Lorentzian and signature changing branes

Abstract

General hypersurface layers are considered in order to describe brane-worlds and shell cosmologies. No restriction is placed on the causal character of the hypersurface which may thus have internal changes of signature. Strengthening the results in our previous letter [1], we confirm that a good, regular and consistent description of signature change is achieved in these brane/shells scenarios, while keeping the hypersurface and the bulk completely regular. Our formalism allows for a unified description of the traditional timelike branes/shells together with the signature-changing, or pure null, ones. This allows for a detailed comparison of the results in both situations. An application to the case of hypersurface layers in static bulks is presented, leading to the general RobertsonWalker geometry on the layer -with a possible signature change. Explicit examples on anti de Sitter bulks are then studied. The permitted behaviours in different settings (Z2-mirror branes, asymmetric shells, signature-changing branes) are analysed in detail. We show in particular that (i) in asymmetric shells there is an upper bound for the energy density, and (ii) that the energy density within the brane vanishes when approaching a change of signature. The description of a signature change as a 'singularity' seen from within the brane is considered. We also find new relations between the fundamental constants in the brane/shell, its tension, and the cosmological and gravitational constants of the bulk, independently of the existence or not of a change of signature.

Equations for general shells

“…The latter formula is the generalization of the Israel equation to general shells, first found in [29] (see also [31,27]), and it reduces to the one found in [5] for null shells when (n · n) = 0, and to the original Israel equation [18] for non-null shells in the canonical gauge:…”

“…I assume that (M, g) has been constructed as the junction of two given spacetimes (M ± , g ± ) across a matching hypersurface Σ, which is the shell, and that the resulting spacetime has a continuous metric g (so that, in appropriate coordinates, g + Σ = g − ). This requires that the first fundamental formsḡ ± inherited by Σ from (M ± , g ± ) agree and, in the case that Σ is null somewhere, furthermore the existence of transversal vectors fields ℓ ± on each side of Σ, ℓ ± having the same scalar products with any given basis of tangent vectors at any x ∈ Σ, and such that ℓ + points towards M + while ℓ − points outwards from M − (see [31], which corrects previous partial statements in [9,29]).…”

“…Some examples of relevant general hypersurfaces are, for instance, the spherically symmetric apparent horizon in Vaidya's spacetime -if the radiation stops at some times; more generally, dynamical horizons which eventually merge with the event horizon in asymptotically flat black-hole spacetimes; any achronal boundary, that is to say, the boundary of any future set [16]; Kerr's stationary limit surface, which is timelike everywhere but tangent to the event horizon at two lines intersecting the axis of symmetry, where it is null [16,48]; the interfaces arising in phase transitions if they occur in a concentrated spatial region and then propagate causally; signature changing braneworlds; and there are many others.Thus, for instance, if one wished to analyze the possibility of placing a surface layer on the stationary limit surface of a Kerr black hole, this should be done on a general, signaturechanging, hypersurface. Similarly, descriptions of signature change in the brane scenario -if we happen to live in a brane of a higher-dimensional world-are described by general shells, see [30,31,32]. The case of achronal boundaries is of particular interest because intuitive arguments lead to the expectation that an initial value problem for the gravitational field equations on such boundaries is well posed, see the interesting discussion in [27].…”

Gravitational double layers

“…For later reference, I write the GR energy-momentum quantities as they would be computed by scientists living on Σ but unaware that this is actually a double layer of a higher dimensional bulk, so that they would use GR as their gravitational theory along the lines explained in more detail in [26,27]. These are denoted by ̺ GR (GR energy density), p GR (GR pressure) and Λ GR (GR 4-dimensional cosmological constant) and given by…”

Nonstandard fractional exponential Lagrangians, fractional geodesic equation, complex general relativity, and discrete gravity

“…In general, Einstein's field equations do not impose any restriction on the space-time signature as the signature is unchanged and remains lorentzian. A signature flip also arises in brane-world scenario [65][66][67] and in loop quantum gravity [68][69][70]. Nevertheless much work remains concerning the signature flip.…”