# The non-integrable mass and the scalar charge

Qiang Wen<EMAIL_ADDRESS>Yau Mathematical Sciences Center,
Tsingshua University, Beijing 100084, China  Perimeter Institute for
Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada  Department of Physics,
Renmin University of China, Beijing 100872, China.

###### Abstract

The non-integrable mass is studied explicitly in this paper. We study
Einstein-scalar gravities with weakened boundary conditions, and calculate the
mass with the Hamiltonian formula and Wald’s formula respectively. We find the
masses calculated by these two formulas are non-integrable. One way to solve
this non-integrability problem is to impose boundary conditions; however, we
find the mass calculated in this way has many other problems. This implies the
macroscopic thermodynamic properties of the scalar hairy black holes should be
described by one more charge beside the mass, which we call a scalar charge.
In fact, the non-integrability of mass will always arise when the matter
fields have charges which is not associate to any diffeomorphisms of
spacetime. We find the mass becomes non-integrable just because Wald’s formula
is used in a wrong way. Based on Wald’s formula and the existence of the
scalar charge, we propose a new definition for mass, with the modification
that we require the variation of the Hamiltonian to have no contribution from
the variation of the other charges. This new definition is also valid for much
more general gravities coupled to matter fields with other charges.

##  I Introduction

Although asymptotically AdS spacetimes (in this paper, when we refer
asymptotically AdS spacetimes we mean spacetimes asymptotically goes to AdS,
whose boundary conditions donot have to preserve all the asymptotic AdS
symmetries) have attracted remarkable interest in recent decades, our
understanding of their conserved charges, like energy, is still not clear
enough, especially when gravity is coupled to matter fields and the boundary
conditions cannot preserve all the asymptotic AdS symmetries. Several method
have been developed to define conserved charges in asymptotically $AdS_{d}$
spacetimes: the Hamiltonian definition developed by Henneaux and Teitelboim
Henneaux:1985tv , the AMD’s (Ashtekar-Magnon-Das) conformal method
Ashtekar:1984zz ; Ashtekar:1999jx , the “counterterm subtraction method”
Henningson:1998gx ; Balasubramanian:1999re ; Skenderis:1999nb ; Bianchi:2001de
; deHaro:2000wj ; Skenderis:2002wp ; de Haro:2000xn ; Skenderis:2000in ;
Bianchi:2001kw ; Papadimitriou:2004ap . There are also other methods like the
KBL method Katz:1996nr ; Deruelle:2004mv , the spinor method Deruelle:2004mv ;
Gibbons:1982jg  and the “pseudotensor” method Abbott:1981ff , which we will
not consider further in this paper. Based on the general “covariant phase
space formalism” developed by Wald et. al. Wald:1993nt ; Iyer:1994ys ,
Hollands, Ishibashi and Marolf developed another method Hollands:2005wt  to
calculate the conserved charges in asymptotically AdS-spacetimes, and made a
comparison between those methods mentioned above. They showed that, with
boundary conditions preserving all the asymptotic AdS symmetries and the
matter fields decreasing fast enough when approaching the boundary, those
methods are all equivalent to each other, except the energy calculated by
“counterterm subtraction method” has an additional constant term in some
cases, which can be interpreted as the Casmir energy of the dual CFT.

These methods all have strict requirements: not only the boundary conditions
should preserve all the asymptotic AdS symmetries, but also the stress-tensor
or matter fields should decrease fast enough when approaching boundary.
However, in general these requirements are not satisfied by the asymptotic
behaviors of the solutions, and thus, the boundary conditions are weakened.
For example, consider gravity minimally coupled to a scalar field $\phi$ with
a scalar mass $m$ above or saturating the BF (Breitenlohner-Freedman) bound BF
, with the Lagrangian (note that in this paper we use the convention $16\pi
G=1$) expressed as

$\mathcal{L}=\sqrt{-g}\left(R-\frac{1}{2}(\partial\phi)^{2}-V(\phi)\right).$
(1)

It is known that, the tachyonic ($m^{2}<0$) scalars AdS spacetime are stable
provided their mass is above the BF bound. In these theories the asymptotic
AdS symmetries cannot be fully preserved by the boundary conditions in general
(which we will see later), and also the scalar field usually decreases slower
than the requirements mentioned above. One can find many examples in
Lu:2014maa , where the asymptotic behaviors of many different solutions are
studied.

If we consider a scalar mass slightly above the BF bound and use the static
spherical metric ansatz,

$ds^{2}=-h(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{d-2}^{2}\,,\qquad\phi=\phi(r)\,,$
(2)

where $d$ is the spacetime dimension and $d\Omega_{d-2}^{2}$ is the metric on
the unit $(d-2)$-sphere, then the asymptotic behaviour of the scalar field is
given by

$\displaystyle\phi(r)=\frac{\phi_{1}}{r^{\lambda_{+}}}+\frac{\phi_{2}}{r^{\lambda_{-}}}+\cdots\,.$
(3)

The two parameter which satisfy $\lambda_{+}>\lambda_{-}$, are determined by
the spacetime dimentsion $d$ and the scalar mass $m^{2}$ (see (53) and (54)).
When $\phi_{1}\neq 0$, the speed requirement for the decreasing of the scalar
field cannot be satisfied. And the asymptotic AdS symmetries cannot be fully
preserved unless we impose a family of special boundary conditions between
$\phi_{1}$ and $\phi_{2}$ Henneaux:2004zi ; Hertog:2004dr ; Henneaux:2006hk ,
which are:

$\displaystyle a)~{}~{}\phi_{1}$ $\displaystyle=0\,,$ (4) $\displaystyle
b)~{}~{}\phi_{2}$ $\displaystyle=0\,,$ (5) $\displaystyle c)~{}~{}\phi_{2}$
$\displaystyle=\mathcal{C}\,\phi_{1}^{\lambda_{-}/\lambda_{+}}\,,$ (6)

where $\mathcal{C}$ is an arbitrary constant with out variation.

With their requirements not satisfied, the methods mentioned above begin to
give different masses. Although they still give finite results in many cases
(for details one should consult Lu:2014maa ). We can not trust their results
in these situations.

Under such weakened boundary conditions, some work Henneaux:2004zi ;
Hertog:2004dr ; Henneaux:2006hk  was done to calculate the conserved charges
in Einstein-scalar gravities using the the Hamiltonian formula (or the
Henneaux-Teitelboim construction). Soon after, the same results Hertog:2005hm
; Amsel:2006uf  were obtained using Wald’s formula Wald:1993nt ; Iyer:1994ys .
The only requirement on boundary conditions for the Hamiltonian formula and
Wald’s formula is to generate finite conserved charges, so it seems their
results can be trusted. Their energy has contributions from both the gravity
sector and the scalar field sector. Since the scalar field doesnot decrease
fast enough when approaching the boundary, both of the two contributions have
divergent terms. However, when we add them together, the divergent terms
cancel with each other, thus, a finite result comes out in the end. In the
next section we give a brief review on the Hamiltonian formula and Wald’s
formula, and show they are equivalent to each other in Einstein-scalar
gravities.

However, the masses calculated in these two ways are non-integrable, or in
other words the variation of the mass cannot be written as a total variation.
This makes the energy ill-defined. We will explain why we call it ill-defined
in section III. In fact the non-integrability of mass also happens in
gravities couples to other matter fields. In some cases, this problem can be
solved by some proper gauge choice, for example the RN-AdS (Reissner-
Nordstrom) black holes, however this does not work for the Einstein-scalar
gravities. We also discuss another problem of these definitions of mass, which
is related to the interpretation of the entropy of some hairy solutions. The
validity of Wald’s formula to calculate the conserved energy of scalar hairy
solutions was discussed in Hertog:2005hm ; Amsel:2006uf ; however, we do not
agree with their conclusions here. We use the RN-AdS black holes as an example
to show that Wald’s formula is in fact not used properly for calculating the
energy of those scalar hairy solutions.

Wald’s formula is a very powerful method to study thermodynamics in gravities,
especially when the gravities have coupled matter fields and weakened boundary
conditions. Based on Wald’s formula and with a little modification, we give a
new definition of mass for these theories including Einstein-scalar gravities.
Our definition is different from Hertog:2005hm ; Amsel:2006uf . In order to
definite the mass in our way, we need to introduce a new charge attached to
the scalar field (section IV), we call this a scalar charge. The existence of
this charge has already been discussed before from the black hole
thermodynamics point of view Gibbons:1996af ; Lu:2013ura ; Lu:2014maa ;
Liu:2013gja . This definition gives the same results with the methods
mentioned in the first paragraph when all their requirements are satisfied.
And when the boundary conditions are weakened, the our results are different
from the definitions given by Henneaux:2004zi ; Hertog:2004dr ;
Henneaux:2006hk ; Hertog:2005hm ; Amsel:2006uf  and do not have the problems
mentioned in the previous paragraph.

We generalize our definition of mass to general gravities couple to matter
fields in the Sec. VI. And in the last section we further discuss the mass in
gravity with boundary conditions.

##  II Hamiltonian formula and Wald’s formula in Einstein-scalar gravities

###  II.1 The Hamiltonian formula

The Hamiltonian formula directly construct the Hamiltonian of the theory
Regge:1974zd , which is the usual Hamiltonian supplemented by an addition of a
surface integral on the boundary

$\textbf{H}=\int_{\Sigma}\xi^{\mu}\mathcal{H}_{\mu}+\text{surface integral
terms}$ (7)

where $\Sigma$ is a spacelike surface and $\mathcal{H}_{\mu}$ are the
constraints. Imposing the constraints

$\mathcal{H}_{\mu}=0,$ (8)

the energy is just given by the surface integral terms. These surface integral
terms are defined on a variational level, and have contributions from both the
gravity sector, $\delta Q_{G}$, and the scalar field sector, $\delta
Q_{\phi}$. Consider the Lagrangian given by (1), we can calculate out these
two contributions:

$\displaystyle\delta Q_{G}=\int
d{S_{i}}\bar{G}^{ijkl}(\xi^{\perp}\bar{D}_{j}\delta h_{kl}-\delta
h_{kl}\bar{D}_{j}\xi^{\perp})\,,$ (9) $\displaystyle\delta Q_{\phi}=-\int
d{S_{i}}~{}\xi^{\perp}\delta\phi D^{i}\phi\,,$ (10)

where $G^{ijkl}=1/2g^{1/2}(g^{ik}g^{jl}+g^{il}g^{jk}-2g^{ij}g^{kl})$,
$h_{ij}=g_{ij}-\bar{g}_{ij}$ is the deviation from the pure AdS spatial metric
$\bar{g}_{ij}$ and $\xi^{\perp}=\xi\cdot n$ with $n$ the unit normal to
$\Sigma$.

Both $\delta Q_{G}$ and $\delta Q_{\phi}$ have divergent terms as the scalar
field decays too slow when approaching the boundary; however, the divergent
terms cancel with each other when adding them together Henneaux:2004zi ;
Hertog:2004dr ; Henneaux:2006hk . Thus we get a finite variation of energy at
last.

