Geodesic motion on the group of boundary diffeomorphisms from Einstein’s
equations

Emine Şeyma Kutluk1 and Dieter Van den Bleeken2,3

1) Middle East Technical University

06800 Çankaya / Ankara, TURKEY

2) Primary address:

Physics Department, Boğaziçi University

34342 Bebek / Istanbul, TURKEY

3) Secondary address:

Institute for Theoretical Physics, KU Leuven

3001 Leuven, Belgium

<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

In [1] it was shown how in an adiabatic limit the vacuum Einstein equations on
a compact spatial region can be re-expressed as geodesic equations on the
group of diffeomorphisms of the boundary. This is reminiscent of the program
initiated by V. Arnold to reformulate models of continuum mechanics in terms
of geodesic motion on diffeomorphism groups. We revisit some of the results of
[1] in this light, pointing out parallels and differences with the typical
examples in geometric continuum mechanics. We work out the case of 2 spatial
dimensions in some detail.

###### Contents

  1. 1 Introduction and overview
  2. 2 Diffeomorphism groups: right invariant metrics and their geodesics
    1. 2.1 Metrics on infinite dimensional manifolds of maps
      1. 2.1.1 The larger perspective: the manifold of embeddings
      2. 2.1.2 The case of interest: the group of diffeomorphisms
    2. 2.2 The geodesic equation on the group of diffeomorphisms
      1. 2.2.1 Derivation of the geodesic equation
      2. 2.2.2 Examples appearing in continuum mechanics
  3. 3 From General Relativity to geodesics on the group of boundary diffeomorphisms
    1. 3.1 An extrinsic metric on the group of boundary diffeomorphisms
    2. 3.2 From superspace to boundary diffeomorphisms
      1. 3.2.1 Pulling back the Wheeler-deWitt metric
      2. 3.2.2 On the Hamiltonian constraint
    3. 3.3 The Case of 2 Spatial Dimensions
      1. 3.3.1 Generic polar curve
      2. 3.3.2 Circle
      3. 3.3.3 Ellipse
  4. 4 Discussion
  5. A Translation of notation

##  1 Introduction and overview

One of the most beautiful and elegant observations of modern mathematical
physics is that various differential equations of physical interest are
equivalent to geodesic equations on Lie groups equipped with a left or right
invariant metric, see e.g. [2] for an introduction. The simplest and most
widely known result is that the free motion of a rigid body can be interpreted
as motion along a geodesic on SO(3), equipped with a metric determined by the
moment of inertia of the body. In [3] V. Arnold generalized this idea, by
reformulating the dynamics of an ideal fluid as geodesic motion on the group
of volume preserving diffeomorphisms. Since then a variety of similar
equivalences have been discovered, for example for Burger’s equation, the
Camassa-Holm equation and others, see e.g. [4]. By today this has developed
into an interesting subfield at the intersection of geometric (continuum)
mechanics and the theory of (infinite dimensional) Lie groups.

Independent of the aforementioned work, in [1] we observed – together with A.
Seraj – that in an adiabatic111This slow motion approximation is alternatively
referred to as the Manton or moduli space approximation. It is best known
through its application in the physics of solitons [5]. We use the
nomenclature of Stuart [6]. approximation the vacuum Einstein equations on a
spacetime manifold of the form $\mathbb{R}\times\bar{M}$ reduce to geodesic
motion on Diff$(\partial\bar{M})$, the group of boundary diffeomorpisms222More
precisely the motion takes place on the coset
Diff$(\partial\bar{M})/$ISO$(\partial\bar{M})$, but for a generic boundary
ISO$(\partial\bar{M})=\\{\mathrm{id}\\}$..

It is the aim of this paper to put the results of [1] into the context of
Arnold’s program in geometric continuum mechanics. We will use some of the
results and language of this field to clarify certain points of [1], while
also highlighting some differences with the class of examples considered so
far in geometric mechanics, such as the pseudo-Riemannian nature of the metric
on the diffeomorphism group in our setup. Furthermore, as we will see, the
right invariant geometry induced on the diffeomorphism group by the dynamics
of general relativity is much more intricate than those in examples so far
considered in geometric continuum mechanics.

We begin our paper in Section 2, by reviewing some key mathematical concepts
and results concerning geodesic motion on diffeomorphism groups and the
corresponding physical dynamics. We present the geodesic equation in a fully
covariant form – a first in the literature as far as we are aware – and
shortly present the classic examples.

In Section 3 we review the key results of [1] expressing them in a
mathematically more elegant form and make the connection with the formalism
introduced in Section 2. The general procedure is then illustrated in the case
of two spatial dimensions, whose detailed treatment is new. This simplest case
already shows the intricate nature of the construction and allows us to point
out some remaining technical challenges. We end the paper with a short
discussion in Section 4.

Note: Our level of rigor when dealing with infinite dimensional analysis will
be that of the typical physics texts in this field, e.g. [2], trusting on the
expert mathematical literare, e.g. [7, 8], to fill some gaps. We are also
somewhat sloppy in our nomenclature. For example we will assume all groups to
be connected to the identity, so that what we call the ‘diffeomorphism group’
is in more detailed and standard nomenclature the ‘identity component of the
diffeomorphism group’, we’ll refer to certain anti-isomorphisms simply as
isomorphisms etc. We hope this will ease reading and it will allow us to focus
on the main points.

##  2 Diffeomorphism groups: right invariant metrics and their geodesics

Although our main interest will lie with Diff($M$), the group of
diffeomorphisms of a manifold $M$ to itself, some parts of the construction
are most clearly exhibited in the slightly larger context of maps between two
manifolds $M$ and $N$. With this motivation we introduce the right invariant
metric on the manifold of embeddings of $M$ into $N$, before specializing to
the case $M=N$, where these embeddings become the diffeomorphisms of $M$.
After we have reviewed this standard construction of right invariant metrics
on the group of diffeomorphisms we recall the derivation of the corresponding
geodesic equation. We end the section by illustrating how for certain choices
of diffeomorphism groups and metrics this equation takes the form of some well
known equations in continuum mechanics.

Our treatment draws from [9, 2, 10].

###  2.1 Metrics on infinite dimensional manifolds of maps

####  2.1.1 The larger perspective: the manifold of embeddings

The set $\mathrm{Emb}(M,N)$ of embeddings of $M$ into $N$ can naturally be
given the structure of an infinite dimensional manifold. In addition there is
a canonical right action of Diff$(M)$ on this manifold, defined via the
composition of maps. Indeed, let $f:M\rightarrow N$ be such an embedding and
$\phi:M\rightarrow M$ a diffeomorphism, then

$R_{\phi}f=f\circ\phi\ .$ (1)

Although this will play no role in our discussion, it is interesting to point
out that this right action makes $\mathrm{Emb}(M,N)$ a principal fibre bundle
with structure group Diff$(M)$.

A vector tangent to $\mathrm{Emb}(M,N)$ can be obtained by considering a curve
in $\mathrm{Emb}(M,N)$, i.e. a one-parameter family of embeddings $f_{s}$. The
vector is then identified with the derivative $\dot{f}=\frac{df_{s}}{ds}$. To
understand this object better, note that given $p\in M$ then $f_{s}(p)$
defines a curve in $N$ and so $\dot{f}(p)=\frac{d}{ds}f_{s}(p)$ defines a
tangent vector in $T_{f(p)}N$. So we see that the tangent vector $\dot{f}$ is
a map from $M$ into the tangent bundle of $N$. In more mathematical term; a
tangent vector to $\mathrm{Emb}(M,N)$ at $f$ is a section of the pull-back
bundle333Note that we can formally identify $M$ with its image $f(M)$ which is
a sub-manifold of $N$, we can then think of $f^{*}TN$ as the restriction of
$TN$ to this sub-manifold. $f^{*}TN$:

$T_{f}\mathrm{Emb}(M,N)=\Gamma_{c}(f^{*}TN)\ .$ (2)

Given $g$, a Riemannian metric on $N$, there exists an associated metric $G$
on $\mathrm{Emb}(M,N)$. First one observes that via the discussion above
$g_{p}(\dot{f}_{1}(p),\dot{f}_{2}(p))$ is well defined for all $p\in M$ and
$\dot{f}_{1},\dot{f}_{2}\in T_{f}\mathrm{Emb}(M,N)$. One then obtains a metric
$G$ on $\mathrm{Emb}(M,N)$ by integrating over $M$:

$G_{f}(\dot{f}_{1},\dot{f}_{2})=\int_{M}g(\dot{f}_{1},\dot{f}_{2})\,\mathrm{vol}f^{*}g\
.$ (3)

Here $\mathrm{vol}f^{*}g$ is the volume element on $M$ determined by the
induced metric $f^{*}g$ on $M$, the pull-back by $f$ of the metric $g$ on $N$.

Since integration over $M$ is diffeomorphism invariant, one finds that the
metric (3) is invariant under the right-action (1):

$G_{R_{\phi}f}(dR_{\phi}(\dot{f}_{1}),dR_{\phi}(\dot{f}_{2}))=G_{f}(\dot{f}_{1},\dot{f}_{2})\
,$ (4)

where one uses that $(dR_{\phi}\dot{f})(p)=\dot{f}(\phi(p))$ and
$\mathrm{vol}(R_{\phi}f)^{*}g=\phi^{*}\mathrm{vol}f^{*}g$.

