# Horizons that Gyre and Gimble:
A Carrollian Hierarchy of Null Hypersurfaces

Samuel Blitz  Department of Mathematics and Statistics, Masaryk University
Building 08, Kotlářská 2, Brno, CZ 61137
<EMAIL_ADDRESS>David McNutt  Department of Mathematics and Statistics,
UiT: The Arctic University of Norway, Tromsø, Norway, 9019
<EMAIL_ADDRESS>
###### Abstract

It is well-known that a null hypersurface in a Lorentzian manifold can be
treated as a Carrollian geometry. Additional structure can be added to this
geometry by choosing a connection which yields a Carrollian manifold. In the
literature various authors have introduced Koszul connections to study the
study the physics on these hypersurfaces. In this paper we examine the various
Carrollian geometries and their relationship to null hypersurface embeddings.
We specify the geometric data required to construct a rigid Carrollian
geometry, and we argue that a connection with torsion is the most natural
object to study Carrollian manifolds. We then use this connection to develop a
hypersurface calculus suitable for a study of intrinsic and extrinsic
differential invariants on embedded null hypersurfaces; motivating examples
are given.

††preprint: APS/123-QED

##  I Introduction

Black holes have captured the imagination of both the relativist and the lay
public for decades. In the last 10 years, observational data of in-spiraling
black holes from the LIGO and Virgo collaborations [1] have sparked greater
interest in dynamical black holes with complicated event horizon structures.
Introduced in 1965 by Levy-Leblond [29], Carrollian geometries have recently
[38, 17, 12, 11, 41] been investigated as an avenue of describing and
characterizing event horizons of many kinds and null hypersurfaces, more
generally. Carrollian geometries have also been studied in other contexts,
such as ultra-relativistic fluid dynamics [14, 13], Carrollian field theories
[7, 5, 6, 8], null infinity [18, 25, 26, 27, 42, 4, 30], and more.

In the literature broadly, Carrollian geometries generically refer to smooth
manifolds with a minimally degenerate metric. Depending on the source, they
might also include additional geometric structures such as a distinguished
generating vector field (sometimes called a rigging vector) or equivalently an
Ehresmann connection [31, 20], and/or an affine connection [13, 12, 11]. One
of the primary goals of this article is to propose a hierarchy of increasing
rigidity of Carrollian geometries and their associated local invariants.

To that end, we now provide formal definitions of the geometries in this
hierarchy, the notions of which were introduced by [40] and [9].

######  Definition 1.

Let $\mathcal{H}^{n}$ be a smooth manifold equipped with a rank-$(n-1)$
negative semi-definite symmetric bilinear form $\bar{g}$. Then the doublet
$(\mathcal{H},\bar{g})$ is called a pre-Carrollian structure. When
$\iota:\mathcal{H}^{n}\hookrightarrow(M,g)$ is a smooth null hypersurface
embedding into a Lorentzian spacetime and $\bar{g}=\iota^{*}g$, we say that
$(\mathcal{H},\bar{g})$ is a null hypersurface structure (NHS).

The latter definition of an NHS was introduced in [36]. It is important to
note that not all pre-Carrollian structures are null hypersurface structures;
however, they can always be identified locally. As we are only interested in
local geometry, for the remainder of this paper, we will implicitly assume all
pre-Carrollian structures can be realized as NHSs.

By specifying a vector that spans the radical of $\bar{g}$, we obtain [9]:

######  Definition 2.

Let $(\mathcal{H}^{n},\bar{g})$ be a pre-Carrollian structure and let a vector
field $\bar{\ell}\in\Gamma(T\mathcal{H})$ span the radical of $\bar{g}$, so
that $\bar{g}(\bar{\ell},\cdot)=0$. Then the triplet
$(\mathcal{H},\bar{g},\bar{\ell})$ is called a Carrollian structure, and the
vector field $\bar{\ell}$ is called its fundamental vector field.

In order to characterize a differential calculus on any geometric object, one
often prescribes a linear connection. In this context, a linear connection is
often constructed from a particular connection 1-form that is dual to
$\bar{\ell}$ and defines a horizontal subbundle $H\mathcal{H}\subset
T\mathcal{H}$. Such a 1-form is called an Ehresmann connection.

Carrollian structures equipped with an Ehresmann connection can be broadly
classified as G-structures using intrinsic torsion [19]. This classification
is not sufficient to determine when two Carrollian structures are equivalent.
However, considered as a manifold equipped with a coframe and a structure
group acting on it, a given Carrollian structure has a wealth of invariants
that can be generated using Cartan’s moving frame approach [37]. This has been
considered for an NHS, where the structure group has been extended [36].
Presumably, a similar application of Cartan’s moving frame approach could be
used for these Carrollian structures to describe their geometries
intrinsically. We leave this potential avenue of classification for another
time and do not consider it further in this article. Instead, we will, as many
others do, rely on a Koszul connection to construct differential invariants.

In the literature, a torsion-free Koszul connection is often employed [14, 18,
17, 11] to describe Carrollian geometries on null hypersurfaces. In some
cases, this is a valid choice, namely when the degenerate metric is invariant
under Lie differentiation with respect to the fundamental vector field.
However, for generic null hypersurfaces, torsion is unavoidable and the
“naturality” of torsion-free connections is put into question. In [14], this
is addressed by introducing additional tensorial objects on the Carrollian
manifold which arise from the metric and Ehresmann connection in order to
capture the information lost by using a torsionless connection.

To avoid that, we implement a torsional Koszul connection introduced by [9]
that is instead compatible with the Carrollian structure. This allows for the
following definition.

######  Definition 3.

Let $(\mathcal{H},\bar{g},\bar{\ell})$ be a Carrollian structure equipped with
a Koszul connection $\bar{\nabla}$ such that $\bar{\nabla}\bar{g}=0$ and
$\bar{\nabla}\bar{\ell}=0$. Then, the quadruplet
$(\mathcal{H},\bar{g},\bar{\ell},\bar{\nabla})$ is a Carrollian manifold and
the connection $\bar{\nabla}$ is termed a Carrollian connection.

In this article, we will denote by $(M^{d},g)$ a $d$-dimensional Lorentzian
spacetime with (mostly negative) signature metric $g$. By abuse of notation,
we will use $\mathcal{H}$ to refer both to an embedded null hypersurface in
$M$ and as an $n$-dimensional manifold in its own right (where $n=d-1$).

Via the above definitions, it is clear that a null hypersurface embedding into
a Lorentzian spacetime $\iota:\mathcal{H}^{n}\hookrightarrow(M^{d},g)$ can be
realized by a unique pre-Carrollian structure, but does not uniquely determine
any stronger Carrollian geometry. Consider, for example, the simplest non-
trivial null hypersurface: the event horizon of a Schwarzschild black hole,
given in Schwarzschild coordinates by $\mathcal{H}:=\\{p\in M\;|\;r(p)=2m\\}$.
Motivated by [17], an incautious reader might conclude that there is a single
Carrollian structure associated with $\mathcal{H}$. Indeed, by setting
$\bar{g}:=\iota^{*}g$, the bilinear form is canonically determined. However,
such an association implicitly assumes a natural choice of $\bar{\ell}$. In
this case, there is a unique Killing vector field that generates the event
horizon (given in Schwarzschild coordinates by $\partial_{t}$) which plays
this role. Then the canonical identification of
$\bar{\ell}:=\partial_{t}|_{\mathcal{H}}$ yields the desired association (see
Section IV for more details). It is important to note, however, that this is a
choice—one may have chosen to identify $\bar{\ell}$ with any vector field
$f(t,\theta,\phi)\partial_{t}|_{\mathcal{H}}$, and the resulting Carrollian
structures would be distinct.

Indeed, given a vector field $\ell\in\Gamma(TM)$ that restricts to the null
vector field $\bar{\ell}\in\Gamma(T\mathcal{H})$, one can consider an
arbitrary null frame $(n,\ell,m_{i})$ such that $n\cdot\ell=1$ and
$i\in\\{3,\ldots,d\\}$. Null boosts given by

$\displaystyle n\mapsto$ $\displaystyle A^{-1}n$ $\displaystyle\ell\mapsto$
$\displaystyle A\ell$

then will, in general, correspond to distinct Carrollian structures, related
by

$(\mathcal{H},\bar{g},\bar{\ell})\mapsto(\mathcal{H},\bar{g},A|_{\mathcal{H}}\bar{\ell})\,.$

It follows that the space of null hypersurface embeddings is much larger than
the space of Carrollian structures. Since the space of pre-Carrollian
structures is so large, it is challenging to invariantly classify. In most
cases, sufficient data can be provided by the embedding geometry to
canonically construct more restrictive Carrollian geometries. As such, this
article will consider (briefly) the case of Carrollian structures and
Carrollian manifolds in Section II, and then we will consider in more depth
the case of special Carrollian manifolds in Section III, which naturally arise
from the induced geometry of embedded null hypersurfaces such as non-extremal
weakly isolated horizons [3].

###  I.1 Notation

Here we provide a brief summary of the notations that we will use throughout
this article. The manifold $M$ represents a $d$-dimensional spacetime which is
endowed with a (mostly negative) Lorentzian signature metric $g$.

Given such a metric, its associated Levi-Civita connection $\nabla$ has
curvature given by

$R(x,y)z=(\nabla_{x}\nabla_{y}-\nabla_{y}\nabla_{x})z-\nabla_{[x,y]}z\,,$

where $x,y,z$ are smooth vector fields on $M$ and $[\cdot,\cdot]$ denotes the
Lie bracket of vector fields. When coordinates are useful, we will use indices
from the Greek alphabet (such as $\alpha$, $\beta$, and $\gamma$ for spacetime
coordinate indices).

To keep track of tensor structure, we will also often use Penrose’s abstract
index notation [39]. These abstract indices will be denoted by Latin letters
at the beginning of the alphabet (such as $a$, $b$, $c$, etc.). As an example,
the above formula for the Riemann curvature tensor is given, in abstract
indices, by

$x^{a}y^{b}R_{ab}{}^{c}{}_{d}z^{d}=x^{a}y^{b}(\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a})z^{c}\,.$

This abstract index notation will also be used to keep track of symmetry
properties of a tensor: for example, we will use round brackets $(\cdot)$ to
represent the symmetrization of a tensor, i.e.
$T_{(ab)}=\tfrac{1}{2}(T_{ab}+T_{ba})$. We denote by the symbol $\odot$ the
operator that maps a tensor to its symmetric part. Similarly, square brackets
$[\cdot]$ will be used to represent the antisymmetrization of a tensor, so
that $T_{[ab]}=\tfrac{1}{2}(T_{ab}-T_{ba})$.

An embedded null hypersurface, denoted by $\mathcal{H}$, will always be
assumed to be smooth. When necessary to avoid overloading of symbols, we will
use overbars $\bar{\bullet}$ to denote objects that belong to the
hypersurface. For example, for a 1-form $\alpha$ on $M$, we might write
$\bar{\alpha}$ for its pullback to $\mathcal{H}$. To eliminate confusion that
might arise when considering null frames, we will use Latin letters to denote
basis vectors, and their corresponding Greek letters to denote the action of
the metric on those basis vectors, i.e. $\nu_{a}=g_{ab}n^{b}$.

When necessary, coordinate indices in $M$ will use letters from the beginning
of the Greek alphabet and coordinate indices in $\mathcal{H}$ will use letters
from the middle of the Greek alphabet, such as $\mu$, $\nu$, and $\rho$. We
will label spacelike coordinates with capital letters from the beginning of
the Latin alphabet, such as $A$, $B$, and $C$. Furthermore, a frame will
sometimes be constructed on $\mathcal{H}$, and the indices used to label the
spacelike vectors or covectors will be letters from the middle of the Latin
alphabet, like $i$, $j$, and $k$. The indices for the whole frame on
$\mathcal{H}$ will be capital letters from the middle of the Latin alphabet,
like $I$, $J$, and $K$.

