A Vacuum of Quantum Gravity That is Ether

: The fact that quantum gravity does not admit a co-variant vacuum state has far-reaching 1 consequences for all physics. It points out that space could not be empty, and we return to the notion 2 of an ether. Such a concept requires a preferred reference frame for, e.g., universe expansion and black 3 holes. Here, we intend to discuss vacuum and quantum gravity from three essential viewpoints: 4 universe expansion, black holes existence, and quantum decoherence.

scale factor a(η, r). Certainly, within the GR frameworks, such a model is not self-consistent [12]. Nevertheless, there exists the (1+1)-dimensional toy model [12] including a scalar field φ(τ, σ) and a scale factor a(τ, σ): where τ is a time variable, σ is a spatial variable, and prime denotes differentiation over τ. Here,30 as in GR, the fields evolve on the curved background a(τ, σ), which is, in turn, determined by the 31 fields (the only scalar field φ(τ, σ) is considered, but there could be a lot of them). The relevant 32 Hamiltonian and momentum constraints, written in the terms of momentums π(σ) ≡ δL δφ (σ) = a 2 φ , obey the constraint algebra similar to GR.

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The action (1) could be rewritten in the form of a string on the curved background [12] where ξ = {τ, σ}. The metric tensor g αβ (ξ) describes the intrinsic geometry of a (1+1)-dimensional manifold and G A B (X(ξ)) describes the geometry of a background 1 space. Taking X A = {a, φ}, and the metric tensors g µν , G A B (X) in the form of where N and N 1 are the lapse and shift functions, respectively, results in (1), which is written in the 36 particular gauge N = 1, N 1 = 0.

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The system (4) manifests invariance relatively to the reparametrization of the variables τ, σ which 38 is analog of the general transformation of coordinates in GR. Here, the (1+1)-infinite string (i.e., an 39 "immersed" space-time τ, σ) corresponds to the observable metric g µν , while the metric G A B has no a 40 physical meaning. The transformations of coordinatesτ =τ(τ, σ),σ =σ(τ, σ) reflects rather physical 41 than gauge symmetry, because one living on a string could perform some experiments allowing to 42 determine the concrete metric g µν .

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Considering quantization of the model (4), let us first imagine that the metric G A B is the Minkowski one, i.e., equals to diag{1, −1}. Then, fixing the gauge by taking g µν as the Minkowski metric results in the equation of motion for X µ in the form of a wave equation: observables, one could also measure some quantities allowing to determine the actual system of 53 reference (i.e., the gauge) 4 .

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3. Vacuum energy problem as a criterion for finding the preferred reference frame 55 In the previous section, we argued that the vacuum state for the model (4) exists when the metric 56 g µν is the Minkowski one. Thus, the vacuum state is not re-parametrically invariant. By analogy, 57 one may think that the vacuum state, which is invariant relatively to the general transformation of 58 coordinates, does not exist in QG.

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The more simple problem is to define the vacuum state on the fixed background. However, even 60 in this case, the exact vacuum state exists only for some particular space-times. In other cases, the 61 vacuum state has only an approximate meaning [15,16] and could be defined, for example, for a slowly 62 expanding universe.

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In this case, a criterion for choosing a preferred reference frame is the cosmological constant problem. This problem arises when one computes the density of vacuum energy of the quantum fields. Using the Pauli hard cutoff of the 3-momentums k max [17,18] allows expanding the zero-point energy density into the different parts The main part of this energy density 1 Here, we use the empirical cutoff of momentums k max , with a hope that some fundamental basis 71 will be found for that in the future (like noncommutative geometry [22][23][24]). On the other hand, 72 theories exist, such as the Horava-Lifshitz theory, resulting in the renormalizability of gravity. The 73 sense of renormalization is that one changes some infinite (or very large, if k max is taken finite) pieces in 74 the operator of the mean values by some finite ones. We hope that in the future, when all the necessary 75 particles will be discovered, the sum rules [25] will provide a mutual compensation of these infinite 76 (or very large) pieces. That is, k max will give the actual UV completions of QG without the need for 77 renormalization.

