Signature Changing Space-times and the New Generalised Functions

A signature changing spacetime is one where an initially Riemannian manifold with Euclidean signature evolves into the Lorentzian universe we see today. This concept is motivated by problems in causality implied by the isotropy and homogeneity of the universe. As initially time and space are indistinguishable in signature change, these problems are removed. There has been some dispute as to the nature of the junction conditions across the signature change, and in particular, whether or not the metric is continuous there. We determine to what extent the Colombeau algebra of new generalised functions resolves this dispute by analysing both types of signature change within its framework. A covariant formulation of the Colombeau algebra is used, in which the usual properties of the new generalised functions are extended. Point values of the new generalised functions are shown to form a field and used in the analysis of signature change. We find that the Colombeau algebra is insufficient to preclude either continuous or discontinuous signature change, and is also unable to settle the dispute over the nature of the junction conditions. 1 Signature Change in Cosmology The topic of signature change is motivated by problems with causality implied by the observed isotropy and homogenity of the universe. Quantum cosmology[10] presents us with the possibility that although space-time is currently Lorentzian (i.e. psuedo-Riemannian), the universe may have evolved from an initial state where space-time was Euclidean (i.e. Riemannian) in nature. These so-called signature changing space-times do not possess an initial singularity. Further, as there is now (initially) no distinction between time and space, the problems involving causality are removed. Signature changing cosmologies are characterised by a division into a Euclidean region and a Lorentzian region. The two regions are separated by a spatial hypersurface. The difficulties in dealing with a signature changing cosmology arise when one comes to the (delicate) matter of analysing quantities on or across the boundary hypersurface. In general, one obtains junction conditions across the hypersurface by requiring that certain Preprint ADP-00-11/M90 quantities are well-defined. Of course, this will depend on what one requires to be welldefined, and what one means by well-defined. Typically, junction conditions are obtained by requiring continuity of a particular field or its derivatives. In the case of signature change, it would seem that requiring a continuous metric might be a natural condition. However, if we demand that the lapse is continuous, then this requires that it vanishes on the boundary hypersurface, and therefore the metric is degenerate at that point (and thus the inverse metric is singular). On the other hand, if we allow a discontinuous lapse, this requires a distributional metric in order for the derivatives of the metric to be well defined. Whether one of these conditions is better or more natural than the other has been the matter of some dispute. There have been various treatments of the subject, some arguing for continuous (strong) signature change [6], some discontinuous (weak) signature change [5], and some regarding both as equally valid. We note that in both continuous and discontinuous signature change it is possible to analyse the situation within a distributional framework. However, in a distributional framework field equations such as the Klein-Gordon or Einstein equations may contain products and quotients of distributions, which are not well defined. This brings us to the subject of Colombeau algebras, also called the new generalised functions. Within the Colombeau framework, rigorous meaning is giving to non-linear operations on distributions[4], which can be extended to tensor distributions[9]. In signature change, where formal calculations involving distributional products have occurred previously, it has been claimed[7] that any difficulties there might be solved by application of the Colombeau algebra. It is our aim to determine the extent to which this is true. We analyse both continuous and discontinuous signature change within the framework of new generalised functions. We begin by defining a version of the new generalised functions suited to our needs. Subsequently we analyse continuous signature change, and are forced to develop a means of dividing by generalised functions. To finish, we perform a similar analysis within discontinuous signature change. Continuity conditions are derived in both cases. 2 Colombeau Algebras Originally, Colombeau[3] developed the space of new generalised functions to deal with products of distributions that occur in quantum field theory. Since then, there have been several variants of the new generalised functions presented. In this section, we construct an algebra of new generalised functions, G, that is based upon one of these presentations. These new generalised functions can be freely summed, multiplied and differentiated. Also, we will show that the smooth functions, continuous functions and distributions can be embedded in G, and their properties are generalised in a consistent fashion. In addition, we construct the field of generalised numbers, where our generalised functions will take point values. The main idea behind G is that each element has some “microscopic” structure, whose description is lacking in the distributions, which allows us to resolve the ambiguity in multiplication. 