# Variational formalism for generic shells in general relativity

Bence Racskó  University of Szeged, Department of Theoretical Physics, Szeged,
Hungary<EMAIL_ADDRESS>
###### Abstract

We investigate the variational principle for the gravitational field in the
presence of thin shells of completely unconstrained signature (generic
shells). Such variational formulations have been given before for shells of
timelike and null signatures separately, but so far no unified treatment
exists. We identify the shell equation as the natural boundary condition
associated with a broken extremal problem along a hypersurface where the
metric tensor is allowed to be nondifferentiable. Since the second order
nature of the Einstein-Hilbert action makes the boundary value problem
associated with the variational formulation ill-defined, regularization
schemes need to be introduced. We investigate several such regularization
schemes and prove their equivalence. We show that the unified shell equation
derived from this variational procedure reproduce past results obtained via
distribution theory by Barrabes and Israel for hypersurfaces of fixed causal
type and by Mars and Senovilla for generic shells. These results are expected
to provide a useful guide to formulating thin shell equations and junction
conditions along generic hypersurfaces in modified theories of gravity.

††: Class. Quantum Grav.

  * May 2021

Keywords: general relativity, thin shells, junction conditions, variational
principles

##  1 Introduction

Thin shells in General Relativity (GR) and field theories in general are weak
(distributional) solutions to the field equations whose pathological behaviour
is concentrated to a single hypersurface (or a series of nonintersecting
hypersurfaces) in spacetime. In GR such solutions may describe energetic
phenomena such as phase transitions, impulsive electromagnetic and
gravitational waves [1, 2]. Thin shells also give rise to junction condition
on the common boundary surface when glueing together spacetime domains.

Thin shells and junction conditions in GR have been considered by Lanczos [3],
Darmois [4], O’Brien and Synge [5], and Lichnerowicz [6], however the most
commonly used formulation has been given by Israel [7]. On a timelike or
spacelike hypersurface partitioning the spacetime manifold into two
subdomains, Israel prescribed the continuity of the induced metric $h_{ab}$
and the Lanczos equation relating the jump of the extrinsic curvature $K_{ab}$
to the surface energy-momentum tensor. In the absence of a material shell, the
Lanczos equation reduces to the continuity of the extrinsic curvature. The
case when the surface energy-momentum tensor does not vanish will be referred
to as a thin shell, and the relations imposed by the vanishing of the surface
energy-momentum tensor as the junction conditions111This terminology is not
universal. Some authors refer to the Lanczos equation itself as a junction
condition, even if the surface energy-momentum tensor does not vanish.. An
advantage of Israel’s formulation is double covariance. For practical
calculations it is often useful to work with coordinate systems adapted to the
subdomains that mismatch along the hypersurface. Differentiability classes of
tensor fields may only be established in coordinate systems whose
differentiability class exceeds that of the tensor field. Israel’s equations
are, however, relations between hypersurface tensors and thus one only has to
ensure that the parametrization of the hypersurface is the same as viewed from
either side and otherwise work with disjoint systems of bulk coordinates in
each spacetime region.

Israel’s formulation breaks down when the hypersurface has null points. At
null points the normal vector field becomes tangential as well and the
extrinsic curvature - which can be seen as the normal derivative of the metric
- becomes an intrinsic tangential quantity that carries no transverse
information. The 3+1 orthogonal decomposition along the shell facilitated by
the normal vector becomes degenerate. To fix terminology, a hypersurface will
be called pure if it is either timelike, spacelike or null, while it will be
referred to as causality-changing, signature-changing or non-pure if its
causal type is not constant. The term generic hypersurface is used when the
causal type is absolutely not fixed and the surface may either be pure or
causality-changing.

Null shells are physically relevant (we refer to [2] for an extensive
treatment of their applications), for example as models for impulsive
electromagnetic and gravitational waves. Generalizations of the Israel
formalism for null shells have been given among others by Clarke and Dray [8],
Barrabes and Israel [9], Mars and Senovilla [10], Poisson [11], Mars [12] and
Senovilla [13]. Out of these, the formalisms of [9, 10, 12, 13] give a unified
prescription valid for generic hypersurfaces222The work by Barrabes and Israel
impose the condition $n\cdot n=\mathrm{const}$, i.e. the length of the normal
vector is constant along the hypersurface. This formally restricts their
formalism to pure hypersurfaces. However the shell equation obtained therein
agrees (after the differences in conventions have been addressed) by that of
eg. Mars and Senovilla, which is valid for generic shells, showing that this
condition is not imperative in the derivation of the shell equation.. The
common point of generalization is that the normal vector field is accompanied
by a transversal vector field which generates a non-orthogonal decomposition
of the spacetime along the hypersurface. The role of the extrinsic curvature
is carried by an analogous quantity built from the transversal vector field. A
hypersurface equipped with a selected transversal vector field is called a
rigged hypersurface. This structure has been investigated by eg. Eisenhart
[14] and Schouten [15] to describe the geometry of subspaces of manifolds with
linear connections. The formalism has been systematically applied to GR by
Mars and Senovilla in [10].

There exists at least four methods of obtaining the timelike or spacelike
shell equation in GR [13]. These will be referred to as i) the “pillbox
integration333This terminology has been borrowed from Misner, Thorne and
Wheeler [16].”, ii) the distributional method, iii) the intrinsic method and
iv) the variational method. Pillbox integration has been employed by Israel in
[7] and involves writing the field equations in Gaussian coordinates adapted
to the hypersurface, separating a normal derivative and integrating the field
equations through the shell as its thickness tends to zero. This approach is
similar to the well-known textbook method [17] to derive the analogous jump
conditions in electrodynamics. The distributional method has been pioneered by
Taub [18], Geroch and Traschen [19]. The metric tensor is taken to be a
$C^{0}$ regular distribution444In $C^{1}$ coordinates, the continuity of the
first fundamental form is equivalent to the continuity of the spacetime metric
on the shell. See Clarke and Dray [8] as well as the comments in [10, 13] for
proof., from which it follows that the connection is allowed to have
discontinuities and the curvature tensor may contain a delta function term.
The field equations then impose a relation between the singular part of the
Einstein tensor and a singular contribution to the energy-momentum tensor,
which is Lanczos’ equation. If the metric tensor were allowed to be
discontinuous, the connection would pick up a delta function term, and the
curvature tensor (quadratic in the connection) would involve products of delta
functions, which are ill-defined. This imposes the continuity of the metric as
a junction condition. The intrinsic method has been used by Mars [12] as an
application of his concept of _hypersurface data_. He abstracted the
properties of rigged hypersurfaces by defining data on an arbitrary
hypersurface which may correspond to data specified by a rigging when the
hypersurface is embedded in a pseudo-Riemannian space. The purpose has been to
open the road for initial value problems in GR for any possible initial
hypersurface, however through the use of the rigged analogues of the usual
constraint equations, it becomes possible to formulate shells in a purely
intrinsic manner with no need for even embedding the hypersurface.

The shell equations have also been obtained via variational methods [20, 21,
22]. This is particularily useful for braneworld scenarios [21], where the
Lanczos equation on the brane is a part of the equations of motion and thus
the brane and bulk dynamics arise from a unified variational principle. When
the variational formulation is followed, the combined shell + bulk dynamics
appear as the broken extremals [23] of a variational problem with the shell
equation being the natural boundary condition on the surface.

For a second order theory described by a first order Lagrangian, this is
straightforward. The Einstein-Hilbert Lagrangian on the other hand is second
order. Since a second order Lagrangian normally produces equations of motion
of order four, the boundary conditions pertinent to the variational problem
are that of a fourth order differential equation and require the fixing of
both the metric and its transverse derivative at the boundary. As the actual
field equations are only second order, fixing the transverse derivative would
overdetermine the field equations and this causes the variational problem to
become ill-defined [24]. When variational principles with only outer
boundaries are considered, a common method of solution [25] is to add a
boundary term (for example the Gibbons-Hawking-York term [26, 27], but other
boundary terms could be introduced at the price of also introducing additional
structures) to the Einstein-Hilbert action such that combined bulk + boundary
action requires the boundary conditions consistent with a first order
Lagrangian, and thus the variation problem becomes well-defined. After Parattu
et al[28] such boundary terms will be referred to as variational counterterms.
A shell may be considered as an interior boundary of the spacetime manifold
thus it is clear that some similar regularization procedure is needed to
obtain the correct results. One such way of regularization is to also add the
Gibbons-Hawking-York counterterms to the action at the shell [21, 22]. Another
which has been employed by Hajicek and Kijowski [20] is to consider the
Lagrangian itself as a distribution. Since the Lagrangian involves a curvature
tensor, it has a delta function term which is proportional the difference of
the Gibbons-Hawking-York terms as calculated from the two sides.

The shell equations and junction conditions for null and generic shells have
been in general derived via the distributional method, which is simple to
generalize. Senovilla [13] has also shown that pillbox integration can also be
adapted to the generic case, and the intrinsic method was already applicable
to generic shells. It seems however that not much attention has been given to
the variational method for hypersurfaces that are not timelike or spacelike.
Jezierski, Kijowski and Czuchry have [29] considered the variational treatment
of null shells, however they did not show that their results agree in the null
limit with the results of eg. Barrabes and Israel [9] or Mars and Senovilla
[10]. There is also an unaddressed issue that has been pointed out by Parattu
et al[30] when investigating counterterms on null boundaries. A variation in
the metric is a variation in the causality, and such variations do not
preserve the nullity of a hypersurface. The underlying reason is that in the
tangent space at a fixed point, null vectors form a topologically closed set
and every neighborhood of each null vector contains both timelike and
spacelike vectors. A general variation will push the initially null surface
off the lightcone. The same issue is not encountered in regards to timelike or
spacelike surfaces as timelike/spacelike vectors form open sets and each such
vector has a neighborhood that consists entirely of timelike/spacelike
vectors. It stands to reason that variational methods involving null surfaces
should be formulated in a way that can accomodate surfaces of arbitrary causal
type.

The purpose of this paper is thus to fill this gap in the literature and
provide a formulation of thin shells and junction conditions for GR through a
variational principle valid for a generic hypersurface. A natural question is
then why should one consider generic shells. One reason is that it is
beneficial to provide a unified formalism capable of encompassing timelike,
spacelike and null shells at the same time, rather than assuming the signature
from the beginning. As the example of the Barrabes-Israel formalism shows
(remarked in Footnote 2), such unified formulations tend to include the case
of non-pure hypersurfaces as well. Moreover, as argued before, even if one is
interested in null shells exclusively, the convenient setting for a
variational treatment of null hypersurfaces is the one which is applicable to
generic hypersurfaces equally. For another reason, non-pure hypersurfaces
themselves can appear in physically interesting situations. Some examples may
be found in [13]. To give one explicitly, the stationary limit surface of a
Kerr black hole is timelike almost everywhere but null at a set of measure
zero. If one wishes to obtain matching conditions for spacetime regions
separated by such hypersurfaces, one must incorporate signature-changing
hypersurfaces. For an application of matching non-pure hypersurfaces, we refer
to the works [31, 32, 33] by Mars, Senovilla and Vera on signature change on
brane worlds.

The primary motivation for the development of this work is the formulation of
thin shell equations in modified theories of gravitation. Thin shells have
already been considered in extended gravitational theories, for example in
[34] thin shells and junction conditions have been examined in Brans-Dicke
type scalar-tensor theories via the distributional formalism with the null and
non-null cases separately. A variational formalism has also been given but
only for the non-null case. In [35, 36], junction conditions have been
formulated in Gauss-Bonnet gravity for applications to Gauss-Bonnet brane
worlds via the variational formulation, but once again only for non-null
cases. Shells in higher order gravity have also been investigated in [37]
through the use of distributions. Higher order theories have qualitatively
different shell behaviour with so-called double layers - energy-momentum terms
proportional to the Dirac delta’s derivative - appearing.

The most general scalar-tensor theory with second-order field equations is
Horndeski’s theory (originally published as [38], but the most common form is
the equivalent ”DGSZ reformulation” [39]). Thin shell equations in Horndeski’s
theory have been found by Padilla and Sivanesan [40] through a variational
method valid only for non-null hyper surfaces.

In [41] we gave a formulation of null shells in a reduced class of Horndeski
theories via the distributional method and the qualitative form of these
equations differed greatly from those obtained by Padilla and Sivanesan. For a
more effective comparision it would have been beneficial to also follow a
variational approach, however no such method was found that would be valid for
generic shells, yet it is a valuable and often-used method for non-null
hypersurfaces. It is thus reasonable to first examine how the variational
formalism works for generic shells in GR before generalizing to more
complicated theories.

The main obstacle for such a formalism seems to be the lack of an appropriate
variational counterterm for generic boundaries, as the Gibbons-Hawking-York
term is valid only for timelike and spacelike surfaces. Counterterms valid for
null boundaries have been considered by Parattu et al[30] and extended to
piecewise smooth boundaries involving corner terms by Lehner et al[42]. This
formalism can be used when the boundary has separate timelike, null and
spacelike pieces but does not allow for a unified treatment or for cases when
the boundary has null points that do not form an entire segment (for example
the null point is isolated or the null points form a line, etc.). An
alternative formulation in terms of tetrads have also been given by Jubb et
al[43], which nonetheless shares the features of the formulation by Lehner et
alin that it is necessary to break the boundary into pieces of pure signatures
instead of giving a fully unified treatment.

However a unified counterterm has been provided recently also by Parattu et
alin [28], which is valid for any boundary hypersurface rigged with a
transversal vector field and reduces to the Gibbons-Hawking-York term in the
appropriate limit. Although the formulation has not been extended to corner
terms, we are mainly interested in smooth shells (as in the hypersurface
corresponding to the shell being smooth) and therefore this limitation of the
formalism does not affect our results. We show that this counterterm properly
regularizes the action at the shell and the equations derived in eg. Barrabes
and Israel [9], Mars and Senovilla [10] and Senovilla [13] via the
distributional method arise as the natural boundary conditions along the
hypersurface. To make contact with the alternative distributional
regularization procedure of Hajicek and Kijowski [20], it is also shown that
the singular part of the Lagrangian supported on a generic surface is
proportional to the difference of the counterterm of Parattu et aland thus it
leads to the same variational principle we obtain by adding the counterterms
manually. Finally, we also derive the correct shell equation via a first order
equivalent to the Einstein-Hilbert action where no regularization is
necessary. This is actually a special case of the regularization by
counterterms as such first order equivalents can be seen as the Einstein-
Hilbert action augmented by a different counterterm.

#### Outline of the paper:

In Section 2 we provide a short summary of the rigged hypersurface formalism
which will be used throughout this paper. In Section 3, several known
variational counterterms for the Einstein-Hilbert action are discussed
including the one recently proposed by Parattu et al[28] valid for generic
hypersurfaces. Some general properties of these counterterms are investigated.
The main part of the paper is Section 4, where the dynamics of thin shells are
formulated as a variational principle via three separate regularization
schemes. Variational counterterms are employed in Subsection 4.1,
distributional regularization is considered in Subsection 4.2 and the shell
equation is also derived from a first order action without the need for
regularization in Subsection 4.3. Some of the longer calculations are given in
A and B.

#### Notation:

The spacetime manifold is $D+1$ dimensional and is denoted $M$. Coordinates on
$M$ are $x^{\mu}$ with the greek indices running $\mu=0,1,...,D$. $\Sigma$ is
a hypersurface in $M$, that is a $D$ dimensional submanifold with coordinates
$y^{a}$ with latin indices $a,b,c,...$ taking the values $1,...,D$. Summation
convention on repeated indices is assumed. The metric tensor in $M$ is
$g_{\mu\nu}$, its determinant is $\mathfrak{g}$ and the volume form determined
by it is

$\mu_{g}=\sqrt{-\mathfrak{g}}dx^{0}\wedge...\wedge dx^{D}.$ (1)

Inner products with respect to the spacetime metric are denoted with dots in
indexless notation, eg. $X\cdot Y=X^{\mu}Y^{\nu}g_{\mu\nu}$. All manifolds are
assumed orientable and oriented.

##  2 Rigged hypersurfaces

In this section we review the formalism of rigged hypersurfaces, establishing
the notation to be used in the rest of the paper. We refer to the exposition
by Mars and Senovilla [10] as well as the works [12, 44] for proofs of the
statements made here. We also recover the limiting cases when the hypersurface
is timelike or spacelike and we derive the null limit.

