arXiv subjects

V. M. Khatsymovsky

Publications and source records attributed to V. M. Khatsymovsky.

At least 19 recordsLinked to original sources

arXiv2026

Discrete gravitational diagram technique and corrections to the Newtonian potential

Starting from simplicial Regge gravity, we use a bell-shaped form of the measure obtained using functional integration over connection. A "hypercubic" structure is considered (some variables are frozen), it is described by the metric $g_{λμ}$ at the sites. The metric is parameterized to make the measure Lebesgue. The linear part of this parametrization leads to a discrete form of standard Feynman diagrams that approximates finite continuum diagrams and is finite for infinite ones; the nonlinear part gives new vertices and diagrams. The maximum of the measure is at the edge length scale $b = b_{\rm s} \sim η^{1 / 2}$, where $η$ defines the free factor like $ ( - \det \| g_{λμ} \| )^{ η/ 2}$ in the measure and should be a large parameter to ensure true action upon integration over connection. For general perturbative expansion (including both that for the measure and S matrix) to be free of increasing powers of $η$, its starting point must be at $b$ sufficiently close to $b_{\rm s}$; this appears to be a dynamic mechanism for establishing $b$ as an optimal starting point of the perturbative expansion. We use a discrete version of the soft synchronous gauge in the principal value type prescription we discuss in a recent paper (with a refined finite-difference form of the action to match the analytical properties of the propagator to the continuum case). This allows one to fix the timelike length scale at a low level for which the measure is known in closed form. This technique is applied to Newton's potential. Some of new diagrams, including potentially large ones, are mutually cancelled. The S matrix expansion is analyzed to consist of standard diagrams. These diagrams form series with a small parameter $η^{- 1}$; one-loop diagrams calculated in the literature represent the leading order.

gr-qc↗

arXiv2026

Towards the consistent perturbative expansion in discrete gravity

We consider correctly defining the perturbative expansion in a discrete gravity (simplicial or Regge calculus) needed to study physical effects like graviton loop corrections to Newton's potential. For the symmetric derivative $Δ^{(s)}_λ=i\sin p_λ$ in the finite-difference action, the propagator has a graviton pole at $\sin^2p_0=\sum^3_{α=1}\sin^2p_α$, or, at small $p_α$, at $p_0$ close to 0 or $\pmπ$. This pole doubling means doubling the result of integration over d$p_0$ compared to the continuum. The usual derivative $Δ_λ=\exp(ip_λ)-1$ leads to a tricky analytical structure of the propagator, since $Δ_λ\neq-\barΔ_λ$, and again to a discrepancy with the continuum. The way out is to use an action $\check{S}_{\rm g}$ with both $Δ^{(s)}_λ$ and $Δ_λ$ and the synchronous gauge $g_{0λ}=g_{0λ}^{(0)}$ (implemented by adding a term bilinear in $n^λ(g_{λμ}-g_{λμ}^{(0)})$, $n^λ=[1,-\varepsilon(Δ^{(s)α}Δ^{(s)}_α)^{-1}Δ^{(s)β}]$, $\varepsilon\to0$, thus removing singularities at $p_0=0$). Given the propagator $\check{G}(n,\bar{n})$, we form a principal value propagator $[\check{G}(n,n)+\check{G}(\bar{n},\bar{n})]/2$ by analytically continuing from real $n=\bar{n}$. Singularities are resolved like $p_0^{-j}\to[(p_0+i\varepsilon)^{-j}+(p_0-i\varepsilon)^{-j}]/2$ leading to separate diagram finiteness at $\varepsilon\to0$. We analyze a 1-parameter family of actions differing in using $Δ_λ$ vs $Δ^{(s)}_λ$, find the only one reproducing convergent continuum diagrams for small external momenta (which is natural to demand from discretization), consider finiteness of the principal value gauge-fixing term and vanishing ghost contribution. The analysis is illustrated by the electromagnetic (Yang-Mills) case.

