<EMAIL_ADDRESS>
# A spinorial quasilocal mass

John Lott
Department of Mathematics
University of California, Berkeley
Berkeley, CA 94720-3840
USA

(Date: December 3, 2023)

###### Abstract.

We define a quasilocal energy of a compact manifold-with-boundary, relative to
a background manifold. The construction uses spinors on one manifold and the
pullback of dual spinors from the other manifold. We prove positivity results
for the quasilocal energy, in both the Riemannian and Lorentzian settings.

##  1\. Introduction

The mass of an asymptotically flat initial data set is a quantity computed
from the asymptotic geometry. It has the physical interpretation of measuring
the gravitational energy within the space. The positive mass theorem says that
under an appropriate positivity assumption on the curvature, the mass is
positive [15, 19].

It is an old problem to localize the measurement of gravitational energy to a
compact region, i.e. to give a good notion of a quasilocal mass. Some survey
articles are [16, 17]. In most of this paper we will refer more precisely to a
quasilocal energy rather than a quasilocal mass.

In view of Witten’s proof of the positive energy theorem for spin manifolds
[19], an attractive approach to define a quasilocal energy is to use spinors.
This idea has been considered by various people, notably Dougan-Mason [5] and
Zhang [20, 21]. There are two basic issues with such an approach. First, the
quasilocal energy is always defined relative to some background space. For
example, in the positive energy theorem, the background space is Euclidean
space. It is not clear how to incorporate the background space into the
definition of the quasilocal energy. The second issue is what boundary
conditions to impose on the spinor fields. Witten’s proof used spinor fields
that asymptotically approach a constant spinor with respect to the model flat
space. It is not evident what the analog should be for a compact manifold-
with-boundary.

We show that both of these issues can be handled by the technique of pulling
back (dual) spinors from the background space. This technique has some history
in the mathematical literature and has been applied to prove sharp comparison
results about scalar curvature by Llarull [10, 11], Goette-Semmelmann [8], the
author [14] and Wang-Xie-Yu [18]. One advantage of the spinorial approach to
the quasilocal energy is that the nonnegativity of the energy, under some
curvature conditions, falls out immediately.

In Section 2 we define a quasilocal energy in the Riemannian setting. Given
compact connected $n$-dimensional Riemannian manifolds-with-boundary
$(N,\partial N)$ and $(M,\partial M)$, suppose that $f:(N,\partial
N)\rightarrow(M,\partial M)$ is a spin map that is an isometry on $\partial
N$. We think of $M$ as the background space and $N$ as the manifold whose
quasilocal energy we want to define, relative to $M$. There is a corresponding
Clifford module $E$ on $N$. For simplicity, suppose that $n$ is even
dimensional and that $N$ and $M$ are spin, in which case $E$ is the twisted
spinor bundle $S_{N}\otimes f^{*}S_{M}^{*}$. Let $D^{N}$ be the corresponding
Dirac-type operator on $C^{\infty}(N;E)$.

On $\partial N$, we have the identifications

$E\Big{|}_{\partial N}\cong(S_{N}\otimes(\partial
f)^{*}S_{M}^{*})\Big{|}_{\partial N}\cong(S_{N}\otimes
S_{N}^{*})\Big{|}_{\partial N}\cong\Lambda^{*}T^{*}N\Big{|}_{\partial
N}\cong\Lambda^{*}T^{*}\partial N\oplus\Lambda^{*}T^{*}\partial N,$ (1.1)

where the last isomorphism is a separation into tangential forms and forms
with a normal component. Let $\pi_{+}:\Lambda^{*}T^{*}N\Big{|}_{\partial
N}\rightarrow\Lambda^{*}T^{*}\partial N$ be projection onto the tangential
component. There is a natural “Dirichlet” boundary condition for sections
$\psi$ of $E$ given by $\pi_{+}\left(\psi\Big{|}_{\partial N}\right)=0$; let
${\mathcal{D}}$ denote the ensuing self-adjoint operator. For simplicity, in
this introduction we mostly consider the case when
$\operatorname{Ker}({\mathcal{D}})=0$.

To set up a boundary value problem, there is a canonical choice of section of
$\Lambda^{*}T^{*}\partial N$, namely the constant function $1$. Let $e_{n}$ be
the inward-pointing unit normal on $\partial N$.

######  Definition 1.2.

If $C$ is a nonempty union of connected components of $\partial N$, let
$\psi\in C^{\infty}(N;E)$ be the (unique) solution to $D^{N}\psi=0$ with
$\pi_{+}\left(\psi\Big{|}_{\partial N}\right)=1_{C}$, the characteristic
function of $C$. The quasilocal energy associated to the boundary subset $C$
is

${\mathcal{E}}_{C}=-\int_{\partial
N}\langle\psi,\nabla^{N}_{e_{n}}\psi\rangle\>\operatorname{dvol}_{\partial
N}.$ (1.3)

Here are some basic properties of the quasilocal energy. Let $R_{N}$ and
$R_{M}$ denote the scalar curvatures of $N$ and $M$, respectively. Define
$|\Lambda^{2}df|$, the distortion of $f$ on $2$-forms, to be the pointwise
norm $\sup_{v\wedge w\neq 0}\frac{|df(v)\wedge df(w)|}{|v\wedge w|}$.

We recall that the boundary map $\partial f$ is an isometry on $\partial N$.

######  Theorem 1.4.

1\. If $f$ is an isometric diffeomorphism then ${\mathcal{E}}_{\partial N}=0$.
2\. If $M$ has nonnegative curvature operator and
$R_{N}\geq|\Lambda^{2}df|(f^{*}R_{M})$ then ${\mathcal{E}}_{C}\geq 0$.

Part 1 of Theorem 1.4 is consistent with the interpretation of ${\mathcal{E}}$
as being a relative energy between $N$ and $M$. Part 2 of Theorem 1.4 is an
immediate consequence of a Bochner-type formula. In particular, if $M$ is flat
and $R_{N}\geq 0$ then ${\mathcal{E}}_{C}\geq 0$.

Although (1.3) is an integral over the boundary of $N$, the solution $\psi$ to
the boundary value problem depends a priori on the interior geometry of $N$
and the map $f$. When $M$ is a domain in ${\mathbb{R}}^{n}$, the quasilocal
energy only depends on $f$ through its boundary restriction $\partial f$.

We give a formula for ${\mathcal{E}}_{C}$ indicating that in the weak field
limit, i.e. when $N$ is a perturbation of $M$, the quasilocal energy is
approximately equal to the Brown-York energy [3]. We expect, but do not prove,
that if one takes an appropriate exhaustion of an asymptotically flat manifold
by compact domains then their quasilocal energies will approach a
normalization constant times the ADM mass. We compute ${\mathcal{E}}_{\partial
N}$ when $N$ is conformally related to $M$ and find that it is exactly the
Brown-York energy. We find a similar statement when $N$ is rotationally
symmetric.

In Section 3 we consider a Lorentzian quasilocal energy. That is, we have
Lorentzian $(n+1)$-dimensional manifolds $\overline{N}$ and $\overline{M}$,
along with compact connected spatial hypersurfaces-with-boundary
$N\subset\overline{N}$ and $M\subset\overline{M}$. Again, we use a comparison
map $f:(N,\partial N)\rightarrow(M,\partial M)$ that is an isometry on
$\partial N$ and look at the Dirac-type operator $D^{N}$ acting on sections of
the twisted spinor bundle $S_{N}\otimes f^{*}S_{M}^{*}$. There is a new
feature that, in general, $D^{N}$ need not be formally self-adjoint on the
interior of $N$. We first look at the case when $M$ is a totally geodesic
hypersurface in $\overline{M}$. Then there is no problem with self-adjointness
and results from Section 2 extend. The curvature condition in part 2 of
Theorem 1.4 gets replaced by
$2\left(T_{00}-\sqrt{-\sum_{\alpha=1}^{n}T_{0\alpha}T^{0\alpha}}\right)\geq|\Lambda^{2}df|(f^{*}R_{M})$,
where
$T_{AB}={R}^{\overline{N}}_{AB}-\frac{1}{2}{R}_{\overline{N}}g^{\overline{N}}_{AB}$
is the Einstein tensor of $\overline{N}$.

We next look at the case when $\overline{M}$ is the flat Minkowski space
${\mathbb{R}}^{n,1}$, but the hypersurface $M$ need not be totally geodesic.
It turns out that $D^{N}$ is formally self-adjoint on the interior of $N$ if
one uses an appropriate weighted $L^{2}$-space. Regarding boundary conditions,
there are two natural choices. The first one does not give a self-adjoint
boundary value problem but nevertheless one can use it to define a quasilocal
energy ${\mathcal{E}}_{C}$. The second one does give a self-adjoint boundary
value problem and gives rise to a quasilocal energy ${\mathcal{E}}^{sa}_{C}$.
They have the following properties.

######  Theorem 1.5.

1\. If $f$ is an isometric diffeomorphism then ${\mathcal{E}}_{\partial N}=0$.
2\. If $\overline{N}$ satisfies the dominant energy condition
$T_{00}\geq\sqrt{-\sum_{\alpha=1}^{n}T_{0\alpha}T^{0\alpha}}$ then
${\mathcal{E}}_{C}\geq 0$ and ${\mathcal{E}}^{sa}_{C}\geq 0$.

Comparing ${\mathcal{E}}_{C}$ and ${\mathcal{E}}^{sa}_{C}$, the first one
${\mathcal{E}}_{C}$ has the advantage, from part 1 of Theorem 1.5, of
vanishing when $C=\partial N$ and $f$ is an isometric diffeomorphism. On the
other hand, ${\mathcal{E}}^{sa}_{C}$ has the advantage of coming from a self-
adjoint boundary value problem.

Finally, we extend the results to the case when $\overline{M}$ is a more
general Lorentzian manifold than Minkowski space. It turns out that it is
enough for $\overline{M}$ to be a product spacetime ${\mathbb{R}}\times X$ for
some Riemannian manifold $X$. This can be compared with Chen-Wang-Wang-Yau’s
construction of a quasilocal energy when the background space is in a static
spacetime [4].

Regarding earlier work about a spinorial approach to quasilocal energy, Dougan
and Mason used the complex structure on an oriented surface to make an
interesting choice of boundary condition [5]. Their approach is clearly
restricted to $n=3$. Zhang used the pure Dirac operator to define a quasilocal
energy when the background space $M$ is a domain in
${\mathbb{R}}^{3}\subset{\mathbb{R}}^{3,1}$ [20]. His boundary condition came
from using constant spinors on ${\mathbb{R}}^{3,1}$; compare with Proposition
3.20 of the present paper. He imposed geometric restrictions to ensure that
the Dirac operator is invertible; compare with Proposition 3.19 of the present
paper. In [21] he also allowed $M$ to be a domain in a hyperbolic submanifold
of ${\mathbb{R}}^{3,1}$.

I thank the Fields Institute for its hospitality while part of this research
was performed, and the referee for helpful comments.

##  2\. Riemannian case

This section is about the quasilocal energy in the Riemannian case. We begin
with even dimensional spaces. Section 2.1 has background information. Section
2.2 has the definition of the quasilocal energy. In Section 2.3 we prove its
basic properties. Section 2.4 computes the quasilocal energy for conformally
related manifolds. Section 2.5 specializes to when the background space $M$ is
a domain in ${\mathbb{R}}^{n}$ and also treats rotationally symmetric
manifolds $N$. Finally, Section 2.6 covers the odd dimensional case.

###  2.1. Background

Let $R$ denote scalar curvature and let $H$ denote mean curvature. With our
convention, $\partial B^{n}$ has $H=n-1$.

Let $N$ and $M$ be compact connected $n$-dimensional Riemannian manifolds with
nonempty boundary. Let $f\>:\>(N,\partial N)\rightarrow(M,\partial M)$ be a
smooth spin map, i.e. $TN\oplus f^{*}TM$ admits a spin structure.
Equivalently, $f^{*}w_{2}(M)=w_{2}(N)$. Let $\partial f\>:\>\partial
N\rightarrow\partial M$ denote the restriction to the boundary. We assume that
for each component $Z$ of $\partial N$, the restriction $\partial
f\Big{|}_{Z}$ is an isometric diffeomorphism from $Z$ to $f(Z)$.

For simplicity, we assume that $N$ and $M$ are spin and that $\partial f$ is a
spin diffeomorphism; the general case is similar. We assume first that $n$ is
even. Then a spinor representation $S^{\pm}$ of $\operatorname{Spin}(n)$ has
complex dimension $2^{\frac{n}{2}-1}$, while the Clifford algebra has a
faithful representation space $S=S^{+}\oplus S^{-}$ of complex dimension
$2^{\frac{n}{2}}$. We let $S^{*}$ denote the complex vector space of complex-
linear functionals on $S$, i.e. no complex conjugation involved. Let $S_{N}$
denote the spinor bundle on $N$, and similarly for $S_{M}$. Put
$E=S_{N}\otimes f^{*}S^{*}_{M}$, a Clifford module on $N$. (This Clifford
module exists in the general case.) We take the inner product
$\langle\cdot,\cdot\rangle$ on $E$ to be ${\mathbb{C}}$-linear in the second
slot and ${\mathbb{C}}$-antilinear in the first slot. As
$\operatorname{End}(S_{N})\cong\Lambda^{*}(T^{*}N)$, we can identify
$E\Big{|}_{\partial N}$ with $\Lambda^{*}(T^{*}N)\Big{|}_{\partial N}$. (We
require this property in the general case, which amounts to a choice of spin
structure on $E\Big{|}_{\partial N}$.)

