Variational formalism for generic shells in general relativity

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

$$\begin{equation}{x}^{\mu }={{\Phi}}^{\mu }\left({y}^{1},\dots ,{y}^{D}\right),\end{equation} \tag{ 2 }$$

where the y^a are the intrinsic coordinates of Σ. The derivatives

$$\begin{equation}{e}_{a}^{\mu }{:=}\frac{\partial {{\Phi}}^{\mu }}{\partial {y}^{a}}\end{equation} \tag{ 3 }$$

are interpreted as the components of the holonomic coordinate frame of Σ, or from a more invariant point of view, the components of the pushforward and pullback operations between Σ and M. Without introducing any extra structure, a vector v^μ defined at a point p ∈ Σ 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^μ 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_μ defined at some point p ∈ Σ 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 ω_μ is a covariant tensor at some p ∈ Σ, its pullback to Σ 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 Σ is the pullback

$$\begin{equation}{h}_{ab}={g}_{\mu \nu }{e}_{a}^{\mu }{e}_{b}^{\nu }.\end{equation} \tag{ 4 }$$

The point p ∈ Σ 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 ℓ^μ along Σ, 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 $\left(\ell ,{e}_{1},\dots ,{e}_{D}\right)$ is then a frame of M along Σ. The choice of rigging selects a unique normal covector field n_μ which satisfies

$$\begin{equation}{n}_{\mu }{\ell }^{\mu }=1.\end{equation} \tag{ 5 }$$

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

$$\begin{equation}{\vartheta }_{\mu }^{a}{\ell }^{\mu }=0,\quad {\vartheta }_{\mu }^{a}{e}_{b}^{\mu }={\delta }_{b}^{a}.\end{equation} \tag{ 6 }$$

Using ${\vartheta }_{\mu }^{a}$, given a hypersurface covector ω_a , we can create a spacetime covector ${\omega }_{\mu }={\vartheta }_{\mu }^{a}{\omega }_{a}$ which satisfies ${\omega }_{a}={e}_{a}^{\mu }{\omega }_{\mu }$ and ω_μ ℓ^μ = 0. Likewise, we can project a spacetime vector v^μ into Σ as ${v}_{{\Vert}}^{a}={v}^{\mu }{\vartheta }_{\mu }^{a}$, and also obtain a direct projection operator ${P}_{\nu }^{\mu }$ by pushing forward ${v}_{{\Vert}}^{a}$, i.e.

$$\begin{equation}{v}_{{\Vert}}^{\mu }={v}_{{\Vert}}^{a}{e}_{a}^{\mu }={v}^{\nu }{\vartheta }_{\nu }^{a}{e}_{a}^{\mu }={P}_{\nu }^{\mu }{v}^{\nu },\end{equation} \tag{ 7 }$$

with

$$\begin{equation}{P}_{\nu }^{\mu }={e}_{a}^{\mu }{\vartheta }_{\nu }^{a}={\delta }_{\nu }^{\mu }-{\ell }^{\mu }{n}_{\nu }.\end{equation} \tag{ 8 }$$

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

$$\begin{align}\hfill {g}_{\mu \nu }& ={\ell }^{2}{n}_{\mu }{n}_{\nu }+{\lambda }_{a}\left({n}_{\mu }{\vartheta }_{\nu }^{a}+{\vartheta }_{\mu }^{a}{n}_{\nu }\right)+{h}_{ab}{\vartheta }_{\mu }^{a}{\vartheta }_{\nu }^{b},\hfill \\ \hfill {g}^{\mu \nu }& ={n}^{2}{\ell }^{\mu }{\ell }^{\nu }+{\nu }^{a}\left({\ell }^{\mu }{e}_{a}^{\nu }+{e}_{a}^{\mu }{\ell }^{\nu }\right)+{h}_{\ast }^{ab}{e}_{a}^{\mu }{e}_{b}^{\nu },\hfill \end{align} \tag{ 9 }$$

where the elements that appear here are given explicitly as

$$\begin{align}\hfill {\ell }^{2}& ={\ell }_{\mu }{\ell }^{\mu }=\ell \cdot \ell ,\hspace{25.0pt}{n}^{2}={n}_{\mu }{n}^{\mu }=n\cdot n,\hfill \\ \hfill {\lambda }_{a}& ={\ell }_{\mu }{e}_{a}^{\mu }=\ell \cdot {e}_{a},\hspace{25.0pt}{\nu }^{a}={n}^{\mu }{\vartheta }_{\mu }^{a}=n\cdot {\vartheta }^{a},\hfill \\ \hfill {h}_{\ast }^{ab}& ={g}^{\mu \nu }{\vartheta }_{\mu }^{a}{\vartheta }_{\nu }^{b}={\vartheta }^{a}\cdot {\vartheta }^{b}.\hfill \end{align} \tag{ 10 }$$

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

The choice of rigging also gives a volume form

$$\begin{equation}{\mu }_{\ell ,g}={\rho }_{\ell ,g}\enspace \mathrm{d}{y}^{1}\wedge \dots \wedge \mathrm{d}{y}^{D}\end{equation} \tag{ 11 }$$

on Σ with

$$\begin{equation}{\rho }_{\ell ,g}=\sqrt{-\mathfrak{g}}{\ell }^{\mu }{\pi }_{\mu {\mu }_{1}\dots {\mu }_{D}}{e}_{1}^{{\mu }_{1}}\dots {e}_{D}^{{\mu }_{D}},\end{equation} \tag{ 12 }$$

where ${\pi }_{\mu {\mu }_{1}\dots {\mu }_{D}}$ is the D + 1 dimensional Levi-Civita symbol. This particular volume element is such that if Ω ⊆ M is a compact D + 1 dimensional domain of integration, whose boundary ∂Ω is rigged with an outward pointing transversal ℓ^μ , Gauss' theorem takes the form

$$\begin{equation}{\int }_{{\Omega}}{\nabla }_{\mu }{X}^{\mu }\enspace {\mu }_{g}={\oint }_{\partial {\Omega}}{n}_{\mu }{X}^{\mu }\enspace {\mu }_{\ell ,g},\end{equation} \tag{ 13 }$$

where n_μ is the normal adapted to the rigging, i.e. n_μ ℓ^μ = 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

$$\begin{equation}{\chi }_{ab}=-{n}_{\nu }{e}_{a}^{\mu }{\nabla }_{\mu }{e}_{b}^{\nu }={e}_{a}^{\mu }{e}_{b}^{\nu }{\nabla }_{\mu }{n}_{\nu },\end{equation} \tag{ 14 }$$

$$\begin{equation}{\varphi }_{a}={n}_{\nu }{e}_{a}^{\mu }{\nabla }_{\mu }{\ell }^{\nu }=-{e}_{a}^{\mu }{\ell }^{\nu }{\nabla }_{\mu }{n}_{\nu },\end{equation} \tag{ 15 }$$

$$\begin{equation}{\psi }_{a}^{b}={e}_{a}^{\mu }{\vartheta }_{\nu }^{b}{\nabla }_{\mu }{\ell }^{\nu }=-{e}_{a}^{\mu }{\ell }^{\nu }{\nabla }_{\mu }{\vartheta }_{\nu }^{b}.\end{equation} \tag{ 16 }$$

These are all hypersurface tensors and χ_ab is symmetric. For thin shells and junction conditions it is also useful to define

$$\begin{equation}{H}_{ab}={e}_{a}^{\mu }{e}_{b}^{\nu }{\nabla }_{\mu }{\ell }_{\nu }={\psi }_{a}^{c}{h}_{bc}+{\varphi }_{a}{\lambda }_{b},\end{equation} \tag{ 17 }$$