####  2.1.2 The case of interest: the group of diffeomorphisms

If we specialize now to the case $N=M$ one obtains
$\mathrm{Emb}(M,M)=\mathrm{Diff}(M)$, which naturally is an infinite
dimensional Lie group. Letting $\mathrm{id}$ denote the identity map we
observe that (2) becomes

$T_{\mathrm{id}}\mathrm{Diff}(M)=\Gamma_{c}(TM)=\mathfrak{X}(M)\ .$ (5)

That is, we recover the well-known fact that the Lie-algebra associated to
diffeomorphisms of a manifold $M$ is naturally isomorphic to the algebra of
vectorfields on $M$. Let us stress that generic tangent vectors $\dot{\phi}\in
T_{\phi}\mathrm{Diff}(M)$ are not in general vector fields on $M$, rather they
are sections of the pullback bundle $\phi^{*}TM$, per (2).

Using the right invariance (4) one can relate the metric (3) on
$\mathrm{Diff}(M)$ to an inner product on $\mathfrak{X}(M)$:

$G_{\phi}(\dot{\phi}_{1},\dot{\phi}_{2})=(dR_{\phi^{-1}}\dot{\phi}_{1},dR_{\phi^{-1}}\dot{\phi}_{2})$
(6)

where

$(\chi_{1},\chi_{2})=\int_{M}g(\chi_{1},\chi_{2})\,\mathrm{vol}\,g\quad\mbox{with}\quad\chi_{1},\chi_{2}\in\mathfrak{X}(M)\
.$ (7)

This reveals the option to define a more general class of inner products, and
also metrics, as follows. First one defines

$\langle\chi_{1},\chi_{2}\rangle=(\chi_{1},\mathbb{D}\chi_{2})=\int_{M}g(\chi_{1},\mathbb{D}\chi_{2})\,\mathrm{vol}\,g\
.$ (8)

This is a proper inner product when $\mathbb{D}$ is a positive definite, self-
adjoint operator on the Hilbert space $\mathfrak{X}(M),(\cdot,\cdot)$.
Replacing the canonical inner product $(\cdot,\cdot)$ with the modified one in
(6), one obtains a new metric on $\mathrm{Diff}(M)$:

$G_{\phi}^{\mathbb{D}}(\dot{\phi}_{1},\dot{\phi}_{2})=\langle
dR_{\phi^{-1}}\dot{\phi}_{1},dR_{\phi^{-1}}\dot{\phi}_{2}\rangle\ .$ (9)

Note that also this new ‘deformed’ metric $G^{\mathbb{D}}$ is right invariant
by construction.

###  2.2 The geodesic equation on the group of diffeomorphisms

####  2.2.1 Derivation of the geodesic equation

Given the construction of the right invariant metrics (9) on the group of
geodesics of a manifold $M$ it is natural to study the geodesics associated to
such a metric. We’ll take a somewhat pedestrian approach inspired by
mechanics, and will define the geodesics associated to a metric
$G^{\mathbb{D}}$ to be those curves $\phi(t)$ that extremize the action

$S=\int\frac{1}{2}G^{\mathbb{D}}_{\phi(t)}(\dot{\phi}(t),\dot{\phi}(t))dt\ ;$
(10)

where $\dot{\phi}(t)=\frac{d}{dt}\phi(t)$, and, as in the previous
subsections, $\phi(t)$ is a diffeomorphism from $M$ to itself for all $t$.

If we choose coordinates $x^{i}$ on $M$ then we can think of the curve
$\phi(t)$ as a one-parameter family of maps $\phi^{i}(x;t)$. Defining

$\chi^{i}(x;t)=(dR_{\phi^{-1}}\dot{\phi})^{i}(x;t)=\partial_{t}\phi^{i}(\phi^{-1}(x;t);t)$
(11)

one sees, via (9), that (10) becomes

$S=\int\frac{1}{2}\langle\chi(t),\chi(t)\rangle dt\ .$ (12)

In physical terms one can think of $\phi^{i}(x;t)$ as defining a fluid flow,
where $\phi^{i}(x,t)$ is the position at time $t$ of a particle that at time
$t=0$ was at $x^{i}$ (assuming the convention $\phi^{i}(x;0)=x^{i}$). The
Lagrangian velocity, i.e. the velocity at time $t$ of the particle which was
at $x^{i}$ at time $t=0$, of the flow is then $\partial_{t}\phi^{i}(x;t)$. In
this picture $\chi^{i}(x;t)$ has the interpretation of the Eulerian velocity
of the flow, i.e. the velocity at time $t$ of the particle which is at $x^{i}$
at that time [2].

In mathematical terms $\partial_{t}\phi^{i}(x;t)\in T_{\phi}\mathrm{Diff}(M)$,
and it gets mapped into $\chi^{i}(x;t)\in T_{\mathrm{id}}\mathrm{Diff}(M)$ by
the right action $dR_{\phi^{-1}}$. This is a coordinate representation of the
concepts we introduced in the previous subsections. As we stressed there,
$\partial_{t}\phi^{i}(x;t)$ is not a vectorfield on $M$, while $\chi^{i}(x;t)$
is. Equivalently $\chi$ is an element of the Lie algebra $\mathfrak{diff}(M)$
(while $\partial_{t}\phi$ is not).

The procedure of rewriting of (10) in terms of (12) together with the rigorous
derivation of the corresponding Euler-Lagrange equations, for an arbitrary Lie
group, is known as Euler-Poincaré reduction, see e.g. [2]. Applied to this
setting, the starting point is the observation that the form (12) of the
action implies that a generic variation $\delta\phi^{i}$ will only affect the
action through $\delta\chi^{i}$. A short computation reveals the relation
between the two variations:

$\displaystyle\delta\chi^{i}$ $\displaystyle=$
$\displaystyle\partial_{t}\delta\phi^{i}(\phi^{-1})+(\partial_{j}\partial_{t}\phi^{i})(\phi^{-1})\,\delta(\phi^{-1})^{j}$
(13) $\displaystyle=$
$\displaystyle\partial_{t}\delta\Phi^{i}+L_{\chi}\delta\Phi^{i}\ .$ (14)

In the last expression we used the notation

$\delta\Phi^{i}(x;t)=\delta\phi^{i}(\phi^{-1}(x;t);t)\ ,$ (15)

as well as the Lie derivative

$L_{\chi}\delta\Phi^{i}=\chi^{j}\partial_{j}\delta\Phi^{i}-\delta\Phi^{j}\partial_{j}\chi^{i}\
.$ (16)

To obtain (14) from (13) one can use the relations

$\displaystyle(\partial_{j}\partial_{t}\phi^{i})(\phi^{-1}(x;t);t)$
$\displaystyle=$
$\displaystyle-\partial_{k}\chi^{i}(x;t)\,\partial_{j}\phi^{k}(\phi^{-1}(x;t);t)\
,$ $\displaystyle\delta(\phi^{-1})^{i}(x;t)$ $\displaystyle=$
$\displaystyle-\partial_{j}(\phi^{-1})^{i}(x;t)\,\delta\Phi^{i}(x;t)\ ,$
$\displaystyle\partial_{t}(\phi^{-1})^{i}(x;t)$ $\displaystyle=$
$\displaystyle-\partial_{j}(\phi^{-1})^{i}(x;t)\,\chi^{i}(x;t)\ ,$ (17)
$\displaystyle\delta^{i}_{j}$ $\displaystyle=$
$\displaystyle\partial_{k}\phi^{i}(\phi^{-1}(x;t);t)\partial_{j}(\phi^{-1})^{k}(x;t)\
,$ $\displaystyle\partial_{t}\delta\Phi^{i}(x;t)$ $\displaystyle=$
$\displaystyle(\partial_{t}\delta\phi^{i})(\phi^{-1}(x;t);t)+\partial_{t}(\phi^{-1})^{j}(x;t)\partial_{j}\delta\Phi^{i}(x;t)\
.$

The variation of the action then becomes

$\displaystyle\delta S$ $\displaystyle=$
$\displaystyle\int\langle\chi,\partial_{t}\delta\Phi^{i}+L_{\chi}\delta\Phi^{i}\rangle
dt$ (18) $\displaystyle=$
$\displaystyle\int\langle-\partial_{t}\chi+L^{\dagger}_{\chi}\chi,\delta\Phi^{i}\rangle
dt\ .$ (19)

One of the key results of the theory of Euler-Poincaré reduction, which we
will use without further argument, is that a generic variation
$\delta\Phi^{i}$, can be obtained from some variation $\delta\phi^{i}$. Since
the inner product is non-degenerate and $\delta\Phi^{i}$ can be a generic
vector one finds that a stationary action implies

$\partial_{t}\chi-L_{\chi}^{\dagger}\chi=0\ .$ (20)

This is then the geodesic equation on $\mathrm{Diff}(M)$. Note that this is a
first order equation, since it is an equation for the velocity $\chi(t)$, to
find the actual corresponding curve of diffeomorphisms $\phi(t)$ one has to
additionally integrate (11).