Let $\\{e_{\alpha}\\}_{\alpha=1}^{n}$ be a local oriented orthonormal framing
on $N$, with dual basis $\\{\tau^{\alpha}\\}_{\alpha=1}^{n}$. Let
$\omega^{\alpha}_{\>\>\beta\gamma}$ be the connection $1$-forms with respect
to $\\{e_{\alpha}\\}_{\alpha=1}^{n}$. Let $\\{\mathaccent 866{e}_{\mathaccent
866{\alpha}}\\}_{\mathaccent 866{\alpha}=1}^{n}$ of $M$ be a local oriented
orthonormal framing of $M$. Let $\mathaccent 866{\omega}^{\mathaccent
866{\alpha}}_{\>\>\mathaccent 866{\beta}\gamma}$ be the pullbacks under $f$ of
the connection $1$-forms with respect to $\\{\mathaccent 866{e}_{\mathaccent
866{\alpha}}\\}_{\mathaccent 866{\alpha}=1}^{n}$.

To make things more symmetric, it will be convenient to do local calculations
on $S_{N}\otimes f^{*}S_{M}$ rather than $S_{N}\otimes f^{*}S_{M}^{*}$. In the
Riemannian setting, $S_{M}^{*}$ is unitarily equivalent to $S_{M}$ and so we
don’t lose anything this way. Let $\\{\gamma^{\alpha}\\}_{\alpha=1}^{n}$ be
Clifford multiplication by $\\{e_{\alpha}\\}_{\alpha=1}^{n}$, satisfying
$\gamma^{\alpha}\gamma^{\beta}+\gamma^{\beta}\gamma^{\alpha}=2\delta^{\alpha\beta}$.
Let $\\{\mathaccent 866{\gamma}^{\mathaccent 866{\alpha}}\\}_{\mathaccent
866{\alpha}=1}^{n}$ be Clifford multiplication by $\\{\mathaccent
866{e}_{\mathaccent 866{\alpha}}\\}_{\mathaccent 866{\alpha}=1}^{n}$. They
both have an odd grading, in the sense that in calculations we take
$\gamma^{\alpha}$ to anticommute with $\mathaccent 866{\gamma}^{\mathaccent
866{\alpha}}$.

The covariant derivative on $E$ has the local form

$\nabla^{N}_{\sigma}=e_{\sigma}+\frac{1}{8}\omega_{\alpha\beta\sigma}[\gamma^{\alpha},\gamma^{\beta}]+\frac{1}{8}\mathaccent
866{\omega}_{\mathaccent 866{\alpha}\mathaccent 866{\beta}\sigma}[\mathaccent
866{\gamma}^{\mathaccent 866{\alpha}},\mathaccent 866{\gamma}^{\mathaccent
866{\beta}}].$ (2.1)

The Dirac operator on $C^{\infty}(N;E)$ is
$D^{N}=-\sqrt{-1}\sum_{\sigma=1}^{n}\gamma^{\sigma}\nabla^{N}_{\sigma}$.

We will take the orthonormal frame $\\{e_{\alpha}\\}$ at a point in $\partial
N$ so that $e_{n}$ is the inward-pointing unit normal vector there. Let
$\operatorname{dvol}_{N}$ denote the Riemannian density on $N$, and similarly
for $\operatorname{dvol}_{\partial N}$. Given $\psi_{1},\psi_{2}\in
C^{\infty}(N;E)$, we have

$\int_{N}\langle
D^{N}\psi_{1},\psi_{2}\rangle\operatorname{dvol}_{N}-\int_{N}\langle\psi_{1},D^{N}\psi_{2}\rangle\operatorname{dvol}_{N}=-\sqrt{-1}\int_{\partial
N}\langle\psi_{1},\gamma^{n}\psi_{2}\rangle\operatorname{dvol}_{\partial N}.$
(2.2)

The Lichnerowicz formula implies

$(D^{N})^{2}=(\nabla^{N})^{*}\nabla^{N}+\frac{R_{N}}{4}-\frac{1}{4}[\gamma^{\sigma},\gamma^{\tau}]\left(\frac{1}{8}\mathaccent
866{R}_{\mathaccent 866{\alpha}\mathaccent 866{\beta}\sigma\tau}[\mathaccent
866{\gamma}^{\mathaccent 866{\alpha}},\mathaccent 866{\gamma}^{\mathaccent
866{\beta}}]\right).$ (2.3)

We now extend some computations in [13, Proof of Lemma 4.1]. Suppose that
$D^{N}\psi=0$. In what follows, summations over Greek letters will go from $1$
to $n$ and summations over Latin letters will go from $1$ to $n-1$. Equation
(2.3) implies that

$\displaystyle 0=$
$\displaystyle\int_{N}|\nabla^{N}\psi|^{2}\>\operatorname{dvol}_{N}+\int_{\partial
N}\langle\psi,\nabla^{N}_{e_{n}}\psi\rangle\operatorname{dvol}_{\partial
N}+\frac{1}{4}\int_{N}R_{N}|\psi|^{2}\>\operatorname{dvol}_{N}-$ (2.4)
$\displaystyle\frac{1}{32}\int_{N}\mathaccent 866{R}_{\mathaccent
866{\alpha}\mathaccent
866{\beta}\sigma\tau}\langle\psi,[\gamma^{\sigma},\gamma^{\tau}][\mathaccent
866{\gamma}^{\mathaccent 866{\alpha}},\mathaccent 866{\gamma}^{\mathaccent
866{\beta}}]\psi\rangle.$

Now $D^{N}\psi=0$ implies that on $\partial N$, we have

$\displaystyle\nabla^{N}_{e_{n}}\psi=$
$\displaystyle-\gamma^{n}\sum_{i=1}^{n-1}\gamma^{i}\nabla^{N}_{i}\psi$ (2.5)
$\displaystyle=$
$\displaystyle-\gamma^{n}\sum_{i=1}^{n-1}\gamma^{i}\left(\nabla^{\partial
N}_{i}\psi+\frac{1}{2}\omega_{nji}\gamma^{n}\gamma^{j}\psi+\frac{1}{2}\mathaccent
866{\omega}_{\mathaccent 866{n}\mathaccent 866{j}i}\mathaccent
866{\gamma}^{\mathaccent 866{n}}\mathaccent 866{\gamma}^{\mathaccent
866{j}}\psi\right)$ $\displaystyle=$ $\displaystyle D^{\partial
N}\psi+\frac{H_{\partial
N}}{2}\psi\>-\>\frac{1}{2}\gamma^{n}\gamma^{i}\mathaccent
866{\gamma}^{\mathaccent 866{n}}\mathaccent 866{\gamma}^{\mathaccent
866{j}}\mathaccent 866{A}_{\mathaccent 866{j}i}\psi,$

where

$D^{\partial N}=-\gamma^{n}\sum_{i=1}^{n-1}\gamma^{i}\nabla^{\partial N}_{i}$
(2.6)

is the Dirac operator on $\partial N$ coupled to $(\partial f)^{*}S^{*}_{M}$,
$\mathaccent 866{A}$ is the second fundamental form of $M$ and $\mathaccent
866{A}_{\mathaccent 866{j}i}=\mathaccent 866{A}(\mathaccent
866{e}_{\mathaccent 866{j}},(\partial f)_{*}(e_{i}))$.

######  Remark 2.7.

The factor of $\frac{1}{2}$ in (2.5) corrects the $\frac{1}{4}$ that appears
in [13, 14].

From [8, Section 1.1], if $M$ has nonnegative curvature operator then

$\frac{1}{32}\mathaccent 866{R}_{\mathaccent 866{\alpha}\mathaccent
866{\beta}\sigma\tau}[\gamma^{\sigma},\gamma^{\tau}][\mathaccent
866{\gamma}^{\mathaccent 866{\alpha}},\mathaccent 866{\gamma}^{\mathaccent
866{\beta}}]\leq\frac{1}{4}|\Lambda^{2}df|(f^{*}R_{M})\operatorname{Id}_{E}.$
(2.8)

(The paper [8] assumes that $|\Lambda^{2}df|\leq 1$ but their argument shows
the result stated in (2.8); c.f. [9].)

######  Proposition 2.9.

If $M$ has nonnegative curvature operator and
$R_{N}\geq|\Lambda^{2}(df)|(f^{*}R_{M})$ then for any $\psi\in
C^{\infty}(N;E)$, we have $-\int_{\partial
N}\langle\psi,\nabla^{N}_{e_{n}}\psi\rangle\operatorname{dvol}_{\partial
N}\geq 0$. If $M$ has nonnegative curvature operator,
$R_{N}\geq|\Lambda^{2}(df)|(f^{*}R_{M})$, $\psi$ is nonzero and
$-\int_{\partial
N}\langle\psi,\nabla^{N}_{e_{n}}\psi\rangle\operatorname{dvol}_{\partial N}=0$
then $R_{N}=|\Lambda^{2}(df)|(f^{*}R_{M})$.

###### Proof.

This follows from (2.4) and (2.8). ∎

###  2.2. Definition of the quasilocal energy

Let $\epsilon$ be the ${\mathbb{Z}}_{2}$-grading operator on $S_{N}$. Put
$\gamma^{0}=i\epsilon$. Let $T$ be the involution on $S_{N}\Big{|}_{\partial
N}$ given by

$T\eta=\gamma^{0}\gamma^{n}\eta.$ (2.10)

Identifying $f^{*}S^{*}_{M}\Big{|}_{\partial N}$ with
$S^{*}_{N}\Big{|}_{\partial N}$, there is an induced involution $T$ on

$\Lambda^{*}T^{*}N\Big{|}_{\partial
N}\cong\operatorname{End}(S_{N})\Big{|}_{\partial N}\cong(S_{N}\otimes
S_{N}^{*})\Big{|}_{\partial N}\cong S_{N}\Big{|}_{\partial N}\otimes
f^{*}S^{*}_{M}\Big{|}_{\partial M}.$ (2.11)

We recall that an element of $\Lambda^{*}T^{*}N$ acts on $S_{N}$ by Clifford
multiplication, realizing the isomorphism
$\Lambda^{*}T^{*}N\cong\operatorname{End}(S_{N})$. Then for
$\omega\in\Lambda^{*}T^{*}N\Big{|}_{\partial N}$ and $\eta\in
S_{N}\Big{|}_{\partial N}$, we have $T(\omega\cdot\eta)=T(\omega)\cdot
T(\eta)$. In terms of the orthonormal frame, $T$ acts on
$T^{*}N\Big{|}_{\partial N}$ as $\gamma^{0}\gamma^{n}\mathaccent
866{\gamma}^{\mathaccent 866{0}}\mathaccent 866{\gamma}^{\mathaccent 866{n}}$.

Let $\tau^{n}$ be the dual covector to $e_{n}$.

######  Lemma 2.12.

Given $\omega\in\Lambda^{*}T^{*}N\Big{|}_{\partial N}$, write
$\omega=\omega_{+}+\tau^{n}\wedge\omega_{-}$ for
$\omega_{\pm}\in\Lambda^{*}T^{*}\partial N$. Then
$T\omega=\omega_{+}-\tau^{n}\wedge\omega_{-}$.

###### Proof.

Given an increasing multi-index $I$ with entries between $1$ and $n-1$, and
$\psi\in S_{N}\Big{|}_{\partial N}$, we have

$T(\gamma^{I}\psi)=(\gamma^{0}\gamma^{n})\gamma^{I}\psi=\gamma^{I}(\gamma^{0}\gamma^{n})\psi=\gamma^{I}T(\psi)$
(2.13)

and

$T(\gamma^{n}\gamma^{I}\psi)=(\gamma^{0}\gamma^{n})\gamma^{n}\gamma^{I}\psi=-\gamma^{n}\gamma^{I}(\gamma^{0}\gamma^{n})\psi=-\gamma^{n}\gamma^{I}T(\psi)$
(2.14)

Using the isomorphism $\Lambda^{*}T^{*}N\Big{|}_{\partial
N}\cong\operatorname{End}(S_{N})\Big{|}_{\partial N}$, the lemma follows. ∎

Let $\pi_{\pm}$ denote orthogonal projection onto the $\pm 1$-eigenspace of
$T$, acting on $\Lambda^{*}T^{*}N\Big{|}_{\partial N}$, so
$\operatorname{Im}(\pi_{+})$ is isomorphic to $\Lambda^{*}T^{*}\partial N$.

Consider the operator $D^{N}$ acting on elements $\psi\in C^{\infty}(N;E)$.
The boundary condition $\pi_{+}\left(\psi\Big{|}_{\partial N}\right)=0$, which
is an analog of Dirichlet boundary conditions, defines an elliptic boundary
condition for $D^{N}$ [2, Section 7.5]. One can check that with the given
boundary condition, $D^{N}$ is formally self-adjoint. Let ${\mathcal{D}}$
denote the ensuing self-adjoint operator, densely defined on $L^{2}(N;E)$ [2,
Chapter 7]. It has compact resolvent. The next proposition gives a sufficient
condition for ${\mathcal{D}}$ to be invertible.

######  Proposition 2.15.

Suppose that

  * •

$M$ has nonnegative curvature operator,

  * •

$\partial M$ has nonnegative second fundamental form,

  * •

$R_{N}\geq|\Lambda^{2}df|(f^{*}R_{M})$,

  * •

$H_{\partial N}\geq(\partial f)^{*}H_{\partial M}$, and

  * •

$R_{N}>|\Lambda^{2}df|(f^{*}R_{M})$ somewhere or $H_{\partial N}>(\partial
f)^{*}H_{\partial M}$ somewhere.

Then $\operatorname{Ker}({\mathcal{D}})=0$.

