Strong gravitational lensing from black holes results in the formation of relativistic images, in particular, relativistic Einstein rings. For objects with event horizons, the radius of the unstable light ring (photon sphere) is the lowest radius at which a relativistic image might be formed. For horizonless ultracompact objects, additional relativistic images and rings can form inside this radius. In this paper, we provide an analytical approach to deal with strong gravitational lensing from such ultracompact objects, which is substantially different from the black hole cases, first reported by Bozza. Here, our analysis indicates that the angular separations and magnifications of relativistic images inside the unstable light ring (photon sphere) might be several orders of magnitude higher compared to the ones outside it. This indicates fundamental differences in the nature of strong gravitational lensing from black holes and ultracompact objects.

In this work, we elaborate on the development of a general relativistic formalism that allows one to analytically express the mass and spin parameters of the Kerr black hole in terms of observational data: the total redshift and blueshift of photons emitted by massive geodesic particles revolving the black hole and their orbital parameters. Thus, we present concise closed formulas for the mass and spin parameters of the Kerr black hole in terms of few directly observed quantities in the case of equatorial circular orbits either when the black hole is static or is moving with respect to a distant observer. Furthermore, we incorporate the gravitational dragging effect generated by the rotating nature of the Kerr black hole into the analysis and elucidate its nontrivial contribution to the expression for the light bending parameter and the frequency shifts of photons emitted by orbiting particles that renders simple symmetric expressions for the kinematic redshift and blueshift. We also incorporate the dependency of the frequency shift on the azimuthal angle, a fact that allows one to express the total redshift/blueshift at any point of the orbit of the revolving particle for the cases when the black hole is both static or moving with respect to us. These formulas allow one to compute the Kerr black hole parameters by applying this general relativistic formalism to astrophysical systems like the megamaser accretion disks orbiting supermassive black holes at the core of active galactic nuclei. Our results open a new window to implement parameter estimation studies to constrain black hole variables, and they can be generalized to black hole solutions beyond Einstein’s gravity.

The $$\gamma $$γ -metric is a static, axially-symmetric singular solution of the vacuum Einstein’s equations without an event horizon. This is a two-parameter family of solutions, generic values of one of which (called $$\gamma $$γ ) measure the deviation from spherical symmetry. We show that this can serve as a black hole mimicker, consistent with current observations. To this end, we first study the shadow cast by this geometry, in order to constrain the $$\gamma $$γ -metric from observations. We find that for $$\gamma < 1/2$$γ < 1 / 2 , there are, in principle, no shadows cast. On the other hand, shadows cast for all values of $$\gamma \ge 1/2$$γ ≥ 1 / 2 are consistent with observations of M $$87^*$$87 ∗ by the Event Horizon Telescope. We also study images of thin accretion disks in the $$\gamma $$γ -metric background. In situations where the $$\gamma $$γ -metric possesses light rings, these qualitatively mimic Schwarzschild black holes with the same ADM mass, while in the absence of such rings, they are drastically different from the black hole case.

Motivated by recent achievements of a full general relativistic method in estimating the mass-to-distance ratio of supermassive black holes hosted at the core of active galactic nuclei, we introduce the new concept redshift rapidity in order to express the Schwarzschild black hole mass and its distance from the Earth just in terms of observational quantities. The redshift rapidity is also an observable relativistic invariant that represents the evolution of the frequency shift with respect to proper time in the Schwarzschild spacetime. We extract concise and elegant analytic formulas that allow us to disentangle mass and distance to black holes in the Schwarzschild background and estimate these parameters separately. This procedure is performed in a completely general relativistic way with the aim of improving the precision in measuring cosmic distances to astrophysical compact objects. Our exact formulas are valid on the midline and close to the line of sight, having direct astrophysical applications for megamaser systems, whereas the general relations can be employed in black hole parameter estimation studies. We also computed the frequency shift and the redshift rapidity for emitter eccentric orbits and calculated their relative error with respect to their numerical exact value for different eccentricities.

We demonstrate a general relativistic approach to model dark matter halos using the Einstein cluster, with the matter stress-energy generated by collisionless particles moving on circular geodesics in all possible angular directions and orbital radii. Such matter, as is known, allows an anisotropic pressure profile with non-zero tangential but zero radial pressure. We use the Einasto density profile for the Einstein cluster. Analytical studies on its properties (metric functions) and stability issues are investigated. Further, to establish this model (with the Einasto profile) as one for a dark matter halo, we use the SPARC galactic rotation curve data and estimate the best-fit values for the model parameters. General relativistic features (beyond the Keplerian velocities) such as the tangential pressure profile, are quantitatively explored. Thus, Einstein clusters with the Einasto profile, which tally well with observations, may be considered as a viable model for dark matter halos.