Let \((\varvec},\varvec)\) be a \(3+1\)-dimensional, smooth, time-oriented Lorentzian manifold.

In this section and throughout the paper, we adopt the bolded notation of [17] in the nonlinear setting.

4.1 Preliminary Definitions

We give some preliminary definitions.

Definition 4.1

(Null frame). A null frame

$$\begin \varvec}=(_}}\,}},_}}\,}},_}}\,}},_}}\,}}) \end$$

on \((\varvec},\varvec)\) is a frame on \(\varvec}\) such that the identities

$$\begin \varvec(_}}\,}},_}}\,}})&=\delta _} ,&\varvec(_}}\,}},_}}\,}})&=-2 , \end$$

(53)

$$\begin \varvec(_}}\,}},_}}\,}})=\varvec(_}}\,}},_}}\,}})&= 0 ,&\varvec(_}}\,}},_}}\,}})=\varvec(_}}\,}},_}}\,}})&=0 \end$$

(54)

hold, with \(\varvec,\varvec=\left\,\varvec\right\} \).

Remark 4.2

In Definition 4.1, the frame vector fields \((_}}\,}},_}}\,}})\) are assumed to be orthonormal.

Remark 4.3

A local null frame on \((\varvec},\varvec)\) is a local frame on \(\varvec}\) such that the identities (53)–(54) hold.

Definition 4.4

(Horizontal distribution). The (local) horizontal distribution associated with a (local) null frame \(\varvec}\) is the (local) distribution

$$\begin \varvec}_}}:=\left\langle _}}\,}},_}}\,}}\right\rangle ^} \end$$

on \(\varvec}\).

Remark 4.5

Strictly speaking, the horizontal distribution is associated to the pair of null vector fields \((_}}\,}},_}}\,}})\) (as opposed to the full frame \(\varvec}\), as phrased for future convenience in Definition 4.4). In fact, null frames that only differ by the choice of orthonormal vector fields \((_}}\,}},_}}\,}})\) induce the same horizontal distribution. We also emphasise that the domain of \(\varvec}\) on which the horizontal distribution is well-defined coincides with that of the null pair \((_}}\,}},_}}\,}})\), and may in principle be larger than the domain on which the null pair \((_}}\,}},_}}\,}})\) can be completed to a null frame. Indeed, in Sect. 5 we will deal with a null pair which is globally well-defined on the manifold considered, but can only locally be completed to a null frame. In such a scenario, the associated horizontal distribution will be global.

Remark 4.6

The horizontal distribution \(\varvec}_}}\) can be regarded as an abstract tensor bundle on \(\varvec}\) or, alternatively, as a sub-bundle \(\varvec}_}}\subset T\varvec}\) on \(\varvec}\) via the inclusion map

$$\begin \iota : \varvec}_}}\hookrightarrow T\varvec} \,. \end$$

Such a distinction will be exploited in Definitions 4.12 and 4.17.

Crucially, the horizontal distribution \(\varvec}_}}\) is allowed to be a non-integrable distribution.Footnote 39

Definition 4.7

(Non-integrable null frame). A non-integrable null frame \(\varvec}\) is a null frame whose associated horizontal distribution \(\varvec}_}}\) is a non-integrable distribution.

Remark 4.8

The vector fields \((_}}\,}},_}}\,}})\) form a local orthonormal basis of \(\varvec}_}}\subset T\varvec}\). By Definition 4.7, the null frame \(\varvec}\) is a non-integrable frame if and only if, for any choice of \((_}}\,}},_}}\,}})\), at least one of the relations

$$\begin \varvec([_}}\,}},_}}\,}}],_}}\,}})&\ne 0 ,&\varvec([_}}\,}},_}}\,}}],_}}\,}})&\ne 0 \end$$

holds.

We now define the main geometric objects of the paper, namely \(\varvec}_}}\) tensors. In the following definition, the horizontal distribution \(\varvec}_}}\) is regarded as an abstract tensor bundle on \(\varvec}\).

