$\displaystyle=\frac{\partial\varepsilon_{i}}{\partial
q_{,\mu}^{j}}+\frac{\partial\varepsilon_{j}}{\partial
q_{,\mu}^{i}}-2d_{\nu}\frac{\partial\varepsilon_{j}}{\partial
q_{,\mu\nu}^{i}}$ $\displaystyle H_{ij}^{\mu\nu}(\varepsilon)$
$\displaystyle=\frac{\partial\varepsilon_{i}}{\partial
q_{,\mu\nu}^{j}}-\frac{\partial\varepsilon_{j}}{\partial q_{,\mu\nu}^{i}}.$
(C.20)

######  Proposition C.2.

Let $E_{i}[q]$ denote an $m$-component differential operator of order $s\leq
2r$ such that there is a Lagrangian function $L[q]$ of order $r$ with
$E_{i}=\delta L/\delta q^{i}$ (then $E$ is called _variational_). Then the
differential operator $E$ satisfies the Helmholtz variationality conditions,
i.e.

$H_{i}(E)=0,\quad\text{or equivalently}\quad
H_{ij}^{\mu_{1}...\mu_{k}}(\varepsilon)=0,\quad 0\leq k\leq s.$ (C.21)

###### Proof.

We work with the action integral

$S[q]=\int_{M}L[q](x)\,d^{n}x$ (C.22)

directly. Ordinarily, variations $\delta q^{i}$ are obtained by first
considering a one-parameter family $q_{s}^{i}(x)$ of fields and taking the
$s$-derivative at $s=0$, i.e. $\delta q^{i}=\left.\partial q_{s}^{i}/\partial
s\right|_{s=0}$. We now consider a _two-parameter_ family $q_{s,t}^{i}(x)$ of
fields with

$\displaystyle\delta_{1}$ $\displaystyle=\left.\frac{\partial}{\partial
s}\right|_{s,t=0},\quad\delta_{2}=\left.\frac{\partial}{\partial
t}\right|_{s,t=0}$ $\displaystyle\delta_{1}\delta_{2}$
$\displaystyle=\delta_{2}\delta_{1}=\left.\frac{\partial^{2}}{\partial
s\partial t}\right|_{s,t=0}.$ (C.23)

Any of the two variations $\delta_{1}q^{i}$ and $\delta_{2}q^{i}$ may be
chosen to have compact support.

Then

$\delta_{1}S=\int_{M}\sum_{k=0}^{r}\frac{\partial L}{\partial
q_{,\mu_{1}...\mu_{k}}^{i}}\delta_{1}q_{,\mu_{1}...\mu_{k}}^{i}\,d^{n}x=\int_{M}\varepsilon_{i}\delta_{1}q^{i}\,d^{n}x,$
(C.24)

where we have integrated by parts and chosen $\delta_{1}q$ to have compact
support, thus

$\delta_{2}\delta_{1}S=\int_{M}\left(\sum_{k=0}^{2r}\frac{\partial\varepsilon_{i}}{\partial
q_{,\mu_{1}...\mu_{k}}^{j}}\delta_{1}q^{i}\delta_{2}q_{,\mu_{1}...\mu_{k}}^{j}+\varepsilon_{i}\delta_{2}\delta_{1}q^{i}\right)d^{n}x.$
(C.25)

As the two variations commute, by substracting $\delta_{1}\delta_{2}S$ from
this, we get zero:

$\displaystyle 0$
$\displaystyle=\delta_{2}\delta_{1}S-\delta_{1}\delta_{2}S=\int_{M}\sum_{k=0}^{2r}\left(\frac{\partial\varepsilon_{i}}{\partial
q_{,\mu_{1}...\mu_{k}}^{j}}\delta_{1}q^{i}\delta_{2}q_{,\mu_{1}...\mu_{k}}^{j}-\frac{\partial\varepsilon_{j}}{\partial
q_{,\mu_{1}...\mu_{k}}^{i}}\delta_{1}q_{,\mu_{1}...\mu_{k}}^{i}\delta_{2}q^{j}\right)d^{n}x$
$\displaystyle=\int_{M}\sum_{k=0}^{2r}\left(\frac{\partial\varepsilon_{i}}{\partial
q_{,\mu_{1}...\mu_{k}}^{j}}\delta_{1}q^{i}\delta_{2}q_{,\mu_{1}...\mu_{k}}^{j}-\left(-1\right)^{k}\frac{\delta\varepsilon_{j}}{\delta
q_{,\mu_{1}...\mu_{k}}^{i}}\delta_{1}q^{i}\delta_{2}q_{,\mu_{1}...\mu_{k}}^{j}\right)d^{n}x,$
(C.26)

where we have used the formula for the formal adjoint along with the
definition of higher variational derivatives. Thus

