# Multipole moments of a charged rotating disc of dust in general relativity

David Rumler<EMAIL_ADDRESS>Theoretisch-Physikalisches Institut,
Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, D-07743 Jena, Germany
Reinhard Meinel  Theoretisch-Physikalisches Institut, Friedrich-Schiller-
Universität Jena, Max-Wien-Platz 1, D-07743 Jena, Germany

###### Abstract

The gravitational and electromagnetic multipole moments of the charged
rotating disc of dust, which is an axisymmetric, stationary solution of the
Einstein-Maxwell equations in terms of a post-Newtonian expansion, are
calculated and discussed. It turns out that the individual mass, angular
momentum, electric and magnetic moments are ordered in the sense that higher
moments have a lower absolute value. There is an interesting conjecture
stating that the absolute values of all higher multipole moments of a
uniformly rotating perfect fluid body are always greater than those of the
corresponding Kerr spacetime, which we generalize to include charged bodies.
We find that for the charged rotating disc of dust the conjecture holds
(within the limits of accuracy of the post-Newtonian expansion).

##  I Introduction

The exterior spacetime of an astrophysical body can be characterized by its
multipole moments. In 1970, Geroch introduced multipole moments in a
coordinate independent way for static, asymptotically flat, vacuum spacetimes
[1] and Hansen later generalized them to stationary (asymptotically flat,
vacuum) spacetimes [2]. In case of axisymmetry, Hansen further showed that the
multipole moment tensors can be represented by scalars $P_{n}$ [2]. Relying on
an asymptotic expansion of the Ernst potential on the symmetry axis, Fodor,
Hoenselaers and Perjés presented, in 1989, an explicit algorithm to compute
those scalar multipole moments [3].

A generalization to stationary, electro-vacuum spacetimes with gravitational
and electromagnetic multipole moments, $P_{n}$ and $Q_{n}$, respectively, was
published by Simon [4] and a corresponding calculation scheme, in the
axisymmetric case, was given by Hoenselaers and Perjés [5]. As it turned out,
[5] contained two non-trivial mistakes, however. The first one was corrected
by Sotiriou and Apostolatos [6] and the second one recently, in 2021, by Fodor
et al. [7].

In the present paper we calculate and discuss the gravitational and
electromagnetic multipole moments of the charged rotating disc of dust (with
constant angular velocity and constant specific charge) using the latest
calculation procedure by Fodor et al. [7]. The charged rotating disc of dust
is an axisymmetric, stationary and physically reasonable solution of the
Einstein-Maxwell equations expressed in terms of a post-Newtonian expansion up
to tenth order [8, 9, 10]. An exact solution to this problem is not available
so far; in particular it is not contained in the solution classes discussed in
[11]. A study of the multipole moments of the exact solution of the uncharged
disc of dust [12, 13, 14] was published by Kleinwächter, Meinel and Neugebauer
[15].

We additionally formulate a generalized version of the multipole conjecture by
Filter and Kleinwächter [16] and test it using the multipole moments of the
charged rotating disc of dust. The conjecture states that the absolute values
of all higher multipole moments of an axisymmetric, stationary, physically
well-defined body are always greater than those of the corresponding multipole
moments of the Kerr-Newman spacetime with the same mass, angular momentum and
charge. We show that the generalized multipole conjecture holds for the
charged rotating disc of dust.

##  II Charged rotating disc of dust

In general relativity it is rather easy to construct purely mathematical
solutions without any physical meaning, however, the charged rotating disc of
dust is also physically relevant. We model the disc by an infinitesimally thin
equilibrium configuration of dust with constant specific charge
$\epsilon\in\left[-1,1\right]$ (electric charge density over baryonic mass
density) that is rigidly rotating around the axis of symmetry with constant
angular velocity $\Omega$. Dust is a perfect fluid with vanishing pressure.
The disc is axisymmetric, stationary and it obeys reflection symmetry [8, 9,
17].