Definition 4.9

(\(\varvec}_}}\) tensors). A \(\varvec}_}}\) (k, q)-tensor field \(\varvec\) on \(\varvec}\), with \(k,q\in \mathbb _0\), is a smooth section

$$\begin \varvec\in \Gamma (\otimes _k \varvec}_}}\otimes _q(\varvec}_}})^}) . \end$$

In particular, a \(\varvec}_}}\) vector field \(\varvec\) on \(\varvec}\) is a smooth section

$$\begin \varvec\in \Gamma (\varvec}_}}) . \end$$

Remark 4.10

The domain of \(\varvec}\) on which a \(\varvec}_}}\) tensor is well-defined coincides with that of the horizontal distribution \(\varvec}_}}\), and may in principle be larger than the domain on which the null frame \(\varvec}\) can be defined. See the related Remark 4.5.

Remark 4.11

We often refer to \(\varvec}_}}\) (0, q)-tensor fields as \(\varvec}_}}\) q-tensors.

Let \(\iota _k\) be the inclusion map

$$\begin \iota _k:\otimes _k \varvec}_}} \hookrightarrow \otimes _k T\varvec} , \end$$

with \(k\in \mathbb \). We denote \(\iota _\) by \(\iota \).

For later convenience, we now define how \(\varvec}_}}\) tensors can be canonically extended to spacetime tensors. We start with \(\varvec}_}}\) vector fields.

Definition 4.12

(Canonical extension of \(\varvec}_}}\) vector fields). Let \(\varvec\) be a \(\varvec}_}}\) vector field on \(\varvec}\). The canonical extension of \(\varvec\) on \(\varvec}\) is the smooth section

$$\begin \varvec^\in \Gamma (T\varvec}) \end$$

such that

$$\begin (p,\varvec_p^)=\iota (p,\varvec_p) \end$$

for all \((p,\varvec_p^)\in T\varvec}\).

Remark 4.13

Let \(\varvec^\) be the canonical extension of a \(\varvec}_}}\) vector field \(\varvec\) on \(\varvec}\). Then, by Definitions 4.4 and 4.12, the identities

$$\begin \varvec(\varvec^,_}}\,}})&=0 ,&\varvec(\varvec^,_}}\,}})&=0 \end$$

hold.

Remark 4.14

With a slight abuse of notation, the frame vector fields \((_}}\,}},_}}\,}})\) will be treated both as \(\varvec}_}}\) vector fields on \(\varvec}\) and as their canonical extensions.

We now define the canonical extension of higher-rank \(\varvec}_}}\) (k, 0)-tensors. For \(k=1\), the following definition and remark reduce to Definition 4.12 and Remark 4.13, respectively.

Definition 4.15

(Canonical extension of \(\varvec}_}}\) (k, 0)-tensors). Let \(\varvec\) be a \(\varvec}_}}\) (k, 0)-tensor on \(\varvec}\). The canonical extension of \(\varvec\) on \(\varvec}\) is the smooth section

$$\begin \varvec^ \in \Gamma (\otimes _k T\varvec}) \end$$

such that

$$\begin (p,\varvec^_p)=\iota _k (p,\varvec_p) \end$$

for all \((p,\varvec^_p)\in \otimes _k T\varvec}\).

Remark 4.16

Let \(\varvec^\) be the canonical extension of a \(\varvec}_}}\) (k, 0)-tensor \(\varvec\) on \(\varvec}\). Then, by Definitions 4.4 and 4.15, the identities

$$\begin \varvec^(\ldots ,\varvec(_}}\,}},\cdot ),\ldots )&=0 ,&\varvec^(\ldots ,\varvec(_}}\,}},\cdot ),\ldots )&=0 \end$$

hold.

We now define the canonical extension of \(\varvec}_}}\) (0, q)-tensors.

Definition 4.17

(Canonical extension of \(\varvec}_}}\) (0, q)-tensors). Let \(\varvec\) be a \(\varvec}_}}\) (0, q)-tensor on \(\varvec}\). The canonical extension of \(\varvec\) on \(\varvec}\) is the smooth section

$$\begin \varvec^ \in \Gamma (\otimes _q(T\varvec})^) \end$$

such that, for any \(\varvec}_}}\) vector fields \(\varvec,\ldots ,\varvec\), the identities

$$\begin \varvec^(\varvec_}^,\ldots ,\varvec_}^)&=\varvec(\varvec,\ldots ,\varvec) ,&\varvec^(\ldots ,_}}\,}},\ldots )&=\varvec^(\ldots ,_}}\,}},\ldots )=0 \end$$

hold.