2.1 The algebra of new generalised functions The formulation of G given here is based upon a simplified presentation given by Colombeau [4]. It is well known that a distribution can be considered as the limit of a sequence of test functions. In a similar fashion, we will make use of the smooth function space, and define Colombeau objects as an ideal limit of a sequence of smooth functions. As we will perform calculations on curved space time, we require that the formulation of the Colombeau algebra be invariant under general coordinate transformations (diffeomorphisms). Space-time is assumed to be an n-dimensional differentiable manifold, M , whose tangent bundle is denoted TM . The set of sections (vector fields) on the tangent bundle is denoted Γ(TM), and the Lie derivative with respect to the vector field V is denoted £V . The elements of the Colombeau algebra, G, are one-parameter families of moderate smooth functions modulo negligible families. For those familiar with other presentations of Colombeau algebras[1, 4], note that we have provided the additional requirement that our families of smooth functions be continuously parameterised. Definition 2.1. The space of moderate functions is the set of continuous one-parameter families of smooth functions defined by C M (M) = {(fǫ)|fǫ ∈ C (M) such that ∀ compact K ⊂ M, ∀ {X1, . . . , Xp}, p ≥ 0 with Xi ∈ Γ(TM) and [Xi, Xj] = 0, ∃ N ∈ N, ∃ η > 0, ∃ c > 0, such that sup x∈K |£X1 . . .£Xpfǫ(x)| ≤ c ǫN for 0 < ǫ < η}. Definition 2.2. The space of negligible functions is the set of continuous one-parameter families of smooth functions defined by C N (M) = {(fǫ)|fǫ ∈ C ∞ M (M) such that ∀ compact K ⊂ M, ∀ {X1, . . . , Xp}, p ≥ 0 with Xi ∈ Γ(TM) and [Xi, Xj] = 0, ∀ q ∈ N, ∃ η > 0, ∃ c > 0, such that sup x∈K |£X1 . . .£Xpfǫ(x)| ≤ cǫ q for 0 < ǫ < η}. Definition 2.3. The space of new generalised functions is defined as G(M) = C M (M) C N (M) . Operations in this formulation of G are relatively straightforward. Addition, subtraction and multiplication of Colombeau objects are simply defined in terms of the corresponding operations upon their representatives. Multiplication by scalars and partial differentiation are similarly defined in the obvious way. We now turn to the embedding of the continuous functions and distributions within G. The embedding is performed by convoluting with a smoothing kernel. The smoothing kernel is an approximate delta-function in order to provide a good generalisation of the classical function product[4]. Specific use of the tangent bundle is made to preserve diffeomorphic invariance[1]. Given coordinates {x} on M , we have an induced basis for TM defined by the coordinate derivative fields. Denote this basis by {(x, ξ)}. The set of test functions is denoted by D and the distributions by D. Definition 2.4. (1) Given q ∈ N, we define the set Aq = {φ ∈ D(R )| ∫ dξ φ(ξ) = 1, and ∫ dξ φ(ξ)ξ = 0 ∀ i ∈ N, with 1 ≤ |i| ≤ q}. (2) Given φ ∈ Aq, x ∈ M , we define the function φǫ,x ∈ Aq by φǫ,x(ξ) = 1 ǫn φ( ξ−x ǫ ). Definition 2.5. Given f ∈ C(M), we define its associated generalised function f̃ ∈ G(M) as having a representative (fǫ)0<ǫ<1 where fǫ(x) = ∫ dξφǫ,x(ξ)f(ξ) = ∫ dξφ(x)f(x+ ǫξ), φ ∈ Aq. Definition 2.6. Given T ∈ D(M), we define its associated generalised function T̃ ∈ G(M) as having a representative (Tǫ)0<ǫ<1 where Tǫ(x) = 〈T, φǫ,x〉. If T has a density, g, then Tǫ(x) ≡ gǫ(x) = ∫ dξφǫ,x(ξ)g(ξ) = ∫ dξφ(x)g(x+ ǫξ), φ ∈ Aq. Smooth functions on M are naturally embedded into G as constant sequences, fǫ = f . But C is a subset of continuous function space, C, and thus subject to the above embedding. We would like the two embeddings to coincide. By Taylor-expanding f in the above formula, it is clear that the moment conditions on φ will guarantee that the first q non-constant terms vanish. A sufficient condition that the two different embeddings coincide for all smooth functions is requiring that φ ∈ A∞. However, we cannot find such a function[4], so rather than choosing a specific smoothing kernel, we allow all functions in Aq, with q arbitrarily large. Distribution space, D, may be extracted from G using an equivalence relation on G called association. Definition 2.7. Given f, g ∈ G(M), if ∀ ψ ∈ D(M), lim ǫ→0 ∫ (fǫ(x)− gǫ(x))ψ(x)dx = 0, then we say the two generalised functions f and g are associated, denoted f ≈ g. Definition 2.8. Given f ∈ G(M), we set 〈f̄ , ψ〉 = limǫ→0 ∫ fǫ(x)ψ(x)dx. If the limit exists ∀ ψ ∈ D(M), then f̄ is a distribution defined by this relation. We say that f has an associated distribution, f̄ . It is clear that given two associated generalised functions, then if they project onto D, they will have the same associated distribution. Furthermore, as an equivalence relation it is clear that the operations of addition, subtraction, derivation and multiplication by scalars are respected by association. That is, acting identically on two associated elements by one of these operations preserves association. Also, multiplication by smooth functions (that do not depend on ǫ) will also preserve association. In this way one can consider association as being equivalent to distributional equality. However, note that multiplication by generalised functions does not preserve association. As C(M) ⊂ D(M), the projection of G onto D defines an indirect projection onto C. However, C is not a subalgebra of G. So if f1, f2 ∈ C(M), then in G, f̃1 · f̃2 6= f̃1f2 in general. However, we do have that f̃1 · f̃2 ≈ f̃1f2. This is apparent when one notices that limǫ→0 fǫ(x) = f(x), for f ∈ C(M). So it is in this way that G provides us with a good generalisation of the classical product. 2.2 Point values and the generalised numbers. In general, distributions in the classical sense have no natural concept of a point value. For example the Dirac delta function has no classical value at the origin. New generalised functions differ from this in that they have a well defined value associated to a point x ∈ M . However, this will not in general be a classical number, but rather a “generalised number”. While the original presentation[3] provided a formal definition for generalised numbers, the presentation which our formulation is based upon[4] did not. We will define generalised numbers in a similar way to generalised functions. Definition 2.9. The space of moderate numbers is the set of continuous one-parameter families of complex numbers defined by CM = {(zǫ)|zǫ ∈ C such that ∃ N ∈ N, ∃ η > 0, ∃ c > 0, such that |zǫ| ≤ c ǫN for 0 < ǫ < η}. Definition 2.10. The space of negligible numbers is the set of moderate numbers defined by CN = {(zǫ)|zǫ ∈ CM such that ∀ q ∈ N, ∃ η > 0, ∃ c > 0, such that |zǫ| ≤ cǫ q for 0 < ǫ < η}. Definition 2.11. The space of generalised numbers is defined as