A short computation, using the definitions of the inner product, reveals that

$L_{\chi_{1}}^{\dagger}\chi_{2}=-\mathbb{D}^{-1}\mathbb{L}_{\chi_{1}}\mathbb{D}\chi_{2}$
(21)

where

$(\mathbb{L}_{\chi_{1}}\chi_{2})^{i}=L_{\chi_{1}}\chi_{2}^{i}+(g^{ik}L_{\chi_{1}}g_{kj}+\delta^{i}_{j}\,g^{-1}L_{\chi_{1}}g)\chi_{2}^{j}\
.$ (22)

This implies we can rewrite the geodesic equation equivalently as

$\partial_{t}\mathbb{D}\chi+\mathbb{L}_{\chi}\mathbb{D}\chi=0\ .$ (23)

Writing this out explicitly one gets

$(\partial_{t}+\chi^{j}\nabla_{j})\mathbb{D}\chi^{i}+(\nabla^{i}\chi_{j}+\delta^{i}_{j}\nabla_{k}\chi^{k})\mathbb{D}\chi^{j}=0\
.$ (24)

This equation provides a covariant version of the geodesic equation as derived
in e.g. [2].

####  2.2.2 Examples appearing in continuum mechanics

There are a number of physical continuum systems whose mechanical dynamics is
described by a geodesic equation on a diffeomorphism group, i.e. a special
case of (24). In this subsection we shortly mention some of the most well
known. In the next section we explain how the Einstein equations in an
adiabatic limit can be added to this list.

  * •

Volume preserving diffeomorphisms and Euler’s equation
If we restrict to the subgroup of volume preserving diffeomorphisms, such that
$L_{\chi}g=\nabla_{i}\chi^{i}=0$, and restrict to the canonical case
$\mathbb{D}=\mathds{1}$, then (24) reduces to

$\partial_{t}\chi^{i}+\chi^{j}\nabla_{j}\chi^{i}+\chi^{j}\nabla^{i}\chi_{j}=0\
.$ (25)

This equation is equivalent to Euler’s equation for the (free) motion of an
incompressible fluid if we identify $\chi^{i}$ with the (Eulerian) velocity of
the fluid. Introducing the material derivative
$D_{t}=\partial_{t}+\chi^{j}\nabla_{j}$ and the kinetic energy of the fluid
$T=\frac{1}{2}g_{ij}\chi^{i}\chi^{j}$ the equation takes the perhaps more
familiar form

$D_{t}\chi^{i}=-\nabla^{i}T\ .$ (26)

  * •

1d diffeomorphisms and Burger’s equation
If we take $M$ to be one-dimensional and consider the canonical choice
$\mathbb{D}=\mathds{1}$, then the geodesic equation (24) takes the form

$\partial_{t}\chi^{x}+3\chi^{x}\nabla_{x}\chi^{x}=0$ (27)

which is the (inviscid) Burger equation444Writing $u=\chi^{x}$ and assuming
the coordinate $x$ to be Cartesian, so that $\nabla_{x}=\partial_{x}$,
reproduces the standard form $\partial_{t}u+3u\partial_{x}u=0$..

  * •

1d diffeomorphisms and the Camassa-Holm equation
When one considers one-dimensional diffeomorphisms, but now with a metric
based on the operator555Note that when $\mathbb{D}=\mathds{1}-\nabla_{x}^{2}$
then the inner product (8) on $\mathfrak{X}(M)$ reproduces the $H^{1}$ Sobolev
norm:
$\langle\chi,\chi\rangle=\int_{M}\sqrt{g}(\chi^{2}+(\nabla\chi)^{2})dx=\|\chi\|^{2}_{H^{1}}\
.$ $\mathbb{D}=\mathds{1}-\nabla_{x}^{2}$, the geodesic equation (24) becomes

$\partial_{t}\chi^{x}-\partial_{t}\nabla_{x}^{2}\chi^{x}+3\chi^{x}\nabla_{x}\chi^{x}-2\nabla_{x}\chi^{x}\nabla_{x}^{2}\chi^{x}-\chi^{x}\nabla_{x}^{3}\chi^{x}=0\
.$ (28)

This equation is the (dispersionless) Camassa-Holm equation666To get a more
familiar form one can again write $u$ for the vector component $\chi^{x}$, and
assume $x$ to be Cartesian so that $\nabla_{x}=\partial_{x}$. describing
shallow water waves, see e.g. [2].

##  3 From General Relativity to geodesics on the group of boundary
diffeomorphisms

In this section we review some of the key results of [1]. To be compatible
with the previous section we change some of the notation and choice of symbols
that were used in [1], see Appendix A for a dictionary.

First we introduce the operator $\mathbb{D}$ that via (8, 9) defines the right
invariant metric on the diffeomorphism group. Only afterwards, in Section 3.2,
will we explain how this particular metric/operator follows from the vacuum
Einstein equations. As an illustration, in the last subsection we then
explicitly compute $\mathbb{D}$ in the simple case of $\dim M=1$.

With respect to [1] the elegant form (31) of $\mathbb{D}$ is new, as well as
the results in Section 3.3.

###  3.1 An extrinsic metric on the group of boundary diffeomorphisms

Let $\bar{M}$ be a compact subset of $\mathbb{R}^{n+1}$, for simplicity
assumed to be homeomorphic to the closed $(n+1)$-ball. We’ll denote it’s
boundary as $M=\partial\bar{M}$ and write $\mathrm{j}:M\rightarrow\bar{M}$ for
the related embedding. The flat Euclidean metric $\bar{g}_{\mathrm{o}}$ on
$\bar{M}$ naturally induces a metric $g=\mathrm{j}^{*}\bar{g}_{\mathrm{o}}$ on
$M$. Abstractly $\bar{M}$ is a manifold with boundary which allows us to
borrow some machinery from the general theory of manifolds with boundary, in
particular related to boundary value problems for differential forms,
following777See also Appendix C in [1] for a summary of the parts most
relevant to our discussion. [11]. We expect much of the discussion below to
generalize to generic Riemannian manifolds with boundary of general topology.

Our aim is to introduce an operator $\mathbb{D}$ on the Hilbert space
$\mathfrak{X}(M),(\cdot,\cdot)$, which in turn – by the discussion of Section
2 – defines a right invariant metric of the type (8, 9) on $\mathrm{Diff}(M)$.
What is crucial is that in the construction of $\mathbb{D}$ we will make use
of the ‘bulk’ space $\bar{M}$ of which $M$ is the boundary, i.e. $\mathbb{D}$
is extrinsic to $M$, rather than intrinsic.

We will now specify the operator $\mathbb{D}$ relevant to us. We start by
considering the map

$\bar{\theta}:\Lambda^{1}(M)\rightarrow\Lambda^{2}(\bar{M}):\omega\mapsto\bar{\theta}_{\omega}\
,$ (29)

defined such that

$\delta\bar{\theta}_{\omega}=d\bar{\theta}_{\omega}=0\,,\qquad\mathrm{j}^{*}\bar{\theta}_{\omega}=d\omega\,.$
(30)

Note that $\bar{\theta}_{\omega}$ is a harmonic field, i.e. a form that is
both closed and co-closed – which on a manifold with boundary is a stronger
condition than just being a harmonic form. Given the boundary 1-form $\omega$,
the bulk two form $\bar{\theta}_{\omega}$, defined through the boundary value
problem above, is guaranteed to exist and be unique888Uniqueness is guaranteed
since, due its trivial topology, there are no non-zero Dirichlet fields –
harmonic fields whose pullback on the boundary vanishes – on $\bar{M}$, except
when $\dim\bar{M}=2$. The case $\dim\bar{M}=2$ needs to be treated separately
and one needs to change the boundary condition on $\bar{\theta}_{\omega}$ –
since $d\omega=0$ trivially on the one dimensional boundary. In Section 3.3 we
will discuss the case $\dim\bar{M}=2$ in detail, see (55) for the modified
boundary value problem defining $\bar{\theta}_{\omega}$ in that case., see
e.g. [11] Thm 3.2.5.

Using the linear map (29) we then define

$\mathbb{D}\,:\,\mathfrak{X}(M)\rightarrow\mathfrak{X}(M)\,:\,\chi\mapsto(\mathrm{j}^{*}(\imath_{n}\bar{\theta}_{\chi^{\flat}})-2\imath_{\chi}K)^{\sharp}\
.$ (31)

Here $\imath_{n}$ is the contraction with $n$, a vectorfield whose restriction
to $M$ is the unit outward normal vector field to $M$. We have used the
musical isomorphism associated to the metric $g$; i.e.
$\flat\,:\,TM\rightarrow T^{*}M\,:\,\chi\mapsto\chi^{\flat}=g(\cdot,\chi)$
with inverse $\sharp:\,T^{*}M\rightarrow TM\,:\omega\mapsto\omega^{\sharp}$.
These are mathematical representations of what most physicists respectively
refer to as lowering and raising indices.

