Well-posedness of (N = 1) classical supergravity
In this paper we investigate whether classical (N = 1) supergravity has a well-posed locally causal Cauchy problem. We define well-posedness to mean that any choice of initial data (from an appropriate function space) which satisfies the supergravity constraint equations and a set of gauge conditions can be continuously developed into a space-time solution of the supergravity field equations around the initial surface. Local causality means that the domains of dependence of the evolution equations coincide with those determined by the light cones. We show that when the fields of classical supergravity are treated as formal objects, the field equations are (under certain gauge conditions) equivalent to a coupled system of quasilinear nondiagonal second-order partial differential equations which is formally nonstrictly hyperbolic (in the sense of Leray--Ohya). Hence, if the fields were numerical valued, there would be an applicable existence theorem leading to well-posedness. We shall observe that well-posedness is assured if the fields are taken to be Grassmann (i.e., exterior algebra) valued, for then the second-order system decouples into the vacuum Einstein equation and a sequence of numerical valued linear diagonal strictly hyperbolic partial differential equations which can be solved successively.
- Research Organization:
- School of Mathematics, The Institute for Advanced Study, Princeton, New Jersey 08540, and Department of Mathematics, University of Houston-University Park, Houston, Texas 77004
- OSTI ID:
- 6210012
- Journal Information:
- J. Math. Phys. (N.Y.); (United States), Vol. 26:2
- Country of Publication:
- United States
- Language:
- English
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GENERAL PHYSICS
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
SUPERGRAVITY
CAUCHY PROBLEM
PARTIAL DIFFERENTIAL EQUATIONS
ALGEBRA
ANALYTICAL SOLUTION
BOSONS
BOUNDARY CONDITIONS
CAUSALITY
EINSTEIN FIELD EQUATIONS
FERMIONS
FIELD EQUATIONS
GAUGE INVARIANCE
GRAVITATION
ITERATIVE METHODS
LIGHT CONE
MATHEMATICAL OPERATORS
RELATIVITY THEORY
SPACE-TIME
SPINORS
SUPERSYMMETRY
DIFFERENTIAL EQUATIONS
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FIELD THEORIES
GENERAL RELATIVITY THEORY
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MATHEMATICS
SYMMETRY
UNIFIED-FIELD THEORIES
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