# The Initial boundary value problem and quasi-local Hamiltonians in general
relativity

Zhongshan An and Michael T. Anderson  Dept. of Mathematics, University of
Connecticut, Storrs, CT 06269<EMAIL_ADDRESS>Dept. of Mathematics,
Stony Brook University, Stony Brook, NY 11790<EMAIL_ADDRESS>
###### Abstract.

We discuss relations between the initial boundary value problem (IBVP) and
quasi-local Hamiltonians in GR. The latter have traditionally been based on
Dirichlet boundary conditions, which however are shown here to be ill-posed
for the IBVP. We present and analyse several other choices of boundary
conditions which are better behaved with respect to the IBVP and carry out a
corresponding Hamiltonian analysis, using the framework of the covariant phase
space method.

##  1\. Introduction

This article is concerned with the initial boundary value problem (IBVP) for
the vacuum Einstein equations and its relation to the existence of quasi-local
Hamiltonians in general relativity. We consider then space-times $M$
topologically of the form $M\cong I\times S$, where $I=(-1,1)$ and $S$ is a
compact 3-manifold with non-empty boundary $\partial S=\Sigma$. Let
${\mathcal{C}}=I\times\Sigma$ denote the boundary $\partial M$ of $M$ with
$\Sigma=\partial S$. The initial boundary value problem is the problem of
finding Lorentz metrics $g$ on $M$ satisfying the vacuum Einstein equations

(1.1) $\mathrm{Ric}_{g}=0,$

together with prescribed boundary conditions along ${\mathcal{C}}$ and initial
conditions along a Cauchy surface (e.g. $S=\\{0\\}\times S\subset M$). More
precisely, one would like to establish existence, uniqueness and stability of
solutions with prescribed initial and boundary data.

The analogous situation for the Cauchy or initial value problem has long been
well-understood and it is worthwhile to recall this briefly to gain the
general perspective. The initial data $(g_{S},K)$ on $S$ consist of a
Riemannian metric $g_{S}$ and symmetric bilinear form $K$ satisfying the
vacuum Einstein constraint equations; these give the induced metric and second
fundamental form of a solution $g$ on $S$.

Let $(V,g)$ denote a vacuum development of an initial data set $(S,g_{S},K)$,
i.e. a globally hyperbolic vacuum spacetime $V$ containing $(S,g_{S},K)$. It
was proved by Choquet-Bruhat [12] that vacuum developments always exist. Two
vacuum developments $(V_{1},g_{2})$, $(V_{2},g_{2})$ are called equivalent if
they contain a common sub-development. Choquet-Bruhat and Geroch [13] further
proved that there is a maximal development $V_{\max}$, unique up to isometry,
i.e. unique up to the action of the group $\mathrm{Diff}_{0}(V_{\max})$ of
diffeomorphisms of $V_{\max}$ fixing $S$ pointwise. Let ${\mathcal{V}}$ be the
space of such isometry classes of maximal solutions.

The main results on the solution of the Cauchy problem can then be summarized
in the following statement: for each Cauchy surface $S$, there is a bijective
correspondence

(1.2) $E_{S}:{\mathcal{V}}\rightarrow\mathcal{I},\ \ g\rightarrow(g_{S},K),$

where $\mathcal{I}$ is the space of all initial data $(g_{S},K)$ satisfying
the vacuum constraint equations. The bijection (1.2) gives an effective
parametrization of the space of maximal solutions of (1.1) by their initial
data. It would appear to be likely that $\mathcal{I}$ and hence
${\mathcal{V}}$ may be given the structure of a smooth infinite dimensional
manifold, possibly away from a small singular set of special data. Further by
work of Fischer, Marsden and Moncrief, cf. [15], ${\mathcal{V}}$ or
$\mathcal{I}$ should carry a naturally defined (non-degenerate) symplectic
form $\Omega$.111Such manifold and symplectic structure results hold when $S$
is a closed or asymptotically flat 3-manifold, cf. [15], but this has not yet
been extended in detail to the situation of manifolds with boundary. Also, for
simplicity, we forgo here any detailed discussion of the specific function
spaces, e.g. Sobolev spaces, and related issues of regularity. The topologies
and all geometric data are assumed to be $C^{\infty}$ in the following.

One would like to obtain similar results and a similar understanding for the
IBVP, and so in particular obtain a bijective correspondence

(1.3) $D_{S}:{\mathcal{V}}\rightarrow\mathcal{I}\times_{c}{\mathcal{B}},\ \
g\rightarrow((g_{S},K),b)$

where ${\mathcal{V}}$ now denotes the space of maximal globally hyperbolic
vacuum spacetimes $(M,g)$ with time-like boundary ${\mathcal{C}}$ and
${\mathcal{B}}$ is a space of boundary data on ${\mathcal{C}}$. The subscript
$c$ denotes compatibility or corner conditions arising in solutions of the
IBVP. Again, a maximal solution should be unique up to isometry, i.e. up to
the action of $\mathrm{Diff}_{0}(M)$ of diffeomorphisms of $M$ fixing $S$ and
${\mathcal{C}}$ pointwise. Of course to even start, this requires finding a
suitable set of boundary data ${\mathcal{B}}$ on ${\mathcal{C}}$.

The issue of the solvability of an IBVP (with certain boundary conditions) was
first addressed in the foundational paper of Friedrich-Nagy [18]. This was
followed by work of Kreiss-Reula-Sarbach-Winicour [25], [26] and more recently
by works of Fournadavlos-Smulevici [16], [17]; we refer to [34] for a general
survey of this topic. Unfortunately, none of these works establishes a well-
posedness result as expressed in (1.3).

As with the Cauchy problem, the most natural boundary conditions are geometric
boundary conditions, i.e. those determined by the
$\mathrm{Diff}_{0}(M)$-invariant Cauchy data at ${\mathcal{C}}$, i.e. the
induced metric $\gamma$ and second fundamental form $A$ of ${\mathcal{C}}$ in
$(M,g)$.

Based on the familiar situation with simpler field theories, e.g. scalar or
Yang-Mills gauge-type fields, it is usually assumed in the physics literature
that the boundary conditions for vacuum gravity are of Dirichlet or Neumann
type. For theories which have a fixed space-time structure (so the metric
field is a background field, not subject to variation), Dirichlet or Neumann
data are well known to be well-behaved and are well-posed for the IBVP, cf.
[29], [36], [41] for instance. In particular one generally has a
correspondence as in (1.3) for such boundary data.

On the other hand, in general relativity there are no fixed background fields
and we will show below that the straightforward analogy does not hold:
Dirichlet boundary data (fixing the induced metric on the boundary
${\mathcal{C}}$) or Neumann boundary data (fixing the second fundamental form
$A$ of ${\mathcal{C}}$ in $M$) are ill-behaved or ill-posed boundary data for
vacuum gravity.222We note that the recent result of [17] for totally geodesic
boundary data $A=0$ does not prove well-posedness for general Neumann boundary
data, which is the situation considered here. This is proved in Proposition
2.1, cf. also Proposition 2.2, below. We conjecture that the conformal-mean
curvature (CH) boundary data

(1.4) $([\gamma],H),$

consisting of the conformal class of the boundary metric
$({\mathcal{C}},\gamma)$ and its mean curvature $H=\mathrm{tr}A$, do lead to a
well-posed IBVP for which (1.3) holds, cf. Conjecture 2.3. Some evidence for
this conjecture is the result, given in Proposition 2.4, that the vacuum
constraint equations are naturally solvable under the CH boundary conditions
(1.4).

However, currently, the only known boundary data for which the IBVP is known
to be well-posed and for which (1.3) holds are the diffeomorphism-invariant
boundary data (called AA data here) recently developed in [2]; these are also
discussed in more detail in §2.

Next, we relate the issues above with a Hamiltonian analysis of GR on finite
manifolds-with-boundary. Hamiltonian analyses of GR on manifolds with boundary
at infinity are well-known and well-developed. The ADM and Regge-Teitelboim
Hamiltonian analysis for asymptotically flat space-times with boundary at
spatial or null infinity are foundational works for the notions of global
energy-momentum and angular momentum (the “conserved charges”) of such space-
times. The case of time-like anti-de-Sitter boundaries at conformal infinity
has also been extensively studied in connection with the AdS-CFT
correspondence. In contrast, there has been much less detailed analysis in the
case of finite, time-like boundaries, although see [6], [14] for instance.

The main concern of this paper is then a more rigorous understanding of the
definition of quasi-local energies (Hamiltonians) through an analysis of the
behavior of the space of solutions of a well-posed IBVP. There is a large
literature on Hamiltonian approaches to a suitable definition of quasi-local
energy, starting with the foundational work of Brown-York [10]; we refer to
[37] for a detailed survey. These are all based on Hamiltonians with Dirichlet
boundary conditions. The focus here is instead on both general boundary data,
and specific boundary data, namely the CH data (1.4) and the AA boundary data
developed in [2].

The traditional canonical phase space approach toward a Hamiltonian analysis
breaks the general covariance of the theory by a choice time evolution and
corresponding time-like vector field $\partial_{t}$. This gives a space-like
foliation or 3+1 decomposition of the space-time, and an associated lapse-
shift pair $(N,X)$, $\partial_{t}=NT+X$. Hamiltonians ${\mathcal{H}}$ thus
depend on this data, ${\mathcal{H}}={\mathcal{H}}_{(N,X)}$, cf. [37], [38].

A very useful alternative to this approach which does not break the full
covariance of the theory is the covariant phase space method, which directly
gives a (pre)-symplectic structure to the space of all vacuum solutions (with
given boundary conditions). We refer to the survey article [24] for a detailed
description of the background and history of this method; important
contributions to this topic are those of Crnkovic-Witten [11] and in
particular Lee-Wald [28], Iyer-Wald [22], [23] and Wald-Zoupas [39].

Here we are interested in the context of space-times $M$ with time-like
boundary ${\mathcal{C}}$ which has been discussed much less in the literature.
Fortunately, an excellent description of the covariant phase space method with
boundary has recently been given by Harlow-Wu [21] and we will generally
follow this description (pointing out however a number of subtleties). In §3,
we provide an overview of this method, as applied to general (not only
Dirichlet) boundary data. We also develop a modification of the method needed
to deal with the diffeomorphism-invariant AA boundary data from [2].

In Propositions 4.1 and 4.3 of §4, we show that the CH boundary data and a
slight modification of the AA boundary data admit Lagrangian descriptions and
have a well-defined variational formulation. In Propositions 4.4 and 4.5, we
provide a detailed description of the corresponding phase space and derive the
associated Hamiltonians.

In §5, we then discuss the basic issue of the choice of zero-point energies
for Hamiltonians. For Dirichlet boundary data, these have been based on
Euclidean (in the case of Brown-York [10]) or Minkowski (in the case of Wang-
Yau [40]) subtraction terms. We consider the situation of general boundary
conditions and a general definition of subtraction term in Definition 5.2,
reminiscent of the Bartnik definition [7] of quasl-local energy.

##  2\. Well-posed boundary data in GR

The primary method to solve an IBVP for the Einstein equations is to find a
gauge in which the reduced Einstein equations form a quasilinear hyperbolic
system and then determine boundary conditions in which reduced system is a
well-posed IBVP. Of course one also needs to ensure that all solutions of the
reduced Einstein equations are true vacuum solutions; this is the issue of
constraint preservation. For such a method of solution, a necessary condition
for well-posedness is that the IBVP localized at a standard flat Minkowski
corner region ${\bf R}=\\{t\geq 0,x^{1}\leq 0\\}\subset\mathbb{R}^{1,3}$ is
well-posed, i.e. the frozen coefficient IBVP is well-posed. All known
existence and uniqueness results for the vacuum IBVP are based on this method.

In this respect, we note the following:

######  Proposition 2.1.

With respect to either Dirichlet or Neumann boundary conditions, the
linearization of the vacuum equations (1.1) at a standard Minkowski background
is not a well-posed IBVP, for any choice of gauge reduction.

###### Proof.

The proof uses only the structure of the vacuum Einstein constraint equations
(not the full Einstein equations) on the boundary ${\mathcal{C}}$;

(2.1) $|A|^{2}-H^{2}-R_{\gamma}=E(\nu,\nu)=0,$ (2.2)
$\mathrm{div}(A-H\gamma)=2E(\nu,\cdot)=0.$

Here $E$ is the Einstein tensor of $g$, $R_{\gamma}$ is the scalar curvature
of the boundary metric $({\mathcal{C}},\gamma)$ and $\mathrm{div}$ is the
divergence operator on symmetric $(2,0)$ tensors on $({\mathcal{C}},\gamma)$;
$\mathrm{div}h=\mathrm{tr}\nabla h$. As above, $A$ is the second fundamental
form of $({\mathcal{C}},\gamma)\subset(M,g)$ and the mean curvature $H$ is its
trace.

Consider first Dirichlet boundary data, where the boundary metric $\gamma$ is
prescribed. The linearization of the Hamiltonian or scalar constraint (2.1) in
the direction of the variation $h$ of $g$ is given by

$2\langle A^{\prime}_{h},A\rangle-2\langle
A^{2},h^{T}\rangle-2HH^{\prime}_{h}-R^{\prime}_{h^{T}}=0,$

where $h^{T}$ is the variation of the induced metric on the boundary. The
linearization of these equations at the standard solid corner region $({\bf
R},g_{0})$ of Minkowski space gives $A=H=0$, so that

$R^{\prime}_{h^{T}}=0.$

Thus, at the linearized level, only scalar flat deformations $h^{T}$ of the
boundary metric $\gamma_{0}=g_{0}|_{{\mathcal{C}}}$,
${\mathcal{C}}=\\{x^{1}=0\\}$, are possible. This is an infinite dimensional
restriction; for example, most conformal deformations are excluded. It follows
that the linearized constraint (2.1) is not solvable for generic variations of
boundary data $h^{T}$; the same applies then to the (more restrictive) full
Einstein equations.