Using axisymmetry and stationarity the corresponding metric can be expressed
globally in terms of Weyl-Lewis-Papapetrou coordinates111We use units in which
$c=G=4\pi\epsilon_{0}=1$.:

$\mathrm{d}s^{2}=f^{-1}\left[h\left(\mathrm{d}\rho^{2}+\mathrm{d}\zeta^{2}\right)+\rho^{2}\mathrm{d}\varphi^{2}\right]-f\left(\mathrm{d}t+a\,\mathrm{d}\varphi\right)^{2}\,.$
(1)

The electromagnetic four-potential $A_{a}$ takes the form:

$A_{a}=(0,0,A_{\varphi},A_{t})\,.$ (2)

In axistationary spacetimes the coupled Einstein-Maxwell equations in electro-
vacuum can be reduced to the Ernst equations [18]:

$\displaystyle\left(\Re\mathcal{E}+\lvert\Phi\rvert^{2}\right)\Delta\mathcal{E}$
$\displaystyle=\left(\nabla\mathcal{E}+2\bar{\Phi}\nabla\Phi\right)\cdot\nabla\mathcal{E}\,,$
(3) $\displaystyle\left(\Re\mathcal{E}+\lvert\Phi\rvert^{2}\right)\Delta\Phi$
$\displaystyle=\left(\nabla\mathcal{E}+2\bar{\Phi}\nabla\Phi\right)\cdot\nabla\Phi\,,$
(4)

where

$\mathcal{E}=\left(f-\lvert\Phi\rvert^{2}\right)+\mathrm{i}b\quad\text{and}\quad\Phi=\alpha+\mathrm{i}\beta$
(5)

are the Ernst potentials. Here $\alpha\coloneqq\Re\Phi=-A_{t}$ and the
potentials $\beta$ and $b$ are defined by

$\beta_{,\rho}=\frac{f}{\rho}\left(A_{\varphi,\zeta}-aA_{t,\zeta}\right)\,,\quad\beta_{,\zeta}=-\frac{f}{\rho}\left(A_{\varphi,\rho}-aA_{t,\rho}\right)$
(6)

and

$\displaystyle b_{,\rho}$
$\displaystyle=-\frac{f^{2}}{\rho}a_{,\zeta}-2\left(\beta
A_{t,\rho}-A_{t}\beta_{,\rho}\right)\,,$ $\displaystyle b_{,\zeta}$
$\displaystyle=\frac{f^{2}}{\rho}a_{,\rho}-2\left(\beta
A_{t,\zeta}-A_{t}\beta_{,\zeta}\right)\,.$ (7)

With the help of reflection symmetry one can formulate a well-defined boundary
value problem to the Ernst equations for the charged rotating disc of dust
which was solved in terms of a post-Newtonian expansion up to eighth order by
Palenta and Meinel [8] and up to tenth by Breithaupt et al. [9]:

$\displaystyle f=1+\sum_{k=1}^{10}f_{2k}g^{2k}\,,\quad
b=\sum_{k=1}^{10}b_{2k+1}g^{2k+1}\,,$ (8)
$\displaystyle\alpha=\sum_{k=1}^{10}\alpha_{2k}g^{2k}\,,\quad\beta=\sum_{k=1}^{10}\beta_{2k+1}g^{2k+1}\,.$
(9)

The coefficient functions $f_{2k}$, $b_{2k+1}$, $\alpha_{2k}$ and
$\beta_{2k+1}$ are functions of the elliptic coordinates $\eta\in[-1,1]$ and
$\nu\in[0,\infty]$, defined via

$\rho=\rho_{0}\sqrt{\left(1-\eta^{2}\right)(1+\nu^{2})}\,,\quad\zeta=\rho_{0}\eta\nu\,.$
(10)

Here, $g\coloneqq\sqrt{\gamma}$ and $\gamma$ is the relativity parameter,
originally introduced by Bardeen and Wagoner [19], defined by:

$\gamma\coloneqq 1-\sqrt{f_{c}}\,,\quad\text{with}\quad f_{c}\coloneqq
f\left(\rho=0,\zeta=0\right)\,.$ (11)

Note that the metric function $f$ can also be written as $f=e^{2U}$, where $U$
can be interpreted as a generalized Newtonian potential. Equivalently, the
relativity parameter can be expressed in terms of the redshift $Z_{c}$ of a
photon travelling from the centre to infinity:

$\gamma=\frac{Z_{c}}{1+Z_{c}}\,.$ (12)

The parameter space of the disc solution is therefore given by $g\in[0,1]$,
$\epsilon\in[0,1]$ and the coordinate radius $\rho_{0}$. For $g\ll 1$ one
obtains a Newtonian solution and $g\to 1$ corresponds to the ultra-
relativistic limit in which we assume black hole formation. Without loss of
generality we restrict to positive charges and $\rho_{0}$ serves as a scaling
parameter. All multipole moments defined in the subsequent section are
functions of $g$, $\epsilon$ and $\rho_{0}$ only.

