A nonlinear theory of distributional geometry

Abstract

This paper builds on the theory of nonlinear generalized functions begun in Nigsch & Vickers (Nigsch, Vickers 2021 Proc. R. Soc. A 20200640 (doi:10.1098/rspa.2020.0640)) and extends this to a diffeomorphism-invariant nonlinear theory of generalized tensor fields with the sheaf property. The generalized Lie derivative is introduced and shown to commute with the embedding of distributional tensor fields and the generalized covariant derivative commutes with the embedding at the level of association. The concept of a generalized metric is introduced and used to develop a non-smooth theory of differential geometry. It is shown that the embedding of a continuous metric results in a generalized metric with well-defined connection and curvature and that for C² metrics the embedding preserves the curvature at the level of association. Finally, we consider an example of a conical metric outside the Geroch–Traschen class and show that the curvature is associated to a delta function.

1. Introduction

This paper is concerned with the definition of a nonlinear theory of distributions with applications to the study of the differential geometry of low-regularity metrics. One of the main motivations we have in mind is the study of low-regularity metrics in general relativity (a topic currently of considerable interest [1–3]) but the theory we describe here has general application to situations where one wants to consider distributional differential geometry. These include singular Yang–Mills connections [4,5] and singular Hamiltonian flows [6].

The definition of the curvature requires second derivatives of the metric; therefore, in classical differential geometry one assumes that the metric is at least C² to ensure that the curvature is well defined and continuous, although mathematically it is in fact sufficient for most purposes to assume that the metric is C^1,1 (i.e. the first derivatives of the metric are Lipschitz). When considering applications in general relativity, Einstein’s equations

$$G_{ab}=8\pi T_{ab}$$

relate the Einstein tensor to the energy–momentum tensor of the matter so that conservation of the energy–momentum, given by ${\mathrm{\nabla }}_a{T}^a{\hspace{0.17em}}_b=0$, is equivalent to the contracted Bianchi identities. Using classical calculus in this context therefore requires the metric to be at least C³. This problem would not arise if one were able to give a distributional interpretation to the curvature since then one can always take the weak derivative; however, because the curvature is a nonlinear function of the metric and its first two derivatives, such a distributional description is in general not available. The curvature tensor

$$R^a{\hspace{0.17em}}_{bcd}={\mathit{\Gamma }}_{bd,c}^a-{\mathit{\Gamma }}_{bc,d}^a+{\mathit{\Gamma }}_{ec}^a{\mathit{\Gamma }}_{bd}^e-{\mathit{\Gamma }}_{ed}^a{\mathit{\Gamma }}_{bc}^e$$

involves terms that are quadratic in the connection and terms that involve derivatives of the connection so one readily sees that one needs the connection to be square integrable and have a (weak) first derivative. Writing down the formula for the connection

$${\mathit{\Gamma }}_{bc}^a=\frac{1}{2}{g}^{ae}(g_{be,c}+g_{ce,b}-g_{bc,e})$$

we see that this in turn imposes conditions on the metric. This led Geroch and Traschen to introduce a class of regular metrics for which the components of the curvature are well defined as a distribution (see [7] for details). Thus it is possible to use classical distributional geometry to describe shells of matter [8,9] and gravitational radiation [10] (see also the review article [11] for more examples of distributional curvature). However, Geroch and Traschen went on to show that in general their class of regular metrics can only have singular support on a submanifold of co-dimension at most 1. Thus a four-dimensional space–time metric representing a shell of matter belongs to this class but a string or particle does not. Despite these limitations there are a number of very interesting situations in which the Geroch–Traschen class of regular metrics can be used to describe the underlying physics. This has resulted in renewed interest in this class of space–times, as demonstrated in a number of papers (see [12,13] and the references therein). In particular in [12] LeFloch and Mardare give a precise description of which differential geometric objects can be well defined using classical (linear) distributions, with metrics of varying degrees of differentiability, and use this to give a coordinate-invariant description of the Geroch–Traschen class of regular metrics. However, although this class contains a number of interesting examples the conditions on the metric are quite restrictive and do not allow one to describe the differential geometry of, for example, general C¹ metrics. In the case of Riemannian metrics the theory of ‘metric geometry’ and in particular the use of length spaces together with the synthetic approach to curvature bounds provide an alternative way of studying spaces with low-regularity metrics [14,15]. However, despite recent progress [16] these methods are not yet well developed for Lorentzian metrics and applications to general relativity. In this paper we will adopt a different approach using the theory of nonlinear generalized functions as described in our previous paper [17].

By going outside conventional distribution theory Colombeau [18] showed that it is possible to construct an associative, commutative differential algebra which contains the space of distributions as a linear subspace and the space of smooth functions as a subalgebra. The basic idea is to represent generalized functions by nets (f_ε)_ε∈(0,1] of smooth functions. Distributions then are embedded essentially by convolution with a net of mollifiers, and smooth functions are embedded as constant nets. However, this description is too fine grained so that one identifies nets which differ by something negligible, i.e. by a net of functions whose derivatives vanish faster than any power of ε on any compact set. Mathematically this requires one to factor out the original space by the set of negligible nets. However, this is not an ideal unless one restricts the basic space to moderate nets, whose derivatives are bounded on compact sets by some positive power of 1/ε. The (special) algebra of generalized functions is then defined to be moderate nets modulo negligible nets; see [19] for more details. The advantage of using such a theory is that it gives a mathematically consistent way of multiplying distributions and provides a framework for understanding and justifying calculations without having to resort to ad hoc regularization procedures. An important feature of these algebras is the notion of association, which gives a correspondence between elements of the algebra and classical (linear) distributions. Applications of Colombeau’s theory to general relativity have included the calculation of nonlinear distributional curvatures which correspond to some metrics of low differentiability, such as those which occur in space–times with thin cosmic strings [20] and Kerr singularities [21], and the calculation of the electromagnetic field tensor of the ultra-relativistic Reissner–Nordström solution [22]. See [11] for a review of further applications to general relativity.

Although Colombeau algebras provide an excellent tool for looking at nonlinear distributions on ${\mathbb{R}}^n$, what is required for a nonlinear theory of distributional geometry is a theory of nonlinear distributions on a manifold M. Because convolution relies on the linear structure on ${\mathbb{R}}^n$ one cannot simply use the theory from ${\mathbb{R}}^n$ to work in a local chart and obtain a diffeomorphism-invariant result. In some particular cases where the final answer is associated to a classical distribution (see for example [23]) one can show that the final result does not depend upon the particular choice of coordinates used, but this is not possible in general. A better solution is to work with a diffeomorphism-invariant Colombeau algebra from the start. Colombeau & Meril [24] and later Jelínek [25] made the first steps toward a diffeomorphism-invariant Colombeau algebra. The local theory was then fully developed in [26], where the first diffeomorphism-invariant full Colombeau algebra on (open sets of) ${\mathbb{R}}^n$ was constructed. However, when considering differential geometric objects it is much better to have a global and manifestly coordinate-free version of the theory rather than simply giving transformation rules for the local theory. Such a construction in the scalar case was first given in [27] but this version turns out to be suboptimal when it comes to applications in differential geometry. In a previous paper [17] the present authors gave a new functional analytic construction of a global theory of generalized functions on a manifold M based on the notion of smoothing operators. The key idea was to replace a non-smooth function f by a net of smooth functions according to

$${\tilde{f}}_{\epsilon }(x)={\int }_{M}f(y){\omega }_{x,\epsilon }(y),$$ 1.1

depending on a suitable family of smoothing kernels ${\omega }_{\epsilon }\in C^{\mathrm{\infty }}(M,{\mathrm{\Omega }}_c^n(M))$ indexed by ε ∈ (0, 1], written in short as (ω_ε)_ε∈(0,1] or just (ω_ε)_ε. For fixed ε these may be treated just like smooth functions on manifolds, so all the standard operations that may be carried out on smooth functions extend to the smoothed functions ${\tilde{f}}_{\epsilon }$. The embedding (1.1) extends to distributions $T\in {\mathcal{D}}^{\prime }(M)$ by defining

$$\tilde{T}({\omega }_{\epsilon })(x):=⟨T,{\omega }_{x,\epsilon }⟩.$$ 1.2

By introducing certain asymptotic conditions on the basic space of generalized functions and adding a locality property needed for obtaining the sheaf property one may define the spaces of moderate generalized functions ${\hat{\mathcal{E}}}_{M,\mathrm{loc}}(M)$ and negligible generalized functions ${\hat{\mathcal{N}}}_{\mathrm{loc}}(M)$ and obtain the Colombeau algebra of generalized functions on M as the quotient ${\hat{\mathcal{G}}}_{\mathrm{loc}}(M)={\hat{\mathcal{E}}}_{M,\mathrm{loc}}(M)/{\hat{\mathcal{N}}}_{\mathrm{loc}}(M)$. See definitions 3.9 and 3.10 of [17] for precise details. The algebra of generalized functions ${\hat{\mathcal{G}}}_{\mathrm{loc}}(M)$ contains the space of smooth functions as a subalgebra and the space of distributions as a canonically embedded linear subspace and is in fact a sheaf. We also introduced both the generalized Lie derivative and the covariant derivative of generalized scalar fields on M. The generalized Lie derivative commutes with the embedding while the covariant derivative commutes at the level of association (see definition 5.12).

For applications of the algebra to differential geometry and general relativity in particular we are interested in calculating the curvature tensor for metrics (both Riemannian and Lorentzian) of low differentiability. This will be done by embedding the metrics into the algebra where one can calculate all the required derivatives and products needed to calculate the curvature. However, metrics are tensorial rather than scalar objects and because the embedding into the algebra does not commute with multiplication (except on the subalgebra of smooth functions) one cannot simply work with the coordinate components of a tensor and use the theory of generalized scalars.

In §2 we show how it is possible to define an algebra of generalized tensor fields on a manifold which contains the algebra of smooth tensor fields as a subalgebra and has a canonical coordinate-independent embedding of the spaces of (r, s)-tensor distributions as linear subspaces. As in [17] we will base our derivation on the notion of smoothing operators, this time applied to tensor fields. The key new observation is remark 3.1. This shows that all such smoothing operators arise as the tensor product of a scalar smoothing operator (as in [17]) and a transport operator $\mathrm{{\rm Y}}(x,y)$, which for fixed x and y is just a linear map from T_xM to T_yM. Such transport operators were first introduced in [28] and used to develop a tensorial theory of nonlinear distributions in [29]; at that time it was not at all clear that these were essential in order to develop the theory. By basing our approach on the notion of smoothing and using a tensorial version of the vector-valued Schwartz kernel theorem (see remark 3.1 for a precise statement) we make it clear in this paper that transport operators are an essential ingredient of the tensorial version of the theory. In order to make contact with previous work we will base our presentation on describing the smoothing of tensor distributions in terms of the equivalent description using transport operators and smoothing kernels. Although we adopt many of the features of [29] it is important to note that our new functional analytic approach to both the scalar and tensor theory leads to a subtantially different theory with different ‘basic spaces’ and better coherence between the scalar and tensor versions of the theory, and through employing the new notion of ‘localization’ we ensure the sheaf property for both scalar and tensor distributions (something that is not given in the theory provided in [29]).