###  2.1 Structures induced by the rigging

We consider a hypersurface $\Sigma$ in the $D+1$ dimensional manifold $M$. The
hypersurface is given locally by the embedding functions

$x^{\mu}=\Phi^{\mu}\left(y^{1},...,y^{D}\right),$ (2)

where the $y^{a}$ are the intrinsic coordinates of $\Sigma$. The derivatives

$e_{a}^{\mu}:=\frac{\partial\Phi^{\mu}}{\partial y^{a}}$ (3)

are interepreted as the components of the holonomic coordinate frame of
$\Sigma$, or from a more invariant point of view, the components of the
pushforward and pullback operations between $\Sigma$ and $M$. Without
introducing any extra structure, a vector $v^{\mu}$ defined at a point
$p\in\Sigma$ is tangential if it can be written in the form
$v^{\mu}=v^{a}e_{a}^{\mu}$ for some intrinsic hypersurface vector $v^{a}$.
Then $v^{\mu}$ is the pushforward of $v^{a}$. Thus, it is possible to decide
whether a contravariant vector (and thus a general contravariant tensor in an
index-by-index basis) is tangential or not. A covector $n_{\mu}$ defined at
some point $p\in\Sigma$ is normal if $n_{\mu}e_{a}^{\mu}=0$, that is it
annihilates all tangential vectors. The space of normal covectors at each
point is one dimensional. Thus it is meaningful to decide if a covariant
vector is normal or not. If $\omega_{\mu}$ is a covariant tensor at some
$p\in\Sigma$, its pullback to $\Sigma$ is the hypersurface covector
$\omega_{a}=\omega_{\mu}e_{a}^{\mu}$ (this notion is extended to all covariant
tensors index-by-index).

The induced metric or first fundamental form on $\Sigma$ is the pullback

$h_{ab}=g_{\mu\nu}e_{a}^{\mu}e_{b}^{\nu}.$ (4)

The point $p\in\Sigma$ is a null point of the hypersurface if and only if
$h_{ab}\left(p\right)$ is a singular matrix. Since we allow for null points
and thus non-invertible induced metrics, we do not raise or lower latin
indices.

To proceed, we need to introduce a vector field $\ell^{\mu}$ along $\Sigma$,
which is nowhere tangential (nor zero). We call this a choice of rigging and
the pair $\left(\Sigma,\ell\right)$ is a rigged hypersurface. The set
$\left(\ell,e_{1},...,e_{D}\right)$ is then a frame of $M$ along $\Sigma$. The
choice of rigging selects a unique normal covector field $n_{\mu}$ which
satisfies

$n_{\mu}\ell^{\mu}=1.$ (5)

Then the set $\left(n,\vartheta^{1},...,\vartheta^{D}\right)$ is the dual
frame of $\left(\ell,e_{1},...,e_{D}\right)$, where the covector fields (along
$\Sigma$) $\vartheta_{\mu}^{a}$ are uniquely determined by the duality
relations

$\vartheta_{\mu}^{a}\ell^{\mu}=0,\quad\vartheta_{\mu}^{a}e_{b}^{\mu}=\delta_{b}^{a}.$
(6)

Using $\vartheta_{\mu}^{a}$, given a hypersurface covector $\omega_{a}$, we
can create a spacetime covector $\omega_{\mu}=\vartheta_{\mu}^{a}\omega_{a}$
which satisfies $\omega_{a}=e_{a}^{\mu}\omega_{\mu}$ and
$\omega_{\mu}\ell^{\mu}=0$. Likewise, we can project a spacetime vector
$v^{\mu}$ into $\Sigma$ as $v_{\parallel}^{a}=v^{\mu}\vartheta_{\mu}^{a}$, and
also obtain a direct projection operator $P_{\ \nu}^{\mu}$ by pushing forward
$v_{\parallel}^{a}$, i.e.

$v_{\parallel}^{\mu}=v_{\parallel}^{a}e_{a}^{\mu}=v^{\nu}\vartheta_{\nu}^{a}e_{a}^{\mu}=P_{\
\nu}^{\mu}v^{\nu},$ (7)

with

$P_{\
\nu}^{\mu}=e_{a}^{\mu}\vartheta_{\nu}^{a}=\delta_{\nu}^{\mu}-\ell^{\mu}n_{\nu}.$
(8)

With respect to these bases, the spacetime metric and inverse metric can be
expressed as

$\displaystyle g_{\mu\nu}$
$\displaystyle=\ell^{2}n_{\mu}n_{\nu}+\lambda_{a}\left(n_{\mu}\vartheta_{\nu}^{a}+\vartheta_{\mu}^{a}n_{\nu}\right)+h_{ab}\vartheta_{\mu}^{a}\vartheta_{\nu}^{b},$
$\displaystyle g^{\mu\nu}$
$\displaystyle=n^{2}\ell^{\mu}\ell^{\nu}+\nu^{a}\left(\ell^{\mu}e_{a}^{\nu}+e_{a}^{\mu}\ell^{\nu}\right)+h_{\ast}^{ab}e_{a}^{\mu}e_{b}^{\nu},$
(9)

where the elements that appear here are given explicitly as

$\displaystyle\ell^{2}$
$\displaystyle=\ell_{\mu}\ell^{\mu}=\ell\cdot\ell,\quad
n^{2}=n_{\mu}n^{\mu}=n\cdot n,$ $\displaystyle\lambda_{a}$
$\displaystyle=\ell_{\mu}e_{a}^{\mu}=\ell\cdot
e_{a},\quad\nu^{a}=n^{\mu}\vartheta_{\mu}^{a}=n\cdot\vartheta^{a},$
$\displaystyle h_{\ast}^{ab}$
$\displaystyle=g^{\mu\nu}\vartheta_{\mu}^{a}\vartheta_{\nu}^{b}=\vartheta^{a}\cdot\vartheta^{b}.$
(10)

In particular, $h_{\ast}^{ab}$ may be seen as a pseudo-inverse to $h_{ab}$.

The choice of rigging also gives a volume form

$\mu_{\ell,g}=\rho_{\ell,g}dy^{1}\wedge...\wedge dy^{D}$ (11)

on $\Sigma$ with

$\rho_{\ell,g}=\sqrt{-\mathfrak{g}}\ell^{\mu}\pi_{\mu\mu_{1}...\mu_{D}}e_{1}^{\mu_{1}}...e_{D}^{\mu_{D}},$
(12)

where $\pi_{\mu\mu_{1}...\mu_{D}}$ is the $D+1$ dimensional Levi-Civita
symbol. This particular volume element is such that if $\Omega\subseteq M$ is
a compact $D+1$ dimensional domain of integration, whose boundary
$\partial\Omega$ is rigged with an outward pointing transversal $\ell^{\mu}$,
Gauss’ theorem takes the form

$\int_{\Omega}\nabla_{\mu}X^{\mu}\,\mu_{g}=\oint_{\partial\Omega}n_{\mu}X^{\mu}\,\mu_{\ell,g},$
(13)

where $n_{\mu}$ is the normal adapted to the rigging, i.e.
$n_{\mu}\ell^{\mu}=1$. Further properties of the volume form may be found in
[10, 44].

Extrinsic curvature-type quantities may be obtained by differentiating the
frame vectors in the tangential directions as

$\displaystyle\chi_{ab}$
$\displaystyle=-n_{\nu}e_{a}^{\mu}\nabla_{\mu}e_{b}^{\nu}=e_{a}^{\mu}e_{b}^{\nu}\nabla_{\mu}n_{\nu},$
(14) $\displaystyle\varphi_{a}$
$\displaystyle=n_{\nu}e_{a}^{\mu}\nabla_{\mu}\ell^{\nu}=-e_{a}^{\mu}\ell^{\nu}\nabla_{\mu}n_{\nu},$
(15) $\displaystyle\psi_{a}^{b}$
$\displaystyle=e_{a}^{\mu}\vartheta_{\nu}^{b}\nabla_{\mu}\ell^{\nu}=-e_{a}^{\mu}\ell^{\nu}\nabla_{\mu}\vartheta_{\nu}^{b}.$
(16)

These are all hypersurface tensors and $\chi_{ab}$ is symmetric. For thin
shells and junction conditions it is also useful to define

$H_{ab}=e_{a}^{\mu}e_{b}^{\nu}\nabla_{\mu}\ell_{\nu}=\psi_{a}^{c}h_{bc}+\varphi_{a}\lambda_{b},$
(17)

which is non-symmetric in general and is not independent of the triplet
$\left(\chi,\psi,\varphi\right)$. However, it will turn out that this quantity
is what most naturally generalizes the extrinsic curvature to thin shells and
will play an important role. For $H_{ab}$ we make an exception to our
convention not to raise latin indices and define

$\displaystyle H_{\ b}^{a}$ $\displaystyle=h_{\ast}^{ac}H_{cb},\quad H_{a}^{\
b}=h_{\ast}^{bc}H_{ac},\quad H^{ab}=h_{\ast}^{ac}h_{\ast}^{bd}H_{cd},$
$\displaystyle H$ $\displaystyle=H_{a}^{a}=H_{ab}h_{\ast}^{ab}.$ (18)

A connection-type quantity $\gamma_{\ ab}^{c}$ is also given on $\Sigma$ by

$\gamma_{\
ab}^{c}=\vartheta_{\nu}^{c}e_{a}^{\mu}\nabla_{\mu}e_{b}^{\nu}=-e_{a}^{\mu}e_{b}^{\nu}\nabla_{\mu}\vartheta_{\nu}^{c}.$
(19)

This connection is torsionless but is not metric compatible in general.

###  2.2 Transformations between riggings

The choice of rigging $\ell^{\mu}$ along a hypersurface $\Sigma\subseteq M$ is
not unique and it may be subjected to two kinds of transformations. The first
is a tangential shift

$\ell^{\prime\mu}=\ell^{\mu}+T^{a}e_{a}^{\mu},$ (20)

where $T^{a}$ is an arbitrary tangent vector field to $\Sigma$, and the second
kind is a rescaling

$\bar{\ell}^{\mu}=\alpha\ell^{\mu},$ (21)

where $\alpha$ is a function along $\Sigma$ that is nowhere vanishing. These
transformations form a group parametrized by $D+1$ functions whose structure
has been analyzed in [44]. The quantities associated with rigged hypersurfaces
transform under the shift as [10, 12]

$\displaystyle\ell^{\prime\mu}$ $\displaystyle=\ell^{\mu}+T^{a}e_{a}^{\mu},$
$\displaystyle\vartheta_{\mu}^{\prime a}$
$\displaystyle=\vartheta_{\mu}^{a}-T^{a}n_{\mu},$ $\displaystyle\ell^{\prime
2}$ $\displaystyle=\ell^{2}+2T^{a}\lambda_{a}+h_{ab}T^{a}T^{b},$
$\displaystyle\lambda_{a}^{\prime}$ $\displaystyle=\lambda_{a}+h_{ab}T^{b},$
$\displaystyle\nu^{\prime a}$ $\displaystyle=\nu^{a}-n^{2}T^{a},$
$\displaystyle h_{\ast}^{\prime ab}$
$\displaystyle=h^{ab}-\nu^{a}T^{b}-T^{a}\nu^{b}+n^{2}T^{a}T^{b},$
$\displaystyle\varphi_{a}^{\prime}$
$\displaystyle=\varphi_{a}-\chi_{ab}T^{b},$ $\displaystyle\psi_{a}^{\prime b}$
$\displaystyle=\psi_{a}^{b}+\varphi_{a}T^{b}+D_{a}T^{b}+\chi_{ac}T^{b}T^{c},$
$\displaystyle\gamma_{ab}^{\prime c}$
$\displaystyle=\gamma_{ab}^{c}+\chi_{ab}T^{c},$ $\displaystyle
H_{ab}^{\prime}$
$\displaystyle=H_{ab}+D_{a}T^{c}h_{bc}-\chi_{ac}\lambda_{b}T^{c},$ (22)

while $e_{a}^{\mu}$, $n_{\mu}$, $h_{ab}$, $\chi_{ab}$ and $\mu_{\ell,g}$ are
invariant. Under a rescaling, the transformations are

$\displaystyle\bar{\ell}^{\mu}$ $\displaystyle=\alpha\ell^{\mu},$
$\displaystyle\bar{n}_{\mu}$ $\displaystyle=\alpha^{-1}n_{\mu},$
$\displaystyle\bar{\ell}^{2}$ $\displaystyle=\alpha^{2}\ell^{2},$
$\displaystyle\bar{n}^{2}$ $\displaystyle=\alpha^{-1}n^{2},$
$\displaystyle\bar{\lambda}_{a}$ $\displaystyle=\alpha\lambda_{a},$
$\displaystyle\bar{\nu}^{a}$ $\displaystyle=\alpha^{-1}\nu^{a}$
$\displaystyle\bar{\chi}_{ab}$ $\displaystyle=\alpha^{-1}\chi_{ab},$
$\displaystyle\bar{\varphi}_{a}$
$\displaystyle=\varphi_{a}+\partial_{a}\ln\alpha,$
$\displaystyle\bar{\psi}_{a}^{b}$ $\displaystyle=\alpha\psi_{a}^{b},$
$\displaystyle\bar{H}_{ab}$ $\displaystyle=\alpha
H_{ab}+\partial_{a}\alpha\lambda_{b},$ $\displaystyle\bar{\mu}_{\ell,g}$
$\displaystyle=\alpha\mu_{\ell,g},$ (23)

while $e_{a}^{\mu}$, $\vartheta_{\mu}^{a}$, $h_{ab}$, $h_{\ast}^{ab}$ and
$\gamma_{ab}^{c}$ are invariant. Note that since the volume element
$\mu_{\ell,g}$ is invariant under shifts, the definition of $\mu_{\ell,g}$
essentially depends on that of the normal $n_{\mu}$ only. Thus if one has a
preferred normal along a hypersurface, the scaling of the normal already fixes
the volume element without the need to choose a rigging explicitly.

###  2.3 Pseudo-Riemannian limit of rigged hypersurfaces

The usual formalism of timelike and spacelike (collectively, pseudo-
Riemannian) hypersurfaces may be obtained from the rigged formalism by making
a particular choice of rigging $\ell^{\mu}$. We assume that $\Sigma$ is
timelike or spacelike and set

$\epsilon=\pm 1=\cases{+1&$\Sigma$ is timelike\\\ -1&$\Sigma$ is spacelike\\\
}$ (24)

to allow for both cases to be considered simultaneously. The the induced
metric $h_{ab}$ is nondegenerate throughout $\Sigma$, its inverse $h^{ab}$
exists and we raise and lower latin indices with $h^{ab}$ and $h_{ab}$
respectively. Normal vectors are everywhere transversal, therefore we take as
the rigging

$\ell^{\mu}=\hat{n}^{\mu}$ (25)

the unit normal (i.e. $\hat{n}\cdot\hat{n}=\epsilon$) to $\Sigma$, which is
unique up to sign. With this particular choice of the rigging, the normal
associated to the rigging is

$n_{\mu}=\epsilon\hat{n}_{\mu}.$ (26)

We will only use $\hat{n}$ and keep track of the $\epsilon$s that appear. The
rest of the quantities become

$\displaystyle\vartheta_{\mu}^{a}$
$\displaystyle=e_{\mu}^{a}=h^{ab}g_{\mu\nu}e_{b}^{\nu},$
$\displaystyle\ell^{2}$ $\displaystyle=n^{2}=\epsilon,$
$\displaystyle\lambda_{a}$ $\displaystyle=\nu^{a}=0,$ $\displaystyle
h_{\ast}^{ab}$ $\displaystyle=h^{ab},$ $\displaystyle\chi_{ab}$
$\displaystyle=\epsilon K_{ab},$ $\displaystyle\varphi_{a}$ $\displaystyle=0,$
$\displaystyle\psi_{a}^{b}$ $\displaystyle=K_{a}^{b},$ $\displaystyle H_{ab}$
$\displaystyle=K_{ab},$ $\displaystyle\mu_{\ell,g}$
$\displaystyle=\mu_{h}=\sqrt{-\epsilon\mathfrak{h}}dy^{1}\wedge...\wedge
dy^{D},$ (27)

where

$K_{ab}=e_{a}^{\mu}e_{b}^{\nu}\nabla_{\mu}\hat{n}_{\nu}=\frac{1}{2}e_{a}^{\mu}e_{b}^{\nu}\mathcal{L}_{\hat{n}}g_{\mu\nu}$
(28)

is the usual extrinsic curvature and $\mathfrak{h}=\det\left(h_{ab}\right)$ is
the determinant of the induced metric. The connection $\gamma_{\ ab}^{c}$
becomes the Levi-Civita connection of the induced metric $h_{ab}$ with

$\gamma_{\
ab}^{c}=\frac{1}{2}h^{cd}\left(\partial_{a}h_{bd}+\partial_{b}h_{ad}-\partial_{d}h_{ab}\right)$
(29)

and Gauss’ theorem takes the form

$\int_{\Omega}\nabla_{\mu}X^{\mu}\,\mu_{g}=\oint_{\partial\Omega}\epsilon\hat{n}_{\mu}X^{\mu}\,\mu_{h},$
(30)

where $\hat{n}^{\mu}$ is the outward pointing unit normal to $\partial\Omega$.

