  Recently we have introduced a nonrelativistic cosmological model (NRCM)
exhibiting a dynamical spatial curvature. For this model the present day cosmic
acceleration is not attributed to a negative pressure (dark energy) but it is
driven by a nontrivial energy flow leading to a negative spatial curvature. In
this paper we generalize the NRCM in two different ways to the relativistic
regime and present analytical solutions of the corresponding Einstein
equations. These relativistic models are characterized by two inequivalent
extensions of the FLWR metric with a time-dependent curvature function $K (t)$
and an expansion scalar $a(t)$. The fluid flow is supposed to be geodesic. The
model V1 is shear-free with isotropic pressure and therefore conformal flat. In
contrast to V1 the second model V2 shows a nontrivial shear and an anisotropic
pressure. For both models the inhomogeneous solutions of the corresponding
Einstein equations will agree in leading order at small distances with the NRCM
if a(t) and K(t) are each identical with those determined in the NCRM. Then the
metric is completely fixed by three constants. The arising energy momentum
tensor contains a nontrivial energy flow vector. Our models violate locally the
weak energy condition. Global volume averaging leads to explicit expressions
for the effective scale factor and the expansion rate $H (z)$. Backreaction
effects cancel each other for the model V2 but they are nonzero and
proportional to the square of the magnitude of the energy flow for the model
V1. The large scale (relativistic) corrections to the NCRM results are small
for the model V2 for a small-sized energy flow. We have reproduced a
corresponding adjustment of the three free constants from [1] to cosmic
chronometer data leading to the prediction of an almost constant, negative
value for the dimensionless curvature function $k(z) \sim - 1$ for redshifts $z
< 2$.