##  III Multipole moments

In order to derive multipole moments, we introduce new potentials

$\Xi\coloneqq\frac{1-\mathcal{E}}{1+\mathcal{E}}\quad\text{and}\quad\Lambda\coloneqq\frac{2\Phi}{1+\mathcal{E}}\,.$
(13)

Those can be expressed as an asymptotic expansion on the upper symmetry axis:

$\Xi_{+}=\frac{1}{\zeta}\sum_{n=0}^{\infty}\frac{m_{n}}{\zeta^{n}}\,,\quad\Lambda_{+}=\frac{1}{\zeta}\sum_{n=0}^{\infty}\frac{q_{n}}{\zeta^{n}}\,.$
(14)

According to Fodor et al. [7] the gravitational and electromagnetic multipole
moments, $P_{n}$ and $Q_{n}$, respectively, of an axisymmetric and stationary
spacetime can be obtained from the coefficients $m_{n}$ and $q_{n}$ by the
following procedure:

$\displaystyle P_{0}$ $\displaystyle=m_{0}\,,$ (15) $\displaystyle P_{1}$
$\displaystyle=m_{1}\,,$ (16) $\displaystyle P_{2}$ $\displaystyle=m_{2}\,,$
(17) $\displaystyle P_{3}$
$\displaystyle=m_{3}+\frac{1}{5}\bar{q}_{0}S_{10}\,,$ (18) $\displaystyle
P_{4}$
$\displaystyle=m_{4}-\frac{1}{7}\bar{m}_{0}M_{20}+\frac{3}{35}\bar{q}_{1}S_{10}+\frac{1}{7}\bar{q}_{0}\left(3S_{20}-2H_{20}\right)\,,$
(19) $\displaystyle P_{5}$
$\displaystyle=m_{5}-\frac{1}{21}\bar{m}_{1}M_{20}-\frac{1}{3}\bar{m}_{0}M_{30}+\frac{1}{21}\bar{q}_{2}S_{10}$
$\displaystyle\quad+\frac{1}{21}\bar{q}_{1}\left(4S_{20}-3H_{20}\right)+\frac{1}{21}\bar{q}_{0}\left(\bar{q}_{0}q_{0}S_{10}\right.$
$\displaystyle\quad\left.-\hskip
2.13387pt\bar{m}_{0}m_{0}S\\!_{10}+14S_{30}+13S\\!_{21}-7H_{30}\right)$ (20)

and

$\displaystyle Q_{0}$ $\displaystyle=q_{0}\,,$ (21) $\displaystyle Q_{1}$
$\displaystyle=q_{1}\,,$ (22) $\displaystyle Q_{2}$ $\displaystyle=q_{2}\,,$
(23) $\displaystyle Q_{3}$
$\displaystyle=q_{3}-\frac{1}{5}\bar{m}_{0}H_{10}\,,$ (24) $\displaystyle
Q_{4}$
$\displaystyle=q_{4}+\frac{1}{7}\bar{q}_{0}Q_{20}-\frac{3}{35}\bar{m}_{1}H_{10}-\frac{1}{7}\bar{m}_{0}\left(3H_{20}-2S_{20}\right)\,,$
(25) $\displaystyle Q_{5}$
$\displaystyle=q_{5}+\frac{1}{21}\bar{q}_{1}Q_{20}+\frac{1}{3}\bar{q}_{0}Q_{30}-\frac{1}{21}\bar{m}_{2}H_{10}$
$\displaystyle\quad-\frac{1}{21}\bar{m}_{1}\left(4H_{20}-3S\\!_{20}\right)+\frac{1}{21}\bar{m}_{0}\left(\bar{m}_{0}m_{0}H_{10}\right.$
$\displaystyle\quad\left.-\hskip
2.13387pt\bar{q}_{0}q_{0}H_{10}-14H_{30}-13H_{21}+7S_{30}\right)\,,$ (26)

where

$\displaystyle M_{ij}=m_{i}m_{j}-m_{i-1}m_{j+1}\,,\quad
Q_{ij}=q_{i}q_{j}-q_{i-1}q_{j+1}\,,$ (27) $\displaystyle
S_{ij}=m_{i}q_{j}-m_{i-1}q_{j+1}\,,\quad H_{ij}=q_{i}m_{j}-q_{i-1}m_{j+1}\,.$
(28)

Higher multipole moments become increasingly complicated. The gravitational
and electromagnetic multipole moments are closely related to each other. In
fact, by interchanging $m_{n}\leftrightarrow q_{n}$ and
$\bar{m}_{n}\leftrightarrow-\bar{q}_{n}$ (correspondingly
$M_{ij}\leftrightarrow Q_{ij}$ and $S_{ij}\leftrightarrow H_{ij}$), $P_{n}$
transforms into $Q_{n}$.