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

$$\begin{align}\hfill {H}_{b}^{a}& ={h}_{\ast }^{ac}{H}_{cb},\hspace{25.0pt}{H}_{a}^{b}={h}_{\ast }^{bc}{H}_{ac},\hspace{25.0pt}{H}^{ab}={h}_{\ast }^{ac}{h}_{\ast }^{bd}{H}_{cd},\hfill \\ \hfill H& ={H}_{a}^{a}={H}_{ab}{h}_{\ast }^{ab}.\hfill \end{align} \tag{ 18 }$$

A connection-type quantity ${\gamma }_{ab}^{c}$ is also given on Σ by

$$\begin{equation}{\gamma }_{ab}^{c}={\vartheta }_{\nu }^{c}{e}_{a}^{\mu }{\nabla }_{\mu }{e}_{b}^{\nu }=-{e}_{a}^{\mu }{e}_{b}^{\nu }{\nabla }_{\mu }{\vartheta }_{\nu }^{c}.\end{equation} \tag{ 19 }$$

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

The choice of rigging ℓ^μ along a hypersurface Σ ⊆ M is not unique and it may be subjected to two kinds of transformations. The first is a tangential shift

$$\begin{equation}{\ell }^{\prime \mu }={\ell }^{\mu }+{T}^{a}{e}_{a}^{\mu },\end{equation} \tag{ 20 }$$

where T^a is an arbitrary tangent vector field to Σ, and the second kind is a rescaling

$$\begin{equation}{\bar{\ell }}^{\mu }=\alpha {\ell }^{\mu },\end{equation} \tag{ 21 }$$

where α is a function along Σ 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]

$$\begin{align}\hfill {\ell }^{\prime \mu }& ={\ell }^{\mu }+{T}^{a}{e}_{a}^{\mu },\hfill \\ \hfill {\vartheta }_{\mu }^{\prime a}& ={\vartheta }_{\mu }^{a}-{T}^{a}{n}_{\mu },\hfill \\ \hfill {\ell }^{\prime 2}& ={\ell }^{2}+2{T}^{a}{\lambda }_{a}+{h}_{ab}{T}^{a}{T}^{b},\hfill \\ \hfill {\lambda }_{a}^{\prime }& ={\lambda }_{a}+{h}_{ab}{T}^{b},\hfill \\ \hfill {\nu }^{\prime a}& ={\nu }^{a}-{n}^{2}{T}^{a},\hfill \\ \hfill {h}_{\ast }^{\prime ab}& ={h}^{ab}-{\nu }^{a}{T}^{b}-{T}^{a}{\nu }^{b}+{n}^{2}{T}^{a}{T}^{b},\hfill \\ \hfill {\varphi }_{a}^{\prime }& ={\varphi }_{a}-{\chi }_{ab}{T}^{b},\hfill \\ \hfill {\psi }_{a}^{\prime b}& ={\psi }_{a}^{b}+{\varphi }_{a}{T}^{b}+{D}_{a}{T}^{b}+{\chi }_{ac}{T}^{b}{T}^{c},\hfill \\ \hfill {\gamma }_{ab}^{\prime c}& ={\gamma }_{ab}^{c}+{\chi }_{ab}{T}^{c},\hfill \\ \hfill {H}_{ab}^{\prime }& ={H}_{ab}+{D}_{a}{T}^{c}{h}_{bc}-{\chi }_{ac}{\lambda }_{b}{T}^{c},\hfill \end{align} \tag{ 22 }$$

while ${e}_{a}^{\mu }$, n_μ , h_ab , χ_ab and μ_ℓ,g are invariant. Under a rescaling, the transformations are

$$\begin{align}\hfill {\bar{\ell }}^{\mu }& =\alpha {\ell }^{\mu },\hfill \\ \hfill {\bar{n}}_{\mu }& ={\alpha }^{-1}{n}_{\mu },\hfill \\ \hfill {\bar{\ell }}^{2}& ={\alpha }^{2}{\ell }^{2},\hfill \\ \hfill {\bar{n}}^{2}& ={\alpha }^{-1}{n}^{2},\hfill \\ \hfill {\bar{\lambda }}_{a}& =\alpha {\lambda }_{a},\hfill \\ \hfill {\bar{\nu }}^{a}& ={\alpha }^{-1}{\nu }^{a}\hfill \\ \hfill {\bar{\chi }}_{ab}& ={\alpha }^{-1}{\chi }_{ab},\hfill \\ \hfill {\bar{\varphi }}_{a}& ={\varphi }_{a}+{\partial }_{a}\enspace \mathrm{ln}\enspace \alpha ,\hfill \\ \hfill {\bar{\psi }}_{a}^{b}& =\alpha {\psi }_{a}^{b},\hfill \\ \hfill {\bar{H}}_{ab}& =\alpha {H}_{ab}+{\partial }_{a}\alpha {\lambda }_{b},\hfill \\ \hfill {\bar{\mu }}_{\ell ,g}& =\alpha {\mu }_{\ell ,g},\hfill \end{align} \tag{ 23 }$$

while ${e}_{a}^{\mu }$, ${\vartheta }_{\mu }^{a}$, h_ab , ${h}_{\ast }^{ab}$ and ${\gamma }_{ab}^{c}$ are invariant. Note that since the volume element μ_ℓ,g is invariant under shifts, the definition of μ_ℓ,g essentially depends on that of the normal n_μ 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.

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 ℓ^μ . We assume that Σ is timelike or spacelike and set

$$\begin{equation}{\epsilon}=\pm 1=\begin{cases}+1\quad \hfill & \mathbf{\Sigma }\enspace \text{is}\;\text{timelike}\hfill \\ -1\quad \hfill & \mathbf{\Sigma }\enspace \text{is}\;\text{spacelike}\hfill \end{cases}\end{equation} \tag{ 24 }$$

to allow for both cases to be considered simultaneously. The induced metric h_ab is nondegenerate throughout Σ, 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

$$\begin{equation}{\ell }^{\mu }={\hat{n}}^{\mu }\end{equation} \tag{ 25 }$$

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

$$\begin{equation}{n}_{\mu }={\epsilon}{\hat{n}}_{\mu }.\end{equation} \tag{ 26 }$$

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

$$\begin{align}\hfill {\vartheta }_{\mu }^{a}& ={e}_{\mu }^{a}={h}^{ab}{g}_{\mu \nu }{e}_{b}^{\nu },\hfill \\ \hfill {\ell }^{2}& ={n}^{2}={\epsilon},\hfill \\ \hfill {\lambda }_{a}& ={\nu }^{a}=0,\hfill \\ \hfill {h}_{\ast }^{ab}& ={h}^{ab},\hfill \\ \hfill {\chi }_{ab}& ={\epsilon}{K}_{ab},\hfill \\ \hfill {\varphi }_{a}& =0,\hfill \\ \hfill {\psi }_{a}^{b}& ={K}_{a}^{b},\hfill \\ \hfill {H}_{ab}& ={K}_{ab},\hfill \\ \hfill {\mu }_{\ell ,g}& ={\mu }_{h}=\sqrt{-{\epsilon}\mathfrak{h}}\enspace \mathrm{d}{y}^{1}\wedge \dots \wedge \mathrm{d}{y}^{D},\hfill \end{align} \tag{ 27 }$$

where

$$\begin{equation}{K}_{ab}={e}_{a}^{\mu }{e}_{b}^{\nu }{\nabla }_{\mu }{\hat{n}}_{\nu }=\frac{1}{2}{e}_{a}^{\mu }{e}_{b}^{\nu }{\mathcal{L}}_{\hat{n}}{g}_{\mu \nu }\end{equation} \tag{ 28 }$$