In the second term we have introduced the extrinsic curvature
$K=\frac{1}{2}\mathrm{j}^{*}(L_{n}\bar{g}_{\mathrm{o}})$ of the boundary as a
hypersurface inside $(\bar{M},\bar{g}_{\mathrm{o}}).$ Note that $K$ is
naturally a symmetric bilinear form on $TM$ and so $\imath_{\chi}K$ defines a
one-form on $M$ for every vectorfield $\chi$ on $M$.

One can show, see the next subsection for an argument, that $\mathbb{D}$ is
symmetric with respect to the inner product

$(\chi_{1},\chi_{2})=\int_{M}g(\chi_{1},\chi_{2})\,\mathrm{vol}\,g\ .$ (32)

The associated right invariant metric on $\mathrm{Diff}(M)$ is then defined as
in (9):

$G_{\phi}^{\mathbb{D}}(\dot{\phi}_{1},\dot{\phi}_{2})=(\chi_{1},\mathbb{D}\chi_{2})$
(33)

where

$\qquad\chi=dR_{\phi^{-1}}\dot{\phi}\,,\mbox{ i.e.
}\chi^{i}(x;t)=\partial_{t}\phi^{i}(\phi^{-1}(x,t);t)\ .$ (34)

As we will also discuss further in the next subsection, the operator
$\mathbb{D}$ defined in (31) is however not positive, so that $G$ is actually
a pseudo-Riemannian metric on $\mathrm{Diff}(M)$.

###  3.2 From superspace to boundary diffeomorphisms

We now review how the geodesic motion on the group of boundary diffeomorphisms
– equipped with the right invariant metric (33) which itself is determined by
the operator (31) – is equivalent to the Einstein equations in an adiabatic
limit. We only summarize the main points and refer to [1] for a detailed
argumentation. In this subsection we switch to the index/component notation
more often used in physics and general relativity (GR).

Before providing some concrete technical details in Section 3.2.1, let us
summarize the main idea in words.

In the geometrodynamic picture, see e.g. [12] for a short review, the vacuum
Einstein equations can be interpreted as the motion of a particle on
superspace – the space of spatial Riemannian metrics – equipped with the
Wheeler-deWitt metric. The Euler-Lagrange equations of such a particle lead to
a geodesic equation on superspace, but one that is sourced by forces arising
from a non-trivial potential on superspace – set by the curvature of the
spatial metric. In a slow velocity approximation, introduced by Manton [5],
one can restrict motion to the subspace of zero potential – referred to as the
space of vacua in [1] – where the motion becomes free, i.e. purely geodesic
with respect to the induced metric. This very general argument thus suggests
that, just like any system described by a natural Lagrangian [13], the
Einstein equations can, in an adiabatic limit, be reduced to purely geodesic
motion. At first sight one could be tempted to think the setup is trivial,
since vanishing potential implies vanishing Ricci curvature of the 3d spatial
metric, and there is, up to diffeomorphism, only one such metric, the flat
one. The subtlety is however that in the presence of a boundary999In this note
and in [1] we focus on the case where space is a compact region with boundary.
On a non-compact space the boundary diffeomorphisms should be replaced by the
so called asymptotic symmetries – see e.g. [14] or [15] more recently – , but
it remains an open problem to understand if some of the methods of [1] can be
applied to that setting as well. not all diffeomorphisms should be thought of
as redundancies describing the metrics that are physically the same. Those
diffeomorphisms that do not vanish on the boundary will generate physically
inequivalent metrics. From this observation it follows, see [1] for a few
extra details, that the zero-potential surface in the superspace of a manifold
with boundary can be identified with the group of boundary diffeomorphisms.
The metric describing the geodesic motion of interest is then nothing but the
pull-back of the Wheeler-deWit metric to this surface.

####  3.2.1 Pulling back the Wheeler-deWitt metric

Let us now make this concrete. Consider the following101010Locally any
Lorentzian metric can be brought in this form, these are so called Gaussian
normal coordinates. Lorentzian metric on $\mathbb{R}\times\bar{M}$

$ds^{2}=-dt^{2}+\bar{g}_{ij}(x;t)dx^{i}dx^{j}\ .$ (35)

We will furthermore assume111111This is the restriction to the zero-potential
surface in superspace. the spatial metric $\bar{g}_{ij}$ to be flat at all
times $t$ – note that this does not imply that the $n+2$ dimensional metric
(35) is flat. The vacuum Einstein equations for the Lorentzian metric (35)
then take the form

$\displaystyle\bar{\nabla}^{i}\left(\dot{\bar{g}}_{ij}-\bar{g}^{kl}\dot{\bar{g}}_{kl}\bar{g}_{ij}\right)$
$\displaystyle=$ $\displaystyle 0\,,$ (36)
$\displaystyle\frac{1}{2}\bar{g}^{ij}\bar{g}^{kl}\dot{\bar{g}}_{i[j}\dot{\bar{g}}_{k]l}$
$\displaystyle=$ $\displaystyle 0\,,$ (37)
$\displaystyle\ddot{\bar{g}}_{ij}+\bar{g}^{kl}(\dot{\bar{g}}_{ik}\dot{\bar{g}}_{jl}-\dfrac{1}{2}\dot{\bar{g}}_{ij}\dot{\bar{g}}_{kl})$
$\displaystyle=$ $\displaystyle 0\,.$ (38)

The first two equations are constraints while the last equation is a dynamical
equation that can be obtained from extremizing the Lagrangian

$\displaystyle
L=\int\frac{1}{2}\bar{\mathrm{G}}_{\bar{g}}(\dot{\bar{g}},\dot{\bar{g}})dt\ .$
(39)

This is a purely kinetic term based on the Wheeler-de Witt metric

$\bar{\mathrm{G}}_{\bar{g}}(\dot{\bar{g}}_{1},\dot{\bar{g}}_{2})=\frac{1}{2}\int_{\bar{M}}\bar{g}^{i[k}g^{j]l}\dot{\bar{g}}_{1\,{ij}}\dot{\bar{g}}_{2\,{kl}}\,\sqrt{\mathrm{det}\bar{g}}\,d^{n+1}x\
.$ (40)

The key point is now that since $\bar{g}_{ij}$ is flat for all $t$ we can
write

$\bar{g}(t)=\bar{\phi}^{*}(t)\bar{g}_{\mathrm{o}}\,,$ (41)

i.e. the time dependent metric is simply the pull-back by a time dependent
diffeomorphism of a fixed, i.e. time-independent, reference flat metric
$\bar{g}_{\mathrm{o}}$. One can then compute121212See e.g. Appendix A in [1].
that

$\dot{\bar{g}}=\bar{\phi}^{*}(L_{\bar{\chi}}\bar{g}_{\mathrm{o}})\,,\quad\mbox{where}\quad\bar{\chi}=\dot{\bar{\phi}}\circ\bar{\phi}^{-1}=dR_{\bar{\phi}^{-1}}\dot{\bar{\phi}}\
.$ (42)

Here the diffeomorphism $\bar{\phi}$ of $\bar{M}$ is assumed to preserve the
boundary $M=\partial\bar{M}$, i.e. $\bar{\phi}(M)=M$, and its restriction to
the boundary then defines a boundary diffeomorphism
$\phi=\left.\bar{\phi}\right|_{M}\,:\,M\rightarrow M$. At the level of
vectorfields this translates to

$\left.\bar{g}_{\mathrm{o}}(n,\bar{\chi})\right|_{M}=0\quad\mbox{such
that}\quad\chi=\left.\bar{\chi}\right|_{M}\ .$ (43)

The condition on the left simply states that the normal part of the bulk
vectorfield that generates the bulk diffeomorphism should vanish at the
boundary, if the bulk diffeomorphism is to preserve the boundary.

In simplifying (40), as a first step it is important to evaluate the momentum
constraint (36) on velocities of the form (42). This leads to

$\bar{\nabla}^{i}_{\\!\\!\mathrm{o}}\partial_{[i}\bar{\chi}_{j]}=0\ .$ (44)

Similarly, evaluating the Wheeler-de Witt metric $\bar{\mathrm{G}}$ on the
velocities (42) one finds – upon partial integration, which produces a bulk
term that vanishes by the momentum constraint (44) so that only the boundary
term remains – that

$\bar{\mathrm{G}}_{\phi^{*}\bar{g}_{\mathrm{o}}}(\phi^{*}(L_{\bar{\chi}_{1}}\bar{g}_{\mathrm{o}}),\phi^{*}(L_{\bar{\chi}_{2}}\bar{g}_{\mathrm{o}}))=\int_{M}\bar{\chi}_{1}^{k}n^{j}\bar{\nabla}_{\\!\\!\mathrm{o}(k}\bar{\chi}_{2\,j)}\sqrt{\det
g}\,d^{n}x\ .$ (45)

One would like to interpret the above in terms of the boundary vector fields
$\chi\in\mathfrak{X}(M)$ only. To do so one needs to determine the expression
inside the integral in terms of $\chi$. First one observes, via a short
computation131313Along the boundary we split the tangent space as
$\left.(\partial_{i})\right|_{M}=\left.(n^{i}\partial_{i},\partial_{a})\right|_{M}$,
so that a vector decomposes as
$\left.(\bar{v}^{i})\right|_{M}=(\left.\bar{v}^{i}\right|_{M}n_{i},v^{a})$. In
this notation (43) gets re-expressed as
$\left.(\bar{\chi}^{i})\right|_{M}=(0,\chi^{a})$. See [1] Appendix B for a
detailed discussion., that

