Actions for signature change

Abstract

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 ({\it weak} signature change) or to vanish ({\it strong} signature change). Led by a Lagrangian point of view, we write down eight candidate action functionals $S_1$,\dots $S_8$ 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 Einstein-Hilbert Lagrangian density $|g{|}^{1/2}R[g]$. The other four actions are constructed as sums of integrals over singe-signature domains. The result is that {\it 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 the ''correct'' one, unless physical predictions are taken into account. From a point of view of naturality dictated by the variational formalism, {\it weak} signature change is slightly favoured over {\it strong} one, because it requires less {\it à priori} restrictions for the class of off-shell metrics. In addition, a proposal for the use of the Lagrangian framework in cosmology is made.

36 pages, LaTeX, no figures; some corrections have been made, several Comments and further references are included and a note has been added.

References in corpus (10)

• The Scalar Field Equation in the Presence of Signature Change
• Failure of Standard Conservation Laws at a Classical Change of Signature
• Hamiltonian quantization of General Relativity with the change of signature
• On Cosmological Isotropy, Quantum Cosmology and the Weyl Curvature Hypothesis
• Black holes, cosmological singularities and change of signature
• The Geometry of classical change of signature
• Junction Conditions for Signature Change
• Signature change induces compactification
• The Patchwork Divergence Theorem
• COMMENT ON ``BOUNDARY CONDITIONS FOR THE SCALAR FIELD IN THE PRESENCE OF SIGNATURE CHANGE''

Cited by in corpus (20)

• Complex actions in two-dimensional topology change
• From Euclidean to Lorentzian General Relativity: The Real Way
• Hypersurface-deformation algebroids and effective space-time models
• Gravity and Signature Change
• Einstein's Equations in the Presence of Signature Change
• Metrics with distributional curvature
• Fundamental units of length and time
• Lagrangian constraints and renormalization of 4D gravity
• Holographic quantization of gravity in a black hole background
• Cosmological perturbations and classical change of signature
• Variational equations on mixed Riemannian-Lorentzian metrics
• Signature change induces compactification
• Comment on `Smooth and Discontinuous Signature Type Change in General Relativity'
• Complex lapse, complex action and path integrals
• Riemann-Cartan gravity with dynamical signature
• On metric-connection compatibility and the signature change of space-time
• Dimensionality, topology, energy, the cosmological constant, and signature change
• Should Metric Signature Matter in Clifford Algebra Formulations of Physical Theories?
• Singularity at the demise of a black hole
• Distinguished solutions for discontinuous signature change with weak junction conditions