Theory of gravity admitting arbitrary choice of the energy density reference level

Five-vectors theory of gravity is proposed, which admits an arbitrary choice of the energy density reference level. This theory is formulated as the constraint theory, where the Lagrange multipliers turn out to be restricted to some class of vector fields unlike the General Relativity (GR), where they are arbitrary. The possible cosmological implication of the model proposed is that the residual vacuum fluctuations dominate during the whole evolution of the universe. That resembles the universe having a nearly linear dependence of scale factor on cosmic time.


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In the first section of the paper, GR in the conformal time gauge will be preliminarily considered.
The modification of GR in this particular gauge will be obtained in the second section. The Schwarzschild solution within the frameworks of FVT will be considered in the third section. The fourth section will demonstrate how the problem of the primary divergence 4 p M in the vacuum energy density could be solved.
Besides, the universe deceleration parameter is discussed.

GR in the conformal time gauge.
The action of a system including one scalar field  and a point particle with the rest-mass 0 m has the form [ The momentums corresponding to N and i N equal zero. Thus, the action of the system considered as the extended Hamiltonian system [19] takes the form: Variation of the action given by (4) should be taken over independently. The Hamiltonian (1) H determines evolution in an arbitrary gauge [19,20] A time derivative of some quantity is given by [19] (1) where the Poisson brackets are In particular  is its inverse metric. For the particular case of the Bianchi model, the corresponding algebra is given in [21].
One can see that the right-hand sides of Eqs. (10) turn to zero on the shell of the constraints ( ) = 0 x and ( ) = 0 x . That is this system is of the first kind in terms of the theory of constraint systems [19,20]. Using the constraint algebra (10) and Eqs.
2. Five-vectors theory of gravity. As was shown in the previous section, the GR equations of motion admit a wider surface of the constraints than that of GR itself. That requires the conformal time gauge. In other gauges, the system moving on this wider surface will leave it if 0  , and thus, one should return to GR which demands 0  . In the conformal time gauge, the system could move on the wider surface const  permanently. One could expect that in some new theory admitting this wider surface of the constraints, the restrictions on the Lagrange multipliers will appear as opposed to GR, where the Lagrange multipliers are arbitrary. Below the version of such a theory will be exposed.
The theory describes an evolution with the time  of a three-geometry defined by the metric tensor  (13) where and i are given by (6). Thus, one has five vectors: three triad vectors where 0  is assumed, otherwise we return to GR. The full system of the constraints . Evolution of the system is governed by the Hamiltonian 5 where H is given by Restrictions on the Lagrange multipliers arise because the system should remain on the shell of constraints during evolution [19,20]: (16) where summation over b is implied. On the constraint surface composed of the Poisson brackets of constraints becomes (3) where it is written in the form of four 3x3 blocks.
gives zeros for all Eqs. (16), (17), (18) result in the restrictions on the Lagrange multipliers that leads to two equations where div consists of conventional partial noncovariant derivatives. Solutions of Eqs. (20), (21) are where f , g are some vector fields, FVT is formulated in the generalized Hamiltonian form (13). To formulate it in the Lagrange form (before Lagrange multipliers fixing), one should vary the action (13) over ij  . That can be done by rewriting the term to avoid appearing the spatial derivatives of the momentums. That results in the equations expressing velocities through momentums. From these equations, one has to express the momentums through velocities and substitute them into (13). As a result, we come to the action given by (1) or (2), but with the lapse function The physical sense of FVT is very simple: the standard Einstein-Hilbert action is varied not over all the possible metrics, but over some restricted class of them. The vacuum energy problem demands that the GR invariance relatively the general coordinate transformations has to be violated. It was found that most of the field theories undergo a violation of symmetry which presents in theory initially [28]. Here we violate the general relativistic invariance restricting the class of the metrics over which the action is varied. Permitted class of the metrics is of the form The result of the restriction is schematically represented in Fig. 1. The Lagrange multipliers in GR are arbitrary, but in FVT they are restricted by (20), (21). At the same time, the constraint surface in FVT is wider than that in GR. It should be noted that the unimodular gravity [15] also uses a mechanism of metric restriction, namely, the 4-metrics with unit determinant are considered. That violates the gauge symmetry, as well.
The expression for the gravity Lagrangian density can be rewritten in terms of triads as 7 2 2 2 2 2 2 4 (3) where (3) R is the three-dimensional curvature, which can be expressed in terms of triads: where the expressions for