##  II  Intrinsic Carrollian Geometries

In a Carrollian structure $(\mathcal{H},\bar{g},\bar{\ell})$, we have very
little structure to employ for classification. Consider some basis of the
tangent space, $\\{\bar{\ell},\tilde{m}_{i}\\}$ with
$\bar{g}(\tilde{m}_{i},\tilde{m}_{i})<0$. Then we could employ the Gram-
Schmidt algorithm to find a new frame, $\\{\bar{\ell},\bar{m}_{i}\\}$ which
yields the following inner-products using the degenerate metric:

$\displaystyle g(\bar{m}_{i},\bar{m}_{j})=-\delta_{ij}.$ (1)

This condition is not just invariant under the action of $SO(n-1)$—there is an
additional frame transformation that will pass between diagonal metrics. If
the Gram-Schmidt algorithm is applied to the following basis
$\\{\bar{\ell},\tilde{m}_{i}+\tilde{c}_{i}\bar{\ell}\\}$, then the output of
the algorithm will be a new frame,
$\\{\bar{\ell},\bar{m}_{i}+c_{i}\bar{\ell}\\}$ where the inner product is
again
$\bar{g}(\bar{m}_{i}+c_{i}\bar{\ell},\bar{m}_{j}+c_{j}\bar{\ell})=-\delta_{ij}$.

There is an equivalent problem in constructing a complete coframe in the
absence of some choice of dual to $\bar{\ell}$. Instead, we can build a
partial coframe using the metric, $\bar{\mu}^{i}=\bar{g}(\bar{m}_{i},\cdot)$.
Note that this mapping from frames to the coframe is many-to-one since
$\bar{\ell}$ belongs to the radical of $\bar{g}$.

In principle, a weak classification of Carrollian structures is possible by
considering the sequence of Lie derivatives of the metric $\bar{g}$. For
example, we can consider $\mathcal{L}_{\bar{\ell}}\bar{g}=:2K$. This tensor is
in some sense horizontal as it is unaffected by shifts of the initial diagonal
frame by $\bar{\ell}$-terms and its components are only affected by the group
$SO(n-1)$. Using the lifted frame, $\hat{m}_{i}=R_{i}^{~{}j}\bar{m}_{j}$ where
$R$ is an arbitrary element in $SO(n-1)$ we can consider directions that
maximize $K(\hat{m}_{i},\hat{m}_{j})$ to pick out geometrically preferred
spatial directions modulo $\bar{\ell}$-terms. Additional invariants could be
determined by considering subsequent Lie derivatives of $K$ with respect to
$\bar{\ell}$ and Lie derivatives of the maximizing spatial directions shifted
by arbitrary $\bar{\ell}$-terms. It is plausible that the degeneracy in the
Gram-Schmidt procedure could also be fixed in some way.

As noted above, specifying a coframe for $T^{*}\mathcal{H}$ requires the
specification of a 1-form $\bar{\nu}$ that is dual to $\bar{\ell}$, i.e.
$\bar{\nu}(\bar{\ell})=1$. This is precisely an Ehresmann connection. As it
defines a horizontal subbundle, it allows us to reduce the frame
transformation group by demanding that $\bar{\nu}(m_{i})=0$, eliminating the
degeneracy in the Gram-Schmidt procedure noted above.

We now return to the context of a Carrollian structure arising from a null
hypersurface embedding in a Lorentzian spacetime. In that case, the
fundamental vector field $\bar{\ell}$ is viewed as arising from a vector field
$\ell\in\Gamma(TM)$. A null frame and coframe pair can then be constructed
(non-uniquely) for $TM|_{\mathcal{H}}$ and $T^{*}M|_{\mathcal{H}}$
respectively, so that they are adapted to $\ell$. Indeed, we may use a null
coframe $\\{\lambda,\nu,\mu^{i}\\}$, where $\lambda:=g(\ell,\cdot)$, so that
the metric takes the form

$\displaystyle\begin{aligned}
g=2\lambda\nu-\delta_{ij}\mu^{i}\mu^{j}.\end{aligned}$ (2)

Having fixed $\lambda$, we have thus excluded both null rotations about $\nu$
and boosts in the $(\lambda,\nu)$ plane. Due to this, the structure group
acting on the coframe adapted to a vector field $\ell$ that is null on a
hypersurface consists of:

  * •

spatial rotations: $\tilde{\mu}^{i}=R^{i}_{~{}j}\mu^{j}$, $R^{i}_{~{}j}\in
SO(n-1)$;

  * •

null rotations about $\lambda$: $\tilde{\nu}=\nu+c_{i}\mu^{i}+|c|^{2}\lambda$
and $\tilde{\mu}^{i}=\mu^{i}+c^{i}\lambda$ for $n-1$ real-valued functions,
$c_{i}$, where $c_{i}=c^{i}$.

Pulling back to the hypersurface, $\iota^{*}\lambda$ vanishes and the
resulting group of transformations are then

  * •

spatial rotations, $\tilde{\bar{\mu}}^{i}=R^{i}_{~{}j}\bar{\mu}^{j}$,
$R^{i}_{~{}j}\in SO(n-1)$;

  * •

Carrollian boosts, $\tilde{\bar{\nu}}=\bar{\nu}+c_{i}\bar{\mu}^{i}$ for $n-1$
real-valued functions, $c_{i}$. We note that on the hypersurface the one-forms
$\mu^{i}$ are unchanged.

This is exactly the structure group for a Carrollian manifold [19].

If we suppose that we no longer have access to the the coframe element $\nu$
then we can write these transformations in terms of the frame basis as

  * •

spatial rotations: $\tilde{\bar{m}}_{i}=R_{i}^{~{}j}\bar{m}_{j}$,
$R_{i}^{~{}j}\in SO(n-1)$;

  * •

Transpose-Carrollian boosts:
$\tilde{\bar{m}}_{i}=\bar{m}_{i}+c_{i}\bar{\ell}$.

This can be viewed as the structure group of a Carrollian structure as it
leaves the vector-field $\bar{\ell}$ and the degenerate metric $\bar{g}$
invariant as given in the definition in section I.

Carrollian structures can appear in the study of null hypersurfaces when there
is no preferred spatial slice and hence a specific transverse vector field
that corresponds to an Ehresmann connection. Examples of this in the
literature appear for non-expanding horizons for a specific choice of $\ell$
and for a extremal weakly isolated horizon where a preferred $\ell$ direction
is determined but no spatial slice arises from the associated structures on
the horizon [3].

In order to work with generic null hypersurfaces, a choice must be made of
either the spatial slice or equivalently the transverse direction in order to
specify an Ehresmann connection. This is exemplified in the construction of
Gaussian null coordinates for an open neighbourhood of a null hypersurface
[35, 21, 10]. In the construction of these coordinates, one must first choose
coordinates for a spatial $(n-1)$-dimensional submanifold in the null
hypersurface along with a vector-field in the direction of the null generator
of the hypersurface. Once these choices are made there is a unique choice of a
transverse null vector-field pointing off of the hypersurface whose
corresponding one-form (via the musical isomorphism) acts as the Ehresmann
connection on the hypersurface. The relationship between the spatial slice,
the Ehresmann connection, and Gaussian null coordinates will be revisited in
Theorem 1 of section III.

Using any Ehresmann connection, it is possible to canonically construct a
Carrollian connection [9]. With this Carrollian connection, we can induce a
Carrollian manifold geometry on the Carrollian structure. Consequently, it is
then possible to consider the derivative of vector-fields in the manifold and
compute the torsion tensor, curvature tensor and their covariant derivatives.
Compared to bare Carrollian structures, Carrollian manifolds have very rigid
structure, allowing for a larger set of geometric invariants.

However, there is a distinguished family of Ehresmann connections that are
more useful: the principal Ehresmann connections. These Ehresmann connections
respect the Carrollian structure in that
$\mathcal{L}_{\bar{\ell}}\bar{\nu}=0$. In the next section, we consider a
specific family of such Carrollian manifolds which naturally arise from null
hypersurface embeddings (which can also be shown to arise from a principal
Ehresmann connection).

##  III  Induced Carrollian manifolds

As discussed in the previous section, general Carrollian structures do not
pick out a unique Ehresmann connection. However, when a Carrollian structure
can be induced from a null hypersurface embedding
$\iota:\mathcal{H}\hookrightarrow(M,g)$ and a distinguished null vector field
$\ell|_{\mathcal{H}}$ along it, the family of such Carrollian structures is
greatly reduced. This is because every null hypersurface can be (locally)
foliated by spatial slices, and thus there exists (at least) one integrable
1-form $\nu$ such that $\iota^{*}\nu\neq 0$. However, this 1-form cannot
annihilate $\bar{\ell}$ anywhere, because then the kernel of $\iota*\nu$ would
have non-constant dimension. So we may demand that
$\bar{\nu}=[\iota^{*}\nu(\bar{\ell})]^{-1}\iota^{*}\nu$, i.e.
$\bar{\nu}(\bar{\ell})=1$. A simple calculation shows $\bar{\nu}$ is
integrable. So, a Carrollian structure arising as an embedded null
hypersurface admits an integrable Ehresmann connection. In fact, the following
results pin down these connections even further:

######  Proposition 1.

On a Carrollian structure, an integrable Ehresmann connection is closed.

###### Proof.

Let $(\mathcal{H},\bar{g},\bar{\ell})$ be a Carrollian structure with
integrable Ehresmann connection $\bar{\nu}$ and let $p\in\mathcal{H}$ be a
point. Now since $\bar{\nu}$ is integrable, in a neighborhood
$U\subset\mathcal{H}$ around $p$ there exists a hypersurface $S\subset U$ such
that $p\in S$ and for every $v\in TS$, $\bar{\nu}(v)=0$. Now let $x^{i}$ be
coordinates on $S$, and extend these coordinate functions transverse to $S$
via parallel transport along $\bar{\ell}$. This coordinatizes $U$ via
$(u,x^{i})$ and yields $\bar{\ell}=\partial_{u}$.

Now because $\bar{\nu}$ is integrable, there exists $f,g\in C^{\infty}U$ such
that $\bar{\nu}=f{\rm{d}}g$. Expressing this in coordinates $(u,x^{i})$ and
imposing the Ehresmann constraint that $\bar{\nu}(\bar{\ell})=1$ and the
horizontality constraint that $\bar{\nu}(\partial_{x^{i}})=0$ forces
$f{\rm{d}}g={\rm{d}}u$. Thus, $\bar{\nu}$ is locally exact and hence closed on
$\mathcal{H}$.

∎

######  Corollary 1.

On a Carrollian structure, an integrable Ehresmann connection is principal.

###### Proof.

From the proposition above, if $\bar{\nu}$ is an integrable Ehresmann
connection on a Carrollian structure $(\mathcal{H},\bar{g},\bar{\ell})$, then
it is closed. From Cartan’s magic formula,

$\mathcal{L}_{\bar{\ell}}\bar{\nu}=i_{\bar{\ell}}{\rm{d}}\bar{\nu}+{\rm{d}}i_{\bar{\ell}}\bar{\nu}=0\,,$

because ${\rm{d}}\bar{\nu}=0$ and $i_{\bar{\ell}}\bar{\nu}=1$. Thus
$\bar{\nu}$ is principal. ∎

As a consequence of Proposition 1 and Corollary 1, it is clear that those
Carrollian structures arising from null hypersurface embeddings should be (and
can always be) prescribed a principal Ehresmann connection.

Recall from the introduction that the Carrollian geometries do not, in
general, have a uniquely defined torsion-free connection [19]. Motivated by
this fact, we instead consider the connection defined by Bekaert and Morand
[9] that respects the Carrollian structure but is agnostic towards torsion.

