Working within the quasi-metric framework (QMF) described elsewhere, we examine the gravitational field exterior respectively interior to a spherically symmetric, isolated body made of perfect fluid. By construction the system is ``metrically static'', meaning that its associated gravitational field is static except for the effects of the global cosmic expansion on the spatial geometry. To ensure that the global cosmic expansion will not induce instabilities in the fluid source and thus violating the metrically static condition, the equation of state of the fluid is required to take a particular form (fulfilled for, e.g., an ideal gas). We set up dynamical equations for the gravitational field and give an exact solution for the exterior part. Furthermore we find equations of motion applying to inertial test particles moving in the exterior gravitational field. The metrically static condition implies that the radius of the source increases and that distances between circular orbits increase according to the Hubble law, but such that circle orbit velocities are unaffected. This means that the dynamically measured mass of the source increases linearly with cosmic scale, or that the gravitational ``constant'' increases in time. We show that, if this model of an expanding gravitational field is taken to represent the gravitational field of the solar system, this has no serious consequences for observational aspects of planetary motion. On the contrary some observational facts of the Earth-Moon system are naturally explained within the QMF.
gr-qc0201097.tex
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\begin{center}
{\large {\bf Spherically Symmetric, Metrically Static, Isolated
Systems in Quasi-Metric Gravity}}
\end{center}
\begin{center}
by
\end{center}
\begin{center}
Dag {\O}stvang \\
{\em Department of Physics, Norwegian University of Science and Technology 
(NTNU) \\
N-7491 Trondheim, Norway}
\end{center}
\begin{abstract}
Working within the quasi-metric framework (QMF) described elsewhere, we 
examine the gravitational field exterior respectively interior to a
spherically symmetric, isolated body made of perfect fluid. By construction 
the system is ``metrically static'', meaning that its associated gravitational 
field is static except for the effects of the global cosmic expansion on the 
spatial geometry. To ensure that the global cosmic expansion will not induce 
instabilities in the fluid source and thus violating the metrically static 
condition, the equation of state of the fluid is required to take a particular 
form (fulfilled for, e.g., an ideal gas). 

We set up dynamical equations for the gravitational field and give an exact 
solution for the exterior part. Furthermore we find equations of motion 
applying to inertial test particles moving in the exterior gravitational 
field. The metrically static condition implies that the radius of the source 
increases and that distances between circular orbits increase according to the 
Hubble law, but such that circle orbit velocities are unaffected. This means 
that the dynamically measured mass of the source increases linearly with 
cosmic scale, or that the gravitational ``constant'' increases in time. We 
show that, if this model of an expanding gravitational field is taken to 
represent the gravitational field of the solar system, this has no serious 
consequences for observational aspects of planetary motion. On the contrary 
some observational facts of the Earth-Moon system are naturally explained 
within the QMF.
\end{abstract}
\section{Introduction}
The idea that the cosmic expansion may possibly be relevant for local systems 
came up many years ago; see, e.g., [1] and references therein. More recently 
there has been renewed interest in this idea; in part because it has become 
clear that there is no compelling observational evidence showing the expected 
deviations from the global Hubble law for galaxies in the vicinity of or even
within the local group of galaxies. See, e.g., [2] and references therein.

Even if there are in principle no direct observations ruling out the relevance
of the global Hubble law on local scales, the generally accepted view is that 
for all practical purposes, local systems may be treated as decoupled from the
cosmic expansion. This view reflects predictions coming from the standard 
framework of metric gravity. That is, it is well-known that metric theory 
predicts that realistic local systems are hardly affected at all by the 
cosmological expansion (its effect should at best be totally negligible, see,
e.g., [1] and references therein). The reason for this prediction is basically 
that in metric theory, the cosmological expansion must be modelled within a 
mathematical framework where space-time is postulated to be a semi-Riemannian 
manifold. However, when analyzing the influence of the cosmological expansion 
on local systems there should be no reason to expect that predictions made 
within the metric framework should continue to hold in a theory where the 
structure of space-time is non-metric.

Recently a review of a new type of non-metric space-time framework, the 
so-called quasi-metric framework (QMF), was presented in [3]. Also presented 
was an alternative relativistic theory of gravity formulated within this 
framework. (A more detailed presentation can be found in [4].) This theory 
correctly predicts the ``classic'' solar system tests in the case where an 
asymptotically Minkowski background is invoked as an approximation and the 
cosmological expansion can be neglected. However, for reasons explained in [3],
the theory is based on a ${\bf S}^3{\times}{\bf R}$-background rather than a 
Minkowski background as the global basic (``prior'') geometry of the Universe. 
According to quasi-metric theory the nature of the cosmological expansion is a 
direct consequence of the existence of this 
${\bf S}^3{\times}{\bf R}$-background and the expansion applies to all systems 
where gravitational dynamics dominates (hereafter called ``gravitational 
systems''), regardless of scale. That is, in quasi-metric gravity the 
mathematical modelling of the Hubble expansion and thus its physical 
interpretation are different than in metric theory, and as a consequence, the 
Hubble expansion is predicted to influence local, gravitationally bound 
systems sufficiently that its effects should be observable in experiments. On 
the other hand, since quantum-mechanical states should be unaffected by the
expansion, quasi-metric theory allows that the global cosmic expansion 
does not apply to quantum-mechanical systems bound by non-gravitational forces 
where gravitational interactions are negligible (hereafter called ``atomic 
systems'') [5].

To find more exactly how the cosmological expansion affects local 
gravitational systems according to the quasi-metric theory, one must first 
calculate the spherically symmetric gravitational field with the 
${\bf S}^3{\times}{\bf R}$-background, both interior and exterior to the 
source. Then one should use the quasi-metric equations of motion to calculate 
how test particles move in the exterior gravitational field. We show in section
3.3 of this paper that the quasi-metric theory predicts that the exterior 
gravitational field should expand according to the Hubble law. This also 
applies to the interior gravitational field if potential instabilities induced 
by the global cosmic expansion can be neglected (see section 3.2). In 
particular, for a source made of ideal gas the cosmic expansion induces no 
instabilities so the radius of a body made of ideal gas is predicted to 
expand. This result may support an interpretation of geological data 
indicating that the Earth is expanding according to the Hubble law, see 
reference [6] and references cited therein. (It is difficult to measure such a 
small expansion rate directly due to the existence of larger local 
displacements of the Earth's surface.) Besides, an expanding Earth should 
cause changes in its spin rate; we show in section 4.2 of this paper that the 
secular spin-down of the Earth as inferred from historical astronomical 
observations may in fact be of cosmological origin and only about half of the 
currently accepted value. Quasi-metric theory also predicts a cosmological 
origin of and different values for the recession of the Moon and its mean 
acceleration, other than those inferred from lunar laser ranging (LLR) 
experiments using standard theory. However, these differences are due to 
model-dependency since the LLR data yields that the recession of the Moon 
follows Hubble's law when analyzed within the QMF, and the quasi-metric 
predictions are consistent with a modern lunar ephemeris.

Finally it is shown that the predicted cosmic expansion of the solar system's 
gravitational field does not lead to easily detected perturbations in the 
observed motion of the planets. However, some less easily detected effects 
should be measurable; in fact a newly discovered secular increase of the 
astronomical unit may be explained by cosmic expansion. Also active mass is
predicted to show a secular increase proportional to the Hubble parameter; in 
section 4.3 we argue that the predicted value of the increase is not in 
conflict with current test experiments. These results are very different from 
their counterparts in metric theory; that is why it is generally believed that 
observations confirm that the solar system is decoupled from the cosmic 
expansion when it is in fact the other way around.
\section{Quasi-metric relativity in brief}
\subsection{General formulae}
In this section we summarize the main features of the QMF and a quasi-metric 
theory of gravity. A considerably more extensive discussion can be found in 
[3] or [4].

The mathematical foundation of the QMF can be described by first considering a
5-dimensional product manifold ${\cal M}{\times}{\bf R}_1$, where
${\cal M}={\cal S}{\times}{\bf R}_2$ is a (globally hyperbolic) Lorentzian
space-time manifold, ${\bf R}_1$ and ${\bf R}_2$ are two copies of the real
line and ${\cal S}$ is a compact Riemannian 3-dimensional manifold (without
boundaries). Then {\em the global time function t} representing the extra
(degenerate) time dimension ${\bf R}_1$ is introduced as a coordinate on 
${\bf R}_1$. Moreover, for $t$ given it is convenient to use a coordinate 
system ${\{}x^{\mu}{\}}$ (${\mu}$ taking values in the interval $0-3$) where 
the ordinary time coordinate $x^0$ on ${\cal M}$ scales like $ct$; this 
ensures that $x^0$ is in some sense a mirror of $t$ and thus a ``preferred'' 
global time coordinate. A coordinate system with a global time coordinate of 
this type we call a {\em global time coordinate system} (GTCS). Hence, 
expressed in a GTCS ${\{}x^{\mu}{\}}$, $x^0$ is interpreted as a global 
coordinate on ${\bf R}_2$ and ${\{}x^j{\}}$ ($j$ taking values in the interval
$1-3$) as spatial coordinates on ${\cal S}$. The class of GTCSs is a set of 
preferred coordinate systems inasmuch as the equations of quasi-metric 
relativity take special forms when expressed in a GTCS. Note that there exist 
infinitely many GTCSs. 

The 4-dimensional quasi-metric space-time manifold ${\cal N}$ can now be 
defined by slicing the sub-manifold $x^0=ct$ (using a GTCS) out of the initial 
5-dimensional space-time manifold. Furthermore, $\cal N$ is equipped with two 
families of Lorentzian space-time metric tensor fields ${\bf {\bar g}}_t$ and 
${\bf g}_t$. The metric family ${\bf {\bar g}}_t$ represents a solution of 
field equations, and from ${\bf {\bar g}}_t$ one can construct the 
``physical'' metric family ${\bf g}_t$ which is used when comparing 
predictions to experiments. It is convenient to think of the metric families 
as single degenerate metrics on (a subset of) ${\cal M}{\times}{\bf R}_1$, 
where the degeneracy manifests itself via the conditions 
${\bf {\bar g}}_t({\frac{\partial}{{\partial}t}},{\cdot}){\equiv}0$,
${\bf g}_t({\frac{\partial}{{\partial}t}},{\cdot}){\equiv}0$. Finally, notice
that ${\cal N}$ differs from a Lorentzian manifold and that this becomes 
evident only when it is equipped with an affine connection (see below).

From the above description we see that within the QMF, the canonical 
description of space-time is taken as fundamental. That is, quasi-metric 
space-time is constructed as consisting of two mutually orthogonal foliations:
on the one hand space-time can be sliced up globally into a family of 
3-dimensional space-like hypersurfaces (called the fundamental hypersurfaces 
(FHSs)) by the global time function $t$, on the other hand space-time can be 
foliated into a family of time-like curves everywhere orthogonal to the FHSs. 
These curves represent the world lines of a family of hypothetical observers 
called the fundamental observers (FOs). There exists a unique relationship 
between $t$ and the proper time as measured by any FO.

Now one characteristic property of quasi-metric theory is that it postulates
the existence of systematic scale changes between gravitational and atomic 
systems (and the main role of $t$ is to describe the global aspects of such
changes). This means that gravitational quantities are postulated to exhibit 
an extra variation when measured in atomic units (and vice versa). One may
think of this as if fixed operationally defined atomic units vary formally in 
space-time. Moreover, since $c$ and Planck's constant ${\hbar}$ by definition 
are not formally variable, the formal variation of time units is equal to that 
of length units and inverse to that of mass units. We now postulate that this 
formal variation of atomic length (or time) units is defined from the geometry 
of the FHSs in $({\cal N},{\bf {\bar g}}_t)$. This means that the form of
${\bf {\bar g}}_t$ must be restricted such that all variation in the spatial 
geometry follows from a spatial scale factor ${\bar F}_t$. Thus by definition, 
measured in atomic units, the formal variability of gravitational quantities 
with the dimension of time or length goes as ${\bar F}_t$, whereas the the 
formal variability of gravitational quantities with the dimension of mass goes 
as ${\bar F}_t^{-1}$.