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

$$\begin{equation}{\gamma }_{ab}^{c}=\frac{1}{2}{h}^{cd}\left({\partial }_{a}{h}_{bd}+{\partial }_{b}{h}_{ad}-{\partial }_{d}{h}_{ab}\right)\end{equation} \tag{ 29 }$$

and Gauss' theorem takes the form

$$\begin{equation}{\int }_{{\Omega}}{\nabla }_{\mu }{X}^{\mu }\enspace {\mu }_{g}{=\oint }_{\partial {\Omega}}{\epsilon}{\hat{n}}_{\mu }{X}^{\mu }\enspace {\mu }_{h},\end{equation} \tag{ 30 }$$

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

Suppose now that Σ is null. There is no universally preferred convention here for the rigging, however the null rigging used by e.g. Poisson [11] is a useful choice and we present it here. If Σ is null then any normal field n_μ is also null and is tangential to Σ. Moreover it satisfies the geodesic equation

$$\begin{equation}{\left({\nabla }_{n}n\right)}^{\mu }=\kappa {n}^{\mu }\end{equation} \tag{ 31 }$$

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

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

and choose a null ℓ^μ rigging which satisfies

$$\begin{equation}{\ell }^{\mu }{\ell }_{\mu }=0,\hspace{25.0pt}{\ell }^{\mu }{n}_{\mu }=1,\hspace{25.0pt}{\ell }_{\mu }{e}_{A}^{\mu }=0,\end{equation} \tag{ 33 }$$

where

$$\begin{equation}{e}_{A}^{\mu }={\left(\frac{\partial }{\partial {\theta }^{A}}\right)}^{\mu }\end{equation} \tag{ 34 }$$

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

$$\begin{equation}{q}_{AB}={e}_{A}^{\mu }{e}_{B}^{\nu }{g}_{\mu \nu }\end{equation} \tag{ 35 }$$

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

$$\begin{equation}\left({h}_{ab}\right)=\left(\begin{matrix}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \left({q}_{AB}\right)\hfill \end{matrix}\right).\end{equation} \tag{ 36 }$$

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 H₁₁, H_1A and H_AB . We have

$$\begin{align}\hfill {H}_{11}& ={n}^{\mu }{n}^{\nu }{\nabla }_{\mu }{\ell }_{\nu }=-{n}^{\mu }{\ell }_{\nu }{\nabla }_{\mu }{n}^{\nu }=-\kappa ,\hfill \\ \hfill {H}_{1A}& ={H}_{A1}={e}_{A}^{\mu }{n}^{\nu }{\nabla }_{\mu }{\ell }_{\nu },\hfill \\ \hfill {H}_{AB}& ={e}_{A}^{\mu }{e}_{B}^{\nu }{\nabla }_{\mu }{\ell }_{\nu }.\hfill \end{align} \tag{ 37 }$$

We may express most of the quantities with H_ab as

$$\begin{align}\hfill {\varphi }_{1}& ={n}^{\mu }{n}_{\nu }{\nabla }_{\mu }{\ell }^{\nu }=-\kappa ={H}_{11},\hfill \\ \hfill {\varphi }_{A}& ={e}_{A}^{\mu }{n}_{\nu }{\nabla }_{\mu }{\ell }^{\nu }={H}_{A1},\hfill \\ \hfill {\psi }_{1}^{1}& ={n}^{\mu }{\ell }_{\nu }{\nabla }_{\mu }{\ell }^{\nu }=\frac{1}{2}{n}^{\mu }{\nabla }_{\mu }\left({\ell }_{\nu }{\ell }^{\nu }\right)=0,\hfill \\ \hfill {\psi }_{A}^{1}& ={e}_{A}^{\mu }{\ell }_{\nu }{\nabla }_{\mu }{\ell }^{\nu }=0,\hfill \\ \hfill {\psi }_{1}^{A}& ={n}^{\mu }{e}_{\nu }^{A}{\nabla }_{\mu }{\ell }^{\nu }={H}_{1}^{A},\hfill \\ \hfill {\psi }_{A}^{B}& ={H}_{A}^{B}.\hfill \end{align} \tag{ 38 }$$

The primary exception is χ_ab , which is

$$\begin{align}\hfill {\chi }_{11}& ={n}^{\mu }{n}^{\nu }{\nabla }_{\mu }{n}_{\nu }=\kappa {n}^{\nu }{n}_{\nu }=0,\hfill \\ \hfill {\chi }_{1A}& ={n}^{\mu }{e}_{A}^{\nu }{\nabla }_{\mu }{n}_{\nu }=\kappa {e}_{A}^{\nu }{n}_{\nu }=0,\hfill \\ \hfill {\chi }_{AB}& ={e}_{A}^{\mu }{e}_{B}^{\nu }{\nabla }_{\mu }{n}_{\nu },\hfill \end{align} \tag{ 39 }$$

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

$$\begin{equation}\left({g}_{\mu \nu }\right)=\left(\begin{matrix}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \left({q}_{AB}\right)\hfill \end{matrix}\right),\end{equation} \tag{ 40 }$$

from which it follows that in this frame

$$\begin{equation}\mathfrak{g}=-\mathfrak{q},\end{equation} \tag{ 41 }$$

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

$$\begin{equation}{\mu }_{\ell ,g}={\mu }_{q}=\sqrt{\mathfrak{q}}\enspace \mathrm{d}r\wedge \enspace \mathrm{d}{\theta }^{2}\dots \wedge \mathrm{d}{\theta }^{D}.\end{equation} \tag{ 42 }$$

Note that while it appears that the volume element μ_q is canonically given, it does depend on the way the manifold Σ has been sliced into spacelike D − 1-surfaces.

The total action will be taken to consist of the gravitational action S_EH, an unspecified bulk matter action S_M and an unspecified thin shell matter action S_TS. Instead of integrating over M at once, we split the integrals into sums of integrals over M⁺ and M⁻. Since Σ is not a part of the outer boundary of the manifold, the usual Dirichlet conditions do not apply to Σ, the metric is not fixed there. We suppose the metric is C⁰ across Σ and at least C³ away from Σ. Since we are varying within this differentiability class, δg_μν 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 Σ is an interior boundary and thus we add the rigged counterterm (56) to the action at both sides of Σ to ensure the proper boundary behaviour of the action. The total action is then

$$\begin{align}\hfill S& =\underset{{S}_{\text{EH}}^{+}}{\underbrace{\frac{1}{2\varkappa }{\int }_{{M}^{+}}{R}^{+}{\mu }_{g}}}+\underset{{S}_{\text{EH}}^{-}}{\underbrace{\frac{1}{2\varkappa }{\int }_{{M}^{-}}{R}^{-}{\mu }_{g}}}+\underset{{S}_{\mathrm{M}}^{+}}{\underbrace{{\int }_{{M}^{+}}{\mathcal{L}}_{\mathrm{M}}^{+}\enspace {\mathrm{d}}^{D+1}x}}+\underset{{S}_{\mathrm{M}}^{-}}{\underbrace{{\int }_{{M}^{-}}{\mathcal{L}}_{\mathrm{M}}^{-}\enspace {\mathrm{d}}^{D+1}x}}\hfill \\ \hfill & \quad \underset{{B}_{\mathrm{R}}^{+}}{\underbrace{-\frac{1}{\varkappa }{\int }_{{\Sigma}}{\left({P}_{\nu }^{\mu }{\nabla }_{\mu }{n}^{\nu }\right)}_{+}{\mu }_{g,\ell }}}+\underset{{B}_{\mathrm{R}}^{-}}{\underbrace{\frac{1}{\varkappa }{\int }_{{\Sigma}}{\left({P}_{\nu }^{\mu }{\nabla }_{\mu }{n}^{\nu }\right)}_{-}{\mu }_{g,\ell }}}+\underset{{S}_{\text{TS}}}{\underbrace{{\int }_{{\Sigma}}{\mathcal{L}}_{\text{TS}}\enspace {\mathrm{d}}^{D}y}}.\hfill \end{align} \tag{ 66 }$$

