From Quantum Amplitudes to Spacetime Geometry: a Multipolar Framework for Black Hole Signatures

Abstract

This thesis develops a unified framework that reconstructs the full classical content of General Relativity from the classical limit of quantum scattering amplitudes. By interpreting the analytic structure of amplitudes as the field-theoretic imprint of spacetime geometry, the work establishes a direct correspondence between quantum processes and classical gravitational observables such as metrics, deflection angles, and multipole moments. Starting from the effective-field-theory description of gravity, the thesis shows that loop amplitudes encode not only quantum corrections but also the nonlinear classical self-interaction of the gravitational field, enabling the systematic derivation of the post-Minkowskian expansion of gravitational quantities by rewriting the Einstein equations in terms of graviton scattering processes. Building upon this foundation, the framework is applied to rotating and charged sources in arbitrary spacetime dimensions. A momentum-space formulation of the energy-momentum tensor is then developed, introducing gravitational form factors and source multipoles that link, for the first time, the internal matter distribution to the external multipolar field in a completely relativistic framework. Furthermore, the thesis completes the transition from the microscopic amplitude picture to the macroscopic description of gravitational sources by engineering a multipole-based framework for black hole mimickers, then applied to build horizon-less compact objects mimicking the multipolar structure of Kerr black holes. Finally, exploiting the Kerr-Schild gauge, the Fourier transforms of rotating black hole metrics are computed in closed form, bridging perturbative and non-perturbative descriptions of gravity, and allowing to probe the multipolar structure of higher-dimensional solutions employing scattering amplitudes.

Figures (12)

• Figure 4.1: Diagrams, with corresponding multiplicity, employed for the computation of the PM expansion of the metric up to 1PM and in dipole approximation. The fermion–fermion squared vertex denotes the Pauli coupling.
• Figure 4.2: Diagrams, with corresponding multiplicity, used in the calculation of the PM expansion of the potential up to 1PM, linear in the charge and in dipole approximation. The fermion–fermion squared vertex denotes the Pauli coupling.
• Figure 4.3: Diagrams including the CS interaction (the three-photon vertex), inserted in five dimensions to compute the electromagnetic potential up to 1PM and dipole order.
• Figure 5.1: On the left the energy density and on the right the $\xi_\phi$ variable for the study of the energy condition of the Gaussian-smeared Israel source for ${a=0.8}$, ${z=0}$ and ${\alpha=0.99}$ in units of ${G_N=m=1}$.
• Figure 5.2: On the left the tangential rotational speed and on the right the sound speed in the $\phi$-direction of the Gaussian-smeared Israel source for ${a=0.8}$, ${z=0}$ and ${\alpha=0.99}$ in units of ${G_N=m=1}$.