$0=\int_{M}\sum_{k=0}^{2r}H_{ij}^{\mu_{1}...\mu_{k}}(\varepsilon)\delta_{1}q^{i}\delta_{2}q_{,\mu_{1}...\mu_{k}}^{j}\,d^{n}x.$
(C.27)

Since $\delta_{1}q^{i}$ is any compactly supported bump function, the usual
argument tells us that this is only possible if

$0=\sum_{k=0}^{2r}H_{ij}^{\mu_{1}...\mu_{k}}(\varepsilon)\delta_{2}q_{,\mu_{1}...\mu_{k}}^{j}.$
(C.28)

Then at any one point $x\in M$ we may set
$\delta_{2}q_{,\mu_{1}...\mu_{k}}^{j}(x)$ to have any fixed value as long as
it is symmetric in the lower indices. As
$H_{ij}^{\mu_{1}...\mu_{k}}(\varepsilon)$ is also symmetric in the greek
indices, we obtain that

$0=H_{ij}^{\mu_{1}...\mu_{k}}(\varepsilon),\quad 0\leq k\leq 2r.$ (C.29)

∎

Stated differently, every Euler-Lagrange differential equation is such that
its linearization is (formally) selfadjoint.

######  Remark.

It is difficult to pinpoint where exactly did the Helmholtz variationality
conditions appear to this degree of generality for the first time.
Formulations of the variational condition in special cases such as second
order ordinary differential equations date back to the 19th century. The
formally self-adjoint nature of the Euler-Lagrange equations have also been
recognized relatively early. We refer to [24, 27, 43] for the Helmholtz
variationality conditions and the citations therein.

##  Appendix D Homotopy operators and integrability conditions

In this section we assume that the zero field $q^{i}(x)=0$ is a valid field
configuration and if $q^{i}(x)$ is allowed, so is $tq^{i}(x)$ for $0\leq t\leq
1$. In other words, the space of field variables is star-shaped with respect
to $0$. For the applications in this paper, this will not be restrictive. If
this assumption is made on the field variables, we are able to derive a number
of powerful integrability relations for variational systems. A _current_ will
be an $n$-component differential operator $K^{\mu}[q]$.

######  Definition D.1.

  1. 1.

Let $L[q]$ be a Lagrangian of order $r$. To $L$ we associate a current
$\mathrm{K}(L)$ defined by

$\mathrm{K}^{\mu}(L)[q]=\int_{0}^{1}\sum_{k=0}^{r-1}P_{i}^{\mu\mu_{1}...\mu_{k}}[tq]q_{,\mu_{1}...\mu_{k}}^{i}\,dt,$
(D.1)

where $f[tq](x):=f(x,tq(x),\dots,tq_{(r)}(x))$.

  2. 2.

Let $\varepsilon_{i}[q]$ be an $m$-component differential operator of order
$s$. To $\varepsilon$ we associate a Lagrangian $\Lambda(\varepsilon)$ as

$\Lambda(\varepsilon)[q]=\int_{0}^{1}\varepsilon_{i}[tq]q^{i}\,dt,$ (D.2)

called the _Vainberg-Tonti Lagrangian_ associated to $\varepsilon$.

  3. 3.

