# On staticity of bifurcate Killing horizons††thanks: Vienna preprint UWThPh
2023-12

Piotr T. Chruściel
Faculty of Physics, University of Vienna
x
Marc Mars
Institute of Fundamental Physics and Mathematics
University of Salamanca Email<EMAIL_ADDRESS>URL
http://homepage.univie.ac.at/piotr.chrusciel/Email<EMAIL_ADDRESS>
###### Abstract

We show that bifurcate Killing horizons with closed torsion form, in
spacetimes of arbitrary dimension satisfying a Ricci-structure condition,
arise from static Killing vectors. The result applies in particular to
$\Lambda$-vacuum spacetimes.

###### Contents

  1. 1 Introduction
  2. 2 The twist form and the quotient space
  3. 3 Rácz-Wald coordinates
  4. 4 Staticity
    1. 4.1 The timelike region
    2. 4.2 The spacelike region
  5. A Appendix: Conformal behavior of the Laplacian on two-forms

##  1 Introduction

Stationary spacetimes are of particular physical importance as they describe
equilibrium configurations of the gravitational field. The static subcase,
where the spacetime is not only time independent but also invariant under time
reversal, plays a particularly important role. It is therefore of interest to
identify conditions that guarantee that a stationary spacetime is static.
Recall that a stationary spacetime with a Killing vector field $\xi$ is
_static_ if the twist form

$\omega:=\frac{1}{2}\mbox{\boldmath$\xi$}\wedge d\mbox{\boldmath$\xi$}\,,$
(1.1)

where $\xi$ is the $1$-form $g(\xi,\cdot)$, vanishes; such Killing vectors
will also be called _static_. In situations where the spacetime is globally
hyperbolic and the Killing vector $\xi$ is timelike everywhere it is immediate
to show that the spacetime is static if and only if the twist form vanishes on
a Cauchy surface. This is a consequence of the fact that $\omega$ is always
Lie constant along the Killing field. However, in black hole configurations
the Killing field is not timelike everywhere and finding necessary and
sufficient conditions for staticity is more delicate.

One of the central notions in the study of stationary black hole spacetimes is
that of Killing horizons [5]. Large families of examples have been constructed
in [11, 10, 15, 16]; see also [8]. This raises the question of staticity in
terms of the properties of Killing horizons. In fact, the vanishing – or not –
of the twist form plays an important role in the classification of stationary
vacuum black holes, cf. e.g. [6, 7] and references therein.

The aim of this work is to give a characterisation of static spacetimes
containing a bifurcate Killing horizon in terms of the data on the horizon. We
note that $\omega$ always vanishes on a Killing horizon $\mycal H$, so a
criterion on a Killing horizon must involve at least one transverse
derivatives on $\mycal H$. Further, both $\omega$ and its first derivatives
vanish at the bifurcation surface of a bifurcate Killing horizon, so a
criterion there must involve at least two transverse derivatives of $\omega$.
The relevant fields are a priori not clear.

We aim at finding such a characterization independently both of the matter
fields involved and of the field equations of the theory, i.e. a purely
geometry characterization valid in full generality. Given that no field
equations are imposed, it is clear that such a characterization will
necessarily involve two types of conditions, namely a bulk condition at the
level of curvature that holds everywhere and a boundary condition that holds
on the horizon. In this paper we deal with bifurcate Killing horizons, so it
is plausible to expect that the boundary conditions can be formulated directly
on the bifurcation surface. We find necessary and sufficient conditions for
staticity both for the bulk and for the boundary conditions. The bulk
condition involves a one-form constructed from the Ricci tensor and the
Killing field. Specifically, it demands that, on all points where the Killing
vector is not null, the one-form, denoted by $g(\xi,\xi)r$, and defined as the
tangential-normal components of the Ricci tensor along the vector $\xi$, is
closed. The boundary condition at the bifurcation surface is that the _torsion
one-form_ is closed.

It turns out that the analysis in the region to the causal future of the
bifurcation surface of the bifurcate Killing horizon, and in its complement,
is essentially different. This is related to the different causal character of
the Killing orbits in these regions. The timelike-orbits region is analysed in
Section 4.1, while the space-like-orbits one in Section 4.2. The precise
statements of our staticity results appear in Theorems 4.2 and 4.4 below. In
addition to being independent of any field equations, our results hold in any
spacetime dimension and for any topology of the bifurcation surface.

An interesting by-product of our just-mentioned staticity theorems is that
whenever the torsion one-form of the bifurcation surface is closed, the fact
that the bulk one-form $g(\xi,\xi)r$ is closed implies that $\xi$ is an
eigenvector of the Ricci tensor (equivalently, that the one-form $g(\xi,\xi)r$
being closed implies that it vanishes everywhere, provided the torsion form of
the bifurcations surface is closed).

Of course, of particular importance is the case where the spacetime is vacuum.
In this case, the characteristic initial data for the spacetime on the
bifurcate Killing horizon consists of the metric induced on the bifurcation
surface $S$ together with the torsion $1$-form of $S$. A special case of our
results is the following theorem, where a non-vanishing cosmological constant
is allowed, with $\mycal H^{-}$ denoting the part of $\mycal H$ lying to the
past of the bifurcation surface:

######  Theorem 1.1.

Consider a $(n+1)$-dimensional vacuum spacetime $(M,g)$, $n\geq 2$, containing
a bifurcate Killing horizon $\mycal H$. Let ${\mycal D}^{-}(\mycal H^{-})$ be
the past domain of dependence of $\mycal H^{-}$ in $M$ and let $\mathring{M}$
denote that connected component of the set

${\mycal D}^{-}(\mycal H^{-})\cup\\{g(\xi,\xi)<0\\}$ (1.2)

which contains ${\mycal D}^{-}(\mycal H^{-})$. Then $g$ is static on
$\mathring{M}$ if and only if the torsion $1$-form of the bifurcation surface
is closed.

######  Remark 1.2.

As already said, we show in fact that a necessary and sufficient condition for
staticity is closedness of the torsion together with

$d\big{(}g(\xi,\xi)\hat{r}\big{)}=0\,,$ (1.3)

where $\hat{r}$ is the quotient-space equivalent of the covector field
$r_{\alpha}dx^{\alpha}$ appearing in the decomposition (2.11) of the Ricci
tensor. This condition will of course hold in vacuum, but it will also hold
for, e.g., a perfect fluid supported in the region exterior to a black hole
and comoving with the Killing flow. $\Box$

######  Remark 1.3.

Our analysis below likewise shows that a vacuum spacetime metric will be
static if and only if (1.3) holds and the first derivatives of $\omega$ vanish
on one of the branches of a bifurcate horizon $\mycal H$. This statement,
however, does not seem to have an obvious formulation in terms of
characteristic Cauchy data on $\mycal H$. $\Box$

At the heart of the proof of Theorem 1.1 lies the derivation of an equation
for a relevant field, to which the unique continuation results of Mazzeo [14]
and the uniqueness results of Fuchsian symmetric hyperbolic equations apply.
Another ingredient is a calculation tying the twist of the Killing vector to
the torsion $1$-form of the horizon. In order to finish the proof we need to
treat separately the region ${\mycal D}(S)$, where the Killing vector is
spacelike, and its complement in $\mathring{M}$. Our claim on ${\mycal
D}(\mycal H)$ is obtained from uniqueness of solutions of Fuchsian hyperbolic
equations; this is the vacuum case of Theorem 4.4 below. The result in the
complement of ${\mycal D}(\mycal H)$ in $\mathring{M}$ is the vacuum case of
Theorem 4.2 below; it is based on the observation, which might have some
interest of its own (cf. Proposition 3.1 below), that the bifurcation surface
of a Killing horizon can be viewed as the conformal boundary at infinity of a
natural rescaling of the quotient space-metric.

We note that in real-analytic spacetimes the result can be established by
Taylor expansion at the horizon; this requires a similar computational effort.
We further note that while a vacuum spacetime metric will be real analytic
near a Killing horizon in many situations of interest [3, 4], Theorems 4.2 and
4.4 neither assume vacuum nor analyticity.

##  2 The twist form and the quotient space

The aim of this section is to derive the key equation (2.26) below, satisfied
by a field closely related to the twist of the Killing vector.