In §3 we look at the embedding of distributional tensor fields into the algebra of generalized tensor fields. In §4 the generalized Lie derivative is introduced and it is shown that it commutes with the embedding. In §5 we use the theory described earlier to develop a nonlinear theory of distributional geometry and briefly look at applications to general relativity in §6. The covariant derivative of a generalized tensor field is introduced and it is shown that this commutes with the canonical embedding at the level of association. We then consider generalized metrics and show that the embedding of a C⁰ metric results in a generalized metric with well-defined connection and curvature. We also show that if one embeds a C² vacuum metric into the Colombeau algebra then its generalized Ricci curvature vanishes at the level of association. Finally we look at an example of a metric for which it is not possible to define the curvature using conventional distribution theory and show that the generalized Einstein tensor of a cone is associated to a distributional energy momentum tensor in a canonical and coordinate-independent manner.

We will continue to use the notation of [17]. In particular, $\mathfrak{X}(M)$ and Ω^p(M) denote the spaces of smooth vector fields and p-forms on M, respectively. A distributional tensor field may be regarded as simply a tensor field with distributional coefficients. However, we prefer to follow [30] and adopt a global description in which type (r, s) tensor distributions are regarded as dual to type (s, r) tensor densities. We denote the space of compactly supported type (s, r) tensor densities ${\tilde{\mathcal{D}}}_r^s(M)$ and denote the space of type (r, s) tensor distributions ${\mathcal{D}}_s^{\prime r}(M)$. We let $\tilde{\mathcal{D}}(M)={\tilde{\mathcal{D}}}_0^0(M)$ denote the space of (compactly supported) densities. Note that on an orientable manifold scalar densities are equivalent to n-forms so in the scalar case what we do here is consistent with [17].

2. The algebra of generalized tensor fields

In this section we will extend the theory of generalized scalar fields on a manifold M presented in [17] to vectors, covectors and more general tensor fields. Before giving the precise definitions we motivate these by looking at the smoothing of continuous (or, more generally, locally integrable) tensor fields on M by integration.

Given a scalar field f ∈ C⁰(M) we may define a smooth scalar field ${\tilde{f}}_{\epsilon }$ by (1.1). Unfortunately this does not make sense if we replace f by a vector field X. One obvious possibility is to work in some local coordinate system and define (leaving out the ε for the moment)

$${\tilde{X}}^a(x)={\int }_M{X}^a(y){\omega }_x(y).$$

However, if we transform to a new coordinate system x′ and then smooth we find

$${\tilde{X}}^{a^{\prime }}(x)={\int }_M\frac{\mathrm{\partial }{x}^{a^{\prime }}}{\mathrm{\partial }{x}^b}(y)X^b(y){\omega }_x(y),$$

which in general is not equal to

$$\frac{\mathrm{\partial }{x^a}^{\prime }}{\mathrm{\partial }{x}^b}(x){\tilde{X}}^b(x)=\frac{\mathrm{\partial }{x^a}^{\prime }}{\mathrm{\partial }{x}^b}(x){\int }_M{X}^b(y){\omega }_x(y).$$

The reason for the problem is that we are attempting to integrate the components of a vector at different points (see [28] for details). To make such an integral well defined in a coordinate-invariant way we need to prescribe some additional structure which enables us to compare tangent spaces at different points of the manifold.

Let $\mathrm{{\rm Y}}(x,y)\in T_{x}M\otimes T_y^{\ast }M$ be a two-point tensor that depends smoothly on x and y. More precisely, $\mathrm{{\rm Y}}$ is an element of $\mathrm{TO}(M):=\mathit{\Gamma }(M\times M,TM⊠T^{\ast }M)$ and will be called a transport operator (see [29,31] for details).

For x, y ∈ M, $\mathrm{{\rm Y}}$ defines a map

$${\mathrm{{\rm Y}}}^{\ast }(x,y):T_x^{\ast }M\to T_y^{\ast }M,$$

which may be written using the abstract index convention of [32] as

$${\beta }_a\mapsto {\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b(x,y){\beta }_a.$$

Contracting ${\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b(x,y)$ with a covector β_a in $T_x^{\ast }M$ hence gives an element of $T_y^{\ast }M$. We may also use $\mathrm{{\rm Y}}$ to define a map

$${\mathrm{{\rm Y}}}_{\ast }(x,y):T_{x}M\to T_{y}M$$

by the assignment

$$X^a\mapsto {\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b(y,x)X^b={\mathrm{{\rm Y}}}_b{\hspace{0.17em}}^a(x,y)X^b,$$

where we set ${\mathrm{{\rm Y}}}_b{\hspace{0.17em}}^a(x,y):={\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b(y,x)$.

By taking suitable tensor products $\mathrm{{\rm Y}}$ may be used to transport arbitrary tensors from x to y. Note that the transport operators we will use in the development of the theory typically satisfy ${\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b={\delta }_b^a$, hence both ${\mathrm{{\rm Y}}}^{\ast }$ and ${\mathrm{{\rm Y}}}_{\ast }$ are the identity on the diagonal and invertible in a neighbourhood of it.

We are now in a position to describe the smoothing of a locally integrable vector field X. Let x ∈ M, $\mathrm{{\rm Y}}$ be a transport operator and let ω ∈ SK(M) be a smoothing kernel; then we define $\tilde{X}(x)$ by its action on covectors $\alpha \in T_x^{\ast }M$ (which may be written using the abstract index convention) as

$${\tilde{X}}^a(x){\alpha }_a={\int }_{y\in M}{\alpha }_a{\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b(x,y)X^b(y){\omega }_x(y).$$

Note that ${\alpha }_a{\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b(x,\cdot )X^b(\cdot )$ is a scalar field on M which may be smoothed by integrating against ω_x.

Similarly, in order to smooth a locally integrable covector field β we consider its action on vectors Y ∈ T_xM and use the transport operator to extend this to a vector field. Thus,

$${\tilde{\beta }}_a(x)Y^a={\int }_{y\in M}{Y}^a{\mathrm{{\rm Y}}}_a{\hspace{0.17em}}^b(x,y){\beta }_b(y){\omega }_x(y).$$

Using the same strategy we can smooth a general locally integrable type (r, s) tensor field S by defining $\tilde{S}$ according to

$$\begin{array}{rl}{\tilde{S}}_{b_1\dots b_s}^{a_1\dots a_r}(x)& ={\int }_M{S}_{d_1\dots d_s}^{c_1\dots c_r}(y){\mathrm{{\rm Y}}}^{a_1}{\hspace{0.17em}}_{c_1}(x,y)\cdots {\mathrm{{\rm Y}}}^{a_r}{\hspace{0.17em}}_{c_r}(x,y)\cdot {\mathrm{{\rm Y}}}_{b_1}{\hspace{0.17em}}^{d_1}(x,y)\cdots {\mathrm{{\rm Y}}}_{b_s}{\hspace{0.17em}}^{d_s}(x,y){\omega }_x(y),\end{array}$$ 2.1

which with some changes of notation is the formula given in [28].

A natural way of obtaining such transport operators is by using a background connection γ. If we choose U to be some geodesically convex neighbourhood for γ (i.e. an open set such that every pair of points in U can be connected by a unique geodesic lying in the set) then we may define ${\mathrm{{\rm Y}}}_{\ast }(x,y)$ to be given by parallel transport of vectors along the geodesic connecting x to y. Note that, for such a transport operator, ${\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b(x,x)={\delta }_b^a$ for x ∈ U (which ensures that ${\tilde{X}}^a(x)\to X^a(x)$ as ε → 0). Unfortunately, such a transport operator is only defined for (x, y) ∈ U × U. However, using a partition of unity we may define a global transport operator which is determined by γ in the above way in a neighbourhood of the diagonal.

We are now in a position to define the basic tensor space that we will use to define generalized tensor field on manifolds.

Definition 2.1 (Basic tensor space) —

The basic space ${\hat{\mathcal{E}}}_s^r(M)$ of type (r, s) generalized tensor fields consists of all maps

$$S:\mathrm{TO}(M)\times \mathrm{SK}(M)\to {\mathcal{T}}_s^r(M),$$

such that $S(\mathrm{{\rm Y}},\omega )$ depends smoothly on $\mathrm{{\rm Y}}\in \mathrm{TO}(M)$ and ω ∈ SK(M), where $\mathrm{SK}(M)=C^{\mathrm{\infty }}(M,\tilde{\mathcal{D}}(M))$ is the space of smoothing kernels [17, definition 3.1] and the smoothness with respect to ω and $\mathrm{{\rm Y}}$ is understood as in [33].

Note that for the sake of presentation we completely omit discussion of the sheaf property; to obtain it we actually would have to restrict the basic space to a somewhat smaller one. For details, we refer to [31,34].

Before going on to define moderate and negligible generalized tensor fields we will look at the properties of the basic space ${\hat{\mathcal{E}}}_s^r(M)$.

3. Embedding distributional tensor fields

In this section we will discuss the embedding of distributional tensor fields into the space of generalized tensor fields. We have already given the basic construction for the embedding of a continuous tensor field S in equation (2.1). We now turn to the case of a distributional tensor field T.