Consider the spherically metric $Ansatz$ (2), we find the variation of mass
defined by this formula $\delta M_{H}=\delta Q_{G}+\delta Q_{\phi}$ is given
by

$\delta M_{H}=-\omega_{d-2}r^{2}\sqrt{\frac{h}{f}}\left(\frac{d-2}{r}\delta
f+f\phi^{\prime}\delta\phi\right)\Big{|}_{r\rightarrow\infty}\,,$ (11)

where $\omega_{d-2}$ is the volume of the unit $(d-2)$-sphere. The $d=4$ case
is given in Hertog:2004bb . One can get the mass by integrating out $\delta
M_{H}$

###  II.2 Wald’s formula

Here we write the covariant Lagrangian as a $d$-form L, and its variation can
always be written in the form

$\delta\textbf{L}=\textbf{E}\cdot\delta\varphi+d\Theta,$ (12)

where $\Theta$ is a $(d-1)$-form, $\varphi$ represents all the metric
functions and the scalar field. And E represents the constraints, which will
vanish when the equations of motion are satisfied. The Noether current defined
by Wald and Iyer is defined as

$\textbf{J}=\Theta-\xi\cdot\textbf{L}\,,$ (13)

which is shown Lee:1990nz  to satisfy

$d\textbf{J}=-\textbf{E}\mathcal{L}_{\xi}\phi\,.$ (14)

This means the current J is conserved when $\varphi$ satisfies the equations
of motion. Hence J is a closed form and can be written as

$\textbf{J}=dQ.$ (15)

It has been shown in Wald:1993nt  that when $\xi$ is a Killing vector, we can
get

$d\delta Q-d(\xi\cdot\Theta)=0\,.$ (16)

This indicates the $(d-2)$-form $\delta Q-(\xi\cdot\Theta)$ is closed.

Stokes’ theorem indicates that the integral of $(\delta
Q_{\xi}-\xi\cdot\Theta)$ over a spherical surface is independent of the radius
of the surface. In other words, the integral on the horizon $S_{r_{0}}$ and on
the boundary $S_{\infty}$ are the same,

$\displaystyle\int_{S_{r_{0}}}(\delta
Q_{\xi}-\xi\cdot\Theta)=\int_{S_{\infty}}(\delta Q_{\xi}-\xi\cdot\Theta).$
(17)

If one is familiar with Wald’s formula, one would know that, the equation
above usually gives the thermodynamic first law of the solutions. For pure
gravity, when we take $\xi=\partial/\partial t$, the right hand side of (17)
gives $\delta M$ while the left hand side gives $T\delta S$, where
$S=\frac{A}{4G}$ is proportional to the area $A$ of the event horizon. Thus,
(17) just gives the standard thermodynamic first law $T\delta S=\delta M$.

The Wald’s-formula definition of mass is just given by

$\delta M_{W}=\int_{S_{\infty}}(\delta
Q_{\xi}-\xi\cdot\Theta)\Big{|}_{\xi=\partial/\partial t},$ (18)

where $M_{W}$ is considered as the conjugate charge associated to the Killing
vector $\xi=\partial/\partial t$. The existence of this mass require the
existence of a ($d-1$)-form B which satisfies

$\delta\int_{S_{\infty}}\xi\cdot\textbf{B}=\int_{S_{\infty}}\xi\cdot\Theta\,,$
(19)

so the variation $\delta M_{W}$ can be written as a total variation. This
makes sure that the mass can be integrated out with the result independent
from the integral path. In Wald:1999wa , this requirement is reinterpreted as,
for any variation $\delta_{1}$ and $\delta_{2}$ tangent to the space of
solutions, the satisfaction of the following equation

$\int_{S_{\infty}}\xi\cdot\left(\delta_{1}\Theta(\varphi,\delta_{2}\varphi)-\delta_{2}\Theta(\varphi,\delta_{1}\varphi)\right)=0\,.$
(20)

Let us now consider the Einstein-scalar gravity (1). We find the $(d-2)$-form
$\xi\cdot\Theta$ also has contributions from both the gravity sector
$\xi\cdot\Theta^{G}$, and the scalar field sector, $\xi\cdot\Theta^{\phi}$.
After some calculations we get

$\displaystyle\xi\cdot\Theta^{G}~{}_{i_{1}\cdots
i_{d-2}}=\epsilon_{i_{1}\cdots i_{d-2}t\mu}(g^{\mu m}g^{\nu n}-g^{\mu n}g^{\nu
m})D_{n}\delta g_{\nu m},$ (21)
$\displaystyle\xi\cdot\Theta^{\phi}~{}_{i_{1}\cdots
i_{d-2}}=-\epsilon_{i_{1}\cdots i_{d-2}t}~{}^{\mu}(D_{\mu}\phi~{}\delta\phi),$
(22) $\displaystyle Q_{i_{1}\cdots i_{d-2}}=\epsilon_{i_{1}\cdots
i_{d-2}}~{}^{\mu t}D_{\mu}\xi_{t},$ (23)

where $\epsilon$ is the Levi-Civita tensor. Applying the static metric ansatz
(2) and choosing the Killing vector to be $\xi=\partial/\partial t$, we have
Lu:2014maa ; Liu:2013gja

$\displaystyle\delta
M_{W}=-\omega_{d-2}r^{d-2}\sqrt{\frac{h}{f}}\left(\frac{d-2}{r}\delta
f+f\phi^{\prime}\delta\phi\right)\big{|}_{r\rightarrow\infty}\,,$ (24)

Compare the two results (24) and (11), we see that, the Wald’s-formula
definition of mass is equivalent with the Hamiltonian-formula definition of
mass in this case.

##  III Ill-defined mass

###  III.1 Problem with a non-integrable mass

Although the surface integrals on the boundary in the above two formulas are
finite fro Einstein-scalar gravities, their definitions of mass have some
fundamental problems. First the variation of their mass are actually non-
integrable.

We use the example in Hertog:2004ns  to show this problem. Consider a
4-dimensional gravity (1) with a scalar potential

$\displaystyle V(\phi)=-2g^{2}(\cosh\phi+2),$ (25)

where $g=1/\ell$. It can be shown that the scalar mass $m^{2}=-2$ is above the
BF bound. By solving the linearized equation of motion for the scalar field,
the large radius decay of the scalar field is given by

$\phi(r)=\frac{\phi_{1}}{r}+\frac{\phi_{2}}{r^{2}}+\cdots.$ (26)

We now impose the metric ansatz (2), and numerically solve the equations of
motion. By varying the initial values of the radius of the horizon $r_{0}$ and
the value of the scalar field on the horizon $\phi(r_{0})$, we get the space
of solutions on the plane of ($\phi_{1},\phi_{2}$), which are described by two
independent parameters, see Fig. (1). Inside this space we can choose boundary
conditions $\phi_{2}(\phi_{1})$ freely. One thing we should keep in mind is
that some points on ($\phi_{1},\phi_{2}$) plane may represent many different
solutions. In other words, it is not a one to one match between the plane and
the space of solutions. For example the origin $(0,0)$ represents all
Schwarzschild-AdS solutions with no scalar hair.

Figure 1:  The solid black line represents the line of soliton solutions with
$r_{0}=0$. As $r_{0}$ increases the line of solutions goes right until to the
dashed blue line which is the line of solutions with an event horizon around
$r_{0}=0.5$. Continuing to increase $r_{0}$, the line of solutions then turns
left, the four dotted lines from right to left representing lines of solutions
with $r_{0}=2,5,10,20$. In the limit $r_{0}\rightarrow\infty$ the line of
solutions approaches the $\phi_{2}$ axis. The orbit of the solution line gives
the space of solutions on ($\phi_{1}$,$\phi_{2}$) plane

The energy $\delta M_{H}$ of these solutions is calculated in Hertog:2004ns
with the Hamiltonian formula. Follow (11), we get

$\delta
M_{H}=4\pi\left(-2\delta\alpha-\delta(\frac{1}{3\ell^{2}}\phi_{1}\phi_{2})+\frac{\phi_{2}\delta\phi_{1}}{\ell^{2}}\right)\,,$
(27)

with $\alpha$ given by the large $r$ bahavior of $h(r)$,

$h(r)=\frac{r^{2}}{\ell^{2}}+1+\frac{\alpha}{r}+\cdots.$ (28)

We see that $\delta M_{H}$ cannot be written as a total variation, hence is
non-integrable. The directest way to solve this problem is assuming a boundary
condition $\phi_{2}(\phi_{1})$ thus giving an integral path. However this way
will lead to controversail results. This means for a specific solution with
specific metric and scalar field we cannot define its mass without imposing a
boundary condition (or integral path) $\phi_{2}(\phi_{1})$. Further more, the
choice of boundary conditions in the space of solutions can be quite arbitrary
and different boundary conditions will give different masses. This implies the
mass of a specific solution is also arbitrary. This contradicts with the usual
understanding of mass.

We can also use Wald’s formula to calculate the mass. Applying to this case,
the variation of mass calculated by Wald’s formula (18) agrees with (27). The
check the validity of the calculation, the holding of Eq. (20) is also checked
in Hertog:2005hm ; Amsel:2006uf , provided we have assumed a boundary
condition $\phi_{2}(\phi_{1})$. Here we just cite the result, which is

$\frac{\partial\phi_{2}(\phi_{1})}{\partial\phi_{1}}(\delta_{1}\phi_{1}\delta_{2}\phi_{1}-\delta_{2}\phi_{1}\delta_{1}\phi_{1})=0.$
(29)

This means Eq. (20) holds and it seems the calculation seems valid. However
this does not mean the energy is independent of the integral path because when
we choose a boundary condition $\phi_{2}(\phi_{1})$, we have already chosen an
integral path in the space of solutions. Hence we do not agree with that the
holding of Eq. (29) after choosing a boundary condition would make the
definition of mass (18) valid.

We argue that, imposing a boundary condition to make a non-integrable mass
integrable may be inappropriate.

To further explain our arguments, we give a more convincing example, which is
the calculation of the mass of four-dimensional electrically charged RN-AdS
black holes (we set the magnetic charge to be zero for simplicity) using
Wald’s formula. The black hole metric is given by

$\displaystyle ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{2}^{2}\,,$
(30) $\displaystyle
f(r)=\frac{r^{2}}{\ell^{2}}+1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\,,$ (31)

with two integration constants $M$ and $Q$, where $M$ is acknowledged as the
black hole mass and $Q$ is the electric charge. We can make a gauge choice
such that the gauge field $A$ vanishes on the horizon. Applying Wald’s formula
we find (for details, see Gao:2003ys ; Lu:2014maa )

$\displaystyle\int_{S_{r_{0}}}(\delta Q_{\xi}-\xi\cdot\Theta)=T\delta S\,,$
(32) $\displaystyle\int_{S_{\infty}}(\delta Q_{\xi}-\xi\cdot\Theta)=\delta
M-\Phi(M,Q)\delta Q=\delta M_{W}\,,$ (33)

where $\Phi$ is the electrical potential. Wald’s formula gives the standard
thermodynamic first law $T\delta S=\delta M-\Phi\delta Q$. And like the case
in Einstein-scalar gravities, $\delta M_{W}$ is also non-integrable here. If
we assume a boundary condition $M(Q)$ again, we can integrate out $M_{W}$ and
(20) will hold. Following the logic of Hertog:2005hm ; Amsel:2006uf , then the
definition of mass (18) seems valid here. However, in fact, the result is

$\displaystyle M_{W}=\int\left(\frac{\partial M(Q)}{\partial
Q}-\Phi(Q)\right)dQ$ (34)

rather than the acknowledged mass $M$.

###  III.2 The entropy problem of some massless solutions

The Hamiltonian formula and Wald’s formula definitions of mass have another
problem, which is the failure to interpret the entropy of some scalar hairy
solutions.

