KEK-TH-2463, RBRC-1339

Energy of boundary of spacetime

Yu Hamada1*** E-mail address<EMAIL_ADDRESS>and Nobuyuki Matsumoto2†††
E-mail address<EMAIL_ADDRESS>

1Theory Center, High Energy Accelerator Research Organization (KEK),
Tsukuba 305-0801, Japan 2RIKEN/BNL Research center, Brookhaven National
Laboratory, Upton, NY 11973, USA

We consider how the energy can be stored in the boundary of spacetime, in
particular in a spherical bubble that can be made by a quantum gravitational
process. Our calculation is performed within the framework of classical
Einstein gravity by identifying the Gibbons-Hawking-York term as the membrane
action. We show that the energy of the bubble can be given consistently with
the Schwarzschild metric. The solution of the consistency condition suggests
positive membrane tension, which explains why we do not observe such
topological defects in ordinary experiences and also gives a mechanism for
suppressing the spacetime with boundary in a dynamical way.

##  1 Introduction

It is widely believed that quantum gravity allows change of topology of
spacetime [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20]. However, we do not observe such topological objects in ordinary
experiences, and thus we can expect that such topology change only occurs at
the size of the Planck length [3, 4, 5, 9, 10] and the objects are difficult
to exist in larger size. It is notable that most studies in the literature
have focused on topology changes without the appearance of the boundary, such
as wormholes and baby universes. As will be referred to in discussion, one of
the reasons to avoid configurations with boundary should be the geodesic
incompleteness. However, such restriction in the summation over all
configurations of spacetime disagrees with the characteristic of quantum
mechanics that we allow all the possible configurations in principle. When we
take such a viewpoint, it is desirable to have a mechanism to suppress the
configurations with boundary at large scale by the dynamics of the gravity.

In this letter, we explore such a mechanism by hypothesizing that the
boundaries, which we observe as the defects of the spacetime, carry energy
[21, 22, 23, 24, 25, 26], and it costs larger energy for larger size. We show
that this picture can be made precise within classical Einstein gravity by a
framework to calculate the energy of the defect. Our calculation is based on
the two observations: (i) Once the topological defect is settled in the form
that can be described by classical gravity, we assign the Gibbons-Hawking-York
term [27, 28] to the added boundary. (ii) The Gibbons-Hawking-York term can be
identified as the Nambu-Goto action [29, 30] describing a membrane whose
tension is given by the extrinsic curvature. This identification then allows
us to analyze the boundary as a dynamical membrane [31] and calculate its
energy from the membrane tension.

As an example, we consider a spherically symmetric bubble whose interior is
outside the spacetime and thus absolute nothing [32]. Our picture is however
different from conventional pictures of the bubble of nothing as we consider
its boundary and interpret boundary term as the membrane action, from which we
calculate the energy of the defect. It should also be noted that we work in
four dimension. At this stage, it is mainly for simplicity to consider this
particular shape of the defect. Nevertheless, since the symmetry greatly
simplifies the argument, we believe that the spherical defect gives a good
testing ground to check the validity of the identification.

Below, we show that the energy of the bubble can be given consistently with
the spacetime described by the Schwarzschild metric. The consistency condition
equates the energy of the bubble to the Schwarzschild mass, which has a unique
solution suggesting that the surface of the boundary be located outside the
event horizon. This means that they are distinguishable from microscopic black
holes [33, 34, 35]. Furthermore, the solution shows the tension is positive,
which indicates that our universe is stable against the creation of such
bubbles as it tends to shrink, not grow and swallow the whole universe [32,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48]. The radius of the defect
will be proportional to the stored energy, and therefore when the bubble is
created by quantum fluctuation of the order of the Planck mass, its size will
be the order of the Planck length. Then, since the bubbles tend to shrink,
they are always smaller than the order of the Planck length. This picture thus
gives a possible explanation why we do not observe such topological defects in
ordinary experiences.