Given a type (r, s) distributional tensor field $T\in {\mathcal{D}}_s^{\prime r}(M)$, a smoothing kernel ω_x(y) and a transport operator $\mathrm{{\rm Y}}\in \mathrm{TO}(M)$ we may define a smooth tensor field $\tilde{T}(\mathrm{{\rm Y}},\omega )\in {\mathcal{T}}_s^r(M)$ according to

$$\tilde{T}(\mathrm{{\rm Y}},\omega )(x)({\alpha }^1,\dots ,{\alpha }^r,Y_1,\dots ,Y_s)=⟨T,{\mathrm{\Psi }}_x⟩,$$ 3.1

where ${\alpha }^j\in T_x^{\ast }M$ for j = 1, …, r, Y_k ∈ T_xM for k = 1, …, s and Ψ_x(y) is the type (s, r) tensor density given by

$$\begin{array}{rl}{\mathrm{\Psi }}_x(y)& =({\mathrm{{\rm Y}}}^{\ast }(x,y){\alpha }^1(x))\otimes \cdots \otimes ({\mathrm{{\rm Y}}}^{\ast }(x,y){\alpha }^r(x))\otimes \\ & ({\mathrm{{\rm Y}}}_{\ast }(x,y)Y_1(x))\otimes \cdots \otimes ({\mathrm{{\rm Y}}}_{\ast }(x,y)Y_s(x))\otimes {\omega }_x(y).\end{array}$$ 3.2

Formula (3.1) therefore gives a canonical embedding

$$\begin{array}{rl}{\iota }_s^r:{\mathcal{D}}_s^{\prime r}(M)& \to {\hat{\mathcal{E}}}_s^r(M),\\ T& \mapsto \tilde{T}.\end{array}$$

It can be shown that $\tilde{T}(\mathrm{{\rm Y}},\omega )$ depends smoothly on the smoothing kernel ω and on the choice of transport operator $\mathrm{{\rm Y}}$. Actually, the seemingly innocuous statement that this mapping is smooth is far from trivial to prove and is considered in detail in [31,35].

It is also clear that if $T\in {\mathcal{T}}_s^r(M)$ is a smooth type (r, s) tensor field then setting

$$\hat{T}{\hspace{0.17em}}_{b_1\dots b_s}^{a_1\dots a_r}(\mathrm{{\rm Y}},\omega )=T{\hspace{0.17em}}_{b_1\dots b_s}^{a_1\dots a_r}$$ 3.3

gives an embedding

$$\begin{array}{rl}{\sigma }_s^r:{\mathcal{T}}_s^r(M)& \to {\hat{\mathcal{E}}}_s^r(M),\\ T& \mapsto \hat{T}.\end{array}$$

Remark 3.1. —

We have seen that by combining a transport operator with a smoothing kernel we may smooth tensor distributions. It is remarkable that all linear and continuous mappings from ${\mathcal{D}}_s^{\prime r}(M)$ into ${\mathcal{T}}_s^r(M)$ are of this form in the following sense: there is an isomorphism of locally convex spaces

$$\begin{array}{rl}& L({\mathcal{D}}_s^{\prime r}(M),{\mathcal{T}}_s^r(M))\cong \mathit{\Gamma }(M\times M,T_s^{r}M⊠T_r^{s}M){\otimes }_{C^{\mathrm{\infty }}(M\times M)}L({\mathcal{D}}^{\prime }(M),C^{\mathrm{\infty }}(M));\end{array}$$

see [35]. For our purposes, elements of $\mathit{\Gamma }(M\times M,T_s^{r}M⊠T_r^{s}M)$ are constructed by taking tensor products of $\mathrm{{\rm Y}}\in \mathit{\Gamma }(M\times M,TM⊠T^{\ast }M)$ as in (2.1).

4. Generalized Lie derivatives

In this section we consider the Lie derivative of generalized tensor fields. Before doing so we review the definition of the Lie derivative of a distributional vector field as given by [30] (see also [36]). We begin by looking at the Lie derivative of a distributional vector field X. If we let θ be an arbitrary smooth 1-form then X(θ) is a distributional scalar field. We now define the distributional Lie derivative of X with respect to a smooth vector field Z to be that given by requiring the Leibniz rule for X(θ) to be satisfied, so that

$$({\mathcal{L}}_{Z}X)(\theta ):={\mathcal{L}}_Z(X(\theta ))-X({\mathcal{L}}_Z\theta )\mathrm{\forall }\theta \in {\mathrm{\Omega }}^1(M).$$

We now define the distributional Lie derivative of a general distributional tensor field S.

Definition 4.1 (Lie derivative of tensor fields) —

Let $S\in {\mathcal{D}}_s^{\prime r}(M)$. The Lie derivative of S with respect to the smooth vector field Z ∈ 𝔰X(M) is the element ${\mathcal{L}}_{Z}S\in {\mathcal{D}}_s^{\prime r}(M)$ given by

$$\begin{array}{rl}⟨({\mathcal{L}}_{Z}S)({\theta }^1,\dots ,{\theta }^r,X_1,\dots ,X_s),\omega ⟩& =-⟨S({\theta }^1,\dots ,{\theta }^r,X_1,\dots ,X_s),{\mathcal{L}}_Z\omega ⟩\\ & -\sum _{i=1}^r⟨S({\theta }^1,\dots ,{\mathcal{L}}_Z{\theta }^i,\dots {\theta }^r,X_1,\dots ,X_s),\omega ⟩\\ & -\sum _{j=1}^s⟨S({\theta }^1,\dots ,{\theta }^r,X_1,\dots ,{\mathcal{L}}_Z{X}_j,\dots ,X_s),\omega ⟩\end{array}$$

for all θ¹, …, θ^r ∈ Ω¹(M), X₁, …, X_s ∈ 𝔰X(M) and $\omega \in {\mathrm{\Omega }}_c^n(M)$.

Note that if we regard $S\in {\mathcal{D}}_s^{\prime r}(M)$ as dual to a type (s, r) tensor density Ψ given by

$$\mathrm{\Psi }={\theta }^1\otimes \cdots \otimes {\theta }^r\otimes X_1\cdots \otimes X_s\otimes \omega$$

then the above formula can be written in the more compact form

$$⟨{\mathcal{L}}_{Z}S,\mathrm{\Psi }⟩=-⟨S,{\mathcal{L}}_Z\mathrm{\Psi }⟩.$$

In [17] we looked at derivatives of a generalized scalar field. This involved defining the Lie derivative ${\mathcal{L}}_X^{\text{SK}}\omega ={\mathcal{L}}_X^{{\mathrm{\Omega }}^n}\omega +{\mathcal{L}}_X^{C^{\mathrm{\infty }}}\omega$ of a smoothing kernel, which we obtained by differentiating the action of a 1-parameter group of diffeomorphisms. For derivatives of a generalized tensor field $T(\mathrm{{\rm Y}},\omega )$ we will also require the Lie derivative of the transport operator $\mathrm{{\rm Y}}$.

In principle we can consider the action of two diffeomorphisms μ and ν, which act separately on the x and y variables. Thinking of $\mathrm{{\rm Y}}$ as a sum of terms of the form V^a(x) ⊗ α_b(y) we can consider the pullback ${\mu }^{\ast }={({\mu }_{\ast })}^{-1}$ by taking the inverse of the pushforward action on the vector V^a(x) and the pullback ${\nu }^{\ast }$ by taking the pullback action on the 1-form α_b(y). This gives us the action

$$({\mu }^{\ast },{\nu }^{\ast }):\mathrm{TO}(M)\to \mathrm{TO}(M).$$

We can also consider two vector fields X and Y with corresponding flows ${\mathrm{Fl}}_t^X$ and ${\mathrm{Fl}}_t^Y$ acting on the x and y variables. This enables us to define the Lie derivative of $\mathrm{{\rm Y}}$ by

$${\mathcal{L}}_{(X,Y)}\mathrm{{\rm Y}}={\left.\frac{\text{d}}{\text{d}t}\right|}_{t=0}({({\mathrm{Fl}}_t^X)}^{\ast },{({\mathrm{Fl}}_t^Y)}^{\ast })\mathrm{{\rm Y}}.$$

Varying the x and y variables separately we have two Lie derivatives ${\mathcal{L}}_{(X,0)}$ and ${\mathcal{L}}_{(0,Y)}$ satisfying

$${\mathcal{L}}_{(X,Y)}\mathrm{{\rm Y}}={\mathcal{L}}_{(X,0)}\mathrm{{\rm Y}}+{\mathcal{L}}_{(0,Y)}\mathrm{{\rm Y}}.$$

The explicit formulae are given by

$${({\mathcal{L}}_{(X,0)}\mathrm{{\rm Y}})}^a{\hspace{0.17em}}_b(x,y)=X^c(x)\frac{\mathrm{\partial }{\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b(x,y)}{\mathrm{\partial }{x}^c}-{\mathrm{{\rm Y}}}^c{\hspace{0.17em}}_b(x,y)\frac{\mathrm{\partial }{X}^a}{\mathrm{\partial }{x}^c}(x)$$

and

$$({\mathcal{L}}_{(0,Y)}{\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b)(x,y)=Y^c(y)\frac{\mathrm{\partial }{\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b(x,y)}{\mathrm{\partial }{y}^c}+{\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_c(x,y)\frac{\mathrm{\partial }{Y}^c}{\mathrm{\partial }{y}^b}(y),$$

so that $({\mathcal{L}}_{(X,0)}\mathrm{{\rm Y}})(x,y)$ corresponds to the Lie derivative of the transport operator with respect to the vector field X at x keeping y fixed (i.e. thinking of $\mathrm{{\rm Y}}(x,y)$ as a vector field at x). Similarly, $({\mathcal{L}}_{(0,Y)}\mathrm{{\rm Y}})(x,y)$ corresponds to the Lie derivative of the transport operator with respect to the vector field Y at y keeping x fixed (i.e. thinking of $\mathrm{{\rm Y}}(x,y)$ as a covector field at y).

Although the above discussion is useful for practical calculations and also for comparison with the situation of [29], in defining the Lie derivative of a transport operator by differentiating the action of a 1-parameter group of diffeomorphisms the same diffeomorphism acts in both variables and hence we abbreviate ${\mathcal{L}}_X^{\text{TO}}\mathrm{{\rm Y}}:={\mathcal{L}}_{(X,X)}\mathrm{{\rm Y}}$.

Having calculated the Lie derivative of the transport operator we are now in a position to look at the Lie derivative of a generalized tensor field. Given a generalized tensor field $T\in {\hat{\mathcal{E}}}_s^r(M)$ then for fixed ω ∈ SK(M) and fixed $\mathrm{{\rm Y}}\in \mathrm{TO}(M)$ we know that $\tilde{T}:=T(\mathrm{{\rm Y}},\omega )$ is a smooth type (r, s) tensor field and hence we may calculate its (ordinary) Lie derivative with respect to a smooth vector field X ∈ 𝔰X(M).

As in the case of the generalized Lie derivative of a scalar field, we find the correct definition for the geometric Lie derivative of a generalized tensor field T with respect to a vector field X by differentiating the pullback of T along the flow of X, i.e. ${({\text{Fl}}_t^X)}^{\ast }T$, at time t = 0. This leads to the following definition.