gr-qc↗

arXiv2024

Soft synchronous gauge: principal value prescription

The synchronous gauge in gravity ($g_{0 λ} = - δ_{0 λ}$) is ill-defined due to the singularity at $p_0 = 0$ in the graviton propagator. Previously we studied "softening" this gauge by considering instead the gauge $n^λg_{λμ} = 0$, $n^λ= (1, - \varepsilon (\partial^j \partial_j )^{- 1} \partial^k ) $ in the limit $\varepsilon \to 0$. We now explore the possibility of using a principal value prescription (not in the standard Cauchy sense), which amounts, roughly speaking, to replacing singularities $p_0^{-j} \Rightarrow [ (p_0 + i \varepsilon )^{-j} + (p_0 - i \varepsilon )^{-j} ] / 2$, which then behave like distributions. We show that such a propagator follows upon adding to the action a gauge-violating term of a general form, which reduces to $ \sim \int f_λΛ^{λμ} f_μ\d^4 x $ with a constant operator $Λ^{λμ}$ depending on $\partial$ and a metric functional $f_λ$. The contribution of the ghost fields to the effective action is analysed. For the required intermediate regularization, the discrete structure of the theory at small distances is implied. It is shown that the ghost contribution can be disregarded in the limit $ \varepsilon \to 0$.

hep-th↗

arXiv2024

Soft synchronous gauge in the perturbative gravity

An attempt to directly use the synchronous gauge ($g_{0 λ} = - δ_{0 λ}$) in perturbative gravity leads to a singularity at $p_0 = 0$ in the graviton propagator. This is similar to the singularity in the propagator for Yang-Mills fields $A^a_λ$ in the temporal gauge ($A^a_0 = 0$). There the singularity was softened, obtaining this gauge as the limit at $\varepsilon \to 0$ of the gauge $n^λA^a_λ= 0$, $n^λ= (1, - \varepsilon (\partial^j \partial_j )^{- 1} \partial^k ) $. Then the singularities at $p_0 = 0$ are replaced by negative powers of $p_0 \pm i \varepsilon$, and thus we bypass these poles in a certain way. Now consider a similar condition on $n^λg_{λμ}$ in perturbative gravity, which becomes the synchronous gauge at $\varepsilon \to 0$. Unlike the Yang-Mills case, the contribution of the Faddeev-Popov ghosts to the effective action is nonzero, and we calculate it. In this calculation, an intermediate regularization is needed, and we assume the discrete structure of the theory at short distances for that. The effect of this contribution is to change the functional integral measure or, for example, to add non-pole terms to the propagator. This contribution vanishes at $\varepsilon \to 0$. Thus, we effectively have the synchronous gauge with the resolved singularities at $p_0 = 0$, where only the physical components $g_{j k}$ are active and there is no need to calculate the ghost contribution.

hep-th↗

arXiv2023

On the discrete version of the Kerr-Newman solution

This paper continues our work on black holes in the framework of the Regge calculus, where the discrete version (with a certain edge length scale $b$ proportional to the Planck scale) of the classical solution emerges as an optimal starting point for the perturbative expansion after functional integration over the connection, with the singularity resolved. An interest in the present discrete Kerr-Newman type solution (with the parameter $a \gg b$) may be to check the classical prediction that the electromagnetic contribution to the metric and curvature on the singularity ring is (infinitely) greater than the contribution of the $δ$-function-like mass distribution, no matter how small the electric charge is. Here we encounter a kind of a discrete diagram technique, but with three-dimensional (static) diagrams and with only a few diagrams, although with modified (extended to complex coordinates) propagators. The metric (curvature) in the vicinity of the former singularity ring is considered. The electromagnetic contribution does indeed have a relative factor that is infinite at $b \to 0$, but, taking into account some existing estimates of the upper bound on the electric charge of known substances, it is not so large for habitual bodies and can only be significant for practically non-rotating black holes.