###  2.4 Null limit of rigged hypersurfaces

Suppose now that $\Sigma$ is null. There is no universally preferred
convention here for the rigging, however the null rigging used by eg. Poisson
[11] is a useful choice and we present it here. If $\Sigma$ is null then any
normal field $n_{\mu}$ is also null and is tangential to $\Sigma$. Moreover it
satisfies the geodesic equation

$\left(\nabla_{n}n\right)^{\mu}=\kappa n^{\mu}$ (31)

for some _non-affinity function_ $\kappa$. We may then set up coordinates
$\left(y^{a}\right)=\left(r,\theta^{A}\right)$ on $\Sigma$ ($A,B,...=2,...,D$)
such that

$n^{\mu}=\left(\frac{\partial}{\partial r}\right)^{\mu},$ (32)

and choose a null $\ell^{\mu}$ rigging which satisfies

$\ell^{\mu}\ell_{\mu}=0,\quad\ell^{\mu}n_{\mu}=1,\quad\ell_{\mu}e_{A}^{\mu}=0,$
(33)

where

$e_{A}^{\mu}=\left(\frac{\partial}{\partial\theta^{A}}\right)^{\mu}$ (34)

are the rest of the basis fields, necessarily spacelike. The functions

$q_{AB}=e_{A}^{\mu}e_{B}^{\nu}g_{\mu\nu}$ (35)

are then the components of the spacelike induced metric on the slices
$r=\mathrm{const}$. They are also the only nonvanishing components of the
induced metric on the entire surface, i.e.

$\left(h_{ab}\right)=\left(\begin{array}[]{*{20}{c}}0&0\\\
0&\left(q_{AB}\right)\end{array}\right).$ (36)

The $D-1$-metric $q_{AB}$ does possess an inverse, denoted $q^{AB}$ and the
capital latin indices are raised and lowered via $q^{AB}$ and $q_{AB}$
respectively. The most important extrinsic curvature quantity in this case is
$H_{ab}$, which is now symmetric and we split it as $H_{11}$, $H_{1A}$ and
$H_{AB}$. We have

$\displaystyle H_{11}$
$\displaystyle=n^{\mu}n^{\nu}\nabla_{\mu}\ell_{\nu}=-n^{\mu}\ell_{\nu}\nabla_{\mu}n^{\nu}=-\kappa,$
$\displaystyle H_{1A}$
$\displaystyle=H_{A1}=e_{A}^{\mu}n^{\nu}\nabla_{\mu}\ell_{\nu},$
$\displaystyle H_{AB}$
$\displaystyle=e_{A}^{\mu}e_{B}^{\nu}\nabla_{\mu}\ell_{\nu}.$ (37)

We may express most of the quantities with $H_{ab}$ as

$\displaystyle\varphi_{1}$
$\displaystyle=n^{\mu}n_{\nu}\nabla_{\mu}\ell^{\nu}=-\kappa=H_{11},$
$\displaystyle\varphi_{A}$
$\displaystyle=e_{A}^{\mu}n_{\nu}\nabla_{\mu}\ell^{\nu}=H_{A1},$
$\displaystyle\psi_{1}^{1}$
$\displaystyle=n^{\mu}\ell_{\nu}\nabla_{\mu}\ell^{\nu}=\frac{1}{2}n^{\mu}\nabla_{\mu}\left(\ell_{\nu}\ell^{\nu}\right)=0,$
$\displaystyle\psi_{A}^{1}$
$\displaystyle=e_{A}^{\mu}\ell_{\nu}\nabla_{\mu}\ell^{\nu}=0,$
$\displaystyle\psi_{1}^{A}$
$\displaystyle=n^{\mu}e_{\nu}^{A}\nabla_{\mu}\ell^{\nu}=H_{1}^{A},$
$\displaystyle\psi_{A}^{B}$ $\displaystyle=H_{A}^{B}.$ (38)

The primary exception is $\chi_{ab}$, which is

$\displaystyle\chi_{11}$
$\displaystyle=n^{\mu}n^{\nu}\nabla_{\mu}n_{\nu}=\kappa n^{\nu}n_{\nu}=0,$
$\displaystyle\chi_{1A}$
$\displaystyle=n^{\mu}e_{A}^{\nu}\nabla_{\mu}n_{\nu}=\kappa
e_{A}^{\nu}n_{\nu}=0,$ $\displaystyle\chi_{AB}$
$\displaystyle=e_{A}^{\mu}e_{B}^{\nu}\nabla_{\mu}n_{\nu},$ (39)

and is thus not expressible with $H_{ab}$. As only tangential derivatives of
tangential vectors are taken, when thin shells are involved, the jump of this
quantity always vanishes.

Finally, with respect to the frame $\left(\ell,n,e_{A}\right)$ the full metric
tensor has components

$\left(g_{\mu\nu}\right)=\left(\begin{array}[]{*{20}{c}}0&1&0\\\ 1&0&0\\\
0&0&\left(q_{AB}\right)\end{array}\right),$ (40)

from which it follows that in this frame

$\mathfrak{g}=-\mathfrak{q},$ (41)

where $\mathfrak{q}=\det\left(q_{AB}\right)$. The volume element can be thus
written as

$\mu_{\ell,g}=\mu_{q}=\sqrt{\mathfrak{q}}dr\wedge d\theta^{2}...\wedge
d\theta^{D}.$ (42)

Note that while it appears that the volume element $\mu_{q}$ is canonically
given, it does depend on the way the manifold $\Sigma$ has been sliced into
spacelike $D-1$-surfaces.

##  3 Variational counterterms

The Einstein-Hilbert action over $M$ is

$S_{\mathrm{EH}}=\frac{1}{2\varkappa}\int_{M}R\,\mu_{g},$ (43)

where $\varkappa=8\pi G$. The integrand is second order in the metric while
its Euler-Lagrange equations are also second order. If we assume the boundary
$\partial M$ has been rigged by an outward pointing vector $\ell^{\mu}$, we
may write its variation in generic form as

$\delta S_{\mathrm{EH}}=-\int_{M}\frac{1}{2\varkappa}G^{\mu\nu}\delta
g_{\mu\nu}\,\mu_{g}+\int_{\partial M}\left(Y^{\mu\nu}\delta
g_{\mu\nu}+Y^{\mu\nu,a}\delta g_{\mu\nu,a}+Y_{\ell}^{\mu\nu}\delta
g_{\mu\nu,\ell}\right)\mu_{g,\ell},$ (44)

where $\delta g_{\mu\nu,a}=e_{a}^{\kappa}\partial_{\kappa}\delta g_{\mu\nu}$
are the tangential derivatives of the metric variation, $\delta
g_{\mu\nu,\ell}=\ell^{\kappa}\partial_{\kappa}\delta g_{\mu\nu}$ is the
transversal derivative and $Y^{\mu\nu}$, $Y^{\mu\nu,a}$ and
$Y_{\ell}^{\mu\nu}$ are the appropriate coefficients that appear on the
boundary. Imposing Dirichlet boundary conditions $\left.\delta
g_{\mu\nu}\right|_{\partial M}=0$ gets rid of the first two terms on the
boundary, but not the third. On the other hand demanding the transversal
derivatives to also vanish would overdetermine the field equations. In order
to make the variational problem well-defined a variational counterterm

$B=\int_{\partial M}\mathcal{B}\left(g,\left(\partial
g\right)_{\parallel},\left(\partial g\right)_{\ell}\right)d^{D}y$ (45)

is added to the action, where $\left(\partial g\right)_{\parallel}$ and
$\left(\partial g\right)_{\ell}$ are schematic notations for the tangential
and transversal derivatives respectively. If the integrand $\mathcal{B}$
satisfies

$\frac{\partial\mathcal{B}}{\partial
g_{\mu\nu,\ell}}=-Y_{\ell}^{\mu\nu}\rho_{g,\ell},$ (46)

then it follows that imposing the usual Dirichlet condition $\left.\delta
g_{\mu\nu}\right|_{\partial M}=0$ on the combined action $S_{\mathrm{EH}}+B$
will get rid of all boundary terms. The variational counterterm is not unique,
however if $\mathcal{B}$ and $\mathcal{B}^{\prime}$ are both integrands of
variational counterterms, their derivatives with respect to $g_{\mu\nu,\ell}$
must be the same function $-Y_{\ell}^{\mu\nu}\rho_{g,\ell}$ and thus the
difference $\mathcal{B}-\mathcal{B}^{\prime}$ is a function of $g$ and
$\left(\partial g\right)_{\parallel}$ only. This result will be of
significance for thin shells.

There are several counterterms known for the Einstein-Hilbert action:

#### The Gibbons-Hawking-York counterterm:

When the boundary $\partial M$ consists of pseudo-Riemannian pieces only, the
appropriate rigging (see Section 2.3) can be chosen. The counterterm is

$B_{\mathrm{GHY}}=\frac{\epsilon}{\varkappa}\int_{\partial M}K\,\mu_{h}.$ (47)

Its validity follows from the variational formula [45]

$\delta S_{\mathrm{EH}}=-\frac{1}{2\varkappa}\int_{M}G^{\mu\nu}\delta
g_{\mu\nu}\,\mu_{g}+\frac{\epsilon}{2\varkappa}\int_{\partial
M}\left(Kh^{ab}-K^{ab}\right)\delta
h_{ab}\,\mu_{h}-\frac{\epsilon}{\varkappa}\delta\left(\int_{\partial
M}K\,\mu_{h}\right).$ (48)

The first boundary term involves only tangential derivatives of the metric and
the second term - which contains normal derivatives - is an exact variation.
Adding this term to the action with an opposite sign will ensure that fixing
the metric without fixing its derivatives on the boundary makes all remaining
boundary terms vanish.

#### The Einstein counterterm:

We assume that the manifold $M$ is covered by the domain of a chosen (and
fixed) coordinate chart $x^{\mu}$. Let us also take an outward pointing normal
$n_{\mu}$ along $\partial M$, and let $\mu_{g,n}$ denote the corresponding
volume element obtained via _any_ rigging $\ell^{\mu}$ which satisfies
$\ell^{\mu}n_{\mu}=1$. The Einstein counterterm is then defined as

$B_{\mathrm{E}}=-\frac{1}{2\varkappa}\int_{\partial
M}n_{\kappa}w^{\kappa}\,\mu_{g,n},$ (49)

where

$w^{\kappa}=\Gamma_{\ \mu\nu}^{\kappa}g^{\mu\nu}+\Gamma_{\
\nu\mu}^{\nu}g^{\kappa\mu}.$ (50)

This expression is naturally defined in the interior of $M$ as well, and by
Gauss’ theorem

$B_{E}=-\frac{1}{2\varkappa}\int_{M}\nabla_{\kappa}w^{\kappa}\,\mu_{g},$ (51)

where the covariant derivative treats $w^{\kappa}$ as if it was a vector field
(the rationale behind this is that we may consider $\partial_{\mu}$ to be a
locally defined connection associated to the chart $x^{\mu}$, and from this
point of view the connection coefficients $\Gamma_{\ \mu\nu}^{\kappa}$ are
tensor components - the components of the difference tensor between $\nabla$
and $\partial$). Decomposing the scalar curvature as

$R=\nabla_{\kappa}\left(\Gamma_{\
\mu\nu}^{\kappa}g^{\mu\nu}+g^{\kappa\mu}\Gamma_{\
\nu\mu}^{\nu}\right)+\left(\Gamma_{\ \mu\rho}^{\kappa}\Gamma_{\
\kappa\nu}^{\rho}-\Gamma_{\ \kappa\rho}^{\kappa}\Gamma_{\
\mu\nu}^{\rho}\right)g^{\mu\nu},$ (52)

one obtains

$S_{\mathrm{EH}}+B_{\mathrm{E}}=\frac{1}{2\varkappa}\int_{M}g^{\mu\nu}\left(\Gamma_{\
\mu\rho}^{\kappa}\Gamma_{\ \kappa\nu}^{\rho}-\Gamma_{\
\kappa\rho}^{\kappa}\Gamma_{\ \mu\nu}^{\rho}\right)\mu_{g},$ (53)

which is Einstein’s first order, noncovariant $\Gamma\Gamma$-action [46, 47].
Since it is first order, fixing the metric at the boundary is sufficient to
eliminate all boundary terms. The Einstein counterterm is not unique in the
sense that different coordinate systems will produce different Einstein
counterterms, as it is clear from the lack of covariance of (50).

#### The background connection counterterm:

We can also introduce an arbitrary torsionless connection
$\bar{\nabla}_{\mu}$. Quantities calculated from $\bar{\nabla}_{\mu}$ are
denoted with an overbar. Let $n_{\mu}$ be any outward pointing normal to the
boundary $\partial M$ and $\mu_{g,n}$ the associated volume element. The
background connection counterterm is

$B_{\mathrm{BC}}=-\frac{1}{2\varkappa}\int_{\partial
M}\left(n_{\kappa}\Delta_{\ \mu\nu}^{\kappa}g^{\mu\nu}+n^{\mu}\Delta_{\
\nu\mu}^{\nu}\right)\mu_{g,n},$ (54)

where $\Delta_{\ \mu\nu}^{\kappa}=\Gamma_{\ \mu\nu}^{\kappa}-\bar{\Gamma}_{\
\mu\nu}^{\kappa}$ is the difference tensor. The Einstein counterterm is
reproduced if $M$ fits into a single coordinate domain and we choose
$\bar{\nabla}_{\mu}=\partial_{\mu}$. The term $\Delta_{\
\mu\nu}^{\kappa}g^{\mu\nu}+\Delta_{\ \nu\mu}^{\nu}g^{\kappa\mu}$ is once again
defined on the entire manifold $M$ and after using Gauss’ theorem we get

$S_{\mathrm{EH}}+B_{\mathrm{BC}}=\frac{1}{2\varkappa}\int_{M}\left[\bar{R}+g^{\mu\nu}\left(\Delta_{\
\mu\rho}^{\kappa}\Delta_{\ \kappa\nu}^{\rho}-\Delta_{\
\kappa\rho}^{\kappa}\Delta_{\ \mu\nu}^{\rho}\right)\mu_{g}\right].$ (55)

Since $\bar{R}=g^{\mu\nu}\bar{R}_{\mu\nu}$ is the scalar curvature of the
nondynamical background connection $\bar{\nabla}_{\mu}$, this action is also
first order, from which immediately follows that fixing the metric at the
boundary removes all boundary terms. Unlike the Einstein counterterm, the
background connection counterterm is globally defined and both the counterterm
and the resulting first order action are covariant. However the counterterm
and action both contain an unphysical background field. This counterterm is
also non-unique, as it depends on the connection chosen as the background.

#### The rigged counterterm:

This counterterm has been introduced by Parattu et al[28] as a generalization
of the Gibbons-Hawking-York term valid for hypersurfaces of arbitrary causal
type. We fix an outward pointing rigging $\ell^{\mu}$ along the boundary
$\partial M$. The rigged counterterm is [28]

$B_{\mathrm{R}}=\frac{1}{\varkappa}\int_{\partial M}P_{\
\nu}^{\mu}\nabla_{\mu}n^{\nu}\,\mu_{\ell,g},$ (56)

where $P_{\ \nu}^{\mu}=\delta_{\nu}^{\mu}-\ell^{\mu}n_{\nu}$ is a tangential
projector that removes the $\ell$-directed parts of vectors. Parattu et aldid
not employ the formalism of rigged hypersurfaces, therefore this counterterm
appeared in terms of spacetime, rather than hypersurface quantities. We
rewrite it via $P_{\ \nu}^{\mu}=e_{a}^{\mu}\vartheta_{\nu}^{a}$ and
$\vartheta^{\mu a}=\nu^{a}\ell^{\mu}+h_{\ast}^{ab}e_{b}^{\mu}$ as

$\displaystyle P_{\ \nu}^{\mu}\nabla_{\mu}n^{\nu}$
$\displaystyle=e_{a}^{\mu}\vartheta_{\nu}^{a}\nabla_{\mu}n^{\nu}=e_{a}^{\mu}\left(h_{\ast}^{ab}e_{b}^{\nu}+\nu^{a}\ell^{\nu}\right)\nabla_{\mu}n_{\nu}$
(57) $\displaystyle=\chi_{ab}h_{\ast}^{ab}-\varphi_{a}\nu^{a},$

thus the counterterm has the equivalent expression

$B_{\mathrm{R}}=\frac{1}{\varkappa}\int_{\partial
M}\left(\chi_{ab}h_{\ast}^{ab}-\varphi_{a}\nu^{a}\right)\mu_{\ell,g},$ (58)

a form resembling the Gibbons-Hawking-York counterterm with $\chi_{ab}$ and
$\varphi_{a}$ playing the role of the extrinsic curvature. From (58) it can be
seen that when the boundary is pseudo-Riemannian, choosing
$\ell^{\mu}=\hat{n}^{\mu}$ gives (see Section 2.3) $\varphi_{a}=0$,
$\chi_{ab}=\epsilon K_{ab}$ and $h_{\ast}^{ab}=h^{ab}$, thus the rigged
counterterm reproduces the Gibbons-Hawking-York term in this limit.

