Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extrapolating the massive fields to future timelike infinity

Wen-Bin Liu, Jiang Long

Abstract

It is well-known that future timelike infinity ($i^+$) in four-dimensional Minkowski spacetime is conformal to the unit three-dimensional hyperboloid ($H^3$). We asymptotically expand massive fields with spin $0,1,2$ near $i^+$ and extrapolate them onto this hyperboloid. These fields oscillate with a frequency equal to their mass and exhibit a universal asymptotic decay $τ^{-3/2}$. The fundamental fields are free and encode the outgoing scattering data. They are local operators defined on the boundary $H^3$ with which we construct the Poincaré charges. The Poincaré algebra can be extended to $\text{MDiff}(H^3)\ltimes C^{\infty}(H^3)$ using smeared operators associated with energy and angular momentum densities. For spinning fields, a spin operator must be included to close the algebra. The extended algebra shares the same form as the five-dimensional intertwined Carrollian diffeomorphism and reduces to the BMS algebra at $i^+$ by restricting the choice of test functions and vectors.

Extrapolating the massive fields to future timelike infinity

Abstract

It is well-known that future timelike infinity () in four-dimensional Minkowski spacetime is conformal to the unit three-dimensional hyperboloid (). We asymptotically expand massive fields with spin near and extrapolate them onto this hyperboloid. These fields oscillate with a frequency equal to their mass and exhibit a universal asymptotic decay . The fundamental fields are free and encode the outgoing scattering data. They are local operators defined on the boundary with which we construct the Poincaré charges. The Poincaré algebra can be extended to using smeared operators associated with energy and angular momentum densities. For spinning fields, a spin operator must be included to close the algebra. The extended algebra shares the same form as the five-dimensional intertwined Carrollian diffeomorphism and reduces to the BMS algebra at by restricting the choice of test functions and vectors.

Paper Structure

This paper contains 43 sections, 274 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Realizing Poincaré algebra through massive fields
  3. Massive scalar
  4. Background spacetime
  5. Massive scalar and its asymptotic expansion
  6. Remark.
  7. Stress tensor and Poincaré charges
  8. Poincaré charge algebra
  9. Proca field
  10. Proca field and its asymptotic expansion
  11. Asymptotic equation of motion
  12. Stress tensor and Poincaré charges
  13. Poincaré charge algebra
  14. Massive Fierz-Pauli field
  15. Foundations
  16. ...and 28 more sections

Figures (2)

  • Figure 1: Two diagrams for the asymptotically Minkowski spacetime. The left one is the standard Penrose diagram, while the right one is schematically an alternative description. In the latter diagram, more structure can be depicted, such as the joint corners of different asymptotic regions. Moreover, two geodesics from bulk to $i^+/\mathcal{I}^+$ are drawn with the colors of blue/brown which describe massive/massless particles, while the red geodesics approach $i^0$. Note that the right diagram is similar to the one in Compere:2023qoa except that we keep the null infinity represented by an oblique line.
  • Figure 2: In this figure, we show two kinds of scattering processes involving massive particles. In the left diagram, $m$ ingoing particles located originally at $y_1,\cdots,y_m$ become $n$ outgoing particles after scattering, and eventually arrive the location $y_{m+1},\cdots,y_{m+n}$ at $i^+$. In the right diagram, we depict a scattering process with input of $m_1=1$ massless particle coming from $\mathcal{I}^-$ and $m_2=1$ massive particle coming from $i^-$, and the outputs are $n_1=1$ massless particle going to $\mathcal{I}^+$ and $n_2=1$ massive particle going to $i^+$.