An expanding locally anisotropic (ELA) metric describing matter in an expanding universe
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: ds2=Zc2dt2-1Z(dcdt)2-r12dΩ, where Z=1-U with U=2GM/(c2r), G is the Newton constant, c is the speed of light, H=H(t)=a˙/a is the time-dependent Hubble rate, dΩ=dθ2+sin2θdφ2 is the solid angle element, a is the universe scale factor and we are employing the coordinates r=ar, being r the radial coordinate for which the RW metric is diagonal. For constant exponent α=α=0 it is retrieved the isotropic McVittie (McV) metric and for α=α=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 α=α>1 exists an event horizon at the SC radius r=2GM/c2 and only for α=α⩾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 α=α>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 r=0 without the presence of more severe singularities it is required a radial coordinate-dependent correction to the exponent α(r)=α+α2GM/(c2r) with a negative coefficient α<0. The energy-momentum density, pressures and equation of state are discussed.
- Publication:
-
Physics Letters B
- Pub Date:
- February 2010
- DOI:
- 10.1016/j.physletb.2010.01.001
- arXiv:
- arXiv:1006.1617
- Bibcode:
- 2010PhLB..684...73C
- Keywords:
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- General Relativity and Quantum Cosmology;
- Astrophysics - Cosmology and Nongalactic Astrophysics;
- High Energy Physics - Theory
- E-Print:
- 6 pages