Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Classical Space-Times from the S Matrix

Duff Neill, Ira Z. Rothstein

Abstract

We show that classical space-times can be derived directly from the S-matrix for a theory of massive particles coupled to a massless spin two particle. As an explicit example we derive the Schwarzchild space-time as a series in $G_N$. At no point of the derivation is any use made of the Einstein-Hilbert action or the Einstein equations. The intermediate steps involve only on-shell S-matrix elements which are generated via BCFW recursion relations and unitarity sewing techniques. The notion of a space-time metric is only introduced at the end of the calculation where it is extracted by matching the potential determined by the S-matrix to the geodesic motion of a test particle. Other static space-times such as Kerr follow in a similar manner. Furthermore, given that the procedure is action independent and depends only upon the choice of the representation of the little group, solutions to Yang-Mills (YM) theory can be generated in the same fashion. Moreover, the squaring relation between the YM and gravity three point functions shows that the seeds that generate solutions in the two theories are algebraically related. From a technical standpoint our methodology can also be utilized to calculate quantities relevant for the binary inspiral problem more efficiently than the more traditional Feynman diagram approach.

Classical Space-Times from the S Matrix

Abstract

We show that classical space-times can be derived directly from the S-matrix for a theory of massive particles coupled to a massless spin two particle. As an explicit example we derive the Schwarzchild space-time as a series in . At no point of the derivation is any use made of the Einstein-Hilbert action or the Einstein equations. The intermediate steps involve only on-shell S-matrix elements which are generated via BCFW recursion relations and unitarity sewing techniques. The notion of a space-time metric is only introduced at the end of the calculation where it is extracted by matching the potential determined by the S-matrix to the geodesic motion of a test particle. Other static space-times such as Kerr follow in a similar manner. Furthermore, given that the procedure is action independent and depends only upon the choice of the representation of the little group, solutions to Yang-Mills (YM) theory can be generated in the same fashion. Moreover, the squaring relation between the YM and gravity three point functions shows that the seeds that generate solutions in the two theories are algebraically related. From a technical standpoint our methodology can also be utilized to calculate quantities relevant for the binary inspiral problem more efficiently than the more traditional Feynman diagram approach.

Paper Structure

This paper contains 9 sections, 23 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Building the S-matrix
  3. Definition of the potential
  4. Tree-level Potential
  5. The 1PN correction to the potential
  6. Extracting the metric by going to the probe limit
  7. Discussion
  8. Acknowledgments
  9. Proof of Two Particle Irreducible Nature of the Classical Contribution

Figures (5)

  • Figure 1: Reconstructing the full scalar-scalar S-matrix by sewing together the scalar-scalar n point on shell scattering amplitudes.
  • Figure 2: Graphs with the correct topology to contribute to the classical potential when asymptotically expanded about the potential region.
  • Figure 3: The time ordered product in the effective theory which must be subtracted from the full theory result. The square vertex is the order $v ^2$ Coulomb potential, while the dot corresponds to the order $v^2$ kinetic term correction. The oval is the order $v^4$ Coulomb potential and the cross is the order $v^4$ correction to the kinetic term. Mirror image diagrams have been suppressed. Diagrams $a$ and $b$ are 1PN while $c-h$ are 2PN and have been included only because we are interested in the metric at order $G^2$.
  • Figure 4: The blob is a classical sub-loop.
  • Figure 5: A generic two particle irreducible contribution. The blob is a classical sub-loop.