The construction of this connection employs a principal Ehresmann connection
$\bar{\nu}$ and a Carrollian structure $(\mathcal{H},\bar{g},\bar{\ell})$. To
do this, first observe that, up to spatial rotations, we have a canonical
frame and coframe, given by $(\bar{\ell},\bar{m}_{i})$ and
$(\bar{\nu},\bar{\mu}^{i})$, respectively, where $\bar{\mu}^{i}(\bar{\ell})=0$
and $\bar{\nu}(\bar{m}_{i})=0$. Furthermore, we have a projector to the
horizontal vector (and form) bundles: namely,
$q_{a}^{b}=\delta_{a}^{b}-\bar{\ell}^{b}\bar{\nu}_{a}$. By construction, we
have that $q^{b}_{~{}a}\bar{\ell}^{a}=0=q^{b}_{~{}a}\bar{\nu}_{b}$.
Furthermore, a partial inverse $\bar{g}^{ab}$ to the degenerate metric
$\bar{g}$ can be constructed via the relations
$\bar{g}^{ab}\bar{g}_{bc}=q^{a}_{~{}b}$. This partial inverse can be used to
raise indices of horizontal 1-forms—that is, if $\omega_{a}\bar{\ell}^{a}=0$,
then we can write $\omega^{a}=\bar{g}^{ab}\omega_{b}$. Using these objects, we
can define the canonical Carrollian connection $\bar{\nabla}$ by the
coordinate expression

$\displaystyle\begin{split}\bar{\Gamma}^{\lambda}_{~{}\rho\sigma}=\bar{\ell}^{\lambda}\partial_{(\rho}\bar{\nu}_{\sigma)}+\tfrac{1}{2}\bar{g}^{\lambda\mu}[\partial_{\rho}\bar{g}_{\sigma\mu}+\partial_{\sigma}\bar{g}_{\rho\mu}-\partial_{\mu}\bar{g}_{\rho\sigma}]\\\
-\tfrac{1}{2}\bar{g}^{\lambda\mu}\bar{\nu}_{\sigma}\mathcal{L}_{\bar{\ell}}\bar{g}_{\rho\mu}\,.\end{split}$
(3)

One nice feature of this Carrollian connection is that $\bar{\nu}$ is
parallel, i.e. $\bar{\nabla}\bar{\nu}=0$.

It is thus clear that when a preferred spatial slice of a null hypersurface is
distinguished from the Lorentzian geometry, an induced Carrollian structure
$(\mathcal{H},\bar{g},\bar{\ell})$ picks out a distinguished principal
Ehresmann connection $\bar{\nu}$ and hence a distinguished Carrollian manifold
$(\mathcal{H},\bar{g},\bar{\ell},\bar{\nabla})$. We call such an induced
Carrollian manifold specified by the tuple
$(\mathcal{H},\bar{g},\bar{\ell},\bar{\nu},\bar{\nabla})$ a special Carrollian
manifold.

######  Remark 1.

Observe that the family of Carrollian manifolds induced from the data
described above is much smaller than the family of all Carrollian manifolds
with the same Carrollian structure. Indeed, the Carrollian manifolds that
arise in this way are specifically those for which there exists a covariantly
constant dual to the fundamental vector field and a suitably minimal torsion.

As discussed in the introduction, a substantial amount of additional data must
be provided to a null hypersurface embedding to canonically pick out a special
Carrollian manifold. To that end, we now we provide an equivalence of three
different ways that this additional data can be specified.

######  Theorem 1.

Let $\iota:\mathcal{H}^{d-1}\hookrightarrow(M^{d},g)$ be a null hypersurface
embedding into a Lorentzian manifold. Let $p\in\mathcal{H}$ and let $U\subset
M$ be a small neighborhood around $p$. Then the following are canonically
equivalent:

  * •

A defining function $r$ for $\mathcal{H}\cap U$ paired with a null vector
$n\in\Gamma(TM)|_{\mathcal{H}\cap U}$ transverse to $\mathcal{H}$ such that
the one-form $\bar{\nu}:=\iota^{*}g(n,\cdot)$ is integrable and
$n(r)|_{\mathcal{H}\cap U}=1$.

  * •

A null vector field $\bar{\ell}\in\Gamma(T(\mathcal{H}\cap U))$ paired with a
spacelike codimension-1 submanifold $\mathcal{S}\subset\mathcal{H}\cap U$
containing $p$;

  * •

A Gaussian null coordinatization of $U$.

###### Proof.

We prove by cycling. We begin by defining
$\ell:=g^{-1}({\rm{d}}r,\cdot)|_{U}$. Since $\mathcal{H}$ is null,
${\rm{d}}r|_{\mathcal{H}\cap U}$ is null, and hence so is
$\ell|_{\mathcal{H}\cap U}$. It then follows that
${\rm{d}}r(\ell)|_{\mathcal{H}\cap U}=0$, and so there exists
$\bar{\ell}\in\Gamma(T(\mathcal{H}\cap U))$ such that $\ell|_{\mathcal{H}\cap
U}=\iota_{*}\bar{\ell}$. Now as a non-vanishing 1-form on $\mathcal{H}\cap U$,
$\bar{\nu}$ has a $d-2$ dimensional kernel. Furthermore, because
$\bar{\nu}(\bar{\ell})=n(r)|_{\mathcal{H}\cap U}=1$, we have that
$\bar{\ell}\not\in\ker\bar{\nu}$, and thus $\ker\bar{\nu}$ is spacelike. Now
observe that because $\bar{\nu}$ is an integrable Ehresmann connection, from
Proposition 1, it is closed. Now let $x,y\in\ker\bar{\nu}$. Then,

$0={\rm{d}}\bar{\nu}(x,y)=x(\bar{\nu}(y))-y(\bar{\nu}(x))-\bar{\nu}([x,y])\,,$

so it follows that $[x,y]\in\ker\bar{\nu}$. Thus we have a spacelike
integrable $d-2$ distribution, which by the Frobenius theorem yields the
desired spacelike submanifold $\mathcal{S}\subset\mathcal{H}\cap U$. So we
have reached the second bullet point.

The third bullet point follows from a direct higher-dimensional generalization
of Friedrich, et. al. [21]. Thus we obtain a set of coordinates on $U$ given
by $(v,u,x^{A})$ such that

${\rm{d}}s^{2}=2{\rm{d}}u{\rm{d}}v+A{\rm{d}}u^{2}+B_{A}{\rm{d}}u{\rm{d}}x^{A}+C_{AB}{\rm{d}}x^{A}{\rm{d}}x^{B}\,,$

where $A|_{\mathcal{H}}=0=B_{A}|_{\mathcal{H}}$.

Finally, given a Gaussian null coordinates $(v,u,x^{A})$ on $U$ as above, we
make the canonical choices $r:=v$ and $n:=\partial_{v}|_{\mathcal{H}}$.
Clearly, $n$ is transverse to $\mathcal{H}$ and null. Furthermore, we find
explicitly that $g(n,\cdot)={\rm{d}}u$, a closed (and hence integrable) one
form. Thus it follows that $\bar{\nu}:=\iota^{*}g(n,\cdot)$ is integrable. We
have thus obtained the first bullet point, which completes the proof. ∎

This theorem shows that these three ways of specifying the additionally
required data are equivalent, and they all provide the same closed (and hence
principal) Ehresmann connection $\bar{\nu}$ as well as a fundamental vector
$\bar{\ell}$ on the null hypersurface with induced (degenerate) metric
$(\mathcal{H},\bar{g})$. This information is sufficient to fully specify a
special Carrollian manifold
$(\mathcal{H},\bar{g},\bar{\ell},\bar{\nu},\bar{\nabla})$, and it is such
induced structures that we will consider going forward.

Before we proceed, it is worthwhile to take stock of what has been
accomplished here. Given a null hypersurface $\mathcal{H}$ embedded in
$(M,g)$, we have canonically induced sufficient structure on $\mathcal{H}$ to
regard it as a Carrollian manifold. This is entirely analogous, if more
complex, than the procedure for spacelike (or timelike) hypersurfaces. There,
one can induce a Riemannian (or pseudo-Riemannian) metric, which in turn
induces a Levi–Civita connection, giving the hypersurface a
(pseudo-)Riemannian structure in its own right. Having done this, one can then
begin to construct intrinsic and extrinsic differential invariants of the
embedding, such as the second fundamental form.

Having suceeded at a similar task in the case of an embedded null
hypersurface, we are now situated to study similarly intrinsic and extrinsic
differential invariants of the embedding. In principle, these invariants can
be used to further study null hypersurface embeddings via classification
methods, such as one similar to the Cartan–Karlhede method.

###  III.1 Intrinsic Differential Invariants

We begin by considering the intrinsic differential invariants associated to a
special Carrollian manifold
$(\mathcal{H},\bar{g},\bar{\ell},\bar{\nu},\bar{\nabla})$. Given the
connection in Equation (3), the torsion tensor is given by

$\bar{T}^{a}_{~{}bc}=-\bar{g}^{ad}\bar{\nu}_{[b}\mathcal{L}_{\bar{\ell}}\bar{g}_{c]d}\,.$

Of particular use is the so-called intrinsic second fundamental form, given by

$K_{ab}=\tfrac{1}{2}\mathcal{L}_{\bar{\ell}}\bar{g}_{ab}\,.$

As noted in Section II, $K$ is horizontal. Thus, in terms of this tensor, we
have that $\bar{T}^{a}_{bc}=2\bar{\nu}_{[b}K^{a}_{c]}$.

Next, we can express the curvature tensor in abstract index notation via

$\bar{R}_{ab}{}^{c}{}_{d}Z^{d}=([\bar{\nabla}_{a},\bar{\nabla}_{b}]+T^{e}_{ab}\bar{\nabla}_{e})Z^{c}\,.$

Rewriting this in terms of the intrinsic second fundamental form, we have that

$\bar{R}_{ab}{}^{c}{}_{d}Z^{d}=([\bar{\nabla}_{a},\bar{\nabla}_{b}]+2\bar{\nu}_{[a}K^{e}_{b]}\bar{\nabla}_{e})Z^{c}\,.$

Now, in more general contexts with less constrained structure, one can extract
invariants by applying invariant operations to those structures. For example,
one might consider producing additional differential invariants by evaluating
the exterior derivative of the Ehresmann connection (these are called the
Carrollian torsion and acceleration in [12]). However, by construction, for a
special Carrollian manifold, the Ehresmann connection is closed, and thus no
additional curvature invariants appear. As noted in [12], this provides
additional confirmation that such a principal Ehresmann connection produces an
integrable horizontal bundle.

Since Lie derivatives with respect to $\bar{\ell}$ can be expressed in terms
of $\bar{\nabla}$ and $\bar{T}$ (which in turn can be expressed in terms of
$\bar{\nu}$ and $K$), it is now clear that any intrinsic differential
invariant of the special Carrollian manifold can be expressed via a tensor
built from the following set of ingredients:

$\displaystyle\\{\bar{g},\bar{\ell},\bar{\nu},\bar{\nabla},\bar{R},K\\}\,.$
(4)

###  III.2 Bundle Decomposition on $\mathcal{H}$

In order to relate spacetime invariants to their hypersurface counterparts, it
is essential that we provide a distinguished decomposition of the tangent and
cotangent bundles at a point $p\in\mathcal{H}\subset M$. We will assume that a
set of data specified by Theorem 1 is given and that it determines the special
Carrollian manifold
$\\{\mathcal{H},\bar{g},\bar{\ell},\bar{\nu},\bar{\nabla}\\}$. Now, consider a
frame of $T_{p}\mathcal{H}$ which is canonically decomposed into vertical and
horizontal components: $\\{\bar{\ell},\bar{m}^{i}\\}$. Here,
$\\{\bar{m}^{i}\\}$ form any orthonormal frame for the horizontal subspace
given by $H\mathcal{H}:=\ker\bar{\nu}\subset T\mathcal{H}$. As $\bar{\nu}$ is
an Ehresmann connection, it is dual to $\bar{\ell}$ and hence, together with
$\bar{g}$, we have a coframe $\\{\bar{\nu},\bar{\mu}_{i}\\}$.