Let $D_{i}[q,\delta q]=\sum_{k=0}^{p}D_{ij}^{\mu_{1}...\mu_{k}}[q]\delta
q_{,\mu_{1}...\mu_{k}}^{j}$ be a linear differential operator on field
variations whose coefficients $D_{ij}^{\mu_{1}...\mu_{k}}[q]$ are themselves
(nonlinear) differential operators acting on the dynamical variables $q^{i}$.
To every such linear operator we associate an $m$-component (nonlinear)
differential operator

$\mathrm{Q}(D)[q]=\int_{0}^{1}\sum_{k=0}^{p}D_{ij}^{\mu_{1}...\mu_{k}}(\varepsilon)[tq]tq_{,\mu_{1}...\mu_{k}}^{j}dt.$
(D.3)

Let us symbolically write $\mathbf{Div}$ for the total divergence, i.e. the
operator which associates to each current $K^{\mu}$ the Lagrangian
$\mathbf{Div}(K):=d_{\mu}K^{\mu}$ and $\mathbf{E}$ for the Euler-Lagrange
operator which associates to each Lagrangian $L$ the Euler-Lagrange
expressions $\mathbf{E}(L):=(\delta L/\delta q^{i})_{i=1}^{m}$. Also write
$\mathbf{H}(\varepsilon):=(H_{i}(\varepsilon))_{i=1}^{m}$ for the symbolic
form of the Helmholtz operator. Finally, if $L$ is a Lagrangian let
$L_{0}(x):=L[0](x)$ be the ordinary function of the independent variables
obtained by evaluating the Lagrangian on the zero field.

######  Proposition D.1.

The operators $\mathbf{Div}$, $\mathbf{E}$, $\mathbf{H}$ and $\mathrm{K}$,
$\Lambda$, $\mathrm{Q}$ satisfy the following relations:

  1. 1.

for each Lagrangian $L$ we have

$L=\Lambda(\mathbf{E}(L))+\mathbf{Div}(\mathrm{K}(L))+L_{0};$ (D.4)

  2. 2.

for each $m$-component differential operator
$\varepsilon=(\varepsilon_{i})_{i=1}^{m}$ we have

$\varepsilon=\mathbf{E}(\Lambda(\varepsilon))+\mathrm{Q}(\mathbf{H}(\varepsilon)).$
(D.5)

###### Proof.

  1. 1.

For deriving concrete formulae, suppose that the order of $L$ is $r$. Then we
have

$\frac{d}{dt}L[tq]=\sum_{k=0}^{r}\frac{\partial L}{\partial
q_{,\mu_{1}...\mu_{k}}^{i}}[tq]q_{,\mu_{1}...\mu_{k}}^{i}=\frac{\delta
L}{\delta
q^{i}}[tq]q^{i}+d_{\mu}\left(\sum_{k=0}^{r-1}P_{i}^{\mu\mu_{1}...\mu_{k}}[tq]q_{,\mu_{1}...\mu_{k}}^{i}\right),$
(D.6)

where we have used the first variation formula (C.8) evaluated at $tq$ with
$\delta q^{i}=q^{i}$. Note that for any operator $f[q]$ we have
$(d_{\mu}f)[tq]=d_{\mu}(f[tq])$ which is clear when the total derivative is
expanded. Integrating the above relation from $0$ to $1$ gives

$L[q]-L_{0}=\int_{0}^{1}\frac{\delta L}{\delta
q^{i}}[tq]q^{i}\,dt+d_{\mu}\left(\sum_{k=0}^{r-1}P_{i}^{\mu\mu_{1}...\mu_{k}}[tq]q_{,\mu_{1}...\mu_{k}}^{i}\right),$
(D.7)

which is just (D.4) written out in more concrete terms.

  2. 2.

Suppose that $\varepsilon_{i}$ has order $s$. Let $L:=\Lambda(\varepsilon)$
denote the Vainberg-Tonti Lagrangian associated to $\varepsilon_{i}$ and
$E_{i}:=\delta L/\delta q^{i}$. We then express $E_{i}$ in terms of
$\varepsilon_{i}$:

