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The Dirac Equation in the Schwarzschild Metric

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Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2025, Tutor: Cristiano Germani

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This thesis studies the possibility of a primordial black hole (PBH) trapping an electron in a stable orbit, potentially allowing detection through the emitted electromagnetic radiation from the accelerated charge. Firstly, the Dirac equation is analysed in the Minkowski metric, generalising afterwards the formulation from flat to curved backgrounds. Using the separation of variables method, it is demonstrated that no non-trivial stationary solutions exist outside the horizon, in accordance with the no-hair theorem. Meanwhile, non-stationary solutions do exist
and are unique for smooth initial conditions. Lastly, the spinor norm’s time evolution is analysed to identify bound states, revealing two quantised circular orbits dependent on the fermion’s mass and angular momentum. The expression for the quantised radius, velocity, and energy has also been derived. With this information, one can calculate the energy spectrum for a trapped electron around a PBH, giving a way of experimentally proving the existence of primordial black holes

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The Dirac Equation in the Schwarzschild Metric