An Expanding Locally Anisotropic (ELA) Metric Describing Matter in an
  Expanding Universe
  It is suggested an expanding locally anisotropic metric (ELA) ansatz
describing matter in a flat expanding universe which interpolates between the
Schwarzschild (SC) metric near point-like central bodies of mass 'M' and the
Robertson-Walker (RW) metric for large radial coordinate: 'ds^2=Z(cdt)2 - 1/Z
(dr1-(Hr1/c) Z^(alpha/2+1/2)(cdt))^2-r1^2 dOmega', where 'Z=1-U' with
'U=2GM/(c^2r1)', 'G' is the Newton constant, 'c' is the speed of light,
'H=H(t)=\dot(a)/a' is the time-dependent Hubble rate,
'dOmega=dtheta^2+sin^2(theta) dvarphi^2' is the solid angle element, 'a' is the
universe scale factor and we are employing the coordinates 'r1=ar', being 'r'
the radial coordinate for which the RW metric is diagonal. For constant
exponent 'alpha=alpha0=0' it is retrieved the isotropic McVittie (McV) metric
and for 'alpha=alpha0=1' it is retrieved the locally anisotropic
Cosmological-Schwarzschild (SCS) metric, both already discussed in the
literature. However it is shown that only for constant exponent 'alpha=alpha0>
1' exists an event horizon at the SC radius 'r1=2GM/c^2' and only for
'alpha=alpha0>= 3' space-time is singularity free for this value of the radius.
These bounds exclude the previous existing metrics, for which the SC radius is
a naked extended singularity. In addition it is shown that for 'alpha=alpha0>5'
space-time is approximately Ricci flat in a neighborhood of the event horizon
such that the SC metric is a good approximation in this neighborhood. It is
further shown that to strictly maintain the SC mass pole at the origin 'r1=0'
without the presence of more severe singularities it is required a radial
coordinate dependent correction to the exponent 'alpha(r1)=alpha0+alpha1
'2GM/(c^2 r1)' with a negative coefficient 'alpha1<0'. The energy-momentum
density, pressures and equation of state are discussed.