We conclude the section with a series of remarks.

Remark 4.18

The canonical extension of a full-rank \(\varvec}_}}\) tensor is not a full-rank tensor.

Remark 4.19

The canonical extension of \(\varvec}_}}\) (0, q)-tensors, as defined in Definition 4.17, differs from the extension induced by the inclusion \(\otimes _q(\varvec}_}})^\hookrightarrow \otimes _q(T\varvec})^\).Footnote 40 We give a concrete example to elucidate this fact. We consider the metric \(\varvec\) in local double-null form (see [18]) and the local null frame \(\varvec}=(_}}\,}},_}}\,}},_}}\,}})=(\partial _}+\varvec}\partial _},\varvec^\partial _},_}}\,}})\). Locally, one has \(\varvec}_}}\equiv T\mathbb ^2_,\varvec}\), with \(\mathbb ^2_,\varvec}:=\left\,\varvec \right\} \times \mathbb ^2\). We consider the \(\varvec}_}}\) one-tensor

$$\begin \varvec=d\varvec \,. \end$$

It is easy to check that the canonical extension of \(\varvec\), as defined in Definition 4.17, reads

$$\begin \varvec^=d\varvec-\varvec}d\varvec \,. \end$$

The extension of \(\varvec\) induced by the inclusion \((\varvec}_}})^\hookrightarrow (T\varvec})^\) is \(d\varvec\in \Gamma ((T\varvec})^)\), which manifestly differs from \(\varvec^\). In particular, we have \(d\varvec(_}}\,}})=\varvec}\ne 0\).

Remark 4.20

In Sect. 7, any \(\varvec \in \Gamma (\otimes _q(T\varvec})^)\) such that \(\varvec(\ldots ,_}}\,}},\ldots )=\varvec(\ldots ,_}}\,}},\ldots )=0\) will be referred to as a \(\varvec}_}}\)-horizontal (0, q)-tensor. Similarly, any \(\varvec \in \Gamma (\otimes _k T\varvec})\) such that \(\varvec(\ldots ,\varvec(_}}\,}},\cdot ),\ldots )=\varvec(\ldots ,\varvec(_}}\,}},\cdot ),\ldots )=0\) will be referred to as a \(\varvec}_}}\)-horizontal (k, 0)-tensor. Using this terminology, the canonical extension of a \(\varvec}_}}\) (0, q)-tensor is a \(\varvec}_}}\)-horizontal (0, q)-tensor. Similarly, the canonical extension of a \(\varvec}_}}\) (k, 0)-tensor is a \(\varvec}_}}\)-horizontal (k, 0)-tensor. Such a terminology is not adopted anywhere else in the paper.

Remark 4.21

All the definitions in this section are manifestly independent of the choice of frame vector fields \((_}}\,}},_}}\,}})\).

4.2 Induced Metric, Connection Coefficients and Curvature Components

Let \(\varvec\) be the Levi-Civita connection associated to \(\varvec\) on \(\varvec}\), \(\varvec\) the Riemann curvature tensor on \((\varvec},\varvec)\) and \(\varvec\) the standard volume form associated to \(\varvec\) on \(\varvec}\). Let \(\varvec}\) be a local null frame on \((\varvec},\varvec)\), according to Definition 4.1. All the definitions in this section are independent of the choice of frame vector fields \((_}}\,}},_}}\,}})\). The time-orientation of \((\varvec},\varvec)\) is such that \(\varvec(_}}\,}},_}}\,}},_}}\,}},_}}\,}})=2\).

We introduce the \(\varvec}_}}\) covariant tensor fields

such thatFootnote 41

Informally, we refer to as the induced metric on \(\varvec}_}}\) and to as the induced volume form on \(\varvec}_}}\).

Remark 4.22

For any \(\varvec}_}}\) vector field \(\varvec\), we have the \(\varvec}_}}\) one-tensor

and the identity

Remark 4.23

For any \(\varvec}_}}\) vector fields \(\varvec, \varvec\), we have the identity

We define the \(\varvec}_}}\) contravariant tensor

as the inverses of the metric . One may view as an induced metric on \((\varvec}_}})^\). We will adopt the notation

where the left hand side will be often written in the more synthetic form . We note that the \(\varvec}_}}\) one-tensors are the dual co-frame to \((_}}\,}},_}}\,}})\) and form a local orthonormal co-frame of \((\varvec}_}})^\).