The relative sign difference between ${B}_{\mathrm{R}}^{+}$ and ${B}_{\mathrm{R}}^{-}$ is caused by the orientation of Σ 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

$$\begin{align}\hfill \delta S& ={\int }_{{M}^{+}}\frac{1}{2}\left({T}_{+}^{\mu \nu }-\frac{1}{\varkappa }{G}_{+}^{\mu \nu }\right)\delta {g}_{\mu \nu }\enspace {\mu }_{g}+{\int }_{{M}^{-}}\frac{1}{2}\left({T}_{-}^{\mu \nu }-\frac{1}{\varkappa }{G}_{-}^{\mu \nu }\right)\delta {g}_{\mu \nu }\enspace {\mu }_{g}\hfill \\ \hfill & \quad +{\int }_{{\Sigma}}\left(\frac{1}{{\rho }_{\ell ,g}}\frac{\delta {S}_{\text{TS}}}{\delta {g}_{\mu \nu }}-{n}_{\kappa }\left[\frac{\partial {L}_{\mathrm{M}}}{\partial {g}_{\mu \nu ,\kappa }}\right]-\frac{1}{2\varkappa }\left[{{\Pi}}^{\mu \nu }\right]\right)\delta {g}_{\mu \nu }\enspace {\mu }_{\ell ,g},\hfill \end{align} \tag{ 67 }$$

where ${L}_{\mathrm{M}}={\mathcal{L}}_{\mathrm{M}}/\sqrt{-\mathfrak{g}}$ is the scalarized matter Lagrangian. The variation of the integral should vanish for all variations δg_μν that are C⁰ across Σ and C³ away from Σ. In particular, we can choose an arbitrary δg_μν which satisfies ${\left.\delta {g}_{\mu \nu }\right\vert }_{{\Sigma}}=0$, which implies that the coefficients of the δg_μν in the bulk integrals should vanish, giving the Einstein field equations in the bulk. It then follows that the surface term ${\int }_{{\Sigma}}\left(\cdots \right)\delta {g}_{\mu \nu }\enspace {\mu }_{\ell ,g}$ should vanish separately even for arbitrary δg_μν , which results in the equation

$$\begin{equation}{S}^{\mu \nu }=\frac{1}{\varkappa }\left[{{\Pi}}^{\mu \nu }\right],\end{equation} \tag{ 68 }$$

where

$$\begin{equation}{S}^{\mu \nu }=\frac{2}{{\rho }_{\ell ,g}}\frac{\delta {S}_{\text{TS}}}{\delta {g}_{\mu \nu }}-2{n}_{\kappa }\left[\frac{\partial {L}_{\mathrm{M}}}{\partial {g}_{\mu \nu ,\kappa }}\right]\end{equation} \tag{ 69 }$$

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_M is the matter action with ${S}_{\mathrm{M}}={\int }_{M}{\mathcal{L}}_{\mathrm{M}}\left(g,\partial g,\psi ,\partial \psi \right){\mathrm{d}}^{D+1}x$, 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

$$\begin{equation}\delta {S}_{\mathrm{M}}={\int }_{M}\left(\left(\frac{\partial {\mathcal{L}}_{\mathrm{M}}}{\partial {g}_{\mu \nu }}-{\partial }_{\kappa }\frac{\partial {\mathcal{L}}_{\mathrm{M}}}{\partial {g}_{\mu \nu ,\kappa }}\right)\delta {g}_{\mu \nu }+{\partial }_{\kappa }\left(\frac{\partial {\mathcal{L}}_{\mathrm{M}}}{\partial {g}_{\mu \nu ,\kappa }}\delta {g}_{\mu \nu }\right)\right){\mathrm{d}}^{D+1}x.\end{equation} \tag{ 70 }$$

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

$$\begin{equation}\frac{\partial {\mathcal{L}}_{\mathrm{M}}}{\partial {g}_{\mu \nu ,\kappa }}=\frac{\partial {L}_{\mathrm{M}}}{\partial {g}_{\mu \nu ,\kappa }}\sqrt{-\mathfrak{g}},\end{equation} \tag{ 71 }$$

and even though g_μν,κ is not a tensor, ∂L_M/∂g_μν,κ is (see [15], chapter 2, section 11). We can therefore write

$$\begin{align}\hfill \delta {S}_{\mathrm{M}}& =\mathrm{B}\mathrm{u}\mathrm{l}\mathrm{k}\enspace \mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{s}+{\int }_{M}{\nabla }_{\kappa }\left(\frac{\partial {L}_{\mathrm{M}}}{\partial {g}_{\mu \nu ,\kappa }}\delta {g}_{\mu \nu }\right){\mu }_{g}\hfill \\ \hfill & =\mathrm{B}\mathrm{u}\mathrm{l}\mathrm{k}\enspace \mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{s}+{\oint }_{\partial M}{n}_{\kappa }\frac{\partial {L}_{\mathrm{M}}}{\partial {g}_{\mu \nu ,\kappa }}\delta {g}_{\mu \nu }{\mu }_{\ell ,g}.\hfill \end{align} \tag{ 72 }$$

If this integral is performed over a spacetime with a shell Σ 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 Π^μν 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 ℓ (i.e. ${\ell }^{\mu }={\left(\partial /\partial \sigma \right)}^{\mu }$ and the y^a are the hypersurface coordinates), giving

$$\begin{align}\hfill {{\Pi}}^{00}& ={\chi }_{ab}\left({n}^{2}{h}_{\ast }^{ab}-{\nu }^{a}{\nu }^{b}\right)+{n}^{2}{\varphi }_{a}{\nu }^{a}-{\left({n}^{2}\right)}^{2}\psi ,\hfill \\ \hfill {{\Pi}}^{0a}& ={\chi }_{cd}\left({\nu }^{a}{h}_{\ast }^{cd}-{\nu }^{c}{h}_{\ast }^{ad}\right)+{n}^{2}{\varphi }_{c}{h}_{\ast }^{ac}-{n}^{2}\psi {\nu }^{a},\hfill \\ \hfill {{\Pi}}^{ab}& ={\chi }_{cd}\left({h}_{\ast }^{ab}{h}_{\ast }^{cd}-{h}_{\ast }^{ac}{h}_{\ast }^{bd}\right)+{\varphi }_{c}\left({\nu }^{a}{h}_{\ast }^{bc}-{h}_{\ast }^{ab}{\nu }^{c}+{h}_{\ast }^{ac}{\nu }^{b}\right)-\psi {\nu }^{a}{\nu }^{b},\hfill \end{align} \tag{ 73 }$$

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

Since the metric is continuous, only the extrinsic curvature-type quantities χ_ab , φ_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

$$\begin{align}\hfill \left[{\psi }_{a}^{b}\right]& =\left[{H}_{ac}\right]{h}_{\ast }^{bc},\hfill \\ \hfill \left[{\varphi }_{a}\right]& =\left[{H}_{ab}\right]{\nu }^{b},\hfill \\ \hfill \left[{\chi }_{ab}\right]& ={n}^{2}\left[{H}_{ab}\right],\hfill \\ \hfill \left[{\gamma }_{ab}^{c}\right]& =-\left[{H}_{ab}\right]{\nu }^{c},\hfill \end{align} \tag{ 74 }$$

where

$$\begin{equation}\left[{H}_{ab}\right]={e}_{a}^{\mu }{e}_{b}^{\nu }\left[{\nabla }_{\mu }{\ell }_{\nu }\right]=-{e}_{a}^{\mu }{e}_{b}^{\nu }\left[{{\Gamma}}_{\mu \nu }^{\kappa }\right]{\ell }_{\kappa },\end{equation} \tag{ 75 }$$

