and the masses of
particles are changing so slowly relative to the time scaler of
human beings that they can be treated as constants approximately.
Thus, Newton's law of gravitation may be considered as a result of
Corollary \ref{Inverse-Square-Law of Gravitation} based on this
hypothesis.

\stepcounter{section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\thesection . \  \label{superposition principle}Superposition principle of gravitational field}
The definition of gravitational field
$\mbox{\upshape\bfseries{G}}$ of a particle with mass $m$ is
 $\mbox{\upshape\bfseries{G}}=\mbox{\upshape\bfseries{F}} / m_{test}$,
 where $m_{test}$ is the mass of a test point mass,
$\mbox{\upshape\bfseries{F}}$ is the gravitational force exerted
on the test point mass by the gravitational field of the particle
with mass $m$. Based on Theorem \ref{motion of a particle} and
Corollary \ref{Inverse-Square-Law of Gravitation} we have
\begin{equation}\label{gravitational field}
\mbox{\upshape\bfseries{G}}=\frac{\rho
q_0}{m_0}\mbox{\upshape\bfseries{u}},
\end{equation}
where $\rho$ is the density of the tao, $m_0$ is the mass of a
monad, $q_0$ is the strength of a monad,
$\mbox{\upshape\bfseries{u}}$ is the velocity of the tao at the
location of the test point mass induced by the particle with mass
$m$. Thus, the superposition principle of gravitational field is
nothing but the superposition theorem of the velocity field of
ideal fluids.

\stepcounter{section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\thesection . \  \label{equation of gravitational field}
Equations of gravitational field of continuously distributed
particles} The definition of the volume density of continuously
distributed sink is
\begin{equation}\label{definition of the volume density of continuously distributed sink}
\rho_s=\lim_{\triangle V \rightarrow 0}\frac{\triangle
Q}{\triangle V},
\end{equation}
where $\triangle V$ is a small volume, $\triangle Q$ is the sum of
strengthes of all the sinks in the volume $\triangle V$.

Now let us to derive the continuity equation of the tao from the
principle of mass conservation. Consider an arbitrary volume $V$
bounded by a closed surface $S$ and fixed in space. Suppose there
are some point sources or sinks continuously distributed in the
volume $V$. The total mass in volume $V$ is
\begin{equation}
M=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int_{V} \rho dV,
\end{equation}
where $\rho$ is the density of the tao.

The rate of increase of the total mass in volume $V$ is
\begin{equation}\label{The rate of increase of the total mass}
 \frac{\partial M}{\partial t}=\frac{\partial }{\partial t}\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int_{V}\rho
 dV.
\end{equation}

The rate of mass outflow through the surface $S$ is
\begin{equation}\label{The rate of mass outflow}
 \int\hspace{-1.9ex}\int_{S}\hspace{-3.27ex}\bigcirc\rho (\mbox{\bfseries{u}} \cdot \mbox{\bfseries{n}}
 )dS = \int\hspace{-1.5ex}\int\hspace{-1.5ex}\int_{V} \nabla \cdot (\rho \mbox{\bfseries{v}})dV,
\end{equation}
where $\mbox{\bfseries{u}}$ is the velocity field of the tao.

The rate of mass created inside the volume $V$ is
\begin{equation}\label{the rate of mass created}
\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int_{V} \rho\rho_{s}dV.
\end{equation}

Now according to the principle of mass conservation, and make use
 of Eq.(\ref{The rate of increase of the total mass}), Eq.(\ref{The rate of mass outflow})
 and  Eq.(\ref{the rate of mass created}), we have
\begin{equation}
 \frac{\partial }{\partial t}\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int_{V} \rho dV
 = \int\hspace{-1.5ex}\int\hspace{-1.5ex}\int_{V} \rho\rho_{s} dV - \int\hspace{-1.5ex}\int\hspace{-1.5ex}\int_{V} \nabla \cdot (\rho \mbox{\bfseries{v}})dV
\end{equation}

Since the volume $V$ is arbitrary, we have
\begin{equation}\label{equation of mass conservation 1}
 \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mbox{\bfseries{v}}) = \rho\rho_{s}.
\end{equation}