Let thus $\xi$ be a Killing vector field (KV) in an $(n+1)$-dimensional
spacetime $({M},g)$, $n\geq 2$. We use $\nabla$ for the Levi-Civita derivative
of $g$. Let $\xi$ be the metrically associated $1$-form, and define the
$2$-form $F:=\frac{1}{2}d\mbox{\boldmath$\xi$}$; recall that the twist
$3$-form $\omega$ has been defined as $\frac{1}{2}\mbox{\boldmath$\xi$}\wedge
d\mbox{\boldmath$\xi$}$. In abstract index notation,

$\displaystyle
F_{\alpha\beta}=\nabla_{\alpha}\xi_{\beta},\qquad\omega_{\alpha\beta\gamma}=\xi_{\alpha}\nabla_{\beta}\xi_{\gamma}+\xi_{\beta}\nabla_{\gamma}\xi_{\alpha}+\xi_{\gamma}\nabla_{\alpha}\xi_{\beta}.$
(2.1)

Recall that $\lambda:=-g(\xi,\xi)$ and define the $2$-form
$\Omega:=i_{\xi}\omega$. Note that, by construction, $\Omega$ is Lie-constant
along $\xi$ and orthogonal to $\xi$. It is immediate that

$\displaystyle\Omega=-\lambda F-\frac{1}{2}\mbox{\boldmath$\xi$}\wedge
d\lambda.$ (2.2)

Define

${M^{*}}:=\\{p\in{M};\lambda(p)\neq 0\\}\,,$

i.e. the open set where $\xi$ is non-null, and assume it to be non-empty. From
the definitions of $\omega$, $F$, and (2.2) we have

$\displaystyle\omega=-\frac{1}{\lambda}\xi\wedge\Omega\qquad\mbox{on
}\quad{M^{*}}\,.$ (2.3)

Note that the left-hand side is smooth on ${M}$, which implies that the right-
hand side extends smoothly across the zero-level set of $\lambda$. In
particular we have:

######  Lemma 2.1.

If ${M^{*}}$ is dense in ${M}$ then the vanishing of $\Omega$ is equivalent to
the vanishing of $\omega$.

Using $F=\frac{1}{2}d\mbox{\boldmath$\xi$}$, the expression (2.2) can be
written as

$\displaystyle\Omega=-\frac{\lambda^{2}}{2}d\left(\lambda^{-1}\mbox{\boldmath$\xi$}\right),\qquad\mbox{on
}\quad{M^{*}}$

from which it immediately follows that

$\displaystyle d\left(\lambda^{-2}\Omega\right)=0,\qquad\mbox{on
}\quad{M^{*}}.$ (2.4)

This is equation (B.10) in Appendix B of [2]. The Killing vector $\xi$
satisfies the standard identity ($R^{\alpha}{}_{\beta\gamma\delta}$ is the
Riemann tensor of $g$)

$\displaystyle\nabla_{\alpha}\nabla_{\beta}\xi_{\gamma}=\xi_{\sigma}R^{\sigma}{}_{\alpha\beta\gamma}.$

Directly from (2.1) (brackets denote antisymmetrisation)

$\displaystyle\nabla_{\mu}\omega_{\alpha\beta\gamma}=F_{\mu\alpha}F_{\beta\gamma}+F_{\mu\beta}F_{\gamma\alpha}+F_{\mu\gamma}F_{\alpha\beta}+3\xi_{\sigma}R^{\sigma}{}_{\mu[\alpha\beta}\xi_{\gamma]}.$

Contracting $\mu$ and $\alpha$ yields $(R_{\alpha\beta}$ is the Ricci tensor
of $g$)

$\displaystyle\nabla^{\mu}\omega_{\mu\beta\gamma}=\xi_{\sigma}\left(\xi_{\beta}R^{\sigma}{}_{\gamma}-\xi_{\gamma}R^{\sigma}{}_{\beta}\right).$
(2.5)

This corresponds to (3.20) in Appendix B of [2] and recovers the well-known
fact that $\omega$ is co-closed when $\xi$ is an eigenvector of the Ricci
tensor.

We wish to compute the codifferential of $\Omega$ on the orbit space of $\xi$.
We assume that the space $\Sigma:={M^{*}}/\sim$ is a smooth manifold, where we
write that $p\sim q$ if and only if the points $p,q\in{M^{*}}$ belong to the
same integral curve of $\xi$. It is well-known that covariant tensors on
${M^{*}}$ which are Lie-constant along $\xi$ and totally orthogonal to $\xi$
descend to the quotient. We use a hat to denote the corresponding object on
$\Sigma$. We only break this rule for scalars, such as

$\lambda:=-g(\xi,\xi)$ (2.6)

and for the tensor

$h_{\alpha\beta}:=g_{\alpha\beta}+\lambda^{-1}\xi_{\alpha}\xi_{\beta}\,,$
(2.7)

which descends to a metric on $\Sigma$. In these last two cases we use the
same symbol for the spacetime and for the quotient objects.

The operation

$\displaystyle
D_{\alpha}T_{\beta_{1}\cdots\beta_{p}}:=h^{\gamma}{}_{\alpha}h^{\delta_{1}}{}_{\beta_{1}}\cdots
h^{\delta_{p}}{}_{\beta_{p}}\nabla_{\gamma}T_{\delta_{1}\cdots\delta_{p}}$

on any $\xi$-Lie-constant and $\xi$-orthogonal covariant tensor $T$, gives a
tensor with the same properties. It therefore induces an operation on
$\Sigma$. It is well-known that this operation is the Levi-Civita covariant
derivative with respect to $h$. We also use $D$ for this derivative on
$\Sigma$.

We already noted that $\Omega$ is $\xi$-Lie-constant and $\xi$-orthogonal. Let
$\widehat{\Omega}$ be the (uniquely defined) corresponding two-form on
$\Sigma$: if we denote by $\pi$ the projection from $M^{*}$ to $M^{*}/\sim$,
then

$\Omega=\pi^{*}\hat{\Omega}\,.$ (2.8)

Exterior differentiation commutes with the operation of descending to the
quotient for forms that descend (this is clearly seen in local coordinates
adapted to the isometries), so (2.4) implies

$\displaystyle d\left(\lambda^{-2}\widehat{\Omega}\right)=0.$

With the aim of computing the codifferential of $\widehat{\Omega}$ we consider
an arbitrary $\xi$-Lie-constant $\xi$-orthogonal $2$-covariant tensor
$P_{\alpha\beta}$ and compute

$\displaystyle D_{\mu}P^{\mu}{}_{\beta}$
$\displaystyle=g^{\mu\alpha}D_{\mu}P_{\alpha\beta}=h^{\mu\alpha}h^{\gamma}{}_{\beta}\nabla_{\mu}P_{\alpha\gamma}=h^{\gamma}{}_{\beta}\left(\nabla^{\mu}P_{\mu\gamma}+\frac{1}{\lambda}\xi^{\mu}\xi^{\alpha}\nabla_{\mu}P_{\alpha\gamma}\right)$
$\displaystyle=h^{\gamma}{}_{\beta}\nabla^{\mu}P_{\mu\gamma}-\frac{1}{2\lambda}P_{\mu\beta}\nabla^{\mu}\lambda.$

Let us define the covector

$\displaystyle
r_{\beta}:=\lambda^{-2}\xi^{\alpha}h^{\gamma}{}_{\beta}\nabla_{\mu}\omega^{\mu}{}_{\alpha\gamma}.$
(2.9)

Then $r$ is orthogonal to $\xi$ and invariant under the flow of $\xi$, hence
there exists a unique covector field $\hat{r}$ on the quotient $M^{*}/\sim$
such that

$r=\pi^{*}\hat{r}\,.$ (2.10)

On ${M^{*}}$ we can decompose the Ricci tensor in components orthogonal to
$\xi$. Using (2.5) it follows that the normal-tangential component of the
Ricci tensor is precisely $r_{\alpha}$. In other words, the decomposition
takes the form

$\displaystyle
R_{\alpha\beta}=r\xi_{\alpha}\xi_{\beta}+r_{\alpha}\xi_{\beta}+r_{\beta}\xi_{\alpha}+r_{\alpha\beta}\quad\mbox{on}\
{M^{*}}\,,$ (2.11)

where $r_{\alpha\beta}$ is symmetric and orthogonal to $\xi$. It follows that
$r_{\alpha}$ vanishes if and only if $\xi$ is an eigenvector of Ricci on
${M^{*}}$. Thus

$\displaystyle D_{\mu}\Omega^{\mu}{}_{\beta}$
$\displaystyle=h^{\gamma}{}_{\beta}\nabla_{\mu}\Omega^{\mu}{}_{\gamma}-\frac{1}{2\lambda}\Omega_{\mu\beta}\nabla^{\mu}\lambda=\nabla_{\mu}\left(\xi^{\alpha}\omega_{\alpha}{}^{\mu}{}_{\gamma}\right)h^{\gamma}{}_{\beta}-\frac{1}{2\lambda}\Omega_{\mu\beta}\nabla^{\mu}\lambda$
$\displaystyle=(\nabla_{\mu}\xi^{\alpha})\omega_{\alpha}{}^{\mu}{}_{\gamma}h^{\gamma}{}_{\beta}+\xi^{\alpha}h^{\gamma}{}_{\beta}\nabla_{\mu}\omega_{\alpha}{}^{\mu}{}_{\gamma}-\frac{1}{2\lambda}\Omega_{\mu\beta}\nabla^{\mu}\lambda$
$\displaystyle=-h^{\gamma}{}_{\beta}\omega_{\alpha\mu\gamma}\nabla^{\alpha}\xi^{\mu}-\lambda^{2}r_{\beta}-\frac{1}{2\lambda}\Omega_{\mu\beta}\nabla^{\mu}\lambda.$