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

$$\begin{align}\hfill \left[{\partial }_{\kappa }{g}_{\mu \nu }\right]& ={\delta }_{\kappa }^{\lambda }\left[{\partial }_{\lambda }{g}_{\mu \nu }\right]=\left({e}_{a}^{\lambda }{\vartheta }_{\kappa }^{a}+{\ell }^{\lambda }{n}_{\kappa }\right)\left[{\partial }_{\lambda }{g}_{\mu \nu }\right]\hfill \\ \hfill & =\left[{\partial }_{a}{g}_{\mu \nu }\right]{\vartheta }_{\kappa }^{a}+\left[{g}_{\mu \nu ,\ell }\right]{n}_{\kappa }=\left[{g}_{\mu \nu ,\ell }\right]{n}_{\kappa },\hfill \end{align} \tag{ 76 }$$

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_μν,ℓ = ℓ^κ ∂_κ g_μν is the transversal derivative. We then have

$$\begin{equation}\left[{{\Gamma}}_{\mu \nu }^{\kappa }\right]=\frac{1}{2}\left({n}_{\mu }{\xi }_{\nu }^{\kappa }+{n}_{\nu }{\xi }_{\mu }^{\kappa }-{n}^{\kappa }{\xi }_{\mu \nu }\right),\end{equation} \tag{ 77 }$$

where ${\xi }_{\mu \nu }{:=}\left[{g}_{\mu \nu ,\ell }\right]$ and it follows that

$$\begin{equation}\left[{H}_{ab}\right]=\frac{1}{2}{e}_{a}^{\mu }{e}_{b}^{\nu }{\xi }_{\mu \nu }=\frac{1}{2}{e}_{a}^{\mu }{e}_{b}^{\nu }\left[{g}_{\mu \nu ,\ell }\right],\end{equation} \tag{ 78 }$$

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

$$\begin{equation}\left[{H}_{ab}\right]=\frac{1}{2}\left[\frac{\partial {h}_{ab}}{\partial \sigma }\right].\end{equation} \tag{ 79 }$$

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

$$\begin{align}\hfill \left[{{\Pi}}^{00}\right]& =\left[{\chi }_{ab}\right]\left({n}^{2}{h}_{\ast }^{ab}-{\nu }^{a}{\nu }^{b}\right)+{n}^{2}\left[{\varphi }_{a}\right]{\nu }^{a}-{\left({n}^{2}\right)}^{2}\left[\psi \right]\hfill \\ \hfill & ={\left({n}^{2}\right)}^{2}\left[H\right]-{\left({n}^{2}\right)}^{2}\left[H\right]+{n}^{2}\left[{H}_{ab}\right]{\nu }^{a}{\nu }^{b}-{n}^{2}\left[{H}_{ab}\right]{\nu }^{a}{\nu }^{b}\hfill \\ \hfill & =0,\hfill \end{align} \tag{ 80 }$$

$$\begin{align}\hfill \left[{{\Pi}}^{0a}\right]& =\left[{\chi }_{cd}\right]\left({\nu }^{a}{h}_{\ast }^{cd}-{\nu }^{c}{h}_{\ast }^{ad}\right)+{n}^{2}\left[{\varphi }_{c}\right]{h}_{\ast }^{ac}-{n}^{2}\left[\psi \right]{\nu }^{a}\hfill \\ \hfill & ={n}^{2}\left[H\right]{\nu }^{a}-{n}^{2}\left[{H}_{cd}\right]{\nu }^{c}{h}_{\ast }^{ad}+{n}^{2}\left[{H}_{cd}\right]{h}_{\ast }^{ac}{\nu }^{d}-{n}^{2}\left[H\right]{\nu }^{a}\hfill \\ \hfill & =0,\hfill \end{align} \tag{ 81 }$$

and

$$\begin{align}\hfill \left[{{\Pi}}^{ab}\right]& =\left[{\chi }_{cd}\right]\left({h}_{\ast }^{ab}{h}_{\ast }^{cd}-{h}_{\ast }^{ac}{h}_{\ast }^{bd}\right)+\left[{\varphi }_{c}\right]\left({\nu }^{a}{h}_{\ast }^{bc}-{h}_{\ast }^{ab}{\nu }^{c}+{h}_{\ast }^{ac}{\nu }^{b}\right)-\left[\psi \right]{\nu }^{a}{\nu }^{b}\hfill \\ \hfill & ={n}^{2}\left(\left[H\right]{h}_{\ast }^{ab}-\left[{H}^{ab}\right]\right)+\left[{H}_{c}^{b}\right]{\nu }^{a}{\nu }^{c}+\left[{H}_{c}^{a}\right]{\nu }^{b}{\nu }^{c}\hfill \\ \hfill & \quad -\left[{H}_{cd}\right]{h}_{\ast }^{ab}{\nu }^{c}{\nu }^{d}-\left[H\right]{\nu }^{a}{\nu }^{b}.\hfill \end{align} \tag{ 82 }$$

It follows that the jump $\left[{{\Pi}}^{\mu \nu }\right]$ is a tangential tensor field along Σ, 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

$$\begin{align}\hfill \varkappa {S}^{ab}& ={n}^{2}\left(\left[H\right]{h}_{\ast }^{ab}-\left[{H}^{ab}\right]\right)+\left[{H}_{c}^{b}\right]{\nu }^{a}{\nu }^{c}+\left[{H}_{c}^{a}\right]{\nu }^{b}{\nu }^{c}-\left[{H}_{cd}\right]{h}_{\ast }^{ab}{\nu }^{c}{\nu }^{d}\hfill \\ \hfill & \quad -\left[H\right]{\nu }^{a}{\nu }^{b}.\hfill \end{align} \tag{ 83 }$$

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 ℓ^μ ↦ αℓ^μ . 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_μ . In the variational principle, both $\left[{{\Pi}}^{ab}\right]$ and S^ab appear as a factor in the expression

$$\begin{equation}\left({S}^{ab}-\frac{1}{\varkappa }\left[{{\Pi}}^{ab}\right]\right)\delta {h}_{ab}\enspace {\mu }_{\ell ,g},\end{equation} \tag{ 84 }$$

and the volume element μ_ℓ,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 Π-tensor ${\mathfrak{P}}^{ab}={{\Pi}}^{ab}{\rho }_{\ell ,g}$, the analogous equation

$$\begin{equation}\varkappa {\mathfrak{S}}^{ab}=\left[{\mathfrak{P}}^{ab}\right]\end{equation} \tag{ 85 }$$

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 Π^μν which has been split into the components Π⁰⁰, Π^0a and Π^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 σ adapted to the rigging ℓ^μ . In the usual formulation, the canonical momentum is the derivative of the Lagrangian with respect to 'time' (which in this case is σ) 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

$$\begin{equation}\frac{1}{2\varkappa }{\int }_{\partial M}{{\Pi}}^{\mu \nu }\delta {g}_{\mu \nu }{\mu }_{\ell ,g}=\frac{1}{2\varkappa }{\int }_{\partial M}\left({{\Pi}}^{00}\delta {\ell }^{2}+{{\Pi}}^{0a}\delta {\lambda }_{a}+{{\Pi}}^{ab}\delta {h}_{ab}\right){\mu }_{\ell ,g},\end{equation} \tag{ 86 }$$

which shows that Π⁰⁰, Π^0a and Π^ab are proportional to the canonical momenta corresponding to the metric degrees of freedom ℓ², λ_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 hypersurface.