Next consider Neumann boundary data, where the second fundamental form $A$ is
prescribed. The linearization of the momentum constraint (2.2) is given by

(2.3) $\mathrm{div}_{\gamma}(A_{h}^{\prime}-H_{h}^{\prime}\gamma-
Hh^{T})=-(\mathrm{div})_{h}^{\prime}(A-H\gamma).$

At the standard solid region of Minkowski space, this gives

(2.4) $\mathrm{div}_{\gamma_{0}}(A^{\prime}_{h}-H^{\prime}_{h}\gamma_{0})=0.$

where $\mathrm{div}_{\gamma_{0}}$ is the divergence operator with respect to
the standard background Lorentz metric $\gamma_{0}$ on $\mathbb{R}^{1,2}$.
Thus the linearization of (2.2) is not solvable for generic $A^{\prime}_{h}$;
the form $A^{\prime}_{h}$ has 6 degrees of freedom while (2.4) gives 3 scalar
conditions on $A^{\prime}_{h}$. This shows again that the linearized Einstein
equations are not generically solvable at $g_{0}$ for prescribed variations
$A^{\prime}_{h}$.

∎

From this, one expects that the space of vacuum solutions having given
boundary metric (Dirichlet boundary data) or given second fundamental form
(Neumann boundary data) is generically empty; the space of boundary data
admitting solutions $g$ is of infinite codimension in the space
${\mathcal{B}}$ of all boundary data. While Proposition 2.1 does not quite
prove this, it does show that any proof of well-posedness cannot be based on
solving a gauged IBVP for a quasi-linear hyperbolic system which localizes.
More generally, one expects the behavior of the space of solutions with
respect to either Dirichlet or Neumann boundary data to be highly unstable;
given one solution with (say) Dirichlet boundary data $\gamma$ on
${\mathcal{C}}$, an open set of nearby solutions have boundary data only in a
subset of infinite codimension in the space
${\mathcal{B}}=\operatorname{Met}({\mathcal{C}})$ of boundary data. In
particular, there is no correspondence as in (1.3).

We also point out that there is a strong failure of uniqueness naturally
associated with the failure of existence above.

######  Proposition 2.2.

In harmonic gauge, solutions to the linearized IBVP for the Einstein equations
with Dirichlet boundary data have an infinite-dimensional kernel.

###### Proof.

Consider the initial boundary value problem for vacuum Einstein equations with
Dirichlet boundary data:

$\begin{split}&\mathrm{Ric}_{g}=0\ \ {\rm in}~{}M,\ \
g|_{S}=g_{S},~{}\tfrac{1}{2}L_{T}g|_{S}=K\ \ {\rm on}~{}S,\ \
g_{{\mathcal{C}}}=\gamma\ \ {\rm on}~{}{\mathcal{C}}.\end{split}$

Let $x^{\alpha}=(t,x^{1},x^{2},x^{3})$ be a local chart near the corner of
$M$, mapping to the corner region ${\bf R}\subset\mathbb{R}^{1,3}$. Assume
$x^{\alpha}$ are harmonic, so that $V^{\alpha}=\Box_{g}x^{\alpha}=0$. The
kernel of the linearization of the system above at the Minkowski corner ${\bf
R}$ is then given by:

(2.5) $\begin{split}&\mathrm{Ric}^{\prime}_{h}+\delta^{*}(V^{\prime}_{h})=0\ \
{\rm in}~{}\mathbf{R}\\\ &h_{\alpha\beta}=0,~{}\partial_{t}h_{\alpha\beta}=0\
\ {\rm on}~{}\\{t=0,x^{1}\leq 0\\}\\\ &V^{\prime}_{h}=0,~{}h^{T}=0\ \ {\rm
on}~{}\\{x^{1}=0\\},\end{split}$

where $V^{\prime}_{h}=-\mathrm{div}h+\frac{1}{2}d\mathrm{tr}h$. We show that
the system (2.5) has an infinite-dimensional space of solutions.

Set $h_{\alpha\beta}=0$ in $\mathbf{R}$ for all $\alpha,\beta$ except
$(\alpha,\beta)=(1,1),(0,1)$ and let

(2.6) $h_{01}=\tfrac{1}{2}h_{11}=f(x^{1}+t)$

for some $f$ determined below. Then clearly
$(\partial_{t}-\partial_{x^{1}})h_{11}=0,\partial_{x^{A}}h_{11}=0$ for
$A=2,3$. It follows that the bulk equation in (2.5) holds:
$(\mathrm{Ric}^{\prime}_{h}+\delta^{*}(V^{\prime}_{h}))_{\alpha\beta}=\Box
h_{\alpha\beta}=0$ in ${\bf R}$, for all $\alpha,\beta$.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be any smooth function such that
$f(x)=0,f^{\prime}(x)=0$ for all $x\leq 0$ but $f(x)\neq 0$ for $x>0$. Then on
the initial surface $\\{t=0,x^{1}\leq 0\\}$,
$h_{01}=h_{11}=\partial_{t}h_{01}=\partial_{t}h_{11}=0$, so that
$h_{\alpha\beta}$ satisfies the initial conditions. It remains to verify the
boundary conditions. The Dirichlet boundary condition $h^{T}=0$, which reads
$h_{00}=h_{0A}=h_{AB}=0$ for $A,B=2,3$, is obviously true. Further simple
calculation also gives $V^{\prime}_{h}=0$, cf. [2, eqn. (3.6)] for example.
(This is the reason for the particular relation between $h_{01}$ and $h_{11}$
in (2.6)).

Thus the deformation $h$ constructed above solves the homogeneous IBVP;
however, $h_{11}\neq 0$ in the region $\\{t+x^{1}>0\\}\subset{\bf R}$. In
other words, incoming wave solutions generated by such $f$ show that solutions
to the linearized IBVP (2.5) are not unique outside the domain of dependence
of the initial surface. ∎

Note that the same result and proof also work well for Neumann boundary data.

This raises the question of whether there are any geometric boundary data on
${\mathcal{C}}$ for which the IBVP could be well-posed. We note that in the
simpler Riemannian or Euclidean setting, it was proved in [3] that neither
Dirichlet nor Neumann boundary data give a well-posed elliptic boundary value
problem for the vacuum Einstein equations, in any gauge. On the other hand, it
was shown there that the data

(2.7) $([\gamma],H),$

consisting of the conformal class $[\gamma]$ of the boundary metric $\gamma$
and the mean curvature $H=trA$, are well-posed elliptic boundary data, in a
suitable gauge. (There exist other choices of well-posed elliptic boundary
data, but the choice (2.7) is perhaps the most natural). This was extended to
the parabolic Ricci-flow setting in [19].

By analogy, it is natural then to consider the same data for the IBVP in the
Lorentzian setting (1.1). We make the following

######  Conjecture 2.3.

The IBVP for the vacuum Einstein equations (1.1) is well-posed with respect to
the mixed boundary data $([\gamma],H)$ on ${\mathcal{C}}$.

There is strong evidence in favor of this conjecture, but a detailed
discussion would be lengthy and is out of place here; we hope to address
Conjecture 2.3 in detail elsewhere.

As a first justification for the conjecture, we note the following result,
which contrasts strongly with Proposition 2.1.

######  Proposition 2.4.

The boundary data $([\gamma],H)$ are well-posed in terms of the constraint
equations (2.1)-(2.2). In particular, the constraint equations on the boundary
${\mathcal{C}}$ can be solved with any prescribed $([\gamma],H)$.

###### Proof.

This is proved by transferring the usual conformal method of Lichnerowicz-
Choquet-Bruhat-York from the space-like elliptic setting to the time-like
hyperbolic setting.

Let $\gamma_{0}$ be a representative metric in $[\gamma]$ (not to be confused
with the Minkowski metric on $\mathbb{R}^{1,2}$ used above), with volume form
$dv_{\gamma_{0}}$ and let $\alpha$ be a background volume form on
${\mathcal{C}}$. Form then the so-called densitized lapse (cf. [31]),

$\mu={\tfrac{1}{2}}\frac{dv_{\gamma_{0}}}{\alpha}.$

Let $\sigma$ be a transverse-traceless (2,0) tensor with respect to
$\gamma_{0}$; $\mathrm{div}\sigma=\mathrm{tr}\sigma=0$. In the conformal
method, cf. [31] again, one then constructs the Cauchy data $(\gamma,A)$ at
${\mathcal{C}}$ by setting

$\gamma=\lambda^{4}\gamma_{0},$
$A=\lambda^{-2}(\sigma+{\tfrac{1}{2\mu}}\widehat{L}_{X}\gamma_{0})+{\tfrac{1}{3}}H\lambda^{4}\gamma_{0},$

where $\widehat{L}_{X}\gamma_{0}$ is the conformal Killing operator with
respect to $\gamma_{0}$:
$\widehat{L}_{X}\gamma_{0}=L_{X}\gamma_{0}-(\frac{2}{3}\mathrm{div}_{\gamma_{0}}X)\gamma_{0}$.
The constraint equations (2.1)-(2.2) in this time-like setting then become a
coupled system of equations for $(\lambda,X)$ which take the form

(2.8)
$\begin{array}[]{c}\mathrm{div}_{\gamma_{0}}({\tfrac{1}{2\mu}}\widehat{L}_{X}\gamma_{0})={\tfrac{2}{3}}\lambda^{6}dH,\\\
8\Box_{\gamma_{0}}\lambda=R_{\gamma_{0}}\lambda-|\sigma+{\tfrac{1}{2\mu}}\widehat{L}_{X}\gamma_{0}|^{2}\lambda^{-7}+{\tfrac{2}{3}}H^{2}\lambda^{5}.\end{array}$

Here $\mathrm{div}_{\gamma_{0}}$ and $\Box_{\gamma_{0}}$ are the divergence
and wave operator with respect to $\gamma_{0}\in[\gamma]$ and $R_{0}$ is the
scalar curvature of $\gamma_{0}$. One has
$\mathrm{div}\widehat{L}_{X}(\gamma_{0})=\mathrm{div}\delta^{*}X-\frac{2}{3}d\mathrm{div}X$
and standard Weitzenbock formulas show that
$\mathrm{div}\widehat{L}_{X}(\gamma_{0})=-D^{*}DX+\frac{1}{2}d\mathrm{div}X+E(X)$,
where $E(X)$ involves only the curvature of $\gamma_{0}$ and no derivatives of
$X$.

The system (2.8) is thus a coupled system of semi-linear wave equations for
$(\lambda,X)$ and so by standard theory is well-posed; one has existence and
uniqueness of solutions for at least some positive time $t\in[0,T)$, given
suitable initial conditions.

It is also easy to see that the York decomposition [42] holds on
$({\mathcal{C}},[\gamma])$; any symmetric bilinear form $h$, (e.g. $h=A$
above) admits a decomposition
$h=\sigma+\frac{\mathrm{tr}h}{3}\gamma_{0}+\widehat{L}_{X}\gamma_{0}$, where
$\sigma$ is transverse-traceless with respect to $\gamma_{0}$. The vector
field $X$ is unique up to choice of initial conditions on a slice $\Sigma$ to
${\mathcal{C}}$. The proof of this is essentially the same as in the
Riemannian or space-like case, using hyperbolic equations in place of elliptic
equations.

∎

Thus the space of solutions of the Einstein constraint equations on a time-
like boundary ${\mathcal{C}}$ is parametrized by the free data
$([\gamma],H,\sigma)$ and initial conditions at $\Sigma$. This time-like
boundary setting is actually significantly better behaved than the analogous
situation for initial data, where the corresponding result is false. In
general, given $([g_{S}],k,\sigma)$, $k=\mathrm{tr}_{S}K$, on an initial slice
$S$, it is not always possible obtain existence of (unique) solutions to the
constraint equations. The conformal method does not always work well for
example when $H$ is far away from the set of constant functions (the far from
CMC regime).

######  Remark 2.5.

We make some further remarks comparing Propositions 2.1 and 2.4, focusing on
the constraint equations with Dirichlet data.

Consider first initial data, i.e. the issue of solving the constraint
equations on a Cauchy surface with given Dirichlet data, i.e. initial metric
$\gamma$. In bulk dimension 3, this is the problem of isometric immersion of a
given Riemannian surface $(S,g_{S})$ into $\mathbb{R}^{1,2}$, by the
fundamental theorem of hypersurfaces in $\mathbb{R}^{1,2}$, (at least when $S$
is simply connected). We are not aware of significant results on this topic.
The analogous isometric immersion problem of a Riemannian surface in Euclidean
space $\mathbb{R}^{3}$ is notoriously hard and largely unresolved except in
the case of positive Gauss curvature $K_{\gamma}>0$; this is the solution of
the Weyl embedding problem by Nirenberg and Pogorelov.333Note that in this
Riemannian situation, the condition $K_{\gamma}>0$ is an open condition, so
that there are open sets of $C^{\infty}$ solutions of the constraint equations
with given Dirichlet data, in strong contrast with Proposition 2.1.
Nevertheless, the constraint equations (2.1)-(2.2) are not a well-posed or
elliptic system, even when $K_{\gamma}>0$. Instead, the main idea is to
quotient the space $\operatorname{Met}(S)$ by the positive definite second
fundamental form $K$, where the problem does become (underdetermined)
elliptic, cf. [20], [3], [41]. This method is thus restricted to the convex
case.

Passing to Dirichlet data on a Cauchy surface in 4 dimensions is the problem
of isometrically immersing a given Riemannian 3-manifold into some vacuum
space-time. Again, we are not aware of significant results on this problem.

It seems even less is known in the setting of time-like boundaries, i.e.
results or a theory of isometrically immersing Lorentzian surfaces into flat
$\mathbb{R}^{1,2}$ or similarly, a theory of isometrically immersing
Lorentzian 3-manifolds into a vacuum spacetime. On the other hand, Proposition
2.4 shows the situation is very well understood for $([\gamma],H)$ data (for
all dimensions).