There are two ways to define the entropy of a black hole. The first is the
definition from the thermodynamic point of view. The variation of the charges
of the black hole satisfies a relationship (here is the Wald’s formula), and
if we consider the black hole as a thermodynamic system and interpret this
relationship as the thermodynamic first law of the black hole, then there is a
term which can play the role of entropy in thermodynamics. And the entropy is
given by a simple formula $S=\frac{A}{4G}$. This entropy is known as the
Bekenstein-Hawking entropy.

The second definition is from the statistical point of view. It is widely
believed that, a black hole is actually a quantum system with discrete energy
levels Susskind:1993if , the total number of states which have the same
macroscopic thermodynamic quantities is the exponential of the entropy. This
definition is believed to be the microscopic origin of the first one, thus,
both of them should give the same results. Inspired by the principle of
holography, the quantum structures of a few black holes, for example the
asymptotically $AdS_{3}$ black holes Brown:1986nw ; Strominger:1997eq  and
extremal Kerr black holes Guica:2008mu , have already been studied, and the
statistical entropy counted by Cardy’s formula Cardy:1986ie  do coincide with
the Bekenstein-Hawking entropy. However, for more general black holes, it is
still extremely hard to study their quantum structures.

In static Einstein-scalar gravities, if we define the mass with the
Hamiltonian formula and Wald’s formula, the mass $M$ is the only charge in the
first law. Then the statistical entropy $S(M)$ is a function of only $M$, and
counts the number of microscopic states which have a the same mass $M$.
However, we find some scalar hairy solutions whose Bekenstein-Hawking entropy
and statistical entropy are not consistent with each other. This indicates
that the definition of mass may be wrong.

The first example is the four-dimensional static spherical soliton solutions
in Hertog:2004ns  (and in Sec. III.1). We set $r_{0}=0$ and give different
initial values for $\phi(0)$, then numerically solve the equations of motion.
We get a series of soliton solutions (not black holes with $r_{0}=0$) which
can be described by the soliton line (the solid black line) in Fig.(1). If we
choose this soliton line as the boundary condition and define $\delta_{s}$ as
the variation along this line, we should have $T\delta_{s}S=0$, since all the
solutions on this line have no horizon. Thus Wald’s formula (17) gives

$\delta_{s}M_{W}=4\pi\left(-2\delta_{s}\alpha-\delta_{s}(\frac{1}{3\ell^{2}}\phi_{1}\phi_{2})+\frac{\phi_{2}\delta_{s}\phi_{1}}{\ell^{2}}\right)=0\,.$
(35)

This indicates the masses $M_{W}$ of all the solitons are the same. As the
soliton line goes through the (0,0) point on ($\phi_{1},\phi_{2}$) plane,
which is the pure vacuum $AdS_{4}$ spacetime when there is no horizon, all the
solitons are massless (if we define the $AdS_{4}$ vacuum as massless).

On one hand, the massless solitons have no horizon, hence no Bekenstein-
Hawking entropy. While on the other hand, the number of states (or soliton
solutions) with zero mass is absolutely more than one, thus their statistical
entropy $S(M=0)$ should be nonzero. The inconsistence of the two entropies
implies that the definition of mass given by Wald’s formula might be wrong.

Another example is given by some 3-dimensional scalar hairy solutions
constructed in Wen:2015xea . Theories (1) with scalar potentials

$\displaystyle
V(\phi)=g^{2}\left(\cosh^{4\mu}\psi\right)\left(\mu\tanh^{2}\psi-2\right),$
(36a) $\displaystyle\psi=\frac{\phi}{2\sqrt{2\mu}},$ (36b)

admit massless (according to the Hamiltonian formula and Wald’s formula) black
hole solutions

$\displaystyle\phi(r)=2\sqrt{2\mu}~{}\text{Arctanh}\frac{1}{\sqrt{1+r/q}}\,,$
(37a) $\displaystyle
ds^{2}=-g^{2}r^{2}dt^{2}+\frac{1}{g^{2}r^{2}(q/r+1)^{2\mu}}dr^{2}+r^{2}d\theta^{2}\,,$
(37b)

with $\mu$ a positive constant which marks different theories and $q$ the only
integration constant which describes the scalar hair. The boundary conditions,
which are chosen as the asymptotic behavior of these solutions, preserve all
the AdS symmetries. We want to point out that solutions described by (37) only
represent a line in the space of solutions. With these boundary conditions, we
can integrate (18) and get the mass defined by Wald’s formula $M_{W}=0$.

Just like the previous soliton solutions, the two entropies are inconsistent.
There is an integration constant $q$ which is not related to the mass, which
means the massless state is degenerate (this conclusion is also made in
Henneaux:2002wm ) and should have a non-zero statistical entropy. On the other
hand, these black hole solutions all have vanishing horizon, thus, have
vanishing Bekenstein-Hawking entropy $S=0$.

To solve this entropy problem, we need another quantity to determine the
macroscopic thermodynamic properties of these solutions, which we call a
scalar charge. Then we find that the mass depend on the scalar hair, in such a
way that these solutions are not all indistinguishable massless black holes.

##  IV Introducing a new scalar charge

###  IV.1 The existence of a new charge attached to scalar field

The existence of the scalar charge has already been discussed not only in
asymptotically AdS spacetimes Lu:2013ura ; Lu:2014maa ; Liu:2013gja , but also
in asymptotically flat spacetimes Gibbons:1996af . The authors of these papers
assume the existence of such a charge because their definitions of mass are
different from the Hamiltonian formula and Wald’s formula, and the
thermodynamic analysis need the scalar field to play the role as a charge.

For example, consider again the four-dimensional theory in section III.1. If
we define the AMD mass Ashtekar:1984zz ; Ashtekar:1999jx  $M_{A}$ as the right
mass (as in Lu:2014maa ), then the variation of the non-integrable mass
$M_{W}$ (or $M_{H}$) can be expressed as

$\delta M_{W}=\delta
M_{A}+\frac{4\pi}{3\ell^{2}}(2\phi_{2}\delta\phi_{1}-\phi_{1}\delta\phi_{2})\,,$
(38)

with $M_{A}$ given by

$\displaystyle M_{A}=-8\pi\alpha\,.$ (39)

This looks just like the non-integrable $\delta M_{W}$ (71) of the RN black
holes. It is thus natural to consider the second term of the right hand side
of Eq. (38) as a contribution from a scalar charge

$\frac{4\pi}{3\ell^{2}}(2\phi_{2}\delta\phi_{1}-\phi_{1}\delta\phi_{2})=-\Phi_{S}\delta
Q_{S}\,,$ (40)

(we will give another definition of the scalar charge later). Wald’s formula
(17), then gives the familiar first law

$T\delta S=\delta M_{A}-\Phi_{S}\delta Q_{S}\,.$ (41)

This have the same formula as the first law of electrically charged RN black
holes.

One interesting fact is that, as mentioned in Lu:2013ura ; Lu:2014maa ;
Anabalon:2014fla , when the boundary conditions preserve all the AdS
symmetries (see (4)), the contribution from the scalar charge $\Phi_{S}\delta
Q_{S}$ in (40) vanishes. We can see this directly from (38) and get $\delta
M_{W}=\delta M_{A}$. It seems that these special boundary conditions makes
$\delta M_{W}$ integrable and trustful, however in these cases the boundary
conditions are also chosen before we calculate the mass, so $M_{W}$ still
depends on the integration path.

The non-integrability of $\delta M_{W}$ not only happens in such Einstein-
scalar gravities and RN-black holes, but also in many other gravities coupled
to other matter fields Lu:2013ura ; Chow:2013gba ; Liu:2014tra ; Liu:2014dva ;
Fan:2014ixa ; Fan:2014ala ; Fan:2015yza . These theories clearly show that
when there exist charges which are not the Noether charge of any
diffeomorphisms of spacetime along Killing vectors, the integral (18) would be
non-integrable. Comparing with these theories, assuming the existence of a
scalar charge is reasonable.

Another reason to assume the existence of the scalar charge $Q_{S}$ is that,
the full space of solutions has two independent integration constants, so it
is hard to believe that energy is the only charge. If the scalar charge
$Q_{S}$ exists, the macroscopic thermodynamic properties of the solutions
should be described by two parameters $M$ and $Q_{S}$, so the entropy
$S(M,Q_{S})$ would count the microscopic states which have mass $M$ and scalar
charge $Q_{S}$. We will see this entropy can be consistent with the
Bekenstein-Hawking entropy.

Before we give a new definition of mass, we need to define the scalar charge
first. Since there is no symmetry corresponding to this charge, it cannot be
defined as a Noether charge, thus, there may be some ambiguities to define the
scalar charge. Here we define the scalar charge $Q_{S}$ as the coefficient of
the leading term of the large $r$ expansion of the scalar field

$\displaystyle Q_{S}=\phi_{1}.$ (42)

It is quite natural to define the scalar charge in this way. First, $\phi_{1}$
is an independent integration constant which describes the scalar hair.
Second, when $\phi_{1}=0$ the scalar field vanishes everywhere, and also the
contribution of the scalar field in $\delta M_{W}$ vanishes. At last, unlike
the definition in (40), when we take $\phi_{1}$ as the scalar charge, its
contribution in $\delta M_{W}$ has the standard form of a charge contribution
$-\Phi_{S}\delta Q_{S}$, where we define the coefficient of $\delta Q_{S}$
multiplied by $-1$ as the conjugate potential $\Phi_{S}$.

As mentioned in Gibbons:1996af , this scalar charge is not conserved. We
cannot construct a conserved current from a scalar field the way we construct
the Bianchi identity from the gauge fields. In Gegenberg:2003jr ; Park:2004yk
; Anabalon:2015ija  the free energy of some exact scalar hairy black-holes has
been calculated and compared with the free energy of the corresponding
Schwarzschild AdS black holes at the same temperature. The results show that
the Schwarzschild AdS black hole solutions are always thermodynamically
preferred, which means the scalar hairy black holes will always decay into
Schwarzchild AdS black holes at last. This is not surprising as we have
mentioned that the scalar charge is not conserved.

###  IV.2 Solutions with a manifest scalar charge

We now show a class of 3-dimensional scalar hairy soliton like solutions (the
metric is regular everywhere, while the scalar field has a $\log$ divergence
at the origin) constructed in Wen:2015xea . These solutions have a manifest
scalar charge term in the thermodynamic first law. The Lagrangian (1) with a
scalar potential

$\displaystyle V(\phi)=\frac{g^{2}\cosh^{6}\psi}{64}\times$
$\displaystyle~{}~{}~{}~{}~{}~{}\big{\\{}\cosh[6\psi]-81\cosh[2\psi]-6(7+\cosh[4\psi])\big{\\}},$
(43a) $\displaystyle\psi=\frac{\phi}{2\sqrt{6}},$ (43b)

admits the following solutions

$\displaystyle ds^{2}=$
$\displaystyle-\frac{g^{2}r^{2}(q^{2}+4qr+2r^{2})}{2(q+r)^{2}}dt^{2}$
$\displaystyle+\frac{2r^{4}}{g^{2}(q+r)^{4}(q^{2}+4qr+2r^{2})}dr^{2}+r^{2}d\theta^{2}\,,$
(44a) $\displaystyle\phi(r)=$ $\displaystyle
2\sqrt{6}~{}\text{Arctanh}\frac{1}{\sqrt{1+r/q}}.$ (44b)

Again, these solutions are not the whole space of solutions, and in fact on a
integral path when we integrate $\delta M_{W}$. Applying Wald’s formula we get
the surface integral on the boundary

$\delta M_{W}=\int_{S_{\infty}}(\delta
Q_{\xi}-\xi\cdot\Theta)=\frac{2\pi}{\ell^{2}}q\delta q,$ (45)

and also the surface integral on the infinitesimal ball around the origin

$\int_{S_{0^{+}}}(\delta Q_{\xi}-\xi\cdot\Theta)=\frac{2\pi}{\ell^{2}}q\delta
q\,.$ (46)

The non-vanishing of the integral (46) usually would not happen in higher
dimensions, but is possible in 3-dimensions as we can see. As the solutions
have no event horizon, the integral (46) cannot be interpreted as $T\delta S$,
so we interpret it as a contribution from scalar charge $\Phi_{S}\delta Q_{S}$
instead. This charge term is not an integral on the boundary, so even if we
can accept a non-integrable mass, it cannot be absorbed by $\delta M_{W}$.
Hence this scalar charge term is always manifest in the first law

$\delta M_{W}=\Phi_{S}\delta Q_{S}.$ (47)

It may be inappropriate to call this equation a thermo- dynamic first law,
since there is no entropy, so the solution is not a thermodynamic system. It
describes how the other charges change with the scalar hair.