Now, on ${M^{*}}$ the expression (2.2) allows us to write
$\nabla_{\alpha}\xi_{\mu}$ in terms of $\Omega_{\alpha\mu}$ as

$\displaystyle\nabla_{\alpha}\xi_{\mu}=-\frac{1}{\lambda}\left(\Omega_{\alpha\mu}+\frac{1}{2}\xi_{\alpha}\nabla_{\mu}\lambda-\frac{1}{2}\xi_{\mu}\nabla_{\alpha}\lambda\right)\,.$
(2.12)

Inserting this into the before-last equation yields

$\displaystyle D_{\mu}\Omega^{\mu}{}_{\beta}$
$\displaystyle=-\lambda^{2}r_{\beta}+\frac{1}{2\lambda}\Omega_{\mu\beta}\nabla^{\mu}\lambda+\frac{1}{\lambda}\omega_{\alpha\mu\gamma}\Omega^{\alpha\mu}h^{\gamma}{}_{\beta}.$
(2.13)

The last term is zero because $\Omega^{\alpha\mu}h^{\gamma}{}_{\beta}$ is
orthogonal to $\xi$ in all contravariant indices, while
$\omega=\mbox{\boldmath$\xi$}\wedge F$ (see (2.1)).

We conclude that the differential and codifferential of $\widehat{\Omega}$ on
$\Sigma$ satisfy

$\displaystyle
d\left(\frac{1}{\lambda^{2}}\widehat{\Omega}\right)=0,\qquad\delta\left(\frac{1}{|\lambda|^{\frac{1}{2}}}\widehat{\Omega}\right)=|\lambda|^{\frac{3}{2}}\widehat{r}\,,$
(2.14)

where we use the sign convention for the codifferential $\delta$ as in [9]:

$(\delta\sigma)_{b_{2}\cdots b_{p}}=-D^{b_{1}}\sigma_{b_{1}b_{2}\cdots
b_{p}}\,.$ (2.15)

Our next task is to derive an identity for $\Delta_{h}\widehat{\Omega}$ using
the Weizenböck formula. Define the two-form

$\widehat{\Xi}:=\lambda^{-2}\widehat{\Omega}\,.$ (2.16)

It will be shown in Section 3 that $\widehat{\Xi}$ extends continuously to
that part of the boundary of the quotient space which corresponds to the
Killing horizon, and is related to the torsion $1$-form of the bifurcation
surface. We will also show that $\widehat{\Xi}$ vanishes if and only the twist
of the Killing vector vanishes.

In order to obtain the promised identity we apply the trivial identity
$\delta(f\alpha)=f\delta\alpha-i_{\mathrm{grad}\ f}\alpha$, valid form any
scalar $f$ and $p$-form $\alpha$, and obtain

$\displaystyle\delta\widehat{\Xi}=\widehat{r}+\frac{3}{2\lambda}i_{\mathrm{grad}\lambda}\widehat{\Xi}.$
(2.17)

Since $d\widehat{\Xi}=0$, we can easily compute the Hodge-de Rahm operator
$\Delta_{\mbox{\tiny dR}}:=(d\circ\delta+\delta\circ d)$ of $\widehat{\Xi}$

$\displaystyle\Delta_{\mbox{\tiny
dR}}\widehat{\Xi}=(d\circ\delta)\,\widehat{\Xi}=d\left(\widehat{r}+\frac{3}{2}i_{\mathrm{grad}(\ln|\lambda|)}\widehat{\Xi}\right)=d\widehat{r}+\frac{3}{2}\pounds_{\mathrm{grad}(\ln|\lambda|)}\widehat{\Xi},$

where in the last equality we used the Cartan identity $\pounds_{X}=i_{X}\circ
d+d\circ i_{X}$. We now apply the Weizenböck identity [19, 9], which for
$2$-forms reads

$\displaystyle(\Delta_{h}\widehat{\Xi})_{ij}=-(\Delta_{\mbox{\tiny
dR}}\widehat{\Xi})_{ij}-2(R^{h})^{k}{}_{i}{}^{\ell}{}_{j}\widehat{\Xi}_{kl}+(R^{h})^{k}{}_{i}\widehat{\Xi}_{kj}-(R^{h})^{k}{}_{j}\widehat{\Xi}_{ki}$
(2.18)

where $\Delta_{h}$ is the rough Laplacian $\Delta_{h}:=D^{i}D_{i}$ and the
Riemann and Ricci tensors of $(\Sigma,h)$ are denoted respectively by
$(R^{h})^{i}{}_{jkl}$ and $(R^{h})_{{j\ell}}$. Thus, the field $\widehat{\Xi}$
satisfies the identity

$\displaystyle\Delta_{h}\widehat{\Xi}_{ij}=$
$\displaystyle-2(R^{h})^{k}{}_{i}{}^{\ell}{}_{j}\widehat{\Xi}_{kl}+(R^{h})^{k}{}_{i}\widehat{\Xi}_{kj}-(R^{h})^{k}{}_{j}\widehat{\Xi}_{ki}-D_{i}\widehat{r}_{j}+D_{j}\widehat{r}_{i}$
$\displaystyle-\frac{3}{2}\pounds_{\mathrm{grad}(\ln|\lambda|)}\widehat{\Xi}_{ij}.$
(2.19)

Recall that $-g(\xi,\xi)\equiv\lambda=\epsilon u^{2}$, $\epsilon=\pm 1$, with
$u>0$. It will be useful to write down an equivalent identity involving the
two-form

$\widehat{\Upsilon}^{(a)}:=u^{a}\,\widehat{\Xi}=u^{a-4}\widehat{\Omega}\,,\quad
a\in\mathbb{R}\,,$

where the subscript $a$ on $\widehat{\Upsilon}^{(a)}$ is _not_ a vector index,
but indicates the power $a$ used in the rescaling. We are mainly interested in
the case

$a=a_{n}:=\frac{n-1}{2}\,,$ (2.20)

but we leave the constant $a$ arbitrary unless explicitly specified otherwise.

A straightforward computation gives

$\displaystyle d\widehat{\Upsilon}^{(a)}$
$\displaystyle=\frac{a}{u}du\wedge\widehat{\Upsilon}^{(a)},$ (2.21)
$\displaystyle\delta\widehat{\Upsilon}^{(a)}$
$\displaystyle=u^{a}\widehat{r}+(3-a)i_{\mathrm{grad}\ln
u}\widehat{\Upsilon}^{(a)},$ (2.22)
$\displaystyle\Delta_{h}\widehat{\Upsilon}^{(a)}_{ij}=$
$\displaystyle-2(R^{h})^{k}{}_{i}{}^{\ell}{}_{j}\widehat{\Upsilon}^{(a)}_{kl}+\left((R^{h})^{k}{}_{i}-\frac{3D_{i}D^{k}u}{u}+\frac{3D^{k}uD_{i}u}{u^{2}}\right)\widehat{\Upsilon}^{(a)}_{kj}$
$\displaystyle-\left((R^{h})^{k}{}_{j}-\frac{3D_{j}D^{k}u}{u}+\frac{3D^{k}uD_{j}u}{u^{2}}\right)\widehat{\Upsilon}^{(a)}_{ki}-2u^{a}D_{[i}\left(\widehat{r}_{j]}\right)$
$\displaystyle+(2a-3)\frac{D^{k}u}{u}D_{k}\widehat{\Upsilon}^{(a)}_{ij}+a\left(\frac{\Delta_{h}u}{u}+(2-a)\frac{|du|^{2}_{h}}{u^{2}}\right)\widehat{\Upsilon}^{(a)}_{ij}.$
(2.23)

It will also be of interest to find an equation in terms of a conformal
metric. Define

$\bar{h}_{ij}:=u^{-2}h_{ij}\,.$ (2.24)

Applying Lemma A.1 in Appendix A to $\Psi=\widehat{\Upsilon}^{(a)}$ and using
(2.21)-(2.22) we find

$\displaystyle\Delta_{\bar{h}}\widehat{\Upsilon}^{(a)}_{ij}=$ $\displaystyle
u^{2}\Delta_{h}\widehat{\Upsilon}^{(a)}_{ij}+2\left(u\Delta_{h}u+(2+a-n)|du|^{2}_{h}\right)\widehat{\Upsilon}^{(a)}_{ij}$
$\displaystyle+(4-n)uD^{k}uD_{k}\widehat{\Upsilon}^{(a)}_{ij}-(n+2)D^{k}uD_{i}u\,\widehat{\Upsilon}^{(a)}_{kj}+(n+2)D^{k}uD_{j}u\,\widehat{\Upsilon}^{(a)}_{ki}$
$\displaystyle-2u^{1+a}\left(D_{i}u\,\widehat{r}_{j}-D_{j}u\,\widehat{r}_{i}\right).$

