###### Abstract

A theorem related to the Newman-Penrose constants is proven. The theorem
states that all the Newman-Penrose constants of asymptotically flat,
stationary, asymptotically algebraically special electrovacuum spacetimes are
zero. Straightforward application of this theorem shows that all the Newman-
Penrose constants of the Kerr-Newman spacetime must vanish.

On Newman-Penrose constants of stationary electrovacuum spacetimes

Xiangdong Zhanga,111e-mail<EMAIL_ADDRESS>Xiaoning
Wub,222e-mail<EMAIL_ADDRESS>and Sijie Gao a,333e-mail<EMAIL_ADDRESS>
a. Department of Physics,

Beijing Normal University,

Beijing, China, 100080.

b. Institute of Applied Mathematics.

Academy of Mathematics and System Science.

Chinese Academy of Sciences,

P.O.Box 2734, Beijing, China, 100080.

PACS number : 04.20.-q, 04.20.Ha

Keywords : Newman-Penrose constants, stationary electrovacuum condition, Kerr-
Newman solution.

##  1 Introduction

Newmen-Penrose(N-P) constants are very interesting and useful quantities in
the study of asymptotic flat space-times. They were first found by E.T.Newman
and R.Penrose in 1968[1] and then discussed by many other authors[2, 3, 4, 5,
6, 7]. Although the N-P constants have been found for forty years, their
physical interpretation remains an open question. One reason is that the
computation of these constants for a general asymptotically flat spacetime is
not easy. In stationary vacuum cases, these constants can be viewed as
combination of multi-pole moments of space-times[8]-[11]. Calculations of the
NP constants for vacuum solutions have been made by many authors [12, 13, 14,
15, 16, 17]. People used to conjecture that the algebraically special
condition (ASC) leads to the vanishing of NP constants. However, Kinnersley
and Walker[12] provided a counterexample. Recently, some authors[15] proposed
the asymptotically algebraically special condition (AASC), and proved that the
N-P constants vanish for vacuum, stationary, asymptotic algebraically special
space-times. In fact, the two conditions are closely related. It is well known
that the ASC implies that the Weyl curvature possesses a multiple principle
null direction. This condition can be expressed in terms of two geometric
invariants $I$ and $J$, defined by
$I=\Psi_{0}\Psi_{4}-4\Psi_{1}\Psi_{3}+3(\Psi_{2})^{2}$ and
$J=\Psi_{4}\Psi_{2}\Psi_{0}+2\Psi_{3}\Psi_{2}\Psi_{1}-(\Psi_{2})^{3}-(\Psi_{3})^{2}\Psi_{0}-(\Psi_{1})^{2}\Psi_{4}$
[18, 21]. A spacetime is said to be algebraically special if
$I^{3}-27J^{2}=0$. It has been shown that a general asymptotically flat,
stationary spacetime satisfies $I^{3}-27J^{2}\sim O(r^{-21})$ near future null
infinity[15]. Thus, $I^{3}-27J^{2}$ will peel off very quickly for a general
stationary vacuum asymptotically flat spacetime although the spacetime may not
be algebraically special. A spacetime is said to be “asymptotically
algebraically special” if $I^{3}-27J^{2}\sim O(r^{-22})$ [15], i.e. one order
faster than general cases. From the geometric point of view, this indicates
that one pair of principle null directions coincides near null infinity. By
imposing this condition, authors of [15] showed that the NP constants vanish
for vacuum, stationary spacetimes. Based on the result of [15], N-P constants
can bee seen as combination of Janis-Newman multi-poles of gravitational
field[24]. An intriguing question is whether the Janis-Newman multi-poles of
matter field will contribute to the N-P constants. In this paper, we extend
the discussion to the electrovacuum case. By imposing the AACS, we show that
the NP constants still vanish in the presence of a stationary Maxwell field.
If the Maxwell field is not stationary, the multi-pole moments of the Maxwell
field will contribute to the N-P constants.

This paper is organized as follows : In section II, we apply the method of
Taylor expansion to a stationary electrovacuum space-time. With the help of
the Killing equation, we reduce the dynamical freedom of gravitational field
into a set of arbitrary constants. Detailed expressions are given up to order
$O(r^{-6})$. We then prove that all the N-P constants of a stationary
asymptotically algebraically special electrovacuum space-time are zero.
Finally, we make some concluding remarks in section III.

##  2 The Newman-Penrose constants of stationary asymptotically algebraically
special electrovacuum space-times

In an asymptotically flat spacetime, the Newman-Penrose constants are defined
by [18]

$\displaystyle G_{m}=\int_{S_{\infty}}{}_{2}Y_{2,m}\Psi^{1}_{0}dS,$

where ${}_{2}Y_{2,m}$ is a spin-weight harmonic function and $\Psi^{1}_{0}$ is
a component of the Weyl tensor. Since the integral is performed on a two-
sphere at infinity, we only need the asymptotic form of the Weyl tensor in the
calculation. According to the peeling off theorem given by Sachs [18], we may
express the Weyl tensor and Maxwell field as :

$\displaystyle\Psi_{n}\sim O(r^{n-5})\,\quad n=0,1,2,3,4,$
$\displaystyle\phi_{m}\sim O(r^{m-3})\quad m=0,1,2.$ (1)

The vacuum case has been studied previously[7, 13, 14, 15]. An interesting
issue is to consider the effect of matter fields on N-P constants. In this
paper, we shall concentrate on the electromagnetic field. Like in the vacuum
case, we require the space-time to be stationary. Obviously, there is no Bondi
energy flux in such a space-time, i.e. ${\dot{\sigma}}^{0}=0$. In this case we
can choose some suitable coordinates, such that the asymptotic shear
$\sigma^{0}$ is zero. Similarly, the stationary condition has eliminated the
freedom of the news function. We also demand the Weyl tensor satisfy the
asymptotically algebraically special condition, which has been discussed
above. The main purpose of this paper is to prove the following theorem:

###### Theorem 1

All the N-P constants of an asymptotically flat, stationary, asymptotically
algebraically special electrovacuum space-time are zero.

Note that Kerr-Newman solution satisfies all the conditions in the theorem. It
follows immediately that the all N-P constants in a Kerr-Newman spacetime must
vanish.

