# On asymptotic structure at null infinity in five dimensions

Kentaro Tanabe  Yukawa Institute for Theoretical Physics, Kyoto University,
Kyoto 606-8502, Japan  Norihiro Tanahashi and Tetsuya Shiromizu  Department of
Physics, Kyoto University, Kyoto 606-8502, Japan

###### Abstract

We discuss the asymptotic structure of null infinity in five dimensional
space-times. Since it is known that the conformal infinity is not useful for
odd higher dimensions, we shall employ the coordinate based method such as the
Bondi coordinate first introduced in four dimensions. Then we will define the
null infinity and identify the asymptotic symmetry. We will also derive the
Bondi mass expression and show its conservation law.

##  I Introduction

Inspired by superstring theory, fundamental studies of higher dimensional
space-time is important. One issue is the asymptotic structure at null
infinity. We often introduce the conformal infinity to discuss the asymptotic
structure at infinities in four dimensions Penrose:1962ij ; Wald . Therein the
space-time is compactified by conformal transformation and embedded into
another space-time. For example, Minkowski space-time is conformally embedded
into the Einstein static universe. Conformally embedded space-time has two
different infinities, i.e., spatial infinity and null infinity. In higher
dimensional space-times, the asymptotic structure at spatial infinity can be
well-defined. The asymptotic symmetry is identified with the Poincare group
and conserved quantities associated with the symmetry are constructed
Shiromizu:2004jt ; Tanabe:2009xb .

On the other hand, the asymptotic structure at null infinity is not completely
understood in higher dimensional space-times Hollands:2003ie ; Hollands:2004ac
; Ishibashi:2007kb . Indeed, the definition of null infinity is given only in
even dimensions Hollands:2003ie ; Ishibashi:2007kb . As seen later, the
difficulty in defining null infinity as compared to spatial infinity is due to
the presence of the gravitational waves at null infinity. Since there are no
gravitational waves at spatial infinity, the asymptotic structure is
“stationary” and the total mass and total angular momentum are conserved. On
the other hand, the asymptotic structure at null infinity might be disturbed
by gravitational waves. Hence, we need a stable definition of null infinity
against gravitational waves. We can give such a definition in even dimensions
if we use a conformal embedding method. But we cannot do this in odd
dimensional space-times because we cannot show the smoothness of the Einstein
equations at null infinity. This non-smoothness would be related to the facts
that in the conformal embedding method we introduce the conformal factor
$\Omega\sim 1/r$, and the behavior of gravitational waves near null infinity
are of the order ${\cal O}(\Omega^{(d-2)/2})$ in $d$ dimensional space-times.
The problem comes from the half-integer power of $\Omega$. From this
smoothness only, however, we cannot show that the boundary conditions at null
infinity for asymptotic flatness in the papers Hollands:2003ie ;
Hollands:2004ac ; Ishibashi:2007kb  are the marginal and weakest conditions
which allow gravitational waves at null infinity to exist. To find such
marginal conditions, which the boundary conditions should be so, we must solve
the Einstein equations near null infinity and clarify the freedom of
gravitational waves.

In this paper, we define null infinity in five dimensional space-time. We do
not use the conformal embedding method, but instead we use the Bondi
coordinate to define null infinity, which was introduced firstly by Bondi and
Sachs in four dimensions Bondi ; Wnicour ; Sachs1 ; Sachs2 . The rest of this
paper is organized as follows. In section 2, we introduce the Bondi coordinate
in five dimensions. In section 3, we define the asymptotic flatness at null
infinity in five dimensions and show the robustness of the definition against
gravitational waves by solving Einstein equations. In this section, we define
the Bondi mass, and obtain the mass loss law in five dimensions. In section 4,
we discuss the asymptotic symmetry at null infinity associated with the
asymptotic flatness defined in section 3. We show that there are no
supertranslations in five dimensions unlike in four dimensions. Finally, in
section 5, we give a discussion and summary.

##  II Bondi coordinate in five dimensions

We consider five dimensional space-time. We introduce the Bondi coordinate to
define asymptotic flatness at null infinity. Suppose there is a function
$u(x^{a})$ which satisfies the equation

$u_{,a}u_{,b}g^{ab}\,=\,0,$ (1)

where Latin indices run from $0$ to $4$ and $u_{,a}=\partial u/\partial
x^{a}$. Then $u=\text{constant}$ surfaces are null hypersurfaces, and we use
this function $u$ as retarded time to construct a coordinate system. Let
$\theta$, $\phi$, $\psi$ be angular coordinates, which are constant along
gradient $u$. The period of each of these coordinates are taken to be $\pi$,
$2\pi$, $2\pi$, respectively. For convenience, we introduce the notation
$(\theta,\phi,\psi)=x^{A}$, where capital Latin indices run from $2$ to $4$.
Now we define the function $r$ by the equation

$r^{6}\sin^{2}\theta\cos^{2}\theta\,=\,{\rm det}(g_{AB}).$ (2)

Using these coordinates (we call them the Bondi coordinates)

$x^{0}=u\,,\,x^{1}=r\,,\,x^{2}=\theta\,,\,x^{3}=\phi\,,\,x^{4}=\psi,$ (3)

we can write down metrics as

$ds^{2}\,=\,-(Ve^{B}/r^{2})du^{2}-2e^{B}dudr+r^{2}h_{AB}(dx^{A}+U^{A}du)(dx^{B}+U^{B}du),$
(4)

where

$h_{AB}\,=\,\begin{pmatrix}e^{C_{1}}&\sin\theta\sinh D_{1}&\cos\theta\sinh
D_{2}\\\ \sin\theta\sinh
D_{1}&e^{C_{2}}\sin^{2}\theta&\sin\theta\cos\theta\sinh D_{3}\\\
\cos\theta\sinh D_{2}&\sin\theta\cos\theta\sinh
D_{3}&e^{C_{3}}\cos^{2}\theta\end{pmatrix}.$ (5)

In the above, $V,B,h_{AB},U^{A},C_{1},C_{2},C_{3},D_{1},D_{2}$ and $D_{3}$ are
functions of $u,r$ and $x^{A}$. In this coordinate, null infinity is
represented by $r\rightarrow\infty$.

From Eq. (2), we have a relation between $C_{1},C_{2},C_{3},D_{1},D_{2}$ and
$D_{3}$ as

$e^{C_{3}}\,=\,\frac{1+e^{C_{2}}\sinh^{2}D_{2}+e^{C_{1}}\sinh^{2}D_{3}-2\sinh
D_{1}\sinh D_{2}\sinh D_{3}}{e^{C_{1}+C_{2}}-\sinh^{2}D_{1}}.$ (6)

As we will realize later, these five independent functions
$C_{1},C_{2},D_{1},D_{2}$ and $D_{3}$ correspond to the degrees of freedom of
gravitational waves in five dimensional space-times.

