An Expanding Locally Anisotropic (ELA) Metric Describing Matter in an Expanding Universe

Details An Expanding Locally Anisotropic (ELA) Metric Describing Matter in an Expanding Universe An Expanding Locally Anisotropic (ELA) Metric Describing Matter in an Expanding Universe

2010-06-08

arxiv.org/abs/1006.1617v1

Phys. Lett.B684:73-76, 2010

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

6 pages; 2 figures; covers some of the derivations in arXiv:0907.0847 with corrected terminology and a new discussion of the e

Scientific paper

10.1016/j.physletb.2010.01.001

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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