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.
Highlights
We present explicit solutions of the time-symmetric initial value constraints, expressed in terms of freely specifiable harmonic functions for examples of supergravity theories, which emerge as effective theories of compactified string theory
We give a direct construction of a general new class of examples of Einstein gravity coupled to a system of scalar fields, and we show how the Einstein-Scalar systems obtained as mappings from systems with Maxwell fields are all special cases within this broader class
Maxwell-Dilaton theory with a general dilation coupling a, which can be expressed in terms of three harmonic functions
Summary
The purpose of this section is to review the formalism for the study of the time-evolution problem for theories depending upon a metric gμν, one or more scalars φα, and one or more closed two-forms, or Maxwell fields, F I = dAI whose equations of motion may be obtained from an action functional S[gμν, φα, AIμ] that is invariant under the semi-direct product of diffeomorphisms and gauge transformations. In subsection 2.1 we present the evolution equations and derive constraints, and in subsection 2.2 give the explicit form of constraints. In the subsection 2.3 we address timesymmetric date and present the well known explicit results for the vacuum Einstein gravity and Einstein-Maxwell gravity. In the subsequent sections we shall focus on new results for an Einstein-Maxwell-Dilaton gravity model and STU models with multiple scalars and Maxwell fields
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