Since the tao is homogeneous, that is
\begin{math}
\frac{\partial\rho}{\partial x}=\frac{\partial\rho}{\partial y}
 =\frac{\partial\rho}{\partial z}=\frac{\partial\rho}{\partial t}=0,
\end{math}
Eq.(\ref{equation of mass conservation 1}) becomes
\begin{equation}\label{equation of mass conservation 2}
 \nabla \cdot  \mbox{\bfseries{u}} = \rho_{s}.
\end{equation}

Thus, from (\ref{gravitational field}), we have
\begin{math}
 \nabla \cdot\mbox{\upshape\bfseries{G}} =\rho q_0 \rho_s / m_0
\end{math}.
Let $\rho_m$ denotes the volume mass density of continuously
distributed particles, then according to Hypothesis \ref{monad},
Eq.(\ref{the mass of a monad is increasing 2})and
Eq.(\ref{definition of the volume density of continuously
distributed sink}),
 we have
\begin{equation}
\rho_m = - \frac{m_0 \rho_s}{q_0}.
\end{equation}
Thus, noticing that the velocity field
$\mbox{\upshape\bfseries{u}}$ of tao is irrotational and $G=\rho
q^2_0/(4\pi m^2_0)$, the equations of gravitational field of
continuously distributed particles can be summarized as
\begin{equation}\label{equations of gravitational field}
\left\{
\begin{array}{ll}
\nabla \times\mbox{\upshape\bfseries{G}}
 =\mbox{\upshape\upshape\bfseries{0}},\\
\nabla \cdot\mbox{\upshape\bfseries{G}}
 =-4\pi G \rho_m.
\end{array}
\right.
\end{equation}

\stepcounter{section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\thesection . \  \label{inertial mass and gravitational mass}
The equivalence of inertial mass and gravitational mass} According
to Hypothesis \ref{monad} and Corollary \ref{Inverse-Square-Law of
Gravitation},
 we have
\begin{math}
m_{inertial}=m_{gravitational}
\end{math},
where $m_{inertial}$ is the inertial mass of a particle,
$m_{gravitational}$ is the inertial mass of the particle.

\stepcounter{section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\thesection . \  \label{gravitational constants}
Time dependence of gravitational constant $G$ and mass} According
to Hypothesis \ref{monad} and Corollary \ref{Inverse-Square-Law of
Gravitation}, we have $G=\rho q^2_0/(4\pi m^2_0(t))$, where
$m_0(t)$ is the mass of monad at time $t$, $-q_0$ is the strength
of a monad, $\rho$ is the density of the tao. The time
 dependence of gravitational mass can be seen from (\ref{the mass of a monad is increasing 3}).

\stepcounter{section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\thesection . \  \label{Conclusion}Conclusion}
There are two features of this theory of gravitation. The first
feature is that the gravitational interactions are transmitted by
a kind of fluidic medium. The second feature is the time
dependence of gravitational constant and gravitational mass.
Newton's law of gravitation is arrived if we introduce a
hypothesis that $G$ and the masses of particles are changing so
slowly that they can be treated as constants. Further, the
equations of gravitational field of continuously distributed
particles and the equivalence of inertial mass and gravitational
mass are derived. As a byproduct, it is shown that there exists a
universal damping force exerted on each particle.

\stepcounter{section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\thesection . \  \label{Discussion}Discussion}
There exists some interesting theoretical, experimental and
applied problems in the fields of fluid mechanics, astronomy,
astrophysics and other related fields involving this theory of
gravitation.

%\begin{itshape}Discussion.\end{itshape}
%\section{\label{Acknowledgments}Acknowledgments}
%\begin{acknowledgments}

%\end{acknowledgments}


%\begin{acknowledgments}
%\end{acknowledgments}
%\appendix

%\newpage %Just because of unusual number of tables stacked at end
%\bibliographystyle{apsrmp}
%\bibliographystyle{bibtex}
%\bibliographystyle{rmp}
%\bibliographystyle{aps}
%\bibliography{nc,cds,sw,connes}
%\bibliography{rmp-sample}
%\bibliography{bibtex}

%\bibliography{gravitation}% Produces the bibliography via BibTeX.
%\bibliography{bib20060912c}% Produces the bibliography via BibTeX.

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\end{thebibliography}


%\end{CJK}{GBK}{song}
%\end{CJK*}

\end{document}
%
% ****** End of file apssamp.tex *********

physics0506062.extracted_bib
\begin{thebibliography}{57}
\expandafter\ifx\csname
natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname bibnamefont\endcsname\relax
  \def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname
urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

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\end{thebibliography}