Remark 4.24

For any \(\varvec}_}}\) one-tensor \(\varvec\), we have the \(\varvec}_}}\) vector field

We will write the expression

(55)

We have the identity

Remark 4.25

For any \(\varvec}_}}\) one-tensors \(\varvec, \varvec\), we have the identity

We define the connection coefficients relative to the null frame \(\varvec}\) as the smooth scalar functions

$$\begin \varvec}\,}}} , \, \varvec}}\,}}} \end$$

and the \(\varvec}_}}\) covariant tensor fields

$$\begin & \varvec\,, \, \varvec}\,}}} \,, \, \varvec \,, \, \varvec}\,}}} \,, \, \varvec\,, \\ & \varvec\,, \, \varvec}\,}}} \end$$

such that, with the frame-index notation \(\varvec(\varvec},\ldots ,\varvec})=\varvec}\), one has

$$\begin \varvec_}&=\varvec(\varvec, \varvec) ,&\varvec}\,}}}_}&=\varvec(\varvec,\varvec) , \\ \varvec_}&= \frac \, \varvec(\varvec,\varvec) ,&\varvec}\,}}}_}&= \frac \, \varvec(\varvec,\varvec) , \\ \varvec_}&= \frac \, \varvec(\varvec,\varvec) ,&\varvec}\,}}}_}&= \frac \, \varvec(\varvec,\varvec) , \\ \varvec}\,}}}&=\frac \, \varvec(\varvec ,\varvec) ,&\varvec}}\,}}}&=\frac \, \varvec(\varvec ,\varvec) , \\ \varvec_}&=\frac \, \varvec(\varvec , \varvec) . \end$$

We define the curvature components relative to the null frame \(\varvec}\) as the smooth scalar functions

$$\begin \varvec , \, \varvec , \end$$

and the \(\varvec}_}}\) covariant tensor fields

$$\begin & \varvec \,, \, \varvec}\,}}} \,, \\ & \varvec\,, \, \varvec}\,}}} \,, \end$$

such that

$$\begin \varvec}&=\varvec(_}}\,}},_}}\,}},_}}\,}},_}}\,}}) ,&\varvec}\,}}_}&= \varvec(_}}\,}},_}}\,}},_}}\,}},_}}\,}}) , \\ \varvec}&= \frac\, \varvec(_}}\,}},_}}\,}},_}}\,}},_}}\,}}) ,&\varvec}\,}}_}&= \frac\, \varvec(_}}\,}},_}}\,}},_}}\,}},_}}\,}}) ,\\ \varvec&= \frac\, \varvec(_}}\,}},_}}\,}},_}}\,}},_}}\,}}) ,&\varvec&=\frac\, \varvec R}(_}}\,}},_}}\,}},_}}\,}},_}}\,}}) , \end$$

where \(\varvec R}\) is the Hodge dual of \(\varvec\) on \((\varvec},\varvec)\).Footnote 42 In view of the symmetries of \(\varvec\), both \(\varvec\) and \(\varvec}\,}}}\) are symmetric \(\varvec}_}}\) tensors.

4.2.1 Decomposition of the Second Fundamental Forms

The following proposition relates the symmetries of the \(\varvec}_}}\) tensors \(\varvec\) and \(\varvec}\,}}}\) and the integrability of the horizontal distribution \(\varvec}_}}\). The proof is an easy check left to the reader.

Proposition 4.26

The distribution \(\varvec}_}}\) is integrable if and only if there exists a choice of \((_}}\,}},_}}\,}})\) such that the identities

$$\begin \varvec}&= \varvec} ,&\varvec}\,}}_}&= \varvec}\,}}_} \end$$

hold. In particular,

$$\begin \varvec([_}}\,}},_}}\,}}],_}}\,}})= 0 \quad&\Leftrightarrow \quad \varvec}= \varvec} , \\ \varvec([_}}\,}},_}}\,}}],_}}\,}})= 0 \quad&\Leftrightarrow \quad \varvec}\,}}_}= \varvec}\,}}_} . \end$$

Since the distribution \(\varvec}_}}\) is allowed to be non-integrable, the \(\varvec}_}}\) tensors \(\varvec\) and \(\varvec}\,}}}\) do not possess, in general, any symmetry.