The existence of these solutions gives further evidence for the existence of
the scalar charge.

##  V A new mass definition

In section III we discussed the problems of the masses defined by the
Hamiltonian formula and Wald’s formula. They all define the mass on a
variational level, and the results are non-integrable when matter fields are
included. Choosing an integral path, $\delta M_{W}$ can be integrated and give
a finite mass, however this mass depends on the integral path (or the boundary
condition) we choose. The entropy problem of some solutions also indicates the
masses defined in these formulas are not right. We think Wald’s formula is not
used properly by giving the definition of mass (18). In this section, based on
Wald’s formula, we propose a new way to define the mass for theories including
matter fields.

We first need to find out why the variation of mass calculated by (18) becomes
non-integrable when we include matter fields with charges, which are not the
Noether charge of any diffeomorphism of spacetime along Killing vectors. Again
we use the electrically charged RN-AdS black holes (30) as an example. Unlike
the energy $M$ and angular momentum $J$, the electric charge $Q$ is not a
Noether charge associated to a Killing vector in spacetime, instead, its
existence and conservation is the result of an internal $U(1)$ gauge symmetry.
When we calculate $\delta M_{W}$, we have chosen $\xi=\partial/\partial t$,
thus other charges, for example the angular momentum $J$, which generate
spacetime deffeomorphisms will not arise. This chosen of $\xi$ equals to stop
the rotation of the solution, however, it is not enough to stop the internal
gauge transformations. As a result, the variation of electric charge $\delta
Q$ will arise in the integration $\delta M_{W}$, thus makes it nonintegrable.
This can be seen clearly in (71).

The scalar charge we defined generates no transformations, so the arising of
$\delta Q_{S}$ in $\delta M_{W}$ will also happen, as we can see from the
previous sectors.

The right mass should be totally independent from not only charges like $J$
which generate spacetime diffeomorphisms, but also charges like $Q$ which are
not attached to a spacetime Killing vector. Hence, when we calculate the
variation of the mass, we should not only stop the other spacetime
deffeomorphisms, but also the internal gauge transformations. In other words,
charges like $Q$ and $Q_{S}$ should be considered as pure numbers without
variation when we calculate the mass.

Although the Hamiltonian formula and Wald’s formula directly constructed the
Hamiltonian, which seems to make their definitions of mass beyond controversy.
However, they both define the mass on a variational level, which means we
still have room to do some modification before we ruin physical fact that the
mass is the Hamiltonian. Following the arguments above, the modification is to
remove the contributions from the variations of all other charges, which are
not associate to a Killing vector, by hand.

This can be done by defining a variation $\bar{\delta}$ which act only along
the mass. By definition we have

$\bar{\delta}Q=0\,,$ (48)

for the electrically charged RN-AdS black holes. The contribution from $\delta
Q$ will be removed if we define the variation of the mass as
$\bar{\delta}M_{W}$, which is given by

$\bar{\delta}M_{W}=\bar{\delta}M-\Phi(M,Q)\bar{\delta}Q=\bar{\delta}M.$ (49)

So after we confine the $\delta$ in (18) to be $\bar{\delta}$, Wald’s formula
can give the right mass $M$ for these RN black holes.

We now arrive at our new definition of mass $M_{N}$,

$\bar{\delta}M_{N}=\int_{S_{\infty}}(\bar{\delta}Q_{\xi}-\xi\cdot\Theta(\varphi,\bar{\delta}\varphi))\Big{|}_{\xi=\partial/\partial
t}.$ (50)

with the variation $\bar{\delta}$ only along along $M_{N}$ in the space of
solutions.

It is quite natural to define the mass in this way, because it uses the fact
that the Hamiltonian should be a totally independent charge. After we
integrate $\bar{\delta}M_{N}$, the result $M_{N}$ is in fact the real
Hamiltonian.

To test this new definition, we can use it to calculate the mass of the scalar
hairy black holes. For these black holes, as we do not agree with the mass
defined by the Hamiltonian formula and Wlad’s formula, we do not know what the
real mass is before calculation. After we substitute into the solutions and
perform some Legendre transformations, we can write the right hand side of
(50) as a total variation plus a charge term $-\Phi_{S}\bar{\delta}Q_{S}$.
Then we apply $\bar{\delta}Q_{S}=0$, so only the total variation remains. This
total variation is the variation of the real mass $\delta M_{N}$.

Based on the analysis in Lu:2014maa , we consider $d$-dimensional gravities
minimally coupled to a scalar field with a scalar mass $m^{2}$ such that

$-\frac{1}{4\ell^{2}}(d-1)^{2}<m^{2}<-\frac{1}{4\ell^{2}}(d-1)^{2}+\frac{1}{4\ell^{2}},$
(51)

and a general scalar potential $V(\phi)$ admitting a Taylor expansion of the
form

$V(\phi)=-\frac{(d-1)(d-2)}{\ell^{2}}+\frac{1}{2}m^{2}\phi^{2}+\gamma_{3}\phi^{3}+\gamma_{4}\phi^{4}+\cdots.$
(52)

Let us apply (2) as the metric ansatz and define

$\sigma=\sqrt{4\ell^{2}m^{2}+(d-1)^{2}}.$ (53)

The asymptotics of the metric functions and scalar field will take the form

$\displaystyle\phi$
$\displaystyle=\frac{\phi_{1}}{r^{(d-1-\sigma)/2}}+\frac{\phi_{2}}{r^{(d-1+\sigma)/2}}+\cdots\,,$
(54a) $\displaystyle h$
$\displaystyle=g^{2}r^{2}+1-\delta_{d,3}+\frac{\alpha}{r^{d-3}}+\cdots\,,$
(54b) $\displaystyle f$
$\displaystyle=g^{2}r^{2}+1-\delta_{d,3}+\frac{b}{r^{d-3-\sigma}}+\frac{\beta}{r^{d-3}}+\cdots\,,$
(54c)

with $\alpha$ and $\phi_{1}$ the only two independent integration constants.
Substituting the expansions (54) into the equations of motion and solving for
the first few coefficients, we find that

$\displaystyle b=\frac{(d-1-\sigma)\,\phi_{1}^{2}}{4(d-2)\ell^{2}}\,,$ (55a)
$\displaystyle\beta=\alpha+\frac{[(d-1)^{2}-\sigma^{2}]\,\phi_{1}\phi_{2}}{2(d-1)(d-2)\ell^{2}}\,.$
(55b)

Using Eq.(24), the integration
$\int_{S_{\infty}}(\bar{\delta}Q_{\xi}-\xi\cdot\Theta(\varphi,\bar{\delta}\varphi))$
on the boundary gives

$\displaystyle\bar{\delta}M_{N}=$
$\displaystyle\omega_{d-2}\Big{[}-(d-2)\bar{\delta}\alpha+\frac{\sigma}{2(d-1)\ell^{2}}\,[(d-1+\sigma)\phi_{2}\bar{\delta}\phi_{1}-(d-1-\sigma)\phi_{1}\bar{\delta}\phi_{2}]\Big{]}$
$\displaystyle=$
$\displaystyle\omega_{d-2}\Big{[}\bar{\delta}\left(-(d-2)\alpha-\frac{\sigma(d-1-\sigma)}{2(d-1)\ell^{2}}\phi_{1}\phi_{2}\right)+\frac{\sigma}{\ell^{2}}\,\phi_{2}\bar{\delta}\phi_{1}\Big{]}$
$\displaystyle=$
$\displaystyle\bar{\delta}\left[\omega_{d-2}\left(-(d-2)\alpha-\frac{\sigma(d-1-\sigma)}{2(d-1)\ell^{2}}\phi_{1}\phi_{2}\right)\right]\,,$
(56)

where we apply $\bar{\delta}\phi_{1}=0$ in the third line of the equation.

The new integrable mass $M_{N}$ is given by

$M_{N}=\omega_{d-2}\left(-(d-2)\alpha-\frac{\sigma(d-1-\sigma)}{2(d-1)\ell^{2}}\phi_{1}\phi_{2}\right).$
(57)

This new mass $M_{N}$ is different from the AMD mass which in these cases is
given by

$M_{A}=-\omega_{d-2}(d-2)\alpha.$ (58)

Our new mass is also different from the mass $M_{C}$ calculated by the
“counterterm subtraction method” . When there is a logarithmic $r$ dependence
in the asymptotic expansions for the metric and scalar field, a logarithmic
divergence will arise in the action. We need a boundary term from the scalar
field

$\displaystyle{\mathcal{L}}_{surf}[\phi]$ $\displaystyle=\frac{\gamma}{16\pi
G}\sqrt{-h}\,n^{\mu}\,\phi\,\partial_{\mu}\phi\,,$ (59)

to remove this divergence and the parameter $\gamma$ can be fixed. Usually the
result $M_{C}$ is also different from $M_{A}$.

However, under the condition (51), no logarithmic function will arise in the
asymptotic expansion for metric and scalar field thus there is no logarithmic
divergence, and the contribution from (59) also become finite. The parameter
$\gamma$ now is free and the result $M_{C}$

$\displaystyle
M_{C}=\frac{\omega_{d-2}}{16\pi}\Big{[}(2-d)\alpha+\frac{\left((d-1)(4\gamma-1)+\sigma\right)\,\sigma}{2(d-1)\ell^{2}}\,\phi_{1}\,\phi_{2}\Big{]}\,$
(60)

depend on it, which means this method cannot give a unique mass. We can
reproduce $M_{A}$ from $M_{C}$ by taking

$\displaystyle\gamma=\frac{d-1-\sigma}{4(d-1)}.$ (61)

And also we can reproduce our new mass $M_{N}$ by just taking

$\displaystyle\gamma=0,$ (62)

which seems to be a more natural choice.

The details for the calculation of $M_{C}$ can be found in Lu:2014maa .