$\displaystyle E_{i}[q]$
$\displaystyle=\sum_{k=0}^{s}\left(-1\right)^{k}d_{\mu_{1}}\dots
d_{\mu_{k}}\frac{\partial}{\partial
q_{,\mu_{1}...\mu_{k}}^{i}}\int_{0}^{1}\varepsilon_{j}[tq]q^{j}\,dt$
$\displaystyle=\int_{0}^{1}\varepsilon_{i}[tq]\,dt+\int_{0}^{1}\sum_{k=0}^{s}\left(-1\right)^{k}d_{\mu_{1}}\dots
d_{\mu_{k}}\left(\frac{\partial\varepsilon_{j}}{\partial
q_{,\mu_{1}...\mu_{k}}^{i}}[tq]tq^{j}\right)dt$
$\displaystyle=\int_{0}^{1}\varepsilon_{i}[tq]\,dt+\int_{0}^{1}\sum_{k=0}^{s}\sum_{l=0}^{s-k}\left(-1\right)^{k+l}d_{\nu_{1}}\dots
d_{\nu_{l}}\frac{\partial\varepsilon_{j}}{\partial
q_{,\mu_{1}...\mu_{k}\nu_{1}...\nu_{l}}^{i}}[tq]tq_{,\mu_{1}...\mu_{k}}^{j}dt$
$\displaystyle=\int_{0}^{1}\varepsilon_{i}[tq]\,dt+\int_{0}^{1}\sum_{k=0}^{s}\left(-1\right)^{k}\frac{\delta\varepsilon_{j}}{\delta
q_{,\mu_{1}...\mu_{k}}^{i}}[tq]tq_{,\mu_{1}...\mu_{k}}^{j}dt$ (D.8)

where we have used the higher order product formula. Inserting here

$\left(-1\right)^{k}\frac{\delta\varepsilon_{j}}{\delta
q_{,\mu_{1}...\mu_{k}}^{i}}[tq]=\frac{\partial\varepsilon_{i}}{\partial
q_{,\mu_{1}...\mu_{k}}^{j}}[tq]-H_{ij}^{\mu_{1}...\mu_{k}}(\varepsilon)[tq],$
(D.9)

we get

$E_{i}[q]=\int_{0}^{1}\left(\varepsilon_{i}[tq]+\sum_{k=0}^{s}\frac{\partial\varepsilon_{i}}{\partial
q_{,\mu_{1}...\mu_{k}}^{j}}[tq]tq_{,\mu_{1}...\mu_{k}}^{j}\right)dt-\int_{0}^{1}\sum_{k=0}^{s}H_{ij}^{\mu_{1}...\mu_{k}}(\varepsilon)[tq]tq_{,\mu_{1}...\mu_{k}}^{j}dt.$
(D.10)

Finally,

$\frac{d}{dt}\left(t\varepsilon_{i}[tq]\right)=\varepsilon_{i}[tq]+\sum_{k=0}^{s}t\frac{\partial\varepsilon_{i}}{\partial
q_{,\mu_{1}...\mu_{k}}^{j}}[tq]q_{,\mu_{1}...\mu_{k}}^{j},$ (D.11)

and thus

$\varepsilon_{i}[q]=E_{i}[q]+\int_{0}^{1}\sum_{k=0}^{s}H_{ij}^{\mu_{1}...\mu_{k}}(\varepsilon)[tq]tq_{,\mu_{1}...\mu_{k}}^{j}dt.$
(D.12)

The latter is just (D.5) when expanded in coordinates.

∎

######  Corollary D.1.

  1. 1.

A Lagrangian $L$ is a null Lagrangian (has vanishing Euler-Lagrange
expressions) if and only if it is the sum of a total divergence and a
functionally constant term.

  2. 2.

An $m$-component differential operator $\varepsilon_{i}$ is variational if and
only if it satisfies the Helmholtz conditions, and then the Vainberg-Tonti
formula gives a Lagrangian for the operator.

###### Proof.

  1. 1.

Take (D.4) and substitute $\mathbf{E}(L)=0$ which then gives

$L=\mathbf{Div}(\mathrm{K}(L))+L_{0}$ (D.13)

as asserted.

  2. 2.

We have already seen that Euler-Lagrange expressions satisfy the Helmholtz
variationality conditions, hence it is sufficient to show that if
$\mathbf{H}(\varepsilon)=0$, then the operator is variational. Substituting
this into (D.5), we get

$\varepsilon=\mathbf{E}(\Lambda(\varepsilon)),$ (D.14)

which shows that $\varepsilon_{i}$ is indeed variational and
$\Lambda(\varepsilon)$ is a Lagrangian for it.