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To avoid at least the main part of the vacuum energy, one could try to build a theory of gravity in which the first term in (6) does not influence universe expansion. That could be done in some particular class of the metrics [7] 3 In the conventional string theory, an observer lives in the external space-time X A , which is 24-or 10-dimensional before compactification [13]. The space is endowed by the metric G A B .
where x µ = {η, x}, η is a conformal time, γ ij is a spatial metric, a = γ 1/6 is a locally defined scale factor, 79 and γ = det γ ij . As one may see, the interval (7)  Eq. (7) determines the preferred reference system in which an Ether "lives." Also, one could 83 suggest that this reference frame coincides with the reference frame in which there is no dipole Other contributors to the vacuum energy density are the terms depending on the derivatives of the universe expansion rate [28][29][30]. These terms cannot be removed by any sum rules, but they have the right order of ρ v ∼ M 2 p H 2 , where H is the Hubble constant, and allow explaining the accelerated expansion of the universe. These energy density and pressure are [28][29][30]: where, S 0 = k 2 max 8π 2 M 2 p is determined by the ultra-violet (UV) cut-off of the comoving momenta. The energy density and pressure of vacuum (8) satisfy a continuity equation and, in the expanding universe, are related by the equation of state p v = w ρ v , as Fig. 1  In Ref.
[39], the spherically symmetric solution of the Einstein equations in the unimodular metric (7) was investigated, and it was found that the finite pressure solution exists for an arbitrary Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 May 2021 doi:10.20944/preprints202105.0576.v1 large mass. As a result, there are no compact objects with an event horizon 5 , that is, an "eicheon" appears instead of a black hole [39]. In Ref.
[39], we have turned from the conformally-unimodular metric (11) to the Schwarzschild's-like metric to demonstrate that a compact object looks like a hollow sphere of the radius greater than the Schwarzschild's ones (see Fig. 2). Here, we intend to (a) A compact object of uncompressible fluid with the radius of r f in the conformally-unimodular metric (11) looks as a shell (b) with the boundaries r g < R i < R f in the Schwarzschild's type metric, where r g is a Schwarzschild's radius.
calculate the radius of a compact object of constant density in the conformally-unimodular metric in dependence on maximum pressure and density. For this aim, it is convenient to proceed, rather, from Schwarzschild's type metric to the conformally-unimodular one. For spherically symmetric space-time, the conformally-unimodular metric (7) is reduced to which looks in the spherical coordinates as ds 2 = e 2α dη 2 − dr 2 e 4λ − e −2λ r 2 dθ 2 + sin 2 θdφ 2 .
Let us compare (11) with a metric of Schwarzschild's type, which has the form The difference between the metrics (10) and (12) is that the metric (12) suggests that the circumference 97 equals 2πR. However, there is no experimental evidence for this fact in the arbitrary spherically 98 symmetric space-time. For the metric (10) the circumference is not equal to 2πr in the close vicinity 99 of the point-like mass, but this metric solves a part of the vacuum energy problem. Coordinate 100 transformation t = η, R = R(r) relates the metrics (11) and (12): Using (13), (14) in (15) to exclude λ and α yields The event horizon is a region of space-time which is causality disjointed from the rest of space-time.
where the reduced Planck mass M p = 3 4πG and r g = 3m 2πM 2 p . Further, as in [39], we will consider a 104 model of the constant density ρ(R) = ρ 0 . In this case Eqs. (17) and (18) can be integrated explicitly 105 that gives and one needs only to find a pressure, which obeys the Tolman-Volkov-Oppenheimer (TVO) equation Here, we measure density and pressure in the units of M 2 p r −2 g , which is convenient, because the mean density of Schwarzschild black hole equals 1/2, while the Tolmen-Oppenheimer-Volkov (TOV) limit R f < 9 8 r g gives the value of ρ 0 = 1 2 8 9 3 ≈ 0.35. Using for solving the TOV equation for pressure, it is possible to find B, and then solve (16) with the initial 107 condition r(R i ) = 0 to find the eicheon radius r f = r(R f ) in the conformally unimodular metric. As for the eicheon radius in Schwarzschild's type metric, it is equal R f = 3 R 3 i + 1 2ρ 0 in the units of r g , where 109 R i is an inner radius, which determines maximum pressure. The closer R i to unity the greater maximal 110 pressure. Let us to plot (see Fig. 3) calculated radius of the eicheon in the conformally unimodular 111 metric in dependence of density ρ 0 and maximum pressure, that is the pressure in the center of a solid 112 ball in the metric (11) .  Under Minkowski's background, one could write a(η, r) = (1 + Φ(η, r)).
As was shown [41], the operator of the gravitational potential in a vacuum has the correlator That means, as is shown schematically in Fig. 4, that fluctuations are similar to a random medium 122 consisting of the sources of a gravitational potential with contact interaction which influences a 123 massive moving particle. As a result, nonrelativistic massive particle waves lose their coherence.

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The difference of particle propagation in QG and QFT is illustrated in Fig. 5