###### Proof.

Suppose that $\psi\in\operatorname{Ker}({\mathcal{D}})$. From (2.4) and (2.5),

$\displaystyle 0=$
$\displaystyle\int_{N}|\nabla^{N}\psi|^{2}\>\operatorname{dvol}_{N}+\frac{1}{4}\int_{N}R_{N}|\psi|^{2}\>\operatorname{dvol}_{N}-\frac{1}{32}\int_{N}\mathaccent
866{R}_{\mathaccent 866{\alpha}\mathaccent
866{\beta}\sigma\tau}\langle\psi,[\gamma^{\sigma},\gamma^{\tau}][\mathaccent
866{\gamma}^{\mathaccent 866{\alpha}},\mathaccent 866{\gamma}^{\mathaccent
866{\beta}}]\psi\rangle+$ (2.16) $\displaystyle\int_{\partial
N}\langle\psi,D^{\partial N}\psi\rangle\>\operatorname{dvol}_{\partial
N}+\frac{1}{2}\int_{\partial N}H_{\partial
N}|\psi|^{2}\>\operatorname{dvol}_{\partial N}-\frac{1}{2}\int_{\partial
N}\langle\psi,\gamma^{n}\gamma^{i}\mathaccent 866{\gamma}^{n}\mathaccent
866{\gamma}^{\mathaccent 866{j}}\mathaccent 866{A}_{\mathaccent
866{j}i}\psi\rangle\>\operatorname{dvol}_{\partial N}.$

As $T$ anticommutes with $D^{\partial N}$, we have

$\langle\psi,D^{\partial N}\psi\rangle=-\langle\psi,D^{\partial
N}T\psi\rangle=\langle\psi,TD^{\partial N}\psi\rangle=\langle
T\psi,D^{\partial N}\psi\rangle=-\langle\psi,D^{\partial N}\psi\rangle,$
(2.17)

so $\langle\psi,D^{\partial N}\psi\rangle=0$. From [14, Lemma 2.1],
$\langle\psi,\gamma^{n}\gamma^{i}\mathaccent 866{\gamma}^{n}\mathaccent
866{\gamma}^{\mathaccent 866{j}}\mathaccent 866{A}_{\mathaccent
866{j}i}\psi\rangle\leq(\partial f)^{*}H_{\partial M}|\psi|^{2}$. Then (2.8)
and (2.16) imply that $\nabla^{N}\psi=0$. Since $\nabla^{N}$ is unitary, it
follows that $|\psi|^{2}$ is constant on the connected manifold $N$. Then
(2.16) implies that $|\psi|^{2}=0$. ∎

######  Remark 2.18.

If the Euler characteristic of $M$ is nonzero and the degree of $f$ is nonzero
then $\operatorname{Ker}({\mathcal{D}})\neq 0$ [14, Section 2.2].

We now discuss the boundary value problem. Given $\sigma\in\Omega^{*}(\partial
N)$ with $\pi_{+}(\sigma)=\sigma$, we wish to find some $\psi\in
C^{\infty}(N;E)$ so that $D^{N}\psi=0$ and $\pi_{+}\left(\psi\Big{|}_{\partial
N}\right)=\sigma$. Suppose that we can do this. If
$\eta\in\operatorname{Ker}({\mathcal{D}})$ then from (2.2), we see that
$\int_{\partial
N}\langle\sigma,\gamma^{n}\eta\rangle\>\operatorname{dvol}_{\partial N}=0$.
(Note that $\eta\Big{|}_{\partial N}\in\operatorname{Im}(\pi_{-})$, so
$\gamma^{n}\eta\Big{|}_{\partial N}\in\operatorname{Im}(\pi_{+})$.)

Conversely, given $\sigma\in\Omega^{*}(\partial N)$ with
$\pi_{+}(\sigma)=\sigma$, if $\int_{\partial
N}\langle\sigma,\gamma^{n}\eta\rangle\>\operatorname{dvol}_{\partial N}=0$ for
all $\eta\in\operatorname{Ker}({\mathcal{D}})$ then there is some $\psi\in
C^{\infty}(N;E)$ so that $D^{N}\psi=0$ and $\pi_{+}\left(\psi\Big{|}_{\partial
N}\right)=\sigma$ [7, Lemma 14]. The section $\psi$ is unique up to addition
by elements of $\operatorname{Ker}({\mathcal{D}})$.

Let $C$ be a nonempty union of connected components of $\partial N$. To define
the quasilocal energy, we would like to take $\sigma$ to be the characteristic
function $1_{C}$. To be sure that $\psi$ exists, we may have to slightly
modify this choice. Let
$P_{\gamma^{n}\operatorname{Ker}({\mathcal{D}})\Big{|}_{\partial N}}$ be
orthogonal projection from $\Omega^{*}(\partial N)$ to
$\gamma^{n}\operatorname{Ker}({\mathcal{D}})\Big{|}_{\partial N}$. We put
$\overline{1}_{C}=1_{C}-P_{\gamma^{n}\operatorname{Ker}({\mathcal{D}})\Big{|}_{\partial
N}}1_{C}$. We then take $\sigma=\overline{1}_{C}$ and construct $\psi$
accordingly.

######  Definition 2.19.

Put

${\mathcal{C}}_{C}=\left\\{\psi\in
C^{\infty}(N;E)\>:\>D^{N}\psi=0,\pi_{+}\left(\psi\Big{|}_{\partial
N}\right)=\overline{1}_{C}\right\\}.$ (2.20)

The quasilocal energy associated to the boundary subset $C$ is

${\mathcal{E}}_{C}=\inf_{\psi\in{\mathcal{C}}_{C}}\left(-\int_{\partial
N}\langle\psi,\nabla^{N}_{e_{n}}\psi\rangle\>\operatorname{dvol}_{\partial
N}\right).$ (2.21)

Note that ${\mathcal{E}}_{C}$ could be $-\infty$.

######  Lemma 2.22.

If $\operatorname{Ker}({\mathcal{D}})=0$ then ${\mathcal{E}}_{C}>-\infty$. In
general, if ${\mathcal{E}}_{C}>-\infty$ then it is realized by some
$\psi_{min}\in{\mathcal{C}}_{C}$.

###### Proof.

We know that ${\mathcal{C}}_{C}$ is a finite dimensional affine space modelled
on $\operatorname{Ker}({\mathcal{D}})$. If
$\operatorname{Ker}({\mathcal{D}})=0$ then ${\mathcal{C}}_{C}$ has a single
element and ${\mathcal{E}}_{C}>-\infty$. If
$\operatorname{Ker}({\mathcal{D}})\neq 0$ then as $-\int_{\partial
N}\langle\psi,\nabla^{N}_{e_{n}}\psi\rangle\>\operatorname{dvol}_{\partial N}$
is a quadratic function on ${\mathcal{C}}_{C}$, the lemma follows. ∎

In some statements that follow we may implicitly assume that
${\mathcal{E}}_{C}>-\infty$. We will see relevant examples where this is the
case in Proposition 2.26, Proposition 2.27, Lemma 2.43 and Proposition 2.55.

###  2.3. Properties of the quasilocal energy

######  Proposition 2.23.

If $f:N\rightarrow M$ is an isometric diffeomorphism then
${\mathcal{E}}_{\partial N}=0$.

###### Proof.

If $f$ is an isometric diffeomorphism then we can take
$E\cong\Lambda^{*}T^{*}N$ and identify $D^{N}$ with $d+d^{*}$. Letting $*$
denote the Hodge duality operator, if $\psi\in C^{\infty}(N;E)$ then

$\displaystyle\int_{N}\langle\left(D^{N}\right)^{2}\psi,\psi\rangle\>\operatorname{dvol}_{N}$
$\displaystyle=\int_{N}\langle(dd^{*}+d^{*}d)\psi,\psi\rangle\>\operatorname{dvol}_{N}$
(2.24)
$\displaystyle=\int_{N}\left(dd^{*}\psi\wedge*\psi+d^{*}d\psi\wedge*\psi\right)$
$\displaystyle=\int_{N}\left(dd^{*}\psi\wedge*\psi+*d^{*}d\psi\wedge*(*\psi)\right)$
$\displaystyle=\int_{N}\left(dd^{*}\psi\wedge*\psi+dd^{*}*\psi\wedge*(*\psi)\right)$
$\displaystyle=\int_{N}\left(|d^{*}\psi|^{2}+|d\psi|^{2}\right)\>\operatorname{dvol}_{N}+\int_{\partial
N}\left(d^{*}\psi\wedge*\psi+d^{*}*\psi\wedge*(*\psi)\right).$

Comparing with equation (2.4) shows

$-\int_{\partial
N}\langle\psi,\nabla^{N}_{e_{n}}\psi\rangle\>\operatorname{dvol}_{\partial
N}=-\int_{\partial
N}\left(d^{*}\psi\wedge*\psi+d^{*}*\psi\wedge*(*\psi)\right),$ (2.25)

There is a solution $\psi$ of $D^{N}\psi=0$ given by $\psi=1$, with
$\pi_{+}(1)=1$. It follows that $1$ is orthogonal to
$\gamma^{n}\operatorname{Ker}({\mathcal{D}})\Big{|}_{\partial N}$, so
$\overline{1}_{\partial N}=1$.

If $i\>:\>\partial N\rightarrow N$ is the boundary inclusion then an element
of $\operatorname{Ker}({\mathcal{D}})$ is a differential form
$\rho\in\Omega^{*}(N)$ satisfying $(d+d^{*})\rho=0$ and the relative
(Dirichlet) boundary condition $i^{*}\rho=0$. Applying $i^{*}$, it follows
that $i^{*}(d^{*}\rho)=0$. Squaring $d+d^{*}$, it follows that
$(dd^{*}+d^{*}d)\rho=0$. Hence $\rho$ is a harmonic form $\alpha$ on $N$
satisfying relative (Dirichlet) boundary conditions. In particular,
$d\rho=d^{*}\rho=0$.

Hence elements $\psi$ of ${\mathcal{C}}_{\partial N}$ are of the form $1+\rho$
for such $\rho$. As $d\psi=d^{*}\psi=0$, equation (2.25) gives
$-\int_{\partial
N}\langle\psi,\nabla^{N}_{e_{n}}\psi\rangle\>\operatorname{dvol}_{\partial
N}=0$, which proves the lemma. ∎

We now give the basic positivity property of the quasilocal energy in the
physically relevant case.

######  Proposition 2.26.

Suppose that $M$ has nonnegative curvature operator. If
$R_{N}\geq|\Lambda^{2}df|(f^{*}R_{M})$ then ${\mathcal{E}}_{C}\geq 0$. If
$R_{N}\geq|\Lambda^{2}df|(f^{*}R_{M})$ and ${\mathcal{E}}_{C}=0$ then
$R_{N}=|\Lambda^{2}df|(f^{*}R_{M})$.

###### Proof.

This follows from (2.4) and (2.8). ∎

######  Proposition 2.27.

If $M$ is flat and $R_{N}\geq 0$ then ${\mathcal{E}}_{C}\geq 0$. If $M$ is
flat, $R_{N}\geq 0$ and ${\mathcal{E}}_{C}=0$ then $N$ is Ricci flat.

###### Proof.

The first statement follows from Proposition 2.26. The second statement
follows as in [14, Proof of Proposition 2.3]. ∎

We now write ${\mathcal{E}}_{C}$ more explicitly as a boundary integral. From
(2.5), we have

$\displaystyle{\mathcal{E}}_{C}=$ $\displaystyle-\int_{\partial
N}\langle\psi_{min},D^{\partial
N}\psi_{min}\rangle\>\operatorname{dvol}_{\partial
N}-\frac{1}{2}\int_{\partial N}H_{\partial
N}|\psi_{min}|^{2}\>\operatorname{dvol}_{\partial N}+$ (2.28)
$\displaystyle\frac{1}{2}\int_{\partial
N}\langle\psi_{min},\gamma^{n}\gamma^{i}\mathaccent 866{\gamma}^{n}\mathaccent
866{\gamma}^{\mathaccent 866{j}}\mathaccent 866{A}_{\mathaccent
866{j}i}\psi_{min}\rangle\>\operatorname{dvol}_{\partial N}.$

We write $\psi_{min}\Big{|}_{\partial N}=\overline{1}_{C}+\tau^{n}\wedge\phi$
for some $\phi\in\Omega^{*}(\partial N)$. Let $E^{i}$ denote exterior
multiplication by $\tau^{i}$ and $I^{i}$ denote interior multiplication by
$e_{i}$, when acting on $\Lambda^{*}T^{*}\partial
N+\tau^{n}\wedge\Lambda^{*}T^{*}\partial N$.

######  Proposition 2.29.

If $\overline{1}_{C}=1_{C}$ then

$\displaystyle{\mathcal{E}}_{C}=$
$\displaystyle-\frac{1}{2}\int_{C}H_{\partial
N}\>\operatorname{dvol}_{\partial N}+\frac{1}{2}\int_{C}(\partial
f)^{*}H_{\partial M}\>\operatorname{dvol}_{\partial N}-$ (2.30)
$\displaystyle\frac{1}{2}\int_{\partial N}H_{\partial
N}|\phi|^{2}\>\operatorname{dvol}_{\partial N}+\int_{\partial
N}\frac{1}{2}\mathaccent
866{A}_{ji}\langle\phi,(E^{i}-I^{i})(E^{j}+I^{j})\phi\rangle\>\operatorname{dvol}_{\partial
N}.$

###### Proof.

On $\partial N$, we can represent $\gamma^{i}$ by $\sqrt{-1}(E^{i}-I^{i})$,
$\gamma^{n}$ by $\sqrt{-1}(E^{n}-I^{n})$, $\mathaccent 866{\gamma}^{i}$ by
$E^{i}+I^{i}$ and $\mathaccent 866{\gamma}^{n}$ by $E^{n}+I^{n}$.