Although we modified the definition of mass, but the holding of Wald’s formula
(17) is always a mathematical fact for on-shell spacetimes and would give the
thermodynamic first law. Replacing $\bar{\delta}$ with the general variation
$\delta$, the second line in (56) is just $\delta M_{W}$. Thus, Wlad’s formula
gives

$\displaystyle T\delta S=$ $\displaystyle\delta
M_{N}+\frac{\omega_{d-2}\sigma}{\ell^{2}}\phi_{2}\delta\phi_{1}$
$\displaystyle=$ $\displaystyle\delta M_{N}-\Phi_{S}\delta Q_{S}\,,$ (63)

with the conjugate potential $\Phi_{S}$ given by

$\displaystyle\Phi_{S}=-\frac{\omega_{d-2}\sigma}{\ell^{2}}\,\phi_{2}\,.$ (64)

The 3-dimensional massless black holes described by (36) and (37) have a
scalar mass
$m^{2}=-\frac{1}{4\ell^{2}}(d-1)^{2}+\frac{1}{4\ell^{2}}=-\frac{3}{4\ell^{2}}$
which is outside (51), but the solutions (37) can still be described by (54)
and do not contain any logarithmic functions, so the general analysis above is
still valid here. The solutions (37) have

$\displaystyle\phi_{1}=2\sqrt{2\mu q}\qquad\phi_{2}=-\frac{\sqrt{2\mu
q^{3}}}{3}\qquad\alpha=0,$ (65)

so according to (57), the masses for different $\mu$ are given by

$M_{N}=-\frac{\pi}{2\ell^{2}}\phi_{1}\phi_{2}=\frac{2\pi\mu
q^{2}}{3\ell^{2}}.$ (66)

Note that, $\mu$ is a parameter in the Lagrangian, so is not an integration
constant and should have no variation. Using our definition of mass (50), we
see the solutions (37) are actually massive black holes, and their masses are
related to the scalar hair. As we have introduced a new macroscopic quantity
for these black holes, the statistical entropy $S(M_{N},Q_{S})$ now counts the
number of micro states with a mass $M_{N}$ and a scalar charge $Q_{S}$. The
solution (37) with

$M_{N}=\frac{2\pi\mu q^{2}}{3\ell^{2}}\qquad Q_{S}=2\sqrt{2\mu q},$ (67)

is now unique, and could be considered as a quantum state, thus have a
vanishing statistical entropy. Which is now consistent with its Bekenstein-
Hawking entropy. This solves the entropy problem we presented in Section III.

The newly defined mass $M_{N}$ is integrable and does not have the entropy
problem mentioned above. The requirement fot the variation $\delta$ in (18) to
be $\bar{\delta}$ is based on the fact that the mass should be an independent
charge of the solution. The consistence conditions (19) and (20) for the
existence of a well defined energy is actually satisfied by
$\bar{\delta}M_{N}$ rather than $\delta M_{W}$. Our definition (50) is in fact
consistent with the spirit of Wald’s construction. The Hamiltonian formula can
also be modified in this way as it also defines the mass on a variational
level.

##  VI Mass in more general gravities coupled to matter fields

Although the initial motivation of this paper is to give a well defined mass
for asymptotically AdS Einstein-scalar gravities, our new definition of mass
is valid for much more general gravities. The only requirements for the
application of our definition can be conclude as: a), The solution should
admit $\partial/\partial t$ as a Killing vector; b), All the terms in Wald’s
formula should be finite; c), All the independent charges can be identified.

Consider, in general, a gravity coupled to matter fields $A_{i}$ with charges
$Q_{i}$ which generate internal symmetry transformations and also matter
fields $\Psi_{j}$ with charges $\psi_{j}$ which do not generate any symmetry
transformation (for example the scalar charge we defined). Further more, if
the Killing vector $\xi_{k}$ of solution have components along other
directions of spacetime, there will be charges $J_{k}$ which generate the
spacetime diffeomorphisms along those directions (for example, the angular
momentum).

Wald’s formula gives a relationship between the variations of all charges, and
when the black hole can be considered as thermodynamic system, Wald’s formula
can be recognised as the thermodynamic first law. The $Q_{i}$ and $J_{k}$
charges are Noether charges and can be determined by the solution, while the
$\psi_{j}$ charges are not. We need to give proper definitions for these
$\psi_{j}$ charges the way we define the scalar charges. For a solution with
all the charges mentioned above, and after some Legendre-like transformations,
Wald’s (17) formula could be written in the following formula

$\displaystyle T\delta S=\delta\mathcal{M}(\varphi)-\sum_{i}\Phi_{i}\delta
Q_{i}-\sum_{i}\Phi_{j}\delta\psi_{j}-\sum_{k}\Phi_{k}\delta J_{k}$ (68)

where the total variation $\delta\mathcal{M}$ is expected to depend on an
extra integration constant which is independent from all the other charges
(for example, the parameter $\alpha$ in (56)), and may also depend on the
other charges. Also we would find that $\delta M_{W}$ becomes non-integrable
and can be written as

$\displaystyle\delta M_{W}=\delta\mathcal{M}(\varphi)-\sum_{i}\Phi_{i}\delta
Q_{i}-\sum_{i}\Phi_{j}\delta\psi_{j}.$ (69)

Terms proportional to $\delta J_{k}$ would not appear in $\delta M_{W}$
because we have chosen $\xi=\partial/\partial t$.

It should be pointed out that, if we make another gauge choice that the gauge
fields $A_{i}$ vanishes asymptotically on the boundary, rather than on the
horizon, the contributions from $\delta Q_{i}$ in $\delta M_{W}$ will be moved
to the surface integral on the horizon. For example, consider again the RN-AdS
black holes, the surface integrals now becomes

$\displaystyle\int_{S_{r_{0}}}(\delta Q_{\xi}-\xi\cdot\Theta)=T\delta
S+\Phi(M,Q)\delta Q\,,$ (70) $\displaystyle\int_{S_{\infty}}(\delta
Q_{\xi}-\xi\cdot\Theta)=\delta M=\delta M_{W}\,.$ (71)

This changing of gauge choice may solve the non-integral mass problem of the
RN black holes in some sence; however, this would not work for the scalar
charges.

Following (50) we get

$\displaystyle\bar{\delta}M_{N}=\bar{\delta}\mathcal{M}(\varphi)$ (72)

where we have apply $\bar{\delta}Q_{i}=0$ and $\bar{\delta}\psi_{j}=0$. This
means $M_{N}=\mathcal{M}(\varphi)$.

These general analysis can be checked by looking at the results of some recent
papers Lu:2013ura ; Liu:2014tra ; Liu:2014dva ; Fan:2014ixa ; Fan:2014ala ;
Fan:2015yza . Using Wald’s formula, these papers study the thermodynamic first
law of gravities coupled to matter fields with both $Q_{i}$ and $\psi_{j}$
type of charges. And the first laws they get are all consistent with (68).

AdS dyonic black holes in gauged supergravities are constructed in Lu:2013ura
with one pair of electric and magnetic charges, and in Chow:2013gba  with more
pairs of charges. The scalar fields in their solutions are totally determined
by the electric and magnetic charges. This means the solutions in both
Lu:2013ura  and Chow:2013gba  are in fact not the full solutions. In our
opinion, their scalar charge have been settled down and does not present as an
independent integration constant. Facing a non-integrable Hamiltonian, authors
of Lu:2013ura  admit the existence of the scalar charge and take the AMD mass
as the mass. While authors of Chow:2013gba  think the mass is ill defined and
does not exist, unless additional consistent boundary conditions are given by
hand, which coincide with the mass defined on boundary conditions
Henneaux:2004zi ; Hertog:2004dr ; Henneaux:2006hk ; Hertog:2005hm ;
Amsel:2006uf .

We do not agree with both of them, because the requirements of the AMD method
are not fully satisfied and the mass defined on the boundary conditions also
have problems we presented in Sec. III.

##  VII Mass in a gravity with holography

According to the AdS/CFT Maldacena:1997re ; Witten:1998qj ; Gubser:1998bc
correspondence, the $(d-1)$-dimensional CFT generating functional of
correlation functions for some operator $\mathcal{O}$ is equivalent to the
partition function of a gravity in $AdS_{d}$ background with some specific
boundary conditions. The Einstein-scalar gravities (1) with a boundary
condition $\phi_{2}(\phi_{1})$ is dual to a CFT with some kind of multi-trace
deformation Berkooz:2002ug ; Sever:2002fk ; Witten:2001ua .

Consider a CFT with a general multi-trace deformation $W(\hat{\mathcal{O}})$

$\displaystyle S=S_{CFT}+\int\rho(x)~{}\hat{\mathcal{O}}+\int
W(\hat{\mathcal{O}}),$ (73)

with $\hat{\mathcal{O}}$ the dual operator of the scalar field and
$W(\hat{\mathcal{O}})$ a general function of $\hat{\mathcal{O}}$. When the
scalar mass satisfies
$-\frac{1}{4}(d-1)^{2}<m^{2}\ell^{2}<-\frac{1}{4}(d-1)^{2}+1$, either the
$\phi_{1}$ and $\phi_{2}$ mode are normalizable, thus, we can take either
$\phi_{1}$ and $\phi_{2}$ as the source $\rho(x)$. If we take $\rho=\phi_{2}$,
then the deformed CFT (73) is dual to the Einstein-scalar gravity with the
following boundary condition

$\displaystyle\phi_{2}(\phi_{1})=\frac{\delta W(\phi_{1})}{\delta\phi_{1}}.$
(74)

As the boundary condition is determined by the boundary CFT (73) and is added
by hand in gravity, it seems a little unnatural to impose a boundary condition
when someone only look at the gravity side. A more comforting way to interpret
a boundary condition is to give additional special boundary terms (which is
also determined by the boundary CFT) to the action of the gravity, thus the
boundary condition is the solution of the boundary equations of motion
Sever:2002fk . From this point of view, the same gravity with different
boundary conditions are actually different gravities. Hence one would expect
that a specific black hole would have different masses when its variation is
confined by different boundary conditions, because the actions changes.

Following the spirit of this paper, we also propose a way to calculate the
mass in a gravity with holography (or with additional boundary conditions). We
should start from the new gravity action with the additional special boundary
terms. The additional boundary terms may change the symplectic structure of
the theory, thus give a different mass if we use our definition of mass
proposed in Sec. V.

We cannot guarantee the validity of our definition of mass in this situation.
The gravity with boundary conditions is still a subtle topic of gravity, since
the boundary conditions cannot be naturally generated from only the gravity
side.

###### Acknowledgements.

I am grateful to Lin-Qing Chen, Si Li, Hai-Shan Liu for helpful discussions,
to Andrzej Banburski, H. Lu, Rob Myers and Chris Pope for helpful comments and
reading the paper, and especially to Wei Song for critical comments and
numerous discussions on holography and boundary conditions. I also would like
to thank Chuan-jie Zhu, the Beijing Municipal Education Committee and the
Perimeter Institute Visiting Graduate Fellows program for financial support.
Research at Perimeter Institute is supported by the Government of Canada and
by the Province of Ontario through the Ministry of Research and Innovation.
This work is also supported in part by NSFC 34112027.

