Narrow Results

In this paper, we investigate the geodesic structure of Clifton–Barrow black hole space–time. Through the numerical analysis of the effective potential and the motion equation, the orbital types of test particles and photons and the corresponding orbital motion diagrams of each orbital types under certain conditions are obtained. We find that angular momentum $b$ and $\delta$ determine the existence of bound orbits and circular orbits. And we also find that the radius of unstable circular orbit decreases with increases in $b$ while the radius of stable circular orbit increases. Furthermore, as $\delta$ increases, the radius of unstable circular orbit increases, while the radius of stable circular orbit decreases. For null geodesic, parameters $b$ and $\delta$ do not affect the types of null orbits. The radius of the unstable circular orbits increases with the increase of $\delta$. However, the radius of the unstable circular orbits remains unchanged as $b$ increases. Also, we show that the precession direction of the bound orbits of the test particles is counterclockwise for $b=2.15$, but clockwise with $b=3.6$. Moreover, different energy values have an effect on the curvature of escape and absorb orbits curve.

The Bardeen model describes a regular spacetime, i.e. a singularity-free black hole spacetime. In this paper, by analyzing the behavior of the effective potential for the particles and photons, we investigate the timelike and null geodesic structures in the Bardeen spacetime. At the same time, all kinds of orbits, which are allowed according to the energy level corresponding to the effective potentials, are numerically simulated in detail. We find many-world bound orbits, two-world escape orbits and escape orbits in this spacetime. We also find that bound orbits precession directions are opposite and their precession velocities are different, the inner bound orbits shift along counter-clockwise with high velocity while the exterior bound orbits shift along clockwise with low velocity.

If there exists a lower bound $l_0$ to spacetime intervals which is Lorentz-invariant, then the effective description of spacetime that incorporates such a lower bound must necessarily be nonlocal. Such a nonlocal description can be derived using standard tools of differential geometry, but using as basic variables certain bi-tensors instead of the conventional metric tensor $g_{ab}(x)$. This allows one to construct a qmetric$q_{ab}(x;y)$, using the Synge’s world function $\mathrm{\Omega }(x,y)$ and the van Vleck determinant $\mathrm{\Delta }(x,y)$, that incorporates the lower bound on spacetime intervals. The same nonanalytic structure of the reconstructed spacetime which renders a perturbative expansion in $l_0$ meaningless, will then also generically leave a non-trivial “relic” in the limit $l_0\to 0$. We present specific results derived from $q_{ab}(x;y)$ where such a relic term manifests, and discuss several implications of the same. Specifically, we will discuss how these results: (i) suggest a description of gravitational dynamics different from the conventional one based on the Einstein–Hilbert Lagrangian, (ii) imply a dimensional reduction to $2$ at small scales and (iii) can be significant for the idea that the cosmological constant itself might be related to some nonlocal vestige of the small-scale structure of spacetime. We will conclude by discussing the ramifications of these ideas in the context of quantum gravity.

We discuss in a critical way the physical foundations of geometric structure of relativistic theories of gravity by the so-called Ehlers–Pirani–Schild formalism. This approach provides a natural interpretation of the observables showing how relate them to General Relativity and to a large class of Extended Theories of Gravity. In particular we show that, in such a formalism, geodesic and causal structures of space-time can be safely disentangled allowing a correct analysis in view of observations and experiment. As specific case, we take into account the case of f(R)-gravity.