Proof of the theorem. We choose the standard Bondi-Sachs’ coordinates and
construct the standard Bondi null tetrad [15, 22]. With the gauge choice in
[18, 19], we can write down the N-P coefficients and null tetrad of the
stationary electrovacuum spacetime. Some low order terms have been calculated
and can be found in [18]. Calculation of the N-P constants requires higher
order terms in the expansions. Consider the following N-P equations

$\displaystyle\delta\lambda-{\bar{\delta}}\mu=\bar{\tau}\mu+(\bar{\alpha}-3\beta)\lambda-\Psi_{3}+\Phi_{21},$
(2)
$\displaystyle\Delta\lambda-{\bar{\delta}}\nu=2\alpha\nu+(\bar{\gamma}-3\gamma-\mu-\bar{\mu})\lambda-\Psi_{4},$
(3)

where $\Phi_{ij}=8\pi\phi_{i}{\bar{\phi}}_{j}$ is the Maxwell stress tensor.
The coefficient of $r^{-2}$ in equation (2) yields $\Psi_{3}^{0}=0$. Expanding
equation (3) up to $O(r^{-3})$, we obtain
$\Psi_{4}^{0}=\Psi_{4}^{1}=\Psi_{4}^{2}=0$.

Now we shall use the Killing equation to reduce other dynamical freedoms and
get a general asymptotic expansion of the stationary electrovacuum space-time.
Write down the time-like Killing vector as

$\displaystyle t^{a}=Tl^{a}+n^{a}+{\bar{A}}m^{a}+A{\bar{m}}^{a}.$

The Killing equations are given by

$\displaystyle-DT+(\gamma+{\bar{\gamma}})+{\bar{\tau}}A+\tau{\bar{A}}=0,$ (4)
$\displaystyle DA+\tau+{\bar{\rho}}A+{\sigma}{\bar{A}}=0,$ (5)
$\displaystyle-{D^{\prime}}T-(\gamma+{\bar{\gamma}})T-\nu
A-{\bar{\nu}}{\bar{A}}=0,$ (6) $\displaystyle-\tau
T+{\bar{\nu}}+{D^{\prime}}A+({\bar{\gamma}}-\gamma)A-\delta T-\tau T-\mu
A-{\bar{\lambda}}{\bar{A}}=0,$ (7)
$\displaystyle-{\sigma}T+{\bar{\lambda}}+\delta A+({\bar{\alpha}}-\beta)A=0,$
(8) $\displaystyle-\rho
T+\mu+\delta{\bar{A}}-({\bar{\alpha}}-\beta){\bar{A}}-{\bar{\rho}}T+{\bar{\mu}}+{\bar{\delta}}A-(\alpha-{\bar{\beta}})A=0.$
(9)

Similarly to the analysis in [15], assuming the asymptotic behaviors of $T$
and $A$ as

$\displaystyle T$ $\displaystyle=$ $\displaystyle
T^{0}+\frac{T^{1}}{r}+\cdots,$ $\displaystyle A$ $\displaystyle=$
$\displaystyle A^{0}+\frac{A^{1}}{r}+\cdots,$ (10)

we can solve the Killing equations order by order. The stationary condition
implies ${\dot{\sigma}}^{0}=0$. It has been found that the Maxwell field does
not change the lowest two powers of $1/r$ in the Killing equations. So the
constant terms in the Killing equations yield the same result as in the vacuum
case, i.e., $T^{0}=\frac{1}{2}$, ${\dot{T}}^{1}=0$, ${\dot{A}}^{1}=0$. The
coefficients of $r^{-1}$ in the Killing equations give rise to
${\sigma}^{0}=0$, $A^{1}=0$, ${\dot{T}}^{2}=0$,
$\Psi^{0}_{2}={\bar{\Psi}}^{0}_{2}$,
$T^{1}=\frac{1}{2}(\Psi^{0}_{2}+{\bar{\Psi}}^{0}_{2})$ ,
${\dot{A}}^{2}=-\frac{1}{2}\eth\Psi^{0}_{2}+\frac{1}{2}\delta_{0}(\Psi^{0}_{2}+{\bar{\Psi}}^{0}_{2})$
and

${\dot{\Psi}}^{0}_{2}=0.$ (11)

From the $r^{-2}$ terms in the N-P equation

$\displaystyle\delta\nu-\Delta\mu=\gamma\mu-2\nu\beta+\bar{\gamma}\mu+\mu^{2}+|\lambda|^{2}+\Phi_{22}\,,$
(12)

we find $8\pi|\phi_{2}^{0}|^{2}={\dot{\Psi}}_{2}^{0}=0$. Hence
$\phi_{2}^{0}=0$. From the $r^{-5}$ terms in the N-P equation

$\displaystyle\delta\rho-\bar{\delta}\sigma=\tau\rho+(\bar{\beta}-3\alpha)\sigma+(\rho-\bar{\rho})\tau-\Psi_{1}+\Phi_{01}\,,$
(13)

we have

$\displaystyle\frac{1}{6}({\bar{\eth}}\Psi_{0}^{0}-40\pi\phi_{0}^{0}{\bar{\phi}}_{1}^{0})+\frac{1}{2}{\bar{\eth}}\Psi_{0}^{0}=-\frac{1}{3}({\bar{\eth}}\Psi_{0}^{0}-40\pi\phi_{0}^{0}{\bar{\phi}}_{1}^{0})+{\bar{\eth}}\Psi_{0}^{0}-16\pi\phi_{0}^{0}{\bar{\phi}}_{1}^{0}$
(14)

which implies

$\displaystyle\phi_{0}^{0}{\bar{\phi}}_{1}^{0}=0$ (15)

This equation will play an important role in our proof, which gives
$\phi_{0}^{0}=0$ or $\phi_{1}^{0}=0$. Now we discuss the two cases
respectively.

1) $\phi_{0}^{0}=0$. Consider the Maxwell equations

$\displaystyle D\phi_{1}-\bar{\delta}\phi_{0}$ $\displaystyle=$
$\displaystyle-2\alpha\phi_{0}+2\rho\phi_{1},$ (16) $\displaystyle
D\phi_{2}-\bar{\delta}\phi_{1}$ $\displaystyle=$
$\displaystyle-\lambda\phi_{0}+\rho\phi_{2}.$ (17)

The coefficients of $r^{-4}$ in these equations yield
$\phi_{1}^{1}=\phi_{2}^{2}=0$.

