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VARIATIONAL EQUATIONS ON MIXED RIEMANNIANLORENTZIAN METRICS
, 2008
"... Abstract. A class of elliptichyperbolic equations is placed in the context of a geometric variational theory, in which the change of type is viewed as a change in the character of an underlying metric. A fundamental example of a metric which changes in this way is the extended projective disc, whic ..."
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Abstract. A class of elliptichyperbolic equations is placed in the context of a geometric variational theory, in which the change of type is viewed as a change in the character of an underlying metric. A fundamental example of a metric which changes in this way is the extended projective disc, which is Riemannian at ordinary points, Lorentzian at ideal points, and singular on the absolute. Harmonic fields on such a metric can be interpreted as the hodograph image of extremal surfaces in Minkowski 3space. This suggests an approach to generalized Plateau problems in 3dimensional spacetime via Hodge theory on the extended projective disc. Analogous variational problems arise on RiemannianLorentzian flow metrics in fiber bundles (twisted nonlinear Hodge equations), and on certain RiemannianLorentzian manifolds which occur in relativity and quantum cosmology. The examples surveyed come with natural gauge theories and Hodge dualities. This paper is mainly a review, but some technical extensions are proven. MSC2000: 35M10, 53A10, 83C80 Key words: signature change, projective disc, Minkowski 3space, equations of mixed type, nonlinear Hodge equations
Actions for signature change
, 1995
"... This is a contribution on the controversy about junction conditions for classical signature change. The central issue in this debate is whether the extrinsic curvature on slices near the hypersurface of signature change has to be continuous (weak signature change) or to vanish (strong signature ch ..."
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This is a contribution on the controversy about junction conditions for classical signature change. The central issue in this debate is whether the extrinsic curvature on slices near the hypersurface of signature change has to be continuous (weak signature change) or to vanish (strong signature change). Led by a Lagrangian point of view, we write down eight candidate action functionals S1,...S8 as possible generalizations of general relativity and investigate to what extent each of these defines a sensible variational problem, and which junction condition is implied. Four of the actions involve an integration over the total manifold. A particular subtlety arises from the precise definition of the EinsteinHilbert Lagrangian density g  1/2 R[g]. The other four actions are constructed as sums of integrals over singesignature domains. The result is that both types of junction conditions occur in different models, i.e. are based on different first principles, none of which can be claimed to represent Work supported by the Austrian Academy of Sciences in the framework of the ”Austrian Programme for Advanced Research and Technology”. 0 the ”correct ” one, unless physical predictions are taken into account. From a point of view of naturality dictated by the variational formalism, weak signature change is slightly favoured over strong one, because it requires less à priori restrictions for the class of offshell metrics. In addition, a proposal for the use of the Lagrangian framework in cosmology is made.
Dimensionality, topology, energy, the cosmological constant, and signature change
, 1995
"... Using the concept of real tunneling configurations (classical signature change) and nucleation energy, we explore the consequences of an alternative minimization procedure for the Euclidean action in multipledimensional quantum cosmology. In both standard HartleHawking type as well as Coleman typ ..."
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Using the concept of real tunneling configurations (classical signature change) and nucleation energy, we explore the consequences of an alternative minimization procedure for the Euclidean action in multipledimensional quantum cosmology. In both standard HartleHawking type as well as Coleman type wormholebased approaches, it is suggested that the action should be minimized among configurations of equal energy. In a simplified model, allowing for arbitrary products of spheres as Euclidean solutions, the favoured spacetime dimension is 4, the global topology of spacelike slices being S 1 ×S 2 (hence predicting a universe of KantowskiSachs type). There is, however, some freedom for a KaluzaKlein scenario, in which case the observed spacelike slices are S 3. In this case, the internal space is a product of twospheres, and the total spacetime dimension is 6, 8, 10 or 12.
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, 2008
"... Distinguished solutions for discontinuous signature change with weak junction conditions ..."
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Distinguished solutions for discontinuous signature change with weak junction conditions