In the presence of a boundary $\partial M$ of any causal type, equipped with a
rigging, the variational formula (48) is replaced by [28]

$\delta S_{\mathrm{EH}}=-\frac{1}{2\varkappa}\int_{M}G^{\mu\nu}\delta
g_{\mu\nu}\mu_{g}+\frac{1}{2\varkappa}\int_{\partial M}\Pi^{\mu\nu}\delta
g_{\mu\nu}\mu_{\ell,g}-\frac{1}{\varkappa}\int_{\partial M}\delta\left(P_{\
\nu}^{\mu}\nabla_{\mu}n^{\nu}\mu_{\ell,g}\right),$ (59)

where

$\Pi^{\mu\nu}=g^{\mu\nu}\left(P_{\
\sigma}^{\rho}\nabla_{\rho}n^{\sigma}\right)-\nabla^{(\mu}n^{\nu)}-\nabla_{\rho}\ell^{\rho}n^{\mu}n^{\nu}.$
(60)

We remark that this quantity involves the covariant derivative of the normal
vector and thus depends on the extension of it to a neighborhood of the
boundary. As the calculations in A show, the expression is independent of the
extension of the normal field. The quantity $\Pi^{\mu\nu}$ will also be
decomposed in terms of hypersurface quantities in Subsection 4.1. The sign of
the last term has been corrected as compared to the corresponding result in
[28]. The boundary term that results from the variation of
$S_{\mathrm{EH}}+B_{\mathrm{R}}$ is proportional to $\delta g_{\mu\nu}$ and
vanishes when the metric is fixed on the boundary. The rigged counterterm is
not unique, different choices of rigging will give different counterterms.

##  4 Variational formalism of thin shells

We assume that the hypersurface $\Sigma$ partitions the spacetime manifold $M$
into two domains $M^{+}$ and $M^{-}$. These domains are manifolds with
boundaries and their interiors are disjoint. For simplicity we assume that $M$
has no outer boundary, which implies that $\partial M^{+}=\partial
M^{-}=\Sigma$ (this notation currently ignores orientations). The formalism
may be equally well used in the presence of outer boundaries, but including
them would needlessly complicate the notation and outer boundaries play no
role in our formalism anyway.

The regions $M^{+}$ and $M^{-}$ are distinct as manifolds with smooth555Or is
at least $C^{3}$ to ensure both the field equations and the Bianchi identities
exist as regular functions. metrics $g_{\mu\nu}^{+}$ and $g_{\mu\nu}^{-}$
respectively. Coordinate systems $x_{+}^{\mu}$ and $x_{-}^{\mu}$ are employed
which need not satisfy any matching conditions at $\Sigma$. As per the
analysis of Clarke and Dray [8] (also comments made in [10, 13]), corrected
and extended for the case of hypersurfaces with null points by Mars, Senovilla
and Vera [32], the conditions for the existence of a $C^{1}$ structure on $M$
is that the induced metrics $h_{ab}^{+}$ and $h_{ab}^{-}$ agree on $\Sigma$,
and in case $\Sigma$ is not timelike or spacelike, there is a pair of rigging
vectors $\ell_{+}^{\mu}$ and $\ell_{-}^{\mu}$ along $\Sigma$ such that
$\ell_{+}$ and $\ell_{-}$ both point towards $M^{+}$ (or both towards $M^{-}$,
depending on one’s choice) and

$\lambda_{a}^{+}=\lambda_{a}^{-},\quad\ell_{+}^{2}=\ell_{-}^{2},$ (61)

where $\lambda_{a}^{\pm}=\ell_{\pm}\cdot e_{a}$ are the projections of the
transverse vectors $\ell^{\mu}_{\pm}$ on the tangent basis of the
hypersurface. This identifies $\ell_{+}$ and $\ell_{-}$ as “being the same”,
and thus generates a $C^{1}$ differentiable structure at $\Sigma$. It follows
that any coordinate system adapted to the rigging $\ell$ , that is a
coordinate system $\left(\sigma,y^{a}\right)$ such that $\sigma=0$ is the
equation for $\Sigma$ and

$\ell=\frac{\partial}{\partial\sigma},$ (62)

is a $C^{1}$ coordinate system. If $\Sigma$ is timelike or spacelike, then the
unit normal $\hat{n}^{\mu}$ always provides a rigging which satisfies the
above conditions, therefore in that case there is no need to find a pair of
matching riggings and it follows that Gaussian normal coordinates are always
$C^{1}$. From this point on we assume that all spacetime coordinate systems
are $C^{1}$ on $\Sigma$ and $C^{4}$ away from $\Sigma$. Since the final
results will be expressed as hypersurface tensors, this does not reduce the
practical applicability of the formalism. In these coordinate systems, the
relation $h_{ab}^{+}=h_{ab}^{-}$ also implies that the spacetime metric
$g_{\mu\nu}$ is continuous, due to expansion (9), which involves only
$\lambda_{a}$, $\ell^{2}$ and $h_{ab}$, which are then all assumed continuous.

We use the notation

$\left[F\right]=\left.F^{+}\right|_{\Sigma}-\left.F^{-}\right|_{\Sigma}$ (63)

for the jump discontinuity of a field $F$ at $\Sigma$ (thus $\left[F\right]$
is a function defined only on $\Sigma$) and

$\bar{F}=F^{+}\theta+F^{-}\left(1-\theta\right)$ (64)

for the “soldering” of a field, where

$\theta(p)=\cases{1&$p\in M^{+}\setminus\Sigma$\\\ 0&$p\in
M^{-}\setminus\Sigma$\\\ \frac{1}{2}&$p\in\Sigma$}$ (65)

is the Heaviside step function associated to $\Sigma$. Any choice of value for
$\theta$ at $\Sigma$ ensures that for a continuous field $F$, $F=\bar{F}$ is a
pointwise identity. The choice $\left.\theta\right|_{\Sigma}=1/2$ is taken for
reasons of symmetry. We imagine that the hypersurface of discontinuity
$\Sigma$ is the limit of a layer of finite thickness, where the field $F$ is
continuous albeit rapidly varying. In the limit of infinitesimal thickness a
value between $F^{+}$ and $F^{-}$ should be picked on $\Sigma$ and taking the
arithmetic average (corresponding to $\left.\theta\right|_{\Sigma}=1/2$) is
the most ”democratic” choice that gives no preference to the field values on
either side of the layer.

To conform to the usual conventions, we also assume that the rigging vector
field $\ell$ points from $M^{-}$ to $M^{+}$. The orientation on $\Sigma$ is
induced by the rigging $\ell$. It follows that $\Sigma$ has the boundary
orientation inherited as the boundary of $M^{-}$ and the opposite to the
boundary orientation inherited from $M^{+}$.

###  4.1 Thin shell equation from the action regularized by counterterms

The total action will be taken to consist of the gravitational action
$S_{\mathrm{EH}}$, an unspecified bulk matter action $S_{\mathrm{M}}$ and an
unspecified thin shell matter action $S_{\mathrm{TS}}$. Instead of integrating
over $M$ at once, we split the integrals into sums of integrals over $M^{+}$
and $M^{-}$. Since $\Sigma$ is not a part of the outer boundary of the
manifold, the usual Dirichlet conditions do not apply to $\Sigma$, the metric
is not fixed there. We suppose the metric is $C^{0}$ across $\Sigma$ and at
least $C^{3}$ away from $\Sigma$. Since we are varying within this
differentiability class, $\delta g_{\mu\nu}$ also inherits these properties.
The equations of motion of the shell arise as the natural boundary conditions
on the shell as the bulk and boundary contributions to the variation of the
action must vanish separately.

The shell hypersurface $\Sigma$ is an interior boundary and thus we add the
rigged counterterm (56) to the action at both sides of $\Sigma$ to ensure the
proper boundary behaviour of the action. The total action is then

$\displaystyle S$
$\displaystyle=\underbrace{\frac{1}{2\varkappa}\int_{M^{+}}R^{+}\mu_{g}}_{S_{\mathrm{EH}}^{+}}+\underbrace{\frac{1}{2\varkappa}\int_{M^{-}}R^{-}\mu_{g}}_{S_{\mathrm{EH}}^{-}}+\underbrace{\int_{M^{+}}\mathcal{L}_{\mathrm{M}}^{+}d^{D+1}x}_{S_{\mathrm{M}}^{+}}+\underbrace{\int_{M^{-}}\mathcal{L}_{\mathrm{M}}^{-}d^{D+1}x}_{S_{\mathrm{M}}^{-}}$
(66) $\displaystyle\underbrace{-\frac{1}{\varkappa}\int_{\Sigma}\left(P_{\
\nu}^{\mu}\nabla_{\mu}n^{\nu}\right)_{+}\mu_{g,\ell}}_{B_{\mathrm{R}}^{+}}+\underbrace{\frac{1}{\varkappa}\int_{\Sigma}\left(P_{\
\nu}^{\mu}\nabla_{\mu}n^{\nu}\right)_{-}\mu_{g,\ell}}_{B_{\mathrm{R}}^{-}}+\underbrace{\int_{\Sigma}\mathcal{L}_{\mathrm{TS}}d^{D}y}_{S_{\mathrm{TS}}}.$

The relative sign difference between $B_{\mathrm{R}}^{+}$ and
$B_{\mathrm{R}}^{-}$ is caused by the orientation of $\Sigma$ being opposite
to the boundary orientation inherited from the domain $M^{+}$. Variation of
this integral with respect to the metric is carried out by applying the
variation formula (59) to both the $+$ and $-$ integrals, giving

$\displaystyle\delta S$
$\displaystyle=\int_{M^{+}}\frac{1}{2}\left(T_{+}^{\mu\nu}-\frac{1}{\varkappa}G_{+}^{\mu\nu}\right)\delta
g_{\mu\nu}\,\mu_{g}+\int_{M^{-}}\frac{1}{2}\left(T_{-}^{\mu\nu}-\frac{1}{\varkappa}G_{-}^{\mu\nu}\right)\delta
g_{\mu\nu}\,\mu_{g}$ (67)
$\displaystyle+\int_{\Sigma}\left(\frac{1}{\rho_{\ell,g}}\frac{\delta
S_{\mathrm{TS}}}{\delta g_{\mu\nu}}-n_{\kappa}\left[\frac{\partial
L_{M}}{\partial
g_{\mu\nu,\kappa}}\right]-\frac{1}{2\varkappa}\left[\Pi^{\mu\nu}\right]\right)\delta
g_{\mu\nu}\,\mu_{\ell,g},$

where $L_{M}=\mathcal{L}_{\mathrm{M}}/\sqrt{-\mathfrak{g}}$ is the scalarized
matter Lagrangian. The variation of the integral should vanish for all
variations $\delta g_{\mu\nu}$ that are $C^{0}$ across $\Sigma$ and $C^{3}$
away from $\Sigma$. In particular, we can choose an arbitrary $\delta
g_{\mu\nu}$ which satisfies $\left.\delta g_{\mu\nu}\right|_{\Sigma}=0$, which
implies that the coefficients of the $\delta g_{\mu\nu}$ in the bulk integrals
should vanish, giving the Einstein field equations in the bulk. It then
follows that the surface term $\int_{\Sigma}\left(\cdots\right)\delta
g_{\mu\nu}\,\mu_{\ell,g}$ should vanish separately even for arbitrary $\delta
g_{\mu\nu}$, which results in the equation

$S^{\mu\nu}=\frac{1}{\varkappa}\left[\Pi^{\mu\nu}\right],$ (68)

where

$S^{\mu\nu}=\frac{2}{\rho_{\ell,g}}\frac{\delta S_{\mathrm{TS}}}{\delta
g_{\mu\nu}}-2n_{\kappa}\left[\frac{\partial L_{\mathrm{M}}}{\partial
g_{\mu\nu,\kappa}}\right]$ (69)

is the surface energy-momentum tensor. The second term is a contribution
coming from the bulk matter Lagrangian if it also depends on the derivatives
of the metric tensor, usually via the connection. It arises precisely as
follows. If $S_{\mathrm{M}}$ is the matter action with
$S_{\mathrm{M}}=\int_{M}\mathcal{L}_{\mathrm{M}}\left(g,\partial
g,\psi,\partial\psi\right)d^{D+1}x$, and scalar Lagrangian function
$L_{\mathrm{M}}=\mathcal{L}_{\mathrm{M}}/\sqrt{-\mathfrak{g}}$, the variation
of the matter action with respect to the metric is

$\delta
S_{\mathrm{M}}=\int_{M}\left(\left(\frac{\partial\mathcal{L}_{\mathrm{M}}}{\partial
g_{\mu\nu}}-\partial_{\kappa}\frac{\partial\mathcal{L}_{\mathrm{M}}}{\partial
g_{\mu\nu,\kappa}}\right)\delta
g_{\mu\nu}+\partial_{\kappa}\left(\frac{\partial\mathcal{L}_{\mathrm{M}}}{\partial
g_{\mu\nu,\kappa}}\delta g_{\mu\nu}\right)\right)d^{D+1}x.$ (70)

The total divergence term here can be expressed in terms of the scalar
Lagrangian and the covariant divergence. Specifically, since the metric
determinant is independent of the metric’s first derivative, we get

$\frac{\partial\mathcal{L}_{\mathrm{M}}}{\partial
g_{\mu\nu,\kappa}}=\frac{\partial L_{\mathrm{M}}}{\partial
g_{\mu\nu,\kappa}}\sqrt{-\mathfrak{g}},$ (71)

and even though $g_{\mu\nu,\kappa}$ is not a tensor, $\partial
L_{\mathrm{M}}/\partial g_{\mu\nu,\kappa}$ is (see [15], Chapter II, §11). We
can therefore write

$\displaystyle\delta S_{\mathrm{M}}$ $\displaystyle=\mathrm{Bulk\
terms}+\int_{M}\nabla_{\kappa}\left(\frac{\partial
L_{\mathrm{M}}}{\partial{g_{\mu\nu,\kappa}}}\delta g_{\mu\nu}\right)\mu_{g}$
(72) $\displaystyle=\mathrm{Bulk\ terms}+\oint_{\partial
M}n_{\kappa}\frac{\partial L_{\mathrm{M}}}{\partial{g_{\mu\nu,\kappa}}}\delta
g_{\mu\nu}\mu_{\ell,g}.$

If this integral is performed over a spacetime with a shell $\Sigma$ we thus
obtain the difference term $-n_{\kappa}\left[\frac{\partial
L_{\mathrm{M}}}{\partial{g_{\mu\nu,\kappa}}}\right]$ on the shell which
contributes to the energy-momentum tensor. Out of the standard model fields,
only the Lagrangian of the Dirac field depends on the connection, however the
Dirac field being spinorial, an alternative formulation based on orthonormal
tetrads would be necessary to incorporate them into the formalism. For some
exotic matter fields (for example the scalar sector of Horndeski’s theory
[38]) this term may be nonvanishing. As far as we are aware, such possible
contributions to the thin shell energy-momentum tensor have not been explored
so far in the literature.

Equation (68) is the equation of motion for the thin shell in unprojected
form. To proceed, we decompose the tensor $\Pi^{\mu\nu}$ in the frame
$\left(\ell,e_{a}\right)$. This is best accomplished by transitioning to a
coordinate system $\left(\sigma,y^{a}\right)$ adapted to the rigging $\ell$
(i.e. $\ell^{\mu}=\left(\partial/\partial\sigma\right)^{\mu}$and the $y^{a}$
are the hypersurface coordinates), giving

$\displaystyle\Pi^{00}$
$\displaystyle=\chi_{ab}\left(n^{2}h_{\ast}^{ab}-\nu^{a}\nu^{b}\right)+n^{2}\varphi_{a}\nu^{a}-\left(n^{2}\right)^{2}\psi,$
$\displaystyle\Pi^{0a}$
$\displaystyle=\chi_{cd}\left(\nu^{a}h_{\ast}^{cd}-\nu^{c}h_{\ast}^{ad}\right)+n^{2}\varphi_{c}h_{\ast}^{ac}-n^{2}\psi\nu^{a},$
$\displaystyle\Pi^{ab}$
$\displaystyle=\chi_{cd}\left(h_{\ast}^{ab}h_{\ast}^{cd}-h_{\ast}^{ac}h_{\ast}^{bd}\right)+\varphi_{c}\left(\nu^{a}h_{\ast}^{bc}-h_{\ast}^{ab}\nu^{c}+h_{\ast}^{ac}\nu^{b}\right)-\psi\nu^{a}\nu^{b},$
(73)

where $\psi=\psi_{a}^{a}$ is the trace. The details of this derivation may be
found in A.