Combining with (2.23) we conclude that $\widehat{\Upsilon}^{(a)}$ satisfies

$\displaystyle L_{a}(\widehat{\Upsilon}^{(a)})=-u^{a}d(u^{2}\widehat{r})\,,$
(2.25)

where $L_{a}$ is the operator

$\displaystyle L_{a}(\Psi)_{ij}:=$
$\displaystyle\Delta_{\bar{h}}\Psi_{ij}+(n-1-2a)uD^{k}u\,D_{k}\Psi_{ij}$
$\displaystyle+2u^{2}(R^{h})^{k}{}_{i}{}^{\ell}{}_{j}\Psi_{kl}+B^{k}{}_{i}\Psi_{kj}-B^{k}{}_{j}\Psi_{ki}\,,$
(2.26) $\displaystyle B^{k}{}_{i}:=$
$\displaystyle-u^{2}((R^{h}))^{k}{}_{i}+3uD^{k}D_{i}u+(n-1)D^{k}uD_{i}u$
$\displaystyle-\left(\frac{2+a}{2}u\Delta_{h}u+\big{(}(2+a-n)+\frac{a(2-a)}{2}\big{)}|du|^{2}_{h}\right)\delta^{k}{}_{i}\,.$
(2.27)

######  Lemma 2.2.

The operator $L_{a}$ with $a=\frac{n-1}{2}$ is formally self-adjoint with
respect to the $L^{2}(d\mu_{\bar{h}})$-scalar product, where $d\mu_{\bar{h}}$
is the Riemannian measure defined by $\bar{h}$.

Proof: For this value of $a$ there are no first order terms in $L_{a}$. The
symmetry of the zero-order terms is the contents of Corollary A.3 in Appendix
A. $\hfill\Box$

##  3 Rácz-Wald coordinates

We wish to tie the quantities introduced above to objects defined on the
bifurcation surface of a bifurcate Killing horizon. For this a convenient set
of coordinates is the one due to Rácz and Wald [17, 18]. Such coordinates can
be introduced in either of the following circumstances:

  1. H1.

The spacetime contains a bifurcate Killing horizon.

  2. H2.

The spacetime has a null boundary ${\mycal N}$ with a causal Killing vector
$\xi$ which is tangent to the generators of ${\mycal N}$, with non-zero
surface gravity. The orbits of $\xi$ are complete and there exists a
hypersurface $\mycal S$ to which $\xi$ is transverse and such that every
orbits of $\xi$ intersects $\mycal S$ precisely once.

In the latter case, which is typical for well-behaved domains of outer
communications, one constructs [17] an auxiliary spacetime which contains a
bifurcate Killing horizon. So, in this sense, it suffices to consider
spacetimes $({M},g)$ which contain a bifurcate Killing horizon, i.e. two
smooth null connected hypersurfaces $\mathcal{H}^{+}$ and $\mathcal{H}^{-}$
intersecting on a spacelike codimension-two surface
$S=\mathcal{H}^{+}\cap\mathcal{H}^{-}$ and such that $({M},g)$ admits a non-
trivial Killing vector $\xi$ which vanishes on $S$. It follows automatically
that $\mathcal{H}^{\pm}\setminus S$ are Killing horizons of $\xi$ with two
connected components each.

A spacetime neighbourhood $\mycal O$ of $S$ can be coordinatised with an atlas
of _Rácz-Wald coordinates_ $\\{U,V,x_{\mathfrak{i}}^{a}\\}$, where
$\\{x^{a}_{\mathfrak{i}},O_{\mathfrak{i}}\\}$ is an atlas of $S$. In such
coordinates, the metric $g$ takes the form

$\displaystyle g=GdU\left(dV+Vq_{a}dx^{a}\right)+\gamma_{ab}dx^{a}dx^{b}\,,$
(3.1)

where all metric coefficients $G,q_{a},\gamma_{ab}$ depend only upon
$\\{UV,x^{a}\\}$, and where $G$ is a positive function. The bifurcation
surface $S$ is the set $\\{U=V=0\\}$. We choose the time orientation so that
$\partial_{U}$ is future pointing and $\partial_{V}$ is past pointing. The
construction of [17] guarantees that there exists a neighborhood of the
bifurcate Killing horizon on which $G$ is strictly positive.

Recall that the torsion $1$-form of a spacelike surface of codimension equal
to two is defined as

$\displaystyle\zeta(X)=g(k,\nabla_{X}\ell)$

where $X$ is a vector field in $S$ and $\\{k,\ell\\}$ are a pair of null
normals to $S$ satisfying the normalization condition $g(k,\ell)=-1$. The null
normals $k$ and $\ell$ can be boosted or interchanged. The boost transforms
the torsion by adding the differential of a function, while the interchange
$k\leftrightarrow\ell$ changes the sign of $\zeta$. Note that the property of
$\zeta$ being closed if not affected by these changes.

In what follows we shall need the explicit form of $\zeta$. We take the normal
basis

$\displaystyle
k=-\sqrt{\frac{2}{G}}\partial_{V}\,,\qquad\ell=\sqrt{\frac{2}{G}}\partial_{U}\,,$

and after a simple calculation based on the metric (3.1) we find

$\displaystyle\zeta=\frac{1}{2}q_{a}|_{\\{UV=0\\}}dx^{a}\,.$ (3.2)

In Rácz-Wald coordinates the Killing vector $\xi$ reads

$\displaystyle\xi=U\partial_{U}-V\partial_{V}.$ (3.3)

and therefore

$\displaystyle\mbox{\boldmath$\xi$}:=g(\xi,\cdot)=\frac{1}{2}G\left(UdV-
VdU+UVq_{a}dx^{a}\right)\,.$ (3.4)

We shall also need the explicit form of the twist $3$-form of $\xi$, recall
that

$\omega:=\frac{1}{2}\mbox{\boldmath$\xi$}\wedge d\mbox{\boldmath$\xi$}\,.$

For this it is simplest to work away from the bifurcate Killing horizon and
write $\xi$ as

$\displaystyle\mbox{\boldmath$\xi$}=\frac{1}{2}GUV\left(d\ln\left|\frac{V}{U}\right|-q_{a}dx^{a}\right)=:\frac{1}{2}GUV\tilde{\mbox{\boldmath$\xi$}}.$

Since $q_{a}=q_{a}(UV,x^{b})$, a direct calculation gives, where a prime
denotes a derivative with respect to $UV$:

$\displaystyle\tilde{\mbox{\boldmath$\xi$}}\wedge
d\tilde{\mbox{\boldmath$\xi$}}=$ $\displaystyle 2q^{\prime}_{c}dU\wedge
dV\wedge dx^{c}+q^{\prime}_{a}q_{c}\left(UdV+VdU\right)\wedge dx^{a}\wedge
dx^{c}$
$\displaystyle-\left(\partial_{a}q_{b}-\partial_{b}q_{a}\right)\left(\frac{dV}{V}-\frac{dU}{U}\right)\wedge
dx^{a}\wedge dx^{b}+q_{c}(\partial_{a}q_{b}-\partial_{b}q_{a})dx^{c}\wedge
dx^{a}\wedge dx^{b}$

and we conclude that

$\displaystyle\omega=\,$
$\displaystyle\frac{1}{8}G^{2}U^{2}V^{2}\tilde{\mbox{\boldmath$\xi$}}\wedge
d\tilde{\mbox{\boldmath$\xi$}}$ $\displaystyle=\,$
$\displaystyle-\frac{1}{8}G^{2}\left(\partial_{a}q_{b}-\partial_{b}q_{a}\right)\left(U^{2}VdV-V^{2}UdU\right)\wedge
dx^{a}\wedge dx^{b}+U^{2}V^{2}\widetilde{\omega}\,,$

where $\widetilde{\omega}$ is a smooth $3$-form. Note that although the
computation has been done away from $U=0$ or $V=0$, the result is valid
everywhere. Observe also that the twist always vanishes on the bifurcate
Killing horizon. This is a consequence of the fact that $\xi$ is a null normal
to the hypersurfaces $U=0$ and $V=0$.

The transverse derivative $\pounds_{\partial_{V}}\omega$ of $\omega$ at the
null hypersurface $\\{V=0\\}$ reads

$\displaystyle\pounds_{\partial_{V}}\mbox{\boldmath$\omega$}|_{\\{V=0\\}}=-\frac{1}{8}G^{2}\left(\partial_{a}q_{b}-\partial_{b}q_{a}\right)|_{UV=0}U^{2}dV\wedge
dx^{a}\wedge dx^{b}.$ (3.5)

This $3$-form vanishes if and only if $q_{a}|_{\\{UV=0\\}}$ is closed, i.e. if
and only if the torsion $1$-form $\zeta$ of the bifurcation is closed. A
similar statement holds for the null hypersurface $\\{U=0\\}$.