## References

  * (1) M. Henneaux and C. Teitelboim, “Asymptotically anti-De Sitter Spaces,” Commun. Math. Phys. 98, 391 (1985).
  * (2) A. Ashtekar and A. Magnon, “Asymptotically anti-de Sitter space-times,” Class. Quant. Grav. 1 (1984) L39.
  * (3) A. Ashtekar and S. Das, “Asymptotically Anti-de Sitter space-times: Conserved quantities,” Class. Quant. Grav. 17, L17 (2000) [hep-th/9911230].
  * (4) M. Henningson and K. Skenderis, “The Holographic Weyl anomaly,” JHEP 9807, 023 (1998) [hep-th/9806087].
  * (5) V. Balasubramanian and P. Kraus, “A Stress tensor for Anti-de Sitter gravity,” Commun. Math. Phys. 208, 413 (1999) [hep-th/9902121].
  * (6) K. Skenderis and S. N. Solodukhin, “Quantum effective action from the AdS / CFT correspondence,” Phys. Lett. B 472, 316 (2000) [hep-th/9910023].
  * (7) M. Bianchi, D. Z. Freedman and K. Skenderis, “How to go with an RG flow,” JHEP 0108, 041 (2001) [hep-th/0105276].
  * (8) S. de Haro, K. Skenderis and S. N. Solodukhin, “Gravity in warped compactifications and the holographic stress tensor,” Class. Quant. Grav. 18, 3171 (2001) [hep-th/0011230].
  * (9) K. Skenderis, “Lecture notes on holographic renormalization,” Class. Quant. Grav. 19, 5849 (2002) [hep-th/0209067].
  * (10) S. de Haro, S. N. Solodukhin and K. Skenderis, “Holographic reconstruction of space-time and renormalization in the AdS / CFT correspondence,” Commun. Math. Phys. 217, 595 (2001) [hep-th/0002230].
  * (11) K. Skenderis, “Asymptotically Anti-de Sitter space-times and their stress energy tensor,” Int. J. Mod. Phys. A 16, 740 (2001) [hep-th/0010138].
  * (12) M. Bianchi, D. Z. Freedman and K. Skenderis, “Holographic renormalization,” Nucl. Phys. B 631, 159 (2002) [hep-th/0112119].
  * (13) I. Papadimitriou and K. Skenderis, “AdS / CFT correspondence and geometry,” hep-th/0404176.
  * (14) J. Katz, J. Bicak and D. Lynden-Bell, “Relativistic conservation laws and integral constraints for large cosmological perturbations,” Phys. Rev. D 55, 5957 (1997) [gr-qc/0504041].
  * (15) N. Deruelle and J. Katz, “On the mass of a Kerr-anti-de Sitter spacetime in D dimensions,” Class. Quant. Grav. 22, 421 (2005) [gr-qc/0410135].
  * (16) G. W. Gibbons, S. W. Hawking, G. T. Horowitz and M. J. Perry, “Positive Mass Theorems for Black Holes,” Commun. Math. Phys. 88, 295 (1983).
  * (17) L. F. Abbott and S. Deser, “Stability of Gravity with a Cosmological Constant,” Nucl. Phys. B 195, 76 (1982).
  * (18) R. M. Wald, “Black hole entropy is the Noether charge,” Phys. Rev. D 48, 3427 (1993) [gr-qc/9307038].
  * (19) V. Iyer and R. M. Wald, “Some properties of Noether charge and a proposal for dynamical black hole entropy,” Phys. Rev. D 50, 846 (1994) [gr-qc/9403028].
  * (20) S. Hollands, A. Ishibashi and D. Marolf, “Comparison between various notions of conserved charges in asymptotically AdS-spacetimes,” Class. Quant. Grav. 22, 2881 (2005) [hep-th/0503045].
  * (21) P. Breitenlohner and D. Z. Freedman, Annals Phys. 144 (1982) 249; Phys. Lett. B 115 (1982) 197
  * (22) H. Lu, C. N. Pope and Q. Wen, “Thermodynamics of AdS Black Holes in Einstein-Scalar Gravity,” arXiv:1408.1514 [hep-th].
  * (23) Q. Wen, “Strategy to Construct Exact Solutions in Einstein-Scalar Gravities,” Phys. Rev. D 92, 104002 (2015) [arXiv:1501.02829 [hep-th]].
  * (24) M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, “Asymptotically anti-de Sitter spacetimes and scalar fields with a logarithmic branch,” Phys. Rev. D 70, 044034 (2004) [hep-th/0404236].
  * (25) T. Hertog and K. Maeda, “Black holes with scalar hair and asymptotics in N = 8 supergravity,” JHEP 0407, 051 (2004) [hep-th/0404261].
  * (26) M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, “Asymptotic behavior and Hamiltonian analysis of anti-de Sitter gravity coupled to scalar fields,” Annals Phys. 322, 824 (2007) [hep-th/0603185].
  * (27) T. Hertog and S. Hollands, “Stability in designer gravity,” Class. Quant. Grav. 22, 5323 (2005) [hep-th/0508181].
  * (28) A. J. Amsel and D. Marolf, “Energy Bounds in Designer Gravity,” Phys. Rev. D 74, 064006 (2006) [Erratum-ibid. D 75, 029901 (2007)] [hep-th/0605101].
  * (29) G. W. Gibbons, R. Kallosh and B. Kol, “Moduli, scalar charges, and the first law of black hole thermodynamics,” Phys. Rev. Lett. 77, 4992 (1996) [hep-th/9607108].
  * (30) H. Lu, Y. Pang and C. N. Pope, “AdS Dyonic Black Hole and its Thermodynamics,” JHEP 1311, 033 (2013) [arXiv:1307.6243 [hep-th]].
  * (31) H. S. Liu and H. Lu, “Scalar Charges in Asymptotic AdS Geometries,” Phys. Lett. B 730, 267 (2014) [arXiv:1401.0010 [hep-th]].
  * (32) T. Regge and C. Teitelboim, “Role of Surface Integrals in the Hamiltonian Formulation of General Relativity,” Annals Phys. 88, 286 (1974).
  * (33) T. Hertog and K. Maeda, “Stability and thermodynamics of AdS black holes with scalar hair,” Phys. Rev. D 71, 024001 (2005) [hep-th/0409314].
  * (34) J. Lee and R. M. Wald, “Local symmetries and constraints,” J. Math. Phys. 31, 725 (1990).
  * (35) T. Hertog and G. T. Horowitz, “Designer gravity and field theory effective potentials,” Phys. Rev. Lett. 94, 221301 (2005) [hep-th/0412169].
  * (36) R. M. Wald and A. Zoupas, “A General definition of ’conserved quantities’ in general relativity and other theories of gravity,” Phys. Rev. D 61, 084027 (2000) [gr-qc/9911095].
  * (37) S. Gao, “The First law of black hole mechanics in Einstein-Maxwell and Einstein-Yang-Mills theories,” Phys. Rev. D 68, 044016 (2003) [gr-qc/0304094].
  * (38) L. Susskind, L. Thorlacius and J. Uglum, “The Stretched horizon and black hole complementarity,” Phys. Rev. D 48, 3743 (1993) [hep-th/9306069].
  * (39) J. D. Brown and M. Henneaux, “Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity,” Commun. Math. Phys. 104, 207 (1986).
  * (40) A. Strominger, “Black hole entropy from near horizon microstates,” JHEP 9802, 009 (1998) [hep-th/9712251].
  * (41) M. Guica, T. Hartman, W. Song and A. Strominger, “The Kerr/CFT Correspondence,” Phys. Rev. D 80, 124008 (2009) [arXiv:0809.4266 [hep-th]].
  * (42) J. L. Cardy, “Operator Content of Two-Dimensional Conformally Invariant Theories,” Nucl. Phys. B 270, 186 (1986).
  * (43) M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, “Black holes and asymptotics of 2+1 gravity coupled to a scalar field,” Phys. Rev. D 65, 104007 (2002) [hep-th/0201170].
  * (44) A. Anabalon, D. Astefanesei and C. Martinez, “Mass of asymptotically anti-de Sitter hairy spacetimes,” Phys. Rev. D 91, no. 4, 041501 (2015) [arXiv:1407.3296 [hep-th]].
  * (45) D. D. K. Chow and G. CompÃ¨re, “Dyonic AdS black holes in maximal gauged supergravity,” Phys. Rev. D 89, no. 6, 065003 (2014) doi:10.1103/PhysRevD.89.065003 [arXiv:1311.1204 [hep-th]].
  * (46) H. S. Liu, H. Lu and C. N. Pope, “Thermodynamics of Einstein-Proca AdS Black Holes,” JHEP 1406, 109 (2014) [arXiv:1402.5153 [hep-th]].
  * (47) H. S. Liu and H. Lu, “Thermodynamics of Lifshitz Black Holes,” JHEP 1412, 071 (2014) [arXiv:1410.6181 [hep-th]].
  * (48) Z. Y. Fan and H. Lu, “SU(2)-Colored (A)dS Black Holes in Conformal Gravity,” JHEP 1502, 013 (2015) [arXiv:1411.5372 [hep-th]].
  * (49) Z. Y. Fan and H. Lu, “Thermodynamical First Laws of Black Holes in Quadratically-Extended Gravities,” Phys. Rev. D 91, no. 6, 064009 (2015) [arXiv:1501.00006 [hep-th]].
  * (50) Z. Y. Fan and H. Lu, “Charged Black Holes in Colored Lifshitz Spacetimes,” Phys. Lett. B 743 (2015) 290 [arXiv:1501.01727 [hep-th]].
  * (51) J. Gegenberg, C. Martinez and R. Troncoso, “A Finite action for three-dimensional gravity with a minimally coupled scalar field,” Phys. Rev. D 67, 084007 (2003) [hep-th/0301190].
  * (52) M. I. Park, “Fate of three-dimensional black holes coupled to a scalar field and the Bekenstein-Hawking entropy,” Phys. Lett. B 597, 237 (2004) [hep-th/0403089].
  * (53) A. Anabalon, D. Astefanesei and D. Choque, “On the thermodynamics of hairy black holes,” Phys. Lett. B 743 (2015) 154 [arXiv:1501.04252 [hep-th]].
  * (54) J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int. J. Theor. Phys. 38, 1113 (1999) [Adv. Theor. Math. Phys. 2, 231 (1998)] doi:10.1023/A:1026654312961 [hep-th/9711200].
  * (55) E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2, 253 (1998) [hep-th/9802150].
  * (56) S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators from noncritical string theory,” Phys. Lett. B 428, 105 (1998) doi:10.1016/S0370-2693(98)00377-3 [hep-th/9802109].
  * (57) M. Berkooz, A. Sever and A. Shomer, “’Double trace’ deformations, boundary conditions and space-time singularities,” JHEP 0205, 034 (2002) doi:10.1088/1126-6708/2002/05/034 [hep-th/0112264].
  * (58) A. Sever and A. Shomer, “A Note on multitrace deformations and AdS/CFT,” JHEP 0207, 027 (2002) doi:10.1088/1126-6708/2002/07/027 [hep-th/0203168].
  * (59) E. Witten, “Multitrace operators, boundary conditions, and AdS / CFT correspondence,” hep-th/0112258.


# Human Fall Detection- Multimodality Approach

Xi Wang  Multimedia
University of Alberta
Edmonton, Canada
<EMAIL_ADDRESS>Ramya Penta  Multimedia
University of Alberta
Edmonton, Canada
<EMAIL_ADDRESS>Bhavya Sehgal  Multimedia
University of Alberta
Edmonton, Canada
<EMAIL_ADDRESS>Dale Chen-Song  Multimedia
University of Alberta
Edmonton, Canada
<EMAIL_ADDRESS>
###### Abstract

Falls have become more frequent in recent years, which has been harmful for
senior citizens.Therefore detecting falls have become important and several
data sets and machine learning model have been introduced related to fall
detection. In this project report, a human fall detection method is proposed
using a multi modality approach. We used the UP-FALL detection data set which
is collected by dozens of volunteers using different sensors and two cameras.
We use wrist sensor with acclerometer data keeping labels to binary
classification, namely fall and no fall from the data set.We used fusion of
camera and sensor data to increase performance. The experimental results shows
that using only wrist data as compared to multi sensor for binary
classification did not impact the model prediction performance for fall
detection.