Consider the other two Maxwell equations

$\displaystyle\delta\phi_{1}-\Delta\phi_{0}$ $\displaystyle=$
$\displaystyle(\mu-2\lambda)\phi_{0}+2\tau\phi_{1}-\sigma\phi_{2},$ (18)
$\displaystyle\delta\phi_{2}-\Delta\phi_{1}$ $\displaystyle=$
$\displaystyle-\nu\phi_{0}+2\mu\phi_{1}+(\tau-2\beta)\phi_{2}.$ (19)

The $r^{-2}$ terms in equation (18) and the $r^{-3}$ terms in equation (19)
yield

$\displaystyle\dot{\phi_{1}^{0}}$ $\displaystyle=$ $\displaystyle 0$ (20)
$\displaystyle\dot{\phi_{0}^{0}}$ $\displaystyle=$
$\displaystyle\eth\phi_{1}^{0}=0$ (21)

where “$\cdot$” denotes ${\frac{\partial}{\partial u}}$. Combining these two
equations, we have $\phi_{1}^{0}=constant$. So from the $r^{-3}$ terms of Eq.
(17), we obtain $\phi_{2}^{1}=-{\bar{\eth}}\phi_{1}^{0}=0$.

2) $\phi_{1}^{0}=0$. Again, from the $r^{-3}$ terms of Eq.(17), we have
$\phi_{2}^{1}=-{\bar{\eth}}\phi_{1}^{0}=0$.

Thus in both cases we have $\phi_{2}^{1}=0$. Note that it is $\Phi_{ij}$,
instead of $\phi_{i}$, that appear in the N-P equations. The fact that
$\phi_{i}=O(r^{-3})$ (except $\phi_{1}\sim O(r^{-2})$ in case 1)) shows that
the presence of the electromagnetic field does not contribute to $r^{-1}$ and
$r^{-2}$ terms. The electromagnetic field makes contribution only to order
$r^{-3}$ and higher orders in the expansions. Combining these results, we
obtain the reduced N-P coefficients

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle-\frac{1}{r}+\frac{8\pi\phi^{0}_{0}\bar{\phi}^{0}_{0}}{3r^{5}}+O(r^{-6}),$
$\displaystyle{\sigma}$ $\displaystyle=$
$\displaystyle-\frac{\Psi^{0}_{0}}{2r^{4}}-\frac{\Psi^{1}_{0}}{3r^{5}}+O(r^{-6}),$
$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{\alpha^{0}}{r}-\frac{{\bar{\alpha}}^{0}{\bar{\Psi}}^{0}_{0}}{6r^{4}}+\frac{\alpha^{0}8\pi\phi^{0}_{0}\bar{\phi}^{0}_{0}-\bar{\alpha}^{0}\bar{\Psi}^{1}_{0}-24\pi(\phi_{1}^{0}{\bar{\phi}}_{0}^{1}+\phi^{1}_{1}\bar{\phi}^{0}_{0})}{12r^{5}}+O(r^{-6}),$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle-\frac{{\bar{\alpha}}^{0}}{r}-\frac{\Psi^{0}_{1}}{2r^{3}}+\frac{\alpha^{0}\Psi^{0}_{0}+2{\bar{\eth}}\Psi^{0}_{0}}{6r^{4}}-\frac{3\Psi^{2}_{1}+8\pi\bar{\alpha}^{0}\phi^{0}_{0}\bar{\phi}^{0}_{0}-\alpha^{0}\Psi^{1}_{0}}{12r^{5}}+O(r^{-6})\,,$
$\displaystyle\tau$ $\displaystyle=$
$\displaystyle-\frac{\Psi^{0}_{1}}{2r^{3}}+\frac{{\bar{\eth}}\Psi^{0}_{0}}{3r^{4}}+\frac{{\bar{\eth}}\Psi^{1}_{0}-8\pi\eth(\phi^{0}_{0}\bar{\phi}^{0}_{0})-48\pi(\phi_{0}^{1}{\bar{\phi}}_{1}^{0}+\phi^{0}_{0}\bar{\phi}^{1}_{1})}{8r^{5}}+O(r^{-6}),$
$\displaystyle{\lambda}$ $\displaystyle=$
$\displaystyle-\frac{{\bar{\Psi}}^{0}_{0}}{12r^{4}}-\frac{3{\bar{\Psi}}^{0}_{0}\Psi^{0}_{2}+{\bar{\Psi}}^{1}_{0}+48\pi\phi^{2}_{2}\bar{\phi}^{0}_{0}}{24r^{5}}+O(r^{-6}),$
$\displaystyle\mu$ $\displaystyle=$
$\displaystyle-\frac{1}{2r}-\frac{\Psi^{0}_{2}}{r^{2}}+\frac{{\bar{\eth}}\Psi^{0}_{1}-16\pi\phi_{1}^{0}{\bar{\phi}}_{1}^{0}}{2r^{3}}-\frac{{\bar{\eth}}^{2}\Psi^{0}_{0}}{6r^{4}}-\frac{6\Psi^{3}_{2}+8\pi\phi^{0}_{0}\bar{\phi}^{0}_{0}}{24r^{5}}+O(r^{-6}),$
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle-\frac{\Psi^{0}_{2}}{2r^{2}}+\frac{2{\bar{\eth}}\Psi^{0}_{1}-48\pi\phi_{1}^{0}{\bar{\phi}}_{1}^{0}+\alpha^{0}\Psi^{0}_{1}-{\bar{\alpha}}^{0}{\bar{\Psi}}^{0}_{1}}{6r^{3}}$
$\displaystyle-\frac{1}{24}\left[2\left(\alpha^{0}{\bar{\eth}}\Psi^{0}_{0}-{\bar{\alpha}}^{0}\eth{\bar{\Psi}}^{0}_{0}\right)+3{\bar{\eth}}^{2}\Psi^{0}_{0}\right]r^{-4}$
$\displaystyle+\frac{1}{20}[\alpha^{0}8\pi(\phi^{0}_{0}\bar{\phi}^{1}_{1}+\phi_{0}^{1}{\bar{\phi}}_{1}^{0})+\alpha^{0}\Psi^{2}_{1}-{\bar{\alpha}}^{0}8\pi(\phi_{1}^{0}{\bar{\phi}}_{0}^{1}+\phi_{1}^{1}\bar{\phi}^{0}_{0})-{\bar{\alpha}}^{0}\bar{\Psi}^{2}_{1}$
$\displaystyle-|\Psi^{0}_{1}|^{2}-4\Psi^{3}_{2}-32\pi(\phi_{1}^{0}{\bar{\phi}}_{1}^{2}+\phi_{1}^{1}\bar{\phi}_{1}^{1}+\phi_{1}^{2}{\bar{\phi}}_{1}^{0})]r^{-5}+O(r^{-6}),$
$\displaystyle\nu$ $\displaystyle=$
$\displaystyle-\frac{1}{12}\left[{\bar{\Psi}}^{0}_{1}+2{\bar{\eth}}^{2}\Psi^{0}_{1}\right]r^{-3}+\frac{1}{24}\left[\eth\bar{\Psi}_{0}^{0}+{\bar{\eth}}^{3}\Psi_{0}^{0}\right]r^{-4}$
(22)
$\displaystyle-\frac{1}{120}[6\Psi_{2}^{1}\bar{\Psi}_{1}^{0}-8\Psi_{2}^{0}\eth\bar{\Psi}_{0}^{0}+24\pi(\phi_{1}^{0}\bar{\phi}_{0}^{1}+\phi_{1}^{1}\bar{\phi}_{0}^{0})+3\bar{\Psi}_{1}^{2}+24\Psi_{3}^{4}$
$\displaystyle+192\pi\phi_{2}^{2}\bar{\phi}_{1}^{1}]r^{-5}+O(r^{-6}).$