##  III Einstein equation at null infinity

As stated in the introduction, the definition of null infinity should be not
disturbed by gravitational waves. The robustness of the null infinity
definition implies that the boundary conditions imposed on the metric (4)
should be compatible with Einstein equations. Hence, in order to define
asymptotic structure at null infinity and show the robustness of this
definition, we should specify the proper boundary conditions by solving
Einstein equations in the Bondi coordinate.

###  III.1 Vacuum Einstein equations

Since equations are very complicated, we will not write down explicit forms
here. We will show the equations in a symbolic way in order to see only the
essential structure of equations. See Appendix A for some details of the
equations which will be used.

From $R_{rr}=0$, we have

$\frac{\partial B}{\partial r}\,=\,\frac{\partial\mathbb{C}^{2}}{\partial r},$
(7)

where $\mathbb{C}$ stands for $C_{1},C_{2},D_{1},D_{2}$ and $D_{3}$. From
$R_{rA}=0$,

$\frac{\partial}{\partial r}\left(r^{5}\frac{\partial U^{A}}{\partial
r}\right)=r^{2}\left(\mathbb{C}+\mathbb{C}^{2}+\cdots\right)$ (8)

The trace and traceless part of $R_{AB}=0$ implies

$\displaystyle\frac{\partial}{\partial
r}(r^{2}e^{-B}V)=\eta(\mathbb{C},U^{A})$ (9)

and

$\displaystyle\frac{\partial^{2}}{\partial u\partial
r}\mathbb{C}=\delta(\mathbb{C},U^{A}),$ (10)

where $\eta$ and $\delta$ are some functions of $\mathbb{C}$ and $U^{A}$.

Since we integrate the equations with respect to $r$ in solving the evolution
equations, some integration functions $f(u,x^{A})$ appear. Constraint
equations describe the evolution of such functions.

In the Bondi coordinates, after integrating the other equations, we can show
that the equation $R_{ur}=0$ would be satisfied trivially.

The evolution equations $R_{uu}=0$ and $R_{uA}=0$ have the following form

$\frac{\partial V}{\partial
u}\,=\,r^{3}(\mathbb{C}^{2}+\mathbb{C}^{3}+\cdots)+r^{2}(\mathbb{C}+\mathbb{C}^{2}+\cdots)$
(11)

and

$r\frac{\partial U^{A}}{\partial u}\,=\,\frac{\partial\mathbb{C}}{\partial
u},$ (12)

respectively.

If $\mathbb{C}$ is given on initial surface $u=u_{0}$, we can determine the
metric function $B$, $U^{A}$,$V$ from Eqs. (7), (8) and (9) except for
integration functions. The evolution of $\mathbb{C}$ and integration functions
are described by Eqs. (10), (11) and (12).

As seen later, the functions $\mathbb{C}=(C_{1},C_{2},D_{1},D_{2},D_{3})$ will
be identified with the freedom of gravitational waves, and Eq. (11) will
govern the evolution of their total mass. Thus, to obtain a stable definition
of null infinity, we should determine the asymptotic behavior of the function
$\mathbb{C}$. Then, using Eqs. (7), (8) and (9), we can obtain the asymptotic
behavior of $B$, $U^{A}$, $V$, and the robustness against gravitational waves
of these boundary conditions would be guaranteed by the evolution equation
(11) and Eq. (12).

###  III.2 Asymptotic behavior of gravitational fields

From now on, we will write down all equations explicitly. When functions
$\mathbb{C}$ corresponds to gravitational waves, $\mathbb{C}$ behaves as $\sim
1/r^{3/2}$ near null infinity. This can been seen from the solutions to the
wave equation and/or the finiteness of the gravitational flux at null
infinity. Therefore, we assume the behavior of $C_{1},C_{2},C_{3},D_{1},D_{2}$
and $D_{3}$ near null infinity such that

$\displaystyle
C_{1}(u,r,x^{A})\,=\,\frac{C_{11}(u,x^{A})}{r^{3/2}}+O(1/r^{2})$ (13)
$\displaystyle
C_{2}(u,r,x^{A})\,=\,\frac{C_{21}(u,x^{A})}{r^{3/2}}+O(1/r^{2})$ (14)
$\displaystyle
C_{3}(u,r,x^{A})\,=\,\frac{C_{31}(u,x^{A})}{r^{3/2}}+O(1/r^{2})$ (15)
$\displaystyle
D_{1}(u,r,x^{A})\,=\,\frac{D_{11}(u,x^{A})}{r^{3/2}}+O(1/r^{2})$ (16)
$\displaystyle
D_{2}(u,r,x^{A})\,=\,\frac{D_{21}(u,x^{A})}{r^{3/2}}+O(1/r^{2})$ (17)
$\displaystyle
D_{3}(u,r,x^{A})\,=\,\frac{D_{31}(u,x^{A})}{r^{3/2}}+O(1/r^{2}).$ (18)

As noted around Eq. (6), $C_{31}$ can be written as

$C_{31}=-(C_{11}+C_{21}).$ (19)

Then, from Eq. (7), we see

$B=B_{1}/r^{3}+O(r^{-7/2})$ (20)

near null infinity, and we obtain a relation

$B_{1}=-\frac{(C_{11}^{2}+C_{11}C_{21}+C_{21}^{2}+D_{11}^{2}+D_{21}^{2}+D_{31}^{2})}{8}.$
(21)

Furthermore, integrating Eqs. (8), we have the relations

$\displaystyle U_{\theta 1}=\frac{2(C_{11}\cot\theta-
C_{21}\cot\theta-2C_{11}\tan\theta-
C_{21}\tan\theta+C_{11,\theta}+D_{11,\phi}\csc\theta+D_{21,\psi}\sec\theta)}{5},$
(22) $\displaystyle U_{\phi 1}\sin\theta=\frac{2(2D_{11}\cot\theta-
D_{11}\tan\theta+D_{11,\theta}+C_{21,\phi}\csc\theta+D_{31,\psi}\sec\theta)}{5}$
(23)

and

$\displaystyle U_{\psi
1}\cos\theta=\frac{2(D_{21}\cot\theta-2D_{21}\tan\theta+D_{21,\theta}+D_{31,\phi}\csc\theta-(C_{11,\psi}+C_{21,\psi})\sec\theta)}{5},$
(24)

where $U_{A1}$ are coefficients of the expansions defined as

$U^{A}=\frac{U_{A1}}{r^{5/2}}+O(1/r^{3}).$ (25)

Then, from Eq. (9), we find

$V\,=\,r^{2}+V_{1}(u,x^{A})r^{1/2}-M(u,x^{A})+O(r^{-1/2}),$ (26)

where

$V_{1}\,=\,-\frac{2}{3}\left(\frac{1}{\sin\theta\cos\theta}(U_{\theta
1}\sin\theta\cos\theta)_{,\theta}+U_{\phi 1,\phi}+U_{\psi 1,\psi}\right)$ (27)

and $M(u,x^{A})$ is an integration constant.