###### Index Terms:

fall detection, multimodal, accelerometer

##  I Introduction

Falling is a serious problem that can occur for elderly people as it can cause
serious injuries, such as fractured hips, traumatic brain injuries, and other
injuries [1]. These injuries can be exasperated if the elderly people are
living alone without the help of a caretaker, as they can stay on the ground
for an extended period of time unable to get up, losing consciousness, getting
hypothermia, and may even lead to death [2]. Therefore, immediate observations
for fall detection of elderly people using fall detection devices are
necessary to prevent such injuries.

There are different systems of monitoring and detecting human falling on a
daily basis. There are two types of classification these systems can fall
under wearable devices and non-wearable devices. Wearable devices include
devices that can be attached to the person, such as accelerometers [3],
gyroscopes [4], and wearable cameras [5] that can monitor fall actions. Some
disadvantages of these types of devices are that they may be cumbersome and
uncomfortable to constantly wear the device and they may need to be recharged
consistently. Furthermore, the devices can be even more uncomfortable if the
subject needs to wear more than one. Non-wearable devices can include optical
cameras [6], microphones [7], microwave radar [8], or depth cameras [9].
However, these also have limitations such as cameras being sensitive to light
intensity, microphones needing a quiet environment to detect sounds
accurately, and microwave characteristics of human action are not robust
enough due to the range and Doppler resolution limits of radar system [10].
Therefore, the best way to mitigate these disadvantages is to combine
different systems using a multimodality approach. However, another challenge
comes in determining which systems should be combined for the multimodality
approach, as some systems may work better with others or be redundant.

Furthermore, there is another challenge to fall detection; the subject needs
to be confirmed falling. There are certain movements or actions that the
devices may detect that may not be actually falling. Therefore, the devices
would need to distinguish between common actions such as walking, laying down,
and standing up, and common falling false alarms such as squatting or picking
up stuff from the ground and then the actual falling itself.

In this report, we will propose a multimodal method to detect human falling
from camera and accelerometer data, using feature fusion and multi-layer
perceptron.

##  II Background

###  II-A Multimodality Approach

As mentioned above, there are many devices that can be used to detect falling.
Therefore, there are many ways to combine these systems for a multimodality
approach. [11] proposes a novel application of deep networks to learn features
over multiple modalities. They presented a series of tasks for multimodal
learning and show how to train deep networks that learn features to address
these tasks. They demonstrated cross-modality feature learning, where better
features for one modality (e.g., video) can be learned if multiple modalities
(e.g., audio and video) are present at feature learning time. Furthermore,
they show how to learn a shared representation between modalities and evaluate
it on a unique task, where the classifier is trained with audio-only data but
tested with video-only data and vice-versa.

The overall task was divided into three phases: feature learning, supervised
training, and testing. A simple linear classifier was used for supervised
training and testing to examine different feature learning models with
multimodal data. In particular, they considered three learning settings:
multimodal fusion, cross-modality learning, and shared representation
learning.

One of the most straightforward approaches to feature learning is to train a
Restricted Boltzmann Machines (RBMs) model separately for audio and video but
it had two issues with it. First, there was no explicit objective for the
models to discover correlations across the modalities; it is possible for the
model to find representations such that some hidden units are tuned only for
audio while others are tuned only for video. Second, the models are clumsy to
use in a cross-modality learning setting where only one modality is present
during supervised training and testing.

Thus, they proposed a deep autoencoder that resolves both issues. cross-
modality learning setting was used where both modalities are present during
feature learning but only a single modality is used for supervised training
and testing. They used the deep autoencoder models in settings where only a
single modality is present at supervised training and testing. They proposed
training the bimodal deep autoencoder using an augmented but noisy dataset
with additional examples that have only a single modality as input. Both
models are pre-trained using sparse RBMs.

[12] In EO, which is earth observation tasks, using single remote sensor data
like hyperspectral data or Lidar data, researchers inevitably meet the
performance bottleneck in identifying and recognizing objects of interest due
to the difficulties in excavating and jointly using the information potential
of multimodal heterogeneous data. And there are some newly developed
approaches have been proven to be effective in fusing multiple remote sensor
data sources, although these methods got limited capabilities. To overcome
this issue, the authors proposed a module called the Cross-Channel
Reconstruction module to obtain a more robust and compact joint feature
representation stored by the encoded latent vector.

For previous traditional modal fusion methods, the most direct way is to
concatenate different features as a fused embedding, depending on the stage in
which the features are concatenated, there were early fusion, middle fusion,
and late fusion. But these naive methods showed limited performance. To
conquer this they have the CCR fusion which uses an encoder-decoder-like
structure to obtain robust feature representation. As you can see in the
detailed illustration the features extracted by the CNN are primarily simply
concatenated by a prescribed order. This concatenated feature is fed into an
encoder to generate a more compact feature encoding. The compressed feature
encoding will be further decoded to a swapped original extracted feature. For
the objective function, the holistic loss is decomposed into a cross-entropy
loss term to judge classification accuracy and a reconstruction loss term to
reflect how well the decoding works.

From [13], the generative model that can classify different types of
information using multimodal Deep Boltzmann Machine. It extracts a unified
representation that fuses modalities together and learns probability density
over the space of multimodal inputs. It can extract this representation even
when some modalities are absent by sampling from the conditional distribution
over them and filling them in. There have been several approaches to learning
from multimodal data. Huiskes showed that using captions, or tags, in addition
to standard low-level image features, significantly improves the
classification accuracy of Support Vector Machines (SVM) and Linear
Discriminant Analysis (LDA) models. Multiple kernel learning framework by
Guillaumin, further demonstrated that an additional text modality could
improve the accuracy of SVMs on various object recognition tasks.

###  II-B Fall Detection

One multimodal approach for fall detection involves combining radar and
optical cameras to detect falling. [10] proposed using convolution neural
networks with multi-sensor fusion, proposed using a combination of radar and
optical camera to detect falling For the radar-returned signals, the time-
frequency (TF) domain is typically used. TF signals of a fall and three common
motions: walking, squatting, and standing up. The authors proposed using the
Alex-based convolutional neural network (CNN) and single shot multi-box
detector (SSD) Net based CNN is adopted to classify the TF images
respectively, and the results of the two CNNs are merged to give the final
detection results. On the other hand, the image sequence captured by the
optical camera is also processed by another SSD-based CNN, and the aspect
ratio sequence of the bounding box of the human is provided to help confirm if
a fall event is occurring. The aspect ratio changing of the bounding boxes
(height divided by width) frame by frame is used to help distinguish the fall
and non-fall events.

[14] aimed at the recognition of and the differentiation between fall
activities and activities of daily living (ADL) using the MobiFall dataset
with the help of machine learning algorithms. They used an accelerometer,
gyroscope, and magnetometer to collect the body movements. The five
classification methods that were implemented are: Naive Bayes,k-nearest
neighbor, artificial neural networks, and least square method

Fall detection classification systems were characterized through five stages:
pre-processing, feature extraction, feature selection, model training, and
classification. The data was pre-processed by a median and low-pass filter.
During the feature extraction stage, a total of 38 features were extracted and
a filter rank-based system was used to eliminate features with no information
and extract the top five features in order to optimize the algorithm’s
dimensionality. (z median and x median which are time…..x mean and y mean in
frequency…..skewness in signal magnitude vector) can be seen in Fig. 1.

Figure 1: Fall detection approach

Receiver Operating Characteristic curves tell that among 5 classifiers, k-
Nearest Neighbors’ algorithm obtained an overall accuracy of 87.5% with a
sensitivity of 90.70%, and a specificity of 83.78%. Sensitivity represents the
capacity to detect falls, specificity represents the capacity to only detect
falls and ignore the non-fall events, and accuracy represents the portion of
true results among the population.

The limitation of this paper is “MobiFall” dataset was recorded in the user’s
pocket, which is below the waist but through research, it was found that the
device at the upper trunk of the body, below the neck, and above the neck
provides the best results. The reason being was that the other locations
contain high movement frequency and complexity.

[15] focuses on fall detection based on deep learning methods. The type of
data used in this approach is accelerometer data collected from wearable
devices. The model is deployed on a fog device, in this specific case, the
authors use Raspberry Pi for the experiment. The authors also proposed a
convolutional architecture called CNN-3B3Conv which composite of 3 blocks.
Finally, this approach reaches an accuracy of 99.86%, while the pure LSTM
model only achieves around 95%. The architecture uses 3 blocks. Block 1 and
Block 2 are in the same shape with 3 convolutional layers each. For block 3
there are 3 fully-connected layers. Finally, the output is a binary label
representing whether the user is falling or not.

##  III Proposed Method

###  III-A Dataset

We use the UP-FALL detection dataset [16] published by Panamerican University.
This dataset is used for human activity recognition and is mainly aimed at
detecting falls of elderly people. The dataset contains data from 17 subjects;
each subject performs 11 activities with three trials for each activity. The
activities were six simple human daily activities (walking, standing, picking
up an object, sitting, jumping, and laying) and five human falls (falling
forward using hands, falling forward using knees, falling backward, falling
sitting in an empty chair, and falling sideways). Devices utilized to collect
data include two cameras (one mounted in front of the subject and one mounted
to the side of the subject); 12 infrared sensors surrounding the test scene as
shown in Fig. 2; a brainwave sensor (electroencephalography helmet), and six
wearable devices bound to different locations on the body to detect
acceleration, angular velocity, and luminosity. There are two types of data in
this dataset. One is the image data captured by the cameras, and the other is
the sensor data stored in a CSV file. The data collected in this dataset for
detecting human behavior is very comprehensive. The raw labels include not
only fall or no fall but also other static behavioral labels such as standing,
laying, etc.

Figure 2: Layout of sensors for the experiment in UP-FALL dataset

###  III-B Preprocessing

The main goal of this project is to binary detect falls or not. The original
labels of the dataset include behavioral labels other than falling or not,
which seems to be redundant at this stage. In the preprocessing, we keep the
falling label unchanged and set all the remaining labels as not falling. This
operation is designed to match the experimental goal of determining whether or
not a fall occurs on a frame-by-frame basis. Furthermore, we used only three
subjects out of the seventeen subjects. In addition, we considered in the
context of real-world usage that although the data collected in the original
dataset was comprehensive, it would be impractical for the elderly to wear up
to six wearable sensors all over their body to detect falls.

Figure 3: Example of accelerometer showing x, y, z axes with FitBit

Figure 4: Preprocessed grayscaled 32 $\times$ 32 images from cameras 1 & 2
with binary labeling of falling or not falling.

Figure 5: Muitimodality approach.

Thus we reduced the number of sensors from six to one, keeping only the
accelerometer on the wrist because the implementation of embedding sensors
into smartwatches is well established and fully commercialized, and wearing a
wristwatch-type device will have minimal impact on the wearers’ daily life
shown in Fig. 3. We also decided not to use angular velocity and luminosity
data for the sensor, keeping only the accelerometer data. Raw accelerometer
data were collected in the x, y, and z directions, forming a three-dimensional
vector. We calculate the magnitudes of the raw accelerometer data by the
following formula,

$SV_{total}=\sqrt{A_{x}^{2}+A_{y}^{2}+A_{z}^{2}}$ (1)

where $A_{x}$, $A_{y}$, and $A_{z}$ denote raw accelerometer data along three
axes, and $SV_{total}$ denotes the computed magnitudes. The rest of the
sensors (brainwave sensor and infrared sensors) were also eradicated, as it is
not practical to deploy specialized equipment in a practical scenario.
Finally, we obtained the preprocessed sensor data, which are the accelerations
in x, y, and z directions collected by the wrist-bounded accelerometers and
their magnitudes.

The images (frames) are converted to grayscale and resized to 32 $\times$ 32\.
A visualization of preprocessed images with corrected labels annotated can be
seen in Fig. 4.