######  Remark 2.6.

Once one solves the time-like constraint equations, if the data are analytic,
then the Cauchy-Kovalevsky theorem can be used to obtain actual vacuum
solutions defined in a neighbhorhood of ${\mathcal{C}}$; this gives local two-
sided solutions, i.e. solutions extending to each side of the boundary
${\mathcal{C}}$. In more detail, given a pair $(\gamma,A)$ of analytic data
satisfying the constraint equations (2.1)-(2.2), one may work in the gauge of
geodesic normal coordinates normal to ${\mathcal{C}}$ where
$g(\nu,\cdot)=\delta_{\nu\,\cdot}$. In this gauge, the Einstein equations are
“evolution equations” off the boundary in the normal $\nu$-direction, and for
analytic data the Cauchy-Kovalevsky theorem gives the existence of analytic
solutions in a neighborhood of ${\mathcal{C}}$. Of course the simple scalar
model here is the Laplace equation $\Delta u=0$ or wave equation $\Box u=0$
with prescribed Cauchy data on ${\mathcal{C}}$.

While this is not a well-posed boundary value problem, it is a delicate issue
to understand whether a solution $g$ defined in a neighborhood of
${\mathcal{C}}$ extends to a smooth solution on $M$.

In the discussion of geometric boundary conditions above, note that the number
of degrees of freedom for the boundary data ($\gamma$, or $A$, or
$([\gamma],H)$) is $6$. This is the correct and expected number for the gauge
group $\mathrm{Diff}_{0}(M)$ of diffeomorphisms fixing ${\mathcal{C}}$
pointwise, i.e. (maximal) solutions with the same initial and boundary data
are expected to be unique up to diffeomorphisms fixing an initial slice $S$
and the boundary ${\mathcal{C}}$. However, the space of all solutions is
invariant under the larger gauge group $\mathrm{Diff}(M)$, which essentially
differs from $\mathrm{Diff}_{0}(M)$ by the group
$\mathrm{Diff}({\mathcal{C}})$ of boundary diffeomorphisms. The group
$\mathrm{Diff}({\mathcal{C}})$ has 3 degrees of freedom and so the number of
true boundary conditions describing non-isometric solutions should be $6-3=3$.
Of these $3$ degrees of freedom, $2$ account for the degrees of freedom of the
gravitational field, and $1$ for the location or evolution of the boundary
${\mathcal{C}}$ off the initial corner surface $\Sigma$. Note that it may not
be easy to find a global slice to (for instance) the space of Dirichlet data
$\operatorname{Met}({\mathcal{C}})/\mathrm{Diff}({\mathcal{C}})$.

Next we discuss the boundary data introduced in [2]. This is based on the
construction of a global preferred harmonic-type gauge for a metric $(M,g)$,
at least near the boundary ${\mathcal{C}}$. The preferred gauge depends on the
space-time $(M,g)$ as well as a vector field $\Theta_{{\mathcal{C}}}$ defined
on ${\mathcal{C}}$; different choices of $\Theta_{{\mathcal{C}}}$ give
different choices of time evolution or space-like foliation of the space-time
$(M,g)$. The gauge is given by a wave map

(2.9) $\begin{array}[]{c}\varphi_{g}:(M,g)\rightarrow(M_{0},g_{R}),\\\
\Box_{g}\varphi_{g}=0,\end{array}$

where for simplicity the target space $M_{0}$ is taken to be a standard solid
cylinder $I\times B^{3}$ in Minkowski space $\mathbb{R}^{1,3}$ with standard
time function $t$. The Riemannian metric $g_{R}$ is the associated flat
Euclidean metric. Initial data for $\varphi_{g}$ on a Cauchy slice $S$ are
given by

(2.10) $\varphi_{g}=E_{g_{S}},\ \ (\varphi_{g})_{*}(T_{g})=T_{g_{R}}\ \ {\rm
on}\ \ S,$

where $T_{g}$ is the $g$-future unit normal to $S$ in $(M,g)$, $T_{g_{R}}$ is
the future unit normal to $\\{t=0\\}$ in $M_{0}$ and $E_{g_{S}}$ is determined
by $g_{S}=g|_{S}$ equivariantly with respect to $\mathrm{Diff}(S)$ near the
boundary $\Sigma$, i.e.

$E_{\psi^{*}(g_{S})}=\psi^{*}E_{g_{S}}=E_{g_{S}}\circ\psi.$

We refer to [2] for details regarding the assignment $g_{S}\rightarrow
E_{g_{S}}$ and for concrete examples of such.

The boundary data for $\varphi_{g}$ are of Sommerfeld type given by

(2.11) $[(\varphi_{g})_{*}(T_{g}+\nu_{g})]^{T}=\Theta_{{\mathcal{C}}},$

together with the (scalar) Dirichlet condition that
$\varphi_{g}:{\mathcal{C}}\rightarrow\partial M_{0}=I\times S^{2}$. Here $\nu$
is the $g$-unit normal of ${\mathcal{C}}\subset M$ and the superscript $T$
denotes orthogonal projection with respect to $g_{R}$ onto $\partial M_{0}$.
Thus, we view $\Theta_{{\mathcal{C}}}$ as a vector field tangent to
${\mathcal{C}}_{0}=\partial M_{0}$.

The equations (2.9) are hyperbolic and initial boundary value problems of this
type are well-understood; it follows from such known results that the IBVP
(2.9)-(2.11) is well-posed. In particular, given a background metric $g$,
there exists a unique solution $\varphi_{g}$ which is a diffeomorphism in a
neighborhood of the initial and time-like boundary $T=S\cup{\mathcal{C}}$.
Moreover (at least near $T$), the solution is equivariant with respect to
diffeomorphisms, so that

(2.12) $\varphi_{\psi^{*}g}=\psi^{*}\varphi_{g},$

for any $\psi\in\mathrm{Diff}(M)$.

Next consider the coupled system $(g,\varphi_{g})$

(2.13) $\begin{array}[]{c}\mathrm{Ric}_{g}=0,\\\
\Box_{g}\varphi_{g}=0,\end{array}$

with initial conditions for $g$ given by

(2.14) $g|_{S}=g_{S},\ \ K_{g}|_{S}=K,$

where $(g_{S},K)$ satisfy the constraint equations (2.1)-(2.2). The boundary
conditions for $g$ are expressed on the target ${\mathcal{C}}_{0}=\partial
M_{0}$ as:

(2.15) $([\gamma^{t}],H^{*},\Theta_{{\mathcal{C}}}).$

Here $\gamma^{t}=(\varphi_{g}^{-1})^{*}g|_{\Sigma_{t}}$, is the metric induced
by the pullback $(\varphi_{g}^{-1})^{*}$ of $g$ to $M_{0}$ restricted to the
slices $\Sigma_{t}=\\{t=constant\\}$ in ${\mathcal{C}}_{0}$ and $[\cdot]$
denotes the conformal class of the 2-metric. The term $H^{*}$ is a combination
of mean curvatures of the form

(2.16)
$H^{*}=\alpha\mathrm{tr}_{{\mathcal{C}}}A|_{\Sigma_{t}}+\beta\mathrm{tr}_{\Sigma_{t}}A+\gamma\mathrm{tr}_{\Sigma_{t}}K,$

for certain values of $\alpha,\beta,\gamma$, (cf. [2] for the exact allowable
coefficients). One admissible choice of $H^{*}$, which also appears in [27],
is $H^{*}=2\mathrm{tr}_{{\mathcal{C}}}A-\mathrm{tr}_{\Sigma_{t}}A$; this
choice will be discussed further in §4. The initial conditions for
$\varphi_{g}$ are as in (2.10).

It is proved in [2] that this is a well-posed IBVP. For any choice of
compatible initial data (2.14) and boundary data (2.15), there is a vacuum
solution $g$ realizing the initial and boundary conditions. Such solutions
have the covariance property

(2.17) $\bar{g}=\psi_{g}^{*}(g)=\psi_{\chi^{*}g}^{*}(\chi^{*}g),$

for any $\chi\in\mathrm{Diff}(M)$, where $\psi_{g}=\varphi_{g}^{-1}$. This
gives a slice (at least near the boundary ${\mathcal{C}}$) to the action of
$\mathrm{Diff}(M)$ on the space of vacuum solutions. Thus, diffeomorphisms no
longer act on the metrics $\bar{g}$; these metrics are evaluated in the fixed
standard coordinates $(t,x^{i})$ on $\mathbb{R}^{1,3}$. The solutions
$\bar{g}$ are unique with given initial and boundary data and one has
stability of solutions and the existence of maximal solutions. Moreover, any
vacuum solution is isometric to such a solution in a preferred gauge. Note
that since the boundary data $([\gamma^{t}],H^{*},\Theta_{{\mathcal{C}}})$
always admit solutions, it follows that the constraint equations (2.1)-(2.2)
on ${\mathcal{C}}$ are automatically satisfied; in fact the form of the
$H^{*}$ term in (2.16) derives partly from the form of the constraint
equations.

An important point is that, by construction, the boundary conditions (2.15)
are invariant under the action of
$\mathrm{Diff}(M)|_{{\mathcal{C}}}=\mathrm{Diff}({\mathcal{C}})$ on the space
of vacuum solutions. However, the choice of $\Theta_{{\mathcal{C}}}$
determines a particular or preferred gauge $\varphi_{g}$, giving a particular
foliation or 3+1 decomposition of the space-time $(M,g)$ in which the boundary
data are measured. The relevant space of (geometric) boundary data is then

(2.18) ${\mathcal{B}}_{geom}=I\times\mathrm{Conf}(\Sigma)\times
C^{\infty}(\Sigma)=\\{(t,[\gamma^{t}]_{\Sigma_{t}},H^{*}_{\Sigma_{t}})\\},$

where $\mathrm{Conf}(\Sigma)$ is the space of pointwise conformal classes
$[\gamma^{t}]$ of smooth metrics on $\Sigma$ and $H^{*}\in
C^{\infty}(\Sigma)$. The space ${\mathcal{B}}_{geom}$ has $3$ degrees of
freedom, corresponding to $\mathrm{Diff}({\mathcal{C}})$ invariant boundary
data. This is in marked contrast with the discussion of geometric boundary
data (Dirichlet, Neumann or $([\gamma],H)$) above, which are not
$\mathrm{Diff}({\mathcal{C}})$ invariant.

The well-posedness of this IBVP then gives a family of effective
parametrizations as in (1.3), i.e.

(2.19)
$D_{S,\Theta_{{\mathcal{C}}}}:{\mathcal{V}}_{geom}\rightarrow\mathcal{I}_{geom}\times_{c}{\mathcal{B}}_{geom},$

depending on the choice of Cauchy surface and boundary gauge
$\Theta_{{\mathcal{C}}}$. The space ${\mathcal{V}}_{geom}$ is now the space of
full isometry classes of solutions (vacuum Einstein metrics), i.e. the space
of all solutions modulo the action of the full diffeomorphism group
$\mathrm{Diff}(M)$. Accordingly $\mathcal{I}_{geom}$ is the space of
equivalence classes of Cauchy data $(g_{S},K)$ on a Cauchy slice, where
$(g_{S},K)\sim(g_{S}^{\prime},K^{\prime})$ if there is a space-time
diffeomorphism $\chi\in\mathrm{Diff}(M)$, $\chi(S)=S$, such that
$(\chi^{*}g_{S}^{\prime},\chi^{*}K^{\prime})=(g_{S},K)$.

A variational description and Hamiltonian analysis of this boundary data will
be discussed in the following sections.

##  3\. The IBVP and Quasi-local Hamiltonians

In this section, we discuss the covariant phase space method applied to vacuum
GR with a time-like boundary, following [21], [28], [22].

Consider Lagrangians with boundary terms describing vacuum gravity,

(3.1)
$I(g)=I_{EH}(g)+\int_{{\mathcal{C}}}\ell(g)=\int_{M}R_{g}dv_{g}+\int_{{\mathcal{C}}}\ell(g).$

(We have chosen units where $16\pi G=1$). The bulk Einstein-Hilbert action
$I_{EH}$ is invariant under the diffeomorphism group $\mathrm{Diff}(M)$; only
boundary actions $\int_{{\mathcal{C}}}\ell$ which are also
$\mathrm{Diff}({\mathcal{C}})$ invariant will be considered. Different choices
of boundary action correspond to different choices of boundary data space
${\mathcal{B}}$. As a simple example, consider the case of a scalar field
$u:M\rightarrow\mathbb{R}$ with action given by

(3.2) $I_{D}(u)=\int_{M}({\tfrac{1}{2}}|du|^{2}+V(u))dv_{g},$

where $g$ is a fixed background globally hyperbolic Lorentz metric on $M$ and
$V(u)$ is a function involving only 0-jet of $u$. Then the variation of
$I_{D}$ in the direction $v=u^{\prime}$ is given by

$\frac{d}{dt}I_{D}(u+tv)|_{t=0}=-\int_{M}v(\Box
u-V^{\prime}(u))-\int_{{\mathcal{C}}}v\partial_{\nu}u-\int_{S^{+}\cup
S^{-}}\Psi_{u}(v),$

where $\Psi_{u}(v)=v\partial_{T}u$ and $T$ is the unit outward time-like
normal to the future and past the slices $S^{\pm}$. This term is ignored in
the variational problem, cf. [21], so that the variational derivative is given
by

$\delta I_{D}(v)=-\int_{M}v(\Box
u-V^{\prime}(u))-\int_{{\mathcal{C}}}v\partial_{\nu}u.$

The boundary condition that $I_{D}$ is stationary $\delta I_{D}=0$ then
requires that $v=0$ at ${\mathcal{C}}$, i.e. the Dirichlet data
$u|_{{\mathcal{C}}}$ of $u$ is fixed. Hence the space ${\mathcal{B}}$ is the
space of Dirichlet boundary data of $u$,
${\mathcal{B}}=C^{\infty}({\mathcal{C}})$. The corresponding equations of
motion are

(3.3) $\Box u+V^{\prime}(u)=0.$

If one adds a boundary action of the form

(3.4)
$I_{N}(u)=\int_{M}({\tfrac{1}{2}}|du|^{2}+V(u))dv_{g}+\int_{{\mathcal{C}}}u\partial_{\nu}u,$

then a similar analysis shows that the boundary condition becomes
$\partial_{\nu}v=0$, i.e. the Neumann boundary data $\partial_{\nu}u$ of $u$
is fixed, with corresponding boundary data space again
${\mathcal{B}}=C^{\infty}({\mathcal{C}})$.

