MSC defense by Tobias Barfod Kristiansen

Tidally deformed extreme mass ratio binary system of Schwarzschild black holes

Tides are ubiquitous in nature. Indeed, any extended object in a non-uniform gravitational field will be subject to tidal effects, whether it be the moon giving Earth its tides or entire galaxies giving each other tidal tails. A third example of particular interest in the age of gravitational wave astronomy, is the tidal interaction between black holes. Although this thesis wont go into detail regarding the effects of tides on gravitational wave signals, we will cover the theoretical foundations of tidal interactions in EMR binary systems of Schwarzschild black holes. We consider a large Schwarzschild black hole, referred to as the background black hole, and a much smaller black, referred to as the (tidally) deformed black hole, in orbit around the background black hole.

As we will see, a vacuum region of an arbitrary spacetime can be described by a set of tidal moments. In particular, the tidal moments of the background spacetime will serve as building blocks for the metric around the tidally deformed black hole. The resulting metric is referred to as the Poisson-Vlasov metric and will serve as the foundation of much of this thesis. We mainly work under the assumption that the deformed black hole follows a radial geodesic in the background spacetime, implying that all magnetic tidal moments and potentials vanish identically. We compute the tidal shifts in the ISCO parameters of a test-particle orbiting the deformed black hole to quadrupole and octupole order. Furthermore, we compute the specific energy of the test-particle as a function of the Euler angles that specify the orientation of the ”deformed black hole + test-particle” binary system with respect to the background black hole. We find that the specific energy is minimized for co-planar orbits, i.e. configurations for which the inclination angle vanishes.

Furthermore, we compute the specific energy of the test-particle to octupole order. In particular, the specific energy is found to be increasing as a function of advanced time. Finally, we study the geometry of the deformed horizon. With the precision maintained in this text, the horizon is located at r = 2m as in the unperturbed case. However, the mass of the deformed black hole now acquires a non-trivial time-dependence. Using the approach of Poisson, we compute the change in m to leading order for a radial infall and for a circular orbit.