The empty universe and Schwarzschild solution
Let us consider an example of a spherically symmetric gravitational field, which includes asymptotically empty universe and embedded Schwarzschild metric. Spherically symmetric metric belonging the class of the FVT metrics is where the prime means the differentiation over  . Eq. (32) To guess the solution of (34), (35), (37)  r is constant of integration. As is shown in Fig. 2(a), the function  is not singular at 0 r  , but the function  describing conformal three geometry is singular. To compare this solution with the canonical Schwarzschild one R r re   (see Fig. 2 (b)). Regarding the canonical Schwarzschild solution, this picture corresponds to the exterior of the Schwarzschild radius, since 0 r  corresponds to g Rr  as it is shown in Fig. 2(b). Let us give some illustrative interpretation (Fig.3) of this fact and consider the inverse transformation () rR from the canonical Schwarzschild to the FVT metric. Let we have "holed" Schwarzschild space initially. The mapping () rR could be considered as shrinking a "hole" g Rr  to a point 0 r  , as it is shown in Fig.3. Thus, FVT repairs a "holed" Schwarzschild spacetime by shrinking a hole edge into a node 0 r  and placing a point-like particle in this node, which corresponds to the delta function term in the Lagrangian (2). The structure of static solution in FVT tell us that delta-functional sources in action have a sense only if situated in the causally reachable part of space.
It is possible to consider a test particle motion in the vicinity of  However, it will be only an academical example because, as will be shown in the next section, the quantum vacuum should be first considered explicitly.

Domination of vacuum fluctuations in the evolution of the universe.
The possibility of an arbitrary choice of the energy reference level allows omitting the huge vacuum energy [30][31][32][33][34]. But the most interesting question is what remains after this omitting [33,34]? It turns out to be that the Milne-type cosmology arises as a result of residual vacuum fluctuations. To demonstrate that fact, let us consider a particular metric: Below, the scale factor () a  will be considered as homogenous whereas the scalar field is inhomogeneous.
It should be noted, that the gravitons contribute to the vacuum energy as two minimally coupled massless scalar fields [33]. Thus, without loss of generality, the only quantum scalar field is considered here.  is the present-day dust matter density. One may see, that the constant in FVT absorbs the leading part of the vacuum energy 4 max k during the whole evolution of universe. On the other hand, its a -dependence is similar to that of radiation density and does not relate to the contribution of  having different a -dependence. In contrast, the unimodular gravity allows arbitrary cosmological constant [15], but that does not solve the vacuum energy problem at a fixed UV cut off of the comoving momentums First of all, one has to note that we handle a theory with the Big Rip occurring at 00 = a a S due to the denominator of (41). That is a higher value of the momentums cut-off results in a longer life of the universe.
There is no a theoretical upper bound on the UV cut-off, but the lower one corresponds to  One may assume that the late time universe acceleration results from the residual vacuum fluctuations of a scalar field. At least one scalar field is already discovered, that is the Higgs boson. Besides, it was shown [33] that gravitational waves should also produce the analogous effect. A linear Milne-like expansion precedes this accelerated stage of universe evolution. It is interesting that the Milne-like universes again retain the great attention. It was shown that the primordial nucleosynthesis is concordant with the 14 observational data within the framework of such models [37]. The other cosmological tests are also under discussion [38][39][40][41][42][43][44].

Conclusion.
On the one hand, the results of the paper could be considered from the abstract point of view as the possibility of a gravity theory admitting an arbitrary choice of the energy density reference level. We have introduced the surface of the constraints =0 i  and =0 i instead of the surface =0 and =0 i , and have found the Hamiltonian, which governs a system evolution along the former surface.
The FVT is completely self-consistent in terms of the theory of constrained systems [19,20]. The price is that time and space are not considered as a single 4 R manifold.
On the other hand, the remarkable property of the theory is that the main part of vacuum energy does not influence the universe evolution. However, there remains an open question regarding the contribution of masses of particles into the vacuum energy. In any case, FVT is a strong argument for the VFD model [34], which predicts accelerated universe expansion at 1 z  , and the Milne-like universe at 1 z  . Следствием теории является утверждение, что основная часть вакуумной плотности энергии не влияет на расширение вселенной, в то время как оставшаяся часть приводит к закону расширения, близкому к линейному, как у вселенной Миллна.