In Eq. (10), $C_{11,u}$, $C_{21,u}$, $D_{11,u}$, $D_{21,u}$ and $D_{31,u}$ do
not appear and then this means that we can fix them freely as initial
conditions. These freedom correspond to the degree of freedom of gravitational
waves in five dimensions.

Finally, from Eq. (11), we obtain the following formula

$\displaystyle\frac{\partial M(u,x^{A})}{\partial u}\,$ $\displaystyle=$
$\displaystyle\,-\frac{1}{3}\Biggl{(}(C_{11,u})^{2}+C_{11,u}C_{21,u}+(C_{21,u})^{2}+(D_{11,u})^{2}+(D_{21,u})^{2}+(D_{31,u})^{2}\Biggr{)}$
(28) $\displaystyle=$
$\displaystyle\,-\frac{1}{3}\Biggl{(}(C_{11,u}+C_{21,u}/2)^{2}+3(C_{21})^{2}/4+(D_{11,u})^{2}+(D_{21,u})^{2}+(D_{31,u})^{2}\Biggr{)}.$

Eq. (28) represents mass loss rate by gravitational waves, and total mass
always decreases as in four dimensions. Then $M(u,x^{A})$ describes the mass
in $u=\text{constant}$ surfaces.

###  III.3 Boundary conditions

As shown in the previous subsection, in the presence of gravitational waves,
asymptotic behaviors of metric (4) in leading order should be

$\displaystyle V=r^{2}+V_{1}(u,x^{A})r^{1/2}-M(u,x^{A})+O(1/r^{1/2})$ (29)
$\displaystyle B=\frac{B_{1}(u,x^{A})}{r^{3}}+O(1/r^{7/2})$ (30)
$\displaystyle U^{A}=\frac{U_{A1}(u,x^{A})}{r^{5/2}}+O(1/r^{3})$ (31)
$\displaystyle\mathbb{C}=\frac{\mathbb{C}_{1}(u,x^{A})}{r^{3/2}}+O(1/r^{2}).$
(32)

In four dimensions, boundary conditions at null infinity for asymptotic
flatness are determined by these leading behavior. As shown below, however, in
higher dimensions than four, further conditions are needed for asymptotic
flatness.

We consider asymptotic behavior in next-to-leading order terms. The function
$\mathbb{C}$ can be expanded as follows

$\mathbb{C}\,=\,\frac{\mathbb{C}_{1}(u,x^{A})}{r^{3/2}}+\frac{\mathbb{A}(u,x^{A})}{r^{2}}+\frac{\mathbb{C}_{2}(u,x^{A})}{r^{5/2}}+\frac{\mathbb{C}_{3}(u,x^{A})}{r^{3}}+O(1/r^{7/2}).$
(33)

The equations $R_{AB}=0$, which describes the evolution of $\mathbb{C}$,
becomes

$\frac{\mathbb{A}_{,u}}{r}+O(r^{-3/2})=0$ (34)

and then we see

$\frac{\partial\mathbb{A}}{\partial u}\,=\,0.$ (35)

At spatial infinity, asymptotic flatness requires that the Weyl tensor on
spatial infinity behave like $\sim r^{-5}$ in order for the Taub-NUT charge to
vanish Tanabe:2009xb . This implies that $\mathbb{A}$ vanishes at spatial
infinity ($u=-\infty$). From Eq. (35), $\mathbb{A}$ should vanish for
asymptotic flatness at null infinity

$\mathbb{A}\,=\,0.$ (36)

Then, from Eqs. (9), (7) and (8), we can show that other metric functions can
be expanded as follows 111$U_{A3}$ terms correspond to angular momentum. This
can be seen from the comparison with the asymptotic behavior of Myers-Perry
solutions MP .

$\displaystyle V=r^{2}+V_{1}(u,x^{A})r^{1/2}-M(u,x^{A})+O(1/r^{1/2}),$ (37)
$\displaystyle
B=\frac{B_{1}(u,x^{A})}{r^{3}}+\frac{B_{2}(u,x^{A})}{r^{4}}+\frac{B_{3}(u,x^{A})}{r^{9/2}}+O(1/r^{5}),$
(38) $\displaystyle
U^{A}=\frac{U_{A1}(u,x^{A})}{r^{5/2}}+\frac{U_{A2}(u,x^{A})}{r^{7/2}}+\frac{U_{A3}(u,x^{A})}{r^{4}}+O(1/r^{9/2}).$
(39)

Thus, boundary conditions at null infinity for asymptotic flatness are

$\mathbb{C}\,=\,\frac{\mathbb{C}_{1}(u,x^{A})}{r^{3/2}}+\frac{\mathbb{C}_{2}(u,x^{A})}{r^{5/2}}+\frac{\mathbb{C}_{3}(u,x^{A})}{r^{3}}+O(1/r^{7/2}),$
(40)

and Eqs. (37), (38), (39).

##  IV Asymptotic symmetry at null infinity

In this section, we consider asymptotic symmetry at null infinity. Asymptotic
symmetry should be defined as transformations preserving the boundary
conditions (40), (37), (38) and (39). By infinitesimal transformation $\xi$,
the metric is transformed as

$\delta g_{ab}\,=\,2\nabla_{(a}\xi_{b)}.$ (41)

To preserve the boundary conditions given in the previous section, the
variation of metric, $\delta g_{ab}$, should satisfy

$\displaystyle\delta g_{rr}\,=\,0\,,\,\delta g_{rA}\,=\,0\,,\,g^{AB}\delta
g_{AB}\,=\,0,$ (42) $\displaystyle\delta g_{uu}\,=\,O(r^{-3/2})\,,\,\delta
g_{ur}\,=\,O(r^{-3})\,,\,\delta g_{uA}\,=\,O(r^{-1/2})\,,\,\delta
g_{AB}\,=\,O(r^{1/2}).$ (43)

Here, as a first step, we will consider leading order terms.

From Eq. (42), we see that the components of infinitesimal transformation
$\xi$ must take the following forms:

$\displaystyle\xi_{r}=f(u,x^{A})e^{B},$ (44)
$\displaystyle\xi_{B}g^{AB}=f^{A}(u,x^{A})-fU^{A}+\int^{\infty}_{r}dr^{\prime}e^{B}f_{,B}g^{AB},$
(45)
$\displaystyle\xi_{u}=-\frac{re^{B}}{3}\left(-\xi_{A,B}+\xi_{C}\Gamma^{C}_{AB}+\xi_{r}\Gamma^{r}_{AB}\right)g^{AB}.$
(46)

The infinitesimal transformation $\xi$ has four free parameters $f,f^{A}$. $f$
and $f^{A}$ corresponds to translations and Lorentz transformation,
respectively. The other components of metric variation become