Definition 4.2 (Generalized Lie derivative of tensors) —

For $T\in {\hat{\mathcal{E}}}_s^r(M)$ and $X\in \mathfrak{X}(M)$ we define

$$({\hat{\mathcal{L}}}_{X}T)(\mathrm{{\rm Y}},\omega ):={\mathcal{L}}_X(T(\mathrm{{\rm Y}},\omega ))-d_{1}T(\mathrm{{\rm Y}},\omega )({\mathcal{L}}_X^{\text{TO}}\mathrm{{\rm Y}})-d_{2}T(\mathrm{{\rm Y}},\omega )({\mathcal{L}}_X^{\text{SK}}\omega ).$$ 4.1

Here, d_i denotes the differential with respect to the ith variable in the sense of [33].

Remark 4.3. —

Formula (4.1) must also be used for scalar fields which have an $\mathrm{{\rm Y}}$ dependence. Such fields may arise, for example, from the contraction of a generalized vector field with a 1-form. For scalar fields with no $\mathrm{{\rm Y}}$ dependence the above formula reduces to that given in [17] for generalized scalar fields.

We now give an explicit formula for the Lie derivative of an embedded vector field Y. For notational ease we first consider the special case where Y is continuous so that we do not have to consider distributional derivatives. The embedded vector field is given by

$${\iota }_0^1(Y){(\mathrm{{\rm Y}},\omega )}^a(x)={\int }_{y\in M}{Y}^b(y){\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b(x,y){\omega }_x(y).$$

Taking the generalized Lie derivative of this according to definition 4.2 gives

$$\begin{array}{rl}{\hat{\mathcal{L}}}_X({\iota }_0^1(Y)){(\mathrm{{\rm Y}},\omega )}^a(x)& ={\int }_{y\in M}{Y}^b(y)(({\mathcal{L}}_{(X,0)}{\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b)(x,y){\omega }_x(y)+{\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b(x,y){({\mathcal{L}}_X^{C^{\mathrm{\infty }}}\omega )}_x(y))\\ & -{\int }_{y\in M}{Y}^b(y)({\mathcal{L}}_X^{\text{TO}}{\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b)(x,y){\omega }_x(y)-{\int }_{y\in M}{Y}^b(y){\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b(x,y){({\mathcal{L}}_X^{\text{SK}}\omega )}_x(y)\\ & =-{\int }_{y\in M}{Y}^b(y)({\mathcal{L}}_{(0,X)}{\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b)(x,y){\omega }_x(y)-{\int }_{y\in M}{Y}^a(y){\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b(x,y){({\mathcal{L}}_X^{{\mathrm{\Omega }}^n}\omega )}_x(y)\\ & ={\int }_{y\in M}{({\mathcal{L}}_{X}Y)}^b(y){\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_b(x,y){\omega }_x(y)\\ & ={\iota }_0^1({\mathcal{L}}_{X}Y){(\omega ,\mathrm{{\rm Y}})}^a(x),\end{array}$$

hence ${\hat{\mathcal{L}}}_X({\iota }_0^1(Y))={\iota }_0^1({\mathcal{L}}_{X}Y)$.

Turning now to the general case a similar calculation shows that for a distributional tensor field $S\in {\mathcal{D}}_s^{\prime r}(M)$ we have ${\hat{\mathcal{L}}}_X({\iota }_s^r(S))={\iota }_s^r({\mathcal{L}}_{X}S)$.

For a smooth type (r, s) tensor field S we have ${\sigma }_s^r(S)(\mathrm{{\rm Y}},\omega )=S$ and since there is no dependence on the smoothing kernel or transport operator the generalized Lie derivative is the same as the ordinary Lie derivative so that ${\hat{\mathcal{L}}}_Z({\sigma }_s^r(S))={\sigma }_s^r({\mathcal{L}}_{Z}S)$.

Combining these two results we have the following.

Proposition 4.4. —

• (a)
  The embedding ${\iota }_s^r$ of distributional tensor fields commutes with the Lie derivative so that

  $${\hat{\mathcal{L}}}_X({\iota }_s^r(S))={\iota }_s^r({\mathcal{L}}_{X}S).$$
• (b)
  The embedding ${\sigma }_s^r$ of smooth tensor fields commutes with the Lie derivative so that

  $${\hat{\mathcal{L}}}_X({\sigma }_s^r(S))={\sigma }_s^r({\mathcal{L}}_{X}S).$$

Remark 4.5. —

The deeper reason for proposition 4.4 comes from looking at the induced action of a diffeomorphism μ : M → N on the space of generalized tensor fields. If $T\in \hat{\mathcal{E}}(N)$ is a generalized tensor field on N then we may pull it back to a generalized tensor field ${\mu }^{\ast }T$ on M by defining

$$({\mu }^{\ast }T)(\mathrm{{\rm Y}},\omega )(x):={(D_{\mu (x)}{\mu }^{-1})}_s^r(T(({\mu }_{\ast },{\mu }_{\ast })\mathrm{{\rm Y}},({\mu }_{\ast },{\mu }_{\ast })\omega )(\mu (x))).$$

It is readily verified that the action of the diffeomorphism commutes with the embedding so that if $S\in {\mathcal{D}}_s^{\prime r}(M)$ is a distributional tensor field then

$${\mu }^{\ast }({\iota }_s^r(S))={\iota }_s^r({\mu }^{\ast }S).$$

If we now take μ to be the flow ${\mathrm{Fl}}_t^X$ of a (complete) vector field then we have

$${({\mathrm{Fl}}_t^X)}^{\ast }({\iota }_s^r(S))={\iota }_s^r({({\mathrm{Fl}}_t^X)}^{\ast }S).$$

Differentiating this with respect to t and using the fact that for any $T\in {\hat{\mathcal{E}}}_s^r(M)$ we have

$${\hat{\mathcal{L}}}_{X}T={\left.\frac{\text{d}}{\text{d}t}\right|}_{t=0}\hat{{({\mathrm{Fl}}_t^X)}^{\ast }}T$$ 4.2

this immediately gives

$${\hat{\mathcal{L}}}_X({\iota }_s^r(S))={\iota }_s^r({\mathcal{L}}_{X}S).$$

Thus, the fact that the generalized Lie derivative commutes with the embedding follows from the fact that the action of a diffeomorphism commutes with the embedding.

5. The quotient construction and the algebra of generalized tensor fields

Having looked at the properties of the basic space ${\hat{\mathcal{E}}}_s^r(M)$ we turn to the definition of generalized tensor fields ${\hat{\mathcal{G}}}_s^r(M)$. These are defined as moderate tensor fields modulo negligible tensor fields.

Similarly to the nets of smoothing kernels (ω_ε)_ε of the scalar case, one needs to introduce a suitable asymptotic structure on nets of transport operators ${({\mathrm{{\rm Y}}}_{\epsilon })}_{\epsilon }$ such that, in the limit ε → 0, ${\mathrm{{\rm Y}}}_{\epsilon }\otimes {\omega }_{\epsilon }$ converges to the identity in the right way. The respective definitions are as follows.

Definition 5.1 (Admissible nets of transport operators) —

A net ${({\mathrm{{\rm Y}}}_{\epsilon })}_{\epsilon }\in \mathrm{TO}{(M)}^{(0,1]}$ is called admissible if

• (i)locally around the diagonal in M × M, ${({\mathrm{{\rm Y}}}_{\epsilon })}_{\epsilon }$ is uniformly bounded for small ε, and
• (ii)${\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_{b,\epsilon }(x,x)={\delta }_b^a$ for all x ∈ M.

The space of all admissible nets of transport operators is denoted $\mathrm{{\rm Y}}(M)$.

Condition (i) means that for each chart U on M and all multi-indices k, l the derivative ${\mathrm{\partial }}_x^{\mathit{k}}{\mathrm{\partial }}_y^{\mathit{l}}{\mathrm{{\rm Y}}}^a{\hspace{0.17em}}_{b,\epsilon }$ is uniformly bounded in a neighbourhood of each point (x, x) of the diagonal; condition (ii) simply means that ${\mathrm{{\rm Y}}}^{\ast }(x,x)$ and ${\mathrm{{\rm Y}}}_{\ast }(x,x)$ are the identity mappings.

The linear space corresponding to this affine space is introduced as

$${\mathrm{{\rm Y}}}_0(M):=\{{({\mathrm{\Xi }}_{\epsilon })}_{\epsilon }\in \mathrm{TO}{(M)}^I\hspace{0.17em}|\hspace{0.17em}{({\mathrm{{\rm Y}}}_{\epsilon })}_{\epsilon }\in \mathrm{{\rm Y}}(M)\Rightarrow {({\mathrm{{\rm Y}}}_{\epsilon })}_{\epsilon }+{({\mathrm{\Xi }}_{\epsilon })}_{\epsilon }\in \mathrm{{\rm Y}}(M)\}.$$

The following definitions are the obvious generalizations of the scalar case (we refer to [31,37] for detailed proofs in a slightly extended setting).

Definition 5.2 (Moderate tensors) —

The tensor field $T\in {\hat{\mathcal{E}}}_s^r(M)$ is called moderate if $\mathrm{\forall }K\subset M$ compact $\mathrm{\forall }j,k,l\in {\mathbb{N}}_0$ $\mathrm{\forall }\mathrm{{\rm Y}}\in \mathrm{{\rm Y}}(M)$, ${\mathrm{{\rm Y}}}_1,\dots ,{\mathrm{{\rm Y}}}_j\in {\mathrm{{\rm Y}}}_0(M)$ $\mathrm{\forall }\omega \in \tilde{\mathcal{A}}(M)$, ${\omega }_1,\dots ,{\omega }_k\in {\tilde{\mathcal{A}}}_0(M)$ $\mathrm{\forall }{X}_1,\dots ,X_l\in \mathfrak{X}\mathfrak{(}\mathfrak{M}\mathfrak{)}$ $\mathrm{\exists }N\in \mathbb{N}$:

$$\underset{x\in K}{sup}||{\mathcal{L}}_{X_1}\dots {\mathcal{L}}_{X_l}({\mathrm{d}}_1^j{\mathrm{d}}_2^{k}T({\mathrm{{\rm Y}}}_{\epsilon },{\omega }_{\epsilon })\hspace{0.17em}({\mathrm{{\rm Y}}}_{1,\epsilon },\dots ,{\mathrm{{\rm Y}}}_{j,\epsilon },{\omega }_{1,\epsilon },\dots ,{\omega }_{k,\epsilon }))(x)||=O({\epsilon }^{-N})(\epsilon \to 0),$$ 5.1

where || · || denotes the norm induced by some background metric and the spaces $\tilde{\mathcal{A}}(M)$ and ${\tilde{\mathcal{A}}}_0(M)$ are defined in definition 3.4 of [17].

