It is suggested a metric ansatz describing matter in an expanding universe , hence interpolating between the Schwarzschild metric near central bodies of mass M and the Friedman-Lemaítre-Robertson-Walker metric for large radial coordinate , given by

\displaystyle ds ^ { 2 } = Z c ^ { 2 } dt ^ { 2 } - \frac { 1 } { Z } \left ( dr _ { 1 } - \frac { H r _ { 1 } } { c } % Z ^ { \frac { \alpha } { 2 } + \frac { 1 } { 2 } } cdt \right ) ^ { 2 } - r _ { 1 } ^ { 2 } d \Omega ,

where Z = 1 - 2 GM / ( c ^ { 2 } r _ { 1 } ) , G is the Newton constant , c is the speed of light , H = H ( t ) is the time-dependent Hubble rate , d \Omega = d \theta ^ { 2 } + \sin ^ { 2 } \theta d \varphi ^ { 2 } is the solid angle element and we are employing Schwarzschild expanding coordinates r _ { 1 } ( also known as physical coordinates for expanding space-time ) .
For constant exponent \alpha = 0 it is retrieved the isotropic McVittie metric and for \alpha = 1 it is retrieved the locally anisotropic Cosmological-Schwarzschild metric , both already discussed in the literature .
It is shown that , for constant exponent \alpha , the event horizon at the Schwarzschild radius r _ { 1 } = 2 GM / c ^ { 2 } is only singularity free for \alpha \geq 3 and space-time is asymptotically flat for \alpha > 5 which excludes these known cases .
Also it is shown that , to strictly maintain the Schwarzschild mass pole at the origin r _ { 1 } = 0 without the presence of more severe singularities , hence describing a complete space-time with finite total mass-energy within a shell of finite radius , it is required a radial coordinate dependent exponent \alpha ( r _ { 1 } ) = \alpha _ { 0 } + \alpha _ { 1 } 2 GM / ( c ^ { 2 } r _ { 1 } ) with a negative coefficient \alpha _ { 1 } < 0 such that at the event horizon , for \alpha _ { 0 } - | \alpha _ { 1 } | \geq 3 space-time is singularity free and for \alpha _ { 0 } - | \alpha _ { 1 } | > 5 space time is asymptotically flat .
This metric may solve the long standing puzzle of describing local matter distributions in an expanding universe firstly addressed by McVittie .

The curvature , curvature invariant and stress-energy tensor are analyzed in detail being derived the allowed bounds for the parameters \alpha _ { 0 } and \alpha _ { 1 } that allow simultaneously space-time to be singularity free ( except for the Schwarzschild mass pole at the origin ) and the mass-energy density to be positive definite outside the event horizon .
It is shown that , although space-time is locally anisotropic near the mass M , isotropy at spatial infinity is maintained .
This characteristic is qualitatively consistent both with the experimental evidence of local anisotropy due to matter structures and global spatial isotropy .
The modified Newton law for this metric is derived being shown that for planetary scales the usual General Relativity Newton law is approximately maintained for the full allowed range of the parameter \alpha _ { 0 } while for galaxy scales and large values of the parameter \alpha _ { 0 } > c ^ { 2 } / \sqrt { q _ { 0 } ( GM H _ { 0 } ) ^ { 2 } } there is a significant deviation from this law which may contribute , for instance , for the flattening of galaxy velocity curves , hence allowing , at least partially , to describe dark matter effects , here interpreted as due to the universe expansion .
Are derived solutions for planetary orbits both in the circular orbit and perturbative static elliptic orbits approximations and estimated the orbital precession , period corrections and time variation of the orbital radius which are well within the existing experimental bounds for the solar system .