Exploration of a singular fluid spacetime

We investigate the properties of a special class of singular solutions for a self-gravitating perfect fluid in general relativity: the singular isothermal sphere. For arbitrary constant equation-of-state parameter $$w=p/\rho $$ w = p / ρ , there exist static, spherically-symmetric solutions with density profile $$\propto 1/r^2$$ ∝ 1 / r 2 , with the constant of proportionality fixed to be a special function of w. Like black holes, singular isothermal spheres possess a fixed mass-to-radius ratio independent of size, but no horizon cloaking the curvature singularity at $$r=0$$ r = 0 . For $$w=1$$ w = 1 , these solutions can be constructed from a homogeneous dilaton background, where the metric spontaneously breaks spatial homogeneity. We study the perturbative structure of these solutions, finding the radial modes and tidal Love numbers, and also find interesting properties in the geodesic structure of this geometry. Finally, connections are discussed between these geometries and dark matter profiles, the double copy, and holographic entropy, as well as how the swampland distance conjecture can obscure the naked singularity.

Geodesic Structure of the Accelerated Stephani Universe

For the spherically symmetric Stephani cosmological model with an accelerated expansion, we investigate the main scenarios of the test particle and photon motion. We show that a comoving observer sees an appropriate picture. In the case of purely radial motion, the radial velocity decreases slightly with time due to the universe expansion. Both particles and photons spiral out of the center when the radial coordinate is constant. In the case of the motion with arbitrary initial velocity, the observable radial distance to the test particle can increase under negative observable radial velocity.

Null geodesics in five-dimensional Reissner–Nordström anti-de Sitter black holes

The study of the motion of photons around massive bodies is one of the most useful tools to find the geodesic structure associated with said gravitational source. In the present work, different possible paths projected in an invariant hyperplane are investigated, considering a five-dimensional Reissner–Nordström anti-de Sitter black hole. Also, we study some observational tests, such as the bending of light and the Shapiro time delay effect. Mainly, we found that the motion of photons follows the hippopede of a Proclus geodesic, which is a new type of trajectory of the second kind, the Limaçon of Pascal being their analog geodesic in four-dimensional Reissner–Nordström anti-de Sitter black hole.

Metrizability of Holonomy Invariant Projective Deformation of Sprays

In this paper, we consider projective deformation of the geodesic system of Finsler spaces by holonomy invariant functions. Starting with a Finsler spray $S$ and a holonomy invariant function ${\mathcal{P}}$ , we investigate the metrizability property of the projective deformation $\widetilde{S}=S-2\unicode[STIX]{x1D706}{\mathcal{P}}{\mathcal{C}}$ . We prove that for any holonomy invariant nontrivial function ${\mathcal{P}}$ and for almost every value $\unicode[STIX]{x1D706}\in \mathbb{R}$ , such deformation is not Finsler metrizable. We identify the cases where such deformation can lead to a metrizable spray. In these cases, the holonomy invariant function ${\mathcal{P}}$ is necessarily one of the principal curvatures of the geodesic structure.

Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces

We study the geometry of Lie groups G with a continuous Finsler metric, in presence of a subgroup K such that the metric is right-invariant for the action of K. We present a systematic study of the metric and geodesic structure of homogeneous spaces M obtained by the quotient {M\simeq G/K}. Of particular interest are left-invariant metrics of G which are then bi-invariant for the action of K. We then focus on the geodesic structure of groups K that admit bi-invariant metrics, proving that one-parameter groups are short paths for those metrics, and characterizing all other short paths. We provide applications of the results obtained, in two settings: manifolds of Banach space linear operators, and groups of maps from compact manifolds.

Geodesic structure of the Clifton–Barrow black hole space–time

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 [Formula: see text] and [Formula: see text] determine the existence of bound orbits and circular orbits. And we also find that the radius of unstable circular orbit decreases with increases in [Formula: see text] while the radius of stable circular orbit increases. Furthermore, as [Formula: see text] increases, the radius of unstable circular orbit increases, while the radius of stable circular orbit decreases. For null geodesic, parameters [Formula: see text] and [Formula: see text] do not affect the types of null orbits. The radius of the unstable circular orbits increases with the increase of [Formula: see text]. However, the radius of the unstable circular orbits remains unchanged as [Formula: see text] increases. Also, we show that the precession direction of the bound orbits of the test particles is counterclockwise for [Formula: see text], but clockwise with [Formula: see text]. Moreover, different energy values have an effect on the curvature of escape and absorb orbits curve.