The set of moderate tensors in ${\hat{\mathcal{E}}}_s^r(M)$ is denoted ${({\hat{\mathcal{E}}}_M)}_s^r(M)$.

Note that the space of moderate tensors does not depend upon the choice of background metric used to define the above norm. The inclusion of the differentials with respect to $\mathrm{{\rm Y}}$ and ω makes the definition look quite complicated but for embedded fields the dependence is at worst linear so that in practice there are no significant complications caused by this.

Definition 5.3 (Negligible tensors) —

The tensor field $\tilde{T}\in {({\hat{\mathcal{E}}}_M)}_s^r(M)$ is negligible if $\mathrm{\forall }K\subset M$ compact $\mathrm{\forall }j,k,l,m\in {\mathbb{N}}_0$ $\mathrm{\forall }\mathrm{{\rm Y}}\in \mathrm{{\rm Y}}(M)$, ${\mathrm{{\rm Y}}}_1,\dots ,{\mathrm{{\rm Y}}}_j\in {\mathrm{{\rm Y}}}_0(M)$ $\mathrm{\forall }\omega \in \tilde{\mathcal{A}}(M)$, ${\omega }_1,\dots ,{\omega }_k\in {\tilde{\mathcal{A}}}_0(M)$ $\mathrm{\forall }{X}_1,\dots ,X_l\in \mathfrak{X}\mathfrak{(}\mathfrak{M}\mathfrak{)}$:

$$\underset{x\in K}{sup}||{\mathcal{L}}_{X_1}\dots {\mathcal{L}}_{X_l}({\mathrm{d}}_1^j{\mathrm{d}}_2^{k}T({\mathrm{{\rm Y}}}_{\epsilon },{\omega }_{\epsilon })\hspace{0.17em}({\mathrm{{\rm Y}}}_{1,\epsilon },\dots ,{\mathrm{{\rm Y}}}_{j,\epsilon },{\omega }_{1,\epsilon },\dots ,{\omega }_{k,\epsilon }))(x)||=O({\epsilon }^{-m})(\epsilon \to 0).$$ 5.2

The set of negligible tensors in ${\hat{\mathcal{E}}}_s^r(M)$ is denoted $\hat{\mathcal{N}}{\hspace{0.17em}}_s^r(M)$.

We should note that the above formulae also apply to type (0, 0) tensor fields (i.e.scalar fields) which depend on $\mathrm{{\rm Y}}$. Such fields arise for example from contraction of higher valence tensors. After taking this point into account it follows from the above definitions that one can test for moderateness and negligibility by looking at the scalar field obtained by contraction.

Proposition 5.4 (Saturation) —

A generalized tensor field $\tilde{T}\in {\hat{\mathcal{E}}}_s^r(M)$ is moderate (respectively negligible) iff for all smooth covector fields θⁱ ∈ Ω¹(M) and smooth vector fields X_j ∈ 𝔰X(M) the generalized scalar field

$$F(\mathrm{{\rm Y}},\omega )(x)=\tilde{T}(\mathrm{{\rm Y}},\omega )(x)({\theta }^1(x)\dots {\theta }^r(x),X_1(x)\dots X_s(x))$$

obtained by contraction is moderate (respectively negligible) when regarded as an element of ${\hat{\mathcal{E}}}_0^0(M)$.

Proposition 5.5. —

${({\hat{\mathcal{E}}}_M)}_s^r(M)$ is an ${({\hat{\mathcal{E}}}_M)}_0^0(M)$-module with ${\hat{\mathcal{N}}}_s^r(M)$ as a submodule.

Proof. —

As with the case of scalar fields the definitions of moderate and negligible may be directly used to establish the following results, which prove the claim:

• (a)$\tilde{f}\in {({\hat{\mathcal{E}}}_M)}_0^0(M)$, $S\in {({\hat{\mathcal{E}}}_M)}_s^r(M)$ ↠ $\tilde{f}S\in {({\hat{\mathcal{E}}}_M)}_s^r(M)$;
• (b)$\tilde{f}\in {\hat{\mathcal{N}}}_0^0(M)$, $S\in {({\hat{\mathcal{E}}}_M)}_s^r(M)$ ↠ $\tilde{f}S\in {\hat{\mathcal{N}}}_s^r(M)$;
• (c)$\tilde{f}\in {({\hat{\mathcal{E}}}_M)}_0^0(M)$, $S\in {\hat{\mathcal{N}}}_s^r(M)$ ↠ $\tilde{f}S\in {\hat{\mathcal{N}}}_s^r(M)$. ▪

We next examine the properties of the embeddings of distributional and smooth tensor fields into the basic space. As an immediate consequence of the analytical properties of the combination of admissible nets of transport operators with test objects [31] one may establish the following proposition.

Proposition 5.6. —

• (a)${\iota }_s^r({{\mathcal{D}}^{\prime }}_s^r(M))\subseteq {({\hat{\mathcal{E}}}_M)}_s^r(M)$.
• (b)${\sigma }_s^r({\mathcal{T}}_s^r(M))\subseteq {({\hat{\mathcal{E}}}_M)}_s^r(M)$.
• (c)$({\iota }_s^r-{\sigma }_s^r)({\mathcal{T}}_s^r(M))\subseteq {\hat{\mathcal{N}}}_s^r(M)$.
• (d)If $T\in {{\mathcal{D}}^{\prime }}_s^r(M)$ and ${\iota }_s^r(T)\in {\hat{\mathcal{N}}}_s^r(M)$, then T = 0.

We may also show that as in the scalar case moderateness and negligibility are stable under the action of the generalized Lie derivative.

Proposition 5.7. —

Let X ∈ 𝔰X(M). Then,

• (a)${\hat{\mathcal{L}}}_X({({\hat{\mathcal{E}}}_M)}_s^r(M))\subseteq {({\hat{\mathcal{E}}}_M)}_s^r(M)$, and
• (b)${\hat{\mathcal{L}}}_X({\hat{\mathcal{N}}}_s^r(M))\subseteq {\hat{\mathcal{N}}}_s^r(M)$.

We are now in a position to define the space of generalized tensor fields, as follows.

Definition 5.8 (Generalized tensor fields) —

We define the $\hat{\mathcal{G}}(M)$-module ${\hat{\mathcal{G}}}_s^r(M)$ of generalized type (r, s) tensor fields by

$${\hat{\mathcal{G}}}_s^r(M)={({\hat{\mathcal{E}}}_M)}_s^r(M)/{\hat{\mathcal{N}}}_s^r(M).$$ 5.3

We now consider tensor operations on ${\hat{\mathcal{G}}}_s^r(M)$. Let $\tilde{S}\in {\hat{\mathcal{E}}}_s^r(M)$ and $\tilde{T}\in {\hat{\mathcal{E}}}_u^t(M)$. Since $\tilde{S}(\mathrm{{\rm Y}},\omega )$ and $\tilde{T}(\mathrm{{\rm Y}},\omega )$ are smooth tensor fields we may define $\tilde{S}\otimes \tilde{T}\in {\hat{\mathcal{E}}}_{s+u}^{r+t}(M)$ by

$$(\tilde{S}\otimes \tilde{T})(\mathrm{{\rm Y}},\omega )=(\tilde{S}(\mathrm{{\rm Y}},\omega ))\otimes (\tilde{T}(\mathrm{{\rm Y}},\omega )).$$ 5.4

In a similar way to the scalar case [17] one may use the definitions of moderateness and negligibility to show that if both $\tilde{T}$ and $\tilde{S}$ are moderate then $\tilde{T}\otimes \tilde{S}$ is moderate and that if either $\tilde{T}$ or $\tilde{S}$ is negligible then so is $\tilde{T}\otimes \tilde{S}$. We may therefore define the tensor product as follows.

Definition 5.9 (Tensor product) —

The tensor product of $[\tilde{S}]\in {\hat{\mathcal{G}}}_s^r(M)$ and $[\tilde{T}]\in {\hat{\mathcal{G}}}_u^t(M)$ is defined by

$$[\tilde{T}]\otimes [\tilde{S}]=[\tilde{T}\otimes \tilde{S}],$$ 5.5

where $\tilde{T}\otimes \tilde{S}$ is given by equation (5.4).

In the same way one can show that if $\tilde{T}$ is obtained from $\tilde{S}$ by contraction on a pair of indices, then $\tilde{T}$ is moderate if $\tilde{S}$ is moderate and that $\tilde{T}$ is negligible if $\tilde{S}$ is negligible; hence, we may define contraction of generalized tensor field as follows.

Definition 5.10 (Contraction) —

Let $[\tilde{S}]\in {\hat{\mathcal{G}}}_s^r(M)$. We may define an element of ${\hat{\mathcal{G}}}_{s-1}^{r-1}(M)$ by making a contraction according to the formula

$${[\tilde{S}]}_{c\dots e\dots d}^{a\dots e\dots b}=[{\tilde{S}}_{c\dots e\dots d}^{a\dots e\dots b}].$$

We now define the generalized tensor algebra as

$$\check{\mathcal{G}}(M)=\underset{r,s\in \mathbb{N}}{⨁}\hat{\mathcal{G}}{\hspace{0.17em}}_s^r(M).$$

From definition 5.9 and definition 5.10 we see that $\check{\mathcal{G}}(M)$ is closed under the operations of tensor product and contraction.

We summarize the properties we have established in the following theorem.

Theorem 5.11. —

• (a)The generalized tensor algebra $\check{\mathcal{G}}(M)$ is an associative differential algebra with product the tensor product ⊗, and derivatives given by the generalized Lie derivatives ${\hat{\mathcal{L}}}_X$ for X ∈ 𝔰X(M).
• (b)The algebra is closed under the action of contraction.
• (c)The space of smooth tensor fields may be embedded by the ‘constant map’ ${\sigma }_s^r$ and the algebra of smooth tensor fields forms a subalgebra of $\check{\mathcal{G}}(M)$.
• (d)For each $r,s\in {\mathbb{N}}_0$ there is a linear map ${\iota }_s^r$ which embeds ${{\mathcal{D}}^{\prime }}_s^r(M)$ as a C^∞(M)-submodule of ${\hat{\mathcal{G}}}_s^r(M)$, and the embedding ${\iota }_s^r$ coincides with ${\sigma }_s^r$ when restricted to smooth tensor fields.
• (e)The embeddings ${\iota }_s^r$ and ${\sigma }_s^r$ commute with the Lie derivative so that ${\hat{\mathcal{L}}}_X({\iota }_s^r(T))={\iota }_s^r({\mathcal{L}}_{X}T)$ and ${\hat{\mathcal{L}}}_X({\sigma }_s^r(T))={\sigma }_s^r({\mathcal{L}}_{X}T)$.

