Recently a simple one-parameter generalization of the Schwarzschild spacetime was suggested by A. M. Visser [JCAP 1902, 042 (2019)] as a toy model describing the regular black hole and traversable wormhole states separated by the border (one-way wormhole) state, depending on the value of the new parameter. Here we will study quasinormal modes of all three states and show that the black-hole/wormhole transition is characterized by echoes, while the memory of the black hole state is kept in the time-domain profile of the wormhole perturbation at the initial stage of the exponential fall off. Calculations of quasinormal modes with the help of the WKB method with Padé expansion and time-domain integration are in a good agreement. An analytical formula for quasinormal modes in the eikonal regime is obtained.

While quasinormal modes of bosonic fields for the non-trivial 4-dimensional Einstein-Gauss-Bonnet theory of gravity suggested in [D. Glavan and C. Lin, Phys. Rev. Lett. 124, 081301 (2020)] have been recently studied, there is no such study for fermionic fields. Here we calculate quasinormal modes of the Dirac field for spherically symmetric asymptotically flat black hole in this novel 4D Einstein-Gauss-Bonnet theory. The values of the quasinormal frequencies, calculated by the 6th order WKB method with Padé approximants and the time-domain integration, show that the real part of the quasinormal modes is considerably increased, while the damping rate is usually decreasing when the coupling constant is growing.

Arbitrarily long lived modes, called quasi-resonances, are known to exist in the spectrum of massive fields for a number of black-hole backgrounds at some discrete values of mass of the field. Here we show that these modes also exist in the background of wormholes, unless a wormhole has a constant red-shift function, that is, tideless in the radial direction. The evidence of quasi-resonances is supported by calculations in the frequency and time domains, which are in a good concordance. At large masses of the field, time-domain profiles of the absolute value of the wave function have peculiar behavior: the long-lived modes dominate in the signal after a long period of power-law tails.

Quasinormal modes in the high frequency (eikonal) regime can be obtained analytically as the Mashhoon-Will-Schiutz WKB formula is exact in this case. This regime is interesting because of the correspondence between eikonal quasinormal modes and null geodesics, as well as due to existence of potential eikonal instabilities in some theories of gravity. At the same time in a number of studies devoted to quasinormal modes of spherically symmetric black holes this opportunity was omitted.Here we find analytical quasinormal modes of black holes in various alternative and extended theories of gravity in the form of the Schwarzschld eikonal quasinormal modes and added corrections due to deviations from Einstein theory. We also deduce a generic formula for analytical calculations of the eikonal quasinormal modes for the class of asymptotically flat metrics in terms of small deviations from the Schwarzschild geometry.

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We present the particular case of the Stephani solution for shear-free perfect fluid with uniform energy density and non-uniform pressure. Such models appeared as possible alternative to the consideration of the exotic forms of matter like dark energy that would cause the acceleration of the universe expansion. These models are characterised by the spatial curvature depending on time. We analyze the properties of the cosmological model obtained on the basis of exact solution of the Stephani class and adopt it to the recent observational data. The spatial geometry of the model is investigated. We show that despite possible singularities, the model can describe the current stage of the universe evolution.

An analytical solution representing traversable asymptotically flat and symmetric wormholes was obtained without adding exotic matter in two different theories independently: in the Einstein-Maxwell-Dirac theory and in the second Randall-Sundrum brane-world model. Further, a smooth normalizable asymmetric wormhole solution has been recently obtained numerically in the Einstein-Maxwell-Dirac theory. Using the time-domain integration method we study quasinormal ringing of all these wormholes with emphasis to the regime of mimicking the near extremal Reissner-Nordström black holes, which is characterised by echoes. In addition, we calculate radius of shadows cast by these wormholes.

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