We would like to align a distinguished frame in the spacetime at the point $p$
with that given for $T_{p}\mathcal{H}$. At any such point, there is a unique
null rigging vector [31] $n\in T_{p}M$ that satisfies
$g(n,\iota_{*}\bar{\ell})=1$ and $g(n,\iota_{*}\bar{m}^{i})=0$ that is
transverse to $\mathcal{H}$. (Note that $\iota^{*}g(n,\cdot)=\bar{\nu}$.) This
vector can be used to provide a distinguished decomposition

$\displaystyle T_{p}M=\langle n_{p}\rangle\oplus T_{p}\mathcal{H}\,.$ (5)

In fact, this decomposition induces a surjection
$T:TM|_{\mathcal{H}}\rightarrow T\mathcal{H}$ so that for any $v\in
TM|_{\mathcal{H}}$, we have that $T(v)\in T\mathcal{H}$. Note that $T$ is the
left-inverse of the pushfoward by $\iota$, in the sense that
$(T\circ\iota_{*})(\bar{v})=\bar{v}$, for any $\bar{v}\in T\mathcal{H}$.
Further, for $v\perp n$ in the sense that under the decomposition given, $v$
has a vanishing coefficient for $n$, we have that $(\iota_{*}\circ T)(v)=v$,
and thus $T$ is a partial right-inverse of $\iota_{*}$.

The decomposition given in Equation (5) can be further refined by the
pushfoward of the frame
$\iota_{*}\\{\bar{\ell},\bar{m}^{i}\\}:=\\{\ell,m^{i}\\}$. We thus achieve

$T_{p}M=\langle n_{p}\rangle\oplus\langle\ell_{p}\rangle\oplus
H_{p}\mathcal{H}\,.$

This decomposition also agrees with the decomposition of $T\mathcal{H}$
because $T(n)=0$, $T(\ell)=\bar{\ell}$, and $T(m^{i})=\bar{m}^{i}$.

A similar distinguished decomposition can be constructed for the cotangent
bundle at a point $p\in\mathcal{H}$. Defining $\lambda:=g(\ell,\cdot)$,
$\nu:=g(n,\cdot)$, and $\mu_{i}:=g(m^{i},\cdot)$, we find a distinguished
decomposition

$T^{*}_{p}M=\langle\lambda_{p}\rangle\oplus\langle\nu_{p}\rangle\oplus
H^{*}_{p}\mathcal{H}\,.$

This decomposition, too, agrees with the decomposition on the special
Carrollian manifold, because $\iota_{*}(\lambda)=0$,
$\iota_{*}(\nu)=\bar{\nu}$, and $\iota_{*}(\mu_{i})=\bar{\mu}_{i}$.

With these decompositions in hand, we can now begin to both relate connections
and construct invariants of the embedding.

###  III.3 Hypersurface Calculus

In principle, every intrinsic invariant of a special Carrollian manifold
$\\{\mathcal{H},\bar{g},\bar{\ell},\bar{\nu},\bar{\nabla}\\}$ can be built
from Set (4). However, as we are interested in studying those that arise from
null hypersurface embeddings and data specified as in Theorem 1, it is
imperative that we determine the relationship between these objects and the
defining objects of the bulk given by $\\{g,\nabla,R,\nu,\ell\\}$. Clearly,
$\iota^{*}g=\bar{g}$, and the relationships for $\nu$ and $\ell$ were given in
the previous section.

We can now consider the first non-trivial calculation, which involves finding
a spacetime expression for $K$. As $T$ is a partial right-inverse of
$\iota_{*}$, we can express $T(\ell)$ as $\iota^{*}\ell$, where $\iota^{*}$
here is the inverse of the pushforward map. Thus, we can write

$\displaystyle K=$
$\displaystyle\tfrac{1}{2}\mathcal{L}_{\iota^{*}\ell}\iota^{*}g$
$\displaystyle=$
$\displaystyle\tfrac{1}{2}({\rm{d}}i_{\iota^{*}\ell}+i_{\iota^{*}\ell}{\rm{d}})\iota^{*}g$
$\displaystyle=$
$\displaystyle\tfrac{1}{2}\iota^{*}({\rm{d}}i_{\ell}+i_{\ell}{\rm{d}})g$
$\displaystyle=$ $\displaystyle\tfrac{1}{2}\iota^{*}\mathcal{L}_{\ell}g$
$\displaystyle=$ $\displaystyle\iota^{*}\odot\nabla\lambda\,.$

Having established a spacetime formula for the intrinsic second fundamental
form, we next turn to establishing the relationship between the spacetime
Levi-Civita connection and the induced Carrollian connection.

###  III.4 Spacetime-Carrollian relationships: the connection

Recall the distinguished decomposition of $T_{p\in\mathcal{H}}M$:

$T_{p}M=\langle n_{p}\rangle\oplus\langle\ell_{p}\rangle\oplus
H_{p}\mathcal{H}\,.$

Restricting the Levi-Civita connection to $\mathcal{H}$, we can write

$\nabla^{\top}:\Gamma(T\mathcal{H})\otimes\Gamma(TM)|_{\mathcal{H}}\rightarrow\Gamma(TM)|_{\mathcal{H}}\,.$

In order to describe the spacetime invariants in terms of intrinsic invariants
of $\mathcal{H}$ and extrinsic invariants of the embedding
$\mathcal{H}\hookrightarrow M$, we must decompose this connection and examine
the projection onto each component. Specifically, for
$\bar{u},\bar{v},\bar{w}\in\Gamma(H\mathcal{H})$, we consider the following
scalars:

$\displaystyle\lambda_{a}$ $\displaystyle\nabla_{\ell}n^{a}\,,\quad$
$\displaystyle\nu_{a}$ $\displaystyle\nabla_{\ell}n^{a}\,,\quad$
$\displaystyle(\iota_{*}\bar{w})_{a}\nabla_{\ell}n^{a}\,,$ (6)
$\displaystyle\lambda_{a}$ $\displaystyle\nabla_{\ell}\ell^{a}\,,\quad$
$\displaystyle\nu_{a}$ $\displaystyle\nabla_{\ell}\ell^{a}\,,\quad$
$\displaystyle(\iota_{*}\bar{w})_{a}\nabla_{\ell}\ell^{a}\,,$
$\displaystyle\lambda_{a}$
$\displaystyle\nabla_{\ell}(\iota_{*}\bar{v})^{a}\,,\quad$
$\displaystyle\nu_{a}$
$\displaystyle\nabla_{\ell}(\iota_{*}\bar{v})^{a}\,,\quad$
$\displaystyle(\iota_{*}\bar{w})_{a}\nabla_{\ell}(\iota_{*}\bar{v})^{a}\,,$
$\displaystyle\lambda_{a}$
$\displaystyle\nabla_{\iota_{*}\bar{u}}n^{a}\,,\quad$ $\displaystyle\nu_{a}$
$\displaystyle\nabla_{\iota_{*}\bar{u}}n^{a}\,,\quad$
$\displaystyle(\iota_{*}\bar{w})_{a}\nabla_{\iota_{*}\bar{u}}n^{a}\,,$
$\displaystyle\lambda_{a}$
$\displaystyle\nabla_{\iota_{*}\bar{u}}\ell^{a}\,,\quad$
$\displaystyle\nu_{a}$
$\displaystyle\nabla_{\iota_{*}\bar{u}}\ell^{a}\,,\quad$
$\displaystyle(\iota_{*}\bar{w})_{a}\nabla_{\iota_{*}\bar{u}}\ell^{a}\,,$
$\displaystyle\lambda_{a}$
$\displaystyle\nabla_{\iota_{*}\bar{u}}(\iota_{*}\bar{v})^{a}\,,$
$\displaystyle\quad\nu_{a}$
$\displaystyle\nabla_{\iota_{*}\bar{u}}(\iota_{*}\bar{v})^{a}\,,\quad$
$\displaystyle(\iota_{*}\bar{w})_{a}\nabla_{\iota_{*}\bar{u}}(\iota_{*}\bar{v})^{a}\,.$

####  III.4.1 Components of $\nabla^{\top}$ in the $n$ direction

We begin by computing, in terms of intrinsic and extrinsic objects, the
derivatives in the left column of Table 6. Observe that, because
$\lambda(n)\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}1$ everywhere on $\mathcal{H}$, we can write
$\lambda_{a}\nabla_{\ell}n^{a}\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}-\nu_{a}\nabla_{\ell}\ell^{a}$. Restricted to the
hypersurface $\mathcal{H}$, this is the projection of $\nabla_{\ell}\ell^{a}$
to the $\ell$ direction. However, because $\ell|_{\mathcal{H}}$ generates a
null hypersurface, we necessarily have that
$\nabla_{\ell}\ell|_{\Sigma}\propto\ell$, with proportionality constant given
by the surface gravity of the hypersurface, denoted by $\kappa$. And so, we
have that

$\lambda_{a}\nabla_{\ell}n^{a}\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}-\kappa\,.$

Next we consider $\lambda_{a}\nabla_{\iota_{*}\bar{u}}n^{a}$. This derivative
measures how the vector $n$ changes in affine length as it is moved along the
horizontal submanifold. In the physics literature, this is called the Hájíček
one-form [23, 24, 22] (or in the mathematics literature, the normal
fundamental form [43, Volume 4]). We write

$\beta_{a}:=q^{c}_{~{}a}\iota^{*}(\lambda_{b}\nabla
n^{b})_{c}\in\Gamma(H\mathcal{H})\,,$

so that

$\lambda_{a}\nabla_{\iota_{*}\bar{u}}n^{a}\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}\bar{u}^{a}\beta_{a}\,.$

We now consider terms involving $\lambda$ with no dependence on $n$—that is,
we consider terms of the form
$\lambda_{a}\nabla_{\iota_{*}\bar{x}}(\iota_{*}\bar{y})^{a}$, where
$\bar{x},\bar{y}\in T\mathcal{H}$, and $x,y$ are their pushforwards,
respectively. As the Levi-Civita connection is linear both in $x$ and $y$, and
we have that

$\lambda_{a}\nabla_{x}fy^{a}=\lambda_{a}\nabla_{fx}y^{a}=f\lambda_{a}\nabla_{x}y^{a}$

for all $f\in C^{\infty}M$, following Spivak [43, Volume 1, Section 4, Theorem
2] and [43, Volume 3, Section 1, Theorem 5], we have that there exists a
symmetric tensor field
$s:\Gamma(T\mathcal{H})\times\Gamma(T\mathcal{H})\rightarrow\Gamma(\langle
n\rangle)$. Hence we can write

$\lambda_{c}s_{~{}ab}^{c}\bar{x}^{a}\bar{y}^{b}=-\lambda_{a}\nabla_{\iota_{*}\bar{x}}(\iota_{*}\bar{y}^{a})\,.$

As $\lambda(\iota_{*}\bar{y})=0$, we can reexpress this tensor as

$\lambda_{c}s^{c}_{~{}ab}=\iota^{*}(\nabla_{(a}\lambda_{b)})\,.$

It follows that

$\lambda_{c}s^{c}_{~{}ab}=K_{ab}\,.$

We have thus established that

$\lambda_{a}\nabla_{\iota_{*}\bar{x}}(\iota_{*}\bar{y})^{a}=-K_{ab}\bar{x}^{a}\bar{y}^{a}\,.$

However, $K$ is horizontal, so we the following four values can be filled into
the left column of Table 6:

