Structure of the compact astrophysical objects in the conformally-unimodular metric

A spherically symmetric solution for a gravitational field is considered in the conformally-unimodular metric. The reason for the study of this particular gauge (i. e., conformally-unimodular metric) is its relation to the vacuum energy problem. That aim connects it to other physical phenomena (including black holes), and one could argue that they should be considered in this particular class of metrics. As the vacuum solutions, so the incompressible liquid ones are investigated. In the last case, the nonsingular «eicheon» appears as a non-point compact static object that possessed different masses and structures. Such objects are a final product of the stellar collapse, with the masses exceeding the Tolman – Oppenheimer – Volkoff limit. The term «eicheon» refers to the fundamental G. Weyl’s paper «Gravitation und Elektrizität», published, in particular in the book «Das Relativitätsprinzip. Eine Sammlung von Originalarbeiten zur Relativitätstheorie Einsteins» (Berlin, 2018), where he introduced the concept of gauge invariance (German Eichtheorie) firstly in its relation to the unified field theory. Using this term to describe the compact nonsingular astrophysical objects emphasizes the decisive role of the gauge fixing by the unimodular metric. Besides, the connotation with Eichel (acorn) stresses the twofold internal structure of an object: as a point-like in the unimodular metric and a surface in the Schwarzschild one. The radial geodesic lines are investigated in the conformally-unimodular metric, as well.