Recall that

$\lambda=-g(\xi,\xi)$

denotes the Lorentzian norm-squared of the Killing vector $\xi$, and that we
write the _quotient-space metric_ as

$h_{\alpha\beta}\equiv g_{\alpha\beta}+\lambda^{-1}\xi_{\alpha}\xi_{\beta}\,.$

Since $\lambda=-\mbox{\boldmath$\xi$}(\xi)=UVG$ a straightforward computation
gives

$\displaystyle
h=\frac{G}{4UV}\left((UdV+VdU)^{2}+U^{2}V^{2}(q_{a}dx^{a})^{2}+2UV(UdV+VdU)q_{a}dx^{a}\right)+\gamma_{ab}dx^{a}dx^{b}.$

We introduce the scalar

$s:=UV$

and note that $\\{s,x^{a}\\}$ descend to local coordinates on the quotient
space (both are Lie-constant along $\xi$). The tensor $h$ takes the form

$\displaystyle
h=\frac{G}{4s}\left(ds+sq_{a}dx^{a}\right)^{2}+\gamma_{ab}dx^{a}dx^{b},\qquad\mbox{and}\
\lambda=sG$

where $G,q_{a},\gamma_{ab}$ are functions of $\\{s,x^{a}\\}$. Let us perform
the coordinate change

$s=\epsilon z^{2}\,,\quad\epsilon=\pm 1\,,\ z>0\,,$

and note that $\epsilon=1$ corresponds to the domain $UV>0$ where the Killing
is timelike, while $\epsilon=-1$ corresponds to the domain $UV<0$, where it is
spacelike.

In the coordinates $\\{z,x^{a}\\}$, the quotient metric is

$\displaystyle h=\epsilon
G\big{(}dz+\frac{1}{2}zq_{a}dx^{a}\big{)}^{2}+\gamma_{ab}dx^{a}dx^{b},\qquad\lambda=\epsilon
z^{2}G$

where $G,q_{a},\gamma_{ab}$ are functions of $\\{\epsilon z^{2},x^{a}\\}$.
This defines a metric smooth-up-to the boundary $z=0$, because $G$ is positive
and $\gamma_{ab}$ positive definite in the domain $\mycal O$. Clearly $h$ is
Riemannian when $\epsilon>0$, i.e. in the region where the Killing vector
field $\xi$ is timelike, and Lorentzian when $\epsilon<0$, where $\xi$ is
spacelike.

Let $u>0$ be defined by

$\lambda=\epsilon u^{2}\,,\quad\epsilon=\pm 1\,.$ (3.6)

The function $u$ in the $\\{z,x^{a}\\}$ coordinates reads $u=z\sqrt{G}$, so it
is smooth-up-to-and-including the boundary, where it has a zero of first
order. Thus $u$ is a defining function for $\partial\Sigma$. The metric

$\bar{h}=u^{-2}h$

appearing in (2.26) provides therefore an example of smoothly conformally
compactifiable metric _à la Penrose_. In the region where the Killing vector
$\xi$ is spacelike the bifurcation surface $S\approx\partial\Sigma$ becomes
the conformal boundary at timelike infinity for the metric $\bar{h}$. This
property will be exploited in what follows.

Perhaps unexpectedly, $|du|^{2}_{h}$ is constant on $\partial\Sigma$. To see
this it is convenient to introduce $P:=\sqrt{G}$ so that $u=zP$, and raise
indices $a,b$ with $\gamma^{ab}$ defined to be the inverse of $\gamma_{ab}$.
The inverse metric $h^{\sharp}$ is given by

$\displaystyle
h^{\sharp}=\left(\begin{array}[]{cc}\frac{\epsilon}{P^{2}}+\frac{1}{4}z^{2}q^{a}q_{a}&-\frac{1}{2}zq^{a}\\\
-\frac{1}{2}zq^{a}&\gamma^{ab}\\\ \end{array}\right).$

Thus

$\displaystyle\frac{1}{z}\mathrm{grad}_{h}z=\frac{\epsilon}{zP^{2}}\partial_{z}+O(1),\qquad\frac{1}{u}\mathrm{grad}_{h}u=\frac{1}{z}\mathrm{grad}_{h}z+\frac{1}{P}\mathrm{grad}_{h}P=\frac{\epsilon}{zP^{2}}\partial_{z}+O(1),$

where $O(1)$ denotes any function that extends smoothly to the boundary
$\partial\Sigma$. Using the fact that $\partial_{z}$ acting on any metric
coefficient $G,q_{a},\gamma_{ab}$ gives a function of the form $zF$, where $F$
is regular up to boundary (because the functions depend on $\\{\epsilon
z^{2},x\\}$) it follows that

$\displaystyle\frac{1}{z}\left<\mathrm{grad}_{h}z,\mathrm{grad}_{h}P\right>=O(1)\,,$

where $\langle\,,\,\rangle$ denote scalar product with $h$. Consequently

$\displaystyle\frac{\langle\mathrm{grad}_{h}u,\mathrm{grad}_{h}u\rangle}{u^{2}}=\frac{\langle\mathrm{grad}_{h}z,\mathrm{grad}_{h}z\rangle}{z^{2}}+O(1)=\frac{\epsilon}{z^{2}P^{2}}+O(1)=\frac{\epsilon}{u^{2}}+O(1)$

and we find that

$|du|^{2}_{h}|_{\partial\Sigma}=\epsilon\,.$ (3.7)

This has the well known geometric interpretation that

######  Proposition 3.1.

In the region $\epsilon>0$ the metric $u^{-2}h$ is asymptotically locally
hyperbolic, i.e., it is smoothly conformally compactifiable, with sectional
curvatures tending to a negative constant when the conformal boundary at
infinity $\partial\Sigma$ of $u^{-2}h$ is approached. $\Box$

We finish this section with a formula for the $2$-form $\widehat{\Xi}$ of
(2.16) in terms of the Rácz-Wald coordinates. A computation of the two-form
$\Omega$ in the metric (3.1) gives

$\displaystyle\Omega=-\frac{1}{4}G^{2}(UV)^{2}dq,\qquad q:=q_{a}dx^{a}\,.$
(3.8)

Hence $\widehat{\Omega}=-\frac{1}{4}u^{4}dq$ and, in the coordinates
$\\{z,x^{a}\\}$,

$\widehat{\Xi}_{ij}=-\frac{1}{4}\left(\partial_{i}q_{j}-\partial_{j}q_{i}\right)\,,\qquad
q_{i}:=\delta^{a}{}_{i}q_{a}\,,$

i.e.

$\widehat{\Xi}_{za}=-\frac{1}{4}\partial_{z}q_{a}\,,\qquad\widehat{\Xi}_{ab}=-\frac{1}{4}\left(\partial_{a}q_{b}-\partial_{b}q_{a}\right)\,.$
(3.9)

Since $q_{a}=q_{a}(z^{2},x)$, all odd $z$-derivatives at $z=0$ vanish.
Combining with the explicit form (3.2) of the torsion $1$-form $\zeta$ of the
bifurcation surface we conclude that:

######  Proposition 3.2.

$\widehat{\Xi}_{ij}=0$ at $\partial\Sigma$ if and only if the torsion $1$-form
$\zeta$ of $S$ is closed. $\Box$

##  4 Staticity

We are ready now to provide necessary and sufficient conditions for staticity
of the spacetime metric. It turns out that the sides of the Killing horizons
where the Killing vector is timelike and these where the Killing vector is
spacelike require separate treatment.

###  4.1 The timelike region

We start by considering the region where the Killing vector is timelike, which
corresponds to $\epsilon=1$. However, all local calculations in this section
apply as long as the Killing vector is not null, and will also be relevant for
Section 4.2.

Using the notation of [12], it follows that $(\Sigma,\bar{h})$ has a compact
conformal boundary at infinity and that $\Delta_{\bar{h}}$ is a uniformly
degenerate elliptic operator. We also use the notion of indicial map
$I_{\sigma}$ as defined in [12]. Namely, given a uniformly degenerate elliptic
operator

$\displaystyle L:\Gamma(E)\longrightarrow\Gamma(E)$

where $\Gamma(E)$ is the set of sections of a tensor bundle $E$ over $\Sigma$,
and using our function $u$ as the defining function for the boundary (which
applies near horizons with non-zero surface gravity, but does not when the
surface gravity vanishes), for $\sigma\in\mathbb{C}$ the indicial map is the
map

$\displaystyle I_{\sigma}:\Gamma(E|_{\partial\Sigma}\otimes\mathbb{C})$
$\displaystyle\longrightarrow\Gamma(E|_{\partial\Sigma}\otimes\mathbb{C})$
$\displaystyle\widehat{\Psi}$ $\displaystyle\longrightarrow
I_{\sigma}(\widehat{\Psi}):=\lim_{u\to
0}\frac{L(u^{\sigma}\Psi)}{u^{\sigma}}\,,$

where $\Gamma(E|_{\partial\Sigma})$ is the set of sections over
$\partial\Sigma$ of the bundle $E$, $E|_{\partial\Sigma}\otimes\mathbb{C}$ is
its complexification and $\Psi\in\Gamma(E\otimes\mathbb{C})$ is any smooth
extension of $\widehat{\Psi}$. The indicial map is a well-defined (i.e.
independent of the choice of extension $\Psi$), and is also independent of the
choice of the defining function. The indicial exponents of $L$ are the values
$\sigma$ for with there exists a point $p\in\partial\Sigma$ and a non-zero
$\widehat{\Psi}$ such that $I_{\sigma}(\widehat{\Psi})|_{p}=0$.