######  Lemma 2.31.

We have

$\int_{\partial N}\langle\psi_{min},D^{\partial
N}\psi_{min}\rangle\>\operatorname{dvol}_{\partial N}=0.$ (2.32)

###### Proof.

As $D^{\partial N}$ anticommutes with $T$, it follows that

$\int_{\partial N}\langle 1_{C},D^{\partial
N}1_{C}\rangle\>\operatorname{dvol}_{\partial N}=\int_{\partial
N}\langle\tau^{n}\wedge\phi,D^{\partial
N}(\tau^{n}\wedge\phi)\rangle\>\operatorname{dvol}_{\partial N}=0.$ (2.33)

Next, since $\nabla^{\partial N}$ restricts to the Riemannian connection on
$\Lambda^{*}T^{*}\partial N$, we know that $\nabla^{\partial N}1_{C}=0$, so

$\int_{\partial N}\langle\tau^{n}\wedge\phi,D^{\partial
N}1_{C}\rangle\>\operatorname{dvol}_{\partial N}=0$ (2.34)

and

$\int_{\partial N}\langle 1_{C},D^{\partial
N}\tau^{n}\wedge\phi\rangle\>\operatorname{dvol}_{\partial N}=\int_{\partial
N}\langle D^{\partial
N}1_{C},\tau^{n}\wedge\phi\rangle\>\operatorname{dvol}_{\partial N}=0$ (2.35)

The lemma follows. ∎

Also

$\int_{\partial N}H_{\partial
N}|\psi_{min}|^{2}\>\operatorname{dvol}_{\partial N}=\int_{C}H_{\partial
N}\>\operatorname{dvol}_{\partial N}+\int_{\partial N}H_{\partial
N}|\phi|^{2}\>\operatorname{dvol}_{\partial N}.$ (2.36)

Next, one can check that

$\langle\psi_{min},\gamma^{n}\gamma^{i}\mathaccent 866{\gamma}^{n}\mathaccent
866{\gamma}^{j}\psi_{min}\rangle=\delta^{ij}1_{C}+\langle\phi,(E^{i}-I^{i})(E^{j}+I^{j})\phi\rangle.$
(2.37)

Hence

$\langle\psi_{min},\gamma^{n}\gamma^{i}\mathaccent 866{\gamma}^{n}\mathaccent
866{\gamma}^{j}\mathaccent 866{A}_{ji}\psi_{min}\rangle=(\partial
f)^{*}H_{\partial M}1_{C}+\mathaccent
866{A}_{ji}\langle\phi,(E^{i}-I^{i})(E^{j}+I^{j})\phi\rangle.$ (2.38)

This proves the proposition. ∎

The first two terms on the right-hand side of (2.30) give the Brown-York
energy. (We have taken a particular normalization of the Brown-York energy.)
In the case when $N$ is a small perturbation of $M$, the function $\phi$ will
also be small. Hence Proposition 2.29 shows that in this weak field limit, to
leading order the quasilocal energy ${\mathcal{E}}_{C}$ is the Brown-York
energy.

######  Remark 2.39.

With reference to Proposition 2.29, in the asymptotically flat case the
geometry on most of a large domain $N$ will be a small perturbation of the
Euclidean metric. Furthermore, it’s known that the Brown-York energy of large
spheres approaches the ADM mass ([6] and references therein). This makes it
plausible that under an appropriate exhaustion of an asymptotically flat
manifold, the quasilocal mass will approach the ADM mass. That this is true in
the rotationally symmetric case follows from Proposition 2.55.

###  2.4. Conformal deformations

In this section we use the conformal covariance of the Dirac operator to say
something about the quasilocal energy; c.f. [20, Section 4].

Put $(N^{\prime},g^{\prime})=(N,e^{2\phi}g)$, where $\phi\Big{|}_{\partial N}$
is locally constant and $\phi\Big{|}_{C}=0$. Let $f:N\rightarrow M$ be as
before. Suppose that $\operatorname{Ker}({\mathcal{D}})=0$. Then
$\overline{1}_{C}=1_{C}$. The pure Dirac operator $Dirac^{N^{\prime}}$ is
related to the pure Dirac operator $Dirac^{N}$ by
$Dirac^{N^{\prime}}=e^{-\frac{n+1}{2}\phi}Dirac^{N}e^{\frac{n-1}{2}\phi}$ [12,
Section II]. Thinking of $S_{N}\otimes f^{*}S_{M}^{*}$ as
$\operatorname{Hom}(f^{*}S_{M},S_{N})$, the same argument gives
$D^{N^{\prime}}=e^{-\frac{n+1}{2}\phi}D^{N}e^{\frac{n-1}{2}\phi}$.

Let $\psi$ be a minimizer for the quasilocal energy of $N$ relative to $M$,
with $\pi_{+}\left(\psi\Big{|}_{\partial N}\right)=1_{C}$. Putting
$\psi^{\prime}=e^{-\frac{n-1}{2}\phi}\psi$, it satisfies
$D^{N^{\prime}}\psi^{\prime}=0$ with
$\pi_{+}\left(\psi^{\prime}\Big{|}_{\partial N^{\prime}}\right)=1_{C}$. As
$\operatorname{Ker}({\mathcal{D}}^{\prime})=0$, it follows that

${\mathcal{E}}^{\prime}_{C}=-\int_{\partial
N^{\prime}}\langle\psi^{\prime},\nabla^{N^{\prime}}_{e_{n}^{\prime}}\psi^{\prime}\rangle\operatorname{dvol}_{g^{\prime}}.$
(2.40)

From [12, Section II],

$\nabla^{N^{\prime}}_{e_{n}^{\prime}}=e^{-\phi}\left(\nabla^{N}_{e_{n}}+\frac{1}{4}(e_{\alpha}\phi)[\gamma^{n},\gamma^{\alpha}]\right).$
(2.41)

Applying this to (2.40), and using the fact that $\phi\Big{|}_{\partial N}$ is
locally constant, one finds

${\mathcal{E}}^{\prime}_{C}-{\mathcal{E}}_{C}=\frac{n-1}{2}\int_{\partial
N}(e_{n}\phi)|\psi|^{2}\>\operatorname{dvol}_{\partial N}.$ (2.42)

We now specialize to the case when $N=M$, $f=\operatorname{Id}$ and
$C=\partial N$. We no longer have $\operatorname{Ker}({\mathcal{D}})=0$.

######  Lemma 2.43.

If $e_{n}\phi\geq 0$ then ${\mathcal{E}}^{\prime}_{\partial
N^{\prime}}=\frac{n-1}{2}\int_{\partial
N}(e_{n}\phi)\>\operatorname{dvol}_{\partial N}$. If $e_{n}\phi<0$ then
${\mathcal{E}}^{\prime}_{\partial N^{\prime}}=-\infty$.

###### Proof.

Note that $\phi\Big{|}_{\partial N}=0$. Given $\psi^{\prime}\in
C^{\infty}(N^{\prime};E^{\prime})$ satisfying $D^{N^{\prime}}\psi^{\prime}=0$
and $\pi_{+}\left(\psi^{\prime}\Big{|}_{\partial
N}\right)=\overline{1}^{\prime}_{\partial N}$, put
$\psi=e^{\frac{n-1}{2}\phi}\psi^{\prime}$. Then $D^{N}\psi=0$ and
$\pi_{+}\left(\psi\Big{|}_{\partial N}\right)=\overline{1}_{\partial N}$. The
proof of Proposition 2.23 implies that $\psi=1+\rho$ for some harmonic form
$\rho$ satisfying relative boundary conditions. As in the proof of Proposition
2.23, we know that $\int_{\partial
N}\langle\psi,\nabla^{N}_{e_{n}}\psi\rangle\operatorname{dvol}_{g}=0$. Since

$-\int_{\partial
N^{\prime}}\langle\psi^{\prime},\nabla^{N^{\prime}}_{e_{n}^{\prime}}\psi^{\prime}\rangle\operatorname{dvol}_{g^{\prime}}+\int_{\partial
N}\langle\psi,\nabla^{N}_{e_{n}}\psi\rangle\operatorname{dvol}_{g}=\frac{n-1}{2}\int_{\partial
N}(e_{n}\phi)|\psi|^{2}\>\operatorname{dvol}_{\partial N},$ (2.44)

if $e_{n}\phi\geq 0$ then we minimize $\frac{n-1}{2}\int_{\partial
N}(e_{n}\phi)|\psi|^{2}\>\operatorname{dvol}_{\partial N}$ by taking $\psi=1$.
As $\operatorname{dvol}_{N}\in\operatorname{Ker}({\mathcal{D}})$, if
$e_{n}\phi<0$ then we can make $\frac{n-1}{2}\int_{\partial
N}(e_{n}\phi)|\psi|^{2}\>\operatorname{dvol}_{\partial N}$ arbitrarily
negative by taking $\psi=1+s\operatorname{dvol}_{N}$ with $s$ large. ∎

One can check that the mean curvatures of $\partial N^{\prime}$ and $\partial
N$ are related by

$H_{\partial N^{\prime}}=H_{\partial N}-(n-1)e_{n}\phi.$ (2.45)

Hence if $e_{n}\phi\geq 0$ then

${\mathcal{E}}^{\prime}_{\partial N^{\prime}}=\frac{1}{2}\int_{\partial
N}\left(H_{\partial N}-H_{\partial
N^{\prime}}\right)\>\operatorname{dvol}_{\partial N}=\frac{1}{2}\int_{\partial
N^{\prime}}\left(H_{\partial N}-H_{\partial
N^{\prime}}\right)\>\operatorname{dvol}_{\partial N^{\prime}},$ (2.46)

showing that the quasilocal energy equals the Brown-York energy.

In this conformal setting, we can also express the quasilocal energy as an
interior integral. If $n>2$ then

$\displaystyle\frac{n-1}{2}\int_{\partial
N}(e_{n}\phi)\>\operatorname{dvol}_{\partial N}$
$\displaystyle=\frac{n-1}{n-2}\int_{\partial
N}\left(e_{n}e^{\frac{(n-2)\phi}{2}}\right)\>\operatorname{dvol}_{\partial N}$
(2.47) $\displaystyle=-\frac{n-1}{n-2}\int_{N}\triangle
e^{\frac{(n-2)\phi}{2}}\>\operatorname{dvol}_{N}$
$\displaystyle=\frac{1}{4}\int_{N}e^{(\frac{n}{2}+1)\phi}\left(R_{g^{\prime}}-e^{-2\phi}R_{g}\right)\operatorname{dvol}_{N}.$

Hence if $e_{n}\phi\geq 0$ then

${\mathcal{E}}^{\prime}_{\partial
N^{\prime}}=\frac{1}{4}\int_{N}e^{(\frac{n}{2}+1)\phi}\left(R_{g^{\prime}}-e^{-2\phi}R_{g}\right)\operatorname{dvol}_{N}$
(2.48)

One can check that this is also true when $n=2$.

As $|\Lambda^{2}(d\operatorname{Id})|=e^{-2\phi}$, for
$\operatorname{Id}\>:(N^{\prime},g^{\prime})\rightarrow(N,g)$, equation (2.48)
is consistent with Proposition 2.26.

As a special case, let $(M,g_{Eucl})$ be a compact connected codimension-zero
submanifold of ${\mathbb{R}}^{2}$ with nonempty boundary. If $\phi\in
C^{\infty}(M)$ vanishes on $\partial M$ and satisfies $e_{n}\phi\geq 0$ there,
put $g^{\prime}=e^{2\phi}g_{Eucl}$. Then the quasilocal energy of
$\left(M,g^{\prime}\right)$, relative to $\left(M,g_{Eucl}\right)$, is

${\mathcal{E}}_{\partial
M}=\frac{1}{4}\int_{M}e^{2\phi}R_{g^{\prime}}\>\operatorname{dvol}_{g_{Eucl}}=\frac{1}{4}\int_{M}R_{g^{\prime}}\>\operatorname{dvol}_{g^{\prime}}.$
(2.49)

###  2.5. Background space in ${\mathbb{R}}^{n}$

In this section we make the quasilocal energy more explicit when the
background space $M$ is a domain in ${\mathbb{R}}^{n}$. We also treat the case
of rotationally symmetric $N$.

From Corollary 2.27, if $M\subset{\mathbb{R}}^{n}$ and $R_{N}\geq 0$ then
${\mathcal{E}}_{C}\geq 0$.

######  Proposition 2.50.

If $M\subset{\mathbb{R}}^{n}$ then two choices of $f:N\rightarrow M$ with the
same boundary restriction $\partial f$ will give the same value for
${\mathcal{E}}_{C}$.

###### Proof.

This is because the construction of ${\mathcal{E}}_{C}$ only involves $f$
through the boundary condition on $\partial f$ and the pullback of the
connection on $S_{M}^{*}$. Since $S_{M}^{*}$ is a trivial bundle with trivial
connection, the pullbacks of $S_{M}^{*}$ under two choices of $f$ will be
equivalent. ∎

Of course, ${\mathcal{E}}_{C}$ still depends on the intrinsic geometry of
$\partial N$, the extrinsic geometry of $\partial N$ (resp. $\partial M$) in
$N$ (resp. $M$), and a priori the interior geometry of $N$.