$\displaystyle\delta g_{uu}$ $\displaystyle=$ $\displaystyle\frac{2r}{3}{\cal
D}_{A}f^{A}_{~{}~{},u}+\frac{2}{3}(3f+{\cal
D}^{2}f)_{,u}+\frac{2}{r^{1/2}}h^{(0)}_{AB}f^{A}_{~{}~{},u}U_{B1}+O(r^{-3/2}),$
(47) $\displaystyle\delta g_{ur}$ $\displaystyle=$
$\displaystyle\frac{1}{3}({\cal
D}_{A}f^{A}+3f_{,u})+\frac{1}{5r^{5/2}}h^{(1)AB}{\cal D}_{A}{\cal
D}_{B}f+O(r^{-3}),$ (48) $\displaystyle\delta g_{uA}$ $\displaystyle=$
$\displaystyle r^{2}h^{(0)}_{AB}f^{B}_{~{}~{},u}+\frac{r}{3}{\cal
D}_{A}(3f_{,u}+{\cal
D}_{B}f^{B})+r^{1/2}h^{(1)}_{AB}f^{B}_{~{}~{},u}+\frac{1}{3}{\cal D}_{A}({\cal
D}^{2}f+3f)+O(r^{-1/2}),$ (49) $\displaystyle\delta g_{AB}$ $\displaystyle=$
$\displaystyle\frac{2r^{2}}{3}(-{\cal D}_{C}f^{C}h^{(0)}_{AB}+3{\cal
D}_{(A}f_{B)})+\frac{2r}{3}(-{\cal D}^{2}fh^{(0)}_{AB}+3{\cal D}_{A}{\cal
D}_{B}f)+T_{AB}(u,x^{A})r^{1/2}+O(r^{-1/2}),$ (50)

where $X_{(AB)}:=(1/2)(X_{AB}+X_{BA})$ for some tensor $X_{AB}$ and $T_{AB}$
is some traceless tensor with respect to $h_{AB}^{(0)}$. In the above, we
expanded the metric $h_{AB}$ as

$h_{AB}\,=\,h^{(0)}_{AB}+\frac{1}{r^{3/2}}h^{(1)}_{AB}+O(r^{-5/2}),$ (51)

and $h^{(1)AB}=h^{(0)AC}h^{(0)BD}h^{(1)}_{CD}.$ In the Bondi coordinate,
$h^{(0)}_{AB}$ and $h^{(1)}_{AB}$ are

$h^{(0)}_{AB}=\begin{pmatrix}1&0&0\\\ 0&\sin^{2}\theta&0\\\
0&0&\cos^{2}\theta\end{pmatrix}$ (52)

and

$h^{(1)}_{AB}=\begin{pmatrix}C_{11}&D_{11}\sin\theta&D_{21}\cos\theta\\\
D_{11}\sin\theta&C_{21}\sin^{2}\theta&D_{31}\sin\theta\cos\theta\\\
D_{21}\cos\theta&D_{31}\sin\theta\cos\theta&-(C_{11}+C_{21})\cos^{2}\theta\end{pmatrix}.$
(53)

Note that $h^{(1)}_{AB}$ is traceless, $h^{(0)AB}h^{(1)}_{AB}=0$.

To satisfy the condition (43), we find that $f^{A}$ and $f$ should satisfy the
following equations:

$\displaystyle f^{A}_{~{}~{},u}=0,$ (54) $\displaystyle{\cal D}_{A}f_{B}+{\cal
D}_{B}f_{A}=-2\frac{\partial f}{\partial u}h_{AB}^{(0)},$ (55)
$\displaystyle{\cal D}_{A}{\cal D}_{B}f=\frac{1}{3}{\cal D}^{2}fh_{AB}^{(0)}.$
(56)

Integrating the trace part of Eq. (55) under the condition (54), we obtain

$f=-\frac{u}{3}F+\alpha(x^{A}),$ (57)

where $F:={\cal D}_{A}f^{A}$ and $\alpha(x^{A})$ is an arbitrary function of
$x^{A}$. Here we can show from Eq. (55) that $F$ satisfies

${\cal D}_{A}{\cal D}_{B}F=\frac{1}{3}{\cal D}^{2}Fh^{(0)}_{AB}$ (58)

and

${\cal D}^{2}F+3F=0.$ (59)

See Appendix B for the derivation. The general solution to these is given by

$F=E_{x}\sin\theta\cos\phi+E_{y}\sin\theta\sin\phi+E_{z}\cos\theta\cos\psi+E_{w}\cos\theta\sin\psi,$
(60)

where $E_{x},E_{y},E_{z},E_{w}$ are constants. Then, from Eqs. (56) and (57),
we see that

${\cal D}_{A}{\cal D}_{B}\alpha=\frac{1}{3}{\cal D}^{2}\alpha h^{(0)}_{AB}$
(61)

holds. The solution to this is

$\alpha=a_{u}+a_{x}\sin\theta\cos\phi+a_{y}\sin\theta\sin\phi+a_{z}\cos\theta\cos\psi+a_{w}\cos\theta\sin\psi,$
(62)

where $a_{u},a_{x},a_{y},a_{z},a_{w}$ are constants. As a summary the general
solution for $f$ is given by

$f\,=\,e_{u}+e_{x}(u)\sin\theta\cos\phi+e_{y}(u)\sin\theta\sin\phi+e_{z}(u)\cos\theta\cos\psi+e_{w}(u)\cos\theta\sin\psi,$
(63)

where $e_{u}:=a_{u}$ is constant,
$e_{x}(u):=-(u/3)E_{x}+a_{x},e_{y}(u):=-(u/3)E_{y}+a_{y},e_{z}(u):=-(u/3)E_{z}+a_{z}$
and $e_{w}(u):=-(u/3)E_{w}+a_{w}$. Now we can show that

${\cal D}_{A}({\cal D}^{2}f+3f)\,=\,0,\qquad\partial_{u}({\cal
D}^{2}f+3f)\,=\,0$ (64)

hold. Using the above equations, at a glance, we see that Eqs. (43) become

$\displaystyle\delta g_{uu}$ $\displaystyle=$ $\displaystyle O(r^{-3/2}),$
(65) $\displaystyle\delta g_{ur}$ $\displaystyle=$
$\displaystyle\frac{1}{5r^{5/2}}h^{(1)AB}{\cal D}_{A}{\cal D}_{B}f+O(r^{-3}),$
(66) $\displaystyle\delta g_{uA}$ $\displaystyle=$ $\displaystyle
O(r^{-1/2}),$ (67) $\displaystyle\delta g_{AB}$ $\displaystyle=$
$\displaystyle\frac{2r^{2}}{3}(-{\cal D}_{C}f^{C}h^{(0)}_{AB}+3{\cal
D}_{(A}f_{B)})+T_{AB}(u,x^{A})r^{1/2}+O(r^{-1/2}).$ (68)

$h^{(1)AB}{\cal D}_{A}{\cal D}_{B}f$ in Eq. (66) vanishes because of ${\cal
D}_{A}{\cal D}_{B}f\propto h_{AB}^{(0)}$ and the trace-free property of
$h^{(1)AB}$. Noting that condition of (55) is rearranged as ${\cal
D}_{A}f_{B}+{\cal D}_{B}f_{A}=(2/3){\cal D}^{C}f_{C}h_{AB}^{(0)}$, we can show
$\delta g_{AB}=O(r^{1/2})$. As a consequence, the transformation satisfying
the conditions of Eqs. (54), (55) and (56) does not disturb the boundary
condition of Eqs. (40), (37), (38) and (39) at leading order. By
straightforward calculation, it can be shown that these transformations keep
the metric satisfying the boundary conditions at next-to-leading order.