and the null tetrad

$\displaystyle l^{a}$ $\displaystyle=$ $\displaystyle{\frac{\partial}{\partial
r}}\,,$ $\displaystyle n^{a}$ $\displaystyle=$
$\displaystyle{\frac{\partial}{\partial
u}}+\left[-\frac{1}{2}-\frac{\Psi^{0}_{2}}{r}+\frac{{\bar{\eth}}\Psi^{0}_{1}+\eth{\bar{\Psi}}^{0}_{1}+64\pi\phi_{1}^{0}{\bar{\phi}}_{1}^{0}}{6r^{2}}\right.-\frac{{\bar{\eth}}^{2}\Psi^{0}_{0}+\eth^{2}{\bar{\Psi}}^{0}_{0}}{24r^{3}}$
$\displaystyle-\frac{1}{20}\left(3|\Psi_{1}^{0}|^{2}+\Psi_{2}^{3}+\bar{\Psi}_{2}^{3}+16\pi(\phi^{0}_{1}\bar{\phi}^{2}_{1}+\phi^{1}_{1}{\bar{\phi}}^{1}_{1}+\phi^{2}_{1}{\bar{\phi}}^{0}_{1})r^{-4}+O(r^{-5})\right]{\frac{\partial}{\partial
r}}$
$\displaystyle+\left[\frac{1+\zeta{\bar{\zeta}}}{6\sqrt{2}r^{3}}\Psi^{0}_{1}-\frac{1+\zeta{\bar{\zeta}}}{12\sqrt{2}r^{4}}{\bar{\eth}}\Psi^{0}_{0}+O(r^{-5})\right]{\frac{\partial}{\partial\zeta}}$
$\displaystyle+\left[\frac{1+\zeta{\bar{\zeta}}}{6\sqrt{2}r^{3}}{\bar{\Psi}}^{0}_{1}-\frac{1+\zeta{\bar{\zeta}}}{12\sqrt{2}r^{4}}\eth{\bar{\Psi}}^{0}_{0}+O(r^{-5})\right]{\frac{\partial}{\partial{\bar{\zeta}}}}\,,$
$\displaystyle m^{a}$ $\displaystyle=$
$\displaystyle\left[-\frac{\Psi^{0}_{1}}{2r^{2}}+\frac{{\bar{\eth}}\Psi^{0}_{0}}{6r^{3}}-\frac{\Psi_{1}^{2}+8\pi(\phi_{0}^{1}{\bar{\phi}}_{1}^{0}+\phi^{0}_{0}\bar{\phi}^{1}_{1})}{12r^{4}}+O(r^{-5})\right]{\frac{\partial}{\partial
r}}$ (23)
$\displaystyle+\left[\frac{1+\zeta{\bar{\zeta}}}{6\sqrt{2}r^{4}}\Psi^{0}_{0}+O(r^{-5})\right]{\frac{\partial}{\partial\zeta}}+\left[\frac{1+\zeta{\bar{\zeta}}}{\sqrt{2}r}+O(r^{-5})\right]{\frac{\partial}{\partial{\bar{\zeta}}}}\,,$

where
$\delta_{0}=\frac{(1+\zeta{\bar{\zeta}})}{\sqrt{2}}\frac{\partial}{\partial{\bar{\zeta}}}$,
$\zeta=e^{i\phi}\cot\frac{\theta}{2}$, $\eth
f=(\delta_{0}+2s{\bar{\alpha}}^{0})f$ ( $s$ is the spin-weight of $f$). The
differential operators $\eth$ and ${\bar{\eth}}$ are defined in [18, 20].

Then the components of the Weyl curvature and the electromagnetic tensor
reduce to

$\displaystyle\Psi_{0}=\frac{\Psi^{0}_{0}}{r^{5}}+\frac{\Psi^{1}_{0}}{r^{6}}+O(r^{-7}),$
$\displaystyle\Psi_{1}=\frac{\Psi^{0}_{1}}{r^{4}}+\frac{\Psi^{1}_{1}}{r^{5}}+\frac{\Psi^{2}_{1}}{r^{6}}+O(r^{-7}),$
$\displaystyle\Psi_{2}=\frac{\Psi^{0}_{2}}{r^{3}}+\frac{\Psi^{1}_{2}}{r^{4}}+\frac{\Psi^{2}_{2}}{r^{5}}+\frac{\Psi^{3}_{2}}{r^{6}}+O(r^{-7}),$
$\displaystyle\Psi_{3}=\frac{\Psi^{2}_{3}}{r^{4}}+\frac{\Psi^{3}_{3}}{r^{5}}+\frac{\Psi^{4}_{3}}{r^{6}}+O(r^{-7}),$
$\displaystyle\Psi_{4}=\frac{\Psi_{4}^{3}}{r^{4}}+\frac{\Psi^{4}_{4}}{r^{5}}+\frac{\Psi^{5}_{4}}{r^{6}}+O(r^{-7}).$
$\displaystyle\phi_{0}=\frac{\phi_{0}^{0}}{r^{3}}+\frac{\phi_{0}^{1}}{r^{4}}+\frac{\phi_{0}^{2}}{r^{5}}+O(r^{-6}),$
$\displaystyle\phi_{1}=\frac{\phi_{1}^{0}}{r^{2}}+\frac{\phi_{1}^{1}}{r^{3}}+\frac{\phi_{1}^{2}}{r^{4}}+\frac{\phi_{1}^{3}}{r^{5}}+O(r^{-6}),$
$\displaystyle\phi_{2}=\frac{\phi_{2}^{2}}{r^{3}}+\frac{\phi_{2}^{3}}{r^{4}}+\frac{\phi_{2}^{4}}{r^{5}}+O(r^{-6}).$
(24)

The Bianchi identity takes the form

$\displaystyle\bar{\delta}\Psi_{0}-D\Psi_{1}+D\Phi_{01}-\delta\Phi_{00}=4\alpha\Psi_{0}-4\rho\Psi_{1}-2\tau\Phi_{00}+2\rho\Phi_{01}+2\sigma\Phi_{10}\,.$
(25)

The coefficient of $r^{-6}$ in equation (25) yields
$\Psi^{1}_{1}=-{\bar{\eth}}\Psi^{0}_{0}$.