$\displaystyle\lambda_{a}\nabla_{\ell}\ell^{a}$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}0$
$\displaystyle\lambda_{a}\nabla_{\ell}(\iota_{*}\bar{v})^{a}$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}0$
$\displaystyle\lambda_{a}\nabla_{\iota_{*}\bar{u}}\ell^{a}$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}0$
$\displaystyle\lambda_{a}\nabla_{\iota_{*}\bar{u}}(\iota_{*}\bar{v})^{a}$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}-K_{ab}\bar{u}^{a}\bar{v}^{b}\,.$

####  III.4.2 Components of $\nabla^{\top}$ in the $\ell$ direction

We next compute the derivatives in the middle column of Table 6 in much the
same way as in the previous subsection. As
$g(n,n)\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}0$, from the Leibniz property and metric compatibility, we
have that

$\nu_{a}\nabla_{b}^{\top}n^{a}\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}0\,.$

By definition of surface gravity, we have that

$\nu_{a}\nabla_{\ell}\ell^{a}\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}\kappa\,.$

Next, consider $\nu_{a}\nabla_{\iota_{*}\bar{u}}\ell^{a}$. By applying the
Leibniz rule to the definition of the Hájíček one-form and the fact that
$\lambda(n)\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}\nu(\ell)\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}1$, we have that

$\displaystyle\nu_{a}\nabla_{\iota_{*}\bar{u}}\ell^{a}\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}-\bar{u}^{a}\beta_{a}\,.$ (7)

Using this, we consider $\nu_{a}\nabla_{\ell}(\iota_{*}\bar{v})^{a}$. We
evaluate below:

$\displaystyle\nu_{a}\nabla_{\ell}(\iota_{*}\bar{v})^{a}$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}\nu_{a}([\iota_{*}\bar{\ell},\iota_{*}\bar{v}]^{a}+\nabla_{\iota_{*}\bar{v}}\ell^{a})$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}(\iota^{*}\nu)([\bar{\ell},\bar{v}])+\nu_{a}\nabla_{\iota_{*}\bar{v}}\ell^{a}$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}\bar{\nu}([\bar{\ell},\bar{v}])-\bar{v}^{a}\beta_{a}$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}\bar{\nu}_{a}(\bar{\nabla}_{\bar{\ell}}\bar{v}^{a}-\bar{\nabla}_{\bar{v}}\bar{\ell}^{a}-\bar{T}^{a}_{bc}\bar{\ell}^{b}\bar{v}^{c})-\bar{v}^{a}\beta_{a}$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}\bar{\nu}_{a}\bar{\nabla}_{\bar{\ell}}\bar{v}^{a}-\bar{v}^{a}\beta_{a}$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}-\bar{v}^{a}\beta_{a}\,.$

In the second equality, we used the standard result that
$\iota_{*}[\bar{u},\bar{v}]=[\iota_{*}\bar{u},\iota_{*}\bar{v}]$, and in the
third equality, we used Equation (7). In the fourth equality, we used the
standard expression for the Lie bracket in terms of a torsionful connection,
and in the fifth equality we used that $\bar{\nabla}\bar{\ell}=0$ and that
$\bar{\nu}_{a}\bar{T}^{a}_{~{}bc}=0$. In the sixth line we used the Leibniz
rule, that $\bar{\nu}(\bar{v})=0$, and that $\bar{\nabla}\bar{\nu}=0$.

We finish this subsection by examining
$\nu_{a}\nabla_{\iota_{*}\bar{u}}(\iota_{*}\bar{v})^{a}$. Similar to the
previous subsection, we observe that for horizontal vector fields
$\bar{u},\bar{v}\in H\mathcal{H}$, we have that
$\nu_{a}\nabla_{\iota_{*}\bar{u}}f(\iota_{*}\bar{v})^{a}=f\nu_{a}\nabla_{\iota_{*}\bar{u}}(\iota_{*}\bar{v})^{a}$
for $f\in C^{\infty}M$. So again following Spivak, there exists a symmetric
tensor field ${\rm
II}:\Gamma(H\mathcal{H})\times\Gamma(H\mathcal{H})\rightarrow\Gamma(\langle\ell\rangle)$,
labelled the extrinsic second fundamental form (or sometimes just the second
fundamental form, for short). So, we can write

$\displaystyle{\rm II}_{ab}\bar{u}^{a}\bar{v}^{b}:=$
$\displaystyle-\nu_{a}\nabla_{\iota_{*}\bar{u}}(\iota_{*}\bar{v}^{a})$
$\displaystyle=$
$\displaystyle(\iota_{*}\bar{v})^{a}(\iota_{*}\bar{u})^{b}\nabla_{(a}\nu_{b)}$
$\displaystyle=$
$\displaystyle\tfrac{1}{2}(\iota_{*}\bar{v})^{a}(\iota_{*}\bar{u})^{b}\mathcal{L}_{n}g_{ab}$
$\displaystyle=$
$\displaystyle\tfrac{1}{2}(\iota^{*}\mathcal{L}_{n}g_{ab})(\bar{u},\bar{v})\,.$

Thus, we have that

${\rm II}_{ab}=\tfrac{1}{2}\iota^{*}\mathcal{L}_{n}g_{ab}\,.$

With this tensor in hand, we thus have that

$\nu_{a}\nabla_{\iota_{*}\bar{u}}(\iota_{*}\bar{v})^{a}=-{\rm
II}_{ab}\bar{u}^{a}\bar{v}^{b}\,.$

####  III.4.3 Components of $\nabla^{\top}$ in the horizontal direction

We finish this series of computations by considering the last column in Table
6. The simplest of these derivatives follows from the geodesicity of $\ell$,
namely $(\iota_{*}\bar{w})_{a}\nabla_{\ell}\ell^{a}=0$. Beyond this, two
others are already easily computed using the Leibniz rule and methods already
established:

$(\iota_{*}\bar{w})_{a}\nabla_{\iota_{*}\bar{u}}n^{a}={\rm
II}_{ab}\bar{u}^{a}\bar{w}^{b}$

and

$(\iota_{*}\bar{w})_{a}\nabla_{\iota_{*}\bar{u}}\ell^{a}=K_{ab}\bar{u}^{a}\bar{w}^{b}\,.$

Next consider $(\iota_{*}\bar{w})_{a}\nabla_{\ell}n^{a}$. Since
$\nabla_{\ell}\nu(\iota_{*}\bar{w})\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}0$, we have that

$(\iota_{*}\bar{w})_{a}\nabla_{\ell}n^{a}=\bar{w}^{a}\beta_{a}\,.$

Now consider terms of the form
$(\iota_{*}\bar{w})_{a}\nabla_{\iota_{*}\bar{u}}(\iota_{*}\bar{v})^{a}$. To do
so, we generalize a similar result found in [43, Volume 3], and hence consider
the sum:

$\displaystyle\bar{g}_{jk}\bar{\nabla}_{\bar{u}}\bar{v}^{j}\bar{w}^{k}+\bar{g}_{jk}\bar{\nabla}_{\bar{v}}\bar{u}^{j}\bar{w}^{k}-\bar{g}_{jk}\bar{\nabla}_{\bar{w}}\bar{u}^{j}\bar{v}^{k}=$
$\displaystyle\bar{g}_{jk}\bar{w}^{k}(\bar{\nabla}_{\bar{u}}\bar{v}^{j}+\bar{\nabla}_{\bar{v}}\bar{u}^{j})+\bar{g}_{jk}\bar{v}^{k}(\bar{\nabla}_{\bar{u}}\bar{w}^{j}-\bar{\nabla}_{\bar{w}}\bar{u}^{j})+\bar{g}_{jk}\bar{u}^{k}(\bar{\nabla}_{\bar{v}}\bar{w}^{j}-\bar{\nabla}_{\bar{w}}\bar{v}^{j})$
$\displaystyle=$
$\displaystyle\bar{g}_{jk}\bar{w}^{k}(2\bar{\nabla}_{\bar{v}}\bar{u}^{j}+\bar{T}^{j}_{~{}im}\bar{u}^{i}\bar{v}^{m}+[\bar{u},\bar{v}]^{j})+\bar{g}_{jk}\bar{v}^{k}(\bar{T}^{j}_{~{}im}\bar{u}^{i}\bar{w}^{m}+[\bar{u},\bar{w}]^{j})$
$\displaystyle+\bar{g}_{jk}\bar{u}^{k}(\bar{T}^{j}_{~{}im}\bar{v}^{i}\bar{w}^{m}+[\bar{v},\bar{w}]^{j})\,.$

Now let $u_{p}=\iota_{*}\bar{u}_{p}$, $v_{p}=\iota_{*}\bar{v}_{p}$, and
$w_{p}=\iota_{*}\bar{w}_{p}$, so that $u,v,w\in TM|_{\mathcal{H}}$. As before,
we can use $\iota^{*}$ as the inverse of the pushforward map here because
these are horizontal vectors. (Note that for what follows, we do not need to
know how $u,v,w$ extend into $M$ because we are only taking derivatives in
directions tangent to $\mathcal{H}$.) Performing the same computation with the
spacetime Levi-Civita connection, a similar computation follows except the
torsion vanishes. However, we also have that

$\displaystyle\bar{g}_{jk}\bar{\nabla}_{\bar{u}}\bar{v}^{j}\bar{w}^{k}=$
$\displaystyle\bar{u}\langle\bar{v},\bar{w}\rangle_{\bar{g}}$ $\displaystyle=$
$\displaystyle i_{\bar{u}}{\rm{d}}\langle\bar{v},\bar{w}\rangle_{\bar{g}}$
$\displaystyle=$ $\displaystyle
i_{\iota^{*}u}{\rm{d}}\langle\iota^{*}v,\iota^{*}w\rangle_{\iota^{*}g}$
$\displaystyle=$ $\displaystyle i_{\iota^{*}u}{\rm{d}}\iota^{*}\langle
v,w\rangle_{g}$ $\displaystyle=$ $\displaystyle\iota^{*}i_{u}{\rm{d}}\langle
v,w\rangle_{g}$ $\displaystyle=$ $\displaystyle
g_{ab}\nabla_{u}v^{a}w^{b}|_{\mathcal{H}}\,.$

We need to compute a similar result for the Lie brackets. However, note that
because $\mathcal{H}$ is a hypersurface, we have that
$[\bar{v},\bar{w}]=:\bar{x}\in T\mathcal{H}$. So in particular there exists
$x\in TM|_{\mathcal{H}}$ such that $\iota^{*}x=\bar{x}$. Viewing $\iota$ as a
diffeomorphism $\mathcal{H}\rightarrow M|_{\mathcal{H}}$, we have that

$x=\iota_{*}\bar{x}=\iota_{*}[\bar{v},\bar{w}]=[\iota_{*}\bar{v},\iota_{*}\bar{w}]=[v,w]\,.$

We now can compute at $p\in\mathcal{H}\subset M$:

$\displaystyle\langle\bar{u},[\bar{v},\bar{w}]\rangle_{\bar{g}_{p}}=$
$\displaystyle\langle\iota^{*}u,\iota^{*}[v,w]\rangle_{\iota^{*}g_{p}}$
$\displaystyle=$ $\displaystyle(\iota^{*}g)_{p}(\iota^{*}u,\iota^{*}[v,w])$
$\displaystyle=$ $\displaystyle g_{p}(u,[v,w])$ $\displaystyle=$
$\displaystyle\langle u,[v,w]\rangle_{g_{p}}\,.$

