The Kottler-Whittaker Metric of the Homogeneous Gravitational Field and the Assumption of the Conservation of Spacetime Areas

In the Kottler-Whittaker metric of the homogeneous gravitational field the spacetime volume is conserved. The spacetime area is conserved, as well, if only the space dimension parallel to the field is taken into account. The latter property leads to an intuitive account of geodesic motion. Starting from the so-called Rindler hyperbolae, which describe constant proper acceleration in special relativity, the coordinate transformations for the equivalent homogeneous gravitational field are developed and applied to some elementary gravitational scenarios. The geodesic equations of planar fields are analyzed and compared for the two cases of volume conserving metrics and "isotropic" metrics conserving spacetime areas. An isotropic counterpart of the Schwarzschild metric is given by Broekaert's "scalar gravitation model", which is shortly discussed.