∎

For completeness and clarity, we also write down again the concrete forms of
(D.4) and (D.5):

$L[q]=\int_{0}^{1}\frac{\delta L}{\delta
q^{i}}[tq]q^{i}\,dt+d_{\mu}\left(\sum_{k=0}^{r-1}P_{i}^{\mu\mu_{1}...\mu_{k}}[tq]q_{,\mu_{1}...\mu_{k}}^{i}\right)+L_{0}$
(D.15)

and

$\varepsilon_{i}[q]=\frac{\delta}{\delta
q^{i}}\int_{0}^{1}\varepsilon_{i}[tq]q^{i}\,dt+\int_{0}^{1}\sum_{k=0}^{s}H_{ij}^{\mu_{1}...\mu_{k}}(\varepsilon)[tq]tq_{,\mu_{1}...\mu_{k}}^{j}dt.$
(D.16)

Note that the topological condition stated in the beginning of this section
(the space of field variables is star-shaped) is paramount for the existence
of the homotopy operators $\mathrm{K}$, $\Lambda$ and $\mathrm{Q}$ and
therefore Proposition D.1 and Corollary D.1 are valid only under these
circumstances.

In the main part of the paper, we need formulae (D.15) and (D.16) when the
objects involved depend on multiple sets of fields, eg. $L[q,p]$ where
$q=(q^{i})_{i=1}^{m}$ and $p=(p^{\alpha})_{\alpha=1}^{m^{\prime}}$. Clearly
the operators
$\mathbf{Div},\mathbf{E},\mathbf{H},\mathrm{K},\Lambda,\mathrm{Q}$ can be
defined for each field variable separately such that the other field variables
are treated as part of the dependence on the independent variables. The only
noteworthy difference is in (D.15) which then becomes

$L[q,p]=\int_{0}^{1}\frac{\delta L}{\delta
q^{i}}[tq,p]q^{i}\,dt+d_{\mu}\left(\sum_{k=0}^{r-1}P_{i}^{\mu\mu_{1}...\mu_{k}}[tq,p]q_{,\mu_{1}...\mu_{k}}^{i}\right)+L[0,p],$
(D.17)

where the Lagrangian momenta

$P_{i}^{\mu_{1}...\mu_{k}}[q,p]=\sum_{l=0}^{r-k}\left(-1\right)^{l}d_{\nu_{1}}\dots
d_{\nu_{l}}\frac{\partial L}{\partial
q_{,\mu_{1}...\mu_{k}\nu_{1}...\nu_{l}}^{i}}[q,p]$ (D.18)

are taken with respect to the variables $q^{i}$ only. Note that the term
$L[0,p]$ at the end is no longer functionally constant and still depends on
the other field variables $p^{\alpha}$. Actually if in addition to the
condition on the space of the field variables we had also assumed the space of
independent variables to be star-shaped then we would be able to write in
(D.15) $L_{0}(x)=d_{\mu}\int_{0}^{1}t^{n-1}L_{0}(tx)x^{\mu}\,dt$ and hence
find that under these more restrictive conditions every null Lagrangian is a
total divergence without the functionally constant term. But the same result
does not extend to the multi-field case (D.17) as $L[0,p]$ still depends
functionally on $p$.

######  Remark.

The original solution of the total divergence problem (i.e. the current (D.1))
is due to Horndeski [46]. The explicit solution (D.2) of the inverse
variational problem has been derived by Vainberg [47] and Tonti [48]. In the
modern context, the contents of this Appendix are considered as part of the
differential calculus on jet spaces [27, 29, 43].

##  Appendix E Rigged surfaces

###  E.1 Elementary definitions

Here we review the notion of a _rigged_(_hyper_ -)_surface_ in a pseudo-
Riemannian space. The concept has been originally introduced by Schouten [49]
and have been employed in GR to great effect by Mars [11] and Mars and
Senovilla [8]. For detailed discussion and proofs we refer to the latter.
Previously the author has also used rigged surfaces to derive a general set of
junction conditions in GR through a variational principle [40], the notations
introduced here are similar.

