Bending the Bruhat-Tits tree. Part I. Tensor network and emergent Einstein equations

As an extended companion paper to [1], we elaborate in detail how the tensor network construction of a p-adic CFT encodes geometric information of a dual geometry even as we deform the CFT away from the fixed point by finding a way to assign distances to the tensor network. In fact we demonstrate that a unique (up to normalizations) emergent graph Einstein equation is satisfied by the geometric data encoded in the tensor network, and the graph Einstein tensor automatically recovers the known proposal in the mathematics literature, at least perturbatively order by order in the deformation away from the pure Bruhat-Tits Tree geometry dual to pure CFTs. Once the dust settles, it becomes apparent that the assigned distance indeed corresponds to some Fisher metric between quantum states encoding expectation values of bulk fields in one higher dimension. This is perhaps a first quantitative demonstration that a concrete Einstein equation can be extracted directly from the tensor network, albeit in the simplified setting of the p-adic AdS/CFT.

PSEUDO-RIEMANNIAN SPACES WITH A SPECIAL RIEMANN TENSOR

The paper considers pseudo-Riemannian spaces, the Riemann tensor of which has a special structure. The structure of the Riemann tensor is given as a combination of special symmetric and obliquely symmetric tensors. Tensors are selected so that the results can be applied in the theory of geodetic mappings, the theory of holomorphic-projective mappings of Kähler spaces, as well as other problems arising in differential geometry and its application in general relativity, mechanics and other fields. Through the internal objects of pseudo-Riemannian space, others are determined, which are studied depending on what problems are solved in the study of pseudo-Riemannian spaces. By imposing algebraic or differential constraints on internal objects, we obtain special spaces. In particular, if constraints are imposed on the metric we will have equidistant spaces. If on the Ricci tensor, we obtain spaces that allow φ (Ric)-vector fields, and if on the Einstein tensor, we have almost Einstein spaces. The paper studies pseudo-Riemannian spaces with a special structure of the curvature tensor, which were introduced into consideration in I. Mulin paper. Note that in his work these spaces were studied only with the requirement of positive definiteness of the metric. The proposed approach to the specialization of pseudo-Riemannian spaces is interesting by combining algebraic requirements for the Riemann tensor with differential requirements for its components. In this paper, the research is conducted in tensor form, without restrictions on the sign of the metric. Depending on the structure of the Riemann tensor, there are three special types of pseudo-Riemannian spaces. The properties which, if necessary, satisfy the Richie tensors of pseudoriman space and the tensors which determine the structure of the curvature tensor are studied. In all cases, it is proved that special tensors satisfy the commutation conditions together with the Ricci tensor. The importance and usefulness of such conditions for the study of pseudo-Riemannian spaces is widely known. Obviously, the results can be extended to Einstein tensors. Proven theorems allow us to effectively investigate spaces with constraints on the Ricci tensor.

Novel thick brane solutions with U(1) symmetry breaking

In this work, using two scalar fields ($$\phi $$ ϕ , $$\psi $$ ψ ) coupled to 4 + 1 dimensional gravity, we construct novel topological brane solutions through an explicit U(1) symmetry breaking term. The potential of this model is constructed so that two distinct degenerate vacua in the $$\phi $$ ϕ field exist, in analogy to the $$\phi ^{4}$$ ϕ 4 potential. Therefore, brane solutions appear due to the vacuum structure of the $$\phi $$ ϕ field. However, the topology and vacuum structure in the $$\psi $$ ψ direction depends on the symmetry breaking parameter $$\beta ^{2}$$ β 2 , which leads to different types of branes. As a result, one can interpret the present model as a combination of a $$\phi ^{4}$$ ϕ 4 brane with an auxiliary field, which leads to deviations from the $$\phi ^{4}$$ ϕ 4 system with the brane achieving a richer internal structure. Furthermore, we analyse in detail the behaviour of the superpotentials, the warp factors, the Ricci and Kretschmann scalars and the Einstein tensor components. In addition to this, we explore the stability of the brane in terms of the free parameters of the model. The analysis presented here complements previous work and is sufficiently novel to be interesting.

Construction of the cosmological model with periodically distributed inhomogeneities with growing amplitude

We construct an approximate solution to the cosmological perturbation theory around Einstein–de Sitter background up to the fourth-order perturbations. This could be done with the help of the specific symmetry condition imposed on the metric, from which follows that the model density forms an infinite, cubic lattice. To verify the convergence of the perturbative construction, we express the resulting metric as a polynomial in the perturbative parameter and calculate the exact Einstein tensor. In our model, it seems that physical quantities averaged over large scales overlap with the respective Einstein–de Sitter prediction, while local observables could differ significantly from their background counterparts. As an example, we analyze the behavior of the local measurements of the Hubble constant and compare them with the Hubble constant of the homogeneous background model. A difference between these quantities is important in the context of a current Hubble tension problem.