Combining these three computations, we have that

$\displaystyle g_{ab}w^{b}\nabla_{v}u^{a}=\bar{g}_{ab}($
$\displaystyle\bar{w}^{b}\bar{\nabla}_{\bar{v}}\bar{u}^{a}+\tfrac{1}{2}\bar{w}^{b}\bar{T}^{a}_{~{}cd}\bar{u}^{c}\bar{v}^{d}$
$\displaystyle+\tfrac{1}{2}\bar{v}^{b}\bar{T}^{a}_{~{}cd}\bar{u}^{c}\bar{w}^{d}+\tfrac{1}{2}\bar{u}^{b}\bar{T}^{a}_{~{}cd}\bar{v}^{c}\bar{w}^{d})\,.$

Expressing this relationship in terms of the intrinsic second fundamental form
and noting that $\bar{\nu}(\bar{u})=\bar{\nu}(\bar{v})=\bar{\nu}(\bar{w})=0$,
we have that

$(\iota_{*}\bar{w})_{a}\nabla_{\iota_{*}\bar{u}}(\iota_{*}\bar{v})^{a}=\bar{w}_{a}\bar{\nabla}_{\bar{u}}\bar{v}^{a}\,.$

We conclude this subsection by considering terms of the form
$(\iota_{*}\bar{w})_{a}\nabla_{\ell}(\iota_{*}\bar{v})^{a}$. To do so, we will
consider the intrinsic derivative
$\bar{w}_{a}\bar{\nabla}_{\bar{\ell}}\bar{v}^{a}$. As
$\bar{\nabla}\bar{\ell}=0$ and $\bar{g}(\bar{\ell},\cdot)=0$, we can express
this derivative in terms of a Lie bracket and the torsion tensor:

$\displaystyle\bar{w}_{a}\bar{\nabla}_{\bar{\ell}}\bar{v}^{a}=$
$\displaystyle\bar{w}_{a}[\bar{\ell},\bar{v}]^{a}+\bar{w}_{a}\bar{\ell}^{c}\bar{v}^{d}\bar{T}^{a}_{~{}cd}$
$\displaystyle=$
$\displaystyle\bar{w}_{a}[\bar{\ell},\bar{v}]^{a}+2\bar{\nu}_{[c}K_{~{}d]}^{a}\bar{\ell}^{c}\bar{v}^{d}$
$\displaystyle=$
$\displaystyle\bar{w}_{a}[\bar{\ell},\bar{v}]^{a}+\bar{w}^{a}K_{ab}\bar{v}^{b}\,.$

Now for the same reasons as before, we have that
$\iota_{*}[\bar{\ell},\bar{v}]=[\ell,v]$, where $v=\iota_{*}\bar{v}$. As
before, let $w=\iota_{*}\bar{w}$. Thus, we have that

$\bar{w}_{a}[\bar{\ell},\bar{v}]^{a}=w_{a}[\ell,v]^{a}\,.$

Now calculating in $M$ on $\mathcal{H}$, we have that

$\displaystyle w_{a}[\ell,v]^{a}=$ $\displaystyle
w_{a}\nabla_{\ell}v^{a}-w^{a}v^{b}\nabla_{b}\lambda_{a}$ $\displaystyle=$
$\displaystyle
w_{a}\nabla_{\ell}v^{a}-w^{a}v^{b}\nabla_{(a}\lambda_{b)}-w^{a}v^{b}({\rm{d}}\lambda)_{ab}$
$\displaystyle=$ $\displaystyle
w_{a}\nabla_{\ell}v^{a}-(\odot\nabla\lambda)(\iota_{*}\bar{v},\iota_{*}\bar{w})-({\rm{d}}\lambda)(\iota_{*}\bar{w},\iota_{*}\bar{v})$
$\displaystyle=$ $\displaystyle
w_{a}\nabla_{\ell}v^{a}-(\iota^{*}\odot\nabla\lambda)(\bar{v},\bar{w})-(\iota^{*}{\rm{d}}\lambda)(\bar{w},\bar{v})$
$\displaystyle=$ $\displaystyle w_{a}\nabla_{\ell}v^{a}-K(\bar{v},\bar{w})\,.$

So, putting this all together, we have that

$(\iota_{*}\bar{w})_{a}\nabla_{\ell}(\iota_{*}\bar{v})^{a}=\bar{w}_{a}\bar{\nabla}_{\bar{\ell}}\bar{v}^{a}\,.$

####  III.4.4 Summary of decomposition

For the purposes of clarity, we fill in Table 6:

$\displaystyle\lambda_{a}$ $\displaystyle\nabla_{\ell}n^{a}=-\kappa\,,\quad$
$\displaystyle\nu_{a}$ $\displaystyle\nabla_{\ell}n^{a}=0\,,\quad$
$\displaystyle(\iota_{*}\bar{w})_{a}\nabla_{\ell}n^{a}=\bar{w}^{a}\beta_{a}\,,$
(8) $\displaystyle\lambda_{a}$ $\displaystyle\nabla_{\ell}\ell^{a}=0\,,\quad$
$\displaystyle\nu_{a}$ $\displaystyle\nabla_{\ell}\ell^{a}=\kappa\,,\quad$
$\displaystyle(\iota_{*}\bar{w})_{a}\nabla_{\ell}\ell^{a}=0\,,$
$\displaystyle\lambda_{a}$
$\displaystyle\nabla_{\ell}(\iota_{*}\bar{v})^{a}=0\,,\quad$
$\displaystyle\nu_{a}$
$\displaystyle\nabla_{\ell}(\iota_{*}\bar{v})^{a}=-\bar{v}^{a}\beta_{a}\,,\quad$
$\displaystyle(\iota_{*}\bar{w})_{a}\nabla_{\ell}(\iota_{*}\bar{v})^{a}=\bar{w}_{a}\bar{\nabla}_{\bar{\ell}}\bar{v}^{a}\,,$
$\displaystyle\lambda_{a}$
$\displaystyle\nabla_{\iota_{*}\bar{u}}n^{a}=\bar{u}^{a}\beta_{a}\,,\quad$
$\displaystyle\nu_{a}$ $\displaystyle\nabla_{\iota_{*}\bar{u}}n^{a}=0\,,\quad$
$\displaystyle(\iota_{*}\bar{w})_{a}\nabla_{\iota_{*}\bar{u}}n^{a}={\rm
II}_{ab}\bar{u}^{a}\bar{w}^{b}\,,$ $\displaystyle\lambda_{a}$
$\displaystyle\nabla_{\iota_{*}\bar{u}}\ell^{a}=0\,,\quad$
$\displaystyle\nu_{a}$
$\displaystyle\nabla_{\iota_{*}\bar{u}}\ell^{a}=-\bar{u}^{a}\beta_{a}\,,\quad$
$\displaystyle(\iota_{*}\bar{w})_{a}\nabla_{\iota_{*}\bar{u}}\ell^{a}=K_{ab}\bar{u}^{a}\bar{w}^{b}\,,$
$\displaystyle\lambda_{a}$
$\displaystyle\nabla_{\iota_{*}\bar{u}}(\iota_{*}\bar{v})^{a}=-K_{ab}\bar{u}^{a}\bar{v}^{b}\,,$
$\displaystyle\quad\nu_{a}$
$\displaystyle\nabla_{\iota_{*}\bar{u}}(\iota_{*}\bar{v})^{a}=-{\rm
II}_{ab}\bar{u}^{a}\bar{v}^{b}\,,\quad$
$\displaystyle(\iota_{*}\bar{w})_{a}\nabla_{\iota_{*}\bar{u}}(\iota_{*}\bar{v})^{a}=\bar{w}_{a}\bar{\nabla}_{\bar{u}}\bar{v}^{a}\,.$

Observe that the difference between the spacetime connection along
$\mathcal{H}$ and the induced connection on $\mathcal{H}$ is quantified by the
extrinsic data specified by $\kappa$, $\beta$, and ${\rm II}$, as well as the
intrinsic tensor $K$. A subset of these invariants were also noted in [28].

###  III.5 Spacetime-Carrollian relationships: the curvature

Having established relationships between $\nabla^{\top}$ and $\bar{\nabla}$,
we now have the tools to rewrite the various projections of the spacetime
Riemann curvature to $\mathcal{H}$ in terms of extrinsic data (such as
$\kappa$, $\beta$, and ${\rm II}$) and intrinsic tensors (such as $\bar{R}$
and $K$).

We first consider the tangential component of the spacetime Riemann curvature.
Since $T\mathcal{H}$ can be decomposed into $\langle\ell\rangle\oplus
H\mathcal{H}$, we will need to consider projections into each component,
resulting in (naively) 16 projections to compute. As before, we will assume
that $\bar{t},\bar{u},\bar{v},\bar{w}\in\Gamma(H\mathcal{H})$ and their
pushforwards are denoted, respectively, by
$t,u,v,w\in\Gamma(TM)|_{\mathcal{H}}$.

For arbitrary vector fields $T,U,V,W\in\Gamma(TM)$, we have that

$T^{a}U^{b}V^{c}W^{d}R_{abcd}=V_{c}(\nabla_{T}\nabla_{U}W^{c}-\nabla_{U}\nabla_{T}W^{c}-\nabla_{[T,U]}W^{c})\,.$

Using this identity and Table 8, we wish to rewrite various projections of
spacetime curvatures in terms of the intrinsic curvatures $\bar{R}$ and $K$
and extrinsic curvatures $\beta$ and ${\rm II}$. The resulting family of
identities is:

$\displaystyle t^{a}u^{b}v^{c}w^{d}R_{abcd}$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}\bar{t}^{a}\bar{u}^{b}\bar{v}^{c}\bar{w}^{d}(\bar{R}_{abcd}+{\rm
II}_{ad}K_{bc}-{\rm II}_{ac}K_{bd}-{\rm II}_{bd}K_{ac}+{\rm II}_{bc}K_{ad})$
$\displaystyle t^{a}u^{b}\ell^{c}w^{d}R_{abcd}$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}2\bar{t}^{a}\bar{u}^{b}\bar{w}^{d}(\bar{\nabla}_{[b}K_{a]d}+\beta_{[b}K_{a]d})$
$\displaystyle t^{a}u^{b}n^{c}w^{d}R_{abcd}$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}2\bar{t}^{a}\bar{u}^{b}\bar{w}^{d}(\bar{\nabla}_{[b}{\rm
II}_{a]d}-\beta_{[b}{\rm II}_{a]d})$ $\displaystyle
t^{a}u^{b}n^{c}\ell^{d}R_{abcd}$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}2\bar{t}^{a}\bar{u}^{b}(K_{c[a}{\rm
II}^{c}_{b]}-\bar{\nabla}_{[a}\beta_{b]})$
$\displaystyle\ell^{a}u^{b}\ell^{c}w^{d}R_{abcd}$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}-\bar{u}^{b}\bar{w}^{d}(\bar{\nabla}_{\bar{\ell}}K_{bd}+K^{2}_{bd}-\kappa
K_{bd})$ $\displaystyle\ell^{a}u^{b}n^{c}w^{d}R_{abcd}$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}\bar{u}^{b}\bar{w}^{d}(\bar{\nabla}_{b}\beta_{d}-\bar{\nabla}_{\bar{\ell}}{\rm
II}_{bd}-\beta_{b}\beta_{d}-K_{b}^{a}{\rm II}_{ad}-\kappa{\rm II}_{bd})$
$\displaystyle\ell^{a}u^{b}n^{c}\ell^{d}R_{abcd}$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\mathcal{H}{\hskip
0.85358pt}}}{{=}}}-\bar{u}^{b}(\bar{\nabla}_{\bar{\ell}}\beta_{b}+\bar{\nabla}_{b}\kappa+2\beta_{a}K_{b}^{a})$

The remaining projections can be found by application of the first Bianchi
identity. However, note that the first Bianchi is not the traditional
statement that $R_{[abc]d}=0$. Indeed, because the torsion of $\bar{\nabla}$
is non-vanishing, we have a more complicated first Bianchi identity:

$\bar{R}_{[abc]d}-T_{[ab}{}^{e}T_{|e|c]d}-\bar{\nabla}_{[a}T_{bc]d}=0\,,$

where $T_{abc}=\bar{g}_{cd}T^{d}_{ab}$. Given the formula for $T$ in terms of
the Ehresmann connection and the intrinsic second fundamental form, we end up
with

$\bar{R}_{[abc]d}+2\bar{\nu}_{[a}\bar{\nabla}_{b}K_{c]d}=0\,.$

For example, we can use this identity to compute
$\bar{\ell}^{a}\bar{R}_{a}bcd$.