Similarly, the other components of the Bianchi identity and the Maxwell
equations lead to

$\displaystyle\phi_{1}^{1}=-{\bar{\eth}}\phi_{0}^{0},\quad\phi_{1}^{2}=-\frac{1}{2}{\bar{\eth}}\phi_{0}^{1},\quad\phi_{1}^{3}=-\frac{1}{3}{\bar{\eth}}\phi_{0}^{2}-\frac{1}{2}\bar{\Psi}_{1}^{0}\phi_{0}^{0}.$
$\displaystyle\phi_{2}^{2}=\frac{1}{2}{\bar{\eth}}^{2}\phi_{0}^{0},\quad\phi_{2}^{3}=\frac{1}{6}{\bar{\eth}}^{2}\phi_{0}^{1},$
$\displaystyle\phi_{2}^{4}=\frac{1}{12}{\bar{\eth}}^{2}\phi_{0}^{2}+\frac{1}{12}\eth\bar{\Psi}_{0}^{0}+\frac{1}{2}\bar{\Psi}_{1}^{0}{\bar{\eth}}\phi_{0}^{0}$
$\displaystyle\Psi^{1}_{1}=-{\bar{\eth}}\Psi^{0}_{0},\quad\Psi^{2}_{1}=-\frac{1}{2}{\bar{\eth}}\Psi^{1}_{0}+16\pi(\phi_{0}^{0}\bar{\phi}_{1}^{1}+\phi_{0}^{1}{\bar{\phi}}_{1}^{0})+4\pi\eth(\phi_{0}^{0}\bar{\phi}_{0}^{0}),$
$\displaystyle\Psi^{1}_{2}=-{\bar{\eth}}\Psi^{0}_{1}+16\pi\phi_{1}^{0}{\bar{\phi}}_{1}^{0},\quad\Psi^{2}_{2}=\frac{1}{2}{\bar{\eth}}^{2}\Psi^{0}_{0},$
$\displaystyle\Psi^{3}_{2}=-\frac{2}{3}|\Psi^{0}_{1}|^{2}-\frac{1}{3}{\bar{\eth}}\Psi^{2}_{1}+\frac{16}{9}\pi\eth(\phi_{1}^{0}{\bar{\phi}}_{0}^{1}+\phi_{1}^{1}\bar{\phi}_{0}^{0})-\frac{8}{9}\pi{\bar{\eth}}(\phi_{0}^{0}\bar{\phi}_{1}^{1}+\phi_{0}^{1}{\bar{\phi}}_{1}^{0})-\frac{20}{9}\pi\phi_{0}^{0}\bar{\phi}_{0}^{0}$
$\displaystyle\quad\quad\quad+\frac{80}{9}\pi(\phi_{1}^{0}{\bar{\phi}}_{1}^{2}+\phi_{1}^{1}{\bar{\phi}}_{1}^{1}+\phi_{1}^{2}{\bar{\phi}}_{1}^{0})+\frac{8}{9}\pi{\frac{\partial}{\partial
u}}(\phi_{0}^{0}{\bar{\phi}}_{0}^{1}+\phi_{0}^{1}{\bar{\phi}}_{0}^{0}),$
$\displaystyle\Psi^{2}_{3}=\frac{1}{2}{\bar{\eth}}^{2}\Psi^{0}_{1},\quad\Psi^{3}_{3}=-\frac{1}{2}{\bar{\Psi}}^{0}_{1}\Psi^{0}_{2}-\frac{1}{6}{\bar{\eth}}^{3}\Psi^{0}_{0},$
$\displaystyle\Psi^{4}_{3}=-\frac{1}{4}{\bar{\eth}}\Psi_{2}^{3}+\frac{1}{8}\Psi_{2}^{0}\eth\Psi_{0}^{0}+\frac{1}{2}{\bar{\Psi}}_{1}^{0}{\bar{\eth}}\Psi_{1}^{0}+\frac{1}{12}k\eth(\phi_{2}^{2}{\bar{\phi}}_{0}^{0})$
$\displaystyle\quad\quad\quad-\frac{4}{3}\pi{\bar{\eth}}(\phi_{1}^{0}{\bar{\phi}}_{1}^{2}+\phi_{1}^{1}{\bar{\phi}}_{1}^{1}\phi_{1}^{2}{\bar{\phi}}_{1}^{0})+4\pi(\phi_{2}^{2}{\bar{\phi}}_{1}^{1}+\phi_{2}^{3}{\bar{\phi}}_{1}^{0})+4\pi(\phi_{1}^{0}{\bar{\phi}}_{0}^{1}+\phi_{1}^{1}{\bar{\phi}}_{0}^{0})$
$\displaystyle\quad\quad\quad+4\pi\bar{\Psi}_{1}^{0}\phi_{1}^{0}{\bar{\phi}}_{1}^{0}+\frac{4}{3}\pi{\frac{\partial}{\partial
u}}(\phi_{1}^{1}{\bar{\phi}}_{0}^{1}+\phi_{1}^{2}{\bar{\phi}}_{0}^{0}),$
$\displaystyle\Psi^{3}_{4}=-\frac{1}{6}{\bar{\eth}}^{3}\Psi_{1}^{0},\quad\Psi^{4}_{4}=-\frac{1}{24}{\bar{\eth}}^{4}\Psi_{0}^{0},$
$\displaystyle\Psi^{5}_{4}=-\frac{1}{5}{\bar{\eth}}\Psi_{3}^{4}-\frac{8}{5}\pi{\bar{\eth}}(\phi_{2}^{2}{\bar{\phi}}_{1}^{1}+\phi_{2}^{3}{\bar{\phi}}_{1}^{0})-\frac{1}{5}{\bar{\Psi}}_{1}^{0}{\bar{\eth}}^{2}\Psi_{1}^{0}-\frac{1}{20}\Psi_{2}^{0}\bar{\Psi}_{0}^{0}$
$\displaystyle\quad\quad\quad+4\pi(\phi_{2}^{2}{\bar{\phi}}_{0}^{0})+\frac{8}{5}\pi{\frac{\partial}{\partial
u}}(\phi_{2}^{2}{\bar{\phi}}_{0}^{1}+\phi_{2}^{3}{\bar{\phi}}_{0}^{0}).$ (26)

Similarly to the treatment in [15], the $r^{-3}$ terms in the Killing
equations lead to $\eth\Psi_{1}^{0}=0$. Thus we have

$\displaystyle\Psi^{0}_{1}$ $\displaystyle=$
$\displaystyle\sum^{1}_{m=-1}B_{m}{\ }_{1}Y_{1,m},$
$\displaystyle\Psi^{0}_{2}$ $\displaystyle=$ $\displaystyle C.$ (27)

The coefficient of $r^{-3}$ in Eq. (2) gives
$\Psi_{3}^{1}={\bar{\delta}}\Psi_{2}^{0}=0$.