gr-qc↗

arXiv2023

On the gravitational diagram technique in the discrete setup

This article is in the spirit of our work on the consequences of the Regge calculus, where some edge length scale arises as an optimal initial point of the perturbative expansion after functional integration over connection. Now consider the perturbative expansion itself. To obtain an algorithmizable diagram technique, we consider the simplest periodic simplicial structure with a frozen part of the variables ("hypercubic"). After functional integration over connection, the system is described by the metric $g_{λμ}$ at the sites. We parameterize $g_{λμ}$ so that the functional measure becomes Lebesgue. The discrete diagrams are free from ultraviolet divergences and reproduce (for ordinary, non-Planck external momenta) those continuum counterparts that are finite. We give the parametrization of $g_{λμ}$ up to terms, providing, in particular, additional three-graviton and two-graviton-two-matter vertices, which can give additional one-loop corrections to the Newtonian potential. The edge length scale is $\sim \sqrt{ η}$, where $η$ defines the free factor $ ( - \det \| g_{λμ} \| )^{ η/ 2}$ in the measure and should be a large parameter to ensure the true action after integration over connection. We verify the important fact that the perturbative expansion does not contain increasing powers of $η$ if its initial point is chosen close enough to the maximum point of the measure, thus justifying this choice. Discrete propagators depend on the Barbero-Immirzi parameter $γ$, which determines the ratio of timelike and spacelike elementary length scales. The existing estimates of $γ$ allow the propagator poles to have real energy for any (real) spatial momenta.

gr-qc↗

arXiv2021

On the discrete version of the Reissner-Nordström solution

This paper generalizes our previous paper on the discrete Schwarzschild type solution in the Regge calculus, the simplicial electrodynamics earlier considered in the literature is incorporated in the case of the presence of a charge. Validity of the path integral approach is assumed, of which the only consequence used here is a loose fixation of edge lengths around a finite nonzero scale (we have considered the latter earlier). In essence, the problem of determining the optimal background metric and electromagnetic field for the perturbative expansion generated by the functional integral is considered, for which the skeleton Regge and electrodynamic equations are analyzed. For the Regge equations, as we have earlier found, the Regge action on the simplest periodic simplicial structure and in the leading order over metric variations between 4-simplices can be substituted by a finite-difference form of the Hilbert-Einstein action (the piecewise constant metric there is defined by providing the vertices with coordinates). Thus we get the absence of the singularity inherent in the continuous solution. At the same time, the discrete solution is close to the continuum Reissner-Nordström one at large distances.

gr-qc↗

arXiv2021

On the discrete version of the Kerr geometry

A Kerr type solution in the Regge calculus is considered. It is assumed that the discrete general relativity, the Regge calculus, is quantized within the path integral approach. The only consequence of this approach used here is the existence of a length scale at which edge lengths are loosely fixed, as considered in our earlier paper. In addition, we previously considered the Regge action on a simplicial manifold on which the vertices are coordinatized and the corresponding piecewise constant metric introduced, and found that for the simplest periodic simplicial structure and in the leading order over metric variations between 4-simplices, this reduces to a finite-difference form of the Hilbert-Einstein action. The problem of solving the corresponding discrete Einstein equations (classical) with a length scale (having a quantum nature) arises as the problem of determining the optimal background metric for the perturbative expansion generated by the functional integral. Using an one-complex-function ansatz for the metric, which reduces to the Kerr-Schild metric in the continuum, we find a discrete metric that approximates the continuum one at large distances and is nonsingular on the (earlier) singularity ring. The effective curvature $R_{λννρ}$, including where $R_{λμ} \neq 0$ (gravity sources), is analyzed with a focus on the vicinity of the singularity ring.

gr-qc↗

arXiv2021

On the Kerr metric in a synchronous reference frame

The Kerr metric is considered in a synchronous frame of reference obtained by using proper time and initial conditions for particles that freely move along a certain set of trajectories as coordinates. Modifying these coordinates in a certain way (keeping their interpretation as initial values at large distances), we still have a synchronous frame and the direct analogue of the Lemaitre metric, the singularities of which are exhausted by the physical Kerr singularity (the singularity ring).

gr-qc↗

arXiv2020

On the discrete version of the Schwarzschild problem

We consider a Schwarzschild type solution in the discrete Regge calculus formulation of general relativity quantized within the path integral approach. Earlier, we found a mechanism of a loose fixation of the background scale of Regge lengths. This elementary length scale is defined by the Planck scale and some free parameter of such a quantum extension of the theory. Besides, Regge action was reduced to an expansion over metric variations between the tetrahedra and, in the main approximation, is a finite-difference form of the Hilbert-Einstein action. Using for the Schwarzschild problem a priori general non-spherically symmetrical ansatz, we get finite-difference equations for its discrete version. This defines a solution which at large distances is close to the continuum Schwarzschild geometry, and the metric and effective curvature at the center are cut off at the elementary length scale. Slow rotation can also be taken into account (Lense-Thirring-like metric). Thus we get a general approach to the classical background in the quantum framework in zero order: it is an optimal starting point for the perturbative expansion of the theory; finite-difference equations are classical, the elementary length scale has quantum origin. Singularities, if any, are resolved.