######  Proposition 2.51.

Whenever $\operatorname{Ker}({\mathcal{D}})=0$, the quasilocal energy can be
described as follows. Within the spinor module associated to
${\mathbb{R}}^{n}$, let $\\{\epsilon_{a}\\}_{a=1}^{2^{\frac{n}{2}-1}}$ be an
orthonormal basis for $\operatorname{Ker}(\gamma^{0}\gamma^{n}-I)$ and let
$\\{\epsilon^{\prime}_{a}\\}_{a=1}^{2^{\frac{n}{2}-1}}$ be an orthonormal
basis for $\operatorname{Ker}(\gamma^{0}\gamma^{n}+I)$. Extending
$\epsilon_{a}$ and $\epsilon^{\prime}_{a}$ to constant-valued sections of
$S_{M}$, let $\psi_{a}\in C^{\infty}(N;S_{N})$ be a harmonic spinor on $N$
with boundary value in $(\partial f)^{*}\epsilon_{a}\cdot
1_{C}+\operatorname{Ker}(\gamma^{0}\gamma^{n}+I)$ and let
$\psi^{\prime}_{a}\in C^{\infty}(N;S_{N})$ be a harmonic spinor on $N$ with
boundary value in $(\partial f)^{*}\epsilon^{\prime}_{a}\cdot
1_{C}+\operatorname{Ker}(\gamma^{0}\gamma^{n}-I)$ Then

${\mathcal{E}}_{C}=-2^{-n/2}\sum_{a=1}^{2^{\frac{n}{2}-1}}\int_{\partial
N}\left(\langle\psi_{a},\nabla^{N}_{e_{n}}\psi_{a}\rangle+\langle\psi^{\prime}_{a},\nabla^{N}_{e_{n}}\psi^{\prime}_{a}\rangle\right)\operatorname{dvol}_{\partial
N}.$ (2.52)

In particular, this is true under the hypotheses of Proposition 2.15,

###### Proof.

Let $\epsilon_{a}^{*}\in S_{M}^{*}$ denote inner product with $\epsilon_{a}$,
and similarly for $\epsilon_{a}^{\prime,*}$. Under the isomorphism
$S_{M}\otimes S_{M}^{*}\cong\Lambda^{*}T^{*}M$, we claim that
$\sum_{a}\left(\epsilon_{a}\otimes\epsilon_{a}^{*}+\epsilon^{\prime}_{a}\otimes\epsilon_{a}^{\prime,*}\right)$
corresponds to $2^{n/2}\in\Lambda^{*}T^{*}M$. This is because if $I$ is a
nontrivial increasing multi-index with entries from $\\{1,\ldots,n\\}$ then
$\operatorname{Tr}\left(\gamma^{I}\frac{\operatorname{Id}-\gamma^{0}\gamma^{n}}{2}\right)+\operatorname{Tr}\left(\gamma^{I}\frac{\operatorname{Id}+\gamma^{0}\gamma^{n}}{2}\right)=0$,
while
$\operatorname{Tr}\left(\frac{\operatorname{Id}-\gamma^{0}\gamma^{n}}{2}\right)+\operatorname{Tr}\left(\frac{\operatorname{Id}+\gamma^{0}\gamma^{n}}{2}\right)=2^{n/2}$.

On the other hand, if $\eta_{a}\in\operatorname{Ker}(\gamma^{0}\gamma^{n}+I)$
then $T(\eta_{a}\otimes\epsilon_{a}^{*})=-\eta_{a}\otimes\epsilon_{a}^{*}$,
and similarly for an element of the form
$\eta_{a}^{\prime}\otimes\epsilon_{a}^{\prime,*}$ with
$\eta_{a}^{\prime}\in\operatorname{Ker}(\gamma^{0}\gamma^{n}+I)$. Thus

$\Psi=2^{-n/2}\sum_{a}\left(\psi_{a}\otimes
f^{*}\epsilon_{a}^{*}+\psi^{\prime}_{a}\otimes
f^{*}\epsilon_{a}^{\prime,*}\right)$ (2.53)

satisfies $D^{N}\Psi=0$ and $\pi_{+}\left(\Psi\Big{|}_{\partial
N}\right)=1_{C}$. As $\operatorname{Ker}({\mathcal{D}})=0$, it is the unique
such solution. Hence

$\displaystyle{\mathcal{E}}_{C}=$ $\displaystyle-\int_{\partial
N}\langle\Psi,\nabla^{N}_{e_{n}}\Psi\rangle\>\operatorname{dvol}_{\partial N}$
(2.54) $\displaystyle=$
$\displaystyle-2^{-n/2}\sum_{a=1}^{2^{\frac{n}{2}-1}}\left(\int_{\partial
N}\langle\psi_{a},\nabla^{N}_{e_{n}}\psi_{a}\rangle\operatorname{dvol}_{\partial
N}+\langle\psi^{\prime}_{a},\nabla^{N}_{e_{n}}\psi^{\prime}_{a}\rangle\operatorname{dvol}_{\partial
N}\right).$

This proves the proposition. ∎

We now consider the case when $N$ is diffeomorphic to a disk, with a
rotationally symmetric metric $g_{N}$. Then $(N,g_{N})$ is conformally
equivalent to a disk in ${\mathbb{R}}^{n}$ and by rescaling the disk, we can
assume that the conformal equivalence $f$ is an isometry on the boundary.

######  Proposition 2.55.

In the rotationally symmetric case, if $H_{\partial N}>0$ and the sectional
curvature of $\partial N$ is $\frac{1}{k^{2}}$ then

${\mathcal{E}}_{\partial N}=\frac{1}{2}\int_{\partial
N}\left(\frac{n-1}{k}-H_{\partial N}\right)\>\operatorname{dvol}_{\partial
N}.$ (2.56)

###### Proof.

This follows from (2.46) ∎

###  2.6. Odd dimensional spaces

We now assume that $n$ is odd. The Clifford algebra has a faithful
representation $S$ of complex dimension $2^{\frac{n+1}{2}}$, which breaks up
into two isomorphic spinor representations $S^{+}\oplus S^{-}$ of
$\operatorname{Spin}(n)$. Explicitly, we can write Clifford generators
$\\{\gamma^{\alpha}\\}_{\alpha=1}^{n}$ on $S$ as
$\gamma^{\alpha}=\begin{pmatrix}0&\sigma^{\alpha}\\\
\sigma^{\alpha}&0\end{pmatrix}$, where $\\{\sigma^{\alpha}\\}_{\alpha=1}^{n}$
satisfy the Clifford relations on ${\mathbb{C}}^{2^{\frac{n-1}{2}}}$. We again
let $\epsilon$ be the ${\mathbb{Z}}_{2}$-grading operator on $S$.

With this definition of the spinors $S$, put $E=S_{N}\otimes f^{*}S_{M}^{*}$.
We can identify $E\Big{|}_{\partial
N}\cong\operatorname{End}\left(S_{N}\Big{|}_{\partial N}\right)$ with
$\Lambda^{*}(T^{*}N)\Big{|}_{\partial
N}\oplus\Lambda^{*}(T^{*}N)\Big{|}_{\partial N}$. To realize this
identification explicitly, put $\gamma^{0}=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}$. Let $I$ be an increasing multi-index with entries between
$1$ and $n$. Let $\tau^{0}$ denote a new odd variable. Then an element
$\omega_{I}\tau^{I}+\tau^{0}\wedge\omega^{\prime}_{I}\tau^{I}$ acts on
$S_{N}\Big{|}_{\partial N}$ by sending $\psi$ to
$(\omega_{I}\gamma^{I}+\gamma^{0}\omega^{\prime}_{I}\gamma^{I})\psi$.

Define $T$ as in (2.10). For
$\omega_{1},\omega_{2},\omega^{\prime}_{1},\omega^{\prime}_{2}\in\Lambda^{*}T^{*}\partial
N$, we have

$\displaystyle\left((\omega_{1}+\tau^{n}\wedge\omega_{2})+\tau^{0}\wedge(\omega^{\prime}_{1}+\tau^{n}\wedge\omega^{\prime}_{2})\right)\gamma^{0}\gamma^{n}\psi=$
(2.57)
$\displaystyle\gamma^{0}\gamma^{n}\left((\omega_{1}-\tau^{n}\wedge\omega_{2})+\tau^{0}\wedge(-\omega^{\prime}_{1}+\tau^{n}\wedge\omega^{\prime}_{2})\right)\psi.$

Hence the induced action of $T$ on $E\Big{|}_{\partial
N}\cong\operatorname{End}(S_{N})\Big{|}_{\partial N}$ is

$T\left(\omega_{1}+\tau^{n}\wedge\omega_{2},\omega^{\prime}_{1}+\tau^{n}\wedge\omega^{\prime}_{2}\right)=(\omega_{1}-\tau^{n}\wedge\omega_{2},-\omega^{\prime}_{1}+\tau^{n}\wedge\omega^{\prime}_{2}).$
(2.58)

We consider solutions to the Dirac equation $D^{N}\psi=0$ on sections $\psi\in
C^{\infty}(N;E)$ with the boundary condition that if $\psi\Big{|}_{\partial
N}=\left(\omega_{1}+\tau^{n}\wedge\omega_{2},\omega^{\prime}_{1}+\tau^{n}\wedge\omega^{\prime}_{2}\right)$
then $\omega_{1}=\overline{1}_{C}$ and
$\omega_{2}=\omega^{\prime}_{1}=\omega^{\prime}_{2}=0$, where the overline
denotes an orthogonal projection as before. Then the results of the previous
sections have straightforward extensions.

##  3\. Lorentzian case

This section is about the extension of Section 2 to hypersurfaces in
Lorentzian manifolds. Section 3.1 has background material. Section 3.2
discusses the case when the background space $M$ is a totally geodesic
hypersurface in a Lorentzian manifold $\overline{M}$. Section 3.3 deals with
the case when $M$ is a compact spatial hypersurface-with-boundary in
${\mathbb{R}}^{n,1}$. Finally, in Section 3.4 we describe how the results of
Section 3.3 extend to when $M$ is a compact spatial hypersurface-with-boundary
in a product spacetime ${\mathbb{R}}\times X$.

###  3.1. Background information

Let $\overline{N}$ and $\overline{M}$ be $(n+1)$-dimensional Lorentzian
manifolds with signature $(-1,1,\ldots,1)$. Let $N$ and $M$ be compact
connected $n$-dimensional spacelike submanifolds of $\overline{N}$ and
$\overline{M}$, respectively, with nonempty boundary. Let $f:N\rightarrow M$
be a smooth spin map so that for each connected component $Z$ of $\partial N$,
the map $\partial f$ restricts to an isometric diffeomorphism from $Z$ to
$(\partial f)(Z)$. By shrinking $\overline{N}$ and $\overline{M}$ to suitable
neighborhoods of $N$ and $M$, respectively, we can assume that $f$ is the
restriction of a spin map $\overline{f}:\overline{N}\rightarrow\overline{M}$.
For simplicity, we will assume that $\overline{N}$ and $\overline{M}$ are spin
and that $\partial f$ is a spin diffeomorphism on components of $\partial N$.
We let $S_{\overline{N}}$ be the standard spinor bundle on $\overline{N}$, and
similarly for $S_{\overline{M}}$.

We can identify $S_{\overline{N}}\Big{|}_{N}$ with $S_{N}$, and similarly for
$S_{\overline{M}}$.

######  Remark 3.1.

We could phrase what follows just in terms of $N$, $M$ and their normal
bundles, but it seems more illuminating to include the ambient spaces
$\overline{N}$ and $\overline{M}$.

######  Remark 3.2.

To clarify the relation between the spinor bundle $S_{N}$ considered here and
that considered in Section 2, we mention some facts about spinors. Suppose
first that $n$ is odd. Then $\dim(S_{{\mathbb{R}}^{n,1}})=2^{\frac{n+1}{2}}$
and $\dim(S_{{\mathbb{R}}^{n,1}}\otimes S_{{\mathbb{R}}^{n,1}}^{*})=2^{n+1}$,
which equals $\dim(\Lambda^{*}{\mathbb{R}}^{n,1})$. We can identify
$S_{{\mathbb{R}}^{n,1}}\otimes S_{{\mathbb{R}}^{n,1}}^{*}$ with
$\Lambda^{*}{\mathbb{R}}^{n,1}$. Using a timelike unit vector $e_{0}$, the
latter can be identified with $\Lambda^{*}{\mathbb{R}}^{n}\mathaccent
866{\otimes}\Lambda^{*}{\mathbb{R}}^{1}$. Compare with Section 2.5.

Now suppose that $n$ is even. Then
$\dim(S_{{\mathbb{R}}^{n,1}})=2^{\frac{n}{2}}$ and
$\dim(S_{{\mathbb{R}}^{n,1}}\otimes S_{{\mathbb{R}}^{n,1}}^{*})=2^{n}$. On the
other hand, $\dim(\Lambda^{*}{\mathbb{R}}^{n,1})=2^{n+1}$. It turns out that
one can identify $S_{{\mathbb{R}}^{n,1}}\otimes S_{{\mathbb{R}}^{n,1}}^{*}$,
as a $\operatorname{Spin}(n,1)$-module, with
$\Lambda^{*}({\mathbb{R}}^{n,1})/(\omega\sim*\omega)$. Using a timelike unit
vector $e_{0}$, the latter can be identified with
$\Lambda^{*}{\mathbb{R}}^{n}$. Compare with Section 2.1.