We wish to prove a vanishing theorem for $\widehat{\Upsilon}^{(a)}$, for this
we need to determine the indicial exponents for the operator (2.25).

######  Lemma 4.1.

In the setup where $({M},g)$ admits a bifurcate Killing horizon, consider the
quotient $\Sigma$ in the region where $\xi$ is timelike, i.e. $\epsilon=1$.
Define $L:=L_{a}$ for $a=(n-1)/2$, cf. (2.20). Then, the indicial exponents of
$L$ are

$\displaystyle\sigma\in\left\\{a-4,a-3,a-1,a\right\\}=\left\\{\frac{n-1}{2}-4,\frac{n-1}{2}-3,\frac{n-1}{2}-1,\frac{n-1}{2}\right\\}\,.$

Proof: Define (cf. Corollary A.3, Appendix A)

$\displaystyle\bar{S}_{ij}{}^{kl}:=2u^{2}(R^{h})^{[k}{}_{i}{}^{l]}{}_{j}-\delta^{[k}{}_{j}B^{l]}{}_{i}-B^{[k}{}_{j}\delta^{l]}{}_{i}$

so that $L=\Delta_{\bar{h}}+S$. We apply item (ii) in Lemma A.2. Let

$V^{i}:=D^{i}u|_{\partial\Sigma}$

be the restriction of the gradient of $u$ to $\partial\Sigma$. Given that

$\displaystyle
B^{k}{}_{i}|_{\partial\Sigma}=\big{(}(n-1)V^{k}V_{i}+\frac{n^{2}-2n-7}{8}|V|^{2}\big{)}\delta^{k}_{i}\big{|}_{\partial_{\Sigma}}$

we have

$\displaystyle\bar{S}_{ij}{}^{kl}|_{\partial\Sigma}\widehat{\Psi}_{kl}=(n-1)\left(V^{k}V_{i}\widehat{\Psi}_{kj}-V^{k}V_{j}\widehat{\Psi}_{ki}\right)+\frac{n^{2}-2n-7}{4}\widehat{\Psi}_{ij}$

and the indicial map is

$\displaystyle
I_{\sigma}(\widehat{\Psi})_{ij}=c_{1}|V|^{2}\widehat{\Psi}_{ij}+3\left(V^{k}V_{i}\widehat{\Psi}_{kj}-V_{k}V_{j}\widehat{\Psi}_{ki}\right),\quad
c_{1}$
$\displaystyle:=\left(\sigma-\frac{n-1}{2}\right)\left(\sigma-\frac{n-9}{2}\right).$

Given $p\in\partial\Sigma$ we want to find the values $\sigma\in\mathbb{C}$
and tensors $\widehat{\Psi}_{ij}$ at $p$ such that
$I_{\sigma}(\widehat{\Psi})|_{p}$ is zero. Assuming that
$I_{\sigma}(\widehat{\Psi})|_{p}$ vanishes, a contraction of
$I_{\sigma}(\widehat{\Psi})_{ij}$ with $V^{i}$ gives

$\displaystyle(c_{1}+3)|V|^{2}_{h}V^{i}\widehat{\Psi}_{ij}=0.$

If $c_{1}+3\neq 0$ (i.e. $\sigma\not\in\\{\frac{n-3}{2},\frac{n-7}{2}\\}$)
then necessarily $V^{i}\widehat{\Psi}_{ij}=0$ and the condition
$I_{\sigma}(\widehat{\Psi})=0$ reduces to $c_{1}\widehat{\Psi}_{ij}=0$. If in
addition $c_{1}\neq 0$, the kernel is trivial. If $c_{1}=0$, i.e.
$\sigma=\frac{n-1}{2}$ or $\sigma=\frac{n-9}{2}$ the kernel is non-trivial and
consists of tensor fields taking the form
$\widehat{\Psi}_{ij}=X_{i}Y_{j}-X_{j}Y_{i}$ with covectors $X,Y$ annihilating
$V$; equivalently, the associated vectors are tangent to $\partial\Sigma$.

If $c_{1}+3=0$ then $\widehat{\Psi}_{ij}:=X_{i}V_{j}-X_{j}V_{i}$ (with $X$ as
before) solves $I_{\sigma}(\widehat{\Psi}_{ij})=0$:

$\displaystyle I_{\sigma}(\widehat{\Psi}_{ij})$
$\displaystyle=I_{\sigma}(X_{i}V_{j}-X_{j}V_{i})$
$\displaystyle=c_{1}|V|^{2}_{h}(X_{i}V_{j}-X_{j}V_{i})+3|V|^{2}_{h}X_{i}V_{j}-3|V|^{2}_{h}X_{j}V_{i}$
$\displaystyle=(c_{1}+3)|V|^{2}_{h}\left(X_{i}V_{j}-X_{j}V_{i}\right)=0\,,$

which concludes the proof. $\hfill\Box$

This leads to:

######  Theorem 4.2 (Staticity extension, timelike region).

Assume that $({M},g)$ contains a bifurcate Killing horizon $\mycal H$ and
consider the quotient manifold $\Sigma$ in the region where $\xi$ is timelike.
Assume that the Ricci tensor of $({M},g)$ satisfies

$d\big{(}g(\xi,\xi)\hat{r}\big{)}=0$

(in particular this is true for $\Lambda$-vacuum spacetimes), where the
$1$-form $\hat{r}$ has been defined in (2.9)-(2.10). Then the tensor field
$\widehat{\Xi}_{ij}$ vanishes on $\partial\Sigma$ if and only if the metric is
static in the connected component adjacent to $\mycal H$ of the region where
the Killing vector is timelike; this is equivalent to the vanishing of
$d\zeta$, where $\zeta$ is the torsion $1$-form of $S$.

######  Remark 4.3.

It further follows from (3.5) that, for bifurcate horizons, this is equivalent
to the vanishing of the transverse derivative of $\omega$ at any of the
branches of the horizon. $\Box$

Proof: Clearly $\widehat{\Xi}$ vanishes everywhere when the metric is static
(cf. (2.3), (2.8) and (2.16)), in particular on $\partial\Sigma$. Thus the
condition is necessary.

To prove the converse implication, we start by noting that for any
$a\in\mathbb{R}$ the vanishing of $\widehat{\Xi}$ at $\partial\Sigma$ is
equivalent to $\widehat{\Upsilon}^{(a)}=o(u^{a})$ at $\partial\Sigma$.

As already hinted-to, we choose $a=a_{n}\equiv(n-1)/2$. The operator $L\equiv
L_{(n-1)/2}$ is elliptic, uniformly degenerate and self-adjoint with respect
to the $L^{2}$ norm of the metric $\bar{h}$. For such an operator [1, Theorem
A.14] and [14, Corollary (11)] apply. In particular [1, Theorem A.14 with
$f\equiv 0$] shows that $\widehat{\Upsilon}^{(a)}$ is polyhomogeneous at
$\partial\Sigma$. Matching coefficients in the polyhomogeneous expansion shows
that $\widehat{\Xi}$ vanishes to infinite order at $\partial\Sigma$ because
$\widehat{\Upsilon}^{(a)}=o(u^{a})$ and Lemma 4.1.

The vanishing of $\widehat{\Upsilon}^{(a)}$ (and hence of $\widehat{\Xi}$)
follows now from [14, Corollary (11)].

The relation to the vanishing of $d\zeta$ is the contents of Proposition 3.2.
$\Box$

###  4.2 The spacelike region

We continue our analysis in the region where $\epsilon=-1$, thus $u$ is a time
function there. The aim is to find necessary and sufficient conditions for the
twist to vanish in that region.

We consider a configuration where the fields are smooth to the causal past of
the bifurcation surface, or sufficiently differentiable as needed in the
argument below, where finite differentiability suffices. Such metrics can be
constructed, for example, by solving the characteristic Cauchy problem with
smooth KID data on a bifurcate null hypersurface, as described e.g. in [8, 16,
11].