To determine the form of ${\bar F}_t$ we require that no extra arbitrary scale 
or parameter should be introduced (i.e., no characteristic scale should be 
associated with ${\bar F}_t$). This yields the (rather unique) choice
${\bar F}_t{\equiv}c{\bar N}_tt$, where ${\bar N}_t$ is the lapse function
field family of the FOs in $({\cal N},{\bf {\bar g}}_t)$. With ${\bar F}_t$
given, together with the requirement that the FHSs should be compact and have
a trivial topology, it is now straightforward to set up the general form of 
${\bf {\bar g}}_t$. It thus can be argued [3, 4] that in a GTCS, the most 
general form allowed for the family ${\bf {\bar g}}_t$ may be represented by 
the family of line elements (we use the metric signature $(-+++)$ and 
Einstein's summation convention throughout)
\eqa
{\overline{ds}}^2_t={\bar N}_t^2{\Big [}({\bar N}^i{\bar N}^jS_{ij}-1)(dx^0)^2
+2{\frac{t}{t_0}}{\bar N}^iS_{ij}dx^jdx^0+
{\frac{t^2}{t_0^2}}S_{ij}dx^idx^j{\Big ]},
\ena
where $t_0$ is an arbitrary reference epoch and where $S_{ij}dx^idx^j$ denotes 
the metric of the 3-sphere ${\bf S}^3$ (with a radius equal to $ct_0$).
Moreover, ${\frac{t_0}{t}}{\bar N}^k{\frac{\partial}{{\partial}x^k}}$ is the 
family of shift vector fields of the FOs in $({\cal N},{\bf {\bar g}}_t)$. (We 
may also define the family of shift covector fields ${\frac{t}{t_0}}
{\bar N}_i{\equiv}{\frac{t}{t_0}}{\bar N}_t^2S_{ik}{\bar N}^k$.) Note that,
whereas ${\bar N}_t$ may depend on $t$, equation (1) does not contain any 
further dependence on $t$ than what is shown explicitly. Also note that
equation (1) may be taken as a postulate.

The time evolution of the spatial geometry of the FHSs is given by the 
change of ${\bar F}_t{\equiv}c{\bar N}_tt$ in the hypersurface-orthogonal 
direction. This evolution may be split up according to the definition (where a 
comma denotes partial derivation, the symbol `${\bar {\perp}}$' denotes a 
scalar product with the unit normal vector field $-{\bf {\bar n}}_t$ of the 
FHSs, and where ${\pounds}_{{\bf {\bar n}}_t}$ denotes Lie derivation in the 
direction normal to the FHSs holding $t$ constant)
\eqa
{\bar F}_t^{-1}{\topstar{\pounds}}_{{\bf {\bar n}}_t}
{\bar F}_t{\equiv}{\bar F}_t^{-1}{\Big (}(c{\bar N}_t)^{-1}{\bar F}_t,_t+
{\pounds}_{{\bf {\bar n}}_t}{\bar F}_t{\Big )}=
{\frac{1}{c{\bar N}_tt}}+{\frac{{\bar N}_t,_t}{c{\bar N}_t^2}}
-{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}
{\equiv}c^{-2}{\bar x}_t+c^{-1}{\bar H}_t,
\ena
where $c^{-2}{\bar x}_t$ represents the {\em kinematical contribution} to 
the evolution of the spatial scale factor and $c^{-1}{\bar H}_t$ represents 
the so-called {\em non-kinematical contribution} defined by
\eqa
{\bar H}_t={\frac{1}{{\bar N}_tt}}+{\bar y}_t, \qquad
{\bar y}_t{\equiv}c^{-1}{\sqrt{{\bar a}_{{\cal F}k}{\bar a}_{\cal F}^k}}, 
\qquad c^{-2}{\bar a}_{{\cal F}j}{\equiv}{\frac{{\bar N}_t,_j}{{\bar N}_t}}.
\ena
We see from equation (3) that the non-kinematical evolution (NKE) of the
spatial geometry takes the form of an ``expansion''. Furthermore the NKE 
consists of two terms; the first term ${\frac{1}{{\bar N}_tt}}$ represents the 
{\em global NKE} due to the global curvature of the FHSs, whereas the second 
term ${\bar y}_t$ represents the {\em local NKE} coming from the gravitational
field. This second term is not ``realized'' globally since it is absent in 
equation (2). Besides we see from equation (2) that the evolution of 
${\bar N}_t$ with time may also be written as a sum of one kinematical and one 
non-kinematical term, i.e.
\eqa
{\frac{{\bar N}_t,_t}{c{\bar N}_t^2}}-
{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}=c^{-2}{\bar x}_t+
c^{-1}{\bar y}_t. 
\ena
The split-ups defined in equations (2), (3) and (4) are necessary to be able 
to construct ${\bf g}_t$ from ${\bf {\bar g}}_t$ [3]. Note that the kinematical
evolution (KE) of the spatial scale factor may be positive or negative.

Next, on the quasi-metric space-time manifold ${\cal N}$ two linear, symmetric 
``degenerate" connections ${\,}{\,}{\topstar{{\bf {\bar \nabla}}}}{\,}{\,}$ and
${\,}{\,}{\topstar{\nabla}}{\,}{\,}$ are defined. These connections are called 
degenerate due to the fact that they are essentially connections compatible 
with the 5-dimensional degenerate metrics ${\bf {\bar g}}_t$ and ${\bf g}_t$, 
respectively, on ${\cal M}{\times}{\bf R}_1$ and then just restricted to 
$\cal N$. In the following we describe the connection
${\,}{\,}{\topstar{\nabla}}{\,}{\,}$ since this connection yields the 
quasi-metric equations of motion in $({\cal N},{\bf g}_t)$.
That is, we introduce a torsion-free, metric-compatible 5-dimensional 
connection ${\,}{\,}{\topstar{\nabla}}{\,}{\,}$ with the property that
\eqa
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf g}_t=0, \qquad
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf n}_t=0, \qquad
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf h}_t=0,
\ena
on ${\cal M}{\times}{\bf R}_1$ and consider the restriction of 
${\,}{\,}{\topstar{\nabla}}{\,}{\,}$ to $\cal N$. Here ${\bf h}_t$ is the
spatial metric family intrinsic to FHSs and ${\bf n}_t$ is the unit vector 
family normal to the FHSs in $({\cal N},{\bf g}_t)$. It can be shown [4] that,
expressed in a GTCS where the spatial coordinates do not depend on $t$, the 
components which do not vanish identically of the degenerate connection field 
are given by
\eqa
{\topstar{\Gamma}}^{\alpha}_{{\mu}t}{\equiv}
{\topstar{\Gamma}}^{\alpha}_{t{\mu}}{\equiv}{\frac{1}{t}}
{\delta}^{\alpha}_i{\delta}^i_{\mu}, {\qquad}
{\topstar{\Gamma}}^{\alpha}_{{\nu}{\mu}}{\equiv}{\frac{1}{2}}
g_{(t)}^{{\alpha}{\sigma}}{\Big(}g_{(t){\sigma}{\mu}},_{\nu}
+g_{(t){\nu}{\sigma}},_{\mu}-g_{(t){\nu}{\mu}},_{\sigma}{\Big)}{\equiv}
{\Gamma}^{\alpha}_{(t){\nu}{\mu}}.
\ena
The general equations of motion for test particles are identical to the 
geodesic equation obtained from ${\,}{\,}{\topstar{\nabla}}{\,}{\,}$.
In a GTCS they take the form (see [4] for a derivation)
\eqa
{\frac{d^2x^{\mu}}{d{\lambda}^2}}+{\Big(}
{\topstar{\Gamma}}^{\mu}_{t{\nu}}{\frac{dt}{d{\lambda}}}+
{\Gamma}^{\mu}_{(t){\beta}{\nu}}{\frac{dx^{\beta}}{d{\lambda}}}{\Big)}
{\frac{dx^{\nu}}{d{\lambda}}} 
={\Big(}{\frac{cd{\tau}_t}{d{\lambda}}}{\Big)}^2c^{-2}a_{(t)}^{\mu},
\ena
where $d{\tau}_t$ is the proper time as measured along the curve, ${\lambda}$ 
is some general affine parameter and ${\bf a}_t$ is the 4-acceleration as 
measured along the curve. From equations (6) and (7) we see that quasi-metric 
theory cannot be identified with any metric theory since the affine connection 
compatible with a general metric family is non-metric.

As mentioned above, a basic property of the QMF is that gravitational 
quantities will be formally variable when measured in atomic units. In 
particular this applies to the gravitational ``constant'' $G$. However, for 
convenience we {\em define} $G$ to be constant so that its formal variability
is transferred to the mass (and charge, if any). Thus, we have to separate 
between {\em active mass}, which is a scalar field, and {\em passive mass} 
(passive gravitational mass and inertial mass). By dimensional analysis it is 
found [4] that active mass $m_t$ varies formally as ${\bar F}_t$ measured in 
atomic units (but passive mass does of course not vary). We then get
\eqa
m_t,_t={\Big (}{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}{\big )}m_t, 
{\qquad}
m_t,_{\bar {\perp}}={\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}m_t, \qquad
m_t,_j=c^{-2}{\bar a}_{{\cal F}j}m_t,
\ena 
where ${\bf {\bar a}}_{\cal F}$ is the 4-acceleration of the FOs in the
family ${\bf {\bar g}}_t$. Taking into the account this variation of active
mass in quasi-metric space-time it is possible to find local conservation
laws. These local conservation laws involve the degenerate covariant
divergence ${\,}{\,}{\topstar{{\bf {\bar \nabla}}}}{\,}{\,}{\cdot}{\bf T}_t$
of the active stress-energy tensor ${\bf T}_t$, and they take the form [4]
\eqa
T_{(t){\mu}{\bar *}{\nu}}^{\nu}=2{\frac{{\bar N}_t,_{\nu}}{{\bar N}_t}}
T_{(t){\mu}}^{\nu}-{\frac{2}{c{\bar N}_t}}{\Big (}{\frac{1}{t}}+
{\frac{{\bar N}_t,_t}{{\bar N}_t}}{\Big )}T_{(t){\bar {\perp}}{\mu}},
\ena
where the symbol `${\bar *}$' denotes degenerate covariant derivation 
${\,}{\,}{\topstar{{\bf {\bar \nabla}}}}{\,}{\,}$ in component notation. 
Notice that these local conservation laws imply that inertial observers move 
along geodesics of ${\,}{\,}{\topstar{\bf {\bar{ \nabla}}}}{\,}{\,}$ in 
$({\cal N},{\bf {\bar g}}_t)$, and that this guarantees that inertial
observers move along geodesics of of ${\,}{\,}{\topstar{\bf {\nabla}}}{\,}{\,}$
in $({\cal N},{\bf g}_t)$ as well [3, 4]. This means that the equations of 
motion (7) are consistent with the local conservation laws (9).

It is useful to project these local conservation laws with respect to the 
FHSs. We then get the equations (in content equivalent to equation (9))
\eqa
{\topstar{\cal L}}_{{\bf {\bar n}}_t}T_{(t){\bar {\perp}}{\bar {\perp}}}
={\Big (}{\bar K}_t-2{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}-
{\frac{2}{c{\bar N}_t}}{\Big [}{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}
{\Big ]}{\Big )}T_{(t){\bar {\perp}}{\bar {\perp}}}
+{\bar K}_{(t)ik}{\hat T}_{(t)}^{ik}-{\hat T}^i_{(t){\bar {\perp}}{\mid}i},
\ena
\eqa
{\topstar{\cal L}}_{{\bf {\bar n}}_t}T_{(t)j{\bar {\perp}}}
&=&{\Big (}{\bar K}_t-2{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}-
{\frac{1}{c{\bar N}_t}}{\Big [}{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}
{\Big ]}{\Big )}T_{(t)j{\bar {\perp}}}-c^{-2}{\bar a}_{{\cal F}j}
T_{(t){\bar {\perp}}{\bar {\perp}}}
\nonumber \\
&&+c^{-2}{\bar a}_{{\cal F}i}{\hat T}_{(t)j}^i
+c^{-2}{\bar a}_{{\cal F}j}{\frac{t_0}{t}}
{\frac{{\bar N}^i}{{\bar N}_t}}T_{(t)i{\bar {\perp}}}-{\hat T}^i_{(t)j{\mid}i},
\ena
where ${\topstar{\cal L}}_{{\bf {\bar n}}_t}$ denotes Lie derivative of spatial
objects in the direction normal to the FHSs and $`{\mid}$' denotes spatial 
covariant derivation. (A ``hat'' denotes an object intrinsic to the FHSs.)
See [4] for a derivation of these equations. Also postulated in [4] are the 
field equations (with ${\kappa}{\equiv}{\frac{8{\pi}G}{c^4}}$)
\eqa
2{\bar R}_{(t){\bar {\perp}}{\bar {\perp}}}=
2(c^{-4}{\bar a}_{{\cal F}k}{\bar a}_{\cal F}^k+
c^{-2}{\bar a}^k_{{\cal F}{\mid}k}-{\bar K}_{(t)ik}{\bar K}_{(t)}^{ik}
+{\cal L}_{{\bf {\bar n}}_t}{\bar K}_t)=
{\kappa}(T_{(t){\bar {\perp}}{\bar {\perp}}}+{\hat T}_{(t)i}^i),
\ena
\eqa
{\bar R}_{(t)j{\bar {\perp}}}=
{\bar K}_{(t)j{\mid i}}^i-{\bar K}_t,_j={\kappa}T_{(t)j{\bar {\perp}}},
\ena
where ${\bf {\bar R}}_t$ is the Ricci tensor family and ${\bf {\bar K}}_t$ is 
the extrinsic curvature tensor family of the FHSs corresponding to the metric 
family (1). (${\bar K}_t$ is the trace of ${\bf {\bar K}}_t$.) Notice that
these field equations contain only one dynamical degree of freedom coupled
explicitly to matter (i.e., gravity is essentially {\em scalar} in
$({\cal N},{\bf {\bar g}}_t)$). However, by construction a second, 
{\em implicit} dynamical degree of freedom is added under the transformation 
${\bf {\bar g}}_t{\rightarrow}{\bf g}_t$ [3, 4]. It should also be emphasized 
that, although the field equations are postulated rather than derived, they 
are by no means arbitrary; equation (12) for example, follows naturally from a 
geometrical correspondence with Newton-Cartan theory. Besides, in contrast to 
General Relativity, these field equations represent only a {\em partial} 
coupling between the space-time geometry and ${\bf T}_t$. Therefore, somewhat 
similar to Newtonian theory, the field equations (12), (13) are in principle
quite independent of the local conservation laws (9) (and the equations of 
motion (7)). But this means that postulating the field equations directly is 
in principle not in any way inferior to ``deriving'' them from a postulated 
invariant action principle.