In either case above, one has a well-defined variational problem on the field
or configuration space ${\mathcal{F}}=C_{b}^{\infty}(M)$ of functions
$u:M\rightarrow\mathbb{R}$ with $u$ satisfying a fixed boundary condition
$b\in{\mathcal{B}}$ (Dirichlet or Neumann) at the boundary ${\mathcal{C}}$;
namely, $I_{D}$, resp. $I_{N}$, is a smooth function on $C^{\infty}(M)$ whose
critical points are solutions of the Euler-Lagrange equations (3.3). Moreover,
standard theory of the IBVP for the (non-linear) wave equation (3.3) implies
that the space of solutions $\mathbb{P}$ of (3.3) satisfies the analog of
(1.3),

$\mathbb{P}\simeq\mathcal{I}\times_{c}{\mathcal{B}}.$

The covariant phase space method discussed below defines a natural symplectic
structure on $\mathbb{P}$, with corresponding Hamiltonians, cf.[11], [21] for
further details.

We emphasize here that the issues of having a well-defined variational problem
for a field theory and a well-posed IBVP for the equations of motion are
fundamentally distinct and independent. In particular, the property of having
a well-defined variational problem does not at all imply the property of
having a well-posed IBVP.

Returning to GR and the action (3.1), let
${\mathcal{F}}=\operatorname{Met}(M)$ be the configuration space of all
maximal globally hyperbolic Lorentz metrics on $M$ with time-like boundary
${\mathcal{C}}$. The pre-phase space

$\widetilde{\mathbb{P}}\subset{\mathcal{F}}$

is defined to be the space of globally hyperbolic solutions of the vacuum
Einstein equations on $M$. (The terminology on-shell for data in
$\widetilde{\mathbb{P}}$ and off-shell for data in ${\mathcal{F}}$ is often
used). The bulk diffeomorphism group $\mathrm{Diff}(M)$, mapping
${\mathcal{C}}\rightarrow{\mathcal{C}}$, acts naturally on both spaces by
pullback. Although it is geometrically natural to consider the quotient space
of isometric metrics $\operatorname{Met}(M)/\mathrm{Diff}(M)$, it will be seen
below that the quotient $\widetilde{\mathbb{P}}/\mathrm{Diff}(M)$ does not
generally serve as a suitable candidate for the phase space.

We first discuss the “bare” situation without any boundary terms. For the bare
EH action,

$I_{EH}(g)=\int_{M}R_{g}dv_{g},$

one has $\delta_{g}(R_{g}dv_{g})(h)=-\langle E_{g},h\rangle dv_{g}+d\theta(h)$
where $d\theta(h)=(-\Box trh+\mathrm{div}\mathrm{div}h)dv_{g}$. The 3-form

(3.5) $\theta(h)=-\star(d\mathrm{tr}h-\mathrm{div}h),$

is called the pre-symplectic potential and plays a central role in this
discussion. Note that for any 2-form $C$ on $M$, $d(\theta-dC)=d\theta$. On-
shell, i.e. tangent to $\widetilde{\mathbb{P}}$, $(R_{g}dv_{g})^{\prime}=0$
and $E_{g}=0$, so that, on $T\widetilde{\mathbb{P}}$,

(3.6) $d\theta(h)=0$

pointwise on $M$. In other words, $\theta$ is a closed 3-form on $M$ on-shell.
It follows that

(3.7) $\frac{d}{dt}I_{EH}(g+th)|_{t=0}=0,$

on-shell. Thus (in contrast to the situation in (3.2) or (3.4)) it might
appear that there are no boundary conditions imposed by the requirement of
stationarity. However, (3.7) only gives $\int_{{\partial M}}(\theta-dC)(h)=0$,
i.e.

$\int_{{\mathcal{C}}}(\theta-dC)(h)=\int_{S^{+}}(\theta-
dC)(h)-\int_{S^{-}}(\theta-dC)(h).$

This shows there is a (global) coupling of the variation at the future and
past Cauchy surfaces with the variation at the boundary ${\mathcal{C}}$. For a
well-defined variational problem, one requires instead that

(3.8) $\delta I_{EH}(h)=\int_{{\mathcal{C}}}(\theta-dC)(h)=0,$

for all on-shell variations $h$.

On ${\mathcal{C}}$,
$\theta(h)=-(\nu(\mathrm{tr}h)-\mathrm{div}h(\nu))dv_{\gamma}$ is a gauge,
i.e. $\mathrm{Diff}_{0}(M)$ dependent term. To obtain a geometric or gauge-
invariant boundary condition, recall the basic identity (valid on any
hypersurface)

(3.9)
$-\theta(h)=\big{(}\nu(\mathrm{tr}h)-(\mathrm{div}h)(\nu)\big{)}dv_{\gamma}=\big{(}2H^{\prime}_{h}+\langle
A,h\rangle+\mathrm{div}(h(\nu)^{T})\big{)}dv_{\gamma}.$

It is natural to set

(3.10) $C(h)=\star h(\nu)^{T}\ \ {\rm on}\ \ {\mathcal{C}},$

and we do so for the remainder of the paper; the discussion to follow is
independent of the extension of $C$ off ${\mathcal{C}}$. Also set

(3.11) $\widehat{\theta}=\theta-dC.$

Then by (3.9), (3.8) is equivalent to

(3.12) $\int_{{\mathcal{C}}}(\langle
A,h\rangle+2H^{\prime}_{h})dv_{\gamma}=0,$

This corresponds to geometric boundary conditions $h^{T}=H^{\prime}_{h}=0$.
However, these are now $7$ boundary conditions and so these are over-
determined boundary conditions, (cf. the discussion in §2).

This shows that it is necessary to include boundary Lagrangians $\ell$ to
obtain a well-defined variational problem. To begin, the action (3.1) is
stationary on the subspace of $h\in T\widetilde{\mathbb{P}}$ such that

(3.13) $[(\theta-dC+\delta\ell)|_{{\mathcal{C}}}](h)=0.$

Thus associate to the action (3.1) a space of (geometric) boundary data
${\mathcal{B}}$ and consider the natural evaluation or restriction map

$R:\widetilde{\mathbb{P}}\rightarrow{\mathcal{B}},$

equivariant with respect to the action of diffeomorphisms. Given a choice of
boundary data $b\in{\mathcal{B}}$, the pre-phase space
$\widetilde{\mathbb{P}}_{b}$ is defined as the inverse image

(3.14) $\widetilde{\mathbb{P}}_{b}=R^{-1}(b):$

$\widetilde{\mathbb{P}}_{b}$ is the space of vacuum metrics with boundary data
in $b$. Then (3.13) is equivalent to

$T\widetilde{\mathbb{P}}_{b}={\rm Ker}\,([\theta-
dC+\delta\ell]_{{\mathcal{C}}}).$

Via (3.9), the boundary condition (3.13) is thus equivalent to

(3.15) $-\langle A,h\rangle-2H^{\prime}_{h}+\ell^{\prime}_{h}=0\ \ {\rm on}\ \
{\mathcal{C}}.$

Of course in general $\widetilde{\mathbb{P}}_{b}$ will no longer have an
effective induced action of $\mathrm{Diff}(M)$; the choice of $b$ breaks the
symmetry, i.e. breaks the isometric action of $\mathrm{Diff}(M)$ on
$\operatorname{Met}(M)$ to that of the subgroup $\mathrm{Diff}_{b}(M)$
preserving the boundary condition $b\in{\mathcal{B}}$.

The pre-symplectic current $\omega$ on $\widetilde{\mathbb{P}}_{b}$, or more
generally on $\widetilde{\mathbb{P}}$ or ${\mathcal{F}}$, is given by

(3.16) $\omega=\delta\widehat{\theta},$

where $\delta$ is the exterior derivative on ${\mathcal{F}}$. Thus,

$\omega(h_{1},h_{2})=(\widehat{\theta}(h_{2}))^{\prime}_{h_{1}}-(\widehat{\theta}(h_{1}))^{\prime}_{h_{2}},$

where $h_{1}$, $h_{2}$ are tangents to a 2-parameter deformation in
$\widetilde{\mathbb{P}}_{b}$, (so that the formal correction term
$\theta([h_{1},h_{2}])$ vanishes).

Given a Cauchy surface $S$, the pre-symplectic form is a 2-form on
$\widetilde{\mathbb{P}}_{b}$ defined by

$\widetilde{\Omega}(h_{1},h_{2})=\int_{S}\omega(h_{1},h_{2})=\int_{S}(\widehat{\theta}(h_{2}))^{\prime}_{h_{1}}-(\widehat{\theta}(h_{1}))^{\prime}_{h_{2}}.$

Using (3.9), this gives

$\widetilde{\Omega}(h_{1},h_{2})=-\int_{S}\langle
K^{\prime}_{h_{1}},h_{2}\rangle-\langle
K^{\prime}_{h_{2}},h_{1}\rangle+{\tfrac{1}{2}}[tr_{S}h_{1}(\langle
K,h_{2}\rangle+2k^{\prime}_{h_{2}})-tr_{S}h_{2}(\langle
K,h_{1}\rangle+2k^{\prime}_{h_{1}})],$

where $K$ is the extrinsic curvature of $S$ in $(M,g)$ and $k=tr_{S}K$.
Writing $\pi=(K-kg_{S})dv_{g_{S}}$, where $K-kg_{S}$ is the momentum conjugate
to $g_{S}$, this may be written (in standard form) as

(3.17)
$\widetilde{\Omega}(h_{1},h_{2})=-\int_{S}\langle\pi^{\prime}_{h_{1}},h_{2}\rangle-\langle\pi^{\prime}_{h_{2}},h_{1}\rangle.$

This gives a non-degenerate symplectic form off-shell, i.e. on the
configuration space ${\mathcal{F}}$, cf. [28]. However, the vacuum constraints
and boundary conditions imply that $\widetilde{\Omega}$ is degenerate on-
shell, i.e. on $\widetilde{\mathbb{P}}$ or $\widetilde{\mathbb{P}}_{b}$.

The boundary condition (3.13) implies $\omega=\delta\delta\ell=0$ on
${\mathcal{C}}$. Since $\omega$ is a closed $3$-form on $M$, it follows that
$\widetilde{\Omega}$ is well-defined, independent of the Cauchy surface $S$,
on $\widetilde{\mathbb{P}}_{b}$. This is not the case without the boundary
condition (3.13); thus $\widetilde{\Omega}=\widetilde{\Omega}_{S}$ gives a
pre-symplectic form on the full space $\widetilde{\mathbb{P}}$ which depends
on the Cauchy surface $S$.

Hamiltonians are functions
${\mathcal{H}}:\widetilde{\mathbb{P}}_{b}\rightarrow\mathbb{R}$ for which
there is a vector field $X_{{\mathcal{H}}}$ such that

(3.18) $\widetilde{\Omega}(X_{{\mathcal{H}}},\cdot)=\delta{\mathcal{H}}.$

If $\widetilde{\Omega}$ were non-degenerate, then any smooth function
${\mathcal{H}}:\widetilde{\mathbb{P}}_{b}\rightarrow\mathbb{R}$ is a
Hamiltonian. Namely, setting $\widetilde{\Omega}=\Omega$ in the non-degenerate
situation, the symplectic gradient $X_{{\mathcal{H}}}$ is given by

$X_{{\mathcal{H}}}=\Omega^{-1}(\delta{\mathcal{H}},\cdot),$

where $\Omega$ defines the pairing: $T\mathbb{P}\rightarrow T^{*}\mathbb{P}$.
If, as it is here, $\widetilde{\Omega}$ is degenerate, then Hamiltonians may
not exist. A Hamiltonian ${\mathcal{H}}$ must be invariant under the flow
generated by the degenerate directions ${\rm Ker}\,\widetilde{\Omega}$.
Moreover, ${\mathcal{H}}$ is always conserved, i.e. ${\mathcal{H}}$ is
invariant under the flow of the Hamiltonian vector field $X_{{\mathcal{H}}}$
on $\widetilde{\mathbb{P}}_{b}$.

The actual phase $\mathbb{P}_{b}$ with boundary data $b\in{\mathcal{B}}$ is
given by a standard symplectic reduction process, cf. [30], as the formal
quotient

(3.19) $\mathbb{P}_{b}=\widetilde{\mathbb{P}}_{b}/{\mathcal{G}},$

where ${\mathcal{G}}$ is the Lie group generated by the Lie algebra ${\rm
Ker}\,\widetilde{\Omega}$. Then $\widetilde{\Omega}$ descends to a (non-
degenerate) symplectic form on $\mathbb{P}_{b}$, giving it formally a
symplectic structure.

As the primary example of interest here, consider a vector field $\xi$
generating a flow in $\mathrm{Diff}(M)$ by spacetime diffeomorphisms of $M$.
This generates a flow on $\widetilde{\mathbb{P}}_{b}$ in the usual way by
pullback, with tangent vector $L_{\xi}g\in T_{g}\widetilde{\mathbb{P}}_{b}$.

As shown in [28], [22], [21], the Hamiltonian ${\mathcal{H}}_{\xi}$ is given
by

(3.20) ${\mathcal{H}}_{\xi}=\int_{\Sigma}q_{\xi}+\xi\cdot\ell-C(L_{\xi}g),$

where $q_{\xi}$ is the Noether potential given by

$q_{\xi}=-\star d\xi,$

viewing $\xi$ as a 1-form on ${\mathcal{C}}$. We consider $\ell$ as a scalar
density, $\ell=\ell dv_{\gamma}$ on ${\mathcal{C}}$, so that
$\xi\cdot\ell=\xi\lfloor dv_{{\mathcal{C}}}$. On $\Sigma$, $\star
d\xi=d\xi(T,\nu)=\langle\nabla_{T}\xi,\nu\rangle-\langle\nabla_{\nu}\xi,T\rangle$,
where $\nu$ is as above and $T$ is the unit normal to
$\Sigma\subset{\mathcal{C}}$. Since
$L_{\xi}g(T,\nu)=\langle\nabla_{T}\xi,\nu\rangle+\langle\nabla_{\nu}\xi,T\rangle$
and $\langle\nabla_{T}\xi,\nu\rangle=-A(\xi,T)$, this gives

(3.21)
${\mathcal{H}}_{\xi}=\int_{\Sigma}(-2A(\xi,T)+\ell\langle\xi,T\rangle)dv_{\gamma_{\Sigma}}.$

Note that ${\mathcal{H}}_{\xi}$ is independent of the slice $\Sigma$ since the
variation $h=L_{\xi}g$ satisfies the boundary condition (3.13), i.e.
$L_{\xi}g\in T\widetilde{\mathbb{P}}_{b}$. Otherwise, as with
$\widetilde{\Omega}$, this will not be the case for general $\xi\in
T\mathrm{Diff}(M)$.