Since the metric is continuous, only the extrinsic curvature-type quantities
$\chi_{ab}$, $\varphi_{a}$, $\psi_{a}^{b}$ may suffer jumps, as they in
general involve the metric’s transversal derivatives. The reason for the
introduction of the tensor $H_{ab}$ in (17) has been that as it turns out the
jumps of all such quantities may be related to that of $H_{ab}$. We refer to
Mars and Senovilla for details (equations. (72-76) in [10]) and merely list
the jump relations

$\displaystyle\left[\psi_{a}^{b}\right]$
$\displaystyle=\left[H_{ac}\right]h_{\ast}^{bc},$
$\displaystyle\left[\varphi_{a}\right]$
$\displaystyle=\left[H_{ab}\right]\nu^{b},$
$\displaystyle\left[\chi_{ab}\right]$
$\displaystyle=n^{2}\left[H_{ab}\right],$ $\displaystyle\left[\gamma_{\
ab}^{c}\right]$ $\displaystyle=-\left[H_{ab}\right]\nu^{c},$ (74)

where

$\displaystyle\left[H_{ab}\right]$
$\displaystyle=e_{a}^{\mu}e_{b}^{\nu}\left[\nabla_{\mu}\ell_{\nu}\right]=-e_{a}^{\mu}e_{b}^{\nu}\left[\Gamma_{\
\mu\nu}^{\kappa}\right]\ell_{\kappa},$ (75)

and is always symmetric. The jump of the metric derivatives can be written as

$\displaystyle\left[\partial_{\kappa}g_{\mu\nu}\right]$
$\displaystyle=\delta_{\kappa}^{\lambda}\left[\partial_{\lambda}g_{\mu\nu}\right]=\left(e_{a}^{\lambda}\vartheta_{\kappa}^{a}+\ell^{\lambda}n_{\kappa}\right)\left[\partial_{\lambda}g_{\mu\nu}\right]$
(76)
$\displaystyle=\left[\partial_{a}g_{\mu\nu}\right]\vartheta_{\kappa}^{a}+\left[g_{\mu\nu,\ell}\right]n_{\kappa}=\left[g_{\mu\nu,\ell}\right]n_{\kappa},$

where the jump of the tangential derivative
$\left[\partial_{a}g_{\mu\nu}\right]\vartheta_{\kappa}^{a}$ vanishes because
of the continuity of the metric and
$g_{\mu\nu,\ell}=\ell^{\kappa}\partial_{\kappa}g_{\mu\nu}$ is the transversal
derivative. We then have

$\left[\Gamma_{\
\mu\nu}^{\kappa}\right]=\frac{1}{2}\left(n_{\mu}\xi_{\nu}^{\kappa}+n_{\nu}\xi_{\mu}^{\kappa}-n^{\kappa}\xi_{\mu\nu}\right),$
(77)

where $\xi_{\mu\nu}:=\left[g_{\mu\nu,\ell}\right]$ and it follows that

$\left[H_{ab}\right]=\frac{1}{2}e_{a}^{\mu}e_{b}^{\nu}\xi_{\mu\nu}=\frac{1}{2}e_{a}^{\mu}e_{b}^{\nu}\left[g_{\mu\nu,\ell}\right],$
(78)

which in adapted coordinates is the jump of the transversal derivative of the
induced metric,

$\left[H_{ab}\right]=\frac{1}{2}\left[\frac{\partial
h_{ab}}{\partial\sigma}\right].$ (79)

For this reason it is $\left[H_{ab}\right]$ that carries information about the
discontinuities of the metric’s transversal development.

Inserting the jump relations (74) into (73) gives

$\displaystyle\left[\Pi^{00}\right]$
$\displaystyle=\left[\chi_{ab}\right]\left(n^{2}h_{\ast}^{ab}-\nu^{a}\nu^{b}\right)+n^{2}\left[\varphi_{a}\right]\nu^{a}-\left(n^{2}\right)^{2}\left[\psi\right]$
(80)
$\displaystyle=\left(n^{2}\right)^{2}\left[H\right]-\left(n^{2}\right)^{2}\left[H\right]+n^{2}\left[H_{ab}\right]\nu^{a}\nu^{b}-n^{2}\left[H_{ab}\right]\nu^{a}\nu^{b}$
$\displaystyle=0,$ $\displaystyle\left[\Pi^{0a}\right]$
$\displaystyle=\left[\chi_{cd}\right]\left(\nu^{a}h_{\ast}^{cd}-\nu^{c}h_{\ast}^{ad}\right)+n^{2}\left[\varphi_{c}\right]h_{\ast}^{ac}-n^{2}\left[\psi\right]\nu^{a}$
(81)
$\displaystyle=n^{2}\left[H\right]\nu^{a}-n^{2}\left[H_{cd}\right]\nu^{c}h_{\ast}^{ad}+n^{2}\left[H_{cd}\right]h_{\ast}^{ac}\nu^{d}-n^{2}\left[H\right]\nu^{a}$
$\displaystyle=0,$

and

$\displaystyle\left[\Pi^{ab}\right]$
$\displaystyle=\left[\chi_{cd}\right]\left(h_{\ast}^{ab}h_{\ast}^{cd}-h_{\ast}^{ac}h_{\ast}^{bd}\right)+\left[\varphi_{c}\right]\left(\nu^{a}h_{\ast}^{bc}-h_{\ast}^{ab}\nu^{c}+h_{\ast}^{ac}\nu^{b}\right)-\left[\psi\right]\nu^{a}\nu^{b}$
(82)
$\displaystyle=n^{2}\left(\left[H\right]h_{\ast}^{ab}-\left[H^{ab}\right]\right)+\left[H_{c}^{b}\right]\nu^{a}\nu^{c}+\left[H_{c}^{a}\right]\nu^{b}\nu^{c}$
$\displaystyle-\left[H_{cd}\right]h_{\ast}^{ab}\nu^{c}\nu^{d}-\left[H\right]\nu^{a}\nu^{b}.$

It follows that the jump $\left[\Pi^{\mu\nu}\right]$ is a tangential tensor
field along $\Sigma$, which we may write as
$\left[\Pi^{\mu\nu}\right]=\left[\Pi^{ab}\right]e_{a}^{\mu}e_{b}^{\nu}$.
Following from (68), the surface energy-momentum tensor must also be
tangential with $S^{\mu\nu}=S^{ab}e_{a}^{\mu}e_{b}^{\nu}$ and the shell
equation can be considered as the hypersurface tensor equation

$\varkappa
S^{ab}=n^{2}\left(\left[H\right]h_{\ast}^{ab}-\left[H^{ab}\right]\right)+\left[H_{c}^{b}\right]\nu^{a}\nu^{c}+\left[H_{c}^{a}\right]\nu^{b}\nu^{c}-\left[H_{cd}\right]h_{\ast}^{ab}\nu^{c}\nu^{d}-\left[H\right]\nu^{a}\nu^{b}.$
(83)

Since a contravariant tensor being tangential is an intrinsic notion
independent of any choice of rigging, the components $\left[\Pi^{ab}\right]$
are calculated from $\left[\Pi^{\mu\nu}\right]$ in a way that is independent
of the rigging. Applying the transformation formulae of Section 2.2 to (83)
shows that $\left[\Pi^{ab}\right]$ is invariant under the shift transformation
$\ell^{\mu}\mapsto\ell^{\mu}+T^{a}e_{a}^{\mu}$ of the rigging and changes as
$\left[\Pi^{ab}\right]\mapsto\alpha^{-1}\left[\Pi^{ab}\right]$ under the
rescaling $\ell^{\mu}\mapsto\alpha\ell^{\mu}$. This ambiguity in the shell
equation is related to the fact that for a generic hypersurface there is no
preferred scaling for the normal field $n_{\mu}$. In the variational
principle, both $\left[\Pi^{ab}\right]$ and $S^{ab}$ appear as a factor in the
expression

$\left(S^{ab}-\frac{1}{\varkappa}\left[\Pi^{ab}\right]\right)\delta
h_{ab}\,\mu_{\ell,g},$ (84)

and the volume element $\mu_{\ell,g}$ depends on the scaling of the normal. It
follows that for the densitized surface tensor
$\mathfrak{S}^{ab}=S^{ab}\rho_{\ell,g}$ and densitized $\Pi$-tensor
$\mathfrak{P}^{ab}=\Pi^{ab}\rho_{\ell,g}$, the analogous equation

$\varkappa\mathfrak{S}^{ab}=\left[\mathfrak{P}^{ab}\right]$ (85)

is completely independent of any gauge choices, including the scaling of the
normal. If one wishes to use tensor equations, the scaling ambiguity in the
generic case is unavoidable. For timelike or spacelike hypersurfaces a
canonical choice is given by the unit normal which fixes the scaling of
$\left[\Pi^{ab}\right]$ and $S^{ab}$, while in the null case Poisson [11] gave
a physical interpretation of this ambiguity in terms of observers taking
measurements of the null shell.

The tensor $\Pi^{\mu\nu}$ which has been split into the components $\Pi^{00}$,
$\Pi^{0a}$ and $\Pi^{ab}$ may be identified with the canonical momentum of the
gravitational field, up to scaling and densitization (canonical momenta are
usually taken to be tensor densities). Ordinarily, canonical momenta are
constructed by foliating spacetime into a one-parameter family of spacelike
hypersurfaces [25], but one may equally well consider the analysis of dynamics
decomposed with respect to any foliation of spacetime, including the case when
foliate with respect to the transversal coordinate $\sigma$ adapted to the
rigging $\ell^{\mu}$. In the usual formulation, the canonical momentum is the
derivative of the Lagrangian with respect to “time” (which in this case is
$\sigma$) however it is well-known [48] that the canonical momentum may also
be identified with the coefficients of the field variation on the boundary
when the Dirichlet conditions are not imposed. This is the basis for the so-
called covariant phase space formalism [49, 50]. For the Einstein-Hilbert
action extended with the rigged boundary term, by (59), the boundary part is

$\frac{1}{2\varkappa}\int_{\partial M}\Pi^{\mu\nu}\delta
g_{\mu\nu}\mu_{\ell,g}=\frac{1}{2\varkappa}\int_{\partial
M}\left(\Pi^{00}\delta\ell^{2}+\Pi^{0a}\delta\lambda_{a}+\Pi^{ab}\delta
h_{ab}\right)\mu_{\ell,g},$ (86)

which shows that $\Pi^{00}$, $\Pi^{0a}$ and $\Pi^{ab}$ are proportional to the
canonical momenta corresponding to the metric degrees of freedom $\ell^{2}$,
$\lambda_{a}$ and $h_{ab}$. The shell equation then has the interpretation
that the surface energy-momentum tensor is the jump of the canonical momentum
on the hypersuface.

If the condition $\left.\delta g_{\mu\nu}\right|_{\partial M}=0$ is not
imposed on a boundary (such is the case for thin shells), the vanishing of the
variation of the action forces the coefficients of the $\delta g_{\mu\nu}$ to
vanish on the boundary. Since these coefficients are identified with the
canonical momentum of the field, the canonical momentum must vanish on the
boundary. This is referred to as the natural boundary condition [23], as it
arises without having to impose a boundary condition by hand. We can thus also
see that the shell equation
$\frac{1}{2}S^{ab}-\frac{1}{2\varkappa}\left[\Pi^{ab}\right]=0$ is the natural
boundary condition for the combined gravitation + bulk matter + shell matter
actions on the hypersurface.

Unlike the equations of motions, canonical momenta are not invariant under
equivalence transformations of Lagrangians such as adding total divergences
and - in the case of Einstein-Hilbert type Lagrangians \- they are sensitive
to the specific form of the variational counterterm added to the action.
However as discussed in Section 3, the difference of two variational
counterterms may depend only on the metric tensor and its tangential
derivatives, but never on the transversal derivative. Only the transversal
derivative has nonzero jump, thus while the expressions $\Pi^{00}$, $\Pi^{0a}$
and $\Pi^{ab}$ depend on the choice of counterterm, their jumps (of which only
$\left[\Pi^{ab}\right]$ is nonvanishing) do not. Therefore, the thin shell
equation (83) is actually independent of the choice of counterterm.

If $\Sigma$ is timelike or spacelike and we apply the canonical choice of
rigging presented in Section 2.3, we obtain the equation

$\varkappa
S^{ab}=\epsilon\left(\left[K\right]h^{ab}-\left[K^{ab}\right]\right),$ (87)

which is the well-known Lanczos equation [7]. If instead we take $\Sigma$ to
be null and choose the null rigging adapted to a spacelike foliation of
$\Sigma$ (Section 2.4), we decompose the equation into components $S^{11}$,
$S^{1A}$ and $S^{AB}$, which are respectively

$\displaystyle\varkappa S^{11}$ $\displaystyle=-\left[H_{AB}\right]q^{AB},$
$\displaystyle\varkappa S^{1A}$ $\displaystyle=\left[H_{1B}\right]q^{AB},$
$\displaystyle\varkappa S^{AB}$ $\displaystyle=-\left[H_{11}\right]q^{AB}.$
(88)

These relations agree with those of Poisson [11], who interprets $\mu:=S^{11}$
as the surface energy density, $j^{A}:=S^{1A}$ as the surface current and -
since $S^{AB}$ is diagonal in that it is proportional to the metric -
$p:=-\left[H_{11}\right]$ as the isotropic surface pressure of the null shell.

We conclude this section by comparing the result (83) to the analogous results
in previous works. As mentioned in Footnote 2, Barrabes and Israel [9] assume
$n\cdot n=\mathrm{const}$, which formally excludes causality-changing
hypersurfaces and they use the normalization $n\cdot\ell=\eta^{-1}$, where
$\eta$ is a nowhere vanishing function along $\Sigma$. One this differing
normalization convention is taken into account, equation (31) in [9] agrees
with our shell equation (83). In place of $H_{ab}$, they employ a different
quantity (denoted $\mathcal{K}_{ab}$), the jump of which however coincides
with that of $H_{ab}$ in all cases.

In [10] Mars and Senovilla consider only junction conditions and analyze the
distributional forms of curvature tensors, therefore the shell equation itself
does not appear directly. However since the energy-momentum tensor is
proportional to the Einstein tensor, the singular part of the Einstein tensor
(equation (71) in [10]) agrees with our $\left[\Pi^{ab}\right]$ up to the
appropriate constant factor and projection. This singular part of the Einstein
tensor also appears in explicitly projected form in equation (23) of [13].

###  4.2 Thin shell equation from the action regularized distributionally

Here we explore a different method of regularizing the action integral at the
shell. In the timelike case this method was applied by Hajicek and Kijowski
[20]. We show that it also works for shells of any signature. We can write the
metric tensor as

$g_{\mu\nu}=\bar{g}_{\mu\nu}=g_{\mu\nu}^{+}\theta+g_{\mu\nu}^{-}\left(1-\theta\right),$
(89)

where $\theta$ is the Heaviside step function defined in (65). This relation
is then interpreted distributionally. Reasonably rigorous treatments of tensor
distribution theory on manifolds, can be found in [51, 18, 19, 10, 52],
therefore we only do here a short review.