Let $M$ be an $n+1$ dimensional smooth manifold with a pseudo-Riemannian
metric $g$. Let $\Sigma\subseteq M$ be a surface with embedding
$i:\Sigma\rightarrow M$. The induced metric is $h:=i^{\ast}g$, the pullback of
$g$ to $\Sigma$. It may happen that $h$ fails to be a pseudo-Riemannian
metric. A _null point_ of $\Sigma$ is a point where $h$ is degenerate. Recall
that at any $x\in\Sigma$, the $T_{x}\Sigma$ is canonically a subspace of
$T_{x}M$, hence for any vector, and more generally, a contravariant tensor, we
may ask whether it is tangent to $\Sigma$ or not. However for the dual tangent
spaces we have $T_{x}^{\ast}\Sigma=T_{x}^{\ast}M/T_{x}^{0}\Sigma$, where
$T_{x}^{0}\Sigma$ is the annihilator of $T_{x}\Sigma$, i.e. the set of all
covectors which vanish on vectors tangent to $\Sigma$. Elements of
$T_{x}^{0}\Sigma$ are called _normal_ and this vector space is one
dimensional. It is thus meaningful to ask whether a covector, or more
generally, a covariant tensor, is normal to $\Sigma$.

It should be remarked that contrary to common intuition, it is not meaningful
to talk about a contravariant vector being normal or a covector being tangent
to $\Sigma$, at least not a priori. We may call a vector $N\in T_{x}\Sigma$ to
be normal if $g(N,-)$ is a normal covector, i.e. $N$ is the metric dual of a
normal covector. Let $N_{x}\Sigma\leq T_{x}M$ denote the set of all normal
vectors at $x\in\Sigma$, clearly $\dim N_{x}\Sigma=1$. Then $N_{x}\Sigma$ is
not necessarily disjoint from $T_{x}\Sigma$. One may show that
_$T_{x}M=N_{x}\Sigma\oplus T_{x}\Sigma$ if and only if $x$ is not a null
point_. Whenever $x$ is a null point, we have $N_{x}\Sigma\leq T_{x}\Sigma$.
Thus if $\Sigma$ has null points, then there is no natural complement to
$T\Sigma$ in $\left.TM\right|_{\Sigma}$.

A _rigged surface_ $\left(\Sigma,l\right)$ is a surface $\Sigma$ together with
a vector field $l\in\Gamma(\left.TM\right|_{\Sigma})$ along $\Sigma$, which is
transversal everywhere, i.e. $l_{x}\notin T_{x}\Sigma$ for any $x\in\Sigma$.
Let $\left\langle l\right\rangle_{x}\leq T_{x}M$ be the subspace spanned by
$l$ and $\left\langle l\right\rangle=\bigsqcup_{x\in\Sigma}\left\langle
l\right\rangle_{x}$, in which case we have

$\left.TM\right|_{\Sigma}=\left\langle l\right\rangle\oplus_{\Sigma}T\Sigma.$
(E.1)

The vector field $l$ is called the _rigging vector field_. Note that if $M$
and $\Sigma$ are both orientable (which we suppose), then there is always a
global transversal vector field along $\Sigma$, hence a global rigging. For
any fixed choice of rigging, there is a unique normal covector field
$n\in\Gamma(T^{0}\Sigma)$ satisfying $n(l)=1$.

The rigging is of course not unique, given any rigging $l$, we may subject it
to the transformations

$l^{\ast}=\alpha l,\quad\bar{l}=l+T,$ (E.2)

where $\alpha$ is a smooth, nowhere vanishing function on $\Sigma$ while
$T\in\Gamma(T\Sigma)$ is a vector field along $\Sigma$ tangent to it. The
first kind of transformation preserves the subbundle $\left\langle
l\right\rangle$ and only changes the global section representing it, while the
second class of transformations changes the rigging subbundle. However the
latter transformation preserves the normalization of $n$, i.e.
$n(\bar{l})=n(l)$.

Corresponding to any choice of rigging is the projection operator
$P:\left.TM\right|_{\Sigma}\rightarrow T\Sigma$ acting as $PX=X-n(X)l$ and its
linear adjoint
$P^{\ast}:T^{\ast}\Sigma\rightarrow\left.T^{\ast}M\right|_{\Sigma}$,
$(P^{\ast}\omega)(X)=\omega(PX)$ for any $\omega\in\Gamma(T^{\ast}\Sigma)$ and
$X\in\Gamma(\left.TM\right|_{\Sigma})$. In addition to the induced metric
$h=i^{\ast}g$ we also define $\lambda:=i^{\ast}l_{\flat}$, where $l_{\flat}$
is the metric dual of $l$, $\nu:=Pn^{\sharp}$, where $n^{\sharp}$ is the
metric dual of $n$, $\varepsilon(n)=g(n,n)$ and $\varepsilon(l)=g(l,l)$.