Clearly if $\xi=0$ on ${\mathcal{C}}$ (or on suitable domains in
${\mathcal{C}}$), then ${\mathcal{H}}_{\xi}=\delta{\mathcal{H}}_{\xi}=0$. so
the symplectic form $\widetilde{\Omega}$ is degenerate on vectors tangent to
$\mathrm{Diff}_{0}(M)$. Thus one always has

(3.22) $\mathrm{Diff}_{0}(M)\subset{\mathcal{G}}.$

On the other hand, the existence of non-trivial Hamiltonians generated by
$\xi$ as above shows that generally $\mathrm{Diff}_{b}(M)$ is not contained in
${\mathcal{G}}$.

Finally we note that a Hamiltonian
$H_{\xi}:\mathbb{P}_{b}\rightarrow\mathbb{R}$ as in (3.20) is determined by
(3.18) only up to an additive constant; the constant may depend on the
boundary condition $b\in{\mathcal{B}}$. The choice of the normalizing constant
plays an important role in Hamitonian approaches to the issue of quasi-local
energy and is discussed in more detail in §5.

The covariant phase space method is related to the perhaps more well-known
canonical phase space method by the choice of a Cauchy surface $S$. Namely,
given such an $S$, one has a formal identification

(3.23) $E_{S}:\widehat{\mathbb{P}}_{b}\simeq\mathcal{I},\ \
E_{S}(g)=(g_{S},K),$

where
$\widehat{\mathbb{P}}_{b}=\widetilde{\mathbb{P}}_{b}/\mathrm{Diff}_{0}(M)$ and
$\mathcal{I}$ is the space of initial data $(g_{S},K)$ satisfying the
constraint equations. Here $\mathrm{Diff}_{0}(M)$ is the group of
diffeomorphisms of $M$ fixing $T=S\cup{\mathcal{C}}$. Of course different
choices of Cauchy surface give different parametrizations $E_{S}$. The data
$(g_{S},K)$, or more precisely $(g_{S},\pi)$, serve as the “$q$, $p$
variables” for equivalence classes of solutions in $\widehat{\mathbb{P}}_{b}$
and so such data effectively serve as a choice of coordinates for
$\widehat{\mathbb{P}}_{b}$. Solutions $g\in\widetilde{\mathbb{P}}_{b}$ are
then given by a specific choice of time evolution along a curve of Cauchy
surfaces $S_{t}$ with associated lapse-shift data $(N,X)$.

This brief summary of the covariant phase space method with boundary neglects
however several significant issues. As discussed in §2, there are seemingly
natural situations, e.g. Dirichlet boundary data where
${\mathcal{B}}=\operatorname{Met}({\mathcal{C}})$, for which the
identification (3.23) fails badly. For generic boundary data
$b\in{\mathcal{B}}$, one may have $\widetilde{\mathbb{P}}_{b}=\emptyset$.
Moreover, the structure of $\widetilde{\mathbb{P}}_{b}$ may be highly unstable
with respect to variations of $b\in{\mathcal{B}}$. For the purely formal
considerations outlined above to accurately reflect the behavior of the space
of solutions $\widetilde{\mathbb{P}}$ or $\widetilde{\mathbb{P}}_{b}$ (or
their symplectic quotients), these spaces should be smooth infinite
dimensional manifolds (or with small, understood singular sets) with the
submanifolds $\widetilde{\mathbb{P}}_{b}$ varying smoothly with $b$ (at least
generically). For this, one needs at least the a priori existence of charts
for these spaces effectively parametrizing them locally.

It appears that the only method to do this is to prove that the IBVP
associated to $\widetilde{\mathbb{P}}_{b}$, $b\in{\mathcal{B}}$, is well-
posed. In this case, one has natural parametrizations

(3.24)
$\widehat{\mathbb{P}}=\widetilde{\mathbb{P}}/\mathrm{Diff}_{0}(M)\simeq\mathcal{I}\times_{c}{\mathcal{B}},$

as in (1.3). As discussed in §2, this does not hold for Dirichlet or Neumann
boundary data. Further, based on the analogous result for closed or
asymptotically flat Cauchy surfaces $S$, cf. [15], one would expect that

$\widehat{\mathbb{P}}\simeq\mathbb{P},$

at least generically.

######  Remark 3.1.

An alternate path to the definition of Hamiltonians is the Hamilton-Jacobi
method, used for instance by Brown-York [10], cf. also [37] for a survey. Here
one also considers the space $\widetilde{\mathbb{P}}$ of (maximal) globally
hyperbolic vacuum solutions on $M$ and a corresponding space ${\mathcal{B}}$
of boundary data (Dirichlet data in the case of [10]). Analogous to the
Hamilton-Jacobi method in classical mechanics, a Hamiltonian is defined as the
variation of the on-shell action (3.1) with respect to a choice of time
translation $\partial_{t}$ of the boundary data $b\in{\mathcal{B}}$, expressed
in terms of a lapse-shift $(N,X)$ decomposition of ${\mathcal{C}}$. For this
approach to be effective, one needs to know that the action
$\mathcal{I}:\widetilde{\mathbb{P}}_{b}\rightarrow\mathbb{R}$ is a smooth
function of $b$ and so in particular the space $\widetilde{\mathbb{P}}_{b}$
varies smoothly with $b$. Again, as discussed in §2, this cannot be expected
to be true for Dirichlet (or Neumann) boundary data.

######  Remark 3.2.

A shortcoming of the Hamiltonian approach above is that Hamiltonians
associated to infinitesimal generators $\xi\in T\mathrm{Diff}({\mathcal{C}})$
only exist for $\xi$ preserving the boundary condition, i.e. $\xi\in
T\mathrm{Diff}_{b}({\mathcal{C}})$. For generic boundary data, no such $\xi$
exist and so the theory applies only in very special circumstances.

One would thus like to develop a well-defined Hamiltonian for arbitrary
$\xi\in T\mathrm{Diff}({\mathcal{C}})$. Note first that it follows easily from
the constraint equations that $\delta I(L_{\xi}g)=0$ for any vector field
$\xi$ of compact support on ${\mathcal{C}}$ (for any diffeomorphism invariant
boundary action). More generally, for $h=L_{\xi}g$, the constraint equations
show that the boundary condition (3.15) may be rewritten in the form

(3.25)
$\mathrm{div}(\tau(\xi))+\mathrm{div}(\ell\xi)+\mathrm{div}h(\nu)^{T}-dC(h)=0\
\ {\rm on}\ \ {\mathcal{C}}.$

Hence for infinitesimal diffeomorphisms, the boundary condition (3.13) or
(3.15) is only a condition at the future and past boundaries $\Sigma^{\pm}$.
This suggests that ideally one should seek boundary conditions which are
preserved under the action of $\mathrm{Diff}({\mathcal{C}})$.

Nevertheless, (3.25) does not imply one can construct Hamiltonians with, for
example, diffeomorphism-invariant Dirichlet boundary data, i.e. boundary data
in the quotient space
$\operatorname{Met}({\mathcal{C}})/\mathrm{Diff}({\mathcal{C}})$. For such a
larger space of admissible boundary data, Hamiltonians will no longer be
conserved and it is necessary to work in the context of time-dependent
Hamiltonians. This has been done using extended phase space techniques in work
of Booth [8] and Booth-Fairhurst [9].

The covariant phase space method applied to the
$\mathrm{Diff}({\mathcal{C}})$-invariant boundary data in [2] discussed in §2
is rather different from the discussion above. To explain the situation, as
before let $\widetilde{\mathbb{P}}$ be the space of all (maximal, globally
hyperbolic) vacuum Einstein metrics on $M$. Given a Cauchy surface $S\subset
M$, and a choice of boundary vector field $\Theta_{{\mathcal{C}}}$ on
${\mathcal{C}}_{0}$ as in (2.11), as discussed in §2 there is a unique wave
map $\varphi_{g}:(M,g)\rightarrow(M_{0},g_{R})$ associated to $g$. As in
(2.12), the assignment $g\rightarrow\varphi_{g}$ is equivariant with respect
to the action of $\mathrm{Diff}(M)$ on $\widetilde{\mathbb{P}}$. Let
$\psi_{g}=\varphi_{g}^{-1}$ and let $\chi({\mathcal{C}}_{0})$ be the space of
vector fields on the boundary ${\mathcal{C}}_{0}$. Define then a map

(3.26)
$\begin{array}[]{c}\widetilde{\mathbb{P}}\times\chi({\mathcal{C}}_{0})\rightarrow\bar{\mathbb{P}},\\\
(g,\Theta_{{\mathcal{C}}})\rightarrow\bar{g}=\psi_{g}^{*}g,\end{array}$

where $\bar{\mathbb{P}}$ is the space of vacuum metrics on $M_{0}$ in the
standard $(t,x^{i})$ Minkowski coordinates. The group $\mathrm{Diff}(M)$ acts
on the first factor on the left and acts trivially on the second and the map
(3.26) descends to the quotient to give a bijection

$(\widetilde{\mathbb{P}}\times\chi({\mathcal{C}}_{0}))/\mathrm{Diff}(M)\rightarrow\bar{\mathbb{P}}.$

Variations $\bar{h}$ of $\bar{g}$ are of the form

(3.27)
$\bar{h}=\psi_{g}^{*}h-L_{X_{h}}\bar{g}-L_{X_{\Theta^{\prime}}}\bar{g}\in
T\bar{\mathbb{P}},$

where $h$ is an on-shell variation of $g$ and $X_{h}$, $X_{\Theta^{\prime}}$
are the variations of the wave map $\varphi_{g}$ induced by the variation $h$
of $g$ (with $\Theta_{{\mathcal{C}}}$ fixed) and variations $\Theta^{\prime}$
of $\Theta_{{\mathcal{C}}}$ with $g$ fixed. Regarding the latter, fix a vacuum
solution $g$ and consider variations $\Theta_{{\mathcal{C}}}(t)$ of the
Sommerfeld boundary data $\Theta_{{\mathcal{C}}}$ for $\varphi_{g}$,
satisfying the compatibility conditions at the corner $\Sigma$, holding the
initial conditions fixed. One has unique wave map solutions $\varphi_{g}(t)$
of (2.9)-(2.11) realizing the boundary conditions $\Theta_{{\mathcal{C}}}(t)$.
By uniqueness, the pair $(g,\varphi_{g}(t))$ are solutions of the coupled
system (2.13)-(2.15) satisfying the initial data and boundary data
$([\gamma^{t}],H^{*},\Theta_{{\mathcal{C}}}(t))$. This gives a curve of
gauges, i.e. 3+1 decomposition or space-time foliation of the fixed metric
$g$. Then

(3.28) $X_{\Theta^{\prime}}=\frac{d}{dt}\varphi_{g}(t)|_{t=0},$

where $\Theta^{\prime}=\frac{d}{dt}\Theta_{{\mathcal{C}}}(t)$.

One may now carry out the covariant phase space method on the space
$\bar{\mathbb{P}}$. The definition of the corresponding pre-symplectic current
$\bar{\theta}$ is just as in (3.5), acting on $\bar{h}\in T\bar{\mathbb{P}}$.
Similarly $\bar{\omega}$ is defined as in (3.16), leading to the
(pre)-symplectic form

(3.29)
$\bar{\Omega}(\bar{h}_{1},\bar{h}_{2})=\int_{S_{0}}\langle\pi^{\prime}_{\bar{h}_{1}},\bar{h}_{2}\rangle-\langle\pi^{\prime}_{\bar{h}_{2}},\bar{h}_{1}\rangle,$

where $S_{0}=\\{t=0\\}$. Although $d\bar{\omega}=0$, $\bar{\omega}$ is non-
zero on ${\mathcal{C}}$ in general, without any boundary condition, so that
$\bar{\Omega}$ is not conserved, i.e. independent of the Cauchy slice $S_{0}$.

A boundary action and associated boundary condition is discussed below in §4,
cf. Proposition 4.3. We also introduce there an expanded phase space allowing
for the existence of non-trivial Hamiltonians; see Propositions 4.4 and 4.5
below.

##  4\. Boundary actions in GR

In this section, we consider several boundary actions
$\int_{{\mathcal{C}}}\ell$ for the Einstein-Hilbert action $I_{EH}$ and
discuss the resulting Hamiltonians generated by space-time diffeomorphisms.

The physics literature has long (only) considered the Gibbons-Hawking-York
boundary term $\frac{1}{8\pi}\int_{{\mathcal{C}}}Hdv_{\gamma}$ and the
corresponding action

$I_{GHY}=I_{EH}+2\int_{{\mathcal{C}}}Hdv_{\gamma},$

(where again we set $16\pi G=1$). The variation of the boundary Lagrangian is
$(H^{\prime}_{h}+\frac{1}{2}\langle H\gamma,h^{T}\rangle)dv_{\gamma}$, where
$h^{T}$ is the variation of the boundary metric $\gamma$. Inserting this in
(3.15) gives the well-known formula

(4.1) $\delta I_{GHY}=-2\int_{{\mathcal{C}}}\langle\tau,h^{T}\rangle
dv_{\gamma},$

on $\widetilde{\mathbb{P}}$ where $\tau=A-H\gamma$ is the momentum conjugate
to the boundary metric $\gamma$. The form $\tau$ is also the Brown-York
stress-energy tensor of the boundary ${\mathcal{C}}$, i.e. the variation of
the action with respect to variation of the Dirichlet boundary data, cf. [10],
[21].