Finally, it is convenient to have expressions for the geometry intrinsic to 
the FHSs obtained from equation (1). We may straightforwardly derive the 
formulae
\eqa
{\bar H}_{(t)ij}=c^{-2}{\Big (}{\bar a}^k_{{\cal F}{\mid}k}-
{\frac{1}{({\bar N}_tt)^2}}{\Big )}{\bar h}_{(t)ij}
-c^{-4}{\bar a}_{{\cal F}i}{\bar a}_{{\cal F}j}
-c^{-2}{\bar a}_{{\cal F}i{\mid}j},
\ena
\eqa
{\bar P}_t={\frac{6}{(c{\bar N}_tt)^2}}
+2c^{-4}{\bar a}_{{\cal F}i}{\bar a}_{{\cal F}}^i
-4c^{-2}{\bar a}_{{\cal F}{\mid}i}^i,
\ena
where ${\bf {\bar h}}_t$ is the metric family intrinsic to the FHSs,
${\bf {\bar H}}_t$ is the Einstein tensor family intrinsic to the FHSs and
${\bar P}_t$ is the Ricci scalar family intrinsic to the FHSs in 
$({\cal N},{\bf {\bar g}}_t)$.
\subsection{Special equations of motion}
In this paper we analyze the equations of motion (7) in the case of a 
uniformly expanding, isotropic gravitational field in vacuum exterior to an 
isolated, spherically symmetric source in an isotropic, compact spatial 
background. We also require that the source is at rest with respect to some 
GTCS. Furthermore we require that ${\bar N}_t$ is independent of $x^0$ 
(and $t$); i.e., that the only explicit time dependence is via $t$ in the 
spatial scale factor (using the chosen GTCS). Then it turns out that also the 
FOs must be at rest with respect to the chosen GTCS and consequently the shift 
vector field vanishes. We denote this a ``metrically static'' case. This 
scenario may be taken as a generalization of the analogous case with a 
Minkowski background (that case is analyzed in [4]) and is more realistic
since the Minkowski background is not a part of our theory but rather invoked 
as an approximation being useful in particular cases.

We start by making a specific {\em ansatz} for the form of ${\bf {\bar g}}_t$.
Introducing a spherical GTCS ${\{}x^0,r,{\theta},{\phi}{\}}$, 
where $r$ is a Schwarzschild radial coordinate, we assume that the 
metric families ${\bf {\bar g}}_t$ and ${\bf g}_t$ can be written in a form 
compatible with equation (1) (using the notation $'{\hspace*{2mm}}{\equiv}
{\frac{\partial}{{\partial}r}}$), i.e.,
\eqa
c^2{\overline{d{\tau}}}^2_t={\bar{B}}(r)(dx^0)^2-({\frac{t}{t_0}})^2{\Big(}
{\bar{A}}(r)dr^2+r^2d{\Omega}^2{\Big)}, {\nonumber} \\
c^2d{\tau}^2_t=B(r)(dx^0)^2-({\frac{t}{t_0}})^2{\Big(}
A(r)dr^2+r^2d{\Omega}^2{\Big)},
\ena
\eqa
{\bar A}(r){\equiv}{\frac{{\Big [}1-r{\frac{{\bar B}'(r)}{2{\bar B}(r)}}
{\Big ]}^2}
{1-{\frac{r^2}{{\bar B}(r){\Xi}^2_0}}}},
\ena
where $d{\Omega}^2{\equiv}d{\theta}^2+{\sin}^2{\theta}d{\phi}^2$ and 
${\Xi}_0{\equiv}ct_0$. Note that the spatial coordinate system covers only 
half of ${\bf S}^3$, thus the range of the radial coordinate is $r<{\Xi}_0$ 
only. The function ${\bar{B}}(r)$ may be calculated from the field equations; 
we treat this problem in the next section. The functions $A(r)$ and $B(r)$ may 
then be found from ${\bf {\bar g}}_t$ and ${\bar y}_t$.

We now calculate the metric connection coefficients from the metric family 
${\bf g}_t$ given in equation (16). A straightforward calculation yields 
\eqa
{\Gamma}^r_{(t)rr}&=&{\frac{A'(r)}{2A(r)}}, {\qquad}
{\Gamma}^r_{(t){\theta}{\theta}}=-{\frac{r}{A(r)}}, {\qquad}
{\Gamma}^r_{(t){\phi}{\phi}}={\Gamma}^r_{(t){\theta}{\theta}}{\sin}^2{\theta},
{\nonumber} \\
{\Gamma}^r_{(t)00}&=&({\frac{t_0}{t}})^2{\frac{B'(r)}{2A(r)}}, 
{\qquad}{\Gamma}^{\theta}_{(t)r{\theta}}={\Gamma}^{\theta}_{(t){\theta}r}=
{\frac{1}{r}}, {\qquad} {\Gamma}^{\theta}_{(t){\phi}{\phi}}=
-{\sin}{\theta}{\cos}{\theta}, {\nonumber} \\
{\Gamma}^{\phi}_{(t)r{\phi}}&=&{\Gamma}^{\phi}_{(t){\phi}r}={\frac{1}{r}},
{\qquad}{\Gamma}^{\phi}_{(t){\phi}{\theta}}=
{\Gamma}^{\phi}_{(t){\theta}{\phi}}={\cot}{\theta}, {\qquad}
{\Gamma}^0_{(t)0r}={\Gamma}^0_{(t)r0}={\frac{B'(r)}{2B(r)}}.
\ena
In the following we use the equations of motion (7) to find the paths 
of inertial test particles moving in the metric family ${\bf g}_t$. Since 
${\bf a}_t$ vanishes for inertial test particles we get the relevant equations 
by using equation (6) and inserting the expressions (18) into equation (7). 
This yields (making explicit use of the fact that $cdt=dx^0$ in a GTCS)
\eqa
{\frac{d^{2}r}{d{\lambda}^{2}}} + 
{\frac{A'(r)}{2A(r)}}{\Big(}{\frac{dr}{d{\lambda}}}{\Big)}^2 -
{\frac{r}{A(r)}}{\bigg[}{\Big(}{\frac{d{\theta}}{d{\lambda}}}{\Big)}^{2} + 
{\sin}^{2}{\theta}{\Big(}{\frac{d{\phi}}{d{\lambda}}}{\Big)}^{2}{\bigg]}
{\nonumber} \\
+({\frac{t_0}{t}})^2
{\frac{B'(r)}{2A(r)}}{\Big(}{\frac{dx^0}{d{\lambda}}}{\Big)}^{2} + 
{\frac{1}{ct}}{\frac{dr}{d{\lambda}}}{\frac{dx^0}{d{\lambda}}}=0,
\ena
\eqa
{\frac{d^{2}{\theta}}{d{\lambda}^{2}}} +
{\frac{2}{r}}{\frac{d{\theta}}{d{\lambda}}}{\frac{dr}{d{\lambda}}} -
{\sin}{\theta}{\cos}{\theta}{\Big(}{\frac{d{\phi}}{d{\lambda}}}{\Big)}^{2} + 
{\frac{1}{ct}}{\frac{d{\theta}}{d{\lambda}}}{\frac{dx^0}{d{\lambda}}} = 0,
\ena
\eqa
{\frac{d^{2}{\phi}}{d{\lambda}^{2}}} +
{\frac{2}{r}}{\frac{d{\phi}}{d{\lambda}}}{\frac{dr}{d{\lambda}}} +
2{\cot}{\theta}{\frac{d{\phi}}{d{\lambda}}}{\frac{d{\theta}}{d{\lambda}}} + 
{\frac{1}{ct}}{\frac{d{\phi}}{d{\lambda}}}{\frac{dx^0}{d{\lambda}}} = 0,
\ena
\eqa
{\frac{d^{2}x^0}{d{\lambda}^{2}}} +
{\frac{B'(r)}{B(r)}}{\frac{dx^0}{d{\lambda}}}{\frac{dr}{d{\lambda}}} =0.
\ena
If we restrict the motion to the equatorial plane equation (20) becomes 
vacuous, and equation (21) reduces to
\eqa
{\frac{d^{2}{\phi}}{d{\lambda}^{2}}} +
{\frac{2}{r}}{\frac{d{\phi}}{d{\lambda}}}{\frac{dr}{d{\lambda}}} +
{\frac{1}{ct}}{\frac{d{\phi}}{d{\lambda}}}{\frac{dx^0}{d{\lambda}}} = 0.
\ena
Dividing equation (23) by ${\frac{d{\phi}}{d{\lambda}}}$ we find (assuming
${\frac{d{\phi}}{d{\lambda}}}{\neq}0$)
\eqa
{\frac{d}{d{\lambda}}}{\bigg[}{\ln}{\Big(}{\frac{d{\phi}}{d{\lambda}}}{\Big)} 
+ {\ln}{\Big(}r^{2}{\frac{t}{t_0}}{\Big)}{\bigg]} = 0.
\ena
We thus have a constant of the motion, namely
\eqa
J{\equiv}{\frac{t}{t_0}}r^2{\frac{d{\phi}}{d{\lambda}}}.
\ena
Dividing equation (22) by ${\frac{dx^0}{d{\lambda}}}$ yields
\eqa
{\frac{d}{d{\lambda}}}{\bigg[}{\ln}{\Big(}{\frac{dx^0}{d{\lambda}}}{\Big)} + 
{\ln}B(r){\bigg]}=0.
\ena
Equation (26) yields a constant of the motion which we can absorb
into the definition of ${\lambda}$ such that a solution of equation
(26) is [7]
\eqa
{\frac{dx^0}{d{\lambda}}}={\frac{1}{B(r)}}.
\ena
Multiplying equation (19) by 
${\frac{2t^2A(r)}{t_0^2}}{\frac{dr}{d{\lambda}}}$ and 
using the expressions (25), (27) we find
\eqa
{\frac{d}{d{\lambda}}}{\bigg[}
{\frac{t^2A(r)}{t_0^2}}{\Big(}{\frac{dr}{d{\lambda}}}{\Big)}^2-
{\frac{1}{B(r)}}+{\frac{J^2}{r^2}}{\bigg]}=0,
\ena
thus a constant $E$ of the motion is defined by
\eqa
{\frac{t^2A(r)}{t_0^2}}{\Big(}{\frac{dr}{d{\lambda}}}{\Big)}^2-
{\frac{1}{B(r)}}+{\frac{J^2}{r^2}}{\equiv}-E.
\ena
Equation (29) may be compared to an analogous expression obtained for
the static isotropic gravitational field in the metric framework [7].
Inserting the formulae (25), (27) and (29) into equation (16) and using the 
fact that in a GTCS we can formally write $dx^0=cdt$ when traversing the 
family of metrics, we find
\eqa
c^2d{\tau}_t^2=Ed{\lambda}^2.
\ena
Thus our equations of motion (7) force $d{\tau}_t/d{\lambda}$ to be
constant, quite similarly to the case when the total connection is
metric, as in the metric framework. From equation (30) we see that we must 
have $E=0$ for photons and $E>0$ for material particles.