In order to find more restrictions on $\Psi_{0}$, we need to compute higher
order terms of the Killing equations. The terms of order $r^{-4}$ of the
Killing equations yield

$\displaystyle 3T^{3}+(\gamma^{4}+{\bar{\gamma}}^{4})=0,$ (28) $\displaystyle
4A^{3}=\frac{1}{3}{\bar{\eth}}\Psi^{0}_{0},$ (29)
$\displaystyle{\dot{T}}^{4}+\frac{8}{3}\pi\phi_{1}^{0}{\bar{\phi}}_{1}^{0}=0,$
(30)
$\displaystyle\frac{1}{2}\Psi^{0}_{1}T^{1}-\tau^{4}+{\bar{\nu}}^{4}+{\dot{A}}^{4}+(\Psi^{0}_{2}+{\bar{\Psi}}^{0}_{2})A^{2}+2A^{3}-\delta_{0}T^{3}+\Psi^{0}_{2}A^{2}=0,$
(31) $\displaystyle\frac{1}{6}\Psi^{0}_{0}+\eth A^{3}=0,$ (32) $\displaystyle
2T^{3}+\mu^{4}+{\bar{\mu}}^{4}+\eth{\bar{A}}^{3}+{\bar{\eth}}A^{3}=0\,.$ (33)

Eq.(29) and (32) imply

$\displaystyle\Psi^{0}_{0}=\sum^{2}_{m=-2}A_{m}(u){\ }_{2}Y_{2,m},$ (34)

Eq.(27) and ${\dot{T}}^{3}=0$ ( which comes from the $r^{-3}$ terms in the
Killing equations) imply that $\Psi^{0}_{0}$ is independent of $u$.

Combining Eqs.(24),(26),(27) and (34), one finds

$\displaystyle I^{3}-27J^{2}\sim O(r^{-21}).$ (35)

This result holds for a general asymptotically flat stationary spacetime. As
mentioned in the introduction, the AASC requires

$\displaystyle I^{3}-27J^{2}\sim O(r^{-22}),$ (36)

which is just one order faster than the falloff rate of a general
asymptotically flat spacetime. This means that the AASC is a weak requirement
and as demonstrated at the end of this section, there exist many asymptotic
flat space-times which satisfy this condition.

Our purpose is to calculate the Newman-Penrose constants, which are contained
in the coefficients of $\Psi^{1}_{0}$. From the $r^{-5}$ terms in the Killing
equations, we have

$\displaystyle
4T^{4}+(\gamma^{5}+{\bar{\gamma}}^{5})-\frac{1}{2}{\bar{\Psi}}^{0}_{1}A^{2}-\frac{1}{2}\Psi^{0}_{1}{\bar{A}}^{2}=0,$
(37) $\displaystyle
A^{4}=\frac{1}{5}\tau^{5}=\frac{1}{40}\left[{\bar{\eth}}\Psi_{0}^{1}-48\pi(\phi_{0}^{0}{\bar{\phi}}_{1}^{1}+\phi_{0}^{1}{\bar{\phi}}_{1}^{0})-8\pi\eth(\phi_{0}^{0}{\bar{\phi}}_{0}^{0})\right],$
(38)
$\displaystyle\frac{1}{8}\Psi_{0}^{1}+\frac{3}{8}\Psi_{0}^{0}\Psi_{2}^{0}-2\pi\phi_{0}^{0}{\bar{\phi}}_{2}^{2}-\frac{1}{4}(\Psi_{1}^{0})^{2}+\eth
A^{4}=0,$ (39)
$\displaystyle-\rho^{5}+2T^{4}+(\mu^{5}+{\bar{\mu}}^{5})+\frac{3}{2}\Psi^{0}_{1}{\bar{A}}^{2}+\frac{3}{2}{\bar{\Psi}}^{0}_{1}A^{2}+\eth{\bar{A}}^{4}+{\bar{\eth}}A^{4}=0\,.$
(40)

Eqs. (38) and (39) yield:

$\displaystyle\eth{\bar{\eth}}\Psi^{1}_{0}+5\Psi^{1}_{0}=10(\Psi^{0}_{1})^{2}-15\Psi^{0}_{0}\Psi^{0}_{2}+80\pi\phi_{0}^{0}{\bar{\phi}}_{2}^{2}+48\pi\eth(\phi_{0}^{0}{\bar{\phi}}_{1}^{1}+\phi_{0}^{1}{\bar{\phi}}_{1}^{0})+8\pi\eth^{2}(\phi_{0}^{0}{\bar{\phi}}_{0}^{0}).$
(41)

The terms of $\phi^{i}_{j}$ on the right-hand side of Eq. (41) are the
contribution from the Maxwell field [15]. To simplify this equation, we need
to investigate the electromagnetic field in more detail.

Since the electromagnetic field is stationary, we have $\pounds_{t}F_{ab}=0$,
where $t^{c}$ is the Killing vector. Noting that $\phi_{0}=F_{lm}$ and using
the expansion of $t^{c}$, we have

