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Regge action is represented analogously to how the Palatini action for general relativity (GR) as some functional of the metric and a general connection as independent variables represents the Einstein–Hilbert action.

The piecewise flat (or simplicial) spacetime of Regge calculus is equipped with some world coordinates and some piecewise affine metric which is completely defined by the set of edge lengths and the world coordinates of the vertices. The conjugate variables are the general nondegenerate matrices on the three-simplices which play the role of a general discrete connection. Our previous result on some representation of the Regge calculus action in terms of the local Euclidean (Minkowsky) frame vectors and orthogonal connection matrices as independent variables is somewhat modified for the considered case of the general linear group GL(4, R) of the connection matrices.

As a result, we have some action invariant w.r.t. arbitrary change of coordinates of the vertices (and related GL(4, R) transformations in the four-simplices). Excluding GL(4, R) connection from this action via the equations of motion we have exactly the Regge action for the considered spacetime.

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 with the help of the equations of motion gives exactly the Regge action.

This paper continues our previous work. Now, we include the parity violation term and the analog of the Barbero–Immirzi parameter introduced in the orthogonal connection form of GR. We consider the path integral and the functional integration over the 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 the connection for the analogous 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 the finiteness of a theory on a lattice with an analogous probability distribution of spacings.

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.

The piecewise flat space–time 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.

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 finite-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 finite-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 cutoff at the elementary length and is finite at the source.

In this paper, 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 is introduced, and found that for the simplest periodic simplicial structure and in the leading order over metric variations between four-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 a 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 $R_{\lambda \nu \nu \rho }$, including where $R_{\lambda \mu }\ne 0$ (gravity sources), is analyzed with a focus on the vicinity of the singularity ring.

This paper generalizes our previous work on the discrete Schwarzschild-type solution in Regge calculus to the case of a charge. The known in the literature simplicial electro-dynamics retaining like Regge calculus geometric features of the continuum counterpart is incorporated into the formalism.

The functional integral provides a loose fixation of edge lengths around some scale and a perturbative expansion, for which we consider, in essence, finding the optimal starting (background) metric/field from the skeleton Regge and electro-dynamic equations. The simplest periodic simplicial structure and the expansion over metric/field variations between 4-simplices are considered. In the leading order of this expansion, the electromagnetic action, as we found earlier for the Regge action, is reducible to a finite-difference form of the continuum counterpart.

Instead of infinite continuous metric/field variables at the center, we have finite discrete variables; the discrete metric in the Schwarzschild-type coordinates turns out to change the sign of its variation when approaching the center from the nearest vertices, so that $g_{00}+1$ is positive at the center (the continuum $g_{00}+1$ tends to $-\infty$ at the center). The metric/field in the neighborhood of the center and the curvature and the Kretschmann scalar at the center are estimated.

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 $\delta$-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 nonrotating black holes.

This paper 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 $g_{\lambda \mu }$ at the sites. We parameterize $g_{\lambda \mu }$ 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 $g_{\lambda \mu }$ 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 $\sim \sqrt{\eta }$, where $\eta$ defines the free factor ${(-det\parallel g_{\lambda \mu }\parallel )}^{\eta ∕2}$ 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 $\eta$ 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 $\gamma$, which determines the ratio of timelike and spacelike elementary length scales. The existing estimates of $\gamma$ allow the propagator poles to have real energy for any (real) spatial momenta.

We describe all operators on scalar finite element spaces which appear as the restriction of a second-order (linear) differential operator. More precisely we provide a family of isomorphisms between this space of discrete differential operators and products of some exotic finite element spaces. This provides a unified framework for the Regge calculus of numerical relativity and the Nédélec edge elements of computational electromagnetism.

We apply the "consistent discretization" technique to the Regge action for (Euclidean and Lorentzian) general relativity in arbitrary number of dimensions. The result is a well-defined canonical theory that is free of constraints and where the dynamics is implemented as a canonical transformation. In the Lorentzian case, the framework appears to be naturally free of the "spikes" that plague traditional formulations. It also provides a well-defined recipe for determining the integration measure for quantum Regge calculus.

The 2011 Nobel Prize in Physics was awarded "for the discovery of the accelerating expansion of the universe through observations of distant supernovae." However, it is not the case that the type Ia supernova data necessitates accelerating expansion. Since we do not have a successful theory of quantum gravity, we should not assume general relativity (GR) will survive unification intact, especially on cosmological scales where tests are scarce. We provide a simple example of how GR cosmology may be modified to produce a decelerating Einstein-de Sitter cosmology (EdS) that accounts for the Union2 Compilation data as well as the accelerating ΛCDM (EdS plus a cosmological constant).

We consider spinfoam quantum gravity in the flipped limit, which is the double scaling limit γ → 0, j → ∞ with γj = const., where γ is the Immirzi parameter, j is the spin and γj gives the physical area in Planck units. In this regime the amplitude for a 2-complex becomes effectively an integral over Regge-like metrics and seems to enforce Einstein equations in the semiclassical regime. The Immirzi parameter must be considered as dynamical in the sense that it runs to zero when the fine structure of the foam is averaged. In addition to quantum corrections which vanish for ℏ → 0, we find new corrections due to the discreteness of geometric spectra.

After an overview of variational principles for discrete gravity, and on the basis of the approach to conformal transformations in a simplicial PL setting proposed by Luo and Glickenstein, we present at a heuristic level an improved scheme for addressing the gravitational (Euclidean) path integral and geometrodynamics.

Recent results in Local Regge Calculus are confronted with Spin Foam Formalism. Introducing Barrett-Crane Quantization in Local Regge Calculus makes it possible to associate a unique Spin j_h with an hinge h, fulfilling one of the requirements of Spin Foam definition. It is shown that inter-twiner terms of Spin Foam can follow from the closure constraint in Local Regge Calculus.