We may eliminate the parameter ${\lambda}$ from equations (25), (27), (29) 
and (30) and alternatively use $t$ as a time parameter. This yields
\eqa
{\frac{t}{t_0}}r^2{\frac{d{\phi}}{cdt}}=B(r)J,
\ena
\eqa
({\frac{t}{t_0}})^2A(r)B^{-2}(r){\Big(}{\frac{dr}{cdt}}{\Big)}^2-
{\frac{1}{B(r)}}+{\frac{J^2}{r^2}}{\equiv}-E,
\ena
\eqa
d{\tau}_t^2=EB^2(r)dt^2.
\ena
We may integrate equations (31) and (32) to find the time history 
$(r(t),{\phi}(t))$ along the curve if the functions $A(r)$ and $B(r)$
are known.

For the static vacuum metric case with no global NKE, one can solve the 
geodesic equation for particles orbiting in circles with different radii, and 
from this find the asymptotically Keplerian nature of the corresponding 
rotational curve [7]. In our case we see from equations (31) and (32) that we 
can find circle orbits as solutions; such orbits have the property that the 
orbital speed $w(r)=B^{-1/2}(r){\frac{t}{t_0}}r{\frac{d{\phi}}{dt}}$ is 
independent of $t$. (Here $w(r)$ is the norm of the coordinate 3-velocity
${\bf w}_t={\sqrt{1-{\frac{w^2}{c^2}}}}{\frac{d{\phi}}{d{\tau}_t}}
{\frac{\partial}{{\partial}{\phi}}}$.)
\section{Metrically static, spherically symmetric systems}
\subsection{Perfect fluid sources}
We now seek general solutions of the type (16) of the field equations, and
where the source is modelled as a perfect fluid. Then the active stress-energy
tensor ${\bf T}_t$ takes the form
\eqa
{\bf T}_t=({\tilde{\rho}}_{\text m}+{\tilde p}/c^2){\bf {\bar u}}_t{\otimes}
{\bf {\bar u}}_t+{\tilde p}{\bf {\bar g}}_t,
\ena
where ${\tilde {\rho}}_{\text m}$ is the density of active mass-energy and 
${\tilde p}$ is the active pressure seen in the local rest frame of the 
fluid. Moreover ${\bf {\bar u}}_t$ is the 4-velocity of the fluid in 
$({\cal N},{\bf {\bar g}}_t)$. But what can be measured locally is not 
${\bf T}_t$ but {\em the passive stress-energy tensor} 
${\bf {\bar {\cal T}}}_t$ in $({\cal N},{\bf {\bar g}}_t)$, given by
\eqa
{\bf {\bar {\cal T}}}_t=({\rho}_{\text m}+p/c^2)
{\bf {\bar u}}_t{\otimes}{\bf {\bar u}}_t+p{\bf {\bar g}}_t,
\ena
where ${\rho}_{\text m}$ is the passive mass-energy as measured in the local
rest frame of the fluid and $p$ is the associated passive pressure. Note that
the counterpart ${\cal T}_t$ in $({\cal N},{\bf g}_t)$ of 
${\bf {\bar {\cal T}}}_t$ is given by
\eqa
{\cal T}_t={\sqrt{\frac{{\bar h}_t}{h_t}}}{\Big [}({\rho}_{\text m}+p/c^2)
{\bf u}_t{\otimes}{\bf u}_t+p{\bf g}_t{\Big ]},
\ena
where ${\bar h}_t$ and $h_t$ are the determinants of the spatial metrics
${\bf {\bar h}}_t$ and ${\bf h}_t$, respectively. Besides, the relationship 
between ${\tilde {\rho}}_{\text m}$ and ${\rho}_{\text m}$ is given by
\eqa
{\tilde {\rho}}_{\text m}=
\left\{
\begin{array}{ll}
{\frac{t}{t_0}}{\bar N}_t{\rho}_{\text m}
& \text{for a fluid of material particles,} \\ [1.5ex]
{\frac{t^2}{t_0^2}}{\bar N}_t^2{\rho}_{\text m} & \text{for the 
electromagnetic field,}
\end{array}
\right.
\ena
and similarly for the relationship between ${\tilde p}$ and $p$. In the 
following sections we set up the relevant equations both for the interior and 
the exterior gravitational field. As we shall see the equations valid inside 
the source get quite complex; this makes analytical calculations rather 
impracticable so the equations should be solved numerically. However, we have 
not performed any numerical calculations. On the other hand an exact 
solution may be found for the exterior field.
\subsection{The interior field}
In this section we analyze the gravitational field inside the source. That is,
we do the necessary analytical calculations in order to write the relevant
equations in a form appropriate for numerical treatment. Proceeding 
with this, from the definitions we get 
\eqa
c^{-2}{\bar a}_{{\cal F}r}&=&{\frac{{\bar B}'}{2{\bar B}}}, {\qquad}
c^{-2}{\bar a}_{{\cal F}r{\mid}r}={\frac{{\bar B}''}{2{\bar B}}}-
{\frac{1}{2}}{\Big (}{\frac{{\bar B}'}{{\bar B}}}{\Big )}^2-
{\frac{{\bar A}'{\bar B}'}{4{\bar A}{\bar B}}}, \nonumber \\
c^{-2}{\sin}^{-2}{\theta}{\bar a}_{{\cal F}{\phi}{\mid}{\phi}}&=&
c^{-2}{\bar a}_{{\cal F}{\theta}{\mid}{\theta}}=
{\frac{r{\bar B}'}{2{\bar A}{\bar B}}}, {\qquad}
{\bar P}_t=({\frac{t_0}{t}})^2{\frac{2}{\bar A}}{\Big (}r^{-2}({\bar A}-1)
+{\frac{{\bar A}'}{r{\bar A}}}{\Big )}, 
\ena
\eqa
c^{-2}{\bar a}^k_{{\cal F}{\mid}k}&=&({\frac{t_0}{t}})^2{\Big (}
{\frac{{\bar B}''}{2{\bar A}{\bar B}}}-{\frac{1}{2{\bar A}}}
{\Big (}{\frac{{\bar B}'}{{\bar B}}}{\Big )}^2
-{\frac{{\bar A}'{\bar B}'}{4{\bar A}^2{\bar B}}}
+{\frac{{\bar B}'}{r{\bar A}{\bar B}}}{\Big )}, {\nonumber} \\
{\bar H}_{(t)rr}&=&r^{-2}(1-{\bar A}), {\qquad}
{\sin}^{-2}{\theta}{\bar H}_{(t){\phi}{\phi}}={\bar H}_{(t){\theta}{\theta}}
=-{\frac{r{\bar A}'}{2{\bar A}^2}}.
\ena
Now active mass density varies formally as ${\bar F}_t^{-2}$ when measured 
in atomic units. For reasons of convenience we choose to extract this formal 
variability explicitly. What is left after separating out the formal 
variability from the active mass density is by definition {\em the coordinate 
volume density of active mass} ${\bar {\rho}}_{\text m}$. (The corresponding 
pressure is ${\bar p}$.) For the case when the perfect fluid is co-moving with 
the FOs (i.e., $T_{(t){\bar {\perp}}j}=0$) we find from equation (34)
\eqa
T_{(t){\bar {\perp}}{\bar {\perp}}}={\tilde {\rho}}_{\text m}c^2{\equiv}
{\frac{t_0^2}{t^2}}{\frac{{\bar {\rho}}_{\text m}c^2}{{\bar B}}}, \qquad
T_{(t)r}^r=T_{(t){\theta}}^{\theta}=T_{(t){\phi}}^{\phi}={\tilde p}{\equiv}
{\frac{t_0^2}{t^2}}{\frac{{\bar p}}{{\bar B}}}.
\ena
Furthermore, using equations (40) the local conservation laws (10), (11) 
applied to the metrically static case yield
(with ${\dot {\ }}{\equiv}{\frac{\partial}{{\partial}t}}$)
\eqa
{\dot {\bar {\rho}}}_{\text m}={\dot {\bar p}}=0, \qquad
{\bar p}'=-c^{-2}{\bar a}_{{\cal F}r}({\bar {\rho}}_{\text m}c^2-3{\bar p})=
-({\bar {\rho}}_{\text m}c^2-3{\bar p}){\frac{{\bar B}'}{2{\bar B}}}.
\ena
Equations (41) are valid for any metrically static perfect fluid. But to have 
experimental input we need to specify an equation of state 
$p=p({\rho}_{\text m})$ consistent with the metrically static condition. That 
is, equations (41) are valid only when the explicit dependence
$p({\rho}_{\text m})$ takes the form $p{\propto}{\rho}_{\text m}$ since
otherwise the equation of state will not be consistent with the given time 
evolution. Moreover, once a suitable equation of state is given it is 
necessary to use the expressions (37) and (40) relating 
${\bar {\rho}}_{\text m}$ to the passive mass density ${\rho}_{\text m}$ and 
similarly for a relationship between ${\bar p}$ and the passive pressure $p$.

To find how the active mass $m_t$ varies in space-time, note that we are free 
to choose the background value of the active mass far from the source to be 
$m_0$. We use this to define $G$ as the constant measured in a Cavendish 
experiment far from the source at epoch $t_0$. Then, using equations
(8) and (38) we find
\eqa
m_t(r,t)={\bar B}^{1/2}(r){\frac{t}{t_0}}m_0.
\ena
Now we can insert the equations (38), (39) and (40) into the field equation 
(12). Since ${\bf {\bar K}}_t$ vanishes identically for the metrically static 
case [4], equations (13) become vacuous and equation (12) yields 
\eqa
{\frac{{\bar B}''}{\bar B}}-
{\frac{1}{2}}{\Big (}{\frac{{\bar B}'}{{\bar B}}}{\Big )}^2
-{\frac{{\bar A}'{\bar B}'}{2{\bar A}{\bar B}}}+
{\frac{2{\bar B}'}{r{\bar B}}}={\kappa}{\frac{{\bar A}}{{\bar B}}}
({\bar {\rho}}_{\text m}c^2+3{\bar p}).
\ena
A second equation can be found from the equation (15) for
the spatial curvature. The radial component just gives equation (17) again
but the angular components yield a new equation, which can be combined
with equation (43) to give
\eqa
{\frac{{\bar A}'}{{\bar A}}}
-{\frac{{\bar B}'}{{\bar B}}}-{\frac{r}{2}}
{\Big (}{\frac{{\bar B}'}{{\bar B}}}{\Big )}^2-
{\frac{2r{\bar A}}{{\Xi}_0^2{\bar B}}}=
-r{\kappa}{\frac{{\bar A}}{{\bar B}}}({\bar {\rho}}_{\text m}c^2+3{\bar p}).
\ena
We may now eliminate ${\bar A}(r)$ and ${\bar A}'(r)$ from equation (43)
by inserting equations (17) and (44). The result is
\eqa
{\Big (}1-{\frac{r^2}{{\bar B}(r){\Xi}_0^2}}{\Big )}
{\frac{{\bar B}''(r)}{{\bar B}(r)}}
&-&{\Big (}1-{\frac{2r^2}{{\bar B}(r){\Xi}_0^2}}{\Big )}
{\Big (}{\frac{{\bar B}'(r)}{{\bar B}(r)}}{\Big )}^2 
+{\frac{2}{r}}{\Big (}1-{\frac{3r^2}{2{\bar B}(r){\Xi}_0^2}}{\Big )}
{\frac{{\bar B}'(r)}{{\bar B}(r)}} \nonumber \\
&& \hspace*{-6.5mm}
-{\frac{r}{4}}{\Big (}{\frac{{\bar B}'(r)}{{\bar B}(r)}}{\Big )}^3=
{\frac{{\kappa}}{{\bar B}(r)}}{\Big (}1-
{\frac{r{\bar B}'(r)}{2{\bar B}(r)}}{\Big )}^3
{\Big [}{\bar {\rho}}_{\text m}(r)c^2+3{\bar p}(r){\Big ]}.
\ena
To solve equation (45) numerically for ${\bar B}(r)$ one may proceed as 
follows. First specify the boundary conditions at the center of the body. 
From equation (17) we easily see that ${\bar A}(0)=1$, and moreover we must 
have ${\bar B}'(0)={\bar p}'(0)=0$, so ${\bar A}'(0)=0$ from equation (44). 
Furthermore, noting that $r^{-1}{\bar B}'$ must be stationary near the center 
of the body we have
\eqa
{\bar B}''(0)={\lim}_{r{\rightarrow}0}{\Big [}r^{-1}{\bar B}'(r){\Big ]}, 
\qquad \Rightarrow \qquad
{\bar B}''(0)={\frac{{\kappa}}{3}}{\Big (}{\bar {\rho}}_{\text m}(0)c^2+
3{\bar p}(0){\Big )},
\ena
where the implication follows from equation (43).