We end this section by considering association for tensor fields. The definition of association is much the same as for scalars.

Definition 5.12 (Association) —

We say that $[T]\in {\hat{\mathcal{G}}}_s^r(M)$ is associated to 0 (denoted [T] ≈ 0) if for each $\mathrm{\Psi }\in {\tilde{\mathcal{D}}}_r^s(M)$ we have

$$\underset{\epsilon \to 0}{lim}\int T_{a\dots b}^{c\dots d}({\mathrm{{\rm Y}}}_{\epsilon },{\omega }_{\epsilon })(x){\mathrm{\Psi }}_{c\dots d}^{a\dots b}(x)=0\mathrm{\forall }\omega \in \tilde{\mathcal{A}}(M),\hspace{0.17em}\mathrm{\forall }\mathrm{{\rm Y}}\in \mathrm{{\rm Y}}(M).$$

We say two elements $[S],[T]\in {\hat{\mathcal{G}}}_s^r(M)$ are associated and write [S] ≈ [T] if [S − T] ≈ 0.

Definition 5.13 (Associated distributional tensor field) —

We say $[T]\in {\hat{\mathcal{G}}}_s^r(M)$ admits $S\in {\mathcal{D}}_s^{\prime r}(M)$ as an associated tensor distribution if for each $\mathrm{\Psi }\in {\tilde{\mathcal{D}}}_r^s(M)$ we have

$$\underset{\epsilon \to 0}{lim}\int T_{a\dots b}^{c\dots d}({\mathrm{{\rm Y}}}_{\epsilon },{\omega }_{\epsilon }){\mathrm{\Psi }}_{c\dots d}^{a\dots b}(x)=⟨S,\mathrm{\Psi }⟩\mathrm{\forall }\omega \in \tilde{\mathcal{A}}(M),\hspace{0.17em}\mathrm{\forall }\mathrm{{\rm Y}}\in \mathrm{{\rm Y}}(M).$$

Then just as in the scalar case one has the following proposition (with similar proof).

Proposition 5.14. —

• (a)
  If S is a smooth tensor field in ${\mathcal{T}}_s^r(M)$ and $T\in {\mathcal{D}}_u^{\prime t}(M)$ then

  $$\iota (S)\otimes \iota (T)\approx \iota (S\otimes T).$$ 5.6
• (b)
  If S and T are continuous tensor fields then

  $$\iota (S)\otimes \iota (T)\approx \iota (S\otimes T).$$ 5.7

In the scalar case the nets of smoothing kernels form a delta-net in the sense that as ε → 0, $\iota (f)({\omega }_{\epsilon })\to f$ in ${\mathcal{D}}^{\prime }(M)$ for $f\in {\mathcal{D}}^{\prime }(M)$. The following result shows that this remains true in the tensor case.

Proposition 5.15. —

Given $S\in {\mathcal{D}}^{\prime }{\hspace{0.17em}}_s^r$ then for all $\omega \in {\tilde{\mathcal{A}}}_0(M)$, $\mathrm{{\rm Y}}\in \mathrm{{\rm Y}}(M)$ and $\mathrm{\Psi }\in {\tilde{\mathcal{D}}}_r^s(M)$ we have

$$\underset{\epsilon \to 0}{lim}{\int }_{x\in M}\iota {\hspace{0.17em}}_s^r(S)({\mathrm{{\rm Y}}}_{\epsilon },{\omega }_{\epsilon })(x)\mathrm{\Psi }(x)=⟨S,\mathrm{\Psi }⟩,$$

i.e. $\iota {\hspace{0.17em}}_s^r(S)({\mathrm{{\rm Y}}}_{\epsilon },{\omega }_{\epsilon })\to S$ in ${\mathcal{D}}^{\prime }{\hspace{0.17em}}_s^r(M)$ as ε → 0.

The following is a trivial corollary which generalizes the corresponding scalar result.

Corollary 5.16. —

At the level of association the embedding does not depend upon the transport operator or the smoothing kernel in the sense that given any two admissible nets of transport operators $\mathrm{{\rm Y}}$ and $\widetilde{\mathrm{{\rm Y}}}$ and two delta-nets of smoothing kernels ω and $\tilde{\omega }$ we have

$$\underset{\epsilon \to 0}{lim}{\int }_{x\in M}\iota {\hspace{0.17em}}_s^r(S)({\mathrm{{\rm Y}}}_{\epsilon },{\omega }_{\epsilon })\mathrm{\Psi }(x)=\underset{\epsilon \to 0}{lim}{\int }_{x\in M}\iota {\hspace{0.17em}}_s^r(S)({\widetilde{\mathrm{{\rm Y}}}}_{\epsilon },{\tilde{\omega }}_{\epsilon })(x)\mathrm{\Psi }(x)$$

for $S\in {\mathcal{D}}_s^{\prime r}(M)$ and $\mathrm{\Psi }\in {\tilde{\mathcal{D}}}_r^s(M)$.

This shows that if one regards our Colombeau type theory as a method for calculating with smoothed distributional tensor fields, then the distributional limit as ε → 0 exists for embedded distributions and does not depend on the choice of transport operators or smoothing kernels.

6. Generalized differential geometry and applications to general relativity

In the previous section we established the key structural properties of the generalized tensor algebra where we showed that it is closed under the operations of tensor product and contraction and also closed under the action of the generalized Lie derivative. Furthermore, we showed that there exists a canonical embedding of distributional tensor fields given by ${\iota }_s^r$ and that this embedding commutes with the Lie derivative. However, from the point of view of applications the key property of generalized tensor fields is that if T is an element of the basic space ${\hat{\mathcal{E}}}_s^r(M)$ then for any fixed smoothing kernel ω and any fixed transport operator $\mathrm{{\rm Y}}$ the tensor field $\tilde{T}$ given by

$$\tilde{T}:=T(\mathrm{{\rm Y}},\omega )$$

is a smooth tensor field, so that we may apply all the usual operations of smooth differential geometry to it. In particular, we can calculate the covariant derivative of a generalized tensor field. Moreover, one can apply the ordinary Lie derivative of smooth tensor fields for fixed $\mathrm{{\rm Y}}$ and ω and define $({\tilde{\mathcal{L}}}_{X}T)(\mathrm{{\rm Y}},\omega ):={\mathcal{L}}_X(T(\mathrm{{\rm Y}},\omega ))$.

We now look at the covariant derivative of a generalized tensor field in the basic space. Let $\mathrm{\nabla }$ be a smooth connection and Z a smooth vector field. For $T\in {\hat{\mathcal{E}}}_s^r(M)$ we define the generalized tensor field ${\mathrm{\nabla }}_{Z}T$ to be given by

$$({\mathrm{\nabla }}_{Z}T)(\mathrm{{\rm Y}},\omega ):={\mathrm{\nabla }}_Z(T(\mathrm{{\rm Y}},\omega )).$$

Furthermore, if T is moderate then so is ${\mathrm{\nabla }}_{Z}T$, and if T is negligible then so is ${\mathrm{\nabla }}_{Z}T$, so that we may define the covariant derivative of a generalized tensor field to be given by

$${\mathrm{\nabla }}_Z[T]:=[{\mathrm{\nabla }}_{Z}T].$$

Lemma 6.1. —

Let $S\in {\mathcal{D}}^{\prime }{\hspace{0.17em}}_s^r(M)$ be a distributional type (r, s) tensor field, $T\in {\mathcal{T}}_r^s(M)$ a smooth type (s, r) tensor field and Z ∈ 𝔰X a smooth vector field. Then

$${\tilde{\mathcal{L}}}_Z(\iota {\hspace{0.17em}}_s^r{(S)}^a{T}_a)\approx \iota {\hspace{0.17em}}_0^0({\mathcal{L}}_Z(S^a{T}_a)).$$

Proof. —

We will illustrate this by considering a distributional vector field $X\in {\mathcal{D}}^{\prime }{\hspace{0.17em}}_0^1(M)$ and contracting with θ ∈ Ω¹(M). Let μ be a smooth density of compact support; then by proposition 5.15 we have

$$\begin{array}{rl}\underset{\epsilon \to 0}{lim}\int {\tilde{\mathcal{L}}}_Z({\iota }_0^1{(X)}^a{\theta }_a)(x)\mu (x)& =-\underset{\epsilon \to 0}{lim}\int {\iota }_0^1{(X)}^a(x){\theta }_a(x)({\mathcal{L}}_Z\mu )(x)\\ & =-⟨X^a{\theta }_a,{\mathcal{L}}_Z\mu ⟩=⟨{\mathcal{L}}_Z(X^a{\theta }_a),\mu ⟩.\end{array}$$

On the other hand,

$$\underset{\epsilon \to 0}{lim}\int {\iota }_0^0({\mathcal{L}}_Z(X^a{\theta }_a))(x)\mu (x)=⟨{\mathcal{L}}_Z(X^a{\theta }_a),\mu ⟩,$$

so that

$${\tilde{\mathcal{L}}}_Z({\iota }_0^1{(X)}^a{\theta }_a)\approx {\iota }_0^0({\mathcal{L}}_Z(X^a{\theta }_a)).$$

The general case is seen similarly. ▪

Proposition 6.2. —

Let $S\in {\mathcal{T}}_s^r(M)$ be a C¹ type (r, s) tensor field, Z ∈ 𝔰X(M) a smooth vector field and $\mathrm{\nabla }$ a smooth covariant derivative. Then

$${\mathrm{\nabla }}_Z(\iota {\hspace{0.17em}}_s^r(S))\approx \iota {\hspace{0.17em}}_s^r({\mathrm{\nabla }}_{Z}S).$$ 6.1

Proof. —

This follows directly from continuity of ${\mathrm{\nabla }}_Z:{\mathcal{D}}_s^{\prime r}(M)\to {\mathcal{D}}_s^{\prime r}(M)$. ▪

Remark 6.3. —

• (a)For a smooth tensor field $S\in {\mathcal{T}}_s^r(M)$ equation (6.1) is true with equality rather than association.
• (b)With a suitable definition of the distributional covariant derivative (see [36, section 3.1]) proposition 6.2 is true for $S\in {\mathcal{D}}^{\prime }{\hspace{0.17em}}_s^r(M)$.
• (c)Give any given coordinate system $x^{\mu }$ we can define a covariant derivative which is nothing but the partial derivative in these coordinates. Hence in any given coordinate system the above result is also true if we replace ${\mathrm{\nabla }}_Z$ by the partial derivatives ${\mathrm{\partial }}_{\mu }$.
• (d)The above result is also true for all $S\in {\mathcal{D}}^{\prime }{\hspace{0.17em}}_s^r(M)$ if we replace ${\mathrm{\nabla }}_Z$ by ${\mathcal{L}}_Z$.