##  2 Energy of the spherically symmetric bubble

We consider an appearance of a boundary when spacetime undergoes a topology
change. In the framework of the Einstein gravity, we assign the Gibbons-
Hawking-York term to the spacetime boundary:

$\displaystyle S_{B}\equiv{1\over 8\pi}\int_{\partial V}d^{3}y\,\sqrt{|{\rm
det}\,h_{ab}|}\ \varepsilon K.$ (2.1)

Here $h_{ab}$ is the induced metric on the boundary $\partial V$:

$\displaystyle ds^{2}|_{\partial V}=h_{ab}dy^{a}dy^{b},$ (2.2)

and $K$ is the trace of the extrinsic curvature, which can be written with the
unit outward normal $n_{\mu}$:

$\displaystyle K={n^{\mu}}_{;\mu}.$ (2.3)

The sign factor is defined as

$\displaystyle\varepsilon\equiv
n^{2}=\left\\{\begin{array}[]{cl}-1&\quad(n:{\rm timelike})\\\ 1&\quad(n:{\rm
spacelike})\end{array}\right..$ (2.6)

The main proposal of this paper is to interpret the boundary action (2.1) as a
membrane action with the position-dependent tension $T\equiv-{\varepsilon
K/(8\pi)}$. Writing the embedding function of the membrane as $X^{\mu}(y)$,
which is nothing but the location of the boundary, we have the induced metric:

$\displaystyle
h_{ab}=g_{\mu\nu}\frac{dX^{\mu}}{dy^{a}}\frac{dX^{\nu}}{dy^{b}},$ (2.7)

where $g_{\mu\nu}$ is the bulk spacetime metric. Here $K$ can be considered as
a functional of both $X^{\mu}$ and $g_{\mu\nu}$. The Gibbons-Hawking-York term
then obtains the roles of both the membrane action and the surface term for
the Einstein-Hilbert action.

To see that this identification gives a reasonable picture, we consider the
case that the induced topological defect is spherically symmetric (Fig. 1). To
simplify the situation further, we consider a thought experiment in which we
make the spherical bubble kept fixed with no ordinary matter present. The
spacetime should then be static and spherically symmetric that is described by
the Schwarzschild metric:

$\displaystyle
ds^{2}=-\Big{(}1-\frac{2M}{r}\Big{)}dt^{2}+\Big{(}1-\frac{2M}{r}\Big{)}^{-1}dr^{2}+r^{2}d\Omega^{2},$
(2.8)

where $d\Omega$ is the line element of the unit 2-sphere. In the ordinary
description, we identify $M$ as the total energy of the matter sourcing
gravity [49, 50]. However, in our case, there is only the boundary of
spacetime in the vacuum, and thus we identify the boundary, which we assume is
a membrane, to be the source of gravity.

In the spherically symmetric case, the correspondence between the Gibbons-
Hawking-York term and the membrane action is manifest because $K$, and thus
$T$, is constant over the surface due to the symmetry. Since the spacetime is
static with the Killing vector $\partial_{t}$ where $t$ corresponds to the
Minkowski time in the asymptotically flat region, we define the potential
energy $E_{\rm pot}$ with respect to $t$. Using the fact that the kinetic term
is zero since the membrane is held still, we have

$\displaystyle S_{B}=-\int dt\,E_{\rm pot}={1\over 8\pi}\int
dt\sqrt{|g_{tt}|}\,\varepsilon K\Sigma,$ (2.9)

where $g_{tt}\equiv-(1-2M/r)$ and $\Sigma$ is the two-dimensional area of the
membrane. We do not add a factor of $\sqrt{|g_{tt}|}$ in the left hand side as
in the case of the gravitational Hamiltonian [51, 52]. Again since the
spacetime is static, we arrive at the following relation:

$\displaystyle E_{\rm pot}=-{1\over 8\pi}\sqrt{|g_{tt}|}\,\varepsilon
K\Sigma=\sqrt{|g_{tt}|}\,T\Sigma,$ (2.10)

which is the desired relation giving the energy stored in the boundary of the
spacetime.

Figure 1: Schematic picture of a spherically symmetric bubble in the spacetime
at a time slice $t$. The boundary lies at $r=R$. The region $r<R$ (interior of
the bubble, shaded in the figure) is the outside of the spacetime. The
solution of the consistency condition suggests $R$ be larger than the
Schwarzschild radius $2M$.