To specify any particular model, choose the boundary condition ${\bar p}(0)$ 
at some arbitrary time. Also choose some arbitrary value ${\bar B}(0)$ as an
initial value for iteration. It must be possible to check how well the chosen
${\bar B}(0)$ reproduces the boundary condition for ${\bar B}({\cal R})$ at 
the surface $r={\cal R}$ of the body. That is, to match the exterior (vacuum)
solution we must have (see the next section, equation (54))
\eqa
{\bar B}({\cal R})=1-{\frac{r_{\text s0}}{\cal R}}+
{\frac{r_{\text s0}^2}{2{\cal R}^2}}+{\frac{r_{\text s0}{\cal R}}{2{\Xi}_0^2}}
-{\frac{r_{\text s0}^3}{8{\cal R}^3}}+{\cdots},
\ena
where $r_{\text s0}$ is a constant which may be identified with the 
Schwarzschild radius of the body at the arbitrary time $t_0$. Hence, by 
definition we set
\eqa
r_{\text s0}{\equiv}{\frac{2M_{t_0}G}{c^2}}, \quad
M_t{\equiv}c^{-2}{\int}{\int}{\int}{\bar N}_t{\Big [}
T_{(t){\bar {\perp}}{\bar {\perp}}}+{\hat T}^i_{(t)i}{\Big ]}d{\bar V}_{t}
=4{\pi}{\frac{t}{t_0}}{\int}_{{\!}{\!}0}^{\cal R}
{\frac{{\sqrt{\bar A}}}{\sqrt{\bar B}}}{\Big [}{\bar {\rho}}_{\text m}+
3{\bar p}/c^2{\Big ]}r^2dr,
\ena
where the integration is taken over the whole body. (The particular form of
$M_t$ follows directly from equation (12) applied to the interior of a
spherically symmetric, metrically static source when extrapolated to the exact 
exterior solution given by equations (52) and (53) below.) Note that $M_{t_0}$ 
is defined to be the total (active) mass-energy of the body as measured by 
distant orbiters at epoch $t_0$. Notice in particular that this expression 
depends not only on the mass density of the body but also on the pressure. 

As already mentioned; to have a metrically static system it is necessary to 
specify an equation of state of the type $p{\propto}{\rho}_{\text m}$ 
(potential implicit dependences not included) since this ensures that 
${\bar {\rho}}_{\text m}$ and ${\bar p}$ are independent of $t$. Since the 
equation of state for an ideal gas has the required form and since 
non-degenerate star matter is reasonably well approximated by an ideal gas, it 
should not be too unrealistic to apply the metrically static condition 
to main sequence stars. (The metrically static condition holds in general for
an ideal gas even if the gas is not isothermal; i.e., even if the temperature
depends on the pressure so that the equation of state takes a polytropic 
form.) Once a suitable equation of state has been specified we may use 
equations (37) and (40) to find ${\bar {\rho}}_{\text m}$ and ${\bar p}$ and 
then integrate equation (45) outwards from $r=0$ using equation (41) until the
pressure vanishes. The surface of the body is now reached (that is 
$r={\cal R}$). If the calculated value ${\bar B}({\cal R})$ does not match the
boundary condition (47), add a constant to ${\bar B}$ everywhere and repeat 
the calculation with the new value of ${\bar B}(0)$. Iterate until sufficient 
accuracy is achieved.

Once we have done the above calculations for an arbitrary time we know the 
time evolution of the system from equation (41). That is, a spherical 
gravitationally bound body made of perfect fluid obeying an equation of state 
of type $p{\propto}{\rho}_{\text m}$ will expand according to the Hubble law. 
But for bodies made of perfect fluid obeying other equations of state 
(degenerate star matter for example) the expansion may induce instabilities; 
mass currents will be set up and such systems cannot be metrically static. 
However, for the metrically static case the gravitational field interior to 
the body will expand along with the fluid; this is similar to the expansion of
the exterior gravitational field found in the next section. We are now able to
calculate the family ${\bf {\bar g}}_t$ inside the body. To find the 
corresponding family ${\bf g}_t$ one uses the method described in [3, 4].

We finish this section by estimating how the cosmic expansion will affect a 
spherically symmetric body made of perfect fluid obeying an equation of state 
of the form $p{\propto}{\rho}_{\text m}^{\gamma}$ (e.g., a polytrope made of
degenerate matter). To do that, we assume that the hydrodynamical effects on 
the gravitational field coming from instabilities can be neglected; i.e., we 
assume that the body can be treated as being approximately in hydrostatic 
equilibrium for each epoch $t$. (Effects coming from gravitational heating of 
the body due to contraction are also neglected.) To justify this approximation 
we work in the Newtonian limit. That is, we first take the Newtonian limits of 
equations (41) and (45), getting
\eqa
{\frac{d}{dr}}{\frac{r^2}{{\bar {\rho}}_{\rm m}}}{\bar p}'=
-4{\pi}Gr^2{\bar {\rho}}_{\rm m0},
\ena
where ${\bar {\rho}}_{\rm m0}$ is the density field ${\bar {\rho}}_{\rm m}$ at 
epoch $t_0$. As it stands, equation (49) is valid only for the fixed epoch 
$t_0$ if ${\gamma}{\neq}1$. However, by making the substitution $r{\rightarrow}
({\frac{t_0}{t}})^{\frac{1}{3{\gamma}-4}}r{\equiv}{\bar r}$, 
${\bar {\rho}}_{\rm m0}{\rightarrow}({\frac{t}{t_0}})^{\frac{3}{3{\gamma}-4}}
{\bar {\rho}}_{\rm m0}{\equiv}{\bar {\rho}}_{\rm m}$, $G{\rightarrow}
{\frac{t}{t_0}}G{\equiv}G_t$, equation (49) may effectively be transformed from
epoch $t_0$ to epoch $t$ and it may then be applied to the body at each fixed 
epoch $t$ even if ${\gamma}{\neq}1$. Equation (49) then becomes equivalent to 
its counterpart in Newtonian theory except for a variable $G_t$. Thus the usual
Newtonian analysis of polytropes [7] applies, but with $G_t$ variable. And as 
consequences of this we see that the physical radius of a polytrope will 
actually {\em shrink} with epoch (if ${\gamma}>{\frac{4}{3}}$), and that the 
Chandrasekhar mass limit will decrease with epoch. Thus any white dwarf made 
of degenerate matter is predicted to shrink with epoch and eventually explode 
as a type Ia supernova when the Chandrasekhar mass gets close to the mass of 
the white dwarf. In particular this should happen to isolated white dwarfs, so 
according to quasi-metric theory it is not necessary to invoke mass accretion 
from exterior sources to ignite type Ia supernovae.
\subsection{The exterior field}
In this section we write down one equation which must be solved to find the 
unknown function ${\bar B}(r)$ valid for the exterior gravitational field.
We give an exact solution of this equation and use this solution to find exact
expressions for the functions ${\bar A}(r)$, $A(r)$ and $B(r)$ as well.
Now, from equations (3), (16) and (38) we get (since ${\bar N}_t=
{\sqrt{{\bar B}(r)}}$)
\eqa
{\frac{{\partial}{\bar h}_{(t)ij}}{{\partial}t}}=
{\frac{2}{t}}{\bar h}_{(t)ij}, {\qquad} y_t(t,r)=
{\frac{t_0}{t}}{\frac{c{\bar B}'(r)}{2{\bar B}(r){\sqrt{{\bar A}(r)}}}}.
\ena
Furthermore, equation (45) without source terms reads
\eqa
{\Big(}1-{\frac{r^2}{{\bar B}(r){\Xi}_0^2}}{\Big)}
{\frac{{\bar B}''(r)}{{\bar B}(r)}}&-&
{\Big(}1-{\frac{2r^2}{{\bar B}(r){\Xi}_0^2}}{\Big)}
{\Big (}{\frac{{\bar B}'(r)}{{\bar B}(r)}} {\Big )}^2 {\nonumber} \\ 
&+&{\frac{2}{r}}{\Big(}1-{\frac{3r^2}{2{\bar B}(r){\Xi}_0^2}}{\Big)}
{\frac{{\bar B}'(r)}{{\bar B}(r)}}-{\frac{r}{4}}
{\Big (}{\frac{{\bar B}'(r)}{{\bar B}(r)}}{\Big )}^3=0.
\ena
Notice that this equation may be viewed as a perturbation of the situation 
we get if we take the limit ${\Xi}_0{\rightarrow}{\infty}$. That case
was analyzed in [4], and it was found that the solution is unique. Moreover,
it turns out that the solution of equation (51) is unique as well.

Before we try to solve equation (51), it is important to notice that no
solution of it can exist on a whole FHS (except the trivial solution 
${\bar B}=$constant), according to the maximum principle
applied to a closed Riemannian 3-manifold. The reason for this is the 
particular form of the field equations, see reference [4] and references
therein for justification. This means that {\em in quasi-metric theory, 
isolated systems cannot exist except as an approximation.} 

Even if a non-trivial solution of equation (51) does not exist on a whole
FHS we may try to find a solution valid in some finite region of a FHS.
That is, we want to find a solution in the region ${\cal R}<r<{\Xi}_0$, where 
${\cal R}$ is the coordinate radius of the source. To get a unique solution of 
equation (51) it is sufficient to require that this solution has the correct
correspondence in the limit ${\Xi}_0{\rightarrow}{\infty}$. But what happens 
at the boundary ${\Xi}_0{\equiv}ct_0$ is not very interesting anyway since the 
approximation made by assuming an isolated system is physically reliable only 
if ${\frac{r}{{\Xi}_0}}{\ll}1$. 