This gives a well-defined variational problem, i.e. (3.13) holds, for
Dirichlet boundary conditions $h^{T}=0$, so that
${\mathcal{B}}=Met({\mathcal{C}})$. However, as discussed in §2, Dirichlet
boundary conditions are not well-posed for the IBVP; for generic boundary data
$\gamma$, there is not any vacuum solution realizing $\gamma$, i.e.
$\widetilde{\mathbb{P}}_{\gamma}=\emptyset$.

Nevertheless, one may formally compute the Hamiltonian of a vector field $\xi$
tangent to ${\mathcal{C}}$ preserving the boundary condition, i.e. $\xi$ a
Killing field on $({\mathcal{C}},\gamma)$. Simple computation from (3.21)
gives

(4.2)
${\mathcal{H}}_{\xi}^{Dir}=-2\int_{\Sigma}\tau(\xi,T)dv_{\gamma_{\Sigma}}.$

This is the Dirichlet or Brown-York “bare” Hamiltonian; normalizations of
${\mathcal{H}}_{\xi}^{Dir}$ corresponding to a choice of zero-point energy are
discussed further in §5. One may of course always define (4.2) formally or by
fiat, but the Hamiltonian interpretation is only well-defined in the special
circumstances where the boundary $({\mathcal{C}},\gamma)$ admits a Killing
field, cf. Remark 3.2. Recall also from §3 that in this case the Hamiltonian
is independent of the choice of slice $\Sigma$.

Consider next the boundary conditions $([\gamma],H)$ in (2.7) which are
conjectured to be well-posed for the IBVP, cf. also [4].

######  Proposition 4.1.

The boundary conditions $([\gamma],H)$ have a well-defined variational
formulation. In fact, the action

(4.3) $I_{CH}=\int_{M}Rdv_{g}+\frac{2}{3}\int_{{\mathcal{C}}}Hdv_{\gamma},$

has variational derivative given by

(4.4) $\delta I_{CH}(h)=-\int_{M}\langle E,h\rangle
dv_{g}-\int_{{\mathcal{C}}}(\langle\tau_{0},h^{T}_{0}\rangle+\frac{4}{3}H^{\prime}_{h})dv_{\gamma},$

where the subscript $0$ denotes the trace-free part with respect to
$({\mathcal{C}},\gamma)$.

If $\xi$ is a vector field on ${\mathcal{C}}$ which preserves the boundary
conditions, i.e. $\xi$ is a conformal Killing field preserving $H$, then

(4.5)
${\mathcal{H}}_{\xi}^{CH}=-2\int_{\Sigma}\tau_{0}(\xi,T)dv_{\gamma_{\Sigma}}.$

###### Proof.

The first statement follows from (4.1) and the simple computation
$-\frac{4}{3}(\int
Hdv_{\gamma})_{h}^{\prime}=-\frac{4}{3}\int(H^{\prime}_{h}+\frac{1}{2}\mathrm{tr}h)dv_{\gamma}$.
The second statement also follows easily from (3.21).

∎

######  Remark 4.2.

The on-shell actions $I_{GHY}$ and $I_{CH}$ are essentially the same; one has
$I_{CH}={\tfrac{1}{3}}I_{GHY}$ on $\widetilde{\mathbb{P}}$, so that on-shell

(4.6) $\delta I_{CH}={\tfrac{1}{3}}\delta I_{GHY}.$

However, the expressions (4.2) and (4.5) for the Hamiltonians associated to
$\xi$ are not related in this simple way. The reason for this is the different
boundary conditions; Dirichlet for $I_{GHY}$ and conformal class-mean
curvature for $I_{CH}$. Distinct boundary conditions give different spaces
$\widetilde{\mathbb{P}}_{b}$ and so different symplectic forms and hence
different Hamiltonians.

Suppose $\xi$ is a Killing field on $({\mathcal{C}},\gamma)$ which also
preserves the mean curvature $\xi(H)=0$, so that $h=L_{\xi}g$ preserves both
both boundary conditions, i.e. $h\in T(\widetilde{\mathbb{P}}_{\gamma})\cap
T(\widetilde{\mathbb{P}}_{([\gamma],H)})$. By (3.21), the Hamiltonians
${\mathcal{H}}_{\xi}^{Dir}$ and ${\mathcal{H}}_{\xi}^{CH}$ have the same
relation (4.6) as the actions,

(4.7) ${\mathcal{H}}_{\xi}^{CH}={\tfrac{1}{3}}{\mathcal{H}}_{\xi}^{Dir},$

if and only if the Noether charge vanishes,

(4.8) $Q_{\xi}=\int_{\Sigma}A(\xi,T)dv_{\gamma_{\Sigma}}=0.$

We note that it can be shown that if $\xi$ extends to a (stationary) Killing
field of the bulk solution $(M,g)$, then (4.8) does hold. However, it appears
unlikely that (4.8) holds in general.

Next we point out that the $\mathrm{Diff}({\mathcal{C}})$-invariant boundary
data $([\gamma^{t}],H^{*})$ from [2] may be given a variational formulation
for a natural choice of $H^{*}$, at least when the Sommerfeld boundary
condition (2.11) for the gauge $\Theta_{{\mathcal{C}}}$ is modified to the
Dirichlet boundary condition

(4.9) $(\varphi_{g})_{*}(T_{g}^{c})=\widehat{\Theta}_{{\mathcal{C}}}.$

Here $T_{g}^{c}$ is the future-pointing time-like unit normal to
$\varphi_{g}^{-1}(\Sigma_{t})\subset({\mathcal{C}},g)$ and
$\widehat{\Theta}_{{\mathcal{C}}}$ is a given vector field tangent to
${\mathcal{C}}_{0}$.

Recall from §2 that we work with vacuum solutions in preferred gauge
$\varphi_{g}$ and let $\bar{g}=\psi_{g}^{*}g$, $\psi_{g}=\varphi_{g}^{-1}$,
defined on the standard cylinder $M_{0}$. Let $\mathrm{tr}_{\Sigma}A$ be the
trace of $A$ along the level set
$\Sigma_{t}=\\{t=constant\\}\subset{\mathcal{C}}_{0}$ of the boundary
${\mathcal{C}}_{0}\subset(M_{0},\bar{g})$.

######  Proposition 4.3.

The action

(4.10)
$I_{AA}(\bar{g})=\int_{M_{0}}R_{\bar{g}}dv_{\bar{g}}+\int_{{\mathcal{C}}}\mathrm{tr}_{\Sigma}Adv_{\bar{\gamma}},$

gives a well-defined variational problem for the boundary data
$([\gamma^{t}],H^{*},\widehat{\Theta}_{{\mathcal{C}}})$, where

(4.11) $H^{*}=2\mathrm{tr}_{{\mathcal{C}}}A-\mathrm{tr}_{\Sigma}A.$

Thus, critical points of the action on the field space
${\mathcal{F}}_{b}=\operatorname{Met}_{b}(M_{0})$ with fixed boundary data
$b=([\gamma^{t}],H^{*},\widehat{\Theta}_{{\mathcal{C}}})$ are the vacuum
Einstein metrics with given boundary data $b$.

###### Proof.

Recall from (3.27) that general variations $\bar{h}$ of $\bar{g}$ are of the
form $\bar{h}=\psi^{*}_{g}h-L_{X_{h}}\bar{g}$. The variation of the bulk
Lagrangian is

$\begin{split}&\delta(R_{\bar{g}}dv_{\bar{g}})=\langle-
E_{\bar{g}},\psi^{*}h\rangle dv_{\bar{g}}+\langle
E_{\bar{g}},L_{X}\bar{g}\rangle dv_{\bar{g}}+{\rm div}[{\rm
div}\bar{h}-(d\mathrm{tr}\bar{h})]dv_{\bar{g}}\\\ &=\langle-
E_{\bar{g}},\psi^{*}h\rangle dv_{\bar{g}}+{\rm div}[2E_{\bar{g}}(X)+{\rm
div}\bar{h}-(d\mathrm{tr}\bar{h})]dv_{\bar{g}},\end{split}$

where we have applied the Bianchi identity in the second equality. Thus the
equations of motion generated by $I_{AA}$ are the vacuum Einstein equations.
Using (3.9) or (3.15), the on-shell variation of the action $I_{AA}$ with
respect to deformation $\bar{h}$ is then easily computed to be

(4.12) $\delta I_{AA}(\bar{h})=\int_{{\mathcal{C}}}(-\langle
A,\bar{h}\rangle-2(\mathrm{tr}_{{\mathcal{C}}}A)^{\prime}_{\bar{h}}+(\mathrm{tr}_{\Sigma}A)^{\prime}_{\bar{h}}+\tfrac{1}{2}\mathrm{tr}_{\Sigma}A\mathrm{tr}_{{\mathcal{C}}}\bar{h})dv_{\bar{\gamma}}.$

Let $\bar{h}_{\Sigma}$ be the restriction of $\bar{h}$ to $\Sigma_{t}$ and let
$(\bar{h}_{\Sigma})_{0}$ denote its trace-free part. The first and fourth
terms in (4.12) may then be combined and rewritten as $-\langle
A,\bar{h}\rangle+\tfrac{1}{2}\mathrm{tr}_{\Sigma}A\langle\bar{\gamma},\bar{h}\rangle=-\langle
A,(\bar{h}_{\Sigma})_{0}\rangle-\langle q(T),\bar{h}(T)\rangle$, where
$q(T)=2A(T)_{\Sigma}+(-\tfrac{1}{2}\mathrm{tr}_{\Sigma}A+A(T,T))T$. Also,
setting $H^{*}=2\mathrm{tr}_{{\mathcal{C}}}A-\mathrm{tr}_{\Sigma}A$, i.e.
setting $\alpha=2$, $\beta=-1$, $\gamma=0$ in (2.16), (4.12) may be rewritten
in the form

(4.13) $\delta I_{AA}(\bar{h})=-\int_{{\mathcal{C}}}\langle
A,(\bar{h}_{\Sigma})_{0}\rangle+(H^{*})^{\prime}_{\bar{h}}+\langle
q(T),\bar{h}(T)\rangle)dv_{\bar{\gamma}}$

The vanishing of the terms $(\bar{h}_{\Sigma})_{0}$,
$(H^{*})^{\prime}_{\bar{h}}$ and $\bar{h}(T)$ is equivalent to fixing the
boundary data $([\gamma^{t}],H^{*},\widehat{\Theta}_{{\mathcal{C}}})$.

∎

We note that for fixed $g$, the Dirichlet boundary condition (4.9) still gives
a well-posed IBVP for the wave map $\varphi_{g}$. It remains open however if
the choice of $\widehat{\Theta}_{{\mathcal{C}}}$ in place of
$\Theta_{{\mathcal{C}}}$ leads to a well-posed IBVP for the coupled system
$(g,\varphi_{g})$ as in (2.13)-(2.15). We hope to investigate this elsewhere.

In this setting, the phase space $\bar{\mathbb{P}}_{b}$ may be defined as in
(3.26) with associated pre-symplectic form $\bar{\Omega}$ in (3.29).
Unfortunately, this does not give rise to Hamiltonians associated with
generators $\xi\in T\mathrm{Diff}({\mathcal{C}})$. This is partly due to the
fact that the boundary conditions are $\mathrm{Diff}({\mathcal{C}})$ invariant
but mainly due to the dependence of the preferred gauge $\varphi_{g}$ on the
choice of the Cauchy surface $S$, as in (2.10).

In order to obtain non-trivial Hamiltonians, we consider the same action on an
enlarged space consisting of the pairs $(g,F)$ where $F$ denotes a general
diffeomorphism $F:M\rightarrow M_{0}$ which maps ${\mathcal{C}}$ to
${\mathcal{C}}_{0}$. In the following let $\bar{g}=(F^{-1})^{*}g$ defined on
$M_{0}$. Choose fixed boundary data
$b=\big{(}[\gamma^{t}],H^{*},\widehat{\Theta}_{\mathcal{C}},G\big{)}$, where
$\big{(}[\gamma^{t}],H^{*},\widehat{\Theta}_{\mathcal{C}}\big{)}$ are as above
and $G:{\mathcal{C}}\rightarrow{\mathcal{C}}_{0}$ is an arbitrary but fixed
diffeomorphism. Let ${\bf C}_{b}$ be the configuration space

(4.14) ${\bf C}_{b}=\\{(g,F):~{}\bar{g}=(F^{-1})^{*}g\mbox{ has boundary data
}b=\big{(}[\gamma^{t}],H^{*},\widehat{\Theta}_{\mathcal{C}}\big{)}\ {\rm
with}\ F|_{{\mathcal{C}}}=G\\}.$

######  Proposition 4.4.

The action

(4.15)
$\bar{I}_{AA}(\bar{g})=\int_{M_{0}}R_{\bar{g}}dv_{\bar{g}}+\int_{{\mathcal{C}}_{0}}\mathrm{tr}_{\Sigma}A_{\bar{g}}dv_{\bar{\gamma}},$

gives a well-defined variational problem on ${\bf C}_{b}$, for any boundary
data $b$ with

(4.16) $H^{*}=2\mathrm{tr}_{{\mathcal{C}}}A-\mathrm{tr}_{\Sigma}A.$

Thus, critical points of the action on ${\bf C}_{b}$ are the vacuum Einstein
metrics with given boundary data $b$.

A vector field $\xi$ on ${\mathcal{C}}$ which preserves the boundary
conditions $([\gamma^{t}],H^{*},\widehat{\Theta}_{{\mathcal{C}}})$ has an
associated Hamiltonian

(4.17)
${\mathcal{H}}_{\xi}^{AA}=\int_{\Sigma}-2A_{\bar{g}}(\bar{\xi},T_{\bar{g}})+\langle\bar{\xi},T_{\bar{g}}\rangle\mathrm{tr}_{\Sigma}A_{\bar{g}}dv_{\bar{\gamma}_{\Sigma}},$

where $\bar{\xi}=G_{*}(\xi)$.