Thus, asymptotic symmetry is generated by $f$ and $f^{A}$ satisfying Eqs.
(54), (55) and (56). The part of $f$ not proportional to $u$ generates a
translation group. Since this part has only five degrees of freedom, there is
no supertranslation freedom unlike in four dimensions 222The fact that
supertranslation freedom does not exist in higher dimensions is firstly
pointed out in Hollands:2003ie  by using conformal method in even dimensions.
We show that in five dimensions using the Bondi coordinates. . $f^{A}$
generates a Lorentz transformation group, so asymptotic symmetry at null
infinity in five dimensional space-time is a Poincare group which is semi-
direct of a five dimensional transformation group and Lorentz group.

##  V summary and discussion

In this paper, we define asymptotic flatness at null infinity in five
dimensional space-time by using the Bondi coordinates. In the conformal
embedding method, we cannot show the smoothness of asymptotic structure at
null infinity because the gravitational waves behave like $\Omega^{3/2}\sim
r^{-3/2}$ near null infinity and the regularity of gravitational fields at
null infinity is not guaranteed in general in five dimensions. On the other
hand, in the Bondi coordinates, we can show the robustness of the asymptotic
structure at null infinity which is defined by boundary conditions of Eqs.
(40), (37), (38) and (39). Solving Einstein equations under these boundary
conditions, we find that total mass always decreases by gravitational waves as
in four dimensions. In addition, we show that the asymptotic symmetry at null
infinity would be a Poincare group in five dimensions.

In four dimensions, asymptotic symmetry at null infinity is not a Poincare
group. There are so called supertranslation freedoms, i.e., there are infinite
dimensional translations. This supertranslation freedom comes from the freedom
of the Bondi coordinates. In the Bondi coordinates, coordinate transformation
is described by the parameter $f$, $f^{A}$ in Eqs. (44), (45) and (46), which
corresponds to a translation and Lorentz transformation, respectively. If
$f^{A}=0$ (pure translation) we may expect that $f$ has only four independent
solutions. In general, however, there are conditions that $f$ should be a
functions on a $2$-sphere and then $f$ has functional freedom, i.e., infinite
dimensional degrees of freedom. This infinite dimensional set, which has an
Abelian group structure, is called a supertranslational group.

In four dimensional Minkowski space-time without any physical perturbations,
the term $O(r)$ of $\delta g_{AB}$ should vanish, and this condition reduces
infinite dimensional supertranslation to four dimensional translation. In
general, however, since gravitational waves contribute to $g_{AB}$ with $O(r)$
terms, which is the same order with metric variance, we cannot reduce
supertranslation to translation. That is, we cannot distinguish the
supertranslational ambiguity from gravitational waves. Thus the asymptotic
symmetry at null infinity in four dimensions is not a Poincare group.

On the other hand, as shown in this paper, in five dimensions, there is no
supertranslational freedoms. This is because the behavior of gravitational
waves in five dimensions is $1/r^{3/2}$ and this contributes to $g_{AB}$ with
$O(r^{1/2})$ terms. In asymptotically flat five dimensional space-time, the
term of $O(r)$ in $\delta g_{AB}$, which could be a contribution from the
supertranslation, should vanish to maintain asymptotic flatness. This
condition eliminates supertranslational freedom, and makes the asymptotic
symmetry a Poincare group. Although we have only shown this feature in five
dimensions, we expect that this feature would be common to the higher
dimensional space-time in general, because gravitational waves in $d>4$
dimensions decay $1/r^{(d-2)/2}$ near null infinity faster than
supertranslation $O(1/r)$.

In four dimensions, since there are supertranslations, we cannot choose a
preferred rotational axis. The definition of angular momentum at null infinity
does not have a precise meaning. Supertranslations and gravitational waves
both contribute to angular momentum change, and we cannot distinguish one from
another. However, since there is no supertranslation in five dimensions, we
can define total angular momentum at null infinity and observe the change of
total angular momentum by gravitational waves. Furthermore, as we show the
robustness of null infinity definition in five dimensions using the Bondi
coordinates, it may be possible for us to redefine asymptotic flatness at null
infinity using a conformal embedding method covariantly. This is left for
future issue.

Another remaining issue is the extension of our work to dimensions higher than
seven. Using the Bondi coordinates, we have to go on step by step. It will be
nice to have a systematic analysis for that. This is also our future work.

###### Acknowledgements.

KT is supported by JSPS Grant-Aid for Scientific Research. NT and TS are
partially supported by Grant-Aid for Scientific Research from Ministry of
Education, Science, Sports and Culture of Japan (Nos. 2056381, 20540258,
21111006, and 19GS0219), the Japan-U.K. Research Cooperative Programs. This
work is also supported by the Grant-in-Aid for the Global COE Program “The
Next Generation of Physics, Spun from Universality and Emergence h from the
Ministry of Education, Culture, Sports, Science and Technology (MEXT) of
Japan.

##  Appendix A Einstein equation

We write the components of an Einstein equation explicitly in the expansion
form with $1/r$. If for example, we expand function $C_{1}(u,r,x^{A})$ such
that

$C_{1}\,=\,\frac{C_{11}(u,x^{A})}{r\sqrt{r}}+\frac{A(u,x^{A})}{r^{2}}+O(r^{-5/2}),$
(69)

The Einstein equation $R_{\theta\theta}=0$ becomes

$-\frac{A_{,u}}{2r}+O(r^{-3/2})=0.$ (70)

That is, $\partial A(u,x^{A})/\partial u=0$. In Ref. Tanabe:2009xb , it was
shown that from asymptotic flatness at spatial infinity,
$A(u,x^{A})|_{u=-\infty}=0$. From this fact and $\partial A(u,x^{A})/\partial
u=0$, we find that $A(u,x^{A})=0$ on null infinity. We can show in the same
way that the $O(r^{-2})$ term in $C_{2},D_{1},D_{2},D_{3}$ should also vanish.