We define the symmetric \(\varvec}_}}\) tensors

the antisymmetric \(\varvec}_}}\) tensors

and the symmetric traceless \(\varvec}_}}\) tensors

$$\begin \varvec}\,}}} , \, \varvec}}\,}}} \end$$

such that

where the traces are taken relative to the inverse metric , i.e.

We define the antitraces of \(\varvec\) and \(\varvec}\,}}}\) as the smooth scalar functionsFootnote 43

respectively.

Remark 4.27

The antitraces and encode the (non-)integrability of the horizontal distribution \(\varvec}_}}\), meaning that \(\varvec}_}}\) is integrable if and only if both the antitraces identically vanish (see later formula (73)).

We decompose the \(\varvec}_}}\) tensors \(\varvec\) and \(\varvec}\,}}}\) as

4.3 Tensor Contractions and Products

In this section, we define contractions and products of \(\varvec}_}}\) tensors. As a general caveat, we note that the \(\varvec}_}}\) tensors involved in the expressions are not assumed to possess any symmetry, and thus the order of the frame indices in the formulae is relevant.

Given a \(\varvec}_}}\) one-tensor \(\varvec\) and a \(\varvec}_}}\) two-tensor \(\varvec\), we introduce the notation

$$\begin \varvec^} \,, \, \varvec^} \,, \, \varvec^} \end$$

such that

Remark 4.28

The \(\varvec_}\)-notation allows to keep track of the position of the frame indices when we write tensorial expressions involving tensors which do not possess any symmetry. If \(\varvec\) is symmetric, then \(\varvec^}=\varvec^}\) and we simply write \(\varvec^}\).

We define the full contraction

and the duality relations

By definition, we have \(}}\varsigma }}_}=-\varvec\).

Given the \(\varvec}_}}\) one-tensors \(\varvec,\varvec}\) and the \(\varvec}_}}\) two-tensors \(\varvec,\varvec}\), we define

and

We also have

The \(\varvec}_}}\) two-tensor \(\varvec \,\widehat \,\varvec}\) is symmetric and traceless relative to .

4.4 Differential Operators

This section deals with differential operators acting on \(\varvec}_}}\) tensors. As a general caveat, we note that the \(\varvec}_}}\) tensors involved in the expressions are not assumed to possess any symmetry, and thus the order of the frame indices in the formulae is relevant.

4.4.1 Covariant Derivative

The Levi-Civita connection \(\varvec\) of \((\varvec},\varvec)\) induces a natural connection over the bundle of \(\varvec}_}}\) tensors on \(\varvec}\). In this section, we shall define the induced connection and discuss its two main properties, namely its compatibility with the (already defined) induced metric on \(\varvec}_}}\) and its torsion. The latter property is novel and is tied to the non-integrability of \(\varvec}_}}\).

We start with two definitions.

Definition 4.29

Let \(\varvec\in \Gamma (T\varvec})\). We define the map

between \(\varvec}_}}\) tensors such that

For any smooth scalar function \(\varvec\), we have the smooth scalar function such that

For any \(\varvec}_}}\) vector field \(\varvec\), we have the \(\varvec}_}}\) vector field such thatFootnote 44

(56)

For any \(\varvec}_}}\) k-tensor \(\varvec\) and \(\varvec}_}}\) vector fields \(\varvec,\ldots ,\varvec\), we have the \(\varvec}_}}\) k-tensor such that

Definition 4.30

We define the map

between \(\varvec}_}}\) tensors such that

For any smooth scalar function \(\varvec\), we have the \(\varvec}_}}\) one-tensor such that

for any \(\varvec}_}}\) vector field \(\varvec\);

For any \(\varvec}_}}\) vector field \(\varvec\), we have the \(\varvec}_}}\) one-tensor such that

for any \(\varvec}_}}\) vector field \(\varvec\);

For any \(\varvec}_}}\) k-tensor \(\varvec\), we have the \(\varvec}_}}\) \((k+1)\)-tensor such that

for any \(\varvec}_}}\) vector fields \(\varvec,\varvec,\ldots ,\varvec\).