Up to now we have discussed the covariant derivative of a generalized tensor field with respect to a smooth classical connection $\mathrm{\nabla }$. We now define a generalized version of this. We may do this by writing a generalized covariant derivative as being given by a (smooth) covariant derivative $\stackrel{0}{\mathrm{\nabla }}$ with respect to some background connection γ together with a correction term given by a generalized type (1, 2) tensor field ${\hat{\mathit{\Gamma }}}_{bc}^a$. Thus, the generalized covariant derivative of a generalized vector field X is given by

$${({\mathrm{\nabla }}_{Z}X)}^a={({\stackrel{0}{\mathrm{\nabla }}}_{Z}X)}^a+{\hat{\mathit{\Gamma }}}_{bc}^a{Z}^b{X}^c.$$ 6.2

Note that this does not depend on the choice of background connection if we change the tensorial correction term by the difference in the connection coefficients of the background connections. This leads to the following definition.

Definition 6.4 (Covariant derivative) —

Let $[T]{\hspace{0.17em}}_{b_1\dots b_s}^{a_1\dots a_r}\in {\hat{\mathcal{G}}}_s^r(M)$, let ${[\hat{\mathit{\Gamma }}]}_{bc}^a\in {\hat{\mathcal{G}}}_2^1(M)$ and let Z be a smooth vector field. Then we may define ${\mathrm{\nabla }}_{Z}T\in {\hat{\mathcal{G}}}_s^r(M)$ by

$${\mathrm{\nabla }}_Z[T]=[{\mathrm{\nabla }}_{Z}T],$$

where

$$\begin{array}{rl}({\mathrm{\nabla }}_{Z}T){\hspace{0.17em}}_{b_1\dots b_s}^{a_1\dots a_r}& ={\stackrel{0}{\mathrm{\nabla }}}_{Z}T{\hspace{0.17em}}_{b_1\dots b_s}^{a_1\dots a_r}+Z^c({\hat{\mathit{\Gamma }}}_{dc}^{a_1}T{\hspace{0.17em}}_{b_1\dots b_s}^{d{a}_2\dots a_r}+\cdots +{\hat{\mathit{\Gamma }}}_{dc}^{a_r}T{\hspace{0.17em}}_{b_1\dots b_s}^{a_1\dots a_{r-1}d}-{\hat{\mathit{\Gamma }}}_{b_{1}c}^{d}T{\hspace{0.17em}}_{d{b}_2\dots b_s}^{a_1\dots a_r}-\cdots -{\hat{\mathit{\Gamma }}}_{b_{s}c}^{d}T{\hspace{0.17em}}_{b_1\dots b_{s-1}d}^{a_1\dots a_r}).\end{array}$$ 6.3

We note that the above definition also makes sense if we replace Z by a generalized vector field Z.

We now turn to the definition of a generalized metric. Generalized metrics have been considered in the context of the special algebra of tensor fields by [38]. There, a number of equivalent definitions of a generalized metric are given. We will use the following definition.

Definition 6.5 (Generalized metric) —

We say $g_{ab}\in {\hat{\mathcal{G}}}_2^0(M)$ is a generalized metric if

• (i)g_ab = g_ba, i.e. g is symmetric, and
• (ii)the map $X^a\mapsto X^a{g}_{ab}$ from ${\hat{\mathcal{G}}}_0^1(M)$ into ${\hat{\mathcal{G}}}_1^0(M)$ is bijective.

Proposition 6.6. —

If g_ab is a C⁰ metric then ${\tilde{g}}_{ab}={\iota }_2^0(g_{ab})$ is a generalized metric.

Proof. —

This follows from the fact that if g_ab is continuous then ${\tilde{g}}_{ab}({\mathrm{{\rm Y}}}_{\epsilon },{\omega }_{\epsilon })$ converges uniformly to g_ab on compact subsets, which allows one to define the inverse metric via the cofactor formula along the lines of [31]. ▪

Definition 6.7 (Generalized Levi-Civita connection) —

Given a generalized metric g_ab one may calculate the generalized Levi-Civita connection with respect to a smooth background connection ${\gamma }_{bc}^a$ by defining ${\hat{\mathit{\Gamma }}}_{bc}^a$ according to

$${\hat{\mathit{\Gamma }}}_{bc}^a=\frac{1}{2}{g}^{ad}(g_{bd|c}+g_{cd|b}-g_{bc|d}),$$ 6.4

where g^ab is defined by $g^{ad}{g}_{db}={\delta }_b^a$ and g_bd|c denotes the covariant derivative of g_bd with respect to the background connection ${\gamma }_{bc}^a$.

One now defines the corresponding generalized covariant derivative according to (6.3) using $\hat{\mathit{\Gamma }}$ defined in equation (6.4).

Proposition 6.8. —

If g_ab is a C¹ metric then the generalized connection coefficients ${\hat{\mathit{\Gamma }}}_{bc}^a$ of ${\hat{g}}_{ab}$ with respect to a background connection ${\gamma }_{bc}^a$ are associated to the classical Levi-Civita connection coefficients of g_ab with respect to the background connection ${\gamma }_{bc}^a$ (which are given by $({\mathit{\Gamma }}_{bc}^a-{\gamma }_{bc}^a)$) so that

$${\hat{\mathit{\Gamma }}}_{bc}^a\approx \iota [({\mathit{\Gamma }}_{bc}^a-{\gamma }_{bc}^a)].$$

Proof. —

The proof follows from the fact that for a C¹ metric

$$\iota [g^{ad}]\iota {[g_{bd}]}_{|c}\approx \iota [g^{ad}{g}_{bd|c}].$$ 6.5

 □

We next consider the generalized curvature of a generalized connection.

Definition 6.9 (Generalized curvature) —

Let $\hat{\mathrm{\nabla }}$ be a generalized connection. We may define a type (1, 3) generalized curvature tensor ${\widehat{R^a}}_{bcd}\in {\hat{\mathcal{G}}}_3^1(M)$ by

$$({\hat{\mathrm{\nabla }}}_X{\hat{\mathrm{\nabla }}}_Y-{\hat{\mathrm{\nabla }}}_Y{\hat{\mathrm{\nabla }}}_X-{\hat{\mathrm{\nabla }}}_{[X,Y]})Z=\hat{R}(X,Y)Z,$$

where X, Y and Z are smooth vector fields.

Proposition 6.10. —

Let ${\mathit{\Gamma }}_{bc}^a$ define a differentiable connection $\mathrm{\nabla }$ and let ${\hat{\mathit{\Gamma }}}_{bc}^a$, given by ${\hat{\mathit{\Gamma }}}_{bc}^a=\iota [({\mathit{\Gamma }}_{bc}^a-{\gamma }_{bc}^a)]$, be used in equation (6.2) to define the generalized connection $\hat{\mathrm{\nabla }}$. Then,

$${\hat{R}}^a{\hspace{0.17em}}_{bcd}\approx \iota [{R^a}_{bcd}],$$

i.e. the generalized curvature of the embedded connection $\hat{\mathrm{\nabla }}$ is associated to the embedding of the curvature of $\mathrm{\nabla }$.

Proof. —

For a C² metric the result follows from writing out the formula for ${\widetilde{R^a}}_{bcd}$ in terms of $\hat{\mathit{\Gamma }}$, γ and their derivatives and using the standard properties of association. ▪

Combining this with our earlier result on connections we have the following result.

Proposition 6.11. —

If g_ab is a C² metric then

$${\tilde{R}}^a{\hspace{0.17em}}_{bcd}\approx \iota [{R^a}_{bcd}],$$

where ${\widetilde{R^a}}_{bcd}$ is the generalized curvature of the generalized Levi-Civita connection of ${\tilde{g}}_{ab}$ and $R^a{\hspace{0.17em}}_{bcd}$ is the curvature of the standard Levi-Civita connection of g_ab.

By contraction we may define ${\tilde{R}}_{bd}={\widetilde{R^a}}_{bad}$, $\tilde{R}={\tilde{g}}^{bd}{\tilde{R}}_{bd}$ and ${\tilde{G}}_{ab}={\tilde{R}}_{ab}-\frac{1}{2}{\tilde{g}}_{ab}\tilde{R}$. Then the above result gives the following proposition.

Proposition 6.12. —

If g_ab is a C² solution of the vacuum Einstein equations G_ab = 0 then the embedded generalized metric ${\tilde{g}}_{ab}$ has generalized Einstein tensor ${\tilde{G}}_{ab}$, which satisfies

$${\tilde{G}}_{ab}\approx \iota [G_{ab}]=0$$

and therefore satisfies the Einstein equations at the level of association.

Thus, if we have a C² solution of the vacuum Einstein equations then the embedded metric ${\tilde{g}}_{ab}$ also satisfies the Einstein equations at the level of association (although the Bianchi identities hold at the level of equality). The important thing to note about this is that it suggests that for generalized metrics the appropriate version of the Einstein equations is

$${\tilde{G}}_{ab}\approx 8\pi {\tilde{T}}_{ab},$$

where ${\tilde{T}}_{ab}$ is the embedding of some distributional energy–momentum tensor. This is in the spirit of the ‘coupled calculus’ approach of [18]: one performs the algebraic operations and derivatives in the differential algebra $\check{\mathcal{G}}(M)$, but solves the differential equations at the level of association.

We now consider the case where g_ab is not C² but satisfies the weaker regularity conditions of Geroch & Traschen [7] which guarantee the existence of a distributional curvature $R^a{\hspace{0.17em}}_{bcd}$. It turns out that with some additional technical conditions, which guarantee that ${\iota }_2^0(g_{ab})$ is indeed a generalized metric, ${\tilde{G}}_{ab}$ is associated to the embedding of the distributional energy tensor defined by $R^a{\hspace{0.17em}}_{bcd}$.

Definition 6.13 (Geroch–Traschen regularity) —

A symmetric tensor g_ab is called a gt-regular metric if it is a metric almost everywhere and g_ab and g^ab are in $L_{\mathrm{loc}}^{\mathrm{\infty }}\cap H_{\mathrm{loc}}^1$.