If $T$ is a type $(k,l)$ tensor field on $M$ we say that a type $(l,k)$ tensor
density $\varphi$ of weight $1$ is a _dual density_ to $T$, since then the
contraction
$\langle\varphi,T\rangle=\varphi_{\mu_{1}...\mu_{k}}{}^{\nu_{1}...\nu_{l}}T^{\mu_{1}...\mu_{k}}{}_{\nu_{1}...\nu_{l}}$
is a scalar density of weight $1$ that may be integrated over $D+1$
dimensional regions of $M$. Let us define the vector space $D_{k,l}(M)$ to
consist of smooth compactly supported tensor densities of type $(l,k)$ (called
_test densities_), and the space $D^{\ast}_{k,l}(M)$ to consist of linear
functionals on $D_{k,l}(M)$ that are continuous in the following sense. A
linear functional $\chi:D_{k,l}(M)\rightarrow\mathbb{R}$ is continuous and
thus belongs to $D^{\ast}_{k,l}(M)$ if for each sequence $\varphi_{n}\in
D_{k,l}(M)$ of test densities whose supports are contained in a common compact
set $K\subseteq M$ which is itself located in the domain of a coordinate
chart, and such that the components
$\left(\varphi_{n}\right)_{\mu_{1}...\mu_{k}}{}^{\nu_{1}...\nu_{l}}$ and their
partial derivatives of all orders tend to $0$ uniformly, we have
$\lim_{n\rightarrow\infty}\chi[\varphi_{n}]=0$. Elements of
$D^{\ast}_{k,l}(M)$ are called _tensor distributions_ of type $(k,l)$. A
tensor distribution $\chi\in D^{\ast}_{k,l}(M)$ is _regular_ if there exists a
(locally integrable but otherwise ”rough”) tensor field also denoted $\chi$
such that for any test density $\varphi$ we have
$\chi[\varphi]=\int_{M}\langle\chi,\varphi\rangle d^{D+1}x$. This integral
converges because $\varphi$ has compact support and since the integrand is a
density, no volume form is necessary here. Otherwise the distribution is
_singular_. We remark that it is well-defined to take the tensor product of a
tensor distribution with a _smooth_ tensor field, however products with non-
smooth tensor fields only make sense in limited circumstances.

de Rham [51] refers to a distributional $k$-form in the above sense as a
_current of degree_ $k$ or a $k$-current for short. Since antisymmetric
contravariant tensor densities with $(D+1)-k$ indices can be identified
canonically with $k$-forms, it follows that the dual densities $\varphi$ of
$k$-forms $\omega$ can be canonically identified with $(D+1)-k$-forms under
the pairing map $\langle\varphi,\omega\rangle=\omega\wedge\varphi$, thus
$k$-currents are continuous linear functionals on $(D+1)-k$-forms. de Rham
defines the _boundary_ $\partial\omega$ of a $k$-current $\omega$ by
$\partial\omega[\varphi]=\omega[d\varphi]$, then the (distributional) exterior
derivative by $d\omega=(-1)^{k+1}\partial\omega$.

Finally, a few remarks on notation and local representations are in order. As
de Rham proves666de Rham deals only with currents in [51], not general tensor
distributions. However his arguments are straightforward to generalize to
tensor distributions, in fact to distributions modelled on sections of
arbitrary vector bundles. in [51], distributions have the sheaf property, i.e.
if $D^{\ast}_{k,l}(U)$ denotes the space of type $(k,l)$ tensor distributions
over the open set $U$, and $V\subseteq U$ is an open subset, we have a well-
defined restriction map $\mathrm{res}_{V,U}(\chi)\equiv\left.\chi\right|_{V}$
given by

$\left.\chi\right|_{V}[\varphi]:=\chi[\mathrm{ext}_{U,V}(\varphi)],$ (90)

where $\mathrm{ext}_{U,V}:D_{k,l}(V)\rightarrow D_{k,l}(U)$ extends the tensor
density $\varphi\in D_{k,l}(V)$ defined on $V$ with compact support to a
tensor density defined on $U$ with compact support by taking $\varphi$ to be
zero on $U\setminus V$. This means that the rule $U\mapsto D^{\ast}_{k,l}(U)$
is a presheaf of real vector spaces, and is in fact a sheaf, i.e. if a tensor
distribution vanishes in a neighborhood of each point in $U$, then it vanishes
on $U$, and if compatible local distributions are given on an open cover, they
glue together to give a well-defined tensor distribution on the covered
domain. One may then show that if $U\subseteq M$ is a coordinate domain and
$\chi\in D^{\ast}_{k,l}(U)$ is a tensor distribution of type $(k,l)$, we can
write $\chi$ uniquely as

$\chi=\chi^{\mu_{1}...\mu_{k}}{}_{\nu_{1}...\nu_{l}}\partial_{\mu_{1}}\otimes\dots\partial_{\mu_{k}}\otimes
dx^{\nu_{1}}\otimes\dots\otimes dx^{\nu_{l}},$ (91)

where the components are scalar distributions on $U$, and for distributions
defined on $M$, the entire distribution may be reconstructed from its sets of
components if the manifold is covered by coordinate domains. Moreover, on any
test density $\varphi$ we have

$\chi[\varphi]=\chi^{\mu_{1}...\mu_{k}}{}_{\nu_{1}...\nu_{l}}\varphi_{\mu_{1}...\mu_{k}}{}^{\nu_{1}...\nu_{l}}[1],$
(92)

where the contraction is a distributional scalar density (i.e. $D+1$-form)
interpreted as a $D+1$-current and it acts on the $0$-form $1$. Although the
$1$ function is not compactly supported, one can also show [51] that it makes
sense to let a distribution act - through the use of a partition of unity - on
a non-compactly supported test density and if the distribution itself has
compact support, then this is always convergent, therefore the above
expression is well-defined. If we further denote the action of a $D+1$-current
$\omega$ on $1$ as

$\omega[1]:=\int_{M}\omega,$ (93)

we obtain ”classical” notation for tensor distributions (eg. similar to what
is found in [1]), since 1) it is possible to use index notation with tensor
densities and make local calculations, 2) actions of distributions can be
symbolically denoted by an integral.

We identify the Heaviside step function $\theta$ with the corresponding
$0$-current and define the ($1$-form) Dirac delta $\delta^{\Sigma}_{\ast}$
associated to the hypersurface $\Sigma$ to be the exterior derivative of the
Heaviside current, i.e. we have for any smooth compactly supported $D$-form
(or equivalently, vector density) $\varphi$

$\displaystyle\delta^{\Sigma}_{\ast}[\varphi]$
$\displaystyle=d\theta[\varphi]=-\partial\theta[\varphi]=-\theta[d\varphi]=-\int_{M}\theta
d\varphi$ (94)
$\displaystyle=-\int_{M^{+}}d\varphi=-\int_{-\Sigma}\varphi=\int_{\Sigma}\varphi$

Since $M$ is equipped with a volume form $\mu_{g}$, choosing a rigging
$\ell^{\mu}$ pointing from $M^{-}$ to $M^{+}$ with adapted normal $n_{\mu}$
satisfying $n_{\mu}\ell^{\mu}=1$, also defines the volume element
$\mu_{\ell,g}$ on $\Sigma$. Then it becomes possible to define a scalar
distribution $\delta^{\Sigma}$ evaluated on a test $D+1$-form $\varphi$ as

$\delta^{\Sigma}[\varphi]:=\int_{\Sigma}f\,\mu_{\ell,g},$ (95)

where $f$ is the scalar function uniquely determined777Note that in a shell
spacetime, $\mu_{g}$ is merely continuous on $\Sigma$. This is not a problem
as some distributions (those that can be identified with Radon measures),
including the various Dirac deltas can also be seen to be linear functionals
on _continuous_ , rather than smooth test functions [51, 53]. by
$\varphi=f\mu_{g}$. It is easy to verify the relation ($\delta^{\Sigma}_{\mu}$
are the components of the $1$-current $\delta^{\Sigma}_{\ast}$)

$\delta_{\mu}^{\Sigma}=n_{\mu}\delta^{\Sigma},$ (96)

which shows that the scalar $\delta^{\Sigma}$ depends on the choice of normal
(i.e. it depends on the rigging $\ell$ only up to scaling). For any soldered
quantity $\bar{F}$ we then have distributionally

$\partial_{\mu}\bar{F}=\overline{\partial_{\mu}F}+\left[F\right]\delta_{\mu}^{\Sigma}=\overline{\partial_{\mu}F}+\left[F\right]n_{\mu}\delta^{\Sigma}.$
(97)

Since the jump of the metric vanishes, the connection can be written as

$\Gamma_{\ \mu\nu}^{\kappa}=\bar{\Gamma}_{\ \mu\nu}^{\kappa},$ (98)

and its jump as (77)

$\left[\Gamma_{\
\mu\nu}^{\kappa}\right]=\frac{1}{2}\left(n_{\mu}\xi_{\nu}^{\kappa}+n_{\nu}\xi_{\mu}^{\kappa}-n^{\kappa}\xi_{\mu\nu}\right),$
(99)

where $\xi_{\mu\nu}=\left[g_{\mu\nu,\ell}\right]$. The curvature tensor is
then

$\displaystyle R_{\ \lambda\mu\nu}^{\kappa}$ $\displaystyle=\bar{R}_{\
\lambda\mu\nu}^{\kappa}+\delta_{\mu}^{\Sigma}\left[\Gamma_{\
\nu\lambda}^{\kappa}\right]-\delta_{\nu}^{\Sigma}\left[\Gamma_{\
\mu\lambda}^{\kappa}\right]$ (100) $\displaystyle=\bar{R}_{\
\lambda\mu\nu}^{\kappa}+\left(n_{\mu}\left[\Gamma_{\
\nu\lambda}^{\kappa}\right]-n_{\nu}\left[\Gamma_{\
\mu\lambda}^{\kappa}\right]\right)\delta^{\Sigma}.$

Let $\mathcal{R}_{\ \lambda\mu\nu}^{\kappa}$ denote its singular part, i.e.
the coefficients of $\delta^{\Sigma}$. Expanding gives

$\mathcal{R}_{\kappa\lambda\mu\nu}=n_{\kappa}\left[H_{\lambda\mu}\right]n_{\nu}-n_{\kappa}\left[H_{\lambda\nu}\right]n_{\mu}+n_{\lambda}\left[H_{\kappa\nu}\right]n_{\mu}-n_{\lambda}\left[H_{\kappa\mu}\right]n_{\nu},$
(101)

where
$\left[H_{\mu\nu}\right]=\left[H_{ab}\right]\vartheta_{\mu}^{a}\vartheta_{\nu}^{b}$
(we refer to [10] for details). The scalar curvature is calculated by
contracting the curvature tensor twice as
$R=\bar{R}+\mathcal{R}\delta^{\Sigma}$, where

$\mathcal{R}=2\left(\left[H_{ab}\right]\nu^{a}\nu^{b}-n^{2}\left[H\right]\right).$
(102)

According to the jump relations (74), we may rewrite this as

$\mathcal{R}=-2\left(\left[\chi_{ab}\right]h_{\ast}^{ab}-\left[\varphi_{a}\right]\nu^{a}\right),$
(103)

which is $-2\varkappa$-times the jump of the integrand of the rigged
counterterm (58). The gravitational (scalar) Lagrangian in the presence of a
shell and interpreted as a distribution is then

$L_{\mathrm{EH}}=\frac{1}{2\varkappa}R-\frac{1}{\varkappa}\left(\left[\chi_{ab}\right]h_{\ast}^{ab}-\left[\varphi_{a}\right]\nu^{a}\right)\delta^{\Sigma}.$
(104)

It follows that the Einstein-Hilbert action over $M$ is

$\displaystyle S_{\mathrm{EH}}$
$\displaystyle=\frac{1}{2\varkappa}\int_{M}\bar{R}\,\mu_{g}-\frac{1}{\varkappa}\int_{\Sigma}\left(\left[\chi_{ab}\right]h_{\ast}^{ab}-\left[\varphi_{a}\right]\nu^{a}\right)\mu_{\ell,g}$
(105)
$\displaystyle=\frac{1}{2\varkappa}\int_{M^{+}}R^{+}\,\mu_{g}+\frac{1}{2\varkappa}\int_{M^{-}}R^{-}\,\mu_{g}-\frac{1}{\varkappa}\int_{\Sigma}\left[P_{\
\nu}^{\mu}\nabla_{\mu}n^{\nu}\right]\mu_{\ell,g},$

where we have used that $\chi_{ab}h_{\ast}^{ab}-\varphi_{a}\nu^{a}$ can be
written in the form $P_{\ \nu}^{\mu}\nabla_{\mu}n^{\nu}$. If we add to this
the bulk and thin shell matter actions, we obtain the same variational
principle as given by (66). We have thus shown that if instead of splitting
the action into separate integrals on $M^{+}$ and $M^{-}$ and adding
counterterms, we integrate over $M$ while taking into account the singular
contribution to the Lagrangian, the resulting singular terms give precisely
the difference of the counterterms that otherwise would have had to be added
by hand.

It is interesting to note that there is no a priori reason for the singular
part of the Lagrangian to have the same value as the difference of the
counterterms. The Einstein-Hilbert Lagrangian density can be written in the
form

$\mathcal{L}_{\mathrm{EH}}=P^{\kappa\lambda\mu\nu}\left(g\right)\sqrt{-\mathfrak{g}}\partial_{\kappa}\partial_{\lambda}g_{\mu\nu}+Q\left(g,\partial
g\right)\sqrt{-\mathfrak{g}},$ (106)

where the coefficients are

$P^{\kappa\lambda\mu\nu}\left(g\right)=\frac{1}{2\varkappa}\left(g^{\kappa\mu}g^{\lambda\nu}-g^{\kappa\lambda}g^{\mu\nu}\right),$
(107)

and

$Q\left(g,\partial
g\right)=\frac{1}{2\varkappa}\Gamma_{\kappa\mu\nu}\Gamma^{\kappa\mu\nu}-\frac{1}{2\varkappa}g_{\kappa\lambda}\Gamma_{\ast}^{\kappa}\Gamma_{\ast}^{\lambda},$
(108)

and $\Gamma_{\ast}^{\kappa}=\Gamma_{\ \mu\nu}^{\kappa}g^{\mu\nu}$. Since only
the second derivatives contribute singular terms, the Lagrangian has the
distributional form

$\mathcal{L}_{\mathrm{EH}}=\bar{\mathcal{L}}_{\mathrm{EH}}+P^{\kappa\lambda\mu\nu}\sqrt{-\mathfrak{g}}n_{\kappa}n_{\lambda}\left[g_{\mu\nu,\ell}\right]\delta^{\Sigma},$
(109)

while a variational counterterm is given by

$B=-\oint_{\partial
M}n_{\kappa}n_{\lambda}P^{\kappa\lambda\mu\nu}g_{\mu\nu,\ell}\mu_{\ell,g},$
(110)

which is obtainable by integrating the first term in
$\mathcal{L}_{\mathrm{EH}}$ by parts.

Since $P^{\kappa\lambda\mu\nu}$ is algebraic in the metric and thus does not
depend on the transversal derivatives, it is clear that the jump of the
integrand of $B$ is the same as the singular part of
$\mathcal{L}_{\mathrm{EH}}$. However if $P^{\kappa\lambda\mu\nu}$ were to
depend on the metric’s transversal derivative, the singular part of the
Lagrangian would be mathematically meaningless as $P^{\kappa\lambda\mu\nu}$
would be discontinuous at $\Sigma$ where it is being evaluated. If we choose
$\left.\theta\right|_{\Sigma}=1/2$ as the value of the step function on
$\Sigma$, then the meaning of such an expression can be salvaged as
$P^{\kappa\lambda\mu\nu}$ evaluated on the average value of the metric’s
transversal derivative at the price of taking products of Dirac deltas with
discontinuous functions. Moreover, were $P^{\kappa\lambda\mu\nu}$ to depend on
the metric’s derivatives, the counterterm would have to take a different form
as one could no longer get rid of second derivatives in the action by simple
integrations by parts.

It thus seems that such a simple relation between the singular part of the
Lagrangian and the jump of the counterterms exists if the Lagrangian is affine
in the second derivatives of the field with coefficients that do not depend on
the derivatives of the field, however if these conditions are violated in a
modified gravitational theory the above derivation breaks down and further
analysis would be necessary.

###  4.3 Thin shell equation from a first order action

We mention here for completeness that the correct shell equation may also be
obtained without having to regularize the Einstein-Hilbert action on $\Sigma$
by employing a first order equivalent. To ensure global validity, we choose
the background connection action (55) rather than the noncovariant
$\Gamma\Gamma$-action (53).

As we have shown in Section 3, we may view the first order equivalents as the
Einstein-Hilbert action extended with a particular variational counterterm.
Therefore we may ascertain without any explicit calculations that the first
order action (55) leads to the correct shell equation, as the difference of
two different variational counterterms to not depend on the metric’s
transversal derivative, therefore their jumps always agree. However it is the
jump of the counterterm that appears in the action (66), therefore a first
order action will lead to the same variational principle and thus the same
shell equation.