Thus, function $\mathbb{C}$ is expanded as following

$\displaystyle
C_{1}(u,r,x^{A})\,=\,\frac{C_{11}(u,x^{A})}{r\sqrt{r}}+\frac{C_{12}(u,x^{A})}{r^{2}\sqrt{r}}+\frac{C_{13}(u,x^{A})}{r^{3}}+O(1/r^{7/2})$
(71) $\displaystyle
C_{2}(u,r,x^{A})\,=\,\frac{C_{21}(u,x^{A})}{r\sqrt{r}}+\frac{C_{22}(u,x^{A})}{r^{2}\sqrt{r}}+\frac{C_{23}(u,x^{A})}{r^{3}}+O(1/r^{7/2})$
(72) $\displaystyle
D_{1}(u,r,x^{A})\,=\,\frac{D_{11}(u,x^{A})}{r\sqrt{r}}+\frac{D_{12}(u,x^{A})}{r^{2}\sqrt{r}}+\frac{D_{13}(u,x^{A})}{r^{3}}+O(1/r^{7/2})$
(73) $\displaystyle
D_{2}(u,r,x^{A})\,=\,\frac{D_{21}(u,x^{A})}{r\sqrt{r}}+\frac{D_{22}(u,x^{A})}{r^{2}\sqrt{r}}+\frac{D_{23}(u,x^{A})}{r^{3}}+O(1/r^{7/2})$
(74) $\displaystyle
D_{3}(u,r,x^{A})\,=\,\frac{D_{31}(u,x^{A})}{r\sqrt{r}}+\frac{D_{32}(u,x^{A})}{r^{2}\sqrt{r}}+\frac{D_{33}(u,x^{A})}{r^{3}}+O(1/r^{7/2}).$
(75)

Then we find from the Einstein equation that the other metric function in (4)
should be expanded as follows

$\displaystyle V\,=\,r^{2}+V_{1}(u,x^{A})\sqrt{r}+M(u,x^{A})+O(1/r^{1/2})$
(76) $\displaystyle
B\,=\,\frac{B_{1}(u,x^{A})}{r^{3}}+\frac{B_{2}(u,x^{A})}{r^{4}}+O(1/r^{9/2})$
(77) $\displaystyle
U^{A}\,=\,\frac{U_{A1}(u,x^{A})}{r^{5/2}}+\frac{U_{A2}(u,x^{A})}{r^{7/2}}+\frac{U_{A3}(u,x^{A})}{r^{4}}+O(1/r^{9/2}).$
(78)

Now we can write down each component of the Einstein equation as follows:

$R_{rr}=0$:

$\displaystyle-\frac{9}{8r^{5}}\left(8B_{1}+C_{11}^{2}+C_{11}C_{21}+C_{21}^{2}+D_{11}^{2}+D_{21}^{2}+D_{31}^{2}\right)$
$\displaystyle-\frac{3}{8r^{6}}\left(32B_{2}+10C_{11}C_{12}+5C_{12}C_{21}+5C_{11}C_{22}+10C_{21}C_{22}\right.$
$\displaystyle~{}~{}~{}~{}~{}\left.+10D_{11}D_{12}+10D_{21}D_{22}+10D_{31}D_{32}\right)+O(r^{-13/2})\,=\,0,$
(79)

$R_{r\theta}=0$:

$\displaystyle-\frac{3}{8r^{5/2}}\left(2C_{11}\cot\theta-2C_{21}\cot\theta-4C_{11}\tan\theta-2C_{21}\tan\theta\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-5U_{\theta
1}\left.+2D_{21,\psi}\sec\theta+2D_{11,\phi}\csc\theta+2C_{11,\theta}\right)$
$\displaystyle+\frac{1}{8r^{7/2}}\left(-10C_{12}\cot\theta+10C_{22}\cot\theta+20C_{12}\tan\theta+10C_{22}\tan\theta\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.+7U_{\theta
2}-10D_{22,\psi}\sec\theta-10D_{12,\phi}\csc\theta-10C_{12,\theta}\right)$
$\displaystyle~{}~{}~{}+O(r^{-4})\,=\,0,$ (80)

$R_{r\phi}=0$:

$\displaystyle-\frac{3}{8r^{5/2}}\left(4D_{11}\cos\theta-2D_{11}\sin\theta\tan\theta-5U_{\phi
1}\sin^{2}\theta\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.+2D_{31,\psi}\tan\theta+2C_{21,\phi}+2D_{11,\theta}\sin\theta\right)$
$\displaystyle+\frac{1}{8r^{7/2}}\left(-20D_{12}\cos\theta+10D_{12}\sin\theta\tan\theta+7U_{\phi
2}\sin^{2}\theta\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.-10D_{12,\theta}\sin\theta-10C_{22,\phi}-10D_{32,\psi}\tan\theta\right)$
$\displaystyle~{}~{}~{}~{}+O(r^{-4})=0,$ (81)

$R_{r\psi}=0$:

$\displaystyle\frac{3}{8r^{5/2}}(-2D_{21}\cot\theta\cos\theta+4D_{21}\sin\theta+5U_{\psi
1}\cos^{2}\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+2C_{11,\psi}+2C_{21,\psi}-2D_{31,\phi}\cot\theta-2D_{21,\theta}\cos\theta)$
$\displaystyle+\frac{1}{8r^{7/2}}(-10D_{22}\cos\theta\cot\theta+20D_{22}\sin\theta+7U_{\psi
2}\cos^{2}\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+10C_{12,\psi}+10C_{22,\psi}-10D_{32,\phi}\cot\theta-10D_{22,\theta}\cos\theta)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+O(r^{-9/2})=0,$ (82)

$R_{\theta\theta}=0$:

$\displaystyle\frac{1}{8r^{3/2}}\left(-13C_{11}-8U_{\theta
1}\cot\theta+8U_{\theta
1}\tan\theta-4V_{1}-8D_{21,\psi}\sec\theta\tan\theta\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-8U_{\psi 1,\psi}-8U_{\phi
1,\phi}-4C_{11,\phi\phi}\csc^{2}\theta-4C_{11,\psi\psi}\sec^{2}\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+8D_{11,\phi}\cot\theta\csc\theta+4C_{11,\theta}\cot\theta-12C_{11,\theta}\tan\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.-8C_{21,\theta}\tan\theta-12U_{\theta
1,\theta}+8D_{21,\theta\psi}\sec\theta+8D_{11,\theta\phi}\csc\theta\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.+4C_{11,\theta\theta}-8C_{21,\theta}\cot\theta-8C_{12,u}\right)$
$\displaystyle+\frac{3}{2r^{2}}\left(C_{13,u}-D_{11}D_{11,u}-D_{21}D_{21,u}\right)+O(r^{-5/2})=0,$
(83)

$R_{\theta\phi}=0$:

$\displaystyle\frac{1}{8r^{3/2}}\left(3D_{11}\sin\theta-4D_{31,\psi}-4D_{31,\psi}\tan^{2}\theta-4D_{11,\psi\psi}\sec\theta\tan\theta\right.$
$\displaystyle~{}~{}-4C_{11,\phi}\cot\theta-8C_{11,\phi}\tan\theta-4C_{21,\phi}\cot\theta-4C_{21,\phi}\tan\theta-2U_{\theta
1,\phi}$ $\displaystyle~{}~{}+4D_{21,\phi\psi}\sec\theta-2U_{\phi
1,\theta}\sin^{2}\theta+4D_{31,\theta\psi}\tan\theta+4C_{11,\theta\phi}$
$\displaystyle~{}~{}\left.+4C_{21,\theta\phi}-8D_{12,u}\sin\theta\right)$
$\displaystyle+\frac{3\sin\theta}{4r^{2}}\left(D_{11}C_{11,u}+D_{11}C_{21,u}+C_{11}D_{11,u}+C_{21}D_{11,u}\right.$
$\displaystyle~{}~{}\left.-2D_{13,u}+D_{31}D_{21,u}+D_{21}D_{31,u}\right)+O(r^{-5/2})=0,$
(84)

