  A thorough study and analysis on the conceptual foundations of unimodular
gravity shows that this theory is essentially general relativity disguised as
unimodular relativity in the literature. The main reason for this dilemma is
accepting the Einsteinian paradigm: general relativistic framework, covariant
divergence law for matter energy-momentum tensor, and cosmological constant as
an integration constant, but introducing the artefact of unimodular description
absent in Einstein's work. A new approach is proposed in this paper in which
pure unimodular relativity is defined in terms of equi-projective geodesics
with the fundamental metric tensor having determinant unity and the geometric
tensors constructed from them. Modification of covariant divergence law for the
matter energy-momentum tensor is shown to have two new consequences. In the
conventional unimodular gravity an effective cosmological term comprising of
two variable scalar fields, namely, the unimodular geometric ambiguity and
unimodular matter energy ambiguity, is proposed. A radical departure on the
cosmological constant problem is possible assuming differing evolution of the
two scalars: the Einstein equations emerge when the two ambiguities cancel each
other. Secondly, in the case of pure unimodular relativity the gravitational
field equations are proposed consistent with the unimodular space-time
structure.