The restriction to $N$ of the connection on $S_{\overline{N}}$ has the local
form

$\nabla^{W}_{\sigma}=e_{\sigma}+\frac{1}{8}\omega_{\alpha\beta\sigma}[\gamma^{\alpha},\gamma^{\beta}]+\frac{1}{2}\omega_{0\alpha\sigma}\gamma^{0}\gamma^{\alpha},$
(3.3)

where $\alpha$ and $\beta$ are summed from $1$ to $n$. Note that $\nabla^{W}$
is generally not a unitary connection, because of the last term on the right-
hand side of (3.3). However, the ensuing Dirac-type operator

$D^{W}=-\sqrt{-1}\sum_{\sigma=1}^{n}\gamma^{\sigma}\nabla^{W}_{\sigma}$ (3.4)

is formally self-adjoint. To verify this, one can use the fact that in normal
coordinates around a point of $N$, one has

$[\nabla^{W}_{\sigma},\gamma^{\alpha}]=\omega_{0\alpha\sigma}\gamma^{0}.$
(3.5)

The restriction of $S_{\overline{M}}$ to $M$ is isomorphic to $S_{M}$. There
are corresponding connections on $S_{M}^{*}$. As in Section 2, it will be
convenient to use the isomorphism of $\operatorname{Spin}(n,1)$-modules
$S_{M}^{*}\cong S_{M}$ to transfer the connection from $S_{M}^{*}$ to $S_{M}$.
There is a subtlety in that the isomorphism is not unitary. If
$\sigma\in\operatorname{Spin}(n,1)$ is a transition function for $S_{M}$ then
the corresponding transition function for $S_{M}^{*}$ is $\sigma^{-T}$. This
is related to $\sigma$ by $\sigma^{-T}=C\sigma C^{-1}$, where $C$ is the
charge conjugation matrix. In short, when written on $S_{M}$, one finds that
the connection on $S_{M}^{*}$ takes the local form

$\mathaccent 866{\nabla}^{W}_{\mathaccent 866{\sigma}}=e_{\mathaccent
866{\sigma}}+\frac{1}{8}\mathaccent 866{\omega}_{{\mathaccent
866{\alpha}}{\mathaccent 866{\beta}}{\mathaccent 866{\sigma}}}[\mathaccent
866{\gamma}^{\mathaccent 866{\alpha}},\mathaccent 866{\gamma}^{\mathaccent
866{\beta}}]-\frac{1}{2}\mathaccent 866{\omega}_{\mathaccent 866{0}\mathaccent
866{\alpha}{\mathaccent 866{\sigma}}}\mathaccent 866{\gamma}^{\mathaccent
866{0}}\mathaccent 866{\gamma}^{\mathaccent 866{\alpha}}.$ (3.6)

Note the change in sign in the last term as compared with (3.3).

Let $\nabla^{N}$ be the connection on $E=S_{N}\otimes f^{*}S_{M}^{*}$. It
takes the local form

$\nabla^{N}_{\sigma}=e_{\sigma}+\frac{1}{8}\omega_{\alpha\beta\sigma}[\gamma^{\alpha},\gamma^{\beta}]+\frac{1}{2}\omega_{0\alpha\sigma}\gamma^{0}\gamma^{\alpha}+\frac{1}{8}\mathaccent
866{\omega}_{\mathaccent 866{\alpha}\mathaccent 866{\beta}\sigma}[\mathaccent
866{\gamma}^{\mathaccent 866{\alpha}},\mathaccent 866{\gamma}^{\mathaccent
866{\beta}}]-\frac{1}{2}\mathaccent 866{\omega}_{\mathaccent 866{0}\mathaccent
866{\alpha}\sigma}\mathaccent 866{\gamma}^{\mathaccent 866{0}}\mathaccent
866{\gamma}^{\mathaccent 866{\alpha}}.$ (3.7)

The corresponding Dirac-type operator

$D^{N}=-\sqrt{-1}\sum_{\sigma=1}^{n}\gamma^{\sigma}\nabla^{N}_{\sigma}$ (3.8)

is formally self-adjoint except for the term
$\sqrt{-1}\gamma^{\sigma}\cdot\frac{1}{2}\mathaccent 866{\omega}_{\mathaccent
866{0}\mathaccent 866{\alpha}\sigma}\mathaccent 866{\gamma}^{\mathaccent
866{0}}\mathaccent 866{\gamma}^{\mathaccent 866{\alpha}}$.

###  3.2. Time-symmetric background space

Suppose that $M$ is a totally geodesic subspace of $\overline{M}$. Then
$\mathaccent 866{\omega}_{\mathaccent 866{0}\mathaccent 866{\alpha}\sigma}=0$.
We are in a situation analogous to Section 2, except that $N$ is now a
hypersurface-with-boundary in a Lorentzian manifold. We give the extensions of
results from Section 2.

Let $C$ be a nonempty union of connected components of $\partial N$. We define
the quasilocal energy ${\mathcal{E}}_{C}$ as in Definition 2.19.

######  Lemma 3.9.

If $f:N\rightarrow M$ is an isometric diffeomorphism, and $N$ is totally
geodesic in $\overline{N}$, then ${\mathcal{E}}_{C}=0$.

###### Proof.

The proof is the same as that of Proposition 2.23. ∎

Define
$T_{AB}={R}^{\overline{N}}_{AB}-\frac{1}{2}{R}_{\overline{N}}g^{\overline{N}}_{AB}$,
for $0\leq A,B\leq n$.

######  Proposition 3.10.

Suppose that $M$ has nonnegative curvature operator. If

$2\left(T_{00}-\sqrt{-\sum_{\alpha=1}^{n}T_{0\alpha}T^{0\alpha}}\right)\geq|\Lambda^{2}df|(f^{*}R_{M})$
(3.11)

then ${\mathcal{E}}_{C}\geq 0$.

###### Proof.

Using the calculations in [19], the analog of (2.3) is

$(D^{N})^{2}=(\nabla^{N})^{*}\nabla^{N}+\frac{1}{2}\left(T_{00}+T_{0\alpha}\gamma^{0}\gamma^{\alpha}\right)-\frac{1}{4}[\gamma^{\sigma},\gamma^{\tau}]\left(\frac{1}{8}\mathaccent
866{R}_{\mathaccent 866{\alpha}\mathaccent 866{\beta}\sigma\tau}[\mathaccent
866{\gamma}^{\mathaccent 866{\alpha}},\mathaccent 866{\gamma}^{\mathaccent
866{\beta}}]\right).$ (3.12)

If $D^{N}\psi=0$, we obtain

$\displaystyle-\int_{\partial
N}\langle\psi,\nabla^{E}_{e_{n}}\psi\rangle\>\operatorname{dvol}=\int_{N}|\nabla^{N}\psi|^{2}\>\operatorname{dvol}+$
(3.13)
$\displaystyle\int_{N}\langle\psi,\left(\frac{1}{2}\left(T_{00}+T_{0\alpha}\gamma^{0}\gamma^{\alpha}\right)-\frac{1}{4}[\gamma^{\sigma},\gamma^{\tau}]\left(\frac{1}{8}\mathaccent
866{R}_{\mathaccent 866{\alpha}\mathaccent 866{\beta}\sigma\tau}[\mathaccent
866{\gamma}^{\mathaccent 866{\alpha}},\mathaccent 866{\gamma}^{\mathaccent
866{\beta}}]\right)\right)\psi\rangle\>\operatorname{dvol}.$

Using (2.8), the proposition follows. ∎

We now give an analog of (2.30). If $D^{N}\psi=0$ then the analog of (2.5) is

$\nabla^{N}_{e_{n}}\psi=D^{\partial N}_{Riem}\psi+\frac{H_{\partial
N}}{2}-\frac{1}{2}\gamma^{n}\gamma^{i}\mathaccent 866{\gamma}^{\mathaccent
866{n}}\mathaccent 866{\gamma}^{\mathaccent 866{j}}\mathaccent
866{A}_{\mathaccent
866{j}i}\psi-\frac{1}{2}\omega_{0ii}\gamma^{0}\gamma^{n}\psi+\frac{1}{2}\omega_{0ni}\gamma^{0}\gamma^{i}\psi,$
(3.14)

where $D^{\partial N}_{Riem}$ is the intrinsic Riemannian Dirac-type operator
on $\partial N$. Then

$\displaystyle-\int_{\partial
N}\langle\psi,\nabla^{E}_{e_{n}}\psi\rangle\>\operatorname{dvol}=$
$\displaystyle-\int_{\partial N}\langle\psi,D^{\partial
N}_{Riem}\psi\rangle\>\operatorname{dvol}_{\partial
N}-\frac{1}{2}\int_{\partial N}H_{\partial
N}|\psi|^{2}\>\operatorname{dvol}_{\partial N}+$ (3.15)
$\displaystyle\frac{1}{2}\int_{\partial
N}\langle\psi,\gamma^{n}\gamma^{i}\mathaccent 866{\gamma}^{n}\mathaccent
866{\gamma}^{\mathaccent 866{j}}\mathaccent 866{A}_{\mathaccent
866{j}i}\psi\rangle\>\operatorname{dvol}_{\partial N}+$
$\displaystyle\frac{1}{2}\int_{\partial
N}\langle\psi,\gamma^{0}\gamma^{n}\omega_{0ii}\psi\rangle\>\operatorname{dvol}_{\partial
N}-$ $\displaystyle\frac{1}{2}\int_{\partial
N}\langle\psi,\gamma^{0}\gamma^{i}\omega_{0ni}\psi\rangle\>\operatorname{dvol}_{\partial
N}.$

If $\psi\in{\mathcal{C}}_{C}$ then the last term in (3.15) vanishes, as $T$
anticommutes with $\gamma^{0}\gamma^{i}$,

To analyze the next-to-last term in (3.15), suppose that $n$ is even and
$\overline{1}_{C}=1_{C}$. Writing $\psi_{min}\Big{|}_{\partial
N}=1_{C}+\tau^{n}\wedge\phi$ for $\phi\in\Omega^{*}(\partial N)$, we have
$\gamma^{0}\gamma^{n}\psi_{min}\Big{|}_{\partial
N}=-\gamma^{n}\gamma^{0}(1_{C}+\tau^{n}\wedge\phi)$. The
${\mathbb{Z}}_{2}$-grading operator $\epsilon$ of $S_{N}$, when acting on
$S_{N}\otimes S_{N}^{*}\cong\Lambda^{*}T^{*}N$, is a $4^{th}$ root of $1$
(depending on the congruence class of $n$ modulo $4$) times

$\gamma^{1}\gamma^{2}\ldots\gamma^{n}=(-1)^{\frac{n}{2}}(E^{1}-I^{1})(E^{2}-I^{2})\ldots(E^{n}-I^{n}).$
(3.16)

Applying both sides to $1_{C}$ gives
$\gamma^{0}1_{C}=c1_{C}\tau^{1}\wedge\ldots\tau^{n}$ for a complex constant
$c$ of unit norm, depending on $n$. Then
$\gamma^{n}\gamma^{0}1_{C}=\sqrt{-1}(E^{n}-I^{n})\cdot
c1_{C}(\tau^{1}\wedge\ldots\tau^{n})=c^{\prime}1_{C}\tau^{1}\wedge\ldots\wedge\tau^{n-1}$,
for another unit constant $c^{\prime}$. In particular, the following terms
vanish:

$\langle
1_{C},\gamma^{0}\gamma^{n}\omega_{0ii}1_{C}\rangle=\langle\tau^{n}\wedge\phi,\gamma^{0}\gamma^{n}\omega_{0ii}1_{C}\rangle=\langle
1_{C},\gamma^{0}\gamma^{n}\omega_{0ii}\tau^{n}\wedge\phi\rangle=0.$ (3.17)

As in (2.30), we obtain

${\mathcal{E}}_{C}=-\frac{1}{2}\int_{C}H_{\partial
N}\>\operatorname{dvol}_{\partial N}+\frac{1}{2}\int_{C}(\partial
f)^{*}H_{\partial M}\>\operatorname{dvol}_{\partial N}+Q(\phi),$ (3.18)

where $Q(\phi)$ is an explicit homogeneous expression of order two in $\phi$.
There is a similar discussion when $n$ is odd.

If $K$ denotes the second fundamental form of $N$ in $\overline{N}$ then the
sum $\omega_{0ii}$ on $\partial N$, which will appear in the next proposition,
equals $\operatorname{Tr}(K)-K(e_{n},e_{n})$.

######  Proposition 3.19.

Suppose that

  * •

$M$ is a convex domain in ${\mathbb{R}}^{n}$,

  * •

$T_{00}-\sqrt{-\sum_{\alpha=1}^{n}T_{0\alpha}T^{0\alpha}}\geq 0$,

  * •

$H_{\partial N}-|\omega_{0ii}|\geq(\partial f)^{*}H_{\partial M}$, and

  * •

$T_{00}-\sqrt{-\sum_{\alpha=1}^{n}T_{0\alpha}T^{0\alpha}}>0$ somewhere or
$H_{\partial N}-|\omega_{0ii}|>(\partial f)^{*}H_{\partial M}$ somewhere.

Then $\operatorname{Ker}({\mathcal{D}})=0$.