$\left.\bar{\chi}^{i}_{1}n^{j}\bar{\nabla}_{\\!\\!\mathrm{o}(i}\bar{\chi}_{2\,j)}\right|_{M}=\left.n^{i}\bar{\chi}_{1}^{j}\partial_{[i}\bar{\chi}_{2\,j]}\right|_{M}-2K_{ab}\chi_{1}^{a}\chi_{2}^{b}\
.$ (46)

The second term, which includes the (pull-back of the) extrinsic curvature is
already explicitly written in terms of boundary quantities, so it remains to
determine $\partial_{[i}\bar{\chi}_{j]}$, in terms of $\chi$. Via (44) we can
think of $\chi$ as providing a boundary condition for a boundary value
differential problem for $\bar{\chi}$. If we define $\bar{\theta}=d\bar{\chi}$
then we see that (44) takes the form $\delta\bar{\theta}=0$ while
simultaneously $d\bar{\theta}=0$. Here we recognize then the map introduced in
(29, 30) on our way to defining the operator $\mathbb{D}$ (31). Rephrased this
way we thus get

$\left.\bar{\chi}_{1}^{i}n^{j}\bar{\nabla}_{\\!\\!\mathrm{o}(i}\bar{\chi}_{2\,j)}\right|_{M}=g_{ab}\chi_{1}^{a}(\mathbb{D}\chi_{2})^{b}\
.$ (47)

In the end we thus find the following form for the Wheeler-de Witt metric
$\bar{\mathrm{G}}$, pulled-back to the space of boundary diffeomorphisms:

$\displaystyle\bar{\mathrm{G}}_{\phi^{*}\bar{g}_{\mathrm{o}}}(\phi^{*}(L_{\bar{\chi}_{1}}\bar{g}_{\mathrm{o}}),\phi^{*}(L_{\bar{\chi}_{2}}\bar{g}_{\mathrm{o}}))$
$\displaystyle=$
$\displaystyle\int_{M}g(\chi_{1},\mathbb{D}\chi_{2})\,\mathrm{vol}\,g=G^{\mathbb{D}}_{\phi}(\dot{\phi}_{1},\dot{\phi}_{2})\
.$ (48)

Here we recognize the inner product (8) and the right invariant metric (9)
introduced in Section 2. The dynamic Einstein equation (38) reduces to the
geodesic equation obtained by varying (48) in the adiabatic, or Manton,
approximation.

Note that in our discussion so far we have ignored the Hamiltonian constraint
(37). We’ll discuss its role and interpretation in the next subsection.

####  3.2.2 On the Hamiltonian constraint

It is important to point out that in the derivation of the metric on the group
of boundary diffeomorphisms from the Einstein equations the Hamiltonian
constraint did not play any role. As we will now shortly discuss it has
another, two-fold role to play however.

First, one can easily see that the Hamiltonian constraint (37), evaluated on
$\dot{\bar{g}}$ satisfying the assumption (42) and restricted to $M$, implies
that (48) vanishes. In other words, it says that those $\chi$ related to an
actual solution of the full Einstein equations will be null vectors on
$\mathrm{Diff}(M)$, i.e. $(\chi,\mathbb{D}\chi)=0$. In particular then, the
Hamiltonian constraint forces (adiabatic) solutions to Einstein’s equations to
be null geodesics on the group of boundary diffeomorphisms. That this is
possible is due to the pseudo-Riemannian nature of the metric (33). In [1] it
was shown that on the subspace of exact boundary vector fields141414We say
that a vector field is (co-)exact iff the dual one-form is (co-)exact. the
operator $\mathbb{D}$ is negative, while on another subspace of boundary
vectorfields – those for which the bulk extension $\bar{\chi}$ is co-exact –
$\mathbb{D}$ is positive. This implies that
$\langle\cdot,\cdot\rangle=(\chi,\mathbb{D}\chi)$ has mixed signature and
allows for null vectors151515Note that zero eigenvectors of $\mathbb{D}$ would
make $\langle\cdot,\cdot\rangle$ degenerate. It was argued in [1] – though not
proven – that the kernel of $\mathbb{D}$ corresponds to Killing vectors of the
boundary metric. This would imply that $\langle\cdot,\cdot\rangle$ when
restricted to the coset space $\mathrm{Diff}(M)/\mathrm{ISO}(M)$ is a non-
degenerate, albeit pseudo-Riemannian, metric..

Secondly, since the momentum constraint is only sensitive to
$d\bar{\chi}^{\flat}$, it follows that it determines a bulk extension of the
boundary vector field $\chi$ only up to an exact part. To make this more
precise, one can split $\bar{\chi}^{\flat}$ in an exact and co-exact part:

$\bar{\chi}^{\flat}=\bar{\eta}+d\bar{\alpha}\quad\mbox{where}\quad\delta\bar{\eta}=0\
.$ (49)

It was shown in [1] that $\bar{\eta}$ is uniquely determined once $\chi$ , and
hence $\bar{\theta}_{\chi^{\flat}}=d\bar{\chi}^{\flat}$, is given. One can
then insert the decomposition (49) into the Hamiltonian constraint (37) – via
(42) – and it takes the form

$(\bar{\nabla}_{i}\partial^{i}\bar{\alpha})^{2}-\bar{\nabla}_{i}\partial_{j}\bar{\alpha}\bar{\nabla}^{i}\partial^{j}\bar{\alpha}-2\bar{\nabla}^{i}\bar{\eta}^{j}\bar{\nabla}_{i}\partial_{j}\bar{\alpha}=\bar{\nabla}_{(i}\bar{\eta}_{j)}\bar{\nabla}^{i}\bar{\eta}^{j}\,.$
(50)

Since $\bar{\eta}$ is already determined through the momentum constraint we
can interpret (50) as a differential equation determining $\bar{\alpha}$,
which should be solved under the boundary condition
$d\left(\left.\alpha\right|_{M}\right)=\chi^{\flat}-\mathrm{j}^{*}\bar{\eta}\,,\
\left.n^{i}\partial_{i}\alpha\right|_{M}=-\left.n^{i}\bar{\eta}_{i}\right|_{M}$,
for a given boundary vector field $\chi$. Once $\bar{\alpha}$ is found and
$\bar{\chi}^{\flat}$ is fully known one should integrate (42) to obtain the
family of bulk diffeomorphisms $\bar{\phi}(t)$ and the metrics $\bar{g}(t)$,
which then finally provide an $(n+2)$ dimensional metric (35) that
(approximately) solves the vacuum Einstein equations.

Solving the Hamiltonian constraint in the form (50) remains however a daunting
task, this equation is a non-homogeneous, non-linear PDE resembling the
$(n+1)$-Hessian equation, see e.g. [16]. In the simple case $n=1$, see the
next subsection, (50) can be recast in the form of the homogeneous real Monge-
Ampere equation.

###  3.3 The Case of 2 Spatial Dimensions

In this subsection we focus on the special case $n=1$, i.e. $\dim M=1$ and
$\dim\bar{M}=2$ and will provide a concrete form for the operator $\mathbb{D}$
defined in (31), the associated metric and geodesic equation. We will also
show how the Hamiltonian constraint (50) can be reduced to the homogeneous
real Monge-Ampere equation in this case. We first discuss a large class of
examples in general, where the boundary is a polar curve, and afterwards make
a few comments on the simplest cases, those of circle and ellipse.

####  3.3.1 Generic polar curve

Let us – permitting a slight loss of generality – concentrate on the following
setup. We take $\bar{M}$ to be the simply connected region around the origin
of the flat Eucidean plane, in the interior of the polar curve $r=f(\varphi)$,
see Figure 1, or explicitly

$\displaystyle\bar{M}$ $\displaystyle=$
$\displaystyle\\{(r\cos\varphi,r\sin\varphi)\in\mathbb{R}^{2}\,|\,0\leq r\leq
f(\varphi)\,,\ \varphi\in[0,2\pi)\\}\ ,$ (51) $\displaystyle M$
$\displaystyle=$
$\displaystyle\\{(r\cos\varphi,r\sin\varphi)\in\mathbb{R}^{2}\,|\,r=f(\varphi)\,,\
\varphi\in[0,2\pi)\\},$ (52) $\displaystyle\mathrm{j}(\varphi)$
$\displaystyle=$ $\displaystyle\left(f(\varphi),\varphi\right)\ .$ (53)

Figure 1: The polar curve $f(\varphi)$ determines the boundary $M$ of the
region $\bar{M}$ in $\mathbb{R}^{2}$.