Nonetheless it is instructive to rederive the result via the first order
action from the beginning. The background connection action (55) is

$S_{\nabla}=S_{\mathrm{EH}}+B_{\mathrm{BC}}=\int_{M}L_{\nabla}\,\mu_{g},$
(111)

where

$L_{\nabla}=\frac{1}{2\varkappa}\left\\{\bar{R}+g^{\mu\nu}\left(\Delta_{\
\mu\rho}^{\kappa}\Delta_{\ \kappa\nu}^{\rho}-\Delta_{\
\kappa\rho}^{\kappa}\Delta_{\ \mu\nu}^{\rho}\right)\right\\},$ (112)

and $\Delta_{\ \mu\nu}^{\kappa}=\Gamma_{\ \mu\nu}^{\kappa}-\bar{\Gamma}_{\
\mu\nu}^{\kappa}$. To ensure manifest covariance, we consider $L_{\nabla}$ to
be a function of $g_{\mu\nu}$ and $\bar{\nabla}_{\kappa}g_{\mu\nu}$, the
covariant derivative of the metric with respect to the background connection,
where the relation [25]

$\Delta_{\
\mu\nu}^{\kappa}=\frac{1}{2}g^{\kappa\lambda}\left(\bar{\nabla}_{\mu}g_{\nu\lambda}+\bar{\nabla}_{\nu}g_{\mu\lambda}-\bar{\nabla}_{\lambda}g_{\mu\nu}\right)$
(113)

is relevant. A variation of the action leads symbolically to

$\delta S_{\nabla}=\int_{M}\left(\delta L_{\nabla}+L_{\nabla}g^{\mu\nu}\delta
g_{\mu\nu}\right)\mu_{g},$ (114)

where

$\displaystyle\delta L_{\nabla}$ $\displaystyle=\frac{\partial
L_{\nabla}}{\partial g_{\mu\nu}}\delta g_{\mu\nu}+\frac{\partial
L_{\nabla}}{\partial\bar{\nabla}_{\kappa}g_{\mu\nu}}\delta\bar{\nabla}_{\kappa}g_{\mu\nu}$
(115) $\displaystyle=\frac{\partial L_{\nabla}}{\partial g_{\mu\nu}}\delta
g_{\mu\nu}+\frac{\partial
L_{\nabla}}{\partial\bar{\nabla}_{\kappa}g_{\mu\nu}}\bar{\nabla}_{\kappa}\delta
g_{\mu\nu}.$

Since the background connection $\bar{\nabla}_{\kappa}$ has no a priori
relation with the volume element $\mu_{g}$, we may not use Gauss’ theorem with
it. Therefore we must express $\bar{\nabla}_{\kappa}\delta g_{\mu\nu}$ with
the Levi-Civita connection $\nabla_{\kappa}$. This is accomplished via the
difference formula [25]

$\bar{\nabla}_{\kappa}\delta g_{\mu\nu}=\nabla_{\kappa}\delta
g_{\mu\nu}+\Delta_{\ \kappa\mu}^{\lambda}\delta g_{\lambda\nu}+\Delta_{\
\kappa\nu}^{\lambda}\delta g_{\mu\lambda}.$ (116)

Inserting this into the variation gives

$\delta L_{\nabla}=\left(\frac{\partial L_{\nabla}}{\partial
g_{\mu\nu}}+\frac{\partial
L_{\nabla}}{\partial\bar{\nabla}_{\kappa}g_{\lambda\nu}}\Delta_{\
\kappa\lambda}^{\mu}+\frac{\partial
L_{\nabla}}{\partial\bar{\nabla}_{\kappa}g_{\mu\lambda}}\Delta_{\
\kappa\lambda}^{\nu}-\nabla_{\kappa}\frac{\partial
L_{\nabla}}{\partial\bar{\nabla}_{\kappa}g_{\mu\nu}}\right)\delta
g_{\mu\nu}+\nabla_{\kappa}\left(\frac{\partial
L_{\nabla}}{\partial\bar{\nabla}_{\kappa}g_{\mu\nu}}\delta g_{\mu\nu}\right),$
(117)

thus the variation of the action is

$\displaystyle\delta S_{\nabla}$ $\displaystyle=\int_{M}\left(\frac{\partial
L_{\nabla}}{\partial g_{\mu\nu}}+L_{\nabla}g^{\mu\nu}+\frac{\partial
L_{\nabla}}{\partial\bar{\nabla}_{\kappa}g_{\lambda\nu}}\Delta_{\
\kappa\lambda}^{\mu}+\frac{\partial
L_{\nabla}}{\partial\bar{\nabla}_{\kappa}g_{\mu\lambda}}\Delta_{\
\kappa\lambda}^{\nu}-\nabla_{\kappa}\frac{\partial
L_{\nabla}}{\partial\bar{\nabla}_{\kappa}g_{\mu\nu}}\right)\delta
g_{\mu\nu}\,\mu_{g}$ (118) $\displaystyle+\oint_{\partial
M}n_{\kappa}\frac{\partial
L_{\nabla}}{\partial\bar{\nabla}_{\kappa}g_{\mu\nu}}\delta
g_{\mu\nu}\,\mu_{\ell,g}.$

The bulk terms must be $-\frac{1}{2\varkappa}G^{\mu\nu}$ in disguise, since
the action differs from the Einstein-Hilbert action in a total derivative term
only. For the boundary term we have

$2\varkappa\frac{\partial
L_{\nabla}}{\partial\bar{\nabla}_{\kappa}g_{\mu\nu}}=\Delta^{\kappa\mu\nu}-\frac{1}{2}\left(g^{\lambda\nu}g^{\kappa\mu}+g^{\lambda\mu}g^{\nu\kappa}-g^{\lambda\kappa}g^{\mu\nu}\right)\Delta_{\lambda}-\frac{1}{2}g^{\mu\nu}\Delta_{\ast}^{\kappa},$
(119)

where $\Delta_{\lambda}=\Delta_{\ \mu\lambda}^{\mu}$ and
$\Delta_{\ast}^{\kappa}=\Delta_{\ \mu\nu}^{\kappa}g^{\mu\nu}$. The details of
the analogous derivation for the $\Gamma\Gamma$-action (53) are given in
Appendix 9 of [47] and is therefore omitted here. Let the contraction of the
above expression be denoted

$M^{\mu\nu}:=2\varkappa n_{\kappa}\frac{\partial
L_{\nabla}}{\partial\bar{\nabla}_{\kappa}g_{\mu\nu}}=n_{\kappa}\Delta^{\kappa\mu\nu}-\frac{1}{2}\left(g^{\lambda\nu}n^{\mu}+g^{\lambda\mu}n^{\nu}-n^{\lambda}g^{\mu\nu}\right)\Delta_{\lambda}-\frac{1}{2}g^{\mu\nu}\Delta_{\ast}^{\kappa}n_{\kappa}.$
(120)

We rewrite the variational principle for thin shells as

$\displaystyle
S=\frac{1}{2\varkappa}\int_{M^{+}}\left\\{\bar{R}+g^{\mu\nu}\left(\Delta_{\
\mu\rho}^{\kappa}\Delta_{\ \kappa\nu}^{\rho}-\Delta_{\
\kappa\rho}^{\kappa}\Delta_{\ \mu\nu}^{\rho}\right)\right\\}\mu_{g}$
$\displaystyle+\frac{1}{2\varkappa}\int_{M^{-}}\left\\{\bar{R}+g^{\mu\nu}\left(\Delta_{\
\mu\rho}^{\kappa}\Delta_{\ \kappa\nu}^{\rho}-\Delta_{\
\kappa\rho}^{\kappa}\Delta_{\ \mu\nu}^{\rho}\right)\right\\}\mu_{g}$
$\displaystyle+\int_{M^{+}}\mathcal{L}_{\mathrm{M}}d^{D+1}x+\int_{M^{-}}\mathcal{L}_{\mathrm{M}}d^{D+1}x+\int_{\Sigma}\mathcal{L}_{\mathrm{TS}}d^{D}y,$
(121)

and varying this with respect to the metric gives

$\displaystyle\delta S$
$\displaystyle=\int_{M^{+}}\frac{1}{2}\left(T^{\mu\nu}-\frac{1}{\varkappa}G^{\mu\nu}\right)\delta
g_{\mu\nu}\,\mu_{g}+\int_{M^{-}}\frac{1}{2}\left(T^{\mu\nu}+\frac{1}{\varkappa}G^{\mu\nu}\right)\delta
g_{\mu\nu}\,\mu_{g}$ (122)
$\displaystyle-\frac{1}{2\varkappa}\int_{\Sigma}\left[M^{\mu\nu}\right]\delta
g_{\mu\nu}\,\mu_{\ell,g}+\int_{\Sigma}\frac{1}{2}S^{\mu\nu}\delta
g_{\mu\nu}\,\mu_{\ell,g},$

where $S^{\mu\nu}$ is again defined by (69). Imposing the stationarity of the
action gives the boundary equation

$\varkappa S^{\mu\nu}=\left[M^{\mu\nu}\right].$ (123)

The jump of $M^{\mu\nu}$ can be written as

$\left[M^{\mu\nu}\right]=n_{\kappa}\left[\Gamma^{\kappa\mu\nu}\right]-\frac{1}{2}\left(g^{\lambda\nu}n^{\mu}+g^{\lambda\mu}n^{\nu}-n^{\lambda}g^{\mu\nu}\right)\left[\Gamma_{\lambda}\right]-\frac{1}{2}g^{\mu\nu}\left[\Gamma_{\ast}^{\kappa}\right]n_{\kappa},$
(124)

where $\Gamma_{\lambda}=\Gamma_{\ \lambda\mu}^{\mu}$ and
$\Gamma_{\ast}^{\kappa}=\Gamma_{\ \mu\nu}^{\kappa}g^{\mu\nu}$. The connection
$\bar{\nabla}_{\mu}$ was assumed to be a smooth background structure,
therefore $\left[\Delta_{\ \mu\nu}^{\kappa}\right]=\left[\Gamma_{\
\mu\nu}^{\kappa}\right]-\left[\bar{\Gamma}_{\
\mu\nu}^{\kappa}\right]=\left[\Gamma_{\ \mu\nu}^{\kappa}\right]$. If we
decompose $\left[M^{\mu\nu}\right]$ in the frame $\left(\ell,e_{a}\right)$
(since the calculation is lengthy, the details are in B), we obtain that
$\left[M^{00}\right]=0$, $\left[M^{0a}\right]=0$, and thus
$\left[M^{\mu\nu}\right]$ is tangential with
$\left[M^{\mu\nu}\right]=\left[M^{ab}\right]e_{a}^{\mu}e_{b}^{\nu}$, its
projected components being

$\left[M^{ab}\right]=n^{2}\left(\left[H\right]h_{\ast}^{ab}-\left[H^{ab}\right]\right)+\nu^{a}\left[H_{\
c}^{b}\right]\nu^{c}+\left[H_{\
c}^{a}\right]\nu^{b}\nu^{c}-\left[H_{cd}\right]\nu^{c}\nu^{d}h_{\ast}^{ab}-\left[H\right]\nu^{a}\nu^{b},$
(125)

which is equal to $\left[\Pi^{ab}\right]$. The shell equation is

$\varkappa S^{ab}=\left[M^{ab}\right],$ (126)

where $S^{ab}$ is defined by $S^{\mu\nu}=S^{ab}e_{a}^{\mu}e_{b}^{\nu}$. This
equation agrees with (83), which shows that the first order action indeed
leads to the correct equations.

We remark that the corresponding derivation for the Einstein-Hilbert action
extended with counterterms crucially relied on the variation formula (59)
originally derived by Parattu et al[28], which is a nontrivial result and
difficult to obtain. On the other hand the first order action provided a
straightforward derivation, which is clearly advantageous. The disadvantage of
the first order approach is that sufficiently complicated theories of
gravitation (eg. Horndeski theory [38]) do not admit first order equivalent
Lagrangians, therefore this method cannot always be relied on.

Whether a given modified theory of gravity with second order field equations
can be described in terms of a first order Lagrangian can be determined easily
by looking at the field equations. A first order Lagrangian will produce
Euler-Lagrange equations that have at most an affine dependence on the second
derivatives of the field variables. However it is known [54, 55] that - at
least locally - the converse of this statement is also true, every locally
variational second-order differential equation888Strictly speaking, those
differential equations for which the number of equations agree with the number
of unknown functions, which are referred to as source equations in eg. [56].
Euler-Lagrange equations are always source equations. that is affine in the
second derivatives has a local first order Lagrangian. Thus, a theory of
gravitation specified in terms of a second order Lagrangian with second order
field equations will have a (possibly only local and non-covariant) first
order equivalent if and only if the field equations are affine functions of
the second derivatives.

Looking at the field equations of Horndeski’s theory (presented for example
[40]) one can ascertain that the restrictions $G_{5}(\phi,X)=0$,
$G_{4}(\phi,X)=G_{4}(\phi)$ and $G_{3}(\phi,X)=G_{3}(\phi)$ are necessary to
ensure the existence of a first order equivalent. This includes the Brans-
Dicke type theories where the scalar field Lagrangian is first order and the
non-minimal coupling of the scalar field to gravity does not involve the
scalar field derivatives but excludes the galileon-type models as well as
kinetic gravity braiding where the higher-order nonlinear derivative
interaction of the scalar field prevents the existence of first order
equivalent Lagrangians. Outside Horndeski theories, Gauss-Bonnet gravity is an
example of a theory with no first order Lagrangian, as the field equations are
quadratic in the curvature tensors [35].

##  5 Conclusions

The purpose of this paper was to provide a variational formalism for
spacetimes containing a thin shell of completely unconstrained signature. To
treat shells of arbitrary signature, we have used the formalism of rigged
hypersurfaces, reviewed in Section 2. Shells are incorporated into the
variational principle as interior boundaries and their equations of motion are
the natural boundary conditions on them. The Einstein-Hilbert action needed to
be regularized at the shell to ensure a valid variational principle. We have
investigated multiple possible regularization procedures.

In Subsection 4.1, regularization has been carried out by adding variational
counterterms (reviewed in Section 3) to the action. The shell equation (83)
obtained by varying this modified action reproduces the results obtained
through distributional methods by Barrabes and Israel [9], Mars and Senovilla
[10] and Senovilla [13]. We have shown that the shell equation does not depend
on the choice of the counterterm and have identified the geometric quantity
the jump of which appears in the equations of motion to be the (tensorial)
canonical momentum of the gravitational field, generalized to unconstrained
instead of just spacelike foliations.

We have considered a different regularization process in Subsection 4.2 by
focusing on the singular part of the Lagrangian. We have shown that the
singular term is related to the jump of the counterterm and leads to the same
variational principle. This generalized the procedure employed by eg. Hajicek
and Kijowski [20] to arbitrary shells. We have also argued that a more general
Lagrangian might have a less trivial relationship between the singular parts
and the counterterms.

Finally, in Subsection 4.3, we have obtained the equations of motion of the
shell by employing a first order equivalent Lagrangian. This lead to a simpler
variational procedure, but we have noted that more complicated theories might
not have first order equivalents, rendering this method less adequate for
generalization.

Aside from filling a gap in the literature, we expect that this work would be
useful for formulating thin shells and junction conditions along generic
hypersurfaces in second order modified theories of gravity, such as Horndeski
theory [38]. Second order Lagrangians are capable of producing second order
differential equations at least quadratic in the second derivatives (the
equations of motions associated with the $G_{3}$ term in Horndeski’s theory is
an example), which could lead to ill-defined products of delta functions if
the distributional method were to be followed. Thus it would seem that
variational approaches to thin shells are better behaved for such theories and
always lead to unambigous shell equations.

## Acknowledgements

This research was funded by the Hungarian National Research Development and
Innovation Office (NKFIH) in the form of grant 123996. I am grateful to László
Á. Gergely for valuable discussions on the topics of this paper.