###### Proof.

The proof of the first statement is the same as the proof of Proposition 4.3,
with deformation $h$ of $g$ replaced by the deformation $(h,X)$ of
$(g,F)\in{\bf C}_{b}$ and with corresponding deformation $\bar{h}$ of
$\bar{g}$ on $M_{0}$ replaced by the deformation
$\bar{h}=(F^{-1})^{*}h-L_{X}\bar{g}$ of $\bar{g}$. The same computations as
before show that the equations of motion generated by $\bar{I}_{AA}$ are again
the vacuum Einstein equations and that $\bar{I}_{AA}$ gives a well-defined
variational problem on ${\bf C}_{b}$ for any $b$ satisfying (4.16).

The resulting pre-phase space $\bar{\mathbb{P}}_{b}$ is given by

(4.18) $\bar{\mathbb{P}}_{b}=\\{(g,F)\in{\bf C}_{b}:\mathrm{Ric}_{g}=0\\}.$

The pre-symplectic potential $\theta$ is given by the same formula as in (3.5)
applied to $\bar{h}$:
$\bar{\theta}(\bar{h})=\star_{\bar{g}}[\mathrm{div}\bar{h}-(d\mathrm{tr}\bar{h})]$.
This leads as before to the pre-symplectic form as in (3.17):

(4.19)
$\bar{\Omega}\big{(}(h_{1},X_{1}),(h_{2},X_{2})\big{)}=-\int_{S}\langle\bar{\pi}^{\prime}_{\bar{h}_{1}},\bar{h}_{2}\rangle-\langle\bar{\pi}^{\prime}_{\bar{h}_{2}},\bar{h}_{1}\rangle,$

where $\bar{h}_{i}=(F^{-1})^{*}(h_{i})-L_{X_{i}}\bar{g}$.

Now let $\xi$ be a vector field on $(M,g)$ such that the deformation
$(L_{\xi}g,X)\in T\bar{\mathbb{P}}_{b}$ for some deformation $X$ of $F$, so
that $X=0$ on ${\mathcal{C}}$. Then $\xi$ induces a deformation of the
pullback metric $\bar{g}$ on $M_{0}$ by $\bar{h}=L_{\bar{\xi}}\bar{g}$, where
$\bar{\xi}=F_{*}(\xi)-X$ on $M_{0}$. The Hamiltonian generated by $\xi$ is
given by the same formula as in (3.21) applied to the field $\bar{\xi}$:

${\mathcal{H}}_{\xi}^{AA}=\int_{\Sigma}-2A_{\bar{g}}\big{(}\bar{\xi},T_{\bar{g}}\big{)}+\langle\bar{\xi},T_{\bar{g}}\rangle\mathrm{tr}_{\Sigma}A_{\bar{g}}dv_{\bar{\gamma}_{\Sigma}}.$

This gives (4.17) since $F=G$ and $X=0$ on ${\mathcal{C}}$.

Therefore we obtain a nontrivial Hamiltonian which is well-defined when
$(L_{\xi}g,X)\in T\bar{\mathbb{P}}_{b}$, for some $X$. Since $X=0$ on
${\mathcal{C}}$, this is equivalent to the existence of a vector field $\xi$
on $(M_{0},\bar{g})$ such that

(4.20) $\begin{split}[L_{\xi}\gamma^{t}]_{0}=0,\ \ \xi(H^{*})=0,\ \
L_{\xi}\widehat{\Theta}_{\mathcal{C}}=0\quad{\rm
on}~{}{\mathcal{C}}.\end{split}$

A simple example is when $\xi=\partial_{t}$ and the data
$([\gamma^{t}],H^{*},\widehat{\Theta}_{{\mathcal{C}}})$ are independent of
$t$, arising for instance as boundary data of a stationary vacuum solution.

∎

Note that the action $\bar{I}_{AA}$ imposes no equation of motion on the
auxiliary field $F$. Also $F$ does not contribute to the Hamiltonian beyond
the fixed boundary condition that $F=G$ at ${\mathcal{C}}$. Next we point out
that $F$ disappears when passing to the associated phase space and the
resulting phase space agrees with the (standard) phase space defined in
(3.19).

######  Proposition 4.5.

The phase space

(4.21) $\mathbb{P}_{b}=\bar{\mathbb{P}}_{b}/{\mathcal{G}},$

associated to (4.18) agrees with the vacuum phase space in (3.19).

###### Proof.

First, observe that for any deformation $X$ of $F$ with $X=0$ on
${\mathcal{C}}$, $(0,X)\in{\rm Ker}\,\widetilde{\Omega}$, as in (3.22). Thus
one may first reduce $\bar{\mathbb{P}}$ to the quotient space
$\mathbb{P}_{1}=\bar{\mathbb{P}}/{\mathcal{G}}_{1}$ where ${\mathcal{G}}_{1}$
is generated by vector fields $(0,X)\in T\bar{\mathbb{P}}_{b}$. If
$(g,F)\in\bar{\mathbb{P}}_{b}$ then $(g,F^{\prime})\in\bar{\mathbb{P}}_{b}$
for arbitrary diffeomorphisms $F^{\prime}:(M,\partial
M)\rightarrow(M_{0},\partial M_{0})$ such that $F^{\prime}=G$ on
${\mathcal{C}}$. All of these are in the same equivalence class in
$\mathbb{P}_{1}$.444One should restrict here to diffeomorphisms isotopic to
the identity but we will forgo such distinctions to simplify the discussion. A
natural choice of representative of an equivalence class is given by the pair
$(g,F)$ with $F$ solving

$\begin{split}\Box_{g}F=0\ \ {\rm in}~{}M,\ \ F=E_{0},~{}F_{*}(T_{g})=E_{1}\ \
{\rm on}~{}S,\ \ F=G\ \ {\rm on}~{}{\mathcal{C}},\end{split}$

for a fixed but arbitrary choice of initial data $(E_{0},E_{1})$; compare with
(2.9)-(2.11). This gives (4.21), but it worthwhile to carry the analysis
somewhat further.

From the expression of $\bar{\Omega}$ in (4.19), it is clear that if the
deformation $(h,X)$ preserves the initial data of $\bar{g}$ on $S$, i.e.
$(\bar{g}_{S})^{\prime}_{\bar{h}}=0~{}(K_{\bar{g}})^{\prime}_{\bar{h}}=0$,
then $(h,X)\in{\rm Ker}\,\bar{\Omega}$. Therefore one may further reduce the
space $\bar{\mathbb{P}}_{1}$ to

$\bar{\mathbb{P}}_{2}=\bar{\mathbb{P}}_{1}/{\mathcal{G}}_{2},$

where ${\mathcal{G}}_{2}$ is generated by such deformations $(h,X)$.

Next, assume $[(g_{t},F_{t})]\in\mathbb{P}_{1}$ is a smooth curve of
equivalence classes in $\mathbb{P}_{1}$ where the initial data of $g_{t}$
varies along a curve of diffeomorphisms: $g_{t}|_{S}=\psi_{t}^{*}(g_{0}|_{S})$
and $K_{g_{t}}=\psi_{t}^{*}(K_{g_{0}})$ for $\psi_{t}\in{\rm Diff}_{0}(S)$
with $\psi=\rm{Id}$ on $\Sigma$. Consider a new curve $F^{\prime}_{t}$ such
that $F^{\prime}_{t}$ solves

$\begin{split}\Box_{g_{t}}F^{\prime}_{t}=0\ \ {\rm in}\ \ M,\ \
F^{\prime}_{t}=F_{0}\circ\psi_{t}\ \ {\rm on}\ \ S,\ \ F^{\prime}_{t}=G\ \
{\rm on}\ \ {\mathcal{C}}.\end{split}$

Clearly $(g_{t},F_{t})\sim(g_{t},F^{\prime}_{t})$ in $\mathbb{P}_{1}$. In
addition, corresponding to the family $(g_{t},F^{\prime}_{t})$, the pullback
metrics $\bar{g}_{t}=((F^{\prime}_{t})^{-1})^{*}g_{t}$ with
$(F^{\prime}_{t})^{-1}=\psi_{t}$ have the same initial data, and hence the
infinitesimal deformations along this curve belong to ${\mathcal{G}}_{2}$.
Thus in $\bar{\mathbb{P}}_{2}$, $(g_{1},F_{1})\sim(g_{2},F_{2})$ if $g_{1}$
and $g_{2}$ have equivalent initial data, modulo diffeomorphisms equal to the
identity on ${\mathcal{C}}$.555As above with diffeomorphisms, we assume there
exists a curve $g_{t}$ connecting $g_{1}$ and $g_{2}$. In particular,
$\mathbb{P}_{b}$ is a further quotient (possibly trivial) of
$\bar{\mathbb{P}}_{2}$.

∎

As with (3.23) one would like to establish a one-to-one correspondence between
$\mathbb{P}_{b}$ and $\mathcal{I}/{\rm Diff}_{b}(S)$, for
$b=\big{(}[\gamma^{t}],H^{*},\widehat{\Theta}_{\mathcal{C}})$ as above. This
would require proving the well-posedness of the following IBVP:

(4.22) $\begin{split}&\mathrm{Ric}_{g}=0,~{}\Box_{g}F=0\ \ {\rm in}~{}M,\\\
&g|_{S}=g_{S},~{}K_{g}=K,~{}F=E_{0},~{}F_{*}(T_{g})=E_{1}\ \ {\rm on}~{}S,\\\
&[\bar{g}^{t}]=[\gamma^{t}],~{}-2\mathrm{tr}_{\mathcal{C}}A_{\bar{g}}+\mathrm{tr}_{\Sigma}A_{\bar{g}}=H^{*},~{}T^{c}_{\bar{g}}=\widehat{\Theta}_{\mathcal{C}},~{}F=G\
\ {\rm on}~{}{\mathcal{C}}.\end{split}$

The system (4.22) with Dirichlet boundary condition
$\widehat{\Theta}_{\mathcal{C}}$ replaced by the Sommerfeld boundary condition
$(T_{\bar{g}}+\nu_{\bar{g}})^{T}=\Theta_{{\mathcal{C}}}$ as in (2.11) on
${\mathcal{C}}$ was proved to be well-posed in [2]. As discussed following
Proposition 4.3, we hope to discuss the well-posedness of (4.22) elsewhere.

######  Remark 4.6.

Based on the geometric uniqueness result in [2], if
$(g_{1},F_{1}),(g_{2},F_{2})$ are solutions to (4.22) with the same initial
data and boundary data, and with $\Theta_{{\mathcal{C}}}$ in place of
$\widehat{\Theta}_{{\mathcal{C}}}$, then they are equivalent in the sense that
one is the pullback of the other by a diffeomorphism in $\mathrm{Diff}_{0}(M)$
(fixing $S$ and ${\mathcal{C}}$); in particular $g_{1}$ and $g_{2}$ are
isometric in $\mathrm{Diff}_{0}(M)$. This is consistent with the equivalence
relationship in $\bar{\mathbb{P}}_{b}$.

On the other hand, if $(g_{1},F_{1})$ and $(g_{2},F_{2})$ are solutions to
(4.22) (again with $\Theta_{{\mathcal{C}}}$ in place of
$\widehat{\Theta}_{{\mathcal{C}}}$) with all the same initial and boundary
data except that $g_{1}=\psi^{*}(g_{2})$, $K_{g_{1}}=\psi^{*}(K_{g_{2}})$ on
$S$ for some diffeomorphism $\psi\neq{\rm Id}_{S}$ of $S$, it is unknown
whether $g_{1},g_{2}$ are equivalent, i.e. isometric. This is the motivation
of introducing the preferred wave map in [2]. However, such elements are
equivalent in the phase space $\bar{\mathbb{P}}_{b}$.

##  5\. Normalized Hamiltonians

As noted in the Introduction, several approaches to the definition of quasi-
local energy-momentum or angular momentum of a space-like 2-surface $\Sigma$
are based on a Hamiltonian or the closely related Hamilton-Jacobi approach.
This was initiated by Brown-York [10], with modifications developed by
numerous authors including Epp, Kijowski, Liu-Yau, Wang-Yau and others; we
refer to [37] for a detailed survey and further references. For all of these
definitions, the space ${\mathcal{B}}$ of boundary data is the Dirichlet data
space ${\mathcal{B}}=\operatorname{Met}({\mathcal{C}})$; the various
definitions of quasi-local energies (or momenta) differ by a choice of
normalization term, i.e. a choice of zero-point energy (or momenta).

In general, the starting point of such a Hamiltonian approach is a space-like
surface $\Sigma$ with data on $\Sigma$ obtained by fixing a choice of boundary
data $b\in{\mathcal{B}}$ restricted to $\Sigma$. One assumes then that
$\Sigma$ with such data can be extended as a Cauchy slice in a time-like
hypersurface ${\mathcal{C}}=\partial M$, where $(M,g)$ is a vacuum space-time.
Moreover, one assumes the extension to ${\mathcal{C}}$ has a vector field
$\xi\in T{\mathcal{C}}$ with flow preserving the boundary condition
$b\in{\mathcal{B}}$. For Dirichlet boundary data, this means $\xi$ is a
Killing field of $({\mathcal{C}},\gamma)$, $L_{\xi}\gamma=0$. If $\xi$ is
time-like, then the Hamiltonian ${\mathcal{H}}_{\xi}$ serves as a notion of
bare quasi-local energy.

Recall from (4.2) that the bare Brown-York quasi-local energy is given by

(5.1) ${\mathcal{H}}^{Dir}=-2\int_{\Sigma}\tau(\xi,T)dv_{\gamma_{\Sigma}},$

where $T$ is the unit time-like normal to $\Sigma\subset{\mathcal{C}}$. Here
the data on $\Sigma$ consist of the specification of a time-like 3-metric
$\gamma$ on ${\mathcal{C}}$ (Dirichlet data) restricted to $\Sigma$ for which
$\xi$ is a Killing field at $\Sigma$. The initial choice of Brown-York is
simply to take

(5.2) $\gamma=-dt^{2}+\gamma_{\Sigma},$

with $\xi=T=\partial_{t}$ representing time translation. A simple computation
shows that

(5.3) ${\mathcal{H}}^{Dir}=-2\int_{\Sigma}H_{\Sigma}dv_{\gamma_{\Sigma}},$

where $H_{\Sigma}=\mathrm{tr}_{\Sigma}A$.