##  IV Examples

###  IV.1 Schwarzschild black hole

We first consider the simplest (non-trivial) spacetime with an embedded null
hypersurface: the Schwarzschild black hole with metric (in Schwarzschild
coordinates)

$ds^{2}=(1-\tfrac{2m}{r})dt^{2}-(1-\tfrac{2m}{r})^{-1}dr^{2}-r^{2}d\Omega^{2}\,,$

where $d\Omega^{2}$ is the metric on the round sphere. The event horizon is
given by the null hypersurface defined by $\mathcal{H}:=\\{p\in
M\;|\;r(p)=2m\\}$.

The geometry distinguishes a fundamental vector field that generates
$\mathcal{H}$. Indeed, there is a unique Killing vector field that generates
the null hypersurface: $\partial_{t}$. We thus define
$\bar{\ell}:=\partial_{t}|_{\mathcal{H}}$.

As the spacetime is spherically symmetric, there is also a distinguished
spatial slice of the hypersurface: the round sphere. So on $\mathcal{H}$, we
can foliate by spatial slices

$S_{t_{0}}=\\{p\in\mathcal{H}\;|\;t(p)=t_{0}\\}\,.$

Now following the construction of Friedrich, et. al. [21] with $\bar{\ell}$
and $S_{t_{0}}$, we find that in Gaussian null coordinates, the metric is
given by

$g=\left(1-\frac{2m}{r}\right){\rm{d}}u^{2}+{\rm{d}}u{\rm{d}}r-r^{2}d\Omega^{2},$

with canonical rigging vector $n=\partial_{r}$. In this coordinate system,
$\ell=\partial_{u}$. We also obtain the following coframe

$\displaystyle\nu$ $\displaystyle={\rm{d}}u,$ $\displaystyle\lambda$
$\displaystyle={\rm{d}}r-\left(1-\frac{2m}{r}\right){\rm{d}}u,$ $\displaystyle
m^{1}$ $\displaystyle=r{\rm{d}}\theta,$ $\displaystyle m^{2}$
$\displaystyle=r\sin\theta{\rm{d}}\phi.$

We can now directly compute the invariants of the induced special Carrollian
manifold. First, observe that the degenerate metric $\bar{g}$ on $\mathcal{H}$
is given by

$\bar{g}=-4m^{2}{\rm{d}}\Omega^{2}\,.$

Since $\bar{g}$ is independent of $u$, we have that $K=0$. Furthermore,
because $\bar{\nu}$ is constant under partial differentiation and the torsion
vanishes, we find that $\bar{\Gamma}$ are precisely the Christoffel symbols of
a round sphere with radius $2m$, and so $\bar{R}$ is that of the same.

In Gaussian null coordinates, we can explicitly compute the extrinsic
invariants as well using Table 8:

$\displaystyle\kappa$ $\displaystyle=\frac{1}{4m}\,,$ $\displaystyle\beta$
$\displaystyle=0\,,$ $\displaystyle{\rm II}$
$\displaystyle=\frac{1}{4m}\bar{g}\,.$

###  IV.2 Non-spinning Thakurta metric

The non-spinning Thakurta metric [44] is a time-dependent conformal rescaling
of the Schwarzschild metric:

${\rm{d}}s^{2}=e^{-2U(u)}\left(\left(1-\frac{2m}{r}\right){\rm{d}}u^{2}+{\rm{d}}u{\rm{d}}r-r^{2}d\Omega^{2}{\rm{d}}\phi^{2}\right),$

While the spacetime does not admit a null Killing field that generates the
event horizon at $\mathcal{H}=\\{p\in M\;|\;r(p)=2m\\}$, it does admit a
conformal Killing field which generates the null hypersurface, $\partial_{u}$,
and so we define $\bar{\ell}=\partial_{u}|_{\mathcal{H}}$. Just as in the
Schwarzschild case, it also admits a distinguished spatial foliation of
$\mathcal{H}$ parametrized by $u$ and given by the round spheres $S_{u}$.

In this case, Gaussian null coordinates are not necessary as the canonical
rigging vector is chosen to be $n=e^{2U}\partial_{r}$ which gives the
following coframe

$\displaystyle\nu$ $\displaystyle=du,$ $\displaystyle\lambda$
$\displaystyle=e^{-2U}\left(dr-\left(1-\frac{2m}{r}\right)du\right),$
$\displaystyle m^{1}$ $\displaystyle=e^{U}rd\theta,$ $\displaystyle m^{2}$
$\displaystyle=e^{U}r\sin(\theta)d\phi.$

We now explicitly compute the invariants, both intrinsic and extrinsic. Here,
the intrinsic second fundamental form is non-vanishing. In fact, this is
expected: the conformal rescaling of the Schwarzschild metric, in a way that
depends on the $u$ coordinate, implies that the induced metric on a spatial
slice will not preserve lengths when Lie dragged in the $\ell=\partial_{u}$
direction.

The lowest order intrinsic invariants are then

$\displaystyle K_{ab}$ $\displaystyle=-U^{\prime}\bar{g}_{ab},$
$\displaystyle\bar{R}_{ijkl}$
$\displaystyle=\frac{e^{2U}}{4m^{2}}(\bar{g}_{ik}\bar{g}_{jl}-\bar{g}_{il}\bar{g}_{jk}).$

Using the expressions in subsection III.4.4 the remaining expressions for the
curvature tensor can be computed in terms of these.

The extrinsic curvatures can similarly be computed, yielding

$\displaystyle\kappa$ $\displaystyle=\frac{8U^{\prime}m^{2}-m}{4m^{2}},$
$\displaystyle\beta$ $\displaystyle=0,$ $\displaystyle{\rm II}$
$\displaystyle=\frac{e^{2U}}{4m}\bar{g}.$

Comparing the above invariants with that of the Schwarzschild black hole, we
see that these null hypersurface embeddings are certainly distinct.

###  IV.3 An arbitrary $d-1$ dimensional null hypersurface

In the construction outlined in this article, in order to study a given null
hypersurace in a Lorentzian spacetime one must choose some spatial slice of
the null hypersurface and build coordinates $\\{x^{A}\\}$ on this slice. In
addition, a representative vector field, $\bar{\ell}$, of the generating null
direction of the null hypersurface must be chosen to be normal to this spatial
slice. With these choices, a Gaussian null coordinate system can be
constructed locally and the metric of the spacetime takes the form [21]:

$\displaystyle{\rm{d}}s^{2}=2{\rm{d}}u{\rm{d}}r+rA{\rm{d}}u^{2}+2rB_{A}{\rm{d}}u{\rm{d}}x^{A}+C_{AB}{\rm{d}}x^{\mu}{\rm{d}}x^{\nu}\,,$
(9)

where $A,B$ and $C$ are smooth functions of $u,r,x^{A}$ such that $C_{AB}$ is
a negative definite $(d-2)\times(d-2)$ matrix. In this coordinate system, the
hypersurface, $\mathcal{H}$ is located at $r=0$.

This construction picks out a single null vector-field transverse to the
hypersurface, $n=\partial_{r}$ along with its dual $\nu=g(n,-)={\rm{d}}u$ as
the principal Ehresmann connection for the Carrollian manifold. In the
spacetime, we can construct the coframe:

$\displaystyle\begin{aligned} \nu&={\rm{d}}u,\\\
\lambda&={\rm{d}}r+rA{\rm{d}}u+rB_{C}{\rm{d}}x^{C},\\\
\mu^{i}&=m^{i}_{~{}C}{\rm{d}}x^{C}.\end{aligned}$ (10)

with the dual frame basis,

$\displaystyle\begin{aligned} \ell&=\partial_{u}-rA\partial_{r},\\\
n&=\partial_{r},\\\
m_{i}&=m_{i}^{~{}C}(\partial_{x^{C}}-rB_{C}\partial_{r}).\end{aligned}$ (11)

where $m^{i}_{~{}C}$ is an invertible matrix with inverse $m_{i}^{~{}C}$,
satisfying, $\delta_{ij}m^{i}_{~{}A}m^{j}_{~{}B}=C_{AB}$. On the Carrollian
geometry, the degenerate metric $\bar{g}$ on $\mathcal{H}$ is

$\displaystyle\bar{g}=C_{AB}dx^{A}dx^{B}.$

To investigate the intrinsic and extrinsic invariants of the induced special
Carrollian manifold, we will compute the connection coefficients in the bulk
and restrict to the hypersurface. The lowest order intrinsic invariants are
then

$\displaystyle\begin{aligned} K_{ij}&=-n(m_{[i|C|})m_{j]}^{~{}C}|_{r=0},\\\
\bar{R}_{ijkl}&=2m_{[l}(\Gamma_{|ij|k]})|_{r=0},\\\ \end{aligned}$ (12)

where

$\Gamma_{ijk}=\frac{1}{2}(D_{ijk}-D_{jik}-D_{kji}),$

and

$D^{i}_{~{}jk}=[m_{[k}(m^{i}_{~{}|C|})m_{j]}^{~{}C}].$

All other components of the curvature tensor and torsion tensor can be
reconstructed from these quantities using the expressions in subsection
III.4.4

The extrinsic curvature is

$\displaystyle\begin{aligned} \kappa&=\frac{1}{2}n(rA)_{r=0},\\\
\beta_{i}&=n(rB_{C})m_{j}^{~{}C}|_{r=0},\\\ {\rm
II}_{ij}&=\ell(m_{[i|C|})m_{j]}^{~{}C}|_{r=0}\end{aligned}$ (13)

##  V Classification of Carrollian geometries

The classification of Lorentzian manifolds can, in principle, be accomplished
using an analogue of the Cartan-Karlhede algorithm [2]. In this algorithm, an
invariantly defined frame is determined by specifying canonical forms of the
curvature tensor and its covariant derivatives up to a finite order. The
maximum order of covariant differentiation and the uniqueness of this
invariant frame is explicitly determined by the algorithm.

While null hypersurfaces are embedded in a Lorentzian manifold, we have shown
that the intrinsic geometry of a null hypersurface is not Lorentzian but
instead is Carrollian and hence admit a torsion tensor and a curvature tensor.
Cartan-Karlhede algorithms have been developed for geometries admitting
torsion such as teleparallel geometries and Riemann-Cartan geometries [16].
However, these geometries still rely on the Lorentz group as a structure
group. This motivates the investigation of a Cartan-Karlhede algorithm for
Carrollian geometries equipped with a canonical connection.

In this section we outline an approach for computing all invariants to locally
characeterize a Carrollian manifold using the torsion tensor, the curvature
tensor and its covariant derivatives. This approach will rely on the existence
of a principal Ehresmann connection in the Carrollian manifold, which is
guaranteed in the case of a null hypersurface. Equipped with this preferred
Ehsresmann connection, it is possible to determine the components of the
connection relative to the frame $\\{e_{I}\\}=\\{\ell,m_{i}\\}$ or the coframe
$\\{\theta^{I}\\}=\\{\nu,\mu^{i}\\}$ as

$\displaystyle\Gamma^{I}_{~{}JK}=e^{I}_{~{}\nu}e_{J}^{~{}\mu}(\partial_{\mu}e_{K}^{~{}\nu}+e_{K}^{~{}\lambda}\Gamma^{\nu}_{~{}\mu\lambda})$
(14)

where $e_{I}=e_{I}^{~{}\mu}\partial_{\mu}$ and
$\theta^{I}=e^{J}_{~{}\mu}{\rm{d}}^{\mu}$.

