Super-Geometrodynamics

TL;DR

This paper derives explicit harmonic function-based solutions for initial data in supergravity theories, enabling the study of their time evolution and extending geometrodynamics to supergravity contexts.

Contribution

It provides explicit harmonic function solutions for supergravity initial data, generalizing geometrodynamics to include supergravity theories with multiple fields.

Findings

Solutions expressed in harmonic functions for supergravity initial data.

Analysis of initial data for Einstein-Maxwell-Dilaton and STU models.

Connection between static black hole solutions and harmonic function representations.

Abstract

We present explicit solutions of the time-symmetric initial value constraints, expressed in terms of freely specfiable harmonic functions for examples of supergravity theories, which emerge as effective theories of compactified string theory. These results are a prequisite for the study of the time-evolution of topologically non-trivial initial data for supergravity theories, thus generalising the "Geometrodynamics" program of Einstein-Maxwell theory to that of supergravity theories. Specifically, we focus on examples of multiple electric Maxwell and scalar fields, and analyse the initial data problem for the general Einstein-Maxwell-Dilaton theory both with one and two Maxwell fields, and the STU model. The solutions are given in terms of up to eight arbitrary harmonic functions in the STU model. As a by-product, in order compare our results with known static solutions, the metric in isotropic coordinates and all the sources of the non-extremal black holes are expressed entirely in terms of harmonic functions. We also comment on generalizations to time-nonsymmetric initial data and their relation to cosmological solutions of gauged so-called fake supergravities with positive cosmological constant.