## Appendix

##  Appendix A Decomposition of $\Pi^{\mu\nu}$

In this Appendix we carry out the explicit decomposition of the tensor field

$\Pi^{\mu\nu}=g^{\mu\nu}\left(P_{\
\sigma}^{\rho}\nabla_{\rho}n^{\sigma}\right)-\nabla^{(\mu}n^{\nu)}-\nabla_{\rho}\ell^{\rho}n^{\mu}n^{\nu}$
(127)

defined along the hypersurface $\Sigma$ in the frame
$\left(\ell,e_{a}\right)$. This calculation is best carried out by evaluating
$\Pi^{\mu\nu}$ in adapted coordinates $\left(\sigma,y^{a}\right)$ such that
the $y^{a}$ parametrize $\Sigma$, while

$\ell^{\mu}=\left(\frac{\partial}{\partial\sigma}\right)^{\mu}.$ (128)

In such an adapted coordinate system we have

$\displaystyle\ell^{\mu}$ $\displaystyle=\delta_{0}^{\mu},\quad
n_{\mu}=\delta_{\mu}^{0},$ $\displaystyle g^{\mu\nu}$
$\displaystyle=n^{2}\delta_{0}^{\mu}\delta_{0}^{\nu}+\nu^{a}\left(\delta_{0}^{\mu}\delta_{a}^{\nu}+\delta_{a}^{\mu}\delta_{0}^{\nu}\right)+h_{\ast}^{ab}\delta_{a}^{\mu}\delta_{b}^{\nu},$
$\displaystyle n^{\mu}$ $\displaystyle=g^{\mu
0}=n^{2}\delta_{0}^{\mu}+\nu^{a}\delta_{a}^{\mu},$ (129) $P_{\
\sigma}^{\rho}\nabla_{\rho}n^{\sigma}=\chi_{cd}h_{\ast}^{cd}-\varphi_{c}\nu^{c},$
(130)

and the connection $\Gamma_{\ \mu\nu}^{\kappa}$ has components

$\Gamma_{\ 00}^{0}=U,\quad\Gamma_{\ 00}^{a}=Z^{a},$ (131) $\Gamma_{\
0a}^{0}=\varphi_{a},\quad\Gamma_{\ ab}^{0}=-\chi_{ab},\quad\Gamma_{\
0b}^{a}=\psi_{b}^{a},\quad\Gamma_{\ ab}^{c}=\gamma_{\ ab}^{c}.$ (132)

The elements $U$ and $Z^{a}$ involve transversal derivatives of the frame
vectors and thus are not independent of the way the quantities are extended
off $\Sigma$. Fortunately, they will cancel. We first evaluate what we can
without fixing the free indices as

$\Pi^{\mu\nu}=g^{\mu\nu}\left(\chi_{cd}h_{\ast}^{cd}-\varphi_{c}\nu^{c}\right)+g^{\mu\kappa}g^{\nu\lambda}\Gamma_{\
\kappa\lambda}^{0}-\Gamma_{\ \rho 0}^{\rho}n^{\mu}n^{\nu},$ (133)

then we get

$\displaystyle\Pi^{00}$
$\displaystyle=n^{2}\left(\chi_{ab}h_{\ast}^{ab}-\varphi_{a}\nu^{a}\right)+\left(n^{2}\right)^{2}U+2n^{2}\varphi_{a}\nu^{a}-\chi_{ab}\nu^{a}\nu^{b}$
(134) $\displaystyle-\left(n^{2}\right)^{2}U-\left(n^{2}\right)^{2}\psi$
$\displaystyle=n^{2}\left(\chi_{ab}h_{\ast}^{ab}-\varphi_{a}\nu^{a}\right)+2n^{2}\varphi_{a}\nu^{a}-\chi_{ab}\nu^{a}\nu^{b}-\left(n^{2}\right)^{2}\psi$
$\displaystyle=\left(n^{2}h_{\ast}^{ab}-\nu^{a}\nu^{b}\right)\chi_{ab}+n^{2}\nu^{a}\varphi_{a}-\left(n^{2}\right)^{2}\psi,$
$\displaystyle\Pi^{0a}$
$\displaystyle=\nu^{a}\left(\chi_{bc}h_{\ast}^{bc}-\varphi_{b}\nu^{b}\right)+g^{0\kappa}g^{a\lambda}\Gamma_{\
\kappa\lambda}^{0}-\Gamma_{\ \rho 0}^{\rho}n^{2}\nu^{a}$ (135)
$\displaystyle=\nu^{a}\left(\chi_{bc}h_{\ast}^{bc}-\varphi_{b}\nu^{b}\right)+n^{2}\nu^{a}U+n^{2}h_{\ast}^{ab}\varphi_{b}+\nu^{a}\nu^{b}\varphi_{b}-\nu^{b}h_{\ast}^{ac}\chi_{bc}$
$\displaystyle-n^{2}\nu^{a}U-n^{2}\nu^{a}\psi$
$\displaystyle=\left(\nu^{a}h_{\ast}^{bc}-\nu^{b}h_{\ast}^{ac}\right)\chi_{bc}+n^{2}h_{\ast}^{ab}\varphi_{b}-n^{2}\nu^{a}\psi,$
$\displaystyle\Pi^{ab}$
$\displaystyle=h_{\ast}^{ab}\left(\chi_{cd}h_{\ast}^{cd}-\varphi_{c}\nu^{c}\right)+g^{a\kappa}g^{b\lambda}\Gamma_{\
\kappa\lambda}^{0}-\Gamma_{\ \rho 0}^{\rho}\nu^{a}\nu^{b}$ (136)
$\displaystyle=h_{\ast}^{ab}\left(\chi_{cd}h_{\ast}^{cd}-\varphi_{c}\nu^{c}\right)+\nu^{a}\nu^{b}U+\nu^{a}h_{\ast}^{bc}\varphi_{c}+h_{\ast}^{ac}\nu^{b}\varphi_{c}-h_{\ast}^{ac}h_{\ast}^{bd}\chi_{cd}$
$\displaystyle-U\nu^{a}\nu^{b}-\nu^{a}\nu^{b}\psi$
$\displaystyle=\left(h_{\ast}^{ab}h_{\ast}^{cd}-h_{\ast}^{ac}h_{\ast}^{bd}\right)\chi_{cd}+\left(\nu^{a}h_{\ast}^{bc}+h_{\ast}^{ac}\nu^{b}-h_{\ast}^{ab}\nu^{c}\right)\varphi_{c}-\nu^{a}\nu^{b}\psi.$

##  Appendix B Decomposition of $\left[M^{\mu\nu}\right]$

We now carry out the decomposition in the frame $\left(\ell,e_{a}\right)$ of
the tensor field

$\displaystyle\left[M^{\mu\nu}\right]$
$\displaystyle=n_{\kappa}\left[\Gamma^{\kappa\mu\nu}\right]-\frac{1}{2}\left(g^{\lambda\nu}n^{\mu}+g^{\lambda\mu}n^{\nu}-n^{\lambda}g^{\mu\nu}\right)\left[\Gamma_{\lambda}\right]-\frac{1}{2}g^{\mu\nu}\left[\Gamma_{\ast}^{\kappa}\right]n_{\kappa},$
(137)

defined only along the hypersurface $\Sigma$, given in (124). As we have
argued at (76), we may write the jump of the metric’s derivative as

$\left[\partial_{\kappa}g_{\mu\nu}\right]=n_{\kappa}\xi_{\mu\nu},$ (138)

where $\xi_{\mu\nu}=\left[g_{\mu\nu,\ell}\right]$ is the jump of the
transversal derivative. The jump of the connection is then

$\left[\Gamma_{\
\mu\nu}^{\kappa}\right]=\frac{1}{2}\left(n_{\mu}\xi_{\nu}^{\kappa}+n_{\nu}\xi_{\mu}^{\kappa}-n^{\kappa}\xi_{\mu\nu}\right).$
(139)

This gives

$\left[\Gamma_{\lambda}\right]=\left[\Gamma_{\
\lambda\mu}^{\mu}\right]=\frac{1}{2}\left(n_{\mu}\xi_{\lambda}^{\mu}+n_{\lambda}\xi_{\mu}^{\mu}-n^{\mu}\xi_{\mu\lambda}\right)=\frac{1}{2}n_{\lambda}\xi_{\mu}^{\mu},$
(140)

and

$\left[\Gamma_{\ast}^{\kappa}\right]=\frac{1}{2}\left(2n_{\mu}\xi^{\kappa\mu}-n^{\kappa}\xi_{\mu}^{\mu}\right)=n_{\mu}\xi^{\kappa\mu}-\frac{1}{2}n^{\kappa}\xi_{\mu}^{\mu}.$
(141)

We also have

$n_{\kappa}\left[\Gamma_{\
\mu\nu}^{\kappa}\right]=\frac{1}{2}\left(n_{\mu}\xi_{\nu}^{\kappa}n_{\kappa}+n_{\nu}\xi_{\mu}^{\kappa}n_{\kappa}-n^{2}\xi_{\mu\nu}\right).$
(142)

With these, we can write

$\displaystyle\left[M^{\mu\nu}\right]$
$\displaystyle=\frac{1}{2}\left(n^{\mu}\xi^{\nu\kappa}n_{\kappa}+n^{\nu}\xi^{\mu\kappa}n_{\kappa}-n^{2}\xi^{\mu\nu}\right)-\frac{1}{4}\left(g^{\lambda\nu}n^{\mu}+g^{\lambda\mu}n^{\nu}-n^{\lambda}g^{\mu\nu}\right)n_{\lambda}\xi_{\kappa}^{\kappa}$
(143)
$\displaystyle-\frac{1}{2}g^{\mu\nu}\left(n_{\lambda}\xi^{\kappa\lambda}-\frac{1}{2}n^{\kappa}\xi_{\lambda}^{\lambda}\right)n_{\kappa}$
$\displaystyle=\frac{1}{2}\left(n^{\mu}\xi^{\nu\kappa}n_{\kappa}+n^{\nu}\xi^{\mu\kappa}n_{\kappa}-n^{2}\xi^{\mu\nu}\right)-\frac{1}{4}\left(n^{\nu}n^{\mu}+n^{\mu}n^{\nu}-n^{2}g^{\mu\nu}\right)\xi_{\kappa}^{\kappa}$
$\displaystyle-\frac{1}{2}g^{\mu\nu}n_{\kappa}n_{\lambda}\xi^{\kappa\lambda}+\frac{1}{4}n^{2}g^{\mu\nu}\xi_{\lambda}^{\lambda}.$

Contracting with $n_{\nu}$ gives

$\displaystyle\left[M^{\mu\nu}\right]n_{\nu}$
$\displaystyle=\frac{1}{2}\left(n^{\mu}\xi^{\nu\kappa}n_{\nu}n_{\kappa}+n^{2}\xi^{\mu\kappa}n_{\kappa}-n^{2}\xi^{\mu\nu}n_{\nu}\right)-\frac{1}{4}\left(n^{2}n^{\mu}+n^{2}n^{\mu}-n^{2}n^{\mu}\right)\xi_{\kappa}^{\kappa}$
(144)
$\displaystyle-\frac{1}{2}n^{\mu}n_{\kappa}n_{\lambda}\xi^{\kappa\lambda}+\frac{1}{4}n^{2}n^{\mu}\xi_{\lambda}^{\lambda}$
$\displaystyle=\frac{1}{2}n^{\mu}\xi^{\nu\kappa}n_{\nu}n_{\kappa}-\frac{1}{4}n^{2}n^{\mu}\xi_{\kappa}^{\kappa}-\frac{1}{2}n^{\mu}n_{\kappa}n_{\lambda}\xi^{\kappa\lambda}+\frac{1}{4}n^{2}n^{\mu}\xi_{\lambda}^{\lambda}=0.$

Since $\left[M^{\mu\nu}\right]$ is symmetric, this implies that it is
tangential to $\Sigma$ with
$\left[M^{\mu\nu}\right]=\left[M^{ab}\right]e_{a}^{\mu}e_{b}^{\nu}$, and these
components are given by

$\displaystyle\left[M^{ab}\right]$
$\displaystyle=\vartheta_{\mu}^{a}\vartheta_{\nu}^{b}\left[M^{\mu\nu}\right]$
(145)
$\displaystyle=\frac{1}{2}\left(\nu^{a}\xi^{\nu\kappa}\vartheta_{\nu}^{b}n_{\kappa}+\nu^{b}\xi^{\mu\kappa}\vartheta_{\mu}^{a}n_{\kappa}-n^{2}\xi^{\mu\nu}\vartheta_{\mu}^{a}\vartheta_{\nu}^{b}\right)-\frac{1}{4}\left(2\nu^{a}\nu^{b}-n^{2}h_{\ast}^{ab}\right)\xi_{\kappa}^{\kappa}$
$\displaystyle-\frac{1}{2}h_{\ast}^{ab}n_{\kappa}n_{\lambda}\xi^{\kappa\lambda}+\frac{1}{4}n^{2}h_{\ast}^{ab}\xi_{\lambda}^{\lambda}$
$\displaystyle=\frac{1}{2}\left(\nu^{a}\vartheta_{\mu}^{b}+\nu^{b}\vartheta_{\mu}^{a}\right)\xi^{\mu\nu}n_{\nu}-\frac{1}{2}n^{2}\xi^{\mu\nu}\vartheta_{\mu}^{a}\vartheta_{\nu}^{b}+\frac{1}{2}\left(n^{2}h_{\ast}^{ab}-\nu^{a}\nu^{b}\right)\xi_{\kappa}^{\kappa}$
$\displaystyle-\frac{1}{2}h_{\ast}^{ab}n_{\kappa}n_{\lambda}\xi^{\kappa\lambda}$

In order to proceed, we write

$\displaystyle\xi_{\mu\nu}$
$\displaystyle=\left(\vartheta_{\mu}^{a}e_{a}^{\kappa}+n_{\mu}\ell^{\kappa}\right)\left(\vartheta_{\nu}^{b}e_{b}^{\lambda}+n_{\nu}\ell^{\lambda}\right)\xi_{\kappa\lambda}$
(146)
$\displaystyle=2\left[H_{ab}\right]\vartheta_{\mu}^{a}\vartheta_{\nu}^{b}+\xi_{a}^{\ell}\left(\vartheta_{\mu}^{a}n_{\nu}+n_{\mu}\vartheta_{\nu}^{a}\right)+\xi^{\ell}n_{\mu}n_{\nu},$

where we have used
$\left[H_{ab}\right]=\frac{1}{2}e_{a}^{\mu}e_{b}^{\nu}\xi_{\mu\nu}$ and
defined

$\xi_{a}^{\ell}=\xi_{\mu\nu}e_{a}^{\mu}\ell^{\nu},\quad\xi^{\ell}=\xi_{\mu\nu}\ell^{\mu}\ell^{\nu}.$
(147)

Then

$\displaystyle\xi^{\mu\nu}\vartheta_{\mu}^{a}n_{\nu}$
$\displaystyle=2\left[H_{b}^{a}\right]\nu^{b}+\xi_{b}^{\ell}\left(h_{\ast}^{ab}n^{2}+\nu^{a}\nu^{b}\right)+\xi^{\ell}\nu^{a}n^{2},$
$\displaystyle\xi^{\mu\nu}\vartheta_{\mu}^{a}\vartheta_{\nu}^{b}$
$\displaystyle=2\left[H^{ab}\right]+\xi_{c}^{\ell}\left(h_{\ast}^{ac}\nu^{b}+\nu^{a}h_{\ast}^{bc}\right)+\xi^{\ell}\nu^{a}\nu^{b},$
$\displaystyle\xi^{\mu\nu}n_{\mu}n_{\nu}$
$\displaystyle=2\left[H_{ab}\right]\nu^{a}\nu^{b}+2n^{2}\xi_{a}^{\ell}\nu^{a}+\left(n^{2}\right)^{2}\xi^{\ell},$
$\displaystyle\xi_{\kappa}^{\kappa}$
$\displaystyle=2\left[H\right]+2\xi_{a}^{\ell}\nu^{a}+n^{2}\xi^{\ell},$ (148)

and inserting these back into $\left[M^{ab}\right]$ gives

$\displaystyle\left[M^{ab}\right]$
$\displaystyle=\frac{1}{2}\left(2\nu^{a}\left[H_{c}^{b}\right]\nu^{c}+\xi_{c}^{\ell}\left(\nu^{a}h_{\ast}^{bc}n^{2}+\nu^{a}\nu^{b}\nu^{c}\right)+\xi^{\ell}\nu^{a}\nu^{b}n^{2}\right)$
(149)
$\displaystyle+\frac{1}{2}\left(2\nu^{b}\left[H_{c}^{a}\right]\nu^{c}+\xi_{c}^{\ell}\left(\nu^{b}h_{\ast}^{ac}n^{2}+\nu^{a}\nu^{b}\nu^{c}\right)+\xi^{\ell}\nu^{a}\nu^{b}n^{2}\right)$
$\displaystyle-\frac{1}{2}n^{2}\left(2\left[H^{ab}\right]+\xi_{c}^{\ell}\left(h_{\ast}^{ac}\nu^{b}+\nu^{a}h_{\ast}^{bc}\right)+\xi^{\ell}\nu^{a}\nu^{b}\right)$
$\displaystyle+\frac{1}{2}\left(n^{2}h_{\ast}^{ab}-\nu^{a}\nu^{b}\right)\left(2\left[H\right]+2\xi_{c}^{\ell}\nu^{c}+n^{2}\xi^{\ell}\right)$
$\displaystyle-\frac{1}{2}h_{\ast}^{ab}\left(2\left[H_{cd}\right]\nu^{c}\nu^{d}+2n^{2}\xi_{c}^{\ell}\nu^{c}+\left(n^{2}\right)^{2}\xi^{\ell}\right).$

Here all terms involving $\xi_{a}^{\ell}$ and $\xi^{\ell}$ cancel, and the
remaining terms are

$\left[M^{ab}\right]=n^{2}\left(\left[H\right]h_{\ast}^{ab}-\left[H^{ab}\right]\right)+\nu^{a}\left[H_{c}^{b}\right]\nu^{c}+\nu^{b}\left[H_{c}^{a}\right]\nu^{c}-h_{\ast}^{ab}\left[H_{cd}\right]\nu^{c}\nu^{d}-\left[H\right]\nu^{a}\nu^{b}.$
(150)

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