To find an exact solution of equation (51) it is useful to rewrite it in terms
of an isotropic radial coordinate. One then finds the following exact solution 
(transformed back as a function of $r$)
\eqa
{\bar B}(r)={\Big (}{\sqrt{1+({\frac{r_{\text s0}}{2r}})^2-
{\frac{r^2}{{\Xi}_0^2}}}}-{\frac{r_{\text s0}}{2r}}{\Big )}^2+
{\frac{r^2}{{\Xi}_0^2}},
\ena
which may be inserted into equation (51) to check that it really is a 
solution. Moreover, from equations (52) and (16) we find
\eqa
{\bar A}(r)={\Big [}1+({\frac{r_{\text s0}}{2r}})^2-{\frac{r^2}{{\Xi}_0^2}}
{\Big ]}^{-1}{\bar B}(r).
\ena
For small $r$ we may write expressions (52) and (53) as series expansions, 
i.e., as perturbations around the analogous problem in a Minkowski background. 
But in contrast to the analogous case with a Minkowski background there 
exists the extra scale ${\Xi}_0$ in addition to the Schwarzschild radius 
$r_{\text s0}$ defined in equation (48). To begin with we try to model the 
gravitational field exterior to galactic-sized objects, so we may assume that 
the typical scales involved are determined by 
${\frac{r}{{\Xi}_0}}\raisebox{-2pt}{\,$\stackrel{>}{\sim}$\,}
{\frac{r_{\text s0}}{r}}$; this criterion tells how to compare the importance 
of the different terms of the series expansion. One may straightforwardly show 
that series expansions of equations (52) and (53) yield
\eqa
{\bar B}(r)&=&1-{\frac{r_{\text s0}}{r}}+{\frac{r_{\text s0}^2}{2r^2}}+
{\frac{r_{\text s0}r}{2{\Xi}_0^2}}-{\frac{r_{\text s0}^3}{8r^3}}+{\cdots},
{\qquad} {\Rightarrow} {\nonumber} \\
{\bar A}(r)&=&1-{\frac{r_{\text s0}}{r}}+
{\frac{r_{\text s0}^2}{4r^2}}+{\frac{r^2}{{\Xi}_0^2}}+{\cdots}.
\ena
To construct the family ${\bf g}_t$ as described in [3] we need the quantity
$v(r)$, which for spherically symmetric systems takes the form [4]
\eqa
v(r)={\bar y}_tr{\sqrt{{\bar h}_{(t)rr}}}={\frac{cr}{2}}
{\frac{{\bar B}'(r)}{{\bar B}(r)}}=
{\frac{r_{\text s0}}{2r}}{\frac{c}{{\sqrt{1+({\frac{r_{\text s0}}{2r}})^2-
{\frac{r^2}{{\Xi}_0^2}}}}}}=
{\frac{r_{\text s0}c}{2r}}[1+O({\frac{r_{\text s0}^2}{r^2}})].
\ena
We note that $v(r)$ does not depend on $t$. The functions $A(r)$ and $B(r)$ 
are found from the relations (valid for spherically symmetric systems [4])
\eqa
A(r)={\Big(}{\frac{1+{\frac{v(r)}{c}}}{1-{\frac{v(r)}{c}}}}{\Big)}^2
{\bar A}(r), {\qquad}
B(r)={\Big(}1-{\frac{v^2(r)}{c^2}}{\Big)}^2{\bar B}(r).
\ena
From equations (52), (53) and (56) we then get
\eqa
B(r)={\frac{(1-{\frac{r^2}{{\Xi}_0^2}})^2}{(1+({\frac{r_{\text s0}}{2r}})^2-
{\frac{r^2}{{\Xi}_0^2}})^2}}{\Big [}
{\Big (}{\sqrt{1+({\frac{r_{\text s0}}{2r}})^2-
{\frac{r^2}{{\Xi}_0^2}}}}-{\frac{r_{\text s0}}{2r}}{\Big )}^2+
{\frac{r^2}{{\Xi}_0^2}}{\Big ]},
\ena
\eqa
A(r)={\frac{{\Big (}{\sqrt{1+({\frac{r_{\text s0}}{2r}})^2-
{\frac{r^2}{{\Xi}_0^2}}}}+{\frac{r_{\text s0}}{2r}}{\Big )}^2}{1+
({\frac{r_{\text s0}}{2r}})^2-{\frac{r^2}{{\Xi}_0^2}}}}{\Big {\{ }}1+
{\frac{{\Big (}{\sqrt{1+({\frac{r_{\text s0}}{2r}})^2-
{\frac{r^2}{{\Xi}_0^2}}}}+{\frac{r_{\text s0}}{2r}}{\Big )}^2}{{\Big (}1-
{\frac{r^2}{{\Xi}_0^2}}{\Big )}^2}}{\frac{r^2}{{\Xi}_0^2}}
{\Big {\}}}.
\ena
Notice that although $B(r)$ increases for small $r$, for large $r$ it 
eventually reaches a maximum value and then decreases towards zero when 
$r{\rightarrow}{\Xi}_0$. This is merely a curious effect due to the global 
curvature of space and the unrealistic assumption that an isolated source
determines the gravitational field at cosmological distances. That is, it is 
utterly unrealistic to assume that an isolated source dominates the 
gravitational field over cosmological scales and that this source has been 
present since the beginning of time. Thus the from equation (57) inferred 
gravitational repulsion on cosmological scales is nothing but an unrealistic 
model artefact.

It is useful to have series expansions for $B(r)$ and $A(r)$. Putting
these into a family of line elements we find
\eqa
ds^2_t&=&-{\bigg(}1-{\frac{r_{\text s0}}{r}}
+{\frac{r_{\text s0}r}{2{\Xi}_0^2}}+{\frac{3r_{\text s0}^3}{8r^3}}+{\cdots}
{\bigg)}(dx^0)^2 \nonumber \\
&&+({\frac{t}{t_0}})^2{\bigg(}{\Big\{}1+{\frac{r_{\text s0}}{r}}+
{\frac{r^2}{{\Xi}_0^2}} + {\frac{r^2_{\text s0}}{4r^2}}+{\cdots}
{\Big\}}dr^2+r^2d{\Omega}^2{\bigg)}.
\ena
This expression represents the wanted metric family as a series expansion. 
Note in particular the fact that all spatial dimensions expand whereas the 
corresponding Newtonian potential $-U=-{\frac{c^2r_{\text s0}}{2r}}$ (to
Newtonian order) remains constant for a fixed FO. This means that the true 
radius of any circle orbit (i.e., with $r$ constant) increases but such that 
the orbital speed remains constant. That is, {\em the (active) mass of the 
central object as measured by distant orbiters increases to exactly balance 
the effect on circle orbit velocities of expanding circle radii.} This is not 
as outrageous as it may seem due to the extra formal variation of atomic units
built into our theory. So this result is merely a consequence of the fact that
the coupling between matter and geometry depends directly on the formal 
variation via the field equations.

What is measured by means of distant orbiters is not the ``bare'' mass $M_t$ 
itself but rather the combination $M_tG$. We have, however, {\em defined} $G$ 
to be a constant. And as might be expected, it turns out that the variation of
$M_tG$ with $t$ as inferred from equation (59) is exactly that found directly 
from the formal variation of the active mass $M_t$ with $t$ by using equations
(8) and (48). This means that the dynamically measured mass increase should 
not be taken as an indication of actual particle creation but that the general
dynamically measured mass scale should be taken to change via a linear 
increase of $M_t$ with $t$, and that this is directly reflected in the 
gravitational field of the source. That is, measured in atomic units active 
mass increases linearly with epoch in accordance with equation (48).

The dynamical measurement of the mass of the central object by means of 
distant orbiters does not represent a local test experiment. Nevertheless the 
dynamically measured mass increase thus found is just as ``real" as the 
expansion in the sense that neither should be neglected on extended scales.
This must be so since in quasi-metric relativity, the global scale increase 
and the dynamically measured mass increase are two different aspects of the 
same basic phenomenon.
\section{The effects of cosmic expansion on gravitation} 
\subsection{Shapes of orbits and rotational curves}
We now explore which kinds of free-fall orbits we get from equation (59) 
and the equations of motion. To begin with we find the shape of the rotational
curve as defined from the coordinate 3-velocities ${\bf w}_t$ of the circle 
orbits. (The 4-velocities ${\bf u}_t$ may be split up into pieces respectively
orthogonal to and intrinsic to the FHSs according to the formula
${\sqrt{1-{\frac{w^2}{c^2}}}}{\bf u}_t=c{\bf n}_t+{\bf w}_t$.) Since equation 
(32) has no time dependence for such orbits we can do a standard calculation 
[7] and the result is that orbital speed $w$ varies as
\eqa
w(r)={\frac{t}{t_0}}r{\frac{d{\phi}}{dt}}
{\sqrt{1-{\frac{w^2}{c^2}}}}{\frac{dt}{d{\tau}_t}}=
B^{-1/2}(r){\frac{t}{t_0}}r{\frac{d{\phi}}{dt}}
={\sqrt{\frac{B'(r)r}{2B(r)}}}c,
\ena
where the second step follows from the formula ${\frac{d{\tau}_t}{dt}}=
{\sqrt{B(r)-{\frac{t^2}{t^2_0}}r^2({\frac{d{\phi}}{cdt}})^2}}$ (obtained from 
equation (16) for circular motion) together with a consistency requirement.
However when we apply equation (60) to the metric family (59) we get a result 
essentially identical to the standard Keplerian rotational curve; the only 
effect of the dynamically measured mass increase and the non-kinematical 
expansion is to increase the scale but such that the shape of the rotational 
curve is unaffected. It is true that $B(r)$ as found from equation (59) 
contains a term linear in $r$ in addition to terms falling off with increasing
$r$; in reference [8] it is shown that such a linear term may be successfully 
used to model the asymptotically non-Keplerian rotational curves of spiral 
galaxies. But the numerical value of the linear term found from equation (59) 
is too small by a factor of order $10^{10}$ to be able to match the data. So 
at least the simple model considered in this paper is unable to explain the 
asymptotically non-Keplerian rotational curves of spiral galaxies from first 
principles. 

Another matter is how the time dependence in the equations of motion will
affect the time histories and shapes of more general orbits than the circle 
orbits. Clearly time histories will be affected as can be seen directly from
equation (32). However to see if this is valid for shapes as well we may insert
equation (31) into equation (32) to obtain $r$ as a function of ${\phi}$.
This yields
\eqa
{\frac{A(r)}{r^4}}{\Big(}{\frac{dr}{d{\phi}}}{\Big)}^2+{\frac{1}{r^2}}-
{\frac{1}{J^2B(r)}}=-{\frac{E}{J^2}},
\ena
and this is identical to the equation valid for the case of a single 
spherically symmetric static metric [7]. Thus the shapes of free-fall orbits
are unaffected by the global non-kinematical expansion present in the metric 
family (59).
\subsection{Expanding space and the solar system}
If one neglects the gravitational effects of the galaxy, one may try to apply 
the metric family (59) to the solar system (by using it to describe the
gravitational field of the Sun). But the solar system is not at rest with 
respect to the cosmic rest frame; this follows from the observed dipole in the 
cosmic microwave background radiation. Besides, if we identify the cosmic
rest frame with some GTCS, the FOs will be at rest with respect to this GTCS
only in the limit where the motions of local gravitational sources can be
neglected. Consequently, if we use a GTCS identified with the cosmic rest
frame, it is necessary to introduce a non-vanishing shift vector field to take
into account the motion of the solar system barycenter. And expressed in such
a GTCS the metric family describing the gravitational field of the Sun cannot 
be (approximately) metrically static since the metric coefficients will now
depend on $x^0$. However, as a good approximation, we may neglect the 
gravitational effects of the galaxy and treat the solar system as an isolated
system moving with constant velocity with respect to the cosmic rest frame.
In this case we are always able to find a GTCS in which the solar system
barycenter is at rest and where the metric coefficients do not depend on
$x^0$ by construction (when gravitational fields of other bodies than
the Sun are neglected). Furthermore it turns out that the FOs must be at
rest with respect to this new GTCS, so the shift vector field vanishes and
all calculations we have done in this paper are still valid. Thus, if we treat
the solar system as approximately isolated, we can neglect its motion with
respect to the cosmic rest frame and use the metric family (59) to describe
the gravitational field of the Sun [9]. Moreover, the solar system is so small 
that we can neglect any dependence on ${\Xi}_0$. The errors made by neglecting 
terms depending on ${\Xi}_0$ in equation (59) are insignificant since the 
typical scales involved for the solar system are 
determined by ${\frac{r}{{\Xi}_0}}\raisebox{-2pt}{\,$\stackrel{<}{\sim}$\,}
{\frac{r^3_{\text s0}}{r^3}}$. Equation (59) then takes the form
\eqa
ds^2_t=-{\Big(}1-{\frac{r_{\text s0}}{r}}+
O({\frac{r^3_{\text s0}}{r^3}}){\Big)}(dx^0)^2
+({\frac{t}{t_0}})^2{\Big(}{\{}1+{\frac{r_{\text s0}}{r}}+
O({\frac{r^2_{\text s0}}{r^2}}){\}}dr^2+r^2d{\Omega}^2{\Big)}.
\ena
Equation (61) shows that the shapes of orbits are unaffected by the expansion; 
this means that all the classical solar system tests come in just as for the 
analogous case of a Minkowski background [4]. However, we get at least one 
extra prediction (irrespective of whether or not the galactic gravitational
field can be neglected); from equation (62) we see that the effective distance 
between the Sun and any planet is predicted to have been smaller in the past. 
That is, the spatial coordinates are co-moving rather than static, thus 
distances within the solar system and its gravitational field (measured in 
atomic units) should not be static with respect to the cosmic expansion. For 
example, the distance between the Sun and the Earth at the time of its 
formation may have been about 50{\%} less than today. But since main sequence 
stars are predicted to expand according to quasi-metric theory, a small 
Earth-Sun distance should not be incompatible with paleo-climatic data, since 
the Sun is expected to have been smaller and thus dimmer in the past. 
Actually, since neither the temperature at the center of the Sun (as estimated
from the virial theorem), nor the radiation energy gradient times the mean 
free path length of a photon depend on $t$, the cosmic luminosity evolution of
the Sun should be determined from the cosmic expansion of its surface area as
long as the ideal gas approximation is sufficient. And this luminosity 
evolution exactly balances the effects of an increasing Earth-Sun distance on 
the effective solar radiation received at the Earth.

