Castelo Ferreira, P. (2010), "An Expanding Locally Anisotropic (ELA) Metric Describing Matter in an Expanding Universe", PLB, 684: 73-76.
Abstract: 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.
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