Our first task is to compute the map $\bar{\theta}_{\omega}$, see (29). The
boundary value problem (30) needs to be slightly modified in the special case
$n=1$, since the boundary condition
$\mathrm{j}^{*}\bar{\theta}_{\omega}=d\omega$ becomes contentless as both
sides are trivially zero. To find an alternative boundary condition, let us
recall the argument below (46), which identifies the key relation

$\bar{\theta}_{\omega}=d\bar{\omega}\quad\mbox{and}\quad\mathrm{j}^{*}(\bar{\omega})=\omega\
.$ (54)

Integrating (54) over $\bar{M}$, which is only possible when $n=1$ because
then $\dim\bar{M}=\mathrm{deg}\,\bar{\theta}_{\omega}=2$, provides the
condition $\int_{\bar{M}}\bar{\theta}_{\omega}=\int_{M}\omega$. Adding the
information that we want $\bar{\theta}_{\omega}$ to solve the momentum
constraint then leads to the replacement of definition (30) in the special
case $n=1$ by

$\delta\bar{\theta}_{\omega}=0\
,d\bar{\theta}_{\omega}=0\quad\mbox{and}\quad\int_{\bar{M}}\bar{\theta}_{\omega}=\int_{M}\omega\,.$
(55)

Note that on $\bar{M}$ any two form that is co-closed is of the form
$\bar{\theta}_{\omega}=\bar{c}\,\overline{\mathrm{vol}}$, with $\bar{c}$ a
constant. Let us recall we work with the flat metric on $\bar{M}$, i.e.

$d\bar{s}^{2}=dx^{2}+dy^{2}=dr^{2}+r^{2}d\varphi^{2}\ ,\qquad ds^{2}={\cal
E}_{\varphi}^{2}d\varphi^{2}\quad\mbox{where}\quad{\cal
E}_{\varphi}=\sqrt{f^{\prime 2}+f^{2}}\ ;$ (56)

which define the unit volume forms

$\displaystyle\overline{\mathrm{vol}}$ $\displaystyle=$
$\displaystyle\,dx\wedge dy=\,rdr\wedge d\varphi\ ,$ (57)
$\displaystyle\mathrm{vol}$ $\displaystyle=$ $\displaystyle{\cal
E}_{\varphi}d\varphi\ .\qquad$ (58)

Note that the area of $\bar{M}$ and its circumference – i.e. the length of $M$
– are given as

$A=\int_{\bar{M}}\overline{\mathrm{vol}}=\frac{1}{2}\int_{0}^{2\pi}f^{2}d\varphi\quad\mbox{and}\quad
C=\int_{M}\mathrm{vol}=\int_{0}^{2\pi}{\cal E}_{\varphi}d\varphi\ .$ (59)

It follows that we can solve (55) explicitly as

$\bar{\theta}_{\omega}=p_{\omega}\,\overline{\mathrm{vol}}\quad\mbox{where}\quad
p_{\omega}=A^{-1}\int_{M}\omega\ .$ (60)

Given the map $\bar{\theta}$, the definition (31) of $\mathbb{D}$ remains
unchanged. The period $p_{\omega}$ will appear through the zeromode of the
vectorfield $\chi=\chi^{\varphi}\partial_{\varphi}$:

$\pi(\chi)=p_{\chi^{\flat}}=A^{-1}\int_{0}^{2\pi}{\cal
E}_{\varphi}^{2}\,\chi^{\varphi}d\varphi\ .$ (61)

To work out the action of $\mathbb{D}$ on $\chi$ explicitly, we need some
information on the extrinsic geometry of $M$ as embedded in $\bar{M}$, in
particular the normal vector field and extrinsic curvature

$\displaystyle n$ $\displaystyle=$ $\displaystyle f{\cal
E}_{\varphi}^{-1}\left(\partial_{r}-\frac{f^{\prime}}{f^{2}}\partial_{\varphi}\right)\
,$ (62) $\displaystyle K$ $\displaystyle=$ $\displaystyle{\cal K}\,({\cal
E}_{\varphi}d\varphi)^{2}\,\quad\mbox{where}\quad{\cal
K}=\frac{f^{2}+2f^{\prime 2}-ff^{\prime\prime}}{(f^{\prime 2}+f^{2})^{3/2}}\
.$ (63)

Using these we then see the operator $\mathbb{D}$ acting on a vector field
$\chi$ on $M$ is

$\displaystyle\mathbb{D}\chi$ $\displaystyle=\pi(\chi){\cal
E}_{\varphi}^{-1}\partial_{\varphi}-2{\cal K}\chi\ ,$ (64)

or fully explicitly

$\frac{1}{2}(\mathbb{D}\chi)^{\varphi}=\frac{\int_{0}^{2\pi}(f^{\prime
2}+f^{\prime 2})\chi^{\varphi}d\tilde{\varphi}}{\sqrt{f^{\prime
2}+f^{2}}\int_{0}^{2\pi}f^{2}d\tilde{\varphi}}-\frac{f^{2}+2f^{\prime
2}-ff^{\prime\prime}}{(f^{\prime 2}+f^{2})^{3/2}}\chi^{\varphi}\ .$ (65)

Once we have determined the operator $\mathbb{D}$, the metric (8) on
$\mathfrak{diff}(M)=\mathfrak{X}(M)$ follows:

$\displaystyle\frac{1}{2}\langle\chi_{1},\chi_{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{2\pi}\sqrt{g}\,g_{\varphi\varphi}\,\chi_{1}^{\varphi}(\mathbb{D}\chi_{2})^{\varphi}d\varphi$
$\displaystyle=$ $\displaystyle\left(\frac{A}{2}-\int_{0}^{2\pi}{\cal
E}_{\varphi}^{-1}{\cal F}_{\varphi}^{2}\,{\cal
K}d\varphi\right)\pi(\chi_{1})\pi(\chi_{2})$
$\displaystyle-\int_{0}^{2\pi}{\cal E}_{\varphi}^{-1}{\cal F}_{\varphi}\,{\cal
K}\,\big{(}\pi(\chi_{1})\partial_{\varphi}\alpha_{2}+\pi(\chi_{2})\partial_{\varphi}\alpha_{1}\big{)}d\varphi$
$\displaystyle-\int_{0}^{2\pi}{\cal E}_{\varphi}^{-1}{\cal
K}\,\partial_{\varphi}\alpha_{1}\partial_{\varphi}\alpha_{2}\,d\varphi\ .$

Here we used the following decomposition of the boundary vector field $\chi$:

$\chi^{\varphi}={\cal E}_{\varphi}^{-2}\left(\pi(\chi){\cal
F}_{\varphi}+\partial_{\varphi}\alpha\right)\ .$ (68)

This decomposition is based on the Hodge decomposition of the dual one-form,
$\chi^{\flat}=\frac{A}{C}\pi(\chi)\mathrm{vol}+d\alpha$ – in which case ${\cal
F}_{\varphi}=\frac{A}{C}{\cal E}_{\varphi}$ –, but can be more general. In
particular, as we will see when discussing the Hamiltonian constraint, it can
be useful to choose161616Let us call $\bar{\psi}=\frac{1}{2}(xdy-
ydx)=\frac{r^{2}}{2}d\varphi$, then $d\bar{\psi}=\overline{\mathrm{vol}}$ and
thus $A=\int_{M}\mathrm{j}^{*}\bar{\psi}$. But since that implies
$\int_{M}(\frac{A}{C}\mathrm{vol}-\mathrm{j}^{*}\bar{\psi})=0$ it follows that
the difference between these two forms is exact. In other words, starting from
the Hodge decomposition $\omega=\omega_{0}\,\frac{A}{C}\mathrm{vol}+d\alpha$
one can obtain the alternative decomposition
$\omega=\omega_{0}\,\mathrm{j}^{*}\bar{\psi}+d\tilde{\alpha}$.
$\chi^{\flat}=\frac{1}{2}\pi(\chi)\mathrm{j}^{*}(xdy-ydx)+d\alpha$, in which
case

${\cal F}_{\varphi}=\frac{f^{2}}{2}\ .$ (69)

We will always make this choice for the decomposition of the boundary
vectorfield $\chi$ in the following. Under the decomposition (68) we have
split the vector field $\chi$ into its zeromode $\pi(\chi)$, which is a real
number, and a (periodic) function $\alpha(\varphi)$.

The explicit form of the metric (3.3.1) shows that indeed the operator
$\mathbb{D}$ is symmetric, and that in general the metric can have mixed
signature.

The geodesic equation on $\mathrm{Diff}(M)$ associated to this metric can be
worked out via the generic form (24):

$\partial_{t}\chi^{\varphi}+3\chi^{\varphi}\nabla_{\varphi}\chi^{\varphi}+(\chi^{\varphi})^{2}\partial_{\varphi}\log{\cal
K}=({\cal E}_{\varphi}{\cal
K})^{-1}\left(\frac{1}{2}\pi(\partial_{t}\chi)+\nabla_{\varphi}\chi^{\varphi}\,\pi(\chi)\right)$
(70)

where

$\displaystyle\nabla_{\varphi}\chi^{\varphi}$ $\displaystyle=$
$\displaystyle\partial_{\varphi}\chi^{\varphi}+\Gamma_{\varphi\varphi}^{\varphi}\chi^{\varphi}=\partial_{\varphi}\chi^{\varphi}+\chi^{\varphi}\partial_{\varphi}\log{\cal
E}_{\varphi}\ .$ (71)

In the first two terms on the left hand side of (70) we recognize Burger’s
equation (27), but we see there are additional terms, in particular the right
hand side where an integral operator on $\chi$ appears.