$\displaystyle\pounds_{t}\phi_{0}$ $\displaystyle=$
$\displaystyle\pounds_{t}F_{ab}l^{a}m^{b}$ (42) $\displaystyle=$
$\displaystyle(Tl^{c}+n^{c}+{\bar{A}}m^{c}+A{\bar{m}}^{c})\phi_{0}$
$\displaystyle=$ $\displaystyle F_{ab}l^{a}[t,\quad m]^{b}+F_{ab}m^{b}[t,\quad
l]^{a}$ $\displaystyle=$
$\displaystyle(\gamma+\bar{\gamma}+\bar{A}\bar{\tau}+A\tau)\phi_{0}-(\tau+\bar{A}\sigma+A\rho)(\phi_{1}-{\bar{\phi}}_{1})$
$\displaystyle+\left[T\bar{\varrho}-\mu+\gamma+\bar{\gamma}-A(\bar{\beta}-\alpha)\right]\phi_{0}+\left[T\sigma-\bar{\lambda}-A(\bar{\alpha}-\beta)\right]{\bar{\phi}}_{0}$

where $[t,\ m]^{b}$ denotes the commutator of $t^{c}$ and $m^{b}$. So we
obtain

$\displaystyle(Tl^{c}+n^{c}+{\bar{A}}m^{c}+A{\bar{m}}^{c})\phi_{0}$ (43)
$\displaystyle=$
$\displaystyle(\gamma+\bar{\gamma}+\bar{A}\bar{\tau}+A\tau)\phi_{0}-(\tau+\bar{A}\sigma+A\rho)(\phi_{1}-{\bar{\phi}}_{1})+\left[T\bar{\varrho}-\mu+\gamma+\bar{\gamma}-A(\bar{\beta}-\alpha)\right]\phi_{0}$
$\displaystyle+\left[T\sigma-\bar{\lambda}-A(\bar{\alpha}-\beta)\right]{\bar{\phi}}_{0}$

Substituting (23) into (43) yields:

$\displaystyle{\frac{\partial}{\partial
u}}\phi_{0}+\left[-\frac{1}{2}-\frac{\Psi^{0}_{2}}{r}+O(r^{-2})\right]{\frac{\partial}{\partial
r}}\phi_{0}+\left[\frac{1+\zeta{\bar{\zeta}}}{6\sqrt{2}r^{3}}\Psi^{0}_{1}+O(r^{-4})\right]{\frac{\partial}{\partial\zeta}}\phi_{0}$
(44)
$\displaystyle+\left[\frac{1+\zeta{\bar{\zeta}}}{6\sqrt{2}r^{3}}{\bar{\Psi}}^{0}_{1}+O(r^{-4})\right]{\frac{\partial}{\partial{\bar{\zeta}}}}\phi_{0}+T{\frac{\partial}{\partial
r}}\phi_{0}-\bar{A}\left[\frac{\Psi^{0}_{1}}{2r^{2}}+O(r^{-3})\right]{\frac{\partial}{\partial
r}}\phi_{0}$
$\displaystyle+\bar{A}\left[\frac{1+\zeta{\bar{\zeta}}}{6\sqrt{2}r^{4}}\Psi^{0}_{0}+O(r^{-5})\right]{\frac{\partial}{\partial\zeta}}\phi_{0}+\bar{A}\left[\frac{1+\zeta{\bar{\zeta}}}{\sqrt{2}r}+O(r^{-5})\right]{\frac{\partial}{\partial{\bar{\zeta}}}}\phi_{0}$
$\displaystyle-A\left[\frac{{\bar{\Psi}}^{0}_{1}}{2r^{2}}+O(r^{-3})\right]{\frac{\partial}{\partial
r}}\phi_{0}$
$\displaystyle+A\left[\frac{1+\zeta{\bar{\zeta}}}{6\sqrt{2}r^{4}}{\bar{\Psi}}^{0}_{0}+O(r^{-5})\right]{\frac{\partial}{\partial{\bar{\zeta}}}}\phi_{0}+A\left[\frac{1+\zeta{\bar{\zeta}}}{\sqrt{2}r}+O(r^{-5})\right]{\frac{\partial}{\partial\zeta}}\phi_{0}$
$\displaystyle=$
$\displaystyle(\gamma+\bar{\gamma}+\bar{A}\bar{\tau}+A\tau)\phi_{0}-(\tau+\bar{A}\sigma+A\rho)(\phi_{1}-{\bar{\phi}}_{1})$
$\displaystyle+\left[T\bar{\varrho}-\mu+\gamma+\bar{\gamma}-A(\bar{\beta}-\alpha)\right]\phi_{0}+\left[T\sigma-\bar{\lambda}-A(\bar{\alpha}-\beta)\right]{\bar{\phi}}_{0}$

Again, we compute the $\phi^{i}_{j}$ terms in Eq.(41) in the two cases.

For case 1) $\phi_{0}^{0}=0$, computing the coefficient of $r^{-5}$ in Eq.
(44), we obtain

$\displaystyle\dot{\phi_{0}^{2}}=-3\Psi_{2}^{0}\phi_{0}^{0}=0$ (45)

The coefficient of $r^{-5}$ of equation (18) gives

$\displaystyle\eth\phi_{1}^{1}-\dot{\phi_{0}^{2}}-2\phi_{0}^{1}-3\Psi_{2}^{0}\phi_{0}^{0}=-\frac{1}{2}\phi_{0}^{1}-\Psi_{1}^{0}\phi_{1}^{0}\,.$
(46)

Using $\phi_{0}^{0}=0$ and $\dot{\phi_{0}^{2}}=0$, we get

$\displaystyle\phi_{0}^{1}=\frac{2}{3}\phi_{1}^{0}\Psi_{1}^{0}\,.$ (47)

By taking $\eth$ on both sides and using $\eth\Psi_{1}^{0}=0$, we have
immediately

$\displaystyle\eth\phi_{0}^{1}=\frac{2}{3}\phi_{1}^{0}\eth\Psi_{1}^{0}=0.$
(48)

Then the $\phi^{i}_{j}$ terms in Eq.(41) become

$\displaystyle
80\pi\phi_{0}^{0}{\bar{\phi}}_{2}^{2}+48\pi\eth(\phi_{0}^{0}{\bar{\phi}}_{1}^{1}+\phi_{0}^{1}{\bar{\phi}}_{1}^{0})+8\pi\eth^{2}(\phi_{0}^{0}{\bar{\phi}}_{0}^{0})$
(49) $\displaystyle=$ $\displaystyle
48\pi\eth(\phi_{0}^{1}{\bar{\phi}}_{1}^{0})$ $\displaystyle=$
$\displaystyle(48\pi\eth\phi_{0}^{1}){\bar{\phi}}_{1}^{0}+\phi_{0}^{1}(48\pi\eth{\bar{\phi}}_{1}^{0})$
$\displaystyle=$ $\displaystyle 0\,,$

where Eqs. (21) and (48) have been used in the last step.