In the above definition $L_{\mathrm{loc}}^{\mathrm{\infty }}$ denotes the space of locally bounded functions and $H_{\mathrm{loc}}^1$ denotes the Sobolev space of functions which are locally square integrable and also have a locally square integrable first (weak) derivative. Note that although the above definition appears to be stronger than that in [7] it is actually equivalent to the original one (see [39] for details). The fact that a gt-regular metric is only defined almost everywhere causes some difficulties. In [39] a class of non-degenerate and stable gt-regular metrics was introduced and it was shown that if these are smoothed componentwise by a suitable class of mollifiers then the curvature of the smoothing ${\tilde{g}}_{ab}^{\epsilon }$ tends to the (distributional) curvature of g_ab in ${\mathcal{D}}^{\prime }$. Rather than go into the complications of defining a non-degenerate and stable metric in the present context we will instead follow [7] (especially theorem 4) and work with the slightly larger class of continuous gt-regular metrics. One can then show that given a continuous gt-regular metric we can either derive the (distributional) Riemann curvature Riem[g] of the gt-regular metric g_ab or embed g_ab in the algebra to obtain the generalized metric ${\tilde{g}}_{ab}$. If we then derive its curvature $\mathrm{Riem}[\tilde{g}]$ within the generalized setting we find that it is associated to the distributional curvature Riem[g]. This is depicted in the following diagram:

More precisely, we have the following theorem.

Theorem 6.14 (Compatibility for the Riemann curvature) —

Let g_ab be a continuous gt-regular metric with Riemann tensor Riem[g]. Let ${\tilde{g}}_{ab}:={\iota }_2^0(g_{ab})$ be the generalized metric obtained by embedding in the algebra. Then

$$\mathrm{Riem}[{\iota }_2^0(g)]\approx \mathrm{Riem}[g].$$

Proof. —

Since g_ab is continuous ${\iota }_2^0(g_{ab})$ defines a generalized metric. We may then obtain the estimates used in deriving the corresponding result in [39] by working with the local form of the smoothing kernel and the transport operators together with the fact that $\mathrm{{\rm Y}}(x,x)=\text{id}$. ▪

The following corollary is immediate.

Proposition 6.15. —

If g_ab is a continuous gt-regular metric that satisfies the vacuum Einstein equations then ${\tilde{g}}_{ab}={\iota }_2^0(g_{ab})$ is a generalized metric that satisfies

$${\tilde{G}}_{ab}\approx 0.$$

Moving beyond the class of gt-regular metrics, it is of considerable interest to find the weakest conditions on g_ab which guarantee that ${\tilde{G}}_{ab}$ is associated to a (conventional) distribution, so that the source admits a distributional interpretation. We know from the example of conical singularities [20,23] that it is possible to have metrics which do not satisfy the Geroch and Traschen regularity conditions, but all the same have a distributional energy–momentum tensor. We briefly review this work in the context of the present manifestly coordinate-invariant theory.

In [20] it was shown that if one computed the scalar curvature density of a cone in ${\mathbb{R}}^2$ in Cartesian coordinates it was associated to a delta distribution δ⁽²⁾(x, y) with a numerical factor that depended on the deficit angle. In a subsequent paper (see [23]) it was furthermore shown that if one transforms the metric to a new coordinate system the generalized scalar curvature density is associated to the transformed delta distribution.

In the present paper we have shown that one can embed g_ab into the Colombeau algebra $\check{\mathcal{G}}(M)$ in a manifestly coordinate-invariant way. We now show that, for the case of a two-dimensional cone, the scalar curvature is associated to a delta distribution. We outline the calculation below.

In Cartesian coordinates the metric of the two-dimensional cone with deficit angle 2(1 − A)π may be written as

$$g_{ab}=\frac{1}{2}(1+A^2){\delta }_{ab}+\frac{1}{2}(1-A^2)m_{ab}$$

and

$$m_{ab}=\left(\begin{array}{cc}\frac{x{\hspace{0.17em}}^2-y{\hspace{0.17em}}^2}{x{\hspace{0.17em}}^2+y{\hspace{0.17em}}^2}& \frac{2xy}{x{\hspace{0.17em}}^2+y{\hspace{0.17em}}^2}\\ \frac{2xy}{x{\hspace{0.17em}}^2+y{\hspace{0.17em}}^2}& -\frac{x{\hspace{0.17em}}^2-y{\hspace{0.17em}}^2}{x{\hspace{0.17em}}^2+y{\hspace{0.17em}}^2}\end{array}\right).$$

Since δ_ab is already smooth and A is a constant, the only term we need to smooth for embedding the metric into $\check{\mathcal{G}}(M)$ is m_ab.

To show that the scalar curvature ${\tilde{R}}_{\epsilon }$ of ${\tilde{g}}_{ab,\epsilon }=\tilde{g}({\mathrm{{\rm Y}}}_{\epsilon },{\omega }_{\epsilon })$ converges in the sense needed for association, one writes the pairing with a smooth 2-form of compact support in local coordinates as

$$\int {\tilde{R}}_{\epsilon }\omega (x)\sqrt{\mid {\tilde{g}}_{\epsilon }(x)\mid }\hspace{0.17em}\text{d}x=\omega (0,0)\int {\tilde{R}}_{\epsilon }(x)\sqrt{\mid {\tilde{g}}_{\epsilon }(x)\mid }\hspace{0.17em}\text{d}x+\int {\int }_0^1{\tilde{R}}_{\epsilon }(x)(D\omega )(tx)x\sqrt{\mid {\tilde{g}}_{\epsilon }(x)\mid }\hspace{0.17em}\text{d}t\hspace{0.17em}\text{d}x.$$

While the first integral on the right-hand side can easily be evaluated using the Gauss–Bonnet theorem to give the desired result, we need precise estimates for the components ${\tilde{g}}_{ab,\epsilon }$ of the regularized metric to show that the second integral vanishes for ε → 0. For this one looks at the integrand inside and outside a neighbourhood of zero whose diameter is proportional to ε, say εR₀. In the inside one can directly employ homogeneity of the components of the metric and the L¹-conditions on (ω_ε)_ε (see [17]) to obtain the needed estimate. For the outside, one has to find an expression for the constant C appearing in the estimate (away from the origin)

$$\mid {\mathrm{\partial }}^{\alpha }{\tilde{g}}_{ab,\epsilon }(x)-{\mathrm{\partial }}^{\alpha }{g}_{ab}(x)\mid \le C{\epsilon }^q$$

in terms of derivatives of g_ab, which is again combined with homogeneity of the metric to obtain

$${\tilde{R}}_{\epsilon }=\left\{\begin{array}{ll}O(1/{\epsilon }^2)& \text{if}\hspace{0.17em}r<\epsilon R_0,\\ O(\epsilon /r^3)& \text{if}\hspace{0.17em}r>\epsilon R_0.\end{array}\right.$$ 6.6

With this one obtains that given any smooth 2-form μ of compact support one has

$$\underset{\epsilon \to 0}{lim}\int {\tilde{R}}_{\epsilon }\mu =4\pi (1-A)⟨{\delta }^{(2)},\mu ⟩,$$ 6.7

which shows that the generalized scalar curvature is associated to a delta distribution (see [31] for details). A similar calculation (but requiring more delicate estimates) can be carried out for the Ricci curvature of a four-dimensional cone along the lines of those in Wilson [40], giving the following result.

Proposition 6.16. —

Let g_ab be the conical metric given in standard cylindrical polar coordinates by

$$\text{d}{s}^2=\text{d}{t}^2-\text{d}{r}^2-A^2{r}^2\hspace{0.17em}\text{d}{\varphi }^2-\text{d}{z}^2.$$

Then

$${\tilde{G}}_{ab}\approx 8\pi {\tilde{T}}_{ab},$$ 6.8

where ${\tilde{T}}_{ab}$ is the embedding into the Colombeau algebra of the energy momentum tensor T of a cosmic string with delta-function terms with singular support on the string and with the stress equal to the density μ = 2π(1 − A), i.e. with non-zero components only $T_1^1=T_4^4=2\pi \mu {\delta }_2$, where δ₂ is the two-dimensional delta function with support on r = 0.

7. Conclusion

In this paper we have developed a theory of nonlinear tensor distributions building on the scalar theory of [17]. Our approach has been to embed classical distributional tensor fields into the tensor algebra of families of smooth tensor fields. The most general way of doing this is via families of smoothing operators that become closer and closer to the identity and we show that this is precisely equivalent to using the scalar smoothing used in [17] and the transport operators used in [29]. Although the theory given here has many features in common with [29] we note that it also has a number of important improvements. These include the ability to define a covariant derivative, greater coherence between the scalar and tensor theories owing to the different choice of basic space and the sheaf property of tensor distributions, which is possible through the concept of localization in the current theory.

The generalized tensor algebra $\check{\mathcal{G}}(M)$ we have constructed is an associative differential algebra with respect to the tensor product ⊗, and derivatives given by the generalized Lie derivative. It contains the space of smooth tensor fields (embedded by the ‘constant map’ ${\sigma }_s^r$) as subalgebra of $\check{\mathcal{G}}(M)$ and the distributional tensor fields ${{\mathcal{D}}^{\prime }}_s^r(M)$ as a C^∞(M)-submodule of ${\hat{\mathcal{G}}}_s^r(M)$. The distributional tensor fields are embedded via ${\iota }_s^r$ and this coincides with the embedding via ${\sigma }_s^r$ when restricted to smooth tensor fields. Both these embeddings commute with the Lie derivative so that ${\hat{\mathcal{L}}}_X({\iota }_s^r(T))={\iota }_s^r({\mathcal{L}}_{X}T)$ and ${\hat{\mathcal{L}}}_X({\sigma }_s^r(T))={\sigma }_s^r({\mathcal{L}}_{X}T)$. We also introduced the generalized covariant derivative of a generalized tensor field and used this to develop a nonlinear theory of distributional geometry. In particular we showed how to construct the generalized curvature tensor of a generalized metric. Given any locally integrable (or indeed distributional) metric one can embed this into the algebra of generalized tensor fields and calculate the generalized curvature. One can then show that if the original metric is at least C² then the generalized curvature that one obtains is associated to the curvature tensor calculated in the standard way. Thus, at the level of association the current theory gives the same results as standard calculus for C² metrics. Finally we gave an application where we used the present theory to calculate the generalized curvature of a conical metric and show that this was associated to a delta function.

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Authors' contributions

J.A.V. conceived the initial draft of the manuscript, when many technical details remained open, and made the main contribution to explaining the theory to non-specialists and to the section on applications. E.A.N. did much of the research to resolve the open questions as well as devised the new functional analytic approach. Both authors contributed to writing all sections and revising the whole manuscript. Both authors gave final approval for publication and agree to be held accountable for the work performed herein.

Competing interests

We declare we have no competing interests.

Funding

E.A.N. was supported by the Austrian Science Fund (FWF) grant nos. P26859 and P30233. J.A.V. was supported by STFC grant no. ST/R00045X.