Remark 4.31

It is easy to check that Definition 4.29 defines a covariant derivative acting on \(\varvec}_}}\) tensors (indeed, the map is linear and satisfies the Leibniz rule). The map is a linear connection over the bundle of \(\varvec}_}}\) tensors. We refer to as the induced covariant derivative over \(\varvec}_}}\) tensors and to as the induced connection over the bundle of \(\varvec}_}}\) tensors.

Let \([\cdot ,\cdot ]\) denote the standard Lie brackets of \((\varvec},\varvec)\), i.e.

$$\begin \varvec,\varvec](\varvec)=\varvec(\varvec(\varvec))-\varvec(\varvec(\varvec)) \end$$

for any \(\varvec,\varvec\in \Gamma (T\varvec})\) and \(\varvec\in C^(\varvec})\). The two following propositions address two key properties of the covariant derivative , namely its torsion and compatibility with the metric . In particular, Proposition 4.32 captures a new feature of arising when non-integrable structures are allowed. See also Remark 4.34.

Proposition 4.32

The horizontal distribution \(\varvec}_}}\) is integrable if and only if, for any \(\varvec}_}}\) vector fields \(\varvec,\varvec\), we have

(57)

Proof

We start by noting the identity

$$\begin \varvec^,\varvec^]=\varvec} Y^}-\varvec} X^} \end$$

(58)

for any \(\varvec}_}}\) vector fields \(\varvec,\varvec\), with \(\varvec\) the Levi-Civita (and thus torsion-free) connection of \(\varvec\).

If \(\varvec}_}}\) is integrable, then, for any \(\varvec}_}}\) vector fields \(\varvec,\varvec\), we have

$$\begin \varvec([\varvec^,\varvec^],_}}\,}})&=0 ,&\varvec([\varvec^,\varvec^],_}}\,}})&=0 , \end$$

(59)

where the equalities hold by the definition of integrable distribution. Therefore, we have

where the first and third equalities use (58), the second uses (59) and the last equality holds by definition of . This proves the identity (57).

To now prove the other direction of Proposition 4.32, we assume that, for any \(\varvec}_}}\) vector fields \(\varvec,\varvec\), the identity (57) holds. By Definition 4.29, the quantity is a \(\varvec}_}}\) vector field. Thus, the left hand side of (57) is the canonical extension of a \(\varvec}_}}\) vector field, implying

$$\begin \varvec^,\varvec^]\in \varvec}_}} \,. \end$$

By definition of integrable distribution, one concludes that \(\varvec}_}}\) is integrable.

\(\square \)

Proposition 4.33

Let \(\varvec\in \Gamma (T\varvec})\). We have the identities

Proof

We prove the first identity, the proof of the second identity is analogous. We compute

where the first equality holds by Definition 4.29 and the equality , the second equality by Definition 4.29 and the last equality by the \(\varvec\) being the Levi-Civita (and thus metric compatible) connection of \(\varvec\). \(\square \)

Remark 4.34

In view of Proposition 4.33, the induced connection over the bundle of \(\varvec}_}}\) tensors is compatible with the induced metric . However, Proposition 4.32 implies that the induced connection is torsion-free (and thus Levi-Civita relative to ) if and only if the distribution \(\varvec}_}}\) is integrable. Thus, in general,

The following proposition states that, in fact, the failure of to be Levi-Civita relative to noted in Remark 4.34 is a manifestation of a more general property: If the horizontal distribution \(\varvec}_}}\) is non-integrable, then there exists no Levi-Civita connection associated to over the bundle of \(\varvec}_}}\) tensors.

Proposition 4.35

The horizontal distribution \(\varvec}_}}\) is integrable if and only if there exists a Levi-Civita connection associated to over the bundle of \(\varvec}_}}\) tensors.

Proof

Assume that the horizontal distribution \(\varvec}_}}\) is integrable. Then, we consider the connection over the bundle of \(\varvec}_}}\) tensors from Definition 4.30. By Propositions 4.32 and 4.33, such a connection is Levi-Civita relative to .

To prove the other direction of the proposition, suppose is a Levi-Civita connection associated to over the bundle of \(\varvec}_}}\) tensors. Then, since is torsion-free, we have the identity