The geometrical origin of dark energy

The geometrical formulation of the quantum Hamilton–Jacobi theory shows that the quantum potential is never trivial, so that it plays the rôle of intrinsic energy. Such a key property selects the Wheeler–DeWitt (WDW) quantum potential $$Q[g_{jk}]$$ Q [ g jk ] as the natural candidate for the dark energy. This leads to the WDW Hamilton–Jacobi equation with a vanishing kinetic term, and with the identification $$\begin{aligned} \Lambda =-\frac{\kappa ^2}{\sqrt{{\bar{g}}}}Q[g_{jk}]. \end{aligned}$$ Λ = - κ 2 g ¯ Q [ g jk ] . This shows that the cosmological constant is a quantum correction of the Einstein tensor, reminiscent of the von Weizsäcker correction to the kinetic term of the Thomas–Fermi theory. The quantum potential also defines the Madelung pressure tensor. The geometrical origin of the vacuum energy density, a strictly non-perturbative phenomenon, provides strong evidence that it is due to a graviton condensate. Time independence of the regularized WDW equation suggests that the ratio between the Planck length and the Hubble radius may be a time constant, providing an infrared/ultraviolet duality. We speculate that such a duality is related to the local to global geometry theorems for constant curvatures, showing that understanding the universe geometry is crucial for a formulation of Quantum Gravity.

Compatibility conditions of continua using Riemann–Cartan geometry

The compatibility conditions for generalised continua are studied in the framework of differential geometry, in particular Riemann–Cartan geometry. We show that Vallée’s compatibility condition in linear elasticity theory is equivalent to the vanishing of the three-dimensional Einstein tensor. Moreover, we show that the compatibility condition satisfied by Nye’s tensor also arises from the three-dimensional Einstein tensor, which appears to play a pivotal role in continuum mechanics not mentioned before. We discuss further compatibility conditions that can be obtained using our geometrical approach and apply it to the microcontinuum theories.

Constraints on scalar–tensor theory of gravity by solar system tests

We study the motion of particles in the background of a scalar–tensor theory of gravity in which the scalar field is kinetically coupled to the Einstein tensor. We constrain the value of the derivative parameter z through solar system tests. By considering the perihelion precession we obtain the constraint $$\sqrt{z}/m_{\mathrm{p}} > 2.6\times 10^{12}$$ z / m p > 2.6 × 10 12 m, the gravitational redshift $$\frac{\sqrt{z}}{m_{\mathrm{p}}}>2.7\times 10^{\,10}$$ z m p > 2.7 × 10 10 m, the deflection of light $$\sqrt{z}/m_{\mathrm{p}} > 1.6 \times 10^{11}$$ z / m p > 1.6 × 10 11 m, and the gravitational time delay $$\sqrt{z}/m_{\mathrm{p}} > 7.9 \times 10^{12}$$ z / m p > 7.9 × 10 12 m; thereby, our results show that it is possible to constrain the value of the z parameter in agreement with the observational tests that have been considered.

Some Properties of Generalized Einstein Tensor for a Pseudo-Ricci Symmetric Manifold

The object of the paper is to study some properties of the generalized Einstein tensor GX,Y which is recurrent and birecurrent on pseudo-Ricci symmetric manifolds PRSn. Considering the generalized Einstein tensor GX,Y as birecurrent but not recurrent, we state some theorems on the necessary and sufficient conditions for the birecurrency tensor of GX,Y to be symmetric.

The perturbed FLRW metric on all scales: Newtonian limit and top-hat collapse

The applicability of a linearized perturbed FLRW metric to the late, lumpy universe has been subject to debate. We consider in an elementary way the Newtonian limit of the Einstein equations with this ansatz for the case of structure formation in late-time cosmology, on small and on large scales, and argue that linearizing the Einstein tensor produces only a small error down to arbitrarily small, decoupled scales (e.g. solar system scales). On subhorizon patches, the metric scale factor becomes a coordinate choice equivalent to choosing the spatial curvature, and not a sign that the FLRW metric cannot perturbatively accommodate very different local physical expansion rates of matter; we distinguish these concepts, and show that they merge on large scales for the Newtonian limit to be globally valid. Furthermore, on subhorizon scales, a perturbed FLRW metric ansatz does not already imply assumptions on isotropy, and effects beyond a FLRW background, including those potentially caused by nonlinearities of GR, may be encoded into nontrivial boundary conditions. The corresponding cosmologies have already been developed in a Newtonian setting by Heckmann and Schücking and none of these boundary conditions can explain the accelerated expansion of the universe. Our analysis of the field equations is confirmed on the level of solutions by an example of pedagogical value, comparing a collapsing top-hat overdensity (embedded into a cosmological background) treated in such perturbative manner to the corresponding exact solution of GR, where we find good agreement even in the regimes of strong density contrast

Constant-roll inflation driven by a scalar field with nonminimal derivative coupling

In this work, we study constant-roll inflation driven by a scalar field with nonminimal derivative coupling to gravity, via the Einstein tensor. This model contains a free parameter, [Formula: see text], which quantifies the nonminimal derivative coupling and a parameter [Formula: see text] which characterizes the constant-roll condition. In this scenario, using the Hamilton–Jacobi-like formalism, an ansatz for the Hubble parameter (as a function of the scalar field) and some restrictions on the model parameters, we found new exact solutions for the inflaton potential which include power-law, de Sitter, quadratic hilltop and natural inflation, among others. Additionally, a phase-space analysis was performed and it is shown that the exact solutions associated to natural inflation and a “cosh-type” potential, are attractors.