We formulate the staticity theorem in the past of a bifurcation surface, an
identical result holds of course to the future:

######  Theorem 4.4 (Staticity extension, spacelike region).

Let $\xi$ be a Killing vector defined on the past of an $(n-1)$-dimensional
spacelike surface $S$ in an $(n+1)$-dimensional spacetime $(M,g)$, with
$\xi=0$ on $S$. Assume that the Ricci tensor of $({M},g)$ satisfies

$d\big{(}g(\xi,\xi)\hat{r}\big{)}=0$

(in particular this is true for $\Lambda$-vacuum spacetimes). Then the Killing
vector is static on the past domain of dependence ${\mycal D}^{-}(S)$ if and
only if the tensor field $\widehat{\Xi}_{ij}$ vanishes on $\partial\Sigma$, if
and only if the torsion covector of $S$ is closed.

Proof: The local calculations at $\partial\Sigma$ implicit in the proof of
Theorem 4.2 show that $\widehat{\Upsilon}^{(a)}$ vanishes to all orders at
$S$. In the analytic case the result immediately follows. In the smooth case
one can instead rewrite the Fuchsian wave equation (2.25) as a symmetric
hyperbolic system in a standard way, and use a large negative number
$a\in\mathbb{R}$ in the definition of $\widehat{\Upsilon}^{(a)}$ so that the
positivity condition of [13, Lemma 4.14] holds. $\Box$

## Acknowledgements

M.M. acknowledges financial support under Grant PID2021-122938NB-I00 funded by
MCIN/AEI /10.13039/501100011033 and by “ERDF A way of making Europe” and
RED2022-134301-T funded by MCIN/AEI/10.13039/501100011033.

##  Appendix A Appendix: Conformal behavior of the Laplacian on two-forms

Consider an $n$-dimensional pseudo-Riemannian manifold $(\Sigma,h)$. Let $u$
be a smooth positive function on $\Sigma$ and consider the metric

$\bar{h}:=u^{-2}h\,.$

Objects defined with $\bar{h}$ carry an overline. Indices on such objects are
moved with $\bar{h}$ while objects without an overline are manipulated using
$h$.

The following lemma relates $\Delta_{h}$ and $\Delta_{\bar{h}}$ on two-forms.

######  Lemma A.1.

Let $\Psi$ be a smooth two-form on $\Sigma$. Then

$\displaystyle\Delta_{\bar{h}}\Psi_{ij}$
$\displaystyle=u^{2}\Delta_{h}\Psi_{ij}+2\left(u\Delta_{h}u+(2-n)|du|^{2}_{h}\right)\Psi_{ij}$
$\displaystyle+(4-n)\left(uD^{k}u\,D_{k}\Psi_{ij}+D^{k}uD_{i}u\,\Psi_{kj}-D^{k}uD_{j}u\,\Psi_{ki}\right)$
$\displaystyle-2u\left(D_{i}u(\delta\Psi)_{j}-D_{j}u(\delta\Psi)_{i}\right)+2uD^{k}u(d\Psi)_{kij}.$
(A.1)

Before going into the proof we need some preliminaries. We write $Q$ for the
difference tensor $Q(X,Y):=\bar{D}_{X}Y-D_{X}Y$. Its explicit form is

$\displaystyle Q^{k}{}_{il}$
$\displaystyle=\frac{1}{u}\left(-\delta^{k}{}_{i}D_{l}u-\delta^{k}{}_{l}D_{i}u+D^{k}u\,h_{il}\right).$
(A.2)

Computing $\Delta_{\bar{h}}$ directly in terms of $\Delta_{h}$ and the tensor
$Q$ leads to long computations. We take a shortcut via the Weizenböck
identity. It is convenient to introduce the tensor

$\displaystyle
H^{k}{}_{ilj}=-2(R^{h})^{k}{}_{ilj}+\left(\delta^{k}{}_{l}(R^{h})_{ij}-\delta^{k}{}_{j}(R^{h})_{il}+(R^{h})^{k}{}_{l}h_{ij}-(R^{h})^{k}{}_{j}h_{il}\right)$
(A.3)

and the corresponding tensor $\bar{H}^{k}{}_{ilj}$. The Weizenböck identity
(2.18) reads

$\displaystyle\Delta_{h}\Psi_{ij}=-\Delta_{\mbox{\tiny
dR}}\Psi_{ij}+H^{k}{}_{i}{}^{l}{}_{j}\Psi_{kl}$

so we can relate $\Delta_{\bar{h}}$ and $\Delta_{h}$ by

$\displaystyle\Delta_{\bar{h}}\Psi_{ij}=u^{2}\Delta_{h}\Psi_{ij}+\left(\bar{H}^{k}{}_{i}{}^{l}{}_{j}-u^{2}H^{k}{}_{i}{}^{l}{}_{j}\right)\Psi_{kl}-\left(\bar{\Delta}_{\mbox{\tiny
dR}}\Psi_{ij}-u^{2}\Delta_{\mbox{\tiny dR}}\Psi_{ij}\right).$ (A.4)

Let us momentarily assume $n\geq 3$, we will consider the case $n=2$ shortly.
In terms of the Schouten tensor $L$,

$\displaystyle(R^{h})_{ij}:=(n-2)L_{ij}+Lh_{jl},\quad L:=h^{ij}L_{ij}\,,$

and letting $(C^{h})^{i}{}_{ijk}$ be the Weyl tensor of $h$, we have the
decomposition

$\displaystyle
H^{k}{}_{ilj}=-2(C^{h})^{k}{}_{ilj}-2(4-n)\left(\delta^{k}{}_{l}L_{ij}-\delta^{k}{}_{j}L_{il}+L^{k}{}_{l}h_{ij}-Lh_{il}\right)+2L\left(\delta^{k}{}_{l}h_{ij}-\delta^{k}{}_{j}h_{il}\right).$

Under the conformal rescaling $\bar{h}=u^{-2}h$, the Schouten tensor
transforms as

$\displaystyle\bar{L}_{ij}=L_{ij}+\frac{1}{u}D_{i}D_{j}u-\frac{|du|^{2}_{h}}{2u^{2}}h_{ij}.$

This, combined with conformal invariance of the Weyl tensor leads to

$\displaystyle\bar{H}^{k}{}_{i}{}^{l}{}_{j}-u^{2}H^{k}{}_{i}{}^{l}{}_{j}=h^{kl}W_{ij}-\delta^{k}{}_{j}W^{l}{}_{i}+W^{kl}h_{ij}-W^{k}{}_{j}\delta^{l}{}_{i}\,,$
(A.5) $\displaystyle
W_{ij}:=(n-4)uD_{i}D_{j}u+\left(u\Delta_{h}u+(2-n)|du|^{2}_{h}\right)h_{ij}\,.$
(A.6)

Let us check that these equations also hold when $n=2$. Since the Riemann and
Ricci tensors in two dimensions can be expressed in terms of the Gauss
curvature and the metric, the tensor $H^{k}{}_{ilj}$ defined in (A.3) is
identically zero, and so is the difference left-hand side of (A.5). Next, in
dimension $n=2$ the tensor $W_{ij}$ defined in (A.6) is trace-free. A tensor
with the symmetries of a Riemann tensor in two-dimensions vanishes if and only
if its double trace vanishes. The trace in $\\{ij\\}$ and then $\\{kl\\}$ of
the right-hand side of (A.5) is $2\mbox{\rm tr}\,_{h}W$, hence zero. So, the
right-hand side of (A.5) is also zero and the validity of the expression in
dimension $n=2$ is established.

In order to determine the conformal change of the de Rahm Laplacian we need to
relate the codifferentials $\bar{\delta}$ and $\delta$. Using the expression
$(\delta T)_{i_{2}\cdots i_{p}}=-D^{i_{1}}T_{i_{1}\cdots i_{p}}$ on any
$p$-form, an easy calculation leads to

$\displaystyle\bar{\delta}T=u^{2}(\delta T)+(n-2p)i_{u\mathrm{grad}_{h}u}T.$

So, for a two-form $\Psi$ we have

$\displaystyle(d\circ\bar{\delta})\Psi$
$\displaystyle=d(u^{2}(\delta\Psi)+(n-4)i_{u\mathrm{grad}_{h}u}\Psi)=2udu\wedge(\delta\Psi)+u^{2}(d\circ\delta)\Psi+(n-4)d\circ
i_{u\mathrm{grad}_{h}u}\Psi,$ $\displaystyle(\bar{\delta}\circ d)\Psi$
$\displaystyle=u^{2}(\delta\circ d)\Psi+(n-6)i_{u\mathrm{grad}_{h}u}\circ
d\Psi\,,$

and hence

$\displaystyle\bar{\Delta}_{\mbox{\tiny dR}}\Psi-u^{2}\Delta_{\mbox{\tiny
dR}}\Psi=2du\wedge(\delta\Psi)+(n-4)\pounds_{u\mathrm{grad}_{h}u}\Psi-2i_{u\mathrm{grad}_{h}u}\circ
d\Psi\,,$ (A.7)

after using Cartan’s identity $\pounds_{V}=d\circ i_{V}+i_{V}\circ d$. With
these identities at hand we may proceed with the proof of Lemma A.1.