###### Proof.

The proof is similar to that of Proposition 2.15. ∎

######  Proposition 3.20.

Whenever $\operatorname{Ker}({\mathcal{D}})=0$, the quasilocal energy can be
described as follows. Within the spinor module associated to
${\mathbb{R}}^{n}$, let $\\{\epsilon_{a}\\}_{a=1}^{2^{\frac{n}{2}-1}}$ be an
orthonormal basis for $\operatorname{Ker}(\gamma^{0}\gamma^{n}-I)$ and let
$\\{\epsilon^{\prime}_{a}\\}_{a=1}^{2^{\frac{n}{2}-1}}$ be an orthonormal
basis for $\operatorname{Ker}(\gamma^{0}\gamma^{n}+I)$. Extending
$\epsilon_{a}$ and $\epsilon^{\prime}_{a}$ to constant-valued sections of
$S_{M}$, let $\psi_{a}\in C^{\infty}(N;S_{N})$ be a harmonic spinor on $N$
with boundary value in $(\partial f)^{*}\epsilon_{a}\cdot
1_{C}+\operatorname{Ker}(\gamma^{0}\gamma^{n}+I)$ and let
$\psi^{\prime}_{a}\in C^{\infty}(N;S_{N})$ be a harmonic spinor on $N$ with
boundary value in $(\partial f)^{*}\epsilon^{\prime}_{a}\cdot
1_{C}+\operatorname{Ker}(\gamma^{0}\gamma^{n}-I)$ Then

${\mathcal{E}}_{C}=-2^{-n/2}\sum_{a=1}^{2^{\frac{n}{2}-1}}\int_{\partial
N}\left(\langle\psi_{a},\nabla^{N}_{e_{n}}\psi_{a}\rangle+\langle\psi^{\prime}_{a},\nabla^{N}_{e_{n}}\psi^{\prime}_{a}\rangle\right)\operatorname{dvol}_{\partial
N}.$ (3.21)

In particular, this is true under the hypotheses of Proposition 3.19

###### Proof.

The proof is the same as for Proposition 2.51. ∎

###  3.3. Background space in ${\mathbb{R}}^{n,1}$

We now let the background space be a more general submanifold of Minkowski
space. Let $\overline{M}$ be the Lorentzian ${\mathbb{R}}^{n,1}$ and suppose
that $M$ is a compact connected spacelike hypersurface-with-boundary in
$\overline{M}$. In general, the term $\mathaccent 866{\omega}_{\mathaccent
866{0}\mathaccent 866{\alpha}\sigma}$ in (3.7) need not be zero and there is
an apparent problem with formal self-adjointness of $D^{N}$ on the interior of
$N$. We can get around this by changing the inner product. We use the fact
that $S_{M}$ is the restriction of the trivial bundle $S_{\overline{M}}$ to
$M$. Choose a constant timelike unit vector field ${\mathcal{T}}$ in
$\overline{M}$. There is a corresponding inner product
$\langle\cdot,\cdot\rangle_{\mathcal{T}}$ on the trivial bundle
$S_{\overline{M}}$. Namely, if $(\cdot,\cdot)$ is the indefinite
$\operatorname{Spin}(n,1)$-invariant bilinear form on $S_{\overline{M}}$ then
the inner product on $S_{\overline{M}}$ is given by $\langle
s_{1},s_{2}\rangle_{\mathcal{T}}=(\sqrt{-1}c({\mathcal{T}})s_{1},s_{2})$,
where $c({\mathcal{T}})$ is Clifford multiplication by ${\mathcal{T}}$. We can
then restrict $(S_{\overline{M}},\langle\cdot,\cdot\rangle_{\mathcal{T}})$ to
$M$, dualize to get an inner product $\langle\cdot,\cdot\rangle_{sa}$ on
$S_{M}^{*}$ and pullback to $N$, to obtain an inner product
$\langle\cdot,\cdot\rangle_{E,sa}$ on $E=S_{N}\otimes f^{*}S^{*}_{M}$.

In order to express the boundary conditions, it will be convenient to write
things more intrinsically on $M$. Let $U\subset M$ be a connected open subset
on which an oriented orthonormal frame $\\{\mathaccent 866{e}_{\mathaccent
866{\alpha}}\\}_{\mathaccent 866{\alpha}=1}^{n}$ is defined and let $m_{0}\in
U$ be a basepoint. Since the connection on $S_{M}^{*}$ is flat, we can locally
write the expression $\frac{1}{8}\mathaccent 866{\omega}_{{\mathaccent
866{\alpha}}{\mathaccent 866{\beta}}{\mathaccent 866{\sigma}}}[\mathaccent
866{\gamma}^{\mathaccent 866{\alpha}},\mathaccent 866{\gamma}^{\mathaccent
866{\beta}}]\otimes\mathaccent 866{\tau}^{\mathaccent
866{\sigma}}-\frac{1}{2}\mathaccent 866{\omega}_{\mathaccent 866{0}\mathaccent
866{\alpha}{\mathaccent 866{\sigma}}}\mathaccent 866{\gamma}^{\mathaccent
866{0}}\mathaccent 866{\gamma}^{\mathaccent 866{\alpha}}\otimes\mathaccent
866{\tau}^{\mathaccent 866{\sigma}}$ in (3.6) as $g^{-1}dg$ for some
$g:U\rightarrow\operatorname{Spin}(n,1)$. Let
$\rho:\operatorname{Spin}(n,1)\rightarrow\operatorname{SO}(n,1)^{+}$ be the
double cover. If $\mathaccent 866{e}_{\mathaccent 866{0}}(m_{0})$ is a unit
normal vector to $M$ at $m_{0}$ then we can partially normalize $g$ by
specifying that $g(m_{0})$ sends $\mathaccent 866{e}_{\mathaccent
866{0}}(m_{0})$ to ${\mathcal{T}}$ in $T_{m_{0}}\overline{M}$. The remaining
ambiguity in $g$ will be left multiplication by a constant element of
$\operatorname{Spin}(n)$. On the other hand, a change of oriented orthonormal
frame corresponds to a map $\phi:U\rightarrow\operatorname{SO}(n)$ which
lifts, using the spin structure, to a a map $\mathaccent
869{\phi}:U\rightarrow\operatorname{Spin}(n)$. The effect of the new frame is
to multiply $g$ on the right by $\mathaccent 869{\phi}^{-1}$.

If $U_{1}$ and $U_{2}$ are overlapping such domains then their corresponding
orthonormal frames will be related by a map $\phi_{12}:U_{1}\cap
U_{2}\rightarrow\operatorname{SO}(n)$, which we lift using the spin structure
to $\mathaccent 869{\phi}_{12}:U_{1}\cap
U_{2}\rightarrow\operatorname{Spin}(n)$. Then the maps
$g_{i}:U_{i}\rightarrow\operatorname{Spin}(n,1)$ are related on $U_{1}\cap
U_{2}$ by $g_{1}=u_{12}g_{2}\mathaccent 869{\phi}_{12}^{-1}$, for some
constant element $u_{12}\in\operatorname{Spin}(n)$. In effect, this uses the
fact that $\mathaccent 866{e}_{\mathaccent 866{0}}$ is well-defined on $M$.
Similarly if $U_{1}\cap U_{2}$ intersects $\partial M$ then using the fact
that the unit normal $\mathaccent 866{e}_{\mathaccent 866{n}}$ is well-
defined, we can assume that $\mathaccent 869{\phi}_{12}\Big{|}_{U_{1}\cap
U_{2}\cap\partial M}$ takes values in $\operatorname{Spin}(n-1)$.

On the domain $U$, put $A=g^{*}g$ and define a weighted inner product
$\langle\cdot,\cdot\rangle_{S_{M}^{*},sa}$ by $\langle
s_{1},s_{2}\rangle_{S_{M}^{*},sa}=\langle s_{1},As_{2}\rangle_{S_{M}^{*}}$ for
$s_{1},s_{2}\in S_{M}^{*}$. Left multiplication of $g$ by an element of
$\operatorname{Spin}(n)$ doesn’t change $A$, so $A$ is independent of choices.
The connection (3.6) on $S_{M}^{*}$ is compatible with
$\langle\cdot,\cdot\rangle_{S_{M}^{*},sa}$, since the ambient flat connection
on $S_{\overline{M}}$ is compatible with
$\langle\cdot,\cdot\rangle_{\mathcal{T}}$.

Given a smooth spin map $f:(N,\partial N)\rightarrow(M,\partial M)$ which is
an isometric spin diffeomorphism on each connected component of $\partial N$,
we construct the Clifford module $E=S_{N}\otimes f^{*}S_{M}^{*}$ on $N$, with
the inner product
$\langle\cdot,\cdot\rangle_{E,sa}=\langle\cdot,\cdot\rangle\Big{|}_{S_{N}}\otimes
f^{*}\langle\cdot,\cdot\rangle\Big{|}_{S_{M}^{*},sa}$ The analog of (2.2) is

$\displaystyle\int_{N}\langle
D^{N}\psi_{1},\psi_{2}\rangle_{E,sa}\operatorname{dvol}_{N}-\int_{N}\langle\psi_{1},D^{N}\psi_{2}\rangle_{E,sa}\operatorname{dvol}_{N}=$
(3.22) $\displaystyle-\sqrt{-1}\int_{\partial
N}\langle\psi_{1},\gamma^{n}\psi_{2}\rangle_{E,sa}\operatorname{dvol}_{\partial
N}.$

######  Proposition 3.23.

If $N$ satisfies the dominant energy condition
$T_{00}\geq\sqrt{-\sum_{\alpha=1}^{n}T_{0\alpha}T^{0\alpha}}$ then for any
$\psi\in C^{\infty}(N;E)$ satisfying $D^{N}\psi=0$, we have

$-\int_{\partial
N}\langle\psi,\nabla^{N}_{e_{n}}\psi\rangle_{E,sa}\>\operatorname{dvol}_{\partial
N}\geq 0.$ (3.24)

###### Proof.

On the interior of $N$, we can think of $D^{N}$ as
$D^{W}\otimes\operatorname{Id}_{{\mathbb{C}}^{2^{n/2}}}$. By the calculations
in [19], in this representation we can write

$(D^{N})^{2}=\left((\nabla^{N})^{*}_{sa}\nabla^{N}+\frac{1}{2}(T_{00}+T_{0\alpha}\gamma^{0}\gamma^{\alpha})\right)\otimes\operatorname{Id}_{{\mathbb{C}}^{2^{n/2}}}.$
(3.25)

As $D^{N}\psi=0$, after integrating by parts we obtain from (3.25) that

$\displaystyle-\int_{\partial
N}\langle\psi,\nabla^{N}_{e_{n}}\psi\rangle_{E,sa}\>\operatorname{dvol}_{\partial
N}=$ (3.26)
$\displaystyle\int_{N}\left(\langle\nabla\psi,\nabla\psi\rangle_{E,sa}+\frac{1}{2}\langle\psi,(T_{00}+T_{0\alpha}\gamma^{0}\gamma^{\alpha})\psi\rangle_{E,sa}\right)\operatorname{dvol}.$

The proposition follows. ∎

To follow what was done in Section 2, we put
$T=\gamma^{0}\gamma^{n}\mathaccent 866{\gamma}^{\mathaccent 866{0}}\mathaccent
866{\gamma}^{\mathaccent 866{n}}$, acting on $E\Big{|}_{\partial N}$. We again
have $T1=1$. Note that $T$ may not be self-adjoint with respect to
$\langle\cdot,\cdot\rangle_{E,sa}$.

Given $\sigma\in\Omega^{*}(\partial N)$ with $T\sigma=\sigma$, there are two
natural boundary conditions to impose on the equation $D^{N}\psi=0$; we could
ask that $\psi\Big{|}_{\partial N}\in\sigma+\operatorname{Ker}(T+I)$ or we
could ask that $\psi\Big{|}_{\partial
N}\in\sigma+(\operatorname{Ker}(T-I))^{\perp}$. For concreteness, we take the
second choice.

Consider the operator $D^{N}$ acting on elements $\psi\in C^{\infty}(N;E)$
that satisfy $\psi\Big{|}_{\partial N}\in(\operatorname{Ker}(T-I))^{\perp}$.
This defines an elliptic boundary condition. Let ${\mathcal{D}}$ be the
corresponding operator, densely defined on $C^{\infty}(N;E)$. Here
${\mathcal{D}}$ may not be self-adjoint but it still has discrete spectrum
with eigenspaces of finite multiplicity.

We wish to find some $\psi\in C^{\infty}(N;E)$ so that $D^{N}\psi=0$ and
$\psi\Big{|}_{\partial N}\in\sigma+(\operatorname{Ker}(T-I))^{\perp}$. This is
an elliptic boundary value problem. Suppose that we can do this. Note that
$\gamma^{n}\psi\Big{|}_{\partial
N}\in\gamma^{n}\sigma+(\operatorname{Ker}(T+I))^{\perp}$ Put

${\mathcal{H}}=\left\\{\>\eta\in
C^{\infty}(N;E)\>:\>D^{N}\eta=0,\eta\Big{|}_{\partial
N}\in\operatorname{Ker}(T+I)\right\\}.$ (3.27)

If $\eta\in{\mathcal{H}}$ then from (3.22), we see that $\int_{\partial
N}\langle\sigma,\gamma^{n}\eta\rangle\>\operatorname{dvol}_{\partial N}=0$.
Conversely, given $\sigma\in\Omega^{*}(\partial N)$ with $T\sigma=\sigma$,
Fredholm theory implies that if $\int_{\partial
N}\langle\sigma,\gamma^{n}\eta\rangle\>\operatorname{dvol}_{\partial N}=0$ for
all $\eta\in{\mathcal{H}}$ then there is some $\psi\in C^{\infty}(N;E)$ so
that $D^{N}\psi=0$ and $\psi\Big{|}_{\partial
N}\in\sigma+(\operatorname{Ker}(T-I))^{\perp}$; c.f. [1, Theorem 2.4.5]. The
section $\psi$ is unique up to addition by elements of
$\operatorname{Ker}({\mathcal{D}})$.

As before, $C$ is a nonempty union of connected components of $\partial N$. We
would like to take $\sigma=1_{C}$. To be sure that $\psi$ exists, we may have
to slightly modify this choice. Let $P_{\gamma^{n}{\mathcal{H}}}$ be
orthogonal projection from $\Omega^{*}(\partial N)$ to
$\gamma^{n}{\mathcal{H}}$. We put
$\overline{1}_{C}=1_{C}-P_{\gamma^{n}{\mathcal{H}}}1_{C}$, take
$\sigma=\overline{1}_{C}$ and construct $\psi$ accordingly.

######  Definition 3.28.

Put

${\mathcal{C}}_{C}=\left\\{\psi\in
C^{\infty}(N;E)\>:\>D^{N}\psi=0,\psi\Big{|}_{\partial
N}\in\overline{1}_{C}+(\operatorname{Ker}(T-I))^{\perp}\right\\}.$ (3.29)

The quasilocal energy is

${\mathcal{E}}_{C}=\inf_{\psi\in{\mathcal{C}}}\left(-\int_{\partial
N}\langle\psi,\nabla^{N}_{e_{n}}\psi\rangle_{E,sa}\>\operatorname{dvol}_{\partial
N}\right).$ (3.30)

######  Proposition 3.31.

If $\overline{N}=\overline{M}={\mathbb{R}}^{n,1}$ and $N=M$ then
${\mathcal{E}}_{\partial N}=0$.