If the condition ${\left.\delta {g}_{\mu \nu }\right\vert }_{\partial M}=0$ 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 δg_μν 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 Π⁰⁰, Π^0a and Π^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 Σ is timelike or spacelike and we apply the canonical choice of rigging presented in section 2.3, we obtain the equation

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

which is the well-known Lanczos equation [7]. If instead we take Σ to be null and choose the null rigging adapted to a spacelike foliation of Σ (section 2.4), we decompose the equation into components S¹¹, S^1A and S^AB , which are respectively

$$\begin{align}\hfill \varkappa {S}^{11}& =-\left[{H}_{AB}\right]{q}^{AB},\hfill \\ \hfill \varkappa {S}^{1A}& =\left[{H}_{1B}\right]{q}^{AB},\hfill \\ \hfill \varkappa {S}^{AB}& =-\left[{H}_{11}\right]{q}^{AB}.\hfill \end{align} \tag{ 88 }$$

These relations agree with those of Poisson [11], who interprets μ := S¹¹ 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, Barrabès and Israel [9] assume n ⋅ n = const, which formally excludes causality-changing hypersurfaces and they use the normalization n ⋅ ℓ = η⁻¹, where η is a nowhere vanishing function along Σ. 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].

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

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

where θ 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 [10, 18, 19, 51, 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 φ of weight 1 is a dual density to T, since then the contraction $\langle \varphi ,T\rangle ={{\varphi }_{{\mu }_{1}\dots {\mu }_{k}}}^{{\nu }_{1}\dots {\nu }_{l}}{{T}^{{\mu }_{1}\dots {\mu }_{k}}}_{{\nu }_{1}\dots {\nu }_{l}}$ 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 ${D}_{k,l}^{\ast }(M)$ to consist of linear functionals on D_k,l (M) that are continuous in the following sense. A linear functional $\chi :{D}_{k,l}(M)\to \mathbb{R}$ is continuous and thus belongs to ${D}_{k,l}^{\ast }(M)$ if for each sequence φ_n ∈ D_k,l (M) of test densities whose supports are contained in a common compact set K ⊆ M which is itself located in the domain of a coordinate chart, and such that the components ${{\left({\varphi }_{n}\right)}_{{\mu }_{1}\dots {\mu }_{k}}}^{{\nu }_{1}\dots {\nu }_{l}}$ and their partial derivatives of all orders tend to 0 uniformly, we have lim_n→∞  χ[φ_n ] = 0. Elements of ${D}_{k,l}^{\ast }(M)$ are called tensor distributions of type (k, l). A tensor distribution $\chi \in {D}_{k,l}^{\ast }(M)$ is regular if there exists a (locally integrable but otherwise 'rough') tensor field also denoted χ such that for any test density φ we have χ[φ] = ∫_M ⟨χ, φ⟩ d^D+1 x. This integral converges because φ 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 φ of k-forms ω can be canonically identified with (D + 1) − k-forms under the pairing map ⟨φ, ω⟩ = ω ∧ φ, thus k-currents are continuous linear functionals on (D + 1) − k-forms. de Rham defines the boundary ∂ω of a k-current ω by ∂ω[φ] = ω[dφ], then the (distributional) exterior derivative by dω = (−1)^k+1∂ω.

Finally, a few remarks on notation and local representations are in order. As de Rham proves ⁶ in [51], distributions have the sheaf property, i.e. if ${D}_{k,l}^{\ast }(U)$ denotes the space of type (k, l) tensor distributions over the open set U, and V ⊆ U is an open subset, we have a well-defined restriction map ${\mathrm{r}\mathrm{e}\mathrm{s}}_{V,U}(\chi )\equiv {\left.\chi \right\vert }_{V}$ given by

$$\begin{equation}{\left.\chi \right\vert }_{V}[\varphi ]{:=}\chi [{\mathrm{e}\mathrm{x}\mathrm{t}}_{U,V}(\varphi )],\end{equation} \tag{ 90 }$$

where ext_U,V : D_k,l (V) → D_k,l (U) extends the tensor density φ ∈ D_k,l (V) defined on V with compact support to a tensor density defined on U with compact support by taking φ to be zero on U\V. This means that the rule $U{\mapsto}{D}_{k,l}^{\ast }(U)$ 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 ⊆ M is a coordinate domain and $\chi \in {D}_{k,l}^{\ast }(U)$ is a tensor distribution of type (k, l), we can write χ uniquely as

$$\begin{equation}\chi ={{\chi }^{{\mu }_{1}\dots {\mu }_{k}}}_{ {\nu }_{1}\dots {\nu }_{l}}{\partial }_{{\mu }_{1}}\otimes \dots {\partial }_{{\mu }_{k}}\otimes \mathrm{d}{x}^{{\nu }_{1}}\otimes \cdots \otimes \mathrm{d}{x}^{{\nu }_{l}},\end{equation} \tag{ 91 }$$

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 φ we have

$$\begin{equation}{\chi [\varphi ]={\chi }^{{\mu }_{1}\dots {\mu }_{k}}}_{ {\nu }_{1}\dots {\nu }_{l}}{{\varphi }_{{\mu }_{1}\dots {\mu }_{k}}}^{ {\nu }_{1}\dots {\nu }_{l}}[1],\end{equation} \tag{ 92 }$$

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 ω on 1 as

$$\begin{equation}\omega [1]{:=}{\int }_{M}\omega ,\end{equation} \tag{ 93 }$$

we obtain 'classical' notation for tensor distributions (e.g. 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 θ with the corresponding 0-current and define the (one-form) Dirac delta ${\delta }_{\ast }^{{\Sigma}}$ associated to the hypersurface Σ to be the exterior derivative of the Heaviside current, i.e. we have for any smooth compactly supported D-form (or equivalently, vector density) φ

$$\begin{align}\hfill {\delta }_{\ast }^{{\Sigma}}[\varphi ]& =\mathrm{d}\theta [\varphi ]=-\partial \theta [\varphi ]=-\theta [\mathrm{d}\varphi ]=-{\int }_{M}\theta \enspace \mathrm{d}\varphi \hfill \\ \hfill & =-{\int }_{{M}^{+}}\mathrm{d}\varphi =-{\int }_{-{\Sigma}}\varphi ={\int }_{{\Sigma}}\varphi .\hfill \end{align} \tag{ 94 }$$

Since M is equipped with a volume form μ_g , choosing a rigging ℓ^μ pointing from M⁻ to M⁺ with adapted normal n_μ satisfying n_μ ℓ^μ = 1, also defines the volume element μ_ℓ,g on Σ. Then it becomes possible to define a scalar distribution δ^Σ evaluated on a test D + 1-form φ as

$$\begin{equation}{\delta }^{{\Sigma}}[\varphi ]{:=}{\int }_{{\Sigma}}f\enspace {\mu }_{\ell ,g},\end{equation} \tag{ 95 }$$

where f is the scalar function uniquely determined ⁷ by φ = fμ_g . It is easy to verify the relation (${\delta }_{\mu }^{{\Sigma}}$ are the components of the one-current ${\delta }_{\ast }^{{\Sigma}}$)

$$\begin{equation}{\delta }_{\mu }^{{\Sigma}}={n}_{\mu }{\delta }^{{\Sigma}},\end{equation} \tag{ 96 }$$

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

$$\begin{equation}{\partial }_{\mu }\bar{F}=\bar{{\partial }_{\mu }F}+\left[F\right]{\delta }_{\mu }^{{\Sigma}}=\bar{{\partial }_{\mu }F}+\left[F\right]{n}_{\mu }{\delta }^{{\Sigma}}.\end{equation} \tag{ 97 }$$

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

$$\begin{equation}{{\Gamma}}_{\mu \nu }^{\kappa }={\bar{{\Gamma}}}_{\mu \nu }^{\kappa },\end{equation} \tag{ 98 }$$

and its jump as (77)

$$\begin{equation}\left[{{\Gamma}}_{\mu \nu }^{\kappa }\right]=\frac{1}{2}\left({n}_{\mu }{\xi }_{\nu }^{\kappa }+{n}_{\nu }{\xi }_{\mu }^{\kappa }-{n}^{\kappa }{\xi }_{\mu \nu }\right),\end{equation} \tag{ 99 }$$

