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Spinning Black Holes with Scalar Hair and Horizonless Compact Objects within and beyond General Relativity

The last years have brought upon us a golden age of observational gravitational physics. The several observations by the LIGO/Virgo/KAGRA collaboration about gravitational waves and by the EHT collaboration about the shadow and lensing of light around the supermassive black hole in the centre of M87 will point the scientific community in the correct direction to find an answer to the Kerr hypothesis. In order to follow that direction, the systematic construction and analysis of the physical properties of solutions within General Relativity with additional fields or within modified theories of gravity is necessary. In this thesis, we shall provide such construction and analysis for compact objects within (complex-)Einstein-Klein-Gordon theory with various scalar potentials and within a particular scalar-tensor theory -- the shift-symmetric Horndesky theory. After a brief introduction to some key topics that shall be useful throughout this thesis, we present a discussion about the horizon geometry of Kerr black holes with and without scalar hair. We follow up with the construction and study of the same hairy solutions discussed in the previous chapter but with higher azimuthal harmonic indexes. In the following two chapters, we introduce a different scalar potential based on the Quantum Chromodynamic axion potential and obtain and study both horizonless compact objects and black holes. We then go to the shift-symmetric Horndeski theory, where we perform similar constructions and analyses to the ones already mentioned. Lastly, we derive a relation between the radial stability of light-rings and timelike circular orbits around them. We follow up with a study on how efficient it is the conversion of gravitational energy in radiation as a timelike particle falls towards all compact objects studied throughout this thesis. We end with some conclusions and remarks.