The main conclusion of long-standing discussions concerning the role of solutions with degenerate metric ( g \equiv det ( g _ { \mu \nu } ) = 0 and even with g _ { \mu \nu } = 0 ) was that in the first order formalism they are physically acceptable and must be included in the path integral .
In particular , they may describe topology changes and reduction of ” metrical dimension ” of space-time .
The latter implies disappearance of the volume element \sqrt { - g } d ^ { 4 } x of a 4-D space-time in a neighborhood of the point with g = 0 .
We pay attention that besides \sqrt { - g } , the 4-D space-time differentiable manifold possesses also a ” manifold volume measure ” ( MVM ) described by a 4-form which is sign indefinite and generically independent of the metric .
The first order formalism proceeds with originally independent connection and metric structures of the space-time manifold .
In this paper we bring up the question whether the first order formalism should be supplemented with degrees of freedom of the space-time differentiable manifold itself , e.g .
by means of the MVM .
It turns out that adding the MVM degrees of freedom to the action principle in the first order formalism one can realize very interesting dynamics .
Such Two Measures Field Theory enables radically new approaches to resolution of the cosmological constant problem .
We show that fine tuning free solutions describing a transition to \Lambda = 0 state involve oscillations of g _ { \mu \nu } and MVM around zero .
The latter can be treated as a dynamics involving changes of orientation of the space-time manifold .
As we have shown earlier , in realistic scale invariant models ( SIM ) , solutions formulated in the Einstein frame satisfy all existing tests of General Relativity ( GR ) .
Here we reveal surprisingly that in SIM , all ground state solutions with \Lambda \neq 0 appear to be degenerate either in g _ { 00 } or in MVM .
Sign indefiniteness of MVM in a natural way yields a dynamical realization of a phantom cosmology ( w < -1 ) .
It is very important that for all solutions , the metric tensor rewritten in the Einstein frame has regularity properties exactly as in GR .
We discuss new physical effects which arise from this theory and in particular strong gravity effect in high energy physics experiments .