gr-qc↗

arXiv2020

On the discrete version of the black hole solution

A Schwarzschild type solution in Regge calculus is considered. Earlier, we considered a mechanism of loose fixing of edge lengths due to the functional integral measure arising from integration over connection in the functional integral for the connection representation of the Regge action. The length scale depends on a free dimensionless parameter that determines the final functional measure. For this parameter and the length scale large in Planck units, the resulting effective action is close to the Regge action. Earlier, we considered the Regge action in terms of affine connection matrices as functions of the metric inside the 4-simplices and found that it is a difference form of the Hilbert-Einstein action in the leading order over metric variations between the 4-simplices. Now we take the (continuum) Schwarzschild problem in the form where spherical symmetry is not set a priori and arises just in the solution, take the difference form of the corresponding equations and get the metric (in fact, in the Lemaitre or Painlevé-Gullstrand like frame), which is nonsingular at the origin, just as the Newtonian gravitational potential, obeying the difference Poisson equation with a point source, is cut off at the elementary length and is finite at the source.

gr-qc↗

arXiv2019

On the discrete Christoffel symbols

The piecewise flat spacetime is equipped with a set of edge lengths and vertex coordinates. This defines a piecewise affine coordinate system and a piecewise affine metric in it, the discrete analogue of the unique torsion-free metric-compatible affine connection or of the Levi-Civita connection (or of the standard expression of the Christoffel symbols in terms of metric) mentioned in the literature, and, substituting this into the affine-connection form of the Regge action of our previous work, we get a second order form of the action. This can be expanded over metric variations from simplex to simplex. For a particular periodic simplicial structure and coordinates of the vertices, the leading order over metric variations is found to coincide with a certain finite difference form of the Hilbert-Einstein action.

gr-qc↗

arXiv2019

Discrete Faddeev action for the tetrad fields strongly varying along different coordinates

Faddeev gravity using a $d$-dimensional tetrad (normally $d = 10$) is classically equivalent to general relativity (GR). The discrete Faddeev gravity on the piecewise flat spacetime normally assumes slowly varying metric and tetrad from vertex to vertex. Meanwhile, Faddeev action is finite (although not unambiguously defined) for discontinuous tetrad fields thus allowing, in particular, to consider a surface as consisting of virtually independent elementary triangles, and its area spectrum as the sum of elementary area spectra. In the discrete connection form, area tensors are canonically conjugate to SO(10) connection matrices, and earlier we have found the elementary area spectrum, which is nonsingular just at large connection or the strongly varying fields constituting a kind of "antiferromagnetic" structure. We appropriately define discrete {\it connection} Faddeev action to unambiguously determine the discrete Faddeev action for the strongly varying fields, but weakly varying metric, equivalent in the continuum limit to the GR action with this metric. Previously, we considered large variations in only one direction, now we use an ansatz in some respects less common, but overall, probably the most common. A unified simplicial connection representation is written out (depending on an auxiliary connection) both for the discrete Faddeev action and for the Regge action.

gr-qc↗

arXiv2018

On the non-perturbative graviton propagator

To reduce general relativity to the canonical Hamiltonian formalism and construct the path (functional) integral in a simpler and, especially in the discrete case, less singular way, one extends the configuration superspace, as in the connection representation. Then we perform functional integration over connection. The module of the result of this integration arises in the leading order of the expansion over a scale of the discrete lapse-shift functions and has maxima at finite (Planck scale) areas/lengths and rapidly decreases at large areas/lengths, as we have mainly considered previously; the phase arises in the leading order (Regge action) of the stationary phase expansion. Now we consider the possibility of confining ourselves to these leading terms in a certain region of the parameters of the theory; consider background edge lengths as an optimal starting point for the perturbative expansion of the theory; estimate the background length scale and consider the form of the graviton propagator. In parallel with the general simplicial structure, we consider the simplest periodic simplicial structure with a part of the variables frozen ("hypercubic"), for which also the propagator in the leading approximation over metric variations can be written in a closed form.

