On spacetime models with an isotropic Hubble law

Details On spacetime models with an isotropic Hubble law On spacetime models with an isotropic Hubble law

Aug 1999

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1999cqgra..16.2559h&link_type=abstract

Classical and Quantum Gravity, Volume 16, Issue 8, pp. 2559-2576 (1999).

Physics

7

Scientific paper

We consider a Lorentzian manifold (icons/Journals/Common/calM" ALT="calM" ALIGN="TOP"/>,g) with an observer field (timelike vector field) V. Along each lightlike geodesic the redshift z and the angular diameter distance D are then well defined functions. (Instead of the angular diameter distance one may equivalently use the corrected or uncorrected luminosity distance.) (icons/Journals/Common/calM" ALT="calM" ALIGN="TOP"/>,g,V) is said to admit an isotropic Hubble law at picons/Journals/Common/in" ALT="in" ALIGN="TOP"/>icons/Journals/Common/calM" ALT="calM" ALIGN="TOP"/> if all past-oriented lightlike geodesics issuing from p yield the same z-D relation. For infinitesimally short lightlike geodesics (i.e. to within a linear approximation with respect to the variable D) it is well known that an isotropic Hubble law holds at all points picons/Journals/Common/in" ALT="in" ALIGN="TOP"/>icons/Journals/Common/calM" ALT="calM" ALIGN="TOP"/> if and only if V is freely falling and shear-free. Here we derive the necessary and sufficient conditions for an isotropic Hubble law beyond the linear regime. To that end we expand the z-D relation in a Taylor series (Kristian-Sachs series) and we investigate the validity of an isotropic Hubble law order by order. In particular, we prove that (icons/Journals/Common/calM" ALT="calM" ALIGN="TOP"/>,g,V) admits an isotropic Hubble law of third order at every point picons/Journals/Common/in" ALT="in" ALIGN="TOP"/>icons/Journals/Common/calM" ALT="calM" ALIGN="TOP"/> if and only if (icons/Journals/Common/calM" ALT="calM" ALIGN="TOP"/>,g,V) is either redshift-free or a Robertson-Walker model (locally around every point picons/Journals/Common/in" ALT="in" ALIGN="TOP"/>icons/Journals/Common/calM" ALT="calM" ALIGN="TOP"/>).

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