Proof: Insert (A.5) and (A.7) into (A.4) to get

$\displaystyle\Delta_{\bar{h}}\Psi_{ij}=$ $\displaystyle
u^{2}\Delta_{h}\Psi_{ij}+W^{k}{}_{i}\Psi_{kj}-W^{k}{}_{j}\Psi_{ki}-(n-4)\pounds_{u\mathrm{grad}_{h}u}\Psi_{ij}$
$\displaystyle-2u\left(D_{i}u(\delta\Psi)_{j}-D_{j}u(\delta\Psi)_{i}\right)+2uD^{k}u(d\Psi)_{kij}.$

Expanding the Lie derivative and using the definition (A.6) of $W$ yields
(A.1). $\Box$

To continue, we write down two simple identities. The first one is the well-
known conformal behaviour of the Laplacian on functions

$\displaystyle\Delta_{\bar{h}}f=u^{2}\Delta_{h}f+(2-n)uD^{i}uD_{i}f\quad\Longrightarrow\quad\Delta_{\bar{h}}u=u^{2}\Delta_{h}u+(2-n)u|du|^{2}_{h}.$
(A.8)

The second is an expression for $\bar{D}^{l}u\bar{D}_{l}\Psi_{ij}$. Using

$D^{l}u\,Q^{k}{}_{il}=-u^{-1}|du|^{2}_{h}\delta^{k}{}_{i}$

(which follows directly from (A.2)) we get

$\displaystyle\bar{D}^{l}u\bar{D}_{l}\Psi_{ij}$
$\displaystyle=u^{2}D^{l}u\left(D_{l}\Psi_{ij}-Q^{k}{}_{il}\Psi_{kj}-Q^{k}{}_{jl}\Psi_{ik}\right)$
$\displaystyle=u^{2}D^{l}uD_{l}\Psi_{ij}+2u|du|^{2}_{h}\Psi_{ij}.$ (A.9)

The next lemma analyses self-adjointness of certain second order operators and
their indicial maps. For this result we assume that $\Sigma$ is a manifold
with boundary, that $h$ is positive definite, and that $u$ is a defining
function for the boundary, i.e. $u=0$ and $|du|^{2}_{h}>0$ on
$\partial\Sigma$.

######  Lemma A.2.

With these assumptions, let
$L(\Psi)_{ij}:=\Delta_{\bar{h}}\Psi_{ij}+\bar{S}_{ij}{}^{kl}\Psi_{kl}$, with
$\bar{S}_{ij}{}^{kl}$ smooth up to and including the boundary. Suppose that
$\bar{S}_{ij}{}^{kl}=\bar{S}_{[ij]}{}^{kl}=\bar{S}_{ij}{}^{[kl]}$. Then

  * (i)

$L$ is formally self-adjoint (with respect to the $L^{2}$-scalar product with
volume form defined by $\bar{h}$) if and only if

$\displaystyle\bar{S}\,^{kl}{}_{ij}=\bar{S}_{ij}{}^{kl}.$ (A.10)

  * (ii)

For $\sigma\in\mathbb{C}$, the indicial map
$I_{\sigma}:\Lambda^{2}(\Sigma)|_{\partial\Sigma}\otimes\mathbb{C}\rightarrow\Lambda^{2}(\Sigma)|_{\partial\Sigma}\otimes\mathbb{C}$
is

$\displaystyle I_{\sigma}(\widehat{\Psi})_{ij}=$
$\displaystyle\left(2(2-n)+\sigma(5+\sigma-n)\right)|V|^{2}_{h}\widehat{\Psi}_{ij}$
$\displaystyle+(4-n)\left(V^{k}V_{i}\widehat{\Psi}_{kj}-V^{k}V_{j}\widehat{\Psi}_{ki}\right)+\bar{S}_{ij}{}^{kl}|_{\partial\Sigma}\widehat{\Psi}_{kl}$
(A.11)

where $V^{i}:=D^{i}u|_{\partial\Sigma}$.

Proof: Recall that the formal adjoint $L^{\dagger}$ is the (unique) operator
that for any pair of two-forms $\bar{\Psi}_{ij}$, $\bar{\Phi}_{ij}$ satisfies
(recall also that indices of objects with an overline are moved with
$\bar{h}$)

$\displaystyle\bar{\Psi}^{ij}L^{\dagger}(\bar{\Phi})_{ij}-\bar{\Phi}^{ij}L(\bar{\Psi})_{ij}=\bar{D}_{k}(\bar{F}^{k}(\bar{\Psi},\bar{\Phi},\bar{D}\bar{\Psi},\bar{D}\Phi))$

for some vector $\bar{F}^{l}$. It is immediate that

$\displaystyle
L^{\dagger}(\bar{\Phi})_{ij}=\Delta_{\bar{h}}\bar{\Phi}_{ij}+\bar{S}^{kl}{}_{ij}\bar{\Phi}_{kl}$

because

$\displaystyle\bar{\Psi}^{ij}L^{\dagger}(\bar{\Phi})_{ij}-\bar{\Phi}^{ij}L(\bar{\Psi})_{ij}=\bar{D}_{k}\left(\bar{\Psi}^{ij}\bar{D}^{k}\bar{\Phi}_{ij}-\bar{\Psi}^{ij}\bar{D}^{k}\bar{\Phi}_{ij}\right)\,.$

Consequently $L$ is formally self-adjoint ($L^{\dagger}=L$) if and only if
(A.10) holds. For item (ii) we compute
$u^{-\sigma}\Delta_{\bar{h}}(u^{\sigma}\Psi)$ and get

$\displaystyle\frac{1}{u^{\sigma}}\Delta_{\bar{h}}(u^{\sigma}\Psi)_{ij}$
$\displaystyle=\left(\frac{\sigma(\sigma-1)}{u^{2}}\bar{D}^{k}u\bar{D}_{k}u+\frac{\sigma}{u}\Delta_{\bar{h}}u\right)\Psi_{ij}+\frac{2\sigma}{u}\bar{D}^{k}u\,\bar{D}_{k}\Psi_{ij}+\Delta_{\bar{h}}\Psi_{ij}$
$\displaystyle=\left(\sigma
u\Delta_{h}u+\sigma(5+\sigma-n)|du|^{2}_{h}\right)\Psi_{ij}+2\sigma
uD^{k}u\,D_{k}\Psi_{ij}+\Delta_{\bar{h}}\Psi_{ij}\,,$

where in the second equality we inserted (A.8), (A.9) and
$\bar{D}^{l}u\bar{D}_{l}u=u^{2}|du|^{2}_{h}$. By Lemma A.1 we have

$\displaystyle\Delta_{\bar{h}}\Psi_{ij}|_{u=0}=\left.2(2-n)|du|^{2}_{h}\Psi_{ij}+(4-n)\left(D^{k}uD_{i}u\,\Psi_{kj}-D^{k}uD_{j}u\,\Psi_{ki}\right)\right|_{u=0}.$

The definition of indicial map gives (A.11).

$\hfill\Box$

The following corollary is used in the main text.

######  Corollary A.3.

Let $B_{ij}$ be a symmetric tensor and $c\in\mathbb{R}$. Then, the operator

$\displaystyle
L(\Psi)_{ij}=\Delta_{\bar{h}}\Psi_{ij}+c(R^{h})^{k}{}_{i}{}^{l}{}_{j}\Psi_{kl}+B^{k}{}_{i}\Psi_{kj}-B^{k}{}_{j}\Psi_{ki}$

is formally self-adjoint with respect to the $L^{2}$ norm of $\bar{h}$.

Proof: One easily checks that the tensor

$\displaystyle
S_{ij}{}^{kl}:=c(R^{h})^{[k}{}_{i}{}^{l]}{}_{j}-\delta^{[k}{}_{j}B^{l]}{}_{i}-B^{[k}{}_{j}\delta^{l]}{}_{i}$

is antisymmetric in $ij$ (the antisymmetry in $kl$ is obvious). In terms of
$S$ we can rewrite
$L(\Psi)_{ij}=\Delta_{\bar{h}}\Psi_{ij}+S_{ij}{}^{kl}\Psi_{kl}$ The symmetry
condition (A.10) can be checked with either $\bar{h}$ or with $h$. We use $h$
and find

$\displaystyle
S^{ij}{}_{kl}=c(R^{h})_{[k}{}^{i}{}_{l]}{}^{j}-\delta_{[k}{}^{j}B_{l]}{}^{i}-B_{[k}{}^{j}\delta_{l]}{}^{i}=c(R^{h})^{[i}{}_{k}{}^{j]}{}_{l}-\delta^{[i}{}_{l}B^{j]}{}_{k}-B^{[i}{}_{l}\delta^{j]}{}_{k}=S_{kl}{}^{ij}$

after using the symmetries of the Riemann tensor and of $B$. $\hfill\Box$

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