In this paper we introduce a new approach to the study of the effects that an impulsive wave, containing a mixture of material sources and gravitational waves, has on a geodesic congruence that traverses it. We find that the effect of the wave on the congruence is a discontinuity in the B-tensor of the congruence. Our results thus provide a detector independent and covariant characterization of gravitational memory. We note some similarities between our results and the study of soft gravitons and gravitational memory on I. * Electronic address: martino@fluiditj.com † Electronic address: demhov@bao.sci.am arXiv:1808.04886v3 [hep-th]

We investigate dynamics of probe particles moving in the near-horizon limit of extremal Myers-Perry black holes in arbitrary dimensions. Employing ellipsoidal coordinates we show that this problem is integrable and separable, extending the results of the odd dimensional case discussed by Hakobyan et al.[Phys. Lett. B 772, 586 (2017).]. We find the general solution of the Hamilton-Jacobi equations for these systems and present explicit expressions for the Liouville integrals and discuss Killing tensors and the associated constants of motion. We analyze special cases of the background near-horizon geometry were the system possesses more constants of motion and is hence superintegrable. Finally, we consider a nearhorizon extremal vanishing horizon case which happens for Myers-Perry black holes in odd dimensions and show that geodesic equations on this geometry are also separable and work out its integrals of motion.

We investigate dynamics of probe particles moving in the near-horizon limit of (2N + 1)-dimensional extremal Myers-Perry black hole (in the cases of N = 3, 4, 5) with arbitrary rotation parameters. Very recently it has been shown [1] that in the most general case with nonequal nonvanishing rotational parameters the system admits separation of variables in N -dimensional ellipsoidal coordinates. We wrote down the explicit expressions of Liouville integrals of motion, given in [1] in ellipsoidal coordinates, in initial "Cartesian" coordinates in seven, nine and eleven dimensions, and found that these expressions hold in any dimension. Then, taking the limit where all of the rotational parameters are equal, we reveal that each of these N − 1 integrals of motion results in the Hamiltonian of the spherical mechanics of a (2N + 1)-dimensional MP black hole with equal nonvanishing rotational parameters.

We describe a procedure naturally associating relativistic Klein-Gordon equations in static curved spacetimes to non-relativistic quantum motion on curved spaces in the presence of a potential. Our procedure is particularly attractive in application to (typically, superintegrable) problems whose energy spectrum is given by a quadratic function of the energy level number, since for such systems the spacetimes one obtains possess evenly spaced, resonant spectra of frequencies for scalar fields of a certain mass. This construction emerges as a generalization of the previously studied correspondence between the Higgs oscillator and Anti-de Sitter spacetime, which has been useful for both understanding weakly nonlinear dynamics in Anti-de Sitter spacetime and algebras of conserved quantities of the Higgs oscillator. Our conversion procedure ("Klein-Gordonization") reduces to a nonlinear elliptic equation closely reminiscent of the one emerging in relation to the celebrated Yamabe problem of differential geometry. As an illustration, we explicitly demonstrate how to apply this procedure to superintegrable Rosochatius systems, resulting in a large family of spacetimes with resonant spectra for massless wave equations.

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