Relative to this frame, we can compute the torsion tensor for the Carrollian
manifold:

$\displaystyle\bar{T}^{i}=\bar{T}^{i}_{[1j]}\bar{m}_{i}\otimes\bar{\nu}\wedge\bar{\mu}^{j}.$
(15)

In principle, this tensor allows for the fixing of the Carrollian boost
parameters. If not, the curvature tensor may be computed and the components,
$\bar{R}_{1ijk}$ can be used to fix the Carrollian boost parameters instead.
The remaining $SO(n-1)$ freedom can be used to fix a canonical form for the
components $\bar{R}_{ijkl}$. Further fixing of the Carrollian group parameters
can be accomplished by computing the respective covariant derivatives of the
torsion tensor and the curvature tensor. We note that it may be advantageous
to use Carrollian boosts for which the resulting Ehresmann connection is no
longer principal in order to achieve a canonical form for the torsion tensor
or curvature tensor.

This suggests the following algorithm.

  1. 1.

Set the order of differentiation, $q$ to zero.

  2. 2.

Compute the derivatives of the torsion tensor and curvature tensor up to order
$q$

  3. 3.

Determine a canonical form for the $q$-th derivatives of the torsion tensor
and curvature tensor.

  4. 4.

Fix the frame parameters using these canonical forms and record the residual
frame freedom, denoted as $H_{q}$.

  5. 5.

Find the number of functionally independent components, $t_{q}$ of the torsion
tensor, curvature tensor and their derivatives in the canonical form.

  6. 6.

If $\operatorname{dim}(H_{q})=\operatorname{dim}(H_{q-1})$ and
$t_{q}=t_{q-1}$, set $p+1=q$ and stop. Otherwise, increment $q$ by $1$ and go
to step 2.

Here, the integer $p$ denotes the highest order of differentiation where new
geometric information is introduced.

Given two special Carrollian manifolds, in order to determine their
equivalence it is sufficient to determine the canonical form of the torsion
tensor, curvature tensors and their respective covariant derivatives for one
special Carrolian manifold and attempt to impose the same canonical forms for
the other. This is a necessary condition for equivalence, but not sufficient.
A sufficient condition follows from comparing the components of their
respective canonical forms and solving the resulting equations arising from
equating each invariant in the first special Carrollian manifold with the
corresponding invariant in the second special Carrollian manifold.

##  VI Conclusion

In the paper, we have sought to provide clarity on the types of Carrollian
geometries and how they relate to null hypersurface embeddings. The most
constrained of these, the special Carrollian manifold, was investigated in
greater detail, and examples were explored to showcase the utility of the
developed calculus. In this case, such Carrollian manifolds are equipped with
a choice of null vector-field as the generator of the hypersurface and a
spatial slice. We have also argued that torsion is an intrinsic part of the
null hypersurface geometries and that any connection for such geometries
should necessarily include this.

Using the framework outlined in this article, we expect that interesting and
fruitful classification results will follow naturally. Beyond just the
geometries investigated in this article, elsewhere there has also been some
work on conformal Carrollian structures [18, 12, 26] which may dove-tail
nicely with our constructions. In future work we hope to tie in our
constructions with those structures as well. In addition, we would like to use
this framework to study the effect of conformal transformations on the event
horizon of stationary black holes leading to dynamical solutions [34]. This
question is significant in the context of the geometric horizon conjectures
[15] when compared to the class of conformally covariant invariants
constructed in [33, 32] where there is the possibility of several curvature
invariants able to pick out horizon candidates as their level sets.

###### Acknowledgements.

SB was supported by a COST Action CaLISTA grant (reference number E-COST-
GRANT-CA21109-1cd88727). Further support was given by the Czech Science
Foundation (GACR) grant GA22-00091S and the Operational Programme Research
Development and Education Project No. CZ.02.01.01/00/22-010/0007541. SB would
also like to acknowledge the UiT Department of Mathematics and Statistics for
their gracious hospitality. DM is supported by the Norwegian Financial
Mechanism 2014-2021 (project registration number 2019/34/H/ST1/00636).

## References

  * [1] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, et al.  Observation of gravitational waves from a binary black hole merger.  Physical Review Letters, 116:061102, Feb 2016.
  * [2] J. E. Åman and A. Karlhede.  A computer-aided complete classification of geometries in general relativity. first results.  Physics Letters A, 80(4):229–231, 1980.
  * [3] A. Ashtekar, C. Beetle, and J. Lewandowski.  Geometry of generic isolated horizons.  Classical and Quantum Gravity, 19(6):1195, 2002.
  * [4] A. Bagchi, S. Banerjee, S. Basu, and S. Dutta.  Scattering amplitudes: Celestial and carrollian.  Physical Review Letters, 128(24), jun 2022.
  * [5] A. Bagchi, R. Basu, A. Mehra, and P. Nandi.  Field theories on null manifolds.  Journal of High Energy Physics, 2020(2), feb 2020.
  * [6] A. Bagchi, S. Dutta, K.S. Kolekar, and P. Sharma.  BMS field theories and weyl anomaly.  Journal of High Energy Physics, 2021(7), jul 2021.
  * [7] A. Bagchi, A. Mehra, and P. Nandi.  Field theories with conformal carrollian symmetry.  Journal of High Energy Physics, 2019(5), may 2019.
  * [8] K. Banerjee, R. Basu, A. Mehra, A. Mohan, and A. Sharma.  Interacting conformal carrollian theories: Cues from electrodynamics.  Physical Review D, 103(10), may 2021.
  * [9] X. Bekaert and K. Morand.  Connections and dynamical trajectories in generalised newton-cartan gravity. ii. an ambient perspective.  Journal of Mathematical Physics, 59(7), 2018.
  * [10] I. Booth.  Spacetime near isolated and dynamical trapping horizons.  Physical Review D, 87(2):024008, 2013.
  * [11] V. Chandrasekaran, É. É. Flanagan, I. Shehzad, and A.J. Speranza.  Brown-York charges at null boundaries.  Journal of High Energy Physics, 2022(1), jan 2022.
  * [12] L. Ciambelli, R. G. Leigh, C. Marteau, and P. M. Petropoulos.  Carroll structures, null geometry, and conformal isometries.  Physical Review D, 100(4), aug 2019.
  * [13] L Ciambelli, C. Marteau, A. C. Petkou, P. M. Petropoulos, and K. Siampos.  Covariant galilean versus carrollian hydrodynamics from relativistic fluids.  Classical and Quantum Gravity, 35(16):165001, jul 2018.
  * [14] L. Ciambelli, C. Marteau, A. C. Petkou, P. M. Petropoulos, and K. Siampos.  Flat holography and carrollian fluids.  Journal of High Energy Physics, 2018(7), jul 2018.
  * [15] A. A. Coley, D. D. McNutt, and A. A. Shoom.  Geometric horizons.  Physics Letters B, 771:131–135, 2017.
  * [16] A. A. Coley, R. J. van den Hoogen, and D. D. McNutt.  Symmetry and equivalence in teleparallel gravity.  Journal of Mathematical Physics, 61(7), 2020.
  * [17] L. Donnay and C. Marteau.  Carrollian physics at the black hole horizon.  Classical and Quantum Gravity, 36(16):165002, jul 2019.
  * [18] C. Duval, G. W. Gibbons, and P. A. Horvathy.  Conformal carroll groups and BMS symmetry.  Classical and Quantum Gravity, 31(9):092001, apr 2014.
  * [19] J. Figueroa-O’Farrill.  On the intrinsic torsion of spacetime structures.  arXiv preprint arXiv:2009.01948, 2020.
  * [20] L. Freidel and P. Jai-akson.  Carrollian hydrodynamics and symplectic structure on stretched horizons, 2022.
  * [21] H. Friedrich, I. Racz, and R. M. Wald.  On the rigidity theorem for spacetimes with a stationary event horizon or a compact cauchy horizon.  Communications in mathematical physics, 204:691–707, 1999.
  * [22] E. Gourgoulhon.  Generalized damour-navier-stokes equation applied to trapping horizons.  Physical Review D, 72(10):104007, 2005.
  * [23] P Hájiček.  Exact models of charged black holes: I. geometry of totally geodesic null hypersurface.  Communications in Mathematical Physics, 34:37–52, 1973.
  * [24] P Hájíček.  Stationary electrovacuum spacetimes with bifurcate horizons.  Journal of Mathematical Physics, 16(3):518–522, 1975.
  * [25] J. Hartong.  Gauging the carroll algebra and ultra-relativistic gravity.  Journal of High Energy Physics, 2015(8), aug 2015.
  * [26] Y. Herfray.  Asymptotic shear and the intrinsic conformal geometry of null-infinity.  Journal of Mathematical Physics, 61(7), jul 2020.
  * [27] Y. Herfray.  Carrollian manifolds and null infinity: a view from cartan geometry.  Classical and Quantum Gravity, 39(21):215005, sep 2022.
  * [28] F. Hopfmüller and L. Freidel.  Gravity degrees of freedom on a null surface.  Physical Review D, 95:104006, May 2017.
  * [29] J.M. Lévy-Leblond.  Une nouvelle limite non-relativiste du groupe de poincaré.  Annales De L Institut Henri Poincare-physique Theorique, 3:1–12, 1965.
  * [30] W.B. Liu and J. Long.  Symmetry group at future null infinity: Scalar theory.  Physical Review D, 107(12), jun 2023.
  * [31] M. Mars and J. M. M. Senovilla.  Geometry of general hypersurfaces in spacetime: junction conditions.  Classical and Quantum Gravity, 10(9):1865, 1993.
  * [32] D. D. McNutt.  Curvature invariant characterization of event horizons of four-dimensional black holes conformal to stationary black holes.  Physical Review D, 96(10):104022, 2017.
  * [33] D. D. McNutt and D. N. Page.  Scalar polynomial curvature invariant vanishing on the event horizon of any black hole metric conformal to a static spherical metric.  Physical Review D, 95(8):084044, 2017.
  * [34] M. M. C. Mello, A. Maciel, and V. T. Zanchin.  Evolving black holes from conformal transformations of static solutions.  Physical Review D, 95(8):084031, 2017.
  * [35] V. Moncrief and J. Isenberg.  Symmetries of cosmological cauchy horizons.  Communications in Mathematical Physics, 89:387–413, 1983.
  * [36] P. Nurowski and D. C. Robinson.  Intrinsic geometry of a null hypersurface.  Classical and Quantum Gravity, 17(19):4065, 2000.
  * [37] P. J. Olver.  Equivalence, invariants and symmetry.  Cambridge University Press, 1995.
  * [38] R. F. Penna.  Near-horizon carroll symmetry and black hole love numbers.  arXiv: High Energy Physics - Theory, 2018.
  * [39] R. Penrose.  Structure of space-time.  In Battelle Rencontres, pages 121–235, 1968.
  * [40] R Penrose.  The geometry of impulsive gravitational waves 1972 general relativity, papers in honour of jl synge, edited by l. o’raifeartaigh, 1972\.
  * [41] C. A. Petkou, P. M. Petropoulos, D. Rivera-Betancour, and K. Siampos.  Relativistic fluids, hydrodynamic frames and their galilean versus carrollian avatars.  Journal of High Energy Physics, 2022(9), sep 2022.
  * [42] K. Prabhu.  Twistorial description of bondi-metzner-sachs symmetries at null infinity.  Physical Review D, 105(2), jan 2022.
  * [43] M. Spivak.  A Comprehensive Introduction to Differential Geometry.  Publish or Perish, Inc., 1999.
  * [44] S.N. G. Thakurta.  Kerr metric in an expanding universe.  Indian Journal of Physics, 55(4):304–310, 1981.