Kavli Affiliate: Pau Amaro Seoane
| First 5 Authors: Kostas Tzanavaris, Pau Amaro Seoane, , ,
| Summary:
In the context of mathematical cosmology, the study of necessary and
sufficient conditions for a semi-Riemannian manifold to be a (generalised)
Robertson-Walker space-time is important. In particular, it is a requirement
for the development of initial data to reproduce or approximate the standard
cosmological model. Usually these conditions involve the Einstein field
equations, which change if one considers alternative theories of gravity or if
the coupling matter fields change. Therefore, the derivation of conditions
which do not depend on the field equations is an advantage. In this work we
present a geometric derivation of such a condition. We require the existence of
a unit vector field to distinguish at each point of space two (non-equal)
sectional curvatures. This is equivalent for the Riemann tensor to adopt a
specific form. Our geometrical approach yields a local isometry between the
space and a Robertson-Walker space of the same dimension, curvature and metric
tensor sign (the dimension of the largest subspace on which the metric tensor
is negative definite). Remarkably, if the space is simply-connected, the
isometry is global. Our result generalize to a class of spaces of non-constant
curvature the theorem that spaces of the same constant curvature, dimension and
metric tensor sign must be locally isometric. Because we do not make any
assumptions regarding field equations, matter fields or metric tensor sign, one
can readily use this result to study cosmological models within alternative
theories of gravity or with different matter fields.
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