##  3 Consistency condition and its solution

We denote the location of the bubble as $r=R$. The outward normal of the
bubble is

$\displaystyle
n_{\mu}=\frac{-\varepsilon\,\partial_{\mu}r}{\sqrt{|1-2M/r|}}\Bigg{|}_{r=R},$
(3.1)

where $\varepsilon={\rm sign}(1-2M/R)$. The overall sign of $n_{\mu}$ is
defined in such a way that

$\displaystyle n^{\mu}\partial_{\mu}r=-\sqrt{|1-2M/R|}<0,$ (3.2)

reflecting the fact that $r=0$ is the outside when seen from the spacetime.
From eq. (2.3), we have

$\displaystyle
K=-\varepsilon\cdot\frac{1}{R}\frac{1}{\sqrt{|1-2M/R|}}\Big{(}2-\frac{3M}{R}\Big{)},$
(3.3)

which gives the tension:

$\displaystyle T=\frac{1}{8\pi
R}\frac{1}{\sqrt{|1-2M/R|}}\Big{(}2-\frac{3M}{R}\Big{)}.$ (3.4)

Note that the tension $T$ is positive for $R>3M/2$.

We now have the two explicit expressions for the energy sourcing gravity. One
is simply the mass parameter $M$ in the Schwarzschild metric (2.8), which is
identical to the ADM mass [49] for this spacetime. The other is the internal
energy of the membrane:

$\displaystyle E_{\rm pot}=\frac{R}{2}\Big{(}2-\frac{3M}{R}\Big{)},$ (3.5)

where we set $\Sigma=4\pi R^{2}$, which is obtained by analyzing the boundary
action. For consistency, these two quantities should be equal:

$\displaystyle M=\frac{R}{2}\Big{(}2-\frac{3M}{R}\Big{)},$ (3.6)

from which we obtain the relation between the radius and the total mass of the
bubble:

$\displaystyle R=5M/2.$ (3.7)

This unique solution gives us important consequences: Firstly, the spherical
defect is not covered by the horizon, $R>2M$, and thus it is distinguishable
from microscopic black holes. This is important because it makes us possible
to directly interact with the remnant of a quantum gravitational effect.
Secondly, the tension is positive, and thus classically the bubble tends to
shrink; otherwise, our universe becomes unstable as the bubble can grow until
it swallows the whole universe, which adds another consistency to this
picture. We thus see that the identification of the Gibbons-Hawking-York term
with the membrane action gives a consistent description of the energy stored
in the boundary of spacetime.

##  4 Discussion

In this letter, we considered a framework to calculate the energy of the
boundary of spacetime and showed that the spherically symmetric bubble defect
can be described consistently with the Schwarzschild metric. However, it is
true that many assumptions are made in the argument. For example, as
emphasized in the main text, we thoroughly used the properties of the
Schwarzschild solution in identifying the potential energy, eq. (2.9). The
notion of energy itself is a subtle topic in general relativity (see, e.g.,
[53, 54, 55, 56, 57]) and we may need a more sophisticated formula to apply
the argument for general cases. Another assumption we implicitly made is that
the cosmological constant is irrelevant. Generalization in this direction is
interesting because its small value has been related to the topological nature
of quantum gravity [6, 12].

Another subtlety related to the argument is the choice of boundary action.
Though we used the Gibbons-Hawking-York term because it is the simplest
boundary term for the variational problem of the metric to be well-defined,
the boundary cosmological constant and boundary curvature terms may be added
in principle [28].111 We thank the referee of PTEP for raising this important
point. See also [58] for a discussion on the boundary cosmological constant.
However, it is possible to make an argument for the simplest choice. Let us
consider a general ADM decomposition [49] for a timelike coordinate $t$:

$\displaystyle ds^{2}=-N^{2}dt^{2}+h_{ab}(N^{a}dt+dy^{a})(N^{b}dt+dy^{b}).$
(4.1)

It is known that [27]:

$\displaystyle\sqrt{-g}R=N\sqrt{h}(^{3}R+K_{ab}K^{ab}-K^{2})+2\partial_{t}(\sqrt{h}K)-2\partial_{a}\\{\sqrt{h}(D^{a}N+N^{a}K)\\},$
(4.2)

where ${}^{3}R$ is the Ricci scalar on the $t$-constant surface $\Sigma_{t}$
and $K_{ab}$ the extrinsic curvature. We now consider the spacetime bounded by
the two $t$-constant surfaces $\Sigma_{t_{1}}$, $\Sigma_{t_{2}}$. The second
term of eq. (4.2) suggests the above-mentioned need for the Gibbons-Hawking-
York term on $\Sigma_{t_{1}}$ and $\Sigma_{t_{2}}$ to make the variational
problem well-defined. On the other hand, if we add additional terms such as
curvature or cosmological constant terms on the boundaries, they cannot be
canceled by the total derivatives in the bulk. The situation remains the same
after moving to the Hamiltonian formalism, in which one may interpret the
additional terms as an instant pulse potential for the induced metric at the
initial and final surfaces. Then, in this particular problem, the potential
depends on the artificial times slices. We consider such a theory to be an
unusual extension from the conventional mechanics.