However, an obvious question is if the predicted effect of the expansion on 
the time histories of non-relativistic orbits is compatible with the observed 
motions of the planets. In order to try to answer this question it is 
illustrating to calculate how the orbit period of any planet depends on $t$. 
For simplicity consider a circular orbit $r=R=$constant. Equation (31) then 
yields
\eqa
{\frac{d{\phi}}{dt}}={\frac{t_0}{t}}B(R)R^{-2}Jc.
\ena
Now integrate equation (63) one orbit period $T<<t$ (i.e., from $t$ to $t+T$). 
The result is
\eqa
T(t)=t({\exp}{\Big [}{\frac{T_{\text{GR}}}{t_0}}{\Big ]}-1)=
{\frac{t}{t_0}}T_{\text{GR}}(1+{\frac{T_{\text{GR}}}{2t_0}}+{\cdots}), 
{\qquad} T_{\text{GR}}{\equiv}{\frac{2{\pi}R^2}{cJB(R)}},
\ena
where $T_{\text{GR}}$ is the orbit period as predicted from General Relativity.
From (64) we see that (sidereal) orbit periods are predicted to increase 
linearly with cosmic scale, i.e. 
\eqa
T(t)={\frac{t}{t_0}}T(t_0), {\qquad} {\frac{dT}{dt}}={\frac{T(t_0)}{t_0}},
\ena
and such that any ratio between periods of different orbits remains constant.
In particular equation (65) predicts that the (sidereal) year $T_{\text E}$ 
should be increasing with about 2.5 ms/yr and the martian year
$T_{\text M}$ should be increasing by about 4.7 ms per martian year at the 
present epoch. This should be consistent with observations since the 
observed difference in the synodical periods of Mars and the Earth is accurate 
to about 5 ms. 

To compare predictions coming from equation (62) against timekeeping data, one 
must also take into account the predicted cosmological contribution to the 
spin-down of the Earth. If one assumes that the gravitational source of the 
exterior field (62) is stable with respect to internal collapse (as for a
source made of ideal gas), i.e., that possible instabilities generated by the
expansion can be neglected, one may model this source as a uniformly expanding 
sphere. Due to the increase with time of active mass, the angular momenta of 
test particles moving in the exterior field (62) increase linearly with cosmic
scale. This also applies to the angular momentum $L_{\text s}$ of a spinning
source made of ideal gas [9], that is
\eqa
L_{\text s}(t)={\frac{t}{t_0}}L_{\text s}(t_0), {\qquad} 
{\frac{dL_{\text s}}{dt}}={\frac{1}{t}}L_{\text s}=(1+O(2))HL_{\text s},
\ena
where the term $O(2)$ is of post-Newtonian order and where the locally 
measured Hubble parameter $H$ is defined by $H{\equiv}{\frac{1}{Nt}}$, or 
equivalently (${\tau}_{\cal F}$ is the proper time of the local FO)
\eqa
H{\equiv}{\frac{t_0}{t}}{\frac{d}{d{\tau}_{\cal F}}}({\frac{t}{t_0}})
={\frac{ct_0}{t}}{\Big (}{\sqrt{B(r)}}{\Big )}^{-1}
{\frac{d}{dx^0}}({\frac{t}{t_0}})={\Big (}{\sqrt{B(r)}}t{\Big )}^{-1}.
\ena
Since the moment of inertia $I{\propto}MR_{\text s}^2$, where $M$ is the 
passive mass and $R_{\text s}$ is the measured radius of the sphere, we must 
have (neglecting terms of post-Newtonian order)
\eqa
{\frac{dR_{\text s}}{dt}}=HR_{\text s}, {\qquad} 
{\frac{d{\omega}_{\text s}}{dt}}=-H{\omega}_{\text s}, {\qquad} 
{\frac{dT_{\text s}}{dt}}=HT_{\text s},
\ena
where ${\omega}_{\text s}$ is the spin circle frequency and $T_{\text s}$ is 
the spin period of the sphere. (To show  equation (68), use the definition 
$L_{\text s}=I{\omega}_{\text s}$.) This means that the spin period of a
sphere made of ideal gas increases linearly with $t$ due to the cosmic 
expansion. Does this apply to the Earth as well? The Earth is not made of
ideal gas, so the cosmic expansion may induce instabilities, affecting its
(sidereal) spin period $T_{\rm sE}$. However, here we assume that the Earth's
mantle is made of a material which may be approximately modelled as a perfect
fluid obeying an equation of state close to linear. Then, if this assumption 
holds, the Earth should be expanding close to the Hubble rate according to the 
discussion following equation (49). Moreover, averaged over long time spans, 
shorter timescale effects of instabilities on $T_{\rm sE}$ should be negligible
to a good approximation. We may also assume that there is no significant tidal 
friction since given the cosmic contribution, this would be inconsistent with 
the observed so-called mean acceleration ${\dot n}_{\rm m}$ of the Moon (see 
below). We then get
\eqa
{\frac{dT_{\rm sE}}{dt}}=HT_{\rm sE}, \qquad \Rightarrow \qquad
T_{\rm sE}(t)={\frac{t}{t_0}}T_{\rm sE}(t_0).
\ena
From equation (69) we may estimate a cosmic spin-down of the Earth at the 
present epoch to be about 0.68 ms/cy (using $H{\sim}2.5{\times}
10^{-18}$ s$^{-1}$). To see if this is consistent with the assumption that
the dominant contribution is due to cosmic effects, we may compare to
results obtained from historical observations of eclipses and occultations 
from AD 1000 and onwards. These observations can be used to infer a 
lengthening of the day of about $1.4$ ms/cy [10], whereas an average 
over the last 2700 years shows a value of about $1.70$ ms/cy [11]. 
%(Note that the data show a secular quadratic divergence between the atomic 
%time scale and a time scale based on the rotation of the Earth. This is as 
%expected since the quantity ${\frac{d^2w_{\text s}}{dt^2}}$ is negligible at 
%the present epoch.) 
But the interpretation of the historical data depends on an assumed vale of 
$-26^{{\prime}{\prime}}$/cy$^2$ for the tidal contribution 
${\dot n}_{\rm tid}$ to the mean acceleration ${\dot n}_{\rm m}$ of the Moon
(moreover, other significant contributions to ${\dot n}_{\rm m}$ are neglected
without justification, see below). This value of ${\dot n}_{\rm tid}$ 
corresponds to a {\em calculated} lengthening of the day (using standard 
theory) of about $2.3$ ms/cy [11]; thus the agreement with the values inferred 
from the historical data is not very good without invoking a secular 
shortening of the length of the day of non-tidal origin. On the other hand, 
the QMF yields a value of about $-13.6^{{\prime}{\prime}}$/cy$^2$ for 
${\dot n}_{\rm m}$ (see below). Reinterpreting the historical data using this 
value yields a correction to the lenghtening of the day of about $-0.62$ 
ms/cy, i.e., the observations could indicate a lengthening of the day of 
$0.78$ ms/cy and $1.08$ ms/cy, respectively, rather than the values given 
above. This means that the values obtained from the historical observations 
are theory dependent and that the secular spin-down of the Earth may be only 
about half of the currently accepted value. Note that such a theory dependence 
also affects the comparison of equation (69) to results obtained from 
sedimentary tidal rhythmities [12].

Another quantity that can be calculated from equation (69) is the number of 
days $N_{\text y}$ in one (sidereal) year $T_E$. This is found to be constant
since
\eqa
T_{\text E}=N_{\text y}T_{\rm sE}={\frac{t}{t_0}}T_{\text E}(t_0), {\qquad} 
\Rightarrow \qquad {\frac{dN_{\text y}}{dt}}=0. 
\ena
Moreover, the number of the days $N_{\text m}$ in one (sidereal) month 
$T_{\text m}$ can be calculated similarly. We then get
\eqa
T_{\text m}=N_{\text m}T_{\rm sE}={\frac{t}{t_0}}T_{\text m}(t_0), {\qquad} 
\Rightarrow \qquad
{\frac{dN_{\text m}}{dt}}=0.
\ena
That $N_{\text y}$ and $N_{\text m}$ are predicted to be constant is not
in agreement with standard (theory dependent) interpretations of
paleo-geological data [6, 12]. (The predicted constancy of the ratio 
$N_{\text y}/N_{\text m}$ agrees well with a standard interpretation of the 
data, though.) In addition to the assumption that active masses do not vary
with time, the assumption that $T_{\text E}$ is constant is routinely used in 
the interpretation of tidal rhythmities and fossil coral growth data; in 
particular this applies to [12], where one has explicitly used this assumption 
when calculating $N_{\text y}$ from the data. However, values determined 
directly from the rhythmite record presented in [12] and the predictions given 
here usually agree within two standard deviations when the predicted variable 
length of the year is taken into account.

As mentioned above, the mean acceleration of the Moon, 
${\dot n}_{\text m}{\equiv}{\frac{d}{dt}}n_{\text m}$ (where $n_{\text m}$ 
is the mean geocentric angular velocity of the Moon as observed from its 
motion) is a very important quantity for calculating the evolution of the
Earth-Moon system. From equation (71) we find the quasi-metric prediction
\eqa
n_{\text m}(t)={\frac{d{\phi}_{\text m}}{dt}}=
{\frac{t_0}{t}}n_{\text m}(t_0), \qquad \Rightarrow \qquad
{\dot n}_{\text m}{\equiv}{\frac{d}{dt}}n_{\text m}=-Hn_{\text m},
\ena
and inserting the observed value 0.549$^{{\prime}{\prime}}$/s for 
$n_{\text m}$ at the present epoch, we get the corresponding cosmological 
contribution to ${\dot n}_{\text m}$, namely about 
$-$13.6$^{{\prime}{\prime}}$/cy$^2$. This value may be compared to the value 
$-$13.74$^{{\prime}{\prime}}$/cy$^2$ obtained from fitting LLR data to a model 
based on the lunar theory ELP [13]. Note that this second value is the 
{\em total} mean acceleration, wherein other modelled (positive) contributions 
are included. These other contributions amount to about 
12.12$^{{\prime}{\prime}}$/cy$^2$ and are mainly attributed to the secular 
variation of the solar eccentricity due to (indirect) planetary perturbations 
[13]. When these contributions are removed, one deduces a {\em tidal} 
contribution ${\dot n}_{\rm tid}$ of about 
$-$25.86$^{{\prime}{\prime}}$/cy$^2$ to ${\dot n}_{\text m}$ [13]. Similar 
values for ${\dot n}_{\rm tid}$ as inferred from LLR data have been found in, 
e.g., [14] ($-$25.9$^{{\prime}{\prime}}$/cy$^2$). We see that in absolute 
values, the quasi-metric result is smaller than the tidal term inferred from 
LLR data using standard theory. But since the non-tidal secular contributions 
to ${\dot n}_{\rm m}$ are calculated from the ELP theory [13] and not 
calculated within the quasi-metric framework, and as long as no independent
measurements exist confirming these contributions, they can be treated as 
model-dependent. Thus it is in principle possible to omit both tidal and the
traditional non-tidal secular contributions to ${\dot n}_{\rm m}$ and construct
a quasi-metric model containing only the cosmic contribution. And as shown 
above, such a model fits the data well.

We may also use Hubble's law directly to calculate the secular recession 
${\dot a}_{\rm qmr}$ of the Moon due to the global cosmic expansion; this 
yields about 3.0 cm/yr whereas the value ${\dot a}_{\rm tid}$ inferred 
from LLR using standard theory is (3.82${\pm}$0.07) cm/yr [14]. To see if the 
difference between these results can be easily explained in terms of 
model-dependence, we note that in standard theory, ${\dot n}_{\rm tid}$ 
represents the value ${\dot n}_{\text m}$ would have had if the Earth-Moon 
system were isolated. Therefore ${\dot n}_{\rm tid}$ enters into an expression 
found by taking the time derivative of Kepler's third law. On the other hand, 
the quasi-metric model includes the cosmic contribution ${\dot n}_{\rm qmr}$ 
only, so that quantity enters into a similar expression. Taking into account 
the fact that active masses increase linearly with time according to 
quasi-metric theory, we find the relationship
\eqa
{\dot a}_{\rm tid}={\frac{2}{3}}{\frac{{\dot n}_{\rm tid}}
{{\dot n}_{\rm qmr}}}{\dot a}_{\rm qmr},
\ena
which is quite consistent with the numerical values given above. It thus seems
that there is a simple explanation of the fact that ${\dot a}_{\rm tid}$ as 
inferred from LLR data using standard theory differs from ${\dot a}_{\rm qmr}$ 
as found from Hubble's law. In other words, analyzing the LLR data within the
QMF yields, to within one standard deviation, that the recession of the 
Moon follows Hubble's law.

Note once more that whereas the secular recession of the Moon and its mean 
acceleration have traditional explanations based on tidal friction, these 
explanations are not confirmed by direct evidence. That is, tidal friction is 
of nature a mesoscopic phenomenon and it should in principle be possible to 
measure the tidal energy dissipated in the Earth's oceans. But since no 
mesoscopic measurements confirming the tidal friction scenario exist so far
[15], there are no restrictions on interpreting the secular evolution of the 
Earth-Moon system as due to cosmological effects.