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

$$\begin{align}\hfill {R}_{\lambda \mu \nu }^{\kappa }& ={\bar{R}}_{\lambda \mu \nu }^{\kappa }+{\delta }_{\mu }^{{\Sigma}}\left[{{\Gamma}}_{\nu \lambda }^{\kappa }\right]-{\delta }_{\nu }^{{\Sigma}}\left[{{\Gamma}}_{\mu \lambda }^{\kappa }\right]\hfill \\ \hfill & ={\bar{R}}_{\lambda \mu \nu }^{\kappa }+\left({n}_{\mu }\left[{{\Gamma}}_{\nu \lambda }^{\kappa }\right]-{n}_{\nu }\left[{{\Gamma}}_{\mu \lambda }^{\kappa }\right]\right){\delta }^{{\Sigma}}.\hfill \end{align} \tag{ 100 }$$

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

$$\begin{equation}{\mathcal{R}}_{\kappa \lambda \mu \nu }={n}_{\kappa }\left[{H}_{\lambda \mu }\right]{n}_{\nu }-{n}_{\kappa }\left[{H}_{\lambda \nu }\right]{n}_{\mu }+{n}_{\lambda }\left[{H}_{\kappa \nu }\right]{n}_{\mu }-{n}_{\lambda }\left[{H}_{\kappa \mu }\right]{n}_{\nu },\end{equation} \tag{ 101 }$$

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

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

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

$$\begin{equation}\mathcal{R}=-2\left(\left[{\chi }_{ab}\right]{h}_{\ast }^{ab}-\left[{\varphi }_{a}\right]{\nu }^{a}\right),\end{equation} \tag{ 103 }$$

which is −2ϰ-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

$$\begin{equation}{L}_{\text{EH}}=\frac{1}{2\varkappa }R-\frac{1}{\varkappa }\left(\left[{\chi }_{ab}\right]{h}_{\ast }^{ab}-\left[{\varphi }_{a}\right]{\nu }^{a}\right){\delta }^{{\Sigma}}.\end{equation} \tag{ 104 }$$

It follows that the Einstein–Hilbert action over M is

$$\begin{align}\hfill {S}_{\text{EH}}& =\frac{1}{2\varkappa }{\int }_{M}\bar{R}\enspace {\mu }_{g}-\frac{1}{\varkappa }{\int }_{{\Sigma}}\left(\left[{\chi }_{ab}\right]{h}_{\ast }^{ab}-\left[{\varphi }_{a}\right]{\nu }^{a}\right){\mu }_{\ell ,g}\hfill \\ \hfill & =\frac{1}{2\varkappa }{\int }_{{M}^{+}}{R}^{+}\enspace {\mu }_{g}+\frac{1}{2\varkappa }{\int }_{{M}^{-}}{R}^{-}\enspace {\mu }_{g}-\frac{1}{\varkappa }{\int }_{{\Sigma}}\left[{P}_{\nu }^{\mu }{\nabla }_{\mu }{n}^{\nu }\right]{\mu }_{\ell ,g},\hfill \end{align} \tag{ 105 }$$

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

$$\begin{equation}{\mathcal{L}}_{\text{EH}}={P}^{\kappa \lambda \mu \nu }\left(g\right)\sqrt{-\mathfrak{g}}{\partial }_{\kappa }{\partial }_{\lambda }{g}_{\mu \nu }+Q\left(g,\partial g\right)\sqrt{-\mathfrak{g}},\end{equation} \tag{ 106 }$$

where the coefficients are

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

and

$$\begin{equation}Q\left(g,\partial g\right)=\frac{1}{2\varkappa }{{\Gamma}}_{\kappa \mu \nu }{{\Gamma}}^{\kappa \mu \nu }-\frac{1}{2\varkappa }{g}_{\kappa \lambda }{{\Gamma}}_{\ast }^{\kappa }{{\Gamma}}_{\ast }^{\lambda },\end{equation} \tag{ 108 }$$

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

$$\begin{equation}{\mathcal{L}}_{\text{EH}}={\bar{\mathcal{L}}}_{\text{EH}}+{P}^{\kappa \lambda \mu \nu }\sqrt{-\mathfrak{g}}{n}_{\kappa }{n}_{\lambda }\left[{g}_{\mu \nu ,\ell }\right]{\delta }^{{\Sigma}},\end{equation} \tag{ 109 }$$

while a variational counterterm is given by

$$\begin{equation}B={-\oint }_{\partial M}{n}_{\kappa }{n}_{\lambda }{P}^{\kappa \lambda \mu \nu }{g}_{\mu \nu ,\ell }{\mu }_{\ell ,g},\end{equation} \tag{ 110 }$$

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

Since P^κλμν 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 ${\mathcal{L}}_{\text{EH}}$. However if P^κλμν were to depend on the metric's transversal derivative, the singular part of the Lagrangian would be mathematically meaningless as P^κλμν would be discontinuous at Σ where it is being evaluated. If we choose ${\left.\theta \right\vert }_{{\Sigma}}=1/2$ as the value of the step function on Σ, then the meaning of such an expression can be salvaged as P^κλμν 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^κλμν 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.

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

$$\begin{equation}{S}_{\nabla }={S}_{\text{EH}}+{B}_{\text{BC}}={\int }_{M}{L}_{\nabla }\enspace {\mu }_{g},\end{equation} \tag{ 111 }$$

where

$$\begin{equation}{L}_{\nabla }=\frac{1}{2\varkappa }\left\{\bar{R}+{g}^{\mu \nu }\left({{\Delta}}_{\mu \rho }^{\kappa }{{\Delta}}_{\kappa \nu }^{\rho }-{{\Delta}}_{\kappa \rho }^{\kappa }{{\Delta}}_{\mu \nu }^{\rho }\right)\right\},\end{equation} \tag{ 112 }$$

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

$$\begin{equation}{{\Delta}}_{\mu \nu }^{\kappa }=\frac{1}{2}{g}^{\kappa \lambda }\left({\bar{\nabla }}_{\mu }{g}_{\nu \lambda }+{\bar{\nabla }}_{\nu }{g}_{\mu \lambda }-{\bar{\nabla }}_{\lambda }{g}_{\mu \nu }\right)\end{equation} \tag{ 113 }$$

is relevant. A variation of the action leads symbolically to

$$\begin{equation}\delta {S}_{\nabla }={\int }_{M}\left(\delta {L}_{\nabla }+{L}_{\nabla }{g}^{\mu \nu }\delta {g}_{\mu \nu }\right){\mu }_{g},\end{equation} \tag{ 114 }$$

where

$$\begin{align}\hfill \delta {L}_{\nabla }& =\frac{\partial {L}_{\nabla }}{\partial {g}_{\mu \nu }}\delta {g}_{\mu \nu }+\frac{\partial {L}_{\nabla }}{\partial {\bar{\nabla }}_{\kappa }{g}_{\mu \nu }}\delta {\bar{\nabla }}_{\kappa }{g}_{\mu \nu }\hfill \\ \hfill & =\frac{\partial {L}_{\nabla }}{\partial {g}_{\mu \nu }}\delta {g}_{\mu \nu }+\frac{\partial {L}_{\nabla }}{\partial {\bar{\nabla }}_{\kappa }{g}_{\mu \nu }}{\bar{\nabla }}_{\kappa }\delta {g}_{\mu \nu }.\hfill \end{align} \tag{ 115 }$$

Since the background connection ${\bar{\nabla }}_{\kappa }$ has no a priori relation with the volume element μ_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 ∇_κ . This is accomplished via the difference formula [25]

$$\begin{equation}{\bar{\nabla }}_{\kappa }\delta {g}_{\mu \nu }={\nabla }_{\kappa }\delta {g}_{\mu \nu }+{{\Delta}}_{\kappa \mu }^{\lambda }\delta {g}_{\lambda \nu }+{{\Delta}}_{\kappa \nu }^{\lambda }\delta {g}_{\mu \lambda }.\end{equation} \tag{ 116 }$$