###### Proof.

In this case, the connection forms satisfy $\mathaccent 866{\omega}=\omega$.
Suppose that $n$ is odd. We can identify $E$ with
$\Lambda^{*}(T^{*}N)+\tau^{0}\wedge\Lambda^{*}(T^{*}N)$. Putting
$\gamma^{\alpha}=\sqrt{-1}(E^{\alpha}-I^{\alpha})$, $\mathaccent
866{\gamma}^{\mathaccent 866{\alpha}}=E^{\alpha}+I^{\alpha}$,
$\gamma^{0}=\sqrt{-1}(E^{0}+I^{0})$ and $\mathaccent 866{\gamma}^{\mathaccent
866{0}}=-(E^{0}-I^{0})$, when acting on
$\Omega^{*}(N)\oplus\tau^{0}\wedge\Omega^{*}(N)$ one can check that

$\nabla^{N}_{\sigma}=e_{\sigma}+\omega_{\alpha\beta\sigma}E^{\alpha}I^{\beta}+\omega_{0\alpha\sigma}(E^{0}I^{\alpha}+E^{\alpha}I^{0}).$
(3.32)

In particular, $\nabla^{N}_{\sigma}1=0$, so $D^{N}1=0$. It follows that
$\overline{1}=1$, so $1\in{\mathcal{C}}$, with $-\int_{\partial N}\langle
1,\nabla^{N}_{e_{n}}1\rangle_{E,sa}\>\operatorname{dvol}_{\partial N}=0$.
Proposition 3.23 now implies that ${\mathcal{E}}_{\partial N}=0$. The proof
when $n$ is even is similar. ∎

For an alternative approach that maintains self-adjointness, if $U\subset M$
is an open set as before then on $U\cap\partial M$, we put

$T_{sa}=\left[(\partial
f)^{*}(g^{*}g)\right]^{-\frac{1}{2}}\gamma^{0}\gamma^{n}\mathaccent
866{\gamma}^{\mathaccent 866{0}}\mathaccent 866{\gamma}^{\mathaccent
866{n}}\left[(\partial f)^{*}(g^{*}g)\right]^{\frac{1}{2}},$ (3.33)

which is now a self-adjoint idempotent with respect to
$\langle\cdot,\cdot\rangle_{E,sa}$. To see that this is globally defined, as
mentioned before on an overlap $U_{1}\cap U_{2}\cap\partial M$ we can assume
that $\mathaccent 869{\phi}_{12}$ commutes with $\mathaccent
866{\gamma}^{\mathaccent 866{0}}$ and $\mathaccent 866{\gamma}^{\mathaccent
866{n}}$. As

$((u_{12}g\mathaccent 869{\phi}_{12}^{-1})^{*}u_{12}g\mathaccent
869{\phi}_{12}^{-1})^{\frac{1}{2}}=\mathaccent
869{\phi}_{12}(g^{*}g)^{\frac{1}{2}}\mathaccent 869{\phi}_{12}^{-1},$ (3.34)

we obtain

$\displaystyle\left[(\partial
f)^{*}(g_{1}^{*}g_{1})\right]^{-\frac{1}{2}}\gamma^{0}\gamma^{n}\mathaccent
866{\gamma}^{\mathaccent 866{0}}\mathaccent 866{\gamma}^{\mathaccent
866{n}}\left[(\partial f)^{*}(g_{1}^{*}g_{1})\right]^{\frac{1}{2}}=$ (3.35)
$\displaystyle(\partial f)^{*}\mathaccent
869{\phi}_{12}\left\\{\left[(\partial
f)^{*}(g_{2}^{*}g_{2})\right]^{-\frac{1}{2}}\gamma^{0}\gamma^{n}\mathaccent
866{\gamma}^{\mathaccent 866{0}}\mathaccent 866{\gamma}^{\mathaccent
866{n}}\left[(\partial
f)^{*}(g_{2}^{*}g_{2})\right]^{\frac{1}{2}}\right\\}(\partial
f)^{*}\mathaccent 869{\phi}_{12}^{-1}.$

Taking into account that $\mathaccent 869{\phi}_{12}$ encodes how spinors
transform, it follows that $T_{sa}$ is globally defined on $\partial N$. We
still have $T_{sa}\gamma^{n}+\gamma^{n}T_{sa}=0$.

######  Lemma 3.36.

Consider the operator $D^{N}$ on elements $\psi\in C^{\infty}(N;E)$ satisfying
the boundary condition

$(T_{sa}+I)\psi\Big{|}_{\partial N}=0.$ (3.37)

It is formally self-adjoint.

###### Proof.

If $\psi_{1},\psi_{2}\in C^{\infty}(N;E)$ satisfy
$(T_{sa}+I)\psi_{1}\Big{|}_{\partial N}=(T_{sa}+I)\psi_{2}\Big{|}_{\partial
N}=0$ then

$\displaystyle\int_{\partial
N}\langle\psi_{1},\gamma^{n}\psi_{2}\rangle_{E,sa}\operatorname{dvol}_{\partial
N}$ $\displaystyle=-\int_{\partial
N}\langle\psi_{1},\gamma^{n}T_{sa}\psi_{2}\rangle_{E,sa}\operatorname{dvol}_{\partial
N}$ (3.38) $\displaystyle=\int_{\partial
N}\langle\psi_{1},T_{sa}\gamma^{n}\psi_{2}\rangle_{E,sa}\operatorname{dvol}_{\partial
N}$ $\displaystyle=\int_{\partial N}\langle
T_{sa}\psi_{1},\gamma^{n}\psi_{2}\rangle_{E,sa}\operatorname{dvol}_{\partial
N}$ $\displaystyle=-\int_{\partial
N}\langle\psi_{1},\gamma^{n}\psi_{2}\rangle_{E,sa}\operatorname{dvol}_{\partial
N},$

so $\int_{\partial
N}\langle\psi_{1},\gamma^{n}\psi_{2}\rangle_{E,sa}\operatorname{dvol}_{\partial
N}$ vanishes. Using (3.22), the lemma follows. ∎

With reference to Lemma 3.36, let ${\mathcal{D}}$ be the self-adjoint
extension. It has compact resolvent.

To find the right boundary value problem, we want a section $\sigma$ of
$E\Big{|}_{\partial N}$ so that $T_{sa}\sigma=\sigma$. Identifying
$E\Big{|}_{\partial N}$ with $\operatorname{End}\left(S_{N}\Big{|}_{\partial
N}\right)$, an initial attempt would be the operator
$s_{C}=2^{-n/2}1_{C}\left[(\partial f)^{*}(g^{*}g)\right]^{-\frac{1}{2}}$ when
acting on $S_{N}\Big{|}_{\partial N}$. We note that $(g^{*}g)^{-\frac{1}{2}}$
is an isometry from $(S_{M}^{*},\langle\cdot,\cdot\rangle)$ to
$(S_{M}^{*},\langle\cdot,\cdot\rangle_{sa})$.

More explicitly, at a point $n\in\partial N$, let
$\\{v_{a}\\}_{a=1}^{2^{\frac{n}{2}}}$ be an orthonormal basis of $S_{N}$ and
let $v_{a}^{*}$ denote inner product with respect to $v_{a}$. Then we put

$s_{C}=2^{-n/2}1_{C}(\partial
f)^{*}(g^{*}g)^{-\frac{1}{2}}\cdot\operatorname{Id}=2^{-n/2}1_{C}\sum_{a=1}^{2^{\frac{n}{2}}}v_{a}\otimes(\partial
f)^{*}(g^{*}g)^{-\frac{1}{2}}v_{a}^{*}.$ (3.39)

If we divide $\\{v_{a}\\}_{a=1}^{2^{\frac{n}{2}}}$ into orthonormal bases for
the $\pm 1$-eigenspaces of $\gamma^{0}\gamma^{n}$ then one sees that
$T_{sa}s_{C}=s_{C}$.

Let $\pi_{\pm}$ denote orthogonal projections onto the $\pm 1$-eigenspaces of
$T_{sa}$ acting on sections of $E\Big{|}_{\partial N}$. As before, we may have
to slightly modify the boundary data $s_{C}$ to ensure that we can solve the
boundary value problem. Put
$\overline{s}_{C}=s_{C}-P_{\gamma^{n}\operatorname{Ker}(\mathcal{D})\Big{|}_{\partial
N}}s_{C}$. Then there is some $\psi\in C^{\infty}(N;E)$ so that $D^{N}\psi=0$
and $\pi_{+}\left(\psi\Big{|}_{\partial N}\right)=\overline{s}_{C}$. The
section $\psi$ is unique up to addition by elements of
$\operatorname{Ker}({\mathcal{D}})$. We define ${\mathcal{E}}^{sa}_{C}$ as in
Definition 2.19, using $\langle\cdot,\cdot\rangle_{E,sa}$.

######  Proposition 3.40.

If $N$ satisfies the dominant energy condition then ${\mathcal{E}}_{C}\geq 0$
and ${\mathcal{E}}^{sa}_{C}\geq 0$.

###### Proof.

This follows from Proposition 3.23. ∎

######  Proposition 3.41.

${\mathcal{E}}_{C}$ and ${\mathcal{E}}^{sa}_{C}$ only depend on $f$ through
its boundary value $\partial f$.

###### Proof.

The proof is the same as for Proposition 2.50. ∎

The quasilocal energy ${\mathcal{E}}_{C}$ depends on the choice of constant
timelike unit vector field ${\mathcal{T}}$ in ${\mathbb{R}}^{n,1}$. By
minimizing over such choices, we obtain the quasilocal mass.

######  Definition 3.42.

The quasilocal mass is
${\mathcal{M}}_{C}=\inf_{{\mathcal{T}}}{\mathcal{E}}_{C}$.

###  3.4. Background space in a product manifold

In Section 3.2, when the background space was a subspace of
${\mathbb{R}}^{n,1}$, we effectively performed a gauge transformation in order
to make $\mathaccent 866{\nabla}^{W}$ locally trivial. This had the effect of
making the Dirac-type operator $D^{N}$ formally self-adjoint on the interior
of $N$. Looking at the connection (3.6), one sees that what one really needs
is a gauge transformation to make $\mathaccent 866{\nabla}^{W}$ unitary. This
can be achieved if the connection on the ambient spinor bundle
$S_{\overline{M}}$ is unitary. The easiest situation in which this is
guaranteed is when the Lorentzian manifold $\overline{M}$ is an isometric
product ${\mathbb{R}}\times X$ for some $n$-dimensional Riemannian spin
manifold $X$. Of course, this includes the case when
$\overline{M}={\mathbb{R}}^{n,1}$.

Let $M$ be a spacelike hypersurface with boundary in
$\overline{M}={\mathbb{R}}\times X$. As in Section 3.2, we can find local maps
$g:U\rightarrow\operatorname{Spin}(n)$ on $M$ so that

$g\left(\frac{1}{8}\mathaccent 866{\omega}_{{\mathaccent
866{\alpha}}{\mathaccent 866{\beta}}{\mathaccent 866{\sigma}}}[\mathaccent
866{\gamma}^{\mathaccent 866{\alpha}},\mathaccent 866{\gamma}^{\mathaccent
866{\beta}}]\otimes\mathaccent 866{\tau}^{\mathaccent
866{\sigma}}-\frac{1}{2}\mathaccent 866{\omega}_{\mathaccent 866{0}\mathaccent
866{\alpha}{\mathaccent 866{\sigma}}}\mathaccent 866{\gamma}^{\mathaccent
866{0}}\mathaccent 866{\gamma}^{\mathaccent 866{\alpha}}\otimes\mathaccent
866{\tau}^{\mathaccent 866{\sigma}}\right)g^{-1}+g\>dg^{-1}$ (3.43)

is a $1$-form with values in skew-Hermitian matrices (as opposed to its
vanishing in Section 3.2). We can assume that $g(m_{0})$ sends $\mathaccent
866{e}_{\mathaccent 866{0}}(m_{0})$ to the timelike unit vector $\partial_{t}$
in $T_{m_{0}}\overline{M}$. On an overlap $U_{1}\cap U_{2}$, the maps $g_{1}$
and $g_{2}$ are related by $g_{1}=u_{12}g_{2}\mathaccent 869{\phi}_{12}^{-1}$
for maps $u_{12},\mathaccent 869{\phi}_{12}:U_{1}\cap
U_{2}\rightarrow\operatorname{Spin}(n)$, where $u_{12}$ is covariantly
constant and $\mathaccent 869{\phi}_{12}$ comes from the change of local
orthonormal frame.

On the domain $U\subset M$, we put $A=g^{*}g$ and define an inner product
$\langle\cdot,\cdot\rangle_{S^{*}_{M},sa}$ by $\langle
s_{1},s_{2}\rangle_{S^{*}_{M},sa}=\langle s_{1},As_{2}\rangle_{S^{*}_{M}}$.

######  Proposition 3.44.

Suppose that $X$ has nonnegative curvature operator. If $\psi\in
C^{\infty}(N;E)$ satisfies $D^{N}\psi=0$ and

$2\left(T_{00}-\sqrt{-\sum_{\alpha=1}^{n}T_{0\alpha}T^{0\alpha}}\right)\geq|\Lambda^{2}df|(f^{*}R_{X})$
(3.45)

then $-\int_{\partial N}\langle\psi,\nabla^{E}_{e_{n}}\psi\rangle\geq 0$.

###### Proof.

After making local gauge transformations, the proof is the same as that of
Proposition 3.23. ∎

We can define ${\mathcal{E}}_{C}$ and ${\mathcal{E}}^{sa}_{C}$ as in Section
3.2. The analogs of Propositions 3.40 and 3.41 hold.

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