gr-qc↗

arXiv2017

Simplicial Palatini action

We consider the piecewise flat spacetime and a simplicial analog of the Palatini form of the general relativity (GR) action where the discrete Christoffel symbols are given on the tetrahedra as variables that are independent of the metric. Excluding these variables classically gives exactly the Regge action. This paper continues our previous work. Now we include the parity violation term and the analogue of the Barbero-Immirzi parameter introduced in the orthogonal connection form of GR. We consider the path integral and the functional integration over connection. The result of the latter (for certain limiting cases of some parameters) is compared with the earlier found result of the functional integration over connection for the analogous {\it orthogonal} connection representation of Regge action. These results, mainly as some measures on the lengths/areas, are discussed for the possibility of the diagram technique where the perturbative diagrams for the Regge action calculated using the measure obtained are finite. This finiteness is due to these measures providing elementary lengths being mostly bounded and separated from zero, just as finiteness of a theory on a lattice with an analogous probability distribution of spacings.

gr-qc↗

arXiv2017

First order discrete Faddeev gravity at the strongly varying fields

We consider the Faddeev formulation of general relativity (GR), which can be characterized by a kind of $d$-dimensional tetrad (typically $d$=10) and a non-Riemannian connection. This theory is invariant w. r. t. the global, but not local, rotations in the $d$-dimensional space. There can be configurations with a smooth or flat metric, but with the tetrad that changes abruptly at small distances, a kind of "antiferromagnetic" structure. Previously, we discussed a first order representation for the Faddeev gravity, which uses the orthogonal connection in the $d$-dimensional space as an independent variable. Using the discrete form of this formulation, we considered the spectrum of (elementary) area. This spectrum turns out to be physically reasonable just on a classical background with large connection like rotations by $π$, that is, with such an "antiferromagnetic" structure. In the discrete first order Faddeev gravity, we consider such a structure with periodic cells and large connection and strongly changing tetrad field inside the cell. We show that this system in the continuum limit reduces to a generalization of the Faddeev system. The action is a sum of related actions of the Faddeev type and is still reduced to the GR action.

gr-qc↗

arXiv2017

Spectrum of area in the Faddeev formulation of gravity

Faddeev formulation of general relativity (GR) is considered where the metric is composed of ten vector fields or a ten-dimensional tetrad. Upon partial use of the field equations, this theory results in the usual GR. Earlier we have proposed first-order representation of the minisuperspace model for the Faddeev formulation where the tetrad fields are piecewise constant on the polytopes like 4-simplices or, say, cuboids into which ${\rm R}^4$ can be decomposed, an analogue of the Cartan-Weyl connection-type form of the Hilbert-Einstein action in the usual continuum GR. In the Hamiltonian formalism, the tetrad bilinears are canonically conjugate to the orthogonal connection matrices. We evaluate the spectrum of the elementary areas, functions of the tetrad bilinears. The spectrum is discrete and proportional to the Faddeev analog $γ_{\rm F}$ of the Barbero-Immirzi parameter $γ$. The possibility of the tetrad and metric discontinuities in the Faddeev gravity allows to consider any surface as consisting of a set of virtually independent elementary areas and its spectrum being the sum of the elementary spectra. Requiring consistency of the black hole entropy calculations known in the literature we are able to estimate $γ_{\rm F}$.

gr-qc↗

arXiv2016

GL(4,R) representation of the gravity action on the piecewise flat spacetime

The gravity action on the piecewise flat Riemannian manifold is formulated using the discrete set of the nondegenerate 4$\times$4 matrices on the 3-simplices as some connection type variables. These variables are the discrete counterpart of the affine (Christoffel) connection used as independent variables in the Palatini form of the Einstein gravity action. Excluding these with the help of the equations of motion we get the original discrete gravity action on the piecewise flat spacetime (Regge action). The discrete version of the diffeomorphisms and path integral are briefly discussed.

gr-qc↗