Let $X,Y\in\Gamma(T\Sigma)$ be vector fields along $\Sigma$ tangent to it and
$\omega\in\Gamma(T^{\ast}\Sigma)$ a covector field on $\Sigma$. We define:

$\displaystyle\mathcal{K}(X,Y)$ $\displaystyle:=\left(\nabla_{X}n\right)(Y),$
$\displaystyle\varphi(X)$ $\displaystyle:=n(\nabla_{X}l),$
$\displaystyle\psi(\omega,X)$ $\displaystyle:=\omega(P\nabla_{X}l),$
$\displaystyle D_{X}Y$ $\displaystyle:=P\nabla_{X}Y.$ (E.3)

Then $\mathcal{K}$ is a symmetric type $(0,2)$, $\varphi$ a type $(0,1)$ and
$\psi$ a type $(1,1)$ tensor field on $\Sigma$, while $D$ is a linear
connection. The tensor fields $\mathcal{K},\varphi,\psi$ are the analogues of
the extrinsic curvature for a pseudo-Riemannian surface.

###  E.2 Coordinate formulae

Let $y^{a}$ be a coordinate system on $\Sigma$, $x^{\mu}$ a coordiate system
on $M$ with the inclusion $i:\Sigma\rightarrow M$ described by the parametric
relations $x^{\mu}=x^{\mu}(y)$. Set

$e_{a}^{\mu}:=\frac{\partial x^{\mu}}{\partial y^{a}}.$ (E.4)

The rigging and the associated normal have components $l^{\mu}$ and $n_{\mu}$.
The set of vectors $\left(l^{\mu},e_{1}^{\mu},\dots,e_{n}^{\mu}\right)$ form a
direct frame along $\Sigma$. The corresponding dual frame is
$\left(n_{\mu},\vartheta_{\mu}^{1},\dots,\vartheta_{\mu}^{n}\right)$, where
the $\vartheta_{\mu}^{a}$ are uniquely determined by the duality relations

$\vartheta_{\mu}^{a}e_{b}^{\mu}=\delta_{b}^{a},\quad\vartheta_{\mu}^{a}l^{\mu}=0.$
(E.5)

The projection operator is $P_{\
\nu}^{\mu}=\delta_{\nu}^{\mu}-l^{\mu}n_{\nu}$. A vector field $X$ along
$\Sigma$ is tangent if and only if it can be written in the form
$X^{\mu}=X^{a}e_{a}^{\mu}$, hence in mixed bulk-surface coordinates, the
projection operator is described by the coefficients $\vartheta_{\mu}^{a}$.
The induced metric is

$h_{ab}=e_{a}^{\mu}e_{b}^{\nu}g_{\mu\nu},$ (E.6)

and we have

$\lambda_{a}=l_{\mu}e_{a}^{\mu},\quad\nu^{a}=\vartheta_{\mu}^{a}n^{\mu}$ (E.7)

with the completeness relations

$\displaystyle g^{\mu\nu}$
$\displaystyle=\varepsilon(n)l^{\mu}l^{\nu}+\nu^{a}\left(l^{\mu}e_{a}^{\nu}+e_{a}^{\mu}l^{\nu}\right)+h_{\ast}^{ab}e_{a}^{\mu}e_{b}^{\nu},$
$\displaystyle g_{\mu\nu}$
$\displaystyle=\varepsilon(l)n_{\mu}n_{\nu}+\lambda_{a}\left(n_{\mu}\vartheta_{\nu}^{a}+\vartheta_{\mu}^{a}n_{\nu}\right)+h_{ab}\vartheta_{\mu}^{a}\vartheta_{\nu}^{b},$
(E.8)

where

$h_{\ast}^{ab}=g^{\mu\nu}\vartheta_{\mu}^{a}\vartheta_{\nu}^{b},$ (E.9)

and we sometimes write $n_{\ast}^{\mu}=\nu^{a}e_{a}^{\mu}$ and
$h_{\ast}^{\mu\nu}=h_{\ast}^{ab}e_{a}^{\mu}e_{b}^{\nu}$.