Finally let us comment on the Hamiltonian constraint in this class of
examples. In the decomposition (49) we take

$\bar{\eta}=\pi(\chi)\frac{r^{2}}{2}d\varphi\,,$ (72)

so that indeed $d\bar{\eta}=\bar{\theta}_{\chi^{\flat}}$. Note that
$\bar{\nabla}_{(i}\bar{\eta}_{j)}=0$, reflecting the fact that the vectorfield
$\bar{\eta}^{i}$ is Killing, indeed it generates rotations around the origin
in $\bar{M}$. It follows that the contribution of $\bar{\eta}$ to the
Hamiltonian constraint (50) vanishes so that it simplifies to

$(\bar{\nabla}_{i}\partial^{i}\bar{\alpha})^{2}-\bar{\nabla}_{i}\partial_{j}\bar{\alpha}\bar{\nabla}^{i}\partial^{j}\bar{\alpha}=0\
.$ (73)

In Cartesian coordinates on $\bar{M}$, this reduces to the homogeneous real
Monge-Ampere (HRMA) equation:

$\left(\partial_{x}\partial_{y}\bar{\alpha}\right)^{2}-\partial_{x}^{2}\bar{\alpha}\partial_{y}^{2}\bar{\alpha}=0\,.$
(74)

The Monge-Ampere equation is a well-studied non-linear PDE that has its
historical origins in the geometric problem to find a hypersurface with a
prescribed extrinsic curvature. I.e. one interpretation of (74) is that a two
dimensional surface with the embedding $\left(x,y,\bar{\alpha}(x,y)\right)$ in
$\operatorname{\mathbb{R}}^{3}$ has zero extrinsic curvature. This observation
is probably not all too relevant to our setup, but we will use in an example
below one of the (few) explicit solutions171717See [17] for an overview. to
this equation that are known.

In addition to solving (74), note that $\bar{\alpha}$ needs to satisfy certain
boundary conditions: these can be extracted from the conditions
$\mathrm{j}^{*}\bar{\chi}=\chi$ and $\left.n^{i}\bar{\chi}_{i}\right|_{M}=0$,
which applied to (49) – together with (53, 62, 68, 69, 72) provide

$\left.\bar{\alpha}\right|_{M}=\alpha\quad\mbox{and}\quad\left.n^{i}\partial_{i}\bar{\alpha}\right|_{M}=\frac{ff^{\prime}}{2\sqrt{f^{\prime
2}+f^{2}}}\pi(\chi)\ $ (75)

Interestingly the boundary conditions (75) are of Cauchy type, rather than of
the typical boundary value type used for elliptic equations. Still this is not
necessarily a problem, due to the pseudo-elliptic nature of the HRMA equation,
although it is not obvious that the solution will be free of singularities.

####  3.3.2 Circle

The simplest special case is clearly that of the circle $M=S^{1}$, where
$f(\varphi)=R$ and ${\cal K}=R^{-1}$. One finds that

$\mathbb{D}\chi=R^{-1}\left(\pi(\chi)\partial_{\varphi}-2\chi\right)=-2R^{-3}\partial_{\varphi}\alpha\,\partial_{\varphi}\
.$ (76)

The rightmost expression – which is obtained upon using the decomposition (68)
– shows that the zeromode ,
$\pi(\chi)=\frac{1}{\pi}\int_{0}^{2\pi}\chi^{\varphi}d\varphi$, spans the
kernel of $\mathbb{D}$. This is no surprise, since such a zeromode describes a
vector field of the form $\chi=c\partial_{\varphi}$, which in this special
case is a Killing vector of the boundary. This is in accord with the
conjecture of [1] that the kernel of $\mathbb{D}$ corresponds to isometries of
the boundary.

The inner product (3.3.1) becomes

$\langle\chi_{1},\chi_{2}\rangle=-2R^{-2}\int_{0}^{2\pi}\partial_{\varphi}\alpha_{1}\partial_{\varphi}\alpha_{2}\,d\varphi\
.$ (77)

Since the zeromode $\pi(\chi)$ has disappeared, the corresponding metric on
$\mathrm{Diff}(S^{1})$ is degenerate. Instead it has a natural interpretation
as a right invariant, non-degenerate metric on
$\mathrm{Diff}(S^{1})/\mathrm{SO}(2)$, since the zeromode, which can now be
consistently quotiented out, generates the rotations of the circle.

From the point of view of GR this example is not very relevant, since the
metric (77) is negative definite and will thus not allow for null geodesics.
I.e. there will be no solutions to the Hamiltonian constraint (74), since the
compatibility of the boundary conditions (55) with the differential equation
is exactly the condition of vanishing norm. From a broader mathematical
perspective one can of course consider the geodesics irrespective of a
condition on their tangent vectors. The geodesic equation (70) for the metric
(77) reduces to Burger’s equation for the function
$u=\partial_{\varphi}\alpha$.

####  3.3.3 Ellipse

To provide an example where there are null directions, so that it is of
relevance to GR, one can consider the case where the boundary is an ellipse,
described by the polar curve

$f(\varphi)=\frac{\rho}{2}\frac{\sinh(2\Upsilon)}{\sqrt{\cosh^{2}\Upsilon-\cos^{2}\varphi}}\
.$ (78)

Here the positive constants $\rho$ and $\Upsilon$ set the scale and
eccentricity $e=1/\cosh\Upsilon$, respectively. It is computationally
convenient to replace the polar coordinates we have been using by elliptic
coordinates for this example:

$\displaystyle x$ $\displaystyle=\rho\cosh u\cos v\ ,$ (79) $\displaystyle y$
$\displaystyle=\rho\sinh u\sin v\ ,$ (80)

where $u\geq 0$ and $2\pi>v\geq 0$. In these coordinates 181818Note that the
angular coordinate $v$ along the boundary is related to the one we have used
previously as  $v=\arctan\left(\frac{\tan\varphi}{\tanh\Upsilon}\right)\ .$
the boundary ellipse is simply the curve $u=\Upsilon$, while the flat metric
on $\bar{M}$ and the induced metric on $M$ take the form

$d\bar{s}^{2}=\rho^{2}(\cosh^{2}u-\cos^{2}v)(du^{2}+dv^{2})\quad\mbox{and}\quad
ds^{2}={\cal E}_{v}^{2}dv^{2}\quad\mbox{where}\quad{\cal
E}_{v}=\rho\sqrt{\cosh^{2}\Upsilon-\cos^{2}v}\ .$ (81)

Furthermore one has

$\displaystyle n={\cal E}_{v}^{-1}\partial_{u}\ ,\qquad K={\cal K}({\cal
E}_{v}dv)^{2}\ ,\qquad{\cal K}=\frac{\rho^{2}}{2}{\cal E}_{v}^{-3}\sinh
2\Upsilon\ ,\qquad{\cal F}_{v}=\frac{\rho^{2}}{4}\sinh 2\Upsilon\ .$ (82)

A short computation then provides

$\displaystyle\frac{1}{2}\langle\chi_{1},\chi_{2}\rangle$ $\displaystyle=$
$\displaystyle-\frac{\pi\rho^{2}}{4}e^{-2\Upsilon}\pi(\chi_{1})\pi(\chi_{1})$
$\displaystyle-\frac{\sinh^{2}2\Upsilon}{8}\int_{0}^{2\pi}\frac{\pi(\chi_{1})\partial_{v}\alpha_{2}+\pi(\chi_{2})\partial_{v}\alpha_{1}}{(\cosh^{2}\Upsilon-\cos^{2}v)^{2}}dv$
$\displaystyle-\frac{\sinh
2\Upsilon}{2\rho^{2}}\int_{0}^{2\pi}\frac{\partial_{v}\alpha_{1}\partial_{v}\alpha_{2}}{(\cosh^{2}\Upsilon-\cos^{2}v)^{2}}dv$

where

$\pi(\chi)=\frac{2}{\pi\sinh
2\Upsilon}\int_{0}^{2\pi}(\cosh^{2}\Upsilon-\cos^{2}v)\chi^{v}dv\ .$ (84)

The geodesic equation (24) takes the form

$\sinh 2\Upsilon(\partial_{t}\chi^{v}+3\chi^{v}\partial_{v}\chi^{v})-\sin
2v\,\chi^{v}\pi(\chi)=(\cosh^{2}\Upsilon-\cos^{2}v)\left(\pi(\partial_{t}\chi)+2\partial_{v}\chi^{v}\pi(\chi)\right)\
.$ (85)

Although one could further work out the metric – and possibly the geodesic
equation as well – by applying a Fourier decomposition to $\alpha$, we will
refrain from providing details since the answer appears – to us – not
particularly enlightening.

