Quantum Effects for Black Holes with On-Shell Amplitudes

TL;DR

The paper develops an on-shell amplitude framework for black hole physics, defining Schwarzschild $S$-matrices in the Boulware and Unruh vacua and treating black holes as on-shell states with mass-changing three-point amplitudes. Hawking radiation emerges as a resummation of emission processes within this formalism, and the approach yields a universal description where vacuum choices manifest in amplitude discontinuities. For observables, the KMOC formalism shows that the Hawking spectrum is well captured by three-point amplitudes, and quantum effects in two-body systems are encoded in a universal mass-shift building block, with mean shifts being classical and vacuum-independent while variances depend on the vacuum. The framework offers a gauge-free, scalable route to study black hole quantum effects and binary dynamics, with potential extensions to Kerr and higher-point gravitational observables.

Abstract

We develop a framework based on modern amplitude techniques to analyze emission and absorption effects in black hole physics, including Hawking radiation. We first discuss quantum field theory on a Schwarzschild background in the Boulware and Unruh vacua, and introduce the corresponding $S$-matrices. We use this information to determine on-shell absorptive amplitudes describing processes where a black hole transitions to a different mass state by absorbing or emitting quanta, to all orders in gravitational coupling. This on-shell approach allows for a universal description of black holes, with their intrinsic differences encapsulated in the discontinuities of the amplitudes, without suffering from off-shell ambiguities such as gauge freedom. Furthermore, the absorptive amplitudes serve as building blocks to describe physics beyond that of isolated black holes. As applications, we find that the Hawking thermal spectrum is well understood by three-point processes. We also consider a binary system and compute the mass shift of a black hole induced by the motion of a companion object, including quantum effects. We show that the mean value of the mass shift is classical and vacuum-independent, while its variance differs depending on the vacuum choice. Our results provide confirmation of the validity of the on-shell program in advancing our understanding of black hole physics.

Figures (5)

• Figure 1: Carter-Penrose conformal diagram for Schwarzschild (in the ingoing Eddington-Finkelstein coordinates) and Vaidya spacetimes. The arrows represent the boundary conditions of: in, out, up, down, and dn modes. The Boulware vacuum can be constructed only by the mode outside of the horizon (red), while the Unruh vacuum, which agrees with the vacuum of the Vaidya spacetime, requires the "dn" mode (blue).
• Figure 2: The black hole $X$-state as a composite particle made of the background black hole and the injected quanta in the Boulware vacuum.
• Figure 3: The black hole $X$-state as a composite particle formed by the background black hole together with the injected and pair-created (down-dn) quanta in the Unruh vacuum. Since down–dn pair production preserves the mass while modifying the internal structure, multiple such pair creations are included in the $X$-state.
• Figure 4: Left: the Hawking thermal distribution and the distribution obtained only via the three-point amplitudes. Right: contributions from $3, 4,$ and $5$ point amplitudes.
• Figure 5: Left: the comparison of the cross-sections in Eq. \ref{['eq:effective_cross_section_for_B_and_U']}. Right: relative error of \ref{['eq:effective_cross_section_for_B_and_U']}.