Classical observables of General Relativity from scattering amplitudes

I will be using scattering amplitudes, instead of the Lagrangian of General Relativity (GR), to compute classical observables in GR. In the first part of the seminar I will consider the elastic scattering of two massive particles, describing two black holes, and I will show how to compute the eikonal up to two-loop order, corresponding to third Post-Minkowskian (3PM) order, that contains all the classical information. From it I will compute the first observable that is the classical deflection angle. In the second part of the seminar I will consider inelastic processes with the emission of soft gravitons. In this case the eikonal becomes an operator containing the creation and annihilation operators of the gravitons. The case of soft gravitons can be treated following the Bloch-Nordsieck approach and, in this case, I will be computing two other observables: the zero-frequency limit (ZFL) of the spectrum dE/domega of the emitted radiation and the angular momentum loss at 2PM and 3PM. I will consider also the case in which there are static modes localised at $\omega=0$. In the third part of the seminar I will be discussing soft theorems with one graviton emission, first briefly at tree level, and then at loop level following the approach of Weinberg from 1965 and comparing it with a more recent approach by Laddha, Saha, Sahoo and Sen. I will compute the universal soft terms of the waveform, that are $\frac{1}{\omega}$, $\log \omega$ and $\omega \log^2 \omega$, first at the tree and one-loop level and then also at two-loop level. Then I will present a guess for the terms $\omega^n \log^{n+1} \omega$ for $n=2, 3 …$ and I will give a formula where all these terms are resummed. Finally, if I have time left, I will study the high energy limit. In particular, since the graviton is the massless particle with the highest spin, we expect universality at high energy in any gravitational amplitude. I will show that universality at high energy is satisfied both in the elastic and inelastic case, but this happens in the inelastic case in a very non trivial way. I will end with some conclusions and with a list of open problems.