(a) Evaluation of Mono-Sensor, Wrist Accelerometer, of F1 Performance over 50
epochs

(b) Evaluation of Multi-Sensor of F1 Performance over 50 epochs

Figure 6: Visualization of F1 scores for mono-sensor and multi-sensors

(a) Wrist Sensor Accelerometer Data Unimodality Accuracy Performance over 50
epochs

(b) Wrist Sensor Accelerometer Data Unimodality F1 score Performance over 50
epochs

(c) Camera 1 and 2 Data Unimodality Accuracy Performance over 50 epochs

(d) Camera 1 and 2 Data Unimodality F1 score Performance over 50 epochs

(e) Multimodality Fusion Accuracy Performance over 50 epochs

(f) Multimodality Fusion F1 score Performance over 50 epochs

Figure 7: Visualization of selected metrics for unimodality and multimodality.

###  III-C Multimodality approach

TABLE I: Evaluation metrics of Sensor, Camera 1, Camera 2, fusion of Camera 1 and 2, and Multimodality fusion | Sensor | Camera 1 | Camera 2 | Camera 1 & 2 | Multimodality fusion
---|---|---|---|---|---
Accuracy | 96.47 | 99.26 | 99.45 | 99.60 | 99.72
Precision | 96.47 | 99.26 | 99.45 | 99.60 | 99.72
Recall | 96.47 | 99.26 | 99.45 | 99.60 | 99.72
F1 score | 95.62 | 99.24 | 99.45 | 99.59 | 99.72

We chose an intuitive and concise method of multimodal feature fusion:
directly concatenating two different sets of features. We first extract the
feature maps of the image data and sensor data using convolutional neural
networks and then flatten the different modal features to concatenate them.
The concatenated feature embeddings are fed into a multi-layer perceptron
consisting (MLP) constituted of fully-connection layers. The final layer is
activated by a softmax function to output the corresponding prediction.

####  III-C1 Sensor data

For sensor data, the wrist accelerometer, even though for each sample
(timestamp), there are four data, i.e., $A_{x}$, $A_{y}$, $A_{z}$ and
$SV_{total}$, the dimensionality of the information is mono. We first tried
feature extraction using an MLP for unimodal prediction of falling or not. In
addition, convolutional neural networks have good feature extraction
capability. A 1D convolutional neural network can be used for one-dimensional
data. Thus we also fed the sensor data into a 1D convolutional neural network
for unimodal prediction. The results show that the 1D convolutional neural
network performs better. And since we will apply a 2D convolutional neural
network to the image data, it is obviously better for feature fusion with the
same operation criterion for the sensor data.

####  III-C2 Image data

For feature extraction of image data, the usage of convolutional networks has
been proven to be superior by various studies and applications. Therefore, we
use a 2D convolutional neural network for feature extraction of image data.
For each frame in the video stream, we convert the RGB image into a single-
channel grayscale map and then resize it to a 32 $\times$ 32 size for
inputting into the convolutional neural network.

####  III-C3 Feature fusion

For the sensor data, we feed it into a convolutional block consisting of a 1D
convolutional layer, a 1D max-pooling layer, and a batch normalization layer,
then flatten it into a 1D embedding. For two sets of image data, we first
input a convolutional block consisting of 2D convolutional layers, 2D max-
pooling layers, and batch normalization layers and then flatten it into a 1D
embedding. For three sets of flattened feature embeddings, we concatenate them
together and input them into an MLP consisting of two fully-connection layers
and a dropout layer evading overfitting. A final fully-connection layer
reduces the embedding dimension to two, and the two-dimensional embedding will
be activated by the Softmax function. The obtained probabilistic scores are
converted to output predictions.

##  IV Evaluation

###  IV-A Classification capability

We evaluated the results for unimodal sensor data, unimodal image data, and
multimodal fusion approach, respectively. The unimodal image data already
fused the extracted features from the video streams of camera one and camera
2. As evaluation metrics, we selected the accuracy and F1 scores. Accuracy is
the most basic metric for a classification task, while the F1 score combines
recall and precision for a more comprehensive and intuitive evaluation of a
classification model. Accuracy, precision, recall, and F1 score can be
calculated by the following equations,

$accuracy=\frac{TP+TN}{TP+FN+TN+FP}$ (2)

$precision=\frac{TP}{TP+FP}$ (3)

$recall=\frac{TP}{TP+FN}$ (4)

$F1score=\frac{2*precision*recall}{precision+recall}\textbf{}$ (5)

where $TP$, $FP$, $TN$, and $FN$ denote true positive, false positive, true
negative, and false negative, respectively. The visualization of the
evaluation metrics can be observed in Fig. 7. Detailed numeric comparison can
be seen in Table. I.

TABLE II: Evaluation metrics of mono-sensor (wrist) v.s multi-sensor | Accuracy | Precision | Recall | F1 score
---|---|---|---|---
Mono-sensor (wrist) | 99.72 | 99.72 | 99.72 | 99.72
Multi-sensor | 99.68 | 99.68 | 99.68 | 99.68

###  IV-B Feasibility of data cleansing

Since we reduced the number of sensors during the data preprocessing period,
we experimented with training the binary prediction model using the original
sensor data after implementing our method to ensure that the cleansing
operation did not impact the model’s binary prediction performance for fall
detection. The F1 scores of the two methods are visualized in Fig. 6, and it
can be seen that there is almost no difference in the best F1 scores of the
two methods on the validation set. Although from the images, the model using
multiple sensors performs more consistently and converges faster during
training, the single-sensor model oscillates more. To further verify the
feasibility of the data refinement operation, we further evaluated it on the
test set. Numeric results can be seen in Table II. On the test set, our model
even shows a slight advantage. We can conclude that the cleansing of the
sensor data did not affect the model’s classification performance.

##  V Discussion and Conclusion

###  V-A Conclusion

To conclude,we cleaned the data set reducing seventeen subjects to three with
eleven activities and three trials. Six sensors were reduced to use only wrist
sensor and we excluded angular velocity and luminosity to use only
accelerometer for the prediction. All of the labels were reduced to only two,
namely fall and no fall. For multi modal approach, we concatenated two
different set of features for fall detection. Furthermore, we used fusion of
sensor as well as image data to increase the performance. This data was fed to
multi layer perceptron having fully connected layers and final layer was
activated using the softmax function. By using all these experiments, we
achieved an accuracy,precision, recall and F1 score of 99.72 percent as
comapred to 99.68 percent for multi sensor data.

###  V-B Future Works

There are many options for future work we could do to improve this project.
For example, in the future, we would aim to focus more on feature extraction
to get more understanding related to falling. We would focus on experimenting
with RNN or LSTM neural networks to improve the performance of fall detection
problems. In this, we are aiming to split frames into time windows and perform
time series prediction. Along with that, we will be expanding feature
dimensions after achieving time windows. We realised that even if the accuracy
did not change with using only wrist data as compared combining all the data,
but we focus to try different combination of categories to increase
performance of model in future.

## References

  * [1] P. Kannus, M. Palvanen, S. Niemi, and J. Parkkari, “Alarming Rise in the Number and Incidence of Fall-Induced Cervical Spine Injuries Among Older Adults,” _The Journals of Gerontology: Series A_ , vol. 62, no. 2, pp. 180–183, 02 2007. [Online]. Available: https://doi.org/10.1093/gerona/62.2.180
  * [2] M. E. Tinetti, W.-L. Liu, and E. B. Claus, “Predictors and prognosis of inability to get up after falls among elderly persons,” _Jama_ , vol. 269, no. 1, pp. 65–70, 1993.
  * [3] F. Bagala, C. Becker, A. Cappello, L. Chiari, K. Aminian, J. M. Hausdorff, W. Zijlstra, and J. Klenk, “Evaluation of accelerometer-based fall detection algorithms on real-world falls,” _PloS one_ , vol. 7, no. 5, p. e37062, 2012\.
  * [4] Q. T. Huynh, U. D. Nguyen, L. B. Irazabal, N. Ghassemian, and B. Q. Tran, “Optimization of an accelerometer and gyroscope-based fall detection algorithm,” _Journal of Sensors_ , vol. 2015, 2015.
  * [5] K. Ozcan, A. K. Mahabalagiri, M. Casares, and S. Velipasalar, “Automatic fall detection and activity classification by a wearable embedded smart camera,” _IEEE journal on emerging and selected topics in circuits and systems_ , vol. 3, no. 2, pp. 125–136, 2013.
  * [6] E. Auvinet, L. Reveret, A. St-Arnaud, J. Rousseau, and J. Meunier, “Fall detection using multiple cameras,” in _2008 30th Annual International Conference of the IEEE Engineering in Medicine and Biology Society_. IEEE, 2008, pp. 2554–2557.
  * [7] Y. Li, K. Ho, and M. Popescu, “A microphone array system for automatic fall detection,” _IEEE Transactions on Biomedical Engineering_ , vol. 59, no. 5, pp. 1291–1301, 2012.
  * [8] L. Liu, M. Popescu, M. Skubic, M. Rantz, T. Yardibi, and P. Cuddihy, “Automatic fall detection based on doppler radar motion signature,” in _2011 5th International Conference on Pervasive Computing Technologies for Healthcare (PervasiveHealth) and Workshops_. IEEE, 2011, pp. 222–225.
  * [9] Z.-P. Bian, J. Hou, L.-P. Chau, and N. Magnenat-Thalmann, “Fall detection based on body part tracking using a depth camera,” _IEEE journal of biomedical and health informatics_ , vol. 19, no. 2, pp. 430–439, 2014.
  * [10] X. Zhou, L.-C. Qian, P.-J. You, Z.-G. Ding, and Y.-Q. Han, “Fall detection using convolutional neural network with multi-sensor fusion,” in _2018 IEEE international conference on Multimedia & Expo Workshops (ICMEW)_. IEEE, 2018, pp. 1–5.
  * [11] J. Ngiam, A. Khosla, M. Kim, J. Nam, H. Lee, and A. Y. Ng, “Multimodal deep learning,” pp. 689–696, 2011.
  * [12] X. Wu, D. Hong, and J. Chanussot, “Convolutional neural networks for multimodal remote sensing data classification,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 60, pp. 1–10, 2022.
  * [13] N. Srivastava and R. R. Salakhutdinov, “Multimodal learning with deep boltzmann machines,” in _Advances in Neural Information Processing Systems_ , F. Pereira, C. Burges, L. Bottou, and K. Weinberger, Eds., vol. 25. Curran Associates, Inc., 2012\. [Online]. Available: https://proceedings.neurips.cc/paper/2012/file/af21d0c97db2e27e13572cbf59eb343d-Paper.pdf
  * [14] P. Vallabh, R. Malekian, N. Ye, and D. C. Bogatinoska, “Fall detection using machine learning algorithms,” pp. 1–9, 2016.
  * [15] G. L. Santos, P. T. Endo, K. H. d. C. Monteiro, E. d. S. Rocha, I. Silva, and T. Lynn, “Accelerometer-based human fall detection using convolutional neural networks,” _Sensors_ , vol. 19, no. 7, 2019. [Online]. Available: https://www.mdpi.com/1424-8220/19/7/1644
  * [16] L. Martínez-Villaseñor, H. Ponce, J. Brieva, E. Moya-Albor, J. Núñez-Martínez, and C. Peñafort-Asturiano, “Up-fall detection dataset: A multimodal approach,” _Sensors_ , vol. 19, no. 9, p. 1988, 2019.