The apparent constancy of the sidereal year (as indicated by astronomical 
observations of the Sun and Mercury since about AD 1680) represents the
observational basis for adopting the notion that ephemeris time (i.e., the 
time scale obtained from the observed motion of the Sun) is equal to atomic 
time (plus a conventional constant), but different from so-called universal 
time (any time scale based on the rotation of the Earth). However, from 
equation (65) we see that ephemeris time should be scaled with a factor 
${\frac{t}{t_0}}$ compared to atomic time according to quasi-metric 
relativity. Moreover, from equation (69) we see that averaged over long time 
spans, universal time should also be scaled with a factor ${\frac{t}{t_0}}$
compared to atomic time, as should any conventional constant difference 
between ephemeris time and universal time. Thus the predicted effect of the 
spin-down of the Earth and the expansion of the Earth's orbit is a seemingly 
inconsistency between gravitationally measured time and time measured by an 
atomic clock. But within the Newtonian framework, any secular changes in the 
Earth-Moon system are explained in terms of tidal friction (and external 
perturbations), so seemingly secular inconsistencies between different time 
scales may be blamed on the variable rotation of the Earth. In practice this 
means introducing leap seconds. Given the fact that leap seconds are routinely 
used to adjust the length of the year, the predicted differences between 
gravitational time and atomic time should be consistent with observations. In 
particular, the extra time corresponding to an increasing year as predicted 
from the quasi-metric model may easily be hidden into the declining number of 
days in a year as predicted from standard theory.

As mentioned earlier, the predicted expansion of the Earth's orbit implies 
that the length of the year increases with about $2.5$ ms at the present 
epoch. This corresponds to a heliocentric mean angular acceleration 
${\dot n}_{\rm E}$ of about $-1.0''$/cy$^2$. But this is inconsistent with the 
calculated orbital motion of the Earth-Moon barycenter from the ELP theory, 
which yields ${\dot n}_{\rm E}$ of about $-0.040''$/cy$^2$ [13]. To see if 
this discrepancy can be explained by model-dependent assumptions inherent in 
the ELP theory, we may estimate the expansion ${\dot a}_{\rm E}$ of the 
Earth's orbit radius corresponding to the ELP value of ${\dot n}_{\rm E}$ and 
then compare this estimate to an independent result. We do the estimate by 
comparing the ELP and the quasi-metric models using an equation similar to 
equation (73). We also assume that external perturbations of the Earth's orbit 
can be neglected in the quasi-metric model. From Kepler's third law we find an 
equation similar to equation (73) relating the expansion of the Earth's orbit 
radius ${\dot a}_{\rm E}$ and ${\dot n}_{\rm E}$ as calculated from ELP theory 
to their counterparts as calculated from quasi-metric theory. Using Hubble's 
law to calculate ${\dot a}_{\rm E}$ we find a value of about $1.2{\times}10^3$ 
m/cy from quasi-metric theory. Then, using the values for ${\dot n}_{\rm E}$ 
mentioned above we estimate ${\dot a}_{\rm E}$ as calculated from ELP theory 
to be about $32$ m/cy from the relationship between said quantities. 
Interestingly, an analysis of all available radiometric measurements of 
distances between the Earth and the major planets yields a value of 
$15{\pm}4$ m/cy for ${\dot a}_{\rm E}$ [16], i.e., about half of the value 
calculated above. So, contrary to what is asserted in [16], an explanation of 
this effect based on the cosmic expansion is not at all shown to be inadequate 
since most, if not all, of the substantial difference between 
${\dot a}_{\text E}$ as calculated from Hubble's law on the one hand and that 
inferred from radiometric data on the other hand, could be due to gross 
modelling errors. Further evidence for the existence of modelling errors 
due to local cosmic expansion comes from optical observations of the Sun,
indicating an inconsistency in modern ephemerides which may be interpreted as 
an error of about $1''$/cy in $n_{\rm E}$ [17, 18].

In this section we have seen that the predicted effects of the cosmic expansion
on the Earth-Moon and solar system gravitational fields have a number of
observable consequences, none of which is shown to be in conflict with 
observations so far, even though superficially, it would seem that some are.
That is, in every case where there is an apparent conflict between quasi-metric
predictions and observations, the discrepancies can be explained in terms of 
model-dependent assumptions made when analyzing the data. In the next section
we will see that a similar situation exists for the predicted versus the
observationally inferred time variation of the gravitational ``constant''.
\subsection{The secular increase of active mass} 
In quasi-metric relativity active mass varies throughout space-time (but 
{\em not} in the Newtonian limit of the QMF, since this variation is defined 
in terms of a varying scale factor). In particular there is a secular increase 
as seen from equations (8) and (48). This is equivalent to a secular increase 
of the gravitational ``constant'' $G_{\text C}$ as measured in a local 
gravitational test experiment (e.g., a Cavendish experiment). From equation 
(8) we get the predicted time variation
\eqa
{\frac{G_{\text C},_t}{G_{\text C}}}={\frac{1}{t}}=(1+O(2))H
{\approx}8{\times}10^{-11}{\,}{\rm yr}^{-1},
\ena
for the present epoch. However, laboratory gravitational experiments are 
nowhere near the experimental accuracy needed to test this prediction. On the 
other hand, space experiments in the solar system (e.g., ranging measurements) 
and observational constraints on solar models from helioseismology are claimed 
to rule out any possible fractional time variation
of $G$ larger than about $10^{-12}$ yr$^{-1}$. See e.g. [19-21] and 
references therein. It thus may appear as the prediction (74) is in conflict 
with experiment. But as we shall see in the following, this does not follow.

To illustrate the difference between metric and quasi-metric theory when it 
comes to the effects of a varying $G$ on the equations of motion, we
note that in the weak field limit of metric theory we may set 
$G(t)=G(t_0)+{\dot G}(t_0)(t-t_0)+{\cdots}$ directly into the Newtonian 
equation of motion. For an inertial test particle this yields (using a 
Cartesian coordinate system)
\eqa
{\frac{d^2x^j}{dt^2}}=U{\Big [}t,x^k,G(t_0)(1+
{\frac{{\dot G}(t_0)}{G(t_0)}}(t-t_0)+{\cdots}){\Big ]},_j,
\ena
leading to an extra, time-dependent term in the coordinate acceleration of
objects. It is the presence of such an extra term which is ruled out to a
high degree of accuracy according to the space experiments testing the 
temporal variation of $G$. That is, one tests a combination of the predicted
changes of the solar system scale and orbit periods $T$ which are predicted to 
vary as ${\frac{{\dot T}}{T}}=-2{\frac{{\dot G}}{G}}$. This follows from
Kepler's third law since one requires that the conservation of angular 
momentum takes the form ${\dot n}=-2{\frac{\dot a}{a}}n$ for any object
with mean heliocentric angular velocity $n$, mean angular acceleration
${\dot n}$ and fractional change of orbit radius ${\frac{\dot a}{a}}$. But this
requirement is inconsistent with quasi-metric gravity since we see from 
equation (31) that the conserved quantity is given by 
${\frac{t_0}{t}}{\ell}^2n$ (where ${\ell}{\equiv}{\frac{t}{t_0}}r$ and where
corrections of post-Newtonian order have been neglected), and not by $r^2n$.
This yields ${\dot n}=-{\frac{\dot a}{a}}n$ and thus equation (74) when 
applying Kepler's third law.

Contrary to metric theory, no such extra term as shown in equation (75) is 
present in the weak field limit of quasi-metric theory since $U$ does not 
depend on $t$. An example of this can be seen from equations (59) and (62), 
where $U{\approx}{\frac{M_{t_0}G}{r}}={\frac{M_tG}{{\ell}}}$ does not depend 
on $t$. (On the other hand $U$ may depend on $x^0$, but any variation of 
$M_tG$ with $x^0$ is not (directly) due to cosmology.) However, from equations 
(65) and (74) we see that in quasi-metric theory we have ${\frac{{\dot T}}{T}}=
{\frac{{\dot G_{\text C}}}{G_{\text C}}}$. But as we have seen in section 4.2,
in combination with the predicted scale changes due to the cosmic expansion 
this is not inconsistent with observations. 

In the weak field limit of metric theory one may calculate the effects on
stellar structure coming from a possible variation of $G$. Such effects
are found by putting a variable $G$ directly into the Poisson equation.
That is, a change in $G$ directly induces a change in the Newtonian potential
yielding a change in star luminosity. Such calculated changes in luminosity 
are tightly constrained from their effects on star models, which can be
compared to observations, e.g., data obtained from helioseismology. On the
other hand, in quasi-metric gravity the effect of the secular increase of 
active mass cannot be separated from the cosmic expansion, so their total 
effect is to {\em decrease} the density (of a body made of ideal gas) with 
cosmic epoch but such that the Newtonian potential is unchanged. (To see how 
this works, recall how equation (12) reduces to equation (43) and take its 
Newtonian limit. Discover that the resulting Poisson equation is unaffected by 
the combination of expansion and increasing $G$.) Thus for a main sequence 
star there will be approximately no change in luminosity except for that due 
to the increase of scale (i.e., the increase in luminosity due to the 
increasing surface area of the expanding star).

We conclude that all space experimental tests of the secular variation of $G$ 
are based on the assumption that this variation is present explicitly in the
Newtonian potential. (This also holds for tests based on stellar structure and
in particular restrictions coming from helioseismology.) However, this 
assumption (and in particular equation (75)) does not hold in quasi-metric 
theory. Hence, the interpretations of these tests are explicitly theory 
dependent and the prediction made in equation (74) has not been shown to be
in conflict with current experimental results, despite the variety of tests
apparently showing otherwise. Finally, notice that any cosmological 
constraints on the secular variation of $G$ found within the metric framework 
are utterly irrelevant for quasi-metric theory.
\section{Conclusion}
In this paper we have shown that according to the QMF, gravitationally bound, 
metrically static systems are predicted to expand with the Universe. Interior 
to sources the metrically static condition applies whenever the equation of 
state is of the form $p{\propto}{\rho}_{\rm m}$ (fulfilled, e.g., for an ideal
gas). When it is not the global cosmic expansion is predicted to induce 
instabilities violating hydrostatic equilibrium. For any such source mass, 
currents are set up to compensate and the system cannot be metrically static. 
(An example of this is a body made of degenerate star matter, e.g. a white 
dwarf, which is predicted to {\em shrink} with epoch.)

According to the QMF, the predicted effects on gravitationally bound systems 
of the global cosmic expansion have a number of observable consequences, none 
of which has been shown to be in conflict with observations. That is, it seems
that at this time no model independent evidence exists that may rule out the 
possibility that the size of the solar system (measured in atomic units) 
expands according to the Hubble law; on the contrary the quasi-metric model 
fits some observational data more naturally than traditional models do.
(But note that predictions coming from the QMF fit these data naturally only 
as long as the data are interpreted in a manner consistent with the QMF.)

Some examples of observations being naturally explained within the non-metric 
sector of the QMF have been discussed in this paper; e.g., the spin-down of 
the Earth [10, 11], the recession of the Moon and its mean acceleration 
[13, 14], and the newly discovered secular increase of the astronomical unit 
[16]. Also the so-called ``Pioneer effect'' has a natural explanation within 
the QMF [22]. Thus the non-metric sector of the QMF has considerable 
predictive power in the solar system, since it makes it possible to explain 
from first principles a number of seemingly unrelated phenomena as different 
aspects of the same model. On the other hand, explanations of these phenomena 
coming from standard theory are invariably {\em ad hoc}; such explanations 
always involve free parameters and mechanisms invented to explain each 
phenomenon separately. Such an approach is untenable according to Occam's 
razor. 

So fact is that several observations in the solar system represent evidence 
that space-time is quasi-metric. Moreover, metric gravity (and General 
Relativity in particular) fails to address the challenge represented by these 
observations. And the main reason that this challenge has not been recognized 
as important, is that experimental gravity in the solar system is analyzed 
within a weak field formalism (the so-called PPN-formalism) where it is 
inherently assumed that space-time must be modelled as a Lorentzian manifold,
and consequently that the cosmic expansion will be unmeasurably small at the 
scale of the solar system. This situation may change in the future, when solar 
system gravitational experiments reach a precision level where the 
quasi-metric effects can no longer reasonably be ``explained'' by adding 
{\em ad hoc} hypotheses to metric theory.
\\ [4mm]
{\bf Acknowledgment} \\ [1mm]
I wish to thank Dr. K{\aa}re Olaussen for making a critical review
of the manuscript. \\ [4mm]
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\end{document}