$R_{\theta\psi}=0$:

$\displaystyle\frac{1}{8r^{3/2}}(3D_{21}\cos\theta+4C_{11,\psi}\cot\theta-4C_{21,\psi}\cot\theta-4C_{21,\psi}\tan\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-2U_{\theta
1,\psi}+4D_{31,\phi}+4D_{31,\phi}\cot^{2}\theta+4D_{11,\phi\psi}\csc\theta-4D_{21,\phi\phi}\cot\theta\csc\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-2U_{\psi
1,\theta}\cos^{2}\theta-4C_{21,\theta\psi}+4D_{31,\theta\phi}\cot\theta-8D_{22,u}\cos\theta)$
$\displaystyle+\frac{3\cos\theta}{4r^{2}}\left(D_{21}C_{21,u}-D_{31}D_{11,u}+C_{21}D_{21,u}+D_{23,u}-D_{11}D_{31,u}\right)+O(r^{-5/2})\,=\,0,$
(85)

$R_{\phi\phi}=0$:

$\displaystyle\frac{1}{8r^{3/2}}\left(-16C_{11}\sin^{2}\theta+3C_{21}\sin^{2}\theta-12U_{\theta
1}\cos\theta\sin\theta+8U_{\theta
1}\sin^{2}\theta\tan\theta-4V_{1}\sin^{2}\theta\right.$
$\displaystyle~{}~{}+8D_{21,\psi}\sin\theta-8U_{\psi
1,\psi}\sin^{2}\theta-4C_{21,\psi\psi}\tan^{2}\theta+8D_{11,\phi}\cos\theta-8D_{11,\phi}\sin\theta\tan\theta$
$\displaystyle~{}~{}-12U_{\phi
1,\phi}\sin^{2}\theta+8D_{31,\phi\psi}\tan\theta+4C_{21,\phi\phi}+8C_{11,\theta}\cos\theta\sin\theta-4C_{21,\theta}\cos\theta\sin\theta$
$\displaystyle~{}~{}+4C_{21,\theta}\sin^{2}\theta\tan\theta-8U_{\theta
1,\theta}\sin^{2}\theta+8D_{11,\theta\phi}\sin\theta-4C_{21,\theta\theta}\sin^{2}\theta-8C_{22,u}\sin^{2}\theta\left.\right)$
$\displaystyle-\frac{3\sin^{2}\theta}{2r^{2}}\left(C_{23,u}-D_{11}D_{11,u}-D_{31}D_{31,u}\right)+O(r^{-5/2})=0,$
(86)

$R_{\phi\psi}=0$:

$\displaystyle\frac{1}{8r^{3/2}}(4D_{31}\cos^{2}\theta\cot\theta+11D_{31}\cos\theta\sin\theta+4D_{31}\sin^{2}\theta\tan\theta+8D_{11,\psi}\cos\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+4D_{11,\psi}\sin\theta\tan\theta-2U_{\phi
1,\psi}\sin^{2}\theta-4D_{21,\phi}\cos\theta\cot\theta-8D_{21,\phi}\sin\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-2U_{\psi
1,\phi}\cos^{2}\theta-4C_{11,\phi\psi}-4D_{31,\theta}\cos^{2}\theta+4D_{31,\theta}\sin^{2}\theta+4D_{11,\theta\psi}\sin\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+4D_{21,\theta\phi}\cos\theta-
D_{31,\theta\theta}\cos\theta\sin\theta-8D_{32,u}\cos\theta\sin\theta)$
$\displaystyle-\frac{3\cos\theta\sin\theta}{4r^{2}}\left(D_{31}C_{11,u}-D_{21}D_{11,u}-D_{11}D_{21,u}+C_{11}D_{31,u}+D_{33,u}\right)$
$\displaystyle~{}~{}~{}~{}+O(r^{-5/2})=0,$ (87)

$R_{\psi\psi}=0$:

$\displaystyle\frac{1}{8r^{3/2}}\left(-19C_{11}\cos^{2}\theta-3C_{21}\cos^{2}\theta-8U_{\theta
1}\cos^{2}\theta\cot\theta-4V_{1}\cos^{2}\theta+12U_{\theta
1}\cos\theta\sin\theta\right.$
$\displaystyle~{}~{}~{}~{}~{}+8D_{21,\psi}\cos\theta\cot\theta-8D_{21,\psi}\sin\theta-12U_{\psi
1,\psi}\cos^{2}\theta-4C_{11,\psi\psi}-4C_{21,\psi\psi}$
$\displaystyle~{}~{}~{}~{}~{}-8D_{11,\phi}\cos\theta-8U_{\phi
1,\phi}\cos^{2}\theta+8D_{31,\phi\psi}\cot\theta+4C_{11,\phi\phi}\cot^{2}\theta+4C_{21,\phi\phi}\cot^{2}\theta$
$\displaystyle~{}~{}~{}~{}~{}+4C_{11,\theta}\cos^{2}\theta\cot\theta-12C_{11,\theta}\cos\theta\sin\theta+4C_{21,\theta}\cos^{2}\theta\cot\theta-4C_{21,\theta}\cos\theta\sin\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}-8U_{\theta
1,\theta}\cos^{2}\theta+8D_{21,\theta\psi}\cos\theta+8C_{11,\theta\theta}\cos^{2}\theta+8(C_{12,u}+C_{22,u})\cos^{2}\theta\left.\right)$
$\displaystyle+\frac{3\cos^{2}\theta}{2r^{2}}\left(C_{13,u}+C_{23,u}-2D_{11}D_{11,u}-D_{21}D_{21,u}-D_{31}D_{31,u}\right)+O(r^{-5/2})=0,$
(88)

$R_{uu}=0$:

$\displaystyle\frac{1}{2r^{5/2}}(3V_{1,u}+2(\cot\theta-\tan\theta)U_{\theta
1,u}+2U_{\theta 1,u\theta}+2U_{\phi 1,u\phi}+2U_{\psi 1,u\psi})$
$\displaystyle-\frac{1}{2r^{3}}\left(3M_{,u}-(C_{11,u})^{2}-C_{11,u}C_{21,u}-(C_{21,u})^{2}-(D_{11,u})^{2}\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-(D_{21,u})^{2}-(D_{31,u})^{2}\left.\right)+O(r^{-7/2})=0,$
(89)

$R_{u\theta}=0$:

$\displaystyle-\frac{1}{4r^{3/2}}(2C_{11,u}\cot\theta-4C_{11,u}\tan\theta+2C_{21,u}\cot\theta-2C_{21,u}\tan\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-5U_{\theta
1,u}+D_{21,u\psi}\sec\theta D_{11,u\phi}\csc\theta+2C_{11,u\theta})$
$\displaystyle+\frac{1}{8r^{5/2}}\left(-U_{\theta 1}-8U_{\psi
1,\psi}\tan\theta-4U_{\theta 1,\psi\psi}\sec^{2}\theta+8U_{\phi
1,\phi}\cot\theta\right.$ $\displaystyle~{}~{}~{}~{}~{}~{}-4U_{\theta
1,\phi\phi}\csc^{2}\theta-2V_{1,\theta}+4U_{\psi 1,\theta\psi}+4U_{\phi
1,\theta\phi}+4C_{12,u}\cot\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}-8C_{12,u}\tan\theta-4C_{22,u}\cot\theta-4C_{22,u}\tan\theta-14U_{\theta
2,u}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+4D_{22,u}\sec\theta+4D_{12,u}\csc\theta+4C_{12,u}\left.\right)+O(r^{-4})=0,$
(90)

$R_{u\phi}=0$:

$\displaystyle\frac{1}{r^{3/2}}(4D_{11,u}\cos\theta-2D_{11,u}\sin\theta\tan\theta-5U_{\phi
1,u}\sin^{2}\theta+2D_{31,u\psi}\tan\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+2C_{21,u\phi}+2D_{11,u\theta}\sin\theta)$
$\displaystyle~{}~{}+\frac{1}{8r^{5/2}}(15U_{\phi
1,\psi\psi}\sin^{2}\theta-4U_{\phi 1,\psi\psi}\tan^{2}\theta-4U_{\theta
1,\phi}\cot\theta-4U_{\theta 1,\phi}\tan\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-2V_{1,\phi}+4U_{\psi 1,\phi\psi}-12U_{\phi
1,\theta}\cos\theta\sin\theta+4U_{\phi 3,\theta}\sin^{2}\theta\tan\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+4U_{\theta 1,\theta\phi}-4U_{\phi
3,\theta\theta}\sin^{2}\theta+D_{12,u\theta}\sin\theta-4D_{12,u}\sin\theta\tan\theta-14U_{\phi
2,u}\sin^{2}\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+4D_{32,u\psi}\tan\theta+4C_{22,u\phi}+4D_{12,u\theta}\sin\theta)+O(r^{-3})=0,$
(91)

$R_{u\psi}=0$:

$\displaystyle\frac{1}{4r^{3/2}}(2D_{21,u}\cos\theta\cot\theta-4D_{21,u}\sin\theta-5U_{\psi
1,u}\cos^{2}\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2C_{11,u\psi}-2C_{21,u\psi}+2D_{31,u\phi}\cot\theta+2D_{21,u\theta}\cos\theta)$
$\displaystyle+\frac{1}{8r^{5/2}}\left(15U_{\psi 1}\cos^{2}\theta+4U_{\theta
1,\psi}\cot\theta+4U_{\theta 1,\psi}\tan\theta-2V_{1,\psi}\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+4U_{\phi 3,\phi\psi}-4U_{\psi
1,\phi\phi}\cot^{2}\theta-4U_{\psi 1,\psi}\cos^{2}\theta\cot\theta+12U_{\psi
1,\theta}\cos\theta\sin\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+4U_{\theta
1,\theta\psi}-8D_{22,u}\sin\theta-14U_{\psi
2,u}\cos^{2}\theta-4C_{12,u\psi}-4C_{22,u\psi}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+4D_{32,u\phi}\cot\theta+4D_{22,u\theta}\cos\theta\left.\right)$
$\displaystyle~{}~{}~{}~{}~{}+O(r^{-3})=0.$ (92)

##  Appendix B Derivation for Eqs. (58) and (59)

Here we sketch the derivation for Eqs. (58) and (59). We begin with

$\displaystyle{\cal D}_{B}{\cal D}_{A}F={\cal D}_{B}{\cal D}_{A}{\cal
D}_{C}f^{C}$ $\displaystyle=$ $\displaystyle{\cal D}_{B}({\cal D}_{C}{\cal
D}_{A}f^{C}-{}^{(3)}R_{AC}f^{C})$ (93) $\displaystyle=$ $\displaystyle-{\cal
D}_{B}{\cal D}_{C}{\cal D}^{C}f_{A}-2{\cal D}_{A}{\cal
D}_{B}\partial_{u}f-2{\cal D}_{B}f_{A},$

where ${}^{(3)}R_{AB}$ is the Ricci tensor with respect to $h_{AB}^{(0)}$. In
the second and last line, we used the definition of Riemann tensor and Eq.
(55), respectively. We also used the fact that that Riemann tensor with
respect to $h_{AB}^{(0)}$ is
${}^{(3)}R_{ABCD}=h_{AC}^{(0)}h_{BD}^{(0)}-h_{AD}^{(0)}h_{BC}^{(0)}$. Using
the definition of Riemann tensor two times, the first term in the last line of
the right-hand side becomes

$\displaystyle{\cal D}_{B}{\cal D}_{C}{\cal D}^{C}f_{A}$ $\displaystyle=$
$\displaystyle{\cal D}_{C}{\cal D}_{B}{\cal D}^{C}f_{A}-2{\cal
D}_{B}f_{A}+h_{AB}^{(0)}{\cal D}_{C}f^{C}-{\cal D}_{A}f_{B}$ (94)
$\displaystyle=$ $\displaystyle{\cal D}^{2}{\cal
D}_{B}f_{A}+2h_{AB}^{(0)}{\cal D}_{C}f^{C}-2({\cal D}_{A}f_{B}+{\cal
D}_{B}f_{A}).$

Substituting this into Eq. (93) and using the symmetry of indices $A$ and $B$,
we have

$\displaystyle{\cal D}_{B}{\cal D}_{A}F=\frac{1}{2}\Biggl{[}-{\cal
D}^{2}({\cal D}_{A}f_{B}+{\cal D}_{B}f_{A})-4h_{AB}^{(0)}F+2({\cal
D}_{A}f_{B}+{\cal D}_{B}f_{A})-4{\cal D}_{A}{\cal
D}_{B}\partial_{u}f\Biggr{]}.$ (95)

Then, using Eq. (55) and $f_{,u}=-(1/3)F$, we obtain

$\displaystyle{\cal D}_{B}{\cal D}_{A}F=-\frac{1}{3}h_{AB}^{(0)}{\cal
D}^{2}F-\frac{4}{3}h_{AB}^{(0)}F+\frac{2}{3}{\cal D}_{A}{\cal D}_{B}F.$ (96)

The trace part implies

$\displaystyle{\cal D}^{2}F+3F=0.$ (97)

Using this, Eq. (96) becomes

$\displaystyle{\cal D}_{A}{\cal D}_{B}F=\frac{1}{3}{\cal D}^{2}Fh_{AB}^{(0)}.$
(98)

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