Allowing the boundary degrees of freedom to exist, there are many topics that
can be explored. An interesting question is whether such bubbles can be
stabilized by a quantum effect at the Planck scale because if such stable
bubbles exist, they can be a candidate of dark matter. To see that this is a
viable possibility, let us assume that the bubbles were created and stabilized
by some quantum gravitational process before the inflationary epoch. Since the
relevant scale is the Planck scale $M_{P}$, the number density should be
$n_{b}\sim M_{P}^{3}$ (before the inflation) and the stabilized mass $M\sim
M_{P}$. During the inflationary expansion, the number density is exponentially
diluted by the factor $\exp{(-3N_{e})}$ with $N_{e}$ the number of e-foldings.
After the inflation ends and the universe is reheated, the yield variable is
given by $Y_{b}\equiv n_{b}/s\sim M_{P}^{3}\exp{(-3N_{e})}/(g_{S}T_{RH}^{3})$,
where $T_{RH}$ is the reheating temperature and
$g_{S}(\simeq\mathcal{O}(10^{2}))$ the effective number of relativistic
degrees of freedom. The fraction of the energy density of the bubble to the
critical density at the present universe then becomes

$\displaystyle\Omega_{b}h^{2}\sim Y_{b}\,\frac{M_{P}}{\mathrm{GeV}}\times
2.8\times 10^{8}\sim
0.1\bigg{(}\frac{10^{6}\,\mathrm{GeV}}{T_{RH}}\bigg{)}^{3}\exp(151-3N_{e}),$
(4.3)

where $h$ is the dimensionless Hubble parameter. By identifying eq. (4.3) with
the observed relic abundance of dark matter: $\Omega_{DM}h^{2}\simeq 0.12$
[59] and taking the e-foldings $N_{e}\simeq 50$ [60], we are led to the
prediction $T_{RH}\sim 10^{6}\,\mathrm{GeV}$, whose value is compatible with
leptogenesis scenarios with low reheating temperatures (see, e.g., [61] for a
review).

It is also important to consider the interaction of the bubbles with ordinary
matters [31] for detection. In this regard, the bubbles may be thought of as
particles. Note that geodesics around the bubble cannot be extended into the
$r<R$ region, and in this sense the spacetime is singular. For this, it should
be crucial to specify an appropriate boundary condition to determine the
behavior of the incoming particles (or the fields) at the surface of the
bubbles so that we can obtain predictions for the entire physical processes.
In such investigations, the full dynamical treatment of the bubble should also
be important, in which we simultaneously solve the equations of motion of the
membrane (i.e., of the embedding functions $X^{\mu}(y)$) and the time-
dependent Einstein equation.

Finally, since the bubble creation assumes a non-perturbative dynamics of
quantum gravity, it is intriguing in turn to consider what kind of theory can
provide this process microscopically, and to confirm that the bubble can be
described consistently from its creation to annihilation. Non-perturbative
dynamics of spacetime has been studied extensively in two-dimensional gravity
with matrix models, in which the correspondence between the spacetime
boundaries and $D$-branes (or $D$-instantons) is pointed out [62]. We consider
establishing our picture with such frameworks to be a promising next step
towards probing quantum gravity.

Studies along these lines are in progress and will be reported elsewhere.

## Acknowledgments

The authors thank Yuta Hamada and Taku Izubuchi for valuable discussions. NM
is also grateful to the radiation laboratory of RIKEN for their hospitality
during the initial stage of this work. The authors further thank the referee
of PTEP for valuable comments that helped improve the manuscript. YH is
supported by JSPS KAKENHI Grant Number JP21J0111, and NM is supported by
JP22H01222 and the Special Postdoctoral Researchers Program of RIKEN.

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