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Black hole mass dynamics and renormalization group evolution

Walter D. Goldberger, Andreas Ross, Ira Z. Rothstein

TL;DR

The paper develops a real-time effective field theory for gravitating black-hole systems in the long-wavelength limit using the in-in formalism, promoting the mass parameter to a dynamical degree of freedom due to classical renormalizability. It derives and solves renormalization-group equations for the mass $M$ and the quadrupole $I_{ij}$, showing that logarithmic UV divergences generate running with the RG scale and resumming leading logarithms yields corrections to the gravitational-wave emission amplitude and the conservative energy $E(\Omega)$. The results connect the mass RG flow to the Bondi mass and provide a systematic way to include higher-order logarithms in the energy distribution of radiated gravitons, improving high-precision gravitational-wave templates. The framework also opens paths to extend to higher multipole moments and to quantum black holes, with potential implications for Hawking radiation and the IR/UV structure of gravity in radiative processes.

Abstract

We examine the real-time dynamics of a system of one or more black holes interacting with long wavelength gravitational fields. We find that the (classical) renormalizability of the effective field theory that describes this system necessitates the introduction of a time dependent mass counterterm, and consequently the mass parameter must be promoted to a dynamical degree of freedom. To track the time evolution of this dynamical mass, we compute the expectation value of the energy-momentum tensor within the in-in formalism, and fix the time dependence by imposing energy-momentum conservation. Mass renormalization induces logarithmic ultraviolet divergences at quadratic order in the gravitational coupling, leading to a new time-dependent renormalization group (RG) equation for the mass parameter. We solve this RG equation and use the result to predict heretofore unknown high order logarithms in the energy distribution of gravitational radiation emitted from the system.

Black hole mass dynamics and renormalization group evolution

TL;DR

The paper develops a real-time effective field theory for gravitating black-hole systems in the long-wavelength limit using the in-in formalism, promoting the mass parameter to a dynamical degree of freedom due to classical renormalizability. It derives and solves renormalization-group equations for the mass and the quadrupole , showing that logarithmic UV divergences generate running with the RG scale and resumming leading logarithms yields corrections to the gravitational-wave emission amplitude and the conservative energy . The results connect the mass RG flow to the Bondi mass and provide a systematic way to include higher-order logarithms in the energy distribution of radiated gravitons, improving high-precision gravitational-wave templates. The framework also opens paths to extend to higher multipole moments and to quantum black holes, with potential implications for Hawking radiation and the IR/UV structure of gravity in radiative processes.

Abstract

We examine the real-time dynamics of a system of one or more black holes interacting with long wavelength gravitational fields. We find that the (classical) renormalizability of the effective field theory that describes this system necessitates the introduction of a time dependent mass counterterm, and consequently the mass parameter must be promoted to a dynamical degree of freedom. To track the time evolution of this dynamical mass, we compute the expectation value of the energy-momentum tensor within the in-in formalism, and fix the time dependence by imposing energy-momentum conservation. Mass renormalization induces logarithmic ultraviolet divergences at quadratic order in the gravitational coupling, leading to a new time-dependent renormalization group (RG) equation for the mass parameter. We solve this RG equation and use the result to predict heretofore unknown high order logarithms in the energy distribution of gravitational radiation emitted from the system.

Paper Structure

This paper contains 5 sections, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Renormalization and time dependence
  3. Renormalization group flows
  4. Applications
  5. Conclusions

Figures (3)

  • Figure 1: Leading diagram topologies for the computation of $\left<T^{\mu\nu}\right>$.
  • Figure 2: Leading diagram topologies which yield logarithmically UV divergent contributions to $\left<T^{\mu\nu}\right>$.
  • Figure 3: An on shell graviton is emitted and scattered back ( denoted by the small box) at a distance $\rho$. Observers at different distance would not agree on the value of the mass.