The extrinsic curvature quantities have the coordinate expressions

$\displaystyle\mathcal{K}_{ab}$
$\displaystyle=e_{a}^{\mu}e_{b}^{\nu}\nabla_{\mu}n_{\nu},$
$\displaystyle\varphi_{a}$
$\displaystyle=e_{a}^{\mu}n_{\nu}\nabla_{\mu}l^{\nu},$
$\displaystyle\psi_{b}^{a}$
$\displaystyle=e_{b}^{\mu}\vartheta_{\nu}^{a}\nabla_{\mu}l^{\nu},$ (E.10)

while for the covariant derivative we have

$D_{a}X^{c}\equiv X_{||a}^{c}=\partial_{a}X^{c}+\gamma_{\ ab}^{c}X^{b},$
(E.11)

where

$\gamma_{\ ab}^{c}=e_{a}^{\mu}\vartheta_{\nu}^{c}\nabla_{\mu}e_{b}^{\nu}.$
(E.12)

Let us now smoothly extend $l^{\mu}$ off $\Sigma$ and introduce coordinates
$\left(x^{0},y^{a}\right)$ adapted to $l^{\mu}$ in the sense that the surface
coordinates $y^{a}$ are Lie transported off $\Sigma$ along $l^{\mu}$. We also
extend $n_{\mu}$ such that the relations $l^{\mu}n_{\mu}=1$ are preserved.
Equality in adapted coordinates is denoted $\overset{\ast}{=}$. Then we have

$\displaystyle l^{\mu}$ $\displaystyle\overset{\ast}{=}\delta_{0}^{\mu},\quad
n_{\mu}\overset{\ast}{=}\delta_{\mu}^{0},\quad
n^{0}\overset{\ast}{=}g^{00}\overset{\ast}{=}\varepsilon(n),\quad
l_{0}\overset{\ast}{=}g_{00}\overset{\ast}{=}\varepsilon(l),$ $\displaystyle
n^{a}$ $\displaystyle\overset{\ast}{=}\nu^{a},\quad
l_{a}\overset{\ast}{=}\lambda_{a},\quad g_{ab}\overset{\ast}{=}h_{ab},\quad
g^{ab}\overset{\ast}{=}h_{\ast}^{ab},$ (E.13)

while for the derivative quantities we have

$\mathcal{K}_{ab}\overset{\ast}{=}-\Gamma_{\
ab}^{0},\quad\varphi_{a}\overset{\ast}{=}\Gamma_{\
0a}^{0},\quad\psi_{b}^{a}\overset{\ast}{=}\Gamma_{\ b0}^{a}$ (E.14)

and

$\gamma_{\ ab}^{c}\overset{\ast}{=}\Gamma_{\ ab}^{c}.$ (E.15)

###  E.3 Pseudo-Riemannian limit

Suppose now that $\Sigma$ is timelike or spacelike. In this case we can choose
$l^{\mu}$ to be the (unique up to sign) unit normal of $\Sigma$. Then

$\varepsilon:=\varepsilon(l)=\pm 1=\begin{cases}+1&\Sigma\text{ is
timelike}\\\ -1&\Sigma\text{ is spacelike}\end{cases},$ (E.16)

and we have

$n_{\mu}=\varepsilon l_{\mu},\quad\varepsilon(n)=\varepsilon.$ (E.17)

Recall that

$K_{ab}:=e_{a}^{\mu}e_{b}^{\nu}\nabla_{\mu}l_{\nu}$ (E.18)

is the extrinsic curvature of the surface. With the above made choice of
$l^{\mu}$,

$h_{\ast}^{ab}=h^{ab},\quad\nu^{a}=\lambda_{a}=0,$ (E.19)

where $h^{ab}$ is the inverse of $h_{ab}$ and furthermore

$\mathcal{K}_{ab}=\varepsilon
K_{ab},\quad\varphi_{a}=0,\quad\psi_{b}^{a}=K_{b}^{a},$ (E.20)

while $D$ reduces to the Levi-Civita connection of $h_{ab}$.

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