Inserting this into the variation gives

$$\begin{align}\hfill \delta {L}_{\nabla }& =\left(\frac{\partial {L}_{\nabla }}{\partial {g}_{\mu \nu }}+\frac{\partial {L}_{\nabla }}{\partial {\bar{\nabla }}_{\kappa }{g}_{\lambda \nu }}{{\Delta}}_{\kappa \lambda }^{\mu }+\frac{\partial {L}_{\nabla }}{\partial {\bar{\nabla }}_{\kappa }{g}_{\mu \lambda }}{{\Delta}}_{\kappa \lambda }^{\nu }-{\nabla }_{\kappa }\frac{\partial {L}_{\nabla }}{\partial {\bar{\nabla }}_{\kappa }{g}_{\mu \nu }}\right)\delta {g}_{\mu \nu }\hfill \\ \hfill & \quad +{\nabla }_{\kappa }\left(\frac{\partial {L}_{\nabla }}{\partial {\bar{\nabla }}_{\kappa }{g}_{\mu \nu }}\delta {g}_{\mu \nu }\right),\hfill \end{align} \tag{ 117 }$$

thus the variation of the action is

$$\begin{align}\hfill \delta {S}_{\nabla }& ={\int }_{M}\left(\frac{\partial {L}_{\nabla }}{\partial {g}_{\mu \nu }}+{L}_{\nabla }{g}^{\mu \nu }+\frac{\partial {L}_{\nabla }}{\partial {\bar{\nabla }}_{\kappa }{g}_{\lambda \nu }}{{\Delta}}_{\kappa \lambda }^{\mu }+\frac{\partial {L}_{\nabla }}{\partial {\bar{\nabla }}_{\kappa }{g}_{\mu \lambda }}{{\Delta}}_{\kappa \lambda }^{\nu }\right.\hfill \\ \hfill & \quad \left.-{\nabla }_{\kappa }\frac{\partial {L}_{\nabla }}{\partial {\bar{\nabla }}_{\kappa }{g}_{\mu \nu }}\right)\delta {g}_{\mu \nu }\enspace {\mu }_{g}{+\oint }_{\partial M}{n}_{\kappa }\frac{\partial {L}_{\nabla }}{\partial {\bar{\nabla }}_{\kappa }{g}_{\mu \nu }}\delta {g}_{\mu \nu }\enspace {\mu }_{\ell ,g}.\hfill \end{align} \tag{ 118 }$$

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

$$\begin{equation}2\varkappa \frac{\partial {L}_{\nabla }}{\partial {\bar{\nabla }}_{\kappa }{g}_{\mu \nu }}={{\Delta}}^{\kappa \mu \nu }-\frac{1}{2}\left({g}^{\lambda \nu }{g}^{\kappa \mu }+{g}^{\lambda \mu }{g}^{\nu \kappa }-{g}^{\lambda \kappa }{g}^{\mu \nu }\right){{\Delta}}_{\lambda }-\frac{1}{2}{g}^{\mu \nu }{{\Delta}}_{\ast }^{\kappa },\end{equation} \tag{ 119 }$$

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 ΓΓ-action (53) are given in appendix 9 of [47] and is therefore omitted here. Let the contraction of the above expression be denoted

$$\begin{align}\hfill {M}^{\mu \nu }& {:=}2\varkappa {n}_{\kappa }\frac{\partial {L}_{\nabla }}{\partial {\bar{\nabla }}_{\kappa }{g}_{\mu \nu }}={n}_{\kappa }{{\Delta}}^{\kappa \mu \nu }-\frac{1}{2}\left({g}^{\lambda \nu }{n}^{\mu }+{g}^{\lambda \mu }{n}^{\nu }-{n}^{\lambda }{g}^{\mu \nu }\right){{\Delta}}_{\lambda }\hfill \\ \hfill & \quad -\frac{1}{2}{g}^{\mu \nu }{{\Delta}}_{\ast }^{\kappa }{n}_{\kappa }.\hfill \end{align} \tag{ 120 }$$

We rewrite the variational principle for thin shells as

$$\begin{align}\hfill S& =\frac{1}{2\varkappa }{\int }_{{M}^{+}}\left\{\bar{R}+{g}^{\mu \nu }\left({{\Delta}}_{\mu \rho }^{\kappa }{{\Delta}}_{\kappa \nu }^{\rho }-{{\Delta}}_{\kappa \rho }^{\kappa }{{\Delta}}_{\mu \nu }^{\rho }\right)\right\}{\mu }_{g}\hfill \\ \hfill & \quad +\frac{1}{2\varkappa }{\int }_{{M}^{-}}\left\{\bar{R}+{g}^{\mu \nu }\left({{\Delta}}_{\mu \rho }^{\kappa }{{\Delta}}_{\kappa \nu }^{\rho }-{{\Delta}}_{\kappa \rho }^{\kappa }{{\Delta}}_{\mu \nu }^{\rho }\right)\right\}{\mu }_{g}\hfill \\ \hfill & \quad +{\int }_{{M}^{+}}{\mathcal{L}}_{\mathrm{M}}{\mathrm{d}}^{D+1}x+{\int }_{{M}^{-}}{\mathcal{L}}_{\mathrm{M}}\enspace {\mathrm{d}}^{D+1}x+{\int }_{{\Sigma}}{\mathcal{L}}_{\text{TS}}\enspace {\mathrm{d}}^{D}y,\hfill \end{align} \tag{ 121 }$$

and varying this with respect to the metric gives

$$\begin{align}\hfill \delta S& ={\int }_{{M}^{+}}\frac{1}{2}\left({T}^{\mu \nu }-\frac{1}{\varkappa }{G}^{\mu \nu }\right)\delta {g}_{\mu \nu }\enspace {\mu }_{g}+{\int }_{{M}^{-}}\frac{1}{2}\left({T}^{\mu \nu }+\frac{1}{\varkappa }{G}^{\mu \nu }\right)\delta {g}_{\mu \nu }\enspace {\mu }_{g}\hfill \\ \hfill & \quad -\frac{1}{2\varkappa }{\int }_{{\Sigma}}\left[{M}^{\mu \nu }\right]\delta {g}_{\mu \nu }\enspace {\mu }_{\ell ,g}+{\int }_{{\Sigma}}\frac{1}{2}{S}^{\mu \nu }\delta {g}_{\mu \nu }\enspace {\mu }_{\ell ,g},\hfill \end{align} \tag{ 122 }$$

where S^μν is again defined by (69). Imposing the stationarity of the action gives the boundary equation

$$\begin{equation}\varkappa {S}^{\mu \nu }=\left[{M}^{\mu \nu }\right].\end{equation} \tag{ 123 }$$

The jump of M^μν can be written as

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

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 appendix 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

$$\begin{align}\hfill \left[{M}^{ab}\right]& ={n}^{2}\left(\left[H\right]{h}_{\ast }^{ab}-\left[{H}^{ab}\right]\right)+{\nu }^{a}\left[{H}_{c}^{b}\right]{\nu }^{c}+\left[{H}_{c}^{a}\right]{\nu }^{b}{\nu }^{c}-\left[{H}_{cd}\right]{\nu }^{c}{\nu }^{d}{h}_{\ast }^{ab}\hfill \\ \hfill & \quad -\left[H\right]{\nu }^{a}{\nu }^{b},\hfill \end{align} \tag{ 125 }$$

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

$$\begin{equation}\varkappa {S}^{ab}=\left[{M}^{ab}\right],\end{equation} \tag{ 126 }$$

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 (e.g. 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 equation ⁸ 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 G₅(ϕ, X) = 0, G₄(ϕ, X) = G₄(ϕ) and G₃(ϕ, X) = G₃(ϕ) 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].