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We propose a toy model of a spherical universe made up of an exotic dark gas with temperature T in thermal equilibrium with a black-body in adiabatic expansion. Each particle of this exotic gas mimics a kind of particle of dark energy. So, each dark particle occupies a very small area of space so-called Planck area $L_p^2$, which represents the minimum area of the whole space-time given by the spherical surface with area $4\pi R_H^2$, where $R_H$ is the Hubble radius. We realize that such spherical surface is the surface of the black-body for representing the dark universe as if it were the surface of an expanding balloon. Thus, we are able to derive the law of universal gravitation, thus leading us to understand the cosmological anti-gravity. We estimate the tiny order of magnitude of the cosmological constant and the acceleration of expansion of the dark sphere. In this toy model, as the dark universe can be thought of as a large black body, when we obtain its power and frequency of emission of radiation, we find very low values. We conclude that such radiation and frequency of the black body made up of dark energy is a background gravitational wave with very low frequency in the order of $10^{-17}\text{Hz}$ due to the slight stretching of the fabric of space-time.

Given a minimum measurable length underlying space–time, the latter may be effectively regarded as discrete, at scales of order of the Planck length. A systematic discretization of continuum physics may be effected most efficiently through the umbral deformation. General functionals yielding such deformations at the level of solutions are furnished and illustrated, and broad features of discrete oscillations and wave propagation are outlined.

It is shown that super-thin and super-long gravitational flux tube solutions in the 5D Kaluza–Klein gravity have the Planck scale regions (≈ l_Pl) where the metric signature changes from {+, -, -, -, -} to {-, -, -, -, +}. Such a change occurs too rapidly according to one of the paradigms of quantum gravity which holds that the Planck length is the minimal length in nature and consequently the physical quantities cannot change very quickly over this length scale. To avoid such a dynamic, it is hypothesized that a pure quantum freezing of the dynamics of the 5th dimension takes place. As a continuation of the flux tube metric in the longitudinal direction, the Reissner–Nordström metric is proposed. As a consequence of such a construction one can avoid the appearance of a point-like singularity in the extremal Reissner–Nordström solution.

We present the quantum spin as a novel test tool for probing directly the Planck scale space–time foam of quantum gravity. Quantum fluctuations of spatial support for the electric vector associated with the spin-one photon affect its polarization sufficiently to allow us to gain deep insights and unprecedented constraints on the most important and fundamental aspect of quantum gravity — the fluctuating structure of space–time with a Planck scale three-dimensional web. We show that the survival of strong polarization of X-rays and gamma rays from the Crab Nebula rejects the conventional space–time foam at Planck length scale and constrains the microscopic scale of quantum gravitational fluctuations to below 2 × 10^-8l_P.

The number of classical paths of a given length, connecting any two events in a (pseudo) Riemannian spacetime is, of course, infinite. It is, however, possible to define a useful, finite, measure $N(x_2,x_1;\sigma )$ for the effective number of quantum paths [of length $\sigma$ connecting two events $(x_1,x_2)$] in an arbitrary spacetime. When $x_2=x_1$, this reduces to $C(x,\sigma )$ giving the measure for closed quantum loops of length $\sigma$ containing an event $x$. Both $N(x_2,x_1;\sigma )$ and $C(x,\sigma )$ are well-defined and depend only on the geometry of the spacetime. Various other physical quantities like, for e.g. the effective Lagrangian, can be expressed in terms of $N(x_2,x_1;\sigma )$. The corresponding measure for the total path length contributed by the closed loops, in a spacetime region ${𝒱}$, is given by the integral of $L(\sigma ;x)\equiv \sigma C(\sigma ;x)$ over ${𝒱}$. Remarkably enough $L(0;x)\propto R(x)$, the Ricci scalar; i.e. the measure for the total length contributed by infinitesimal closed loops in a region of spacetime gives us the Einstein–Hilbert action. Its variation, when we vary the metric, can provide a new route towards induced/emergent gravity descriptions. In the presence of a background electromagnetic field, the corresponding expressions for $N(x_2,x_1;\sigma )$ and $C(x,\sigma )$ can be related to the holonomies of the field. The measure $N(x_2,x_1;\sigma )$ can also be used to evaluate a wide class of path integrals for which the action and the measure are arbitrary functions of the path length. As an example, I compute a modified path integral which incorporates the zero-point-length in the spacetime. I also describe several other properties of $N(x_2,x_1;\sigma )$ and outline a few simple applications.

The existence of a minimal and fundamental length scale, say, the Planck length, is a characteristic feature of almost all the models of quantum gravity. The presence of the fundamental length is expected to lead to an improved ultra-violet behavior of the semi-classical propagators. The hypothesis of path integral duality provides a prescription to evaluate the modified propagator of a free, quantum scalar field in a given spacetime, taking into account the existence of the fundamental length in a locally Lorentz invariant manner. We use this prescription to compute the quantum gravitational modifications to the propagators in spacetimes with constant curvature, and show that: (i) the modified propagators are ultra-violet finite, and (ii) the modifications are non-perturbative in the Planck length. We discuss the implications of our results.