As we have pointed out before, any solution to the Hamiltonian constraint –
equivalent here to the HRMA (74) – restricted to the boundary, provides a null
direction for the metric (3.3.3). Interestingly on the ellipse we managed to
find a closed form solution of the HRMA satisfying the correct boundary
conditions. Given that

$\bar{\eta}=\pi(\chi)\frac{\rho^{2}}{4}(\sin 2v\,du+\sinh 2u\,dv)$ (86)

one finds that the boundary conditions for the HRMA in terms of $\chi$ become

$\left.\partial_{u}\bar{\alpha}\right|_{u=\Upsilon}=-\frac{\rho^{2}\,\pi(\chi)}{4}\sin(2v)\quad\mbox{and}\quad\left.\bar{\alpha}\right|_{u=\Upsilon}=\alpha\
,$ (87)

where we have decomposed the boundary vectorfield as in (68). The natural
problem is to find a solution $\bar{\alpha}$ to HRMA satisfying (87) given a
boundary vector field $\chi$ that is null with respect to (3.3.3). But one can
also reverse the logic: any solution $\bar{\alpha}$ to HRMA satisfying the
left condition of (87) will provide a boundary vectorfield via the right
equation of (87) that is null with respect to (3.3.3) – see [1] for an
argument.

One can verify that the following function solves (74)191919This solution is
within the class of solutions to HMRA that are of the form [17]
$\bar{\alpha}(x,y)=y\ \Phi\left(\frac{y}{x}\right)\ .$ and satisfies the left
equality of (87):

$\bar{\alpha}(x,y)=-\frac{\rho^{2}\,\pi(\chi)}{2\cosh\Upsilon}\operatorname{sgn}(x)\,y\,\,_{2}F_{1}\left(\frac{1}{2},\frac{\cosh^{2}\Upsilon}{2};\frac{\cosh^{2}\Upsilon}{2}+1;-\frac{1}{\tanh^{2}\Upsilon}\frac{y^{2}}{x^{2}}\right)\
.$ (88)

The corresponding vectorfield $\bar{\chi}=(\bar{\eta}+d\bar{\alpha})^{\sharp}$
is plotted in Figure 2 in the special case202020If $\cosh\Upsilon=\sqrt{2}$
then
${}_{2}F_{1}\left(\frac{1}{2},1,2,-2\frac{y^{2}}{x^{2}}\right)=\frac{x^{2}}{y^{2}}\left(-1+\sqrt{1+2\frac{y^{2}}{x^{2}}}\right)\,.$
where $\cosh\Upsilon=\sqrt{2}$ and for $\rho^{2}\pi(\chi)=1$.

It is important to point out that this vector field is discontinuous at the
origin, which means that the corresponding solution to GR will have a
singularity there. We’ll comment further on this observation and possible
consequences in the discussion section.

Figure 2: The bulk vector field $\bar{\chi}$ associated to (88), when
$\cosh\Upsilon=\sqrt{2}$ and $\rho^{2}\pi(\chi)=1$.

##  4 Discussion

We reviewed how the $n+2$ dimensional Einstein equations give rise to geodesic
motion on the group of diffeomorphisms of the $n$ dimensional boundary $M$ of
a finite $n+1$ dimensional spatial region $\bar{M}$, as in [1]. The metric one
obtains on $\mathrm{Diff}(M)$ can be interpreted as the pull-back of the
Wheeler-deWitt metric onto an embedding of $\mathrm{Diff}(M)$ in superspace.
It takes the standard form
$G_{\phi}(\dot{\phi}_{1},\dot{\phi}_{2})=\int_{M}g(dR_{\phi^{-1}}\dot{\phi_{1}},\mathbb{D}\,dR_{\phi^{-1}}\dot{\phi_{2}})\mathrm{vol}\,g$
of a right-invariant metric, with $\mathbb{D}$ a symmetric operator on the
space of vector fields $\mathfrak{X}(M)=\mathfrak{diff}(M)$. The geodesic
equation for some simple choices for $\mathbb{D}$ correspond to some well
known equations of continuum mechanics, as we reviewed in Section 2.2.2. In
(31) we provided a new, more elegant, form for the operator $\mathbb{D}$
obtained from the Einstein equations. This operator differs from the standard
examples in continuum mechanics in a number of ways. First of all it is not
determined in terms of data intrinsic to $M$, but rather uses properties that
are extrinsic to $M$ and determined in terms of its embedding as the boundary
of $\bar{M}$. Secondly the operator $\mathbb{D}$ obtained in this way is not
positive, so that the associated metric $G$ on $\mathrm{Diff}(M)$ is pseudo-
Riemannian rather than Riemannian. Finally we should point out that in this
case $\mathbb{D}$ is an integral operator rather than a differential operator.
This has the consequence that the associated geodesic equation is an integro-
differential equation rather than just a PDE such as in the standard continuum
mechanics examples. These features were illustrated in detail in the $n=1$
class of examples discussed in Section 3.3 .

The geodesic motion of our setup describes a one parameter family of boundary
diffeomorphisms; to reconstruct a $n+2$ dimensional Lorentzian metric out of
this, one first of all needs to extend this boundary diffeomorphism into the
bulk space $\bar{M}$. This goes in two steps. The co-exact part of the vector
field generating the bulk diffeomorphism is uniquely determined through the
momentum constraint of GR; this is a linear equation which is well under
control. The exact part however can only be obtained upon solution of the
Hamiltonian constraint, which is a complicated non-linear PDE of Hessian type.
As we showed in Section 3.3, in the case $n=1$ the Hamiltonian constraint
becomes the homeogeneous real Monge-Ampere (HRMA) equation. This equation
should be solved under the Cauchy type boundary conditions (75), which puts
into doubt the existence of a smooth solution. Indeed, in the special case of
an elliptical boundary we managed to find the explicit closed form solution
(88) which leads to a mildly singular – discontinuous at a point – bulk vector
field. As a consequence the corresponding time dependent Lorentzian metric can
be expected to have a point singularity as well. Obtaining such a metric
explicitly remains out of reach however, as – even in this simple case – we
could only solve the HRMA equation for a very special boundary condition, not
for a one-parameter family of boundary conditions corresponding to an actual
geodesic on $\mathrm{Diff}(M)$. In general the three step procedure of

  * 1)

solving the geodesic equation on $\mathrm{Diff}(M)$ for the boundary
vectorfield

  * 2)

solving the Hamiltonian constraint to obtain the bulk extension of the
vectorfield

  * 3)

integrating the bulk vector field to a bulk diffeomorphism

appears too challenging to carry out explicitly. It could be interesting to
explore this numerically.

So far we have only considered the setup of a compact spatial region with
boundary. An interesting alternative would be to consider a non-compact
spatial region with an interior boundary, or possibly even a non compact
region with only an asymptotic boundary. Furthermore one could consider the
inclusion of a cosmological constant or other sources. Although we expect our
method to remain largely unchanged and applicable in such generalizations,
this has not been explored.

Finally let us shortly comment on our work in the context of the general
theory of gauge symmetry in the presence of boundaries, a topic which has seen
much recent activity – see e.g. [18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. The
presence of a boundary affects the phase space, see [28] for a recent in depth
discussion and further references. Such work tries to understand in detail the
kinematics – i.e. symplectic structure – of gauge theories in this setting.
The work [1] on which this paper is based, as well as the earlier results [29,
30] in Yang-Mills theory – see also [31] – are rather a first attempt to gain
an understanding of the dynamics – i.e. Hamiltonian – of that sector. The key
point is that the extra degrees of freedom associated to the boundary also
cover a new region of configuration space equipped with a non-trivial metric
[32, 33] describing the dynamics. Actually the global gauge sector212121With
‘global gauge sector’ we mean those gauge orbits in configuration space which
are not quotiented out, these are typically those generated by gauge
transformations that are non-trivial on the boundary. of configuration space
is in some sense the simplest one, – at least when restricted to orbits of
simple configurations such as the vacuum – as it has by definition a
homogeneous space structure. It is also in this sense that the work discussed
here is quite similar to the continuum mechanics setting, where the
diffeomorphism group plays the role of the configuration space of the
continuum material.

## Acknowledgements

We thank I. Gahramanov, B. Gürel, B. Oblak and A. Seraj for useful
discussions. EŞK is supported by the Research Fund of the Middle East
Technical University, Project Number DOSAP-B-105-2021-10763. DVdB is partially
supported by the Boğaziçi University Research Fund under grant number 21BP2.

##  Appendix A Translation of notation

Notation of [1] Notation here $\displaystyle M$ $\displaystyle\rightarrow$
$\displaystyle\bar{M}$ $\displaystyle\partial M$ $\displaystyle\rightarrow$
$\displaystyle M$ $\displaystyle h_{\mathrm{o}\,}$ $\displaystyle\rightarrow$
$\displaystyle\bar{g}_{\mathrm{o}}$ $\displaystyle h$
$\displaystyle\rightarrow$ $\displaystyle\bar{g}$ $\displaystyle k$
$\displaystyle\rightarrow$ $\displaystyle g$ $\displaystyle\chi$
$\displaystyle\rightarrow$ $\displaystyle\bar{\chi}$ $\displaystyle\zeta$
$\displaystyle\rightarrow$ $\displaystyle\chi$ $\displaystyle D^{\perp}\chi$
$\displaystyle\rightarrow$
$\displaystyle\left(j^{*}(\imath_{n}\bar{\theta}_{\chi^{\flat}})-\imath_{\chi}K\right)^{\sharp}$
$\displaystyle\eta$ $\displaystyle\rightarrow$ $\displaystyle\bar{\eta}$
$\displaystyle\phi$ $\displaystyle\rightarrow$ $\displaystyle\bar{\alpha}$

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