Conserved Charges in Higher Derivative Gravity

One of the very first predictions extracted from general relativity is the existence of black-hole solutions. Initially regarded as mathematical curiosities, over the decades black holes have been taking a more and more central role in the study of gravity, both experimentally and theoretically. From the theoretical point of view, black holes are regarded as the natural testing ground for the fundamental properties of gravity in the classical, semi-classical and even (to the extent that we understand it) quantum regime. This thesis investigates the effective field theory (EFT) extension of gravity, specifically focusing on a theory that includes cubic-order (six-derivative) terms which preserve parity symmetry. Due to the complexity of finding the solutions in this advanced theory, the research concentrates on a particular region of spacetime, the near-horizon extremal geometry (NHEG) of black holes. This thought comes from the Bekenstein-Hawking formula, which relates the black hole entropy to its horizon area, indicating that the thermodynamic properties can be derived from the horizon. The extremality, also, is beneficial since it prepares us with more symmetries than the exact solution. To have a compatible definition of entropy in the presence of higher-order terms, we use the Iyer-Wald entropy. By calculating entropy within this modified theory, the thesis clarifies how the extension of the Lagrangian influences black hole thermodynamics. Additionally, the study develops a generalized Komar integral for this theory, providing valuable insights into the calculation of the angular momentum. The main body of the work involves detailed calculations of entropy and angular momentum within the Einstein gravity framework and the modified gravity theory up to the six-derivative terms. The findings reveal that these corrections impact the structure of the NHEG and modify the relationship between entropy and angular momentum.