This expression is of course non-vanishing for general surfaces
$\Sigma\subset\mathbb{R}^{3}$ and the prescription of Brown-York is to
normalize the bare Hamiltonian by subtracting its Euclidean value:

(5.4)
${\mathcal{H}}^{BY}=2\int_{\Sigma}(H_{\Sigma}^{0}-H_{\Sigma})dv_{\gamma_{\Sigma}},$

where $H_{\Sigma}^{0}$ is the mean curvature of an isometric immersion
$\iota(\Sigma,\gamma_{\Sigma})\rightarrow\mathbb{R}^{3}$. Thus, the
subtraction term is the Brown-York Hamiltonian of
${\mathcal{C}}_{0}=\mathbb{R}\times\iota(\Sigma)\subset\mathbb{R}^{1,3}$ with
respect to the standard time translation $t$ in Minkowski space. At least in
the case of $\gamma_{\Sigma}$ with positive Gauss curvature, such an embedding
exists by fundamental work of Nirenberg and Pogorelov, and is unique (up to
congruence) by fundamental work of Cohn-Vossen. Unfortunately, little is known
about such existence and uniqueness results outside the case of positive Gauss
curvature.

More generally, the Wang-Yau energy [40] considers families of isometric
embeddings $(\Sigma,\gamma_{\Sigma})\rightarrow\mathbb{R}^{1,3}$ and chooses
the reference cylinder ${\mathcal{C}}_{0}$ to be spanned by arbitrary
Minkowski time-translation fields transverse to $\iota(\Sigma)$. This may be
viewed as a space-time generalization of the Brown-York prescription and
requires developing an understanding of the possible isometric immersions
$\Sigma\rightarrow\mathbb{R}^{1,3}$, carried out in detail in [40].

Returning to the general expression (5.1), since Dirichlet data are not well-
posed, it is of course not well-understood when $(\Sigma,\gamma_{\Sigma})$ can
be embedded in $({\mathcal{C}},\gamma)\subset(M,g)$ with time-like Killing
field $\xi$ on ${\mathcal{C}}$, as in (5.2) for instance. Thus, (5.1) can only
be considered as being formally defined, not well-defined.

Note also that even if (5.1) is defined, i.e. one has a vacuum space-time
$(M,g)$ with boundary $({\mathcal{C}},\gamma)$ extending $\Sigma$ and
possessing a time-like Killing field $\xi$, the Euclidean (Brown-York) or
Minkowski (Wang-Yau) subtraction term above may not be defined. To make this
explicit, consider the space $\operatorname{Met}_{0}({\mathcal{C}})$ of
Lorentz metrics on ${\mathcal{C}}$ with a time-like Killing field $\xi$. Such
metrics are parametrized by metrics $\operatorname{Met}(\Sigma)$ on $\Sigma$,
together with a choice of lapse and shift $(N,X)$ on $\Sigma$. This data is
time-independent, i.e. extended to be invariant under the flow of $\xi$, but
the data $(N,X)$ may be arbitrarily chosen over $\Sigma$. However, in
$\mathbb{R}^{1,3}$ there is only a 4-dimensional space of time-translation
Killing fields; thus most Killing fields on $({\mathcal{C}},\gamma)$ cannot be
renormalized to zero-point energy in this way.

######  Remark 5.1.

When the IBVP is well-posed so that (3.23) holds, a Hamiltonian
${\mathcal{H}}_{\xi}:\mathbb{P}_{b}\rightarrow\mathbb{R}$ becomes a function
on the space of initial data
${\mathcal{H}}_{\xi}:\mathcal{I}\rightarrow\mathbb{R}$. The normalization
terms above then correspond to solutions evaluated on flat initial data
$(g_{S},K)$. Recall that $(g_{S},K)$ represent flat initial data if and only
if the flat constraint equations hold; in schematic form

(5.5) $\begin{array}[]{c}dK=0,\\\ R_{g_{S}}+K^{2}=0.\end{array}$

The traces of these equations give the vacuum Einstein constraint equations
(2.1)-(2.2). If (5.5) holds, then by the fundamental theorem of hypersurfaces
in space-forms, $S$ has an isometric immersion into $\mathbb{R}^{1,3}$ unique
up to isometry of $\mathbb{R}^{1,3}$, (at least when $S$ is simply connected).
In the time-symmetric (Brown-York) case, $K=0$ and so $g_{S}$ is a flat metric
on $S\subset\mathbb{R}^{3}$, or more precisely an immersion of $S$ into
$\mathbb{R}^{3}$. However, in general it is not well understood when a given
geometry at the boundary $\Sigma$, i.e. boundary data $b\in{\mathcal{B}}$, has
a fill-in by a flat metric on $S$, or when such fillings are unique.

We consider instead a natural definition of the normalization term that is
formally valid in general:

######  Definition 5.2.

Given a Hamiltonian ${\mathcal{H}}_{\xi}:\mathbb{P}_{b}\rightarrow\mathbb{R}$
associated with a boundary condition $b\in{\mathcal{B}}$, define the
normalized Hamiltonian by

$\widehat{\mathcal{H}}_{\xi}={\mathcal{H}}_{\xi}-\inf_{\mathbb{P}_{b}}{\mathcal{H}}_{\xi}.$

For this to be well-defined, one needs of course

(5.6) $\inf_{\mathcal{I}}{\mathcal{H}}_{\xi}>-\infty.$

When the identification (3.23) holds, so that
$\mathbb{P}_{b}\simeq\mathcal{I}$, this gives the more concrete expression

(5.7)
$\widehat{\mathcal{H}}_{\xi}={\mathcal{H}}_{\xi}-\inf_{\mathcal{I}}{\mathcal{H}}_{\xi}.$

An advantage of this definition is that it immediately implies the positivity
property:

$\widehat{\mathcal{H}}_{\xi}\geq 0\ \ {\rm on}\ \ \mathcal{I}.$

Consider first the time-symmetric case where $K=0$. If there exists a flat
solution with time-symmetric initial data, i.e. flat initial data
$(g_{S}^{0},0)\in\mathcal{I}$ filling in the boundary data
$b\in{\mathcal{B}}$, then it is natural to ask if the infimum
$\inf_{\mathcal{I}}{\mathcal{H}}_{\xi}$ is realized by flat data:

$\inf_{\mathcal{I}}{\mathcal{H}}_{\xi}={\mathcal{H}}_{\xi}(g_{S}^{0},0),$

i.e.

$\widehat{\mathcal{H}}_{\xi}(g_{S}^{0},K^{0})=0.$

It is also natural to consider the corresponding uniqueness or rigidity issue.
For Dirichlet boundary data $\gamma_{\Sigma}$ with positive Gauss curvature
and with $N=1$ on $\Sigma$ (the Brown-York case as in (5.2)-(5.3)), this is
proved to be true by a basic result of Shi-Tam [35]. A generalization of this
result to the setting with non-zero $K$ (the space-time setting) has been
proved in basic work of Wang-Yau [40].

The term $\inf_{\mathcal{I}}{\mathcal{H}}_{\xi}$ bears a formal similarity
with the definition of the Bartnik quasi-local energy [7], which is based on
minimizing the ADM Hamiltonian for an asymptotically flat space-time in a
region exterior to the Cauchy surface $S$; thus the interior minimization
problem considered here is replaced in the Bartnik program by minimization
over a complementary exterior region. Bartnik conjectures that such infima are
realized by stationary vacuum solutions, i.e. vacuum solutions for which the
asymptotic time-like Killing field extends to a time-like Killing field over
the full exterior region. It is of course natural to make the analgous
conjecture in this (interior) setting. This will be discussed in more detail
elsewhere.

In the case that the Killing field $\xi$ on ${\mathcal{C}}$ is hypersurface
orthogonal, i.e. the shift $X$ of $\xi$ satisfies $X=0$, it is proved in [33]
that critical points of ${\mathcal{H}}_{\xi}$ on $\mathcal{I}$ are solutions
of the static vacuum Einstein equations

$u\mathrm{Ric}=D^{2}u,\ \ \Delta u=0,$

on $S$ with $u=N$ on $\Sigma$, partially confirming this conjecture. It is
also proved in [32] that the bound (5.6) holds in this case.

Next we consider analogs of the discussion above for other more well-behaved
boundary conditions.

Consider first the $([\gamma],H)$ data (4.3) which are conjectured to be well-
posed. Let $\Sigma$ be any space-like 2-surface with metric $\gamma_{\Sigma}$
and let $\xi$ be a vector field on ${\mathcal{C}}$ restricted $\Sigma$. For
simplicity, consider here just extensions of this data to ${\mathcal{C}}$ as
in (5.2), so that $\xi=T=\partial_{t}$. Assume also $H$ is independent of $t$.
Then

$[\gamma]=[-dt^{2}+g_{\Sigma}]$

and $\xi$ is a conformal Killing field on ${\mathcal{C}}$ preserving the mean
curvature $\xi(H)=0$, so $\xi$ preserves the boundary condition. Assuming the
IBVP is well-posed for $([\gamma],H)$ (or instead just proceeding formally),
this gives the Hamiltonian

${\mathcal{H}}_{\xi}^{CH}=-2\int_{\Sigma}\langle A-\frac{1}{3}H\gamma,T\cdot
T\rangle dv_{\Sigma}=-2\int_{\Sigma}(A(T,T)+\frac{1}{3}H)dv_{\Sigma}.$

Since $H=-A(T,T)+H_{\Sigma}$ and $H$ is given boundary data, this may be re-
written in the form

(5.8)
${\mathcal{H}}_{\xi}^{CH}=-2\int_{\Sigma}(-\frac{2}{3}H+H_{\Sigma})dv_{\gamma_{\Sigma}}.$

Both $H_{\Sigma}$ and the induced volume form $dv_{\gamma_{\Sigma}}$ are
determined by the solution $g$, i.e. initial data $(g_{S},K)$.

Here it appears more difficult to identify a suitable Euclidean subtraction
term in general. One possibility is as follows; given
$(\Sigma,\gamma_{\Sigma})$, as in the Brown-York case suppose there is a
unique isometric embedding
$\iota:(\Sigma,g_{\Sigma})\rightarrow(\mathbb{R}^{3},g_{Eucl})$ into
$\mathbb{R}^{3}$. Let $H^{0}_{\Sigma}$ be the mean curvature of the image
$\iota(\Sigma)\subset\mathbb{R}^{3}$. Then choose the boundary condition
$H=H^{0}=H^{0}_{\Sigma}$. As above, there is a corresponding conformal
embedding $({\mathcal{C}},[\gamma])\rightarrow(\mathbb{R}^{1,3},g_{Mink})$ and
subtracting the Euclidean Hamilonian from (5.8) gives

$\widetilde{\mathcal{H}}_{\xi}^{CH}=-2\int_{\Sigma}(-\frac{2}{3}(H-H^{0})+(H_{\Sigma}-H^{0}_{\Sigma}))dv_{\gamma_{\Sigma}}.$

Since by construction $H=H^{0}$, this gives

$\widetilde{\mathcal{H}}_{\xi}^{CH}=-2\int_{\Sigma}(H_{\Sigma}-H^{0}_{\Sigma})dv_{\gamma_{\Sigma}},$

which is exactly the Brown-York energy again (without the $1/3$ term as in
(4.7)). One advantage to this approach is that the boundary data
$([\gamma],H)$ appear to be better behaved than Dirichlet boundary data. On
the other hand, it is not clear if there is any suitable Euclidean subtraction
term if $H$ is chosen arbitrarily. In this case, the definition (5.7) may be
more suitable.

Finally consider the $([\gamma^{t}],H^{*})$ data from [2] with
$H^{*}=2\mathrm{tr}_{{\mathcal{C}}}A-\mathrm{tr}_{\Sigma}A$ as in Proposition
4.4. For simplicity, extend given data $([\gamma^{t}],H^{*})$ on $\Sigma$ to
be independent of $t$ on ${\mathcal{C}}$ and set
$\xi=\partial_{t}=T=\widehat{\Theta}_{{\mathcal{C}}}$, $G=Id$. Then the
boundary condition (4.20) holds and the resulting bare Hamiltonian (4.17) is
given by

${\mathcal{H}}_{\xi}^{AA}=\int_{\Sigma}[-2A_{\bar{g}}(T,T)-\mathrm{tr}_{\Sigma}A_{\bar{g}}]dv_{\bar{\gamma}_{\Sigma}}.$

As in the Brown-York case, consider here for simplicity a Euclidean
subtraction term. Thus, given $([\gamma^{t}],H^{*})$ on $\Sigma$ as above, let
$\iota:(\Sigma,[\gamma_{\Sigma}])\rightarrow\mathbb{R}^{3}$ be a conformal
embedding of $\Sigma$ into $\mathbb{R}^{3}$ with prescribed mean curvature
$H^{*}$. A general theory for such embeddings (or more precisely branched
immersions) when $H^{*}>0$ is developed in [5]. Such an embedding problem is
elliptic and in general is much better behaved than the isometric embedding
problem associated to Dirichlet boundary data. As above, one may extend
$\iota$ trivially in the $t$-direction to an embedding of the cylinder
${\mathcal{C}}_{0}$ into $\mathbb{R}^{1,3}$ with time-like Killing field
$\xi=\partial_{t}$. This gives a flat vacuum solution with the same boundary
data $([\gamma^{t}],H^{*})$ and with $H^{*}=H_{\Sigma}^{0}$ (since $A(T,T)=0$
on the flat cylinder). This gives a Euclidean normalized Hamiltonian

$\widetilde{\mathcal{H}}_{\xi}^{AA}=\int_{\Sigma}[-2A_{\bar{g}}(T,T)-\mathrm{tr}_{\Sigma}A_{\bar{g}}]dv_{\bar{\gamma}_{\Sigma}}+\int_{\Sigma}H^{*}dv_{\iota^{*}g_{Eucl}},$

or equivalently

$\widetilde{\mathcal{H}}_{\xi}^{AA}=\int_{\Sigma}H^{*}(dv_{\bar{\gamma}_{\Sigma}}+dv_{\iota^{*}g_{Eucl}})-2\int_{\Sigma}\mathrm{tr}_{\Sigma}A_{\bar{g}}dv_{\bar{\gamma}_{\Sigma}}.$

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