For case 2) $\phi_{1}^{0}=0$, the coefficient of $r^{-4}$ in Eq. (44) leads to

$\displaystyle
0=\dot{\phi_{0}^{1}}-3T^{0}\phi_{0}^{0}+\frac{3}{2}\phi_{0}^{0}=\dot{\phi_{0}^{1}}\,.$
(50)

Because the spinweight of $\phi_{0}$ is 1, we can expand $\phi_{0}^{0}$ as
$\phi_{0}^{0}=\sum_{l=1}^{\infty}\sum_{m=-l}^{l}d_{l,m}{\ }_{1}Y_{l,m}$, where
$d_{l,m}$ are some constants. The $r^{-4}$ terms in (18) yield

$\displaystyle\dot{\phi_{0}^{1}}=-{\bar{\eth}}\eth\phi_{0}^{0}=\frac{1}{2}\sum_{l=1}^{\infty}(l+2)(l-1)\sum_{m=-l}^{l}d_{l,m}\
{}_{1}Y_{l,m}\,.$ (51)

Combining (50) and (51) and using the fact that spin-weight harmonic function
components are linearly independent, we obtain $l=1$. Consequently,

$\displaystyle\phi_{0}^{0}=\sum_{m=-1}^{1}d_{m}{\ }_{1}Y_{1,m}\,,$ (52)

where $d_{m}$ are constants. By expanding $\phi_{0}^{0}$, we find
$\eth\phi_{0}^{0}=0$. The contribution from the Maxwell field in Eq. (41) then
leads to:

$\displaystyle
80\pi\phi_{0}^{0}{\bar{\phi}}_{2}^{2}+48\pi\eth(\phi_{0}^{0}{\bar{\phi}}_{1}^{1})+8\pi\eth^{2}(\phi_{0}^{0}{\bar{\phi}}_{0}^{0})$
(53) $\displaystyle=$ $\displaystyle
40\pi\phi_{0}^{0}\eth^{2}{\bar{\phi}}_{0}^{0}-48\pi\eth(\phi_{0}^{0}\eth{\bar{\phi}}_{0}^{0})+8\pi\eth(\phi_{0}^{0}\eth{\bar{\phi}}_{0}^{0}+{\bar{\phi}}_{0}^{0}\eth\phi_{0}^{0})$
$\displaystyle=$ $\displaystyle
40\pi\phi_{0}^{0}\eth^{2}{\bar{\phi}}_{0}^{0}-48\pi\eth\phi_{0}^{0}\eth{\bar{\phi}}_{0}^{0}-48\pi\phi_{0}^{0}\eth^{2}{\bar{\phi}}_{0}^{0}+8\pi\eth\phi_{0}^{0}\eth{\bar{\phi}}_{0}^{0}$
$\displaystyle+8\pi\eth{\bar{\phi}}_{0}^{0}\eth\phi_{0}^{0}+8\pi\phi_{0}^{0}\eth^{2}{\bar{\phi}}_{0}^{0}+8\pi{\bar{\phi}}_{0}^{0}\eth^{2}\phi_{0}^{0}$
$\displaystyle=$
$\displaystyle-32\pi\eth\phi_{0}^{0}\eth{\bar{\phi}}_{0}^{0}+8\pi{\bar{\phi}}_{0}^{0}\eth^{2}\phi_{0}^{0}$
$\displaystyle=$ $\displaystyle 0$

where we have used $\phi_{1}^{1}=-{\bar{\eth}}\phi_{0}^{0}$ and
$\phi_{2}^{2}=\frac{1}{2}{\bar{\eth}}^{2}\phi_{0}^{0}$. Therefore, the
electromagnetic field makes no contribution to the equation of $\Psi_{0}^{1}$.
So in either case, the equation of $\Psi_{0}^{1}$ reduces to

$\displaystyle\eth{\bar{\eth}}\Psi^{1}_{0}+5\Psi^{1}_{0}=10(\Psi^{0}_{1})^{2}-15\Psi^{0}_{0}\Psi^{0}_{2}\,,$
(54)

which is exactly the same equation as that in the vacuum case. Then by
imposing the AASC, it is shown in [15] that Eq. (54) implies that all the
Newman-Penrose constants must be zero. This completes the proof of our
theorem.

Remark : The asymptotically algebraically special condition has played an
important role in the proof of this paper and in [15]. Obviously, this
condition is satisfied by the Kerr-Newman solution. The following arguments
show that the AASC is a rather weak condition imposed on a general
asymptotically flat spacetime. Note that the Kerr-Newman spacetime is
axisymmetric. Such symmetry is not required in our theorem. From Eq.(27), we
can see that $\Psi^{0}_{1}$ contains ${}_{1}Y_{1,1}$ and ${}_{1}Y_{1,-1}$
components that do not appear in the Kerr-Newman solution. Simple calculation
shows that $span\\{{}_{1}Y_{1,1},{}_{1}Y_{1,0},{}_{1}Y_{1,-1}\\}$ is not a
representative space of $SO(3)$. Thus we cannot cancel such components by a
rotation. Based on the characteristic initial value method[23], it is not
difficult to construct exact solutions with non-zero $B_{1}$ and $B_{-1}$.
Furthermore, the spin-weight components of $\Psi^{k}_{0}$ are just the Janis-
Newman multi-poles of gravitational field[24]. The AASC only gives a
restriction between Janis-Newman’s dipoles and quadrupoles[15]. Since there is
no restriction on higher order multi-poles, it is easy to see that there are
many solutions which satisfy the conditions of our theorem and are not
equivalent to the Kerr-Newman solution.

##  3 Concluding remarks

We have proven that all the N-P constants of an asymptotic flat, stationary,
asymptotically algebraically special electrovacuum space-time are zero. The
Kerr-Newman solution manifestly satisfies all the conditions. So our theorem
implies that all the N-P constants of the Kerr-Newman solution are zero. This
result has been obtained resently[25] by other authors. In the proof of the
theorem, we have assumed that the Maxwell field is stationary. If this
condition is not imposed, ${\dot{\phi}}^{1}_{0}$ will not be zero. Then
Eq.(51) tells us $\phi^{0}_{0}$ will contain other components of the spin-
weight spherical functions. These terms correspond to the Janis-Newman multi-
pole of Maxwell field[24]. In the presence of these terms, the N-P constants
may not vanish. Last but not least, an interesting issue is to single out the
Kerr-Newman solution from solutions which satisfy the conditions of our
theorem. From the discussion of the last section, we find that the AASC is not
enough to uniquely determine the Kerr-Newman solution. It seems that more
restrictions on the Maxwell field are needed. This will be discussed in our
future work.

## Acknowledgement

This work is supported by the Natural Science Foundation of China (NSFC) under
Grant Nos. 10705048, 10605006, 10731080. Authors would like to thank the
referees for helpful comments on the asymptotically algebraically special
condition.

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