As mentioned in section II the disc of dust additionally obeys reflection
symmetry. Expressed on the upper symmetry axis, reflection symmetry is
equivalent to [20, 21, 22, 23, 24]

$\mathcal{E}_{+}\left(\zeta\right)\bar{\mathcal{E}}_{+}\left(-\zeta\right)=1\,,\quad\Phi_{+}\left(\zeta\right)=-\bar{\Phi}_{+}\left(-\zeta\right)\mathcal{E}_{+}\left(\zeta\right)$
(29)

or in terms of the new potentials to

$\Xi_{+}\left(\zeta\right)=-\bar{\Xi}_{+}\left(-\zeta\right)\,,\quad\Lambda_{+}\left(\zeta\right)=-\bar{\Lambda}_{+}\left(-\zeta\right)\,.$
(30)

This means that $P_{n}$ and $Q_{n}$ are real for even $n$ and imaginary for
odd $n$. The real and imaginary parts of $P_{n}$ are called mass and angular
momentum moments and those of $Q_{n}$ are referred to as electric and magnetic
moments. We denote:

$\displaystyle P_{n}$ $\displaystyle=M_{n}+\mathrm{i}J_{n}\,,$ (31)
$\displaystyle Q_{n}$ $\displaystyle=E_{n}+\mathrm{i}B_{n}\,,$ (32)

where due to reflection symmetry,

$\displaystyle\text{for even $n$:}\quad P_{n}=M_{n}\,,\quad Q_{n}=E_{n}\,,$
(33) $\displaystyle\text{for odd $n$:}\quad P_{n}=\mathrm{i}J_{n}\,,\quad
Q_{n}=\mathrm{i}B_{n}\,.$ (34)

Note that on the upper symmetry axis $\eta=1$, $\zeta=\rho_{0}\nu$ and thus

$\Xi_{+}=\frac{1}{\nu}\sum_{n=0}^{\infty}\frac{m_{n}^{\star}}{\nu^{n}}\,,\quad\Lambda_{+}=\frac{1}{\nu}\sum_{n=0}^{\infty}\frac{q_{n}^{\star}}{\nu^{n}}$
(35)

with dimensionless $m_{n}^{\star}\coloneqq\frac{m_{n}}{\rho_{0}^{n+1}}$ and
$q_{n}^{\star}\coloneqq\frac{q_{n}}{\rho_{0}^{n+1}}$. In order to obtain
coordinate independent expressions for the multipole moments, we normalize
$m_{n}$ and $q_{n}$ by the disc’s proper radius
$R_{0}\coloneqq\int_{0}^{\rho_{0}}\\!\sqrt{g_{\rho\rho}}\,\mathrm{d}\rho$,
where $g_{\rho\rho}=f^{-1}h$, see also [25]:

$m_{n}^{\circ}\coloneqq\frac{m_{n}}{R_{0}^{n+1}}\,,\quad
q_{n}^{\circ}\coloneqq\frac{q_{n}}{R_{0}^{n+1}}\,.$ (36)

The first multipole moments are the gravitational mass, $P_{0}=M_{0}=M$, the
angular momentum, $P_{1}/\mathrm{i}=J_{1}=J$, the electric charge,
$Q_{0}=E_{0}=Q$, and the magnetic dipole moment, $Q_{1}/\mathrm{i}=B_{1}=D$.

Inserting the potentials $f$, $b$, $\alpha$ and $\beta$, eqs. 8 and 9, into
$\Xi$ and $\Lambda$, eq. 13, and applying the above mentioned procedure
reveals the multipole moments of the charged rotating disc of dust in terms of
a post-Newtonian expansions up to tenth order. The first multipole moments,
normalized by the disc’s proper radius $R_{0}$, up to third order ($k=3$),
read222Note that the notation with the circle superscript introduced here,
i.e. $P_{n}^{\circ}\coloneqq\frac{P_{n}}{R_{0}^{n+1}}$ and
$Q_{n}^{\circ}\coloneqq\frac{Q_{n}}{R_{0}^{n+1}}$, has a different meaning
than that in [9].:

$\displaystyle M^{\circ}$
$\displaystyle=\frac{4g^{2}}{3\pi}-\frac{4\left(\epsilon^{2}-1\right)g^{4}}{45\pi}$
$\displaystyle\quad+\frac{1}{30240\pi^{3}}\left(\left(2790\epsilon^{4}-23699\epsilon^{2}+20464\right)\pi^{2}\right.$
$\displaystyle\quad\left.-\,35840\epsilon^{4}+247296\epsilon^{2}-211456\right)g^{6}+\mathcal{O}\left(g^{8}\right)\,,$
(37) $\displaystyle J^{\circ}$
$\displaystyle=\sqrt{1-\epsilon^{2}}\,\left[\frac{8g^{3}}{15\pi}-\frac{2\left(34\epsilon^{2}-7\right)g^{5}}{315\pi}\right.$
$\displaystyle\quad+\frac{1}{453600\pi^{3}}\left(\left(59894\epsilon^{4}-40131\epsilon^{2}-126312\right)\pi^{2}\right.$
$\displaystyle\quad\left.\left.-\,322560\epsilon^{4}+207872\epsilon^{2}+1189888\right)g^{7}+\mathcal{O}\left(g^{9}\right)\vphantom{\frac{2\left(34\epsilon^{2}-7\right)g^{5}}{315\pi}}\right]\,,$
(38) $\displaystyle Q^{\circ}$
$\displaystyle=\epsilon\,\left[\frac{4g^{2}}{3\pi}-\frac{16\left(\epsilon^{2}-1\right)g^{4}}{45\pi}\right.$
$\displaystyle\quad+\frac{1}{30240\pi^{3}}\left(\left(5862\epsilon^{4}-32147\epsilon^{2}+25840\right)\pi^{2}\right.$
$\displaystyle\quad\left.\left.-\,35840\epsilon^{4}+247296\epsilon^{2}-211456\right)g^{6}+\mathcal{O}\left(g^{8}\right)\vphantom{\frac{16\left(\epsilon^{2}-1\right)g^{4}}{45\pi}}\right]\,,$
(39) $\displaystyle D^{\circ}$
$\displaystyle=\epsilon\sqrt{1-\epsilon^{2}}\,\left[\frac{4g^{3}}{15\pi}-\frac{\left(34\epsilon^{2}-115\right)g^{5}}{315\pi}\right.$
$\displaystyle\quad+\frac{1}{907200\pi^{3}}\left(\left(58614\epsilon^{4}-177091\epsilon^{2}-200232\right)\pi^{2}\right.$
$\displaystyle\quad\left.\left.-\,322560\epsilon^{4}-867328\epsilon^{2}+3340288\right)g^{7}+\mathcal{O}\left(g^{9}\right)\vphantom{\frac{\left(34\epsilon^{2}-115\right)g^{5}}{315\pi}}\right]\,.$
(40)

Below, we also list the normalized gravitational and electromagnetic
quadrupole and octupole moments (up to $k=3$) of the charged rotating disc of
dust:

$\displaystyle P_{2}^{\circ}$ $\displaystyle=M_{2}^{\circ}$
$\displaystyle=-\frac{4g^{2}}{15\pi}+\frac{4\left(5\epsilon^{2}+8\right)g^{4}}{105\pi}$
$\displaystyle\quad+\frac{1}{453600\pi^{3}}\left(\left(-46258\epsilon^{4}+23337\epsilon^{2}-71376\right)\pi^{2}\right.$
$\displaystyle\quad\left.+\,107520\epsilon^{4}+68096\epsilon^{2}-175616\right)g^{6}+\mathcal{O}\left(g^{8}\right)\,,$
(41) $\displaystyle\frac{P_{3}^{\circ}}{\mathrm{i}}$
$\displaystyle=J_{3}^{\circ}$
$\displaystyle=\sqrt{1-\epsilon^{2}}\,\left[-\frac{8g^{3}}{35\pi}+\frac{2\left(68\epsilon^{2}+171\right)g^{5}}{945\pi}\right.$
$\displaystyle\quad-\frac{1}{4989600\pi^{3}}\left(\left(449242\epsilon^{4}+1015547\epsilon^{2}-256536\right)\pi^{2}\right.$
$\displaystyle\quad\left.\left.-\,1520640\epsilon^{4}-4764672\epsilon^{2}+10543104\right)g^{7}\right.$
$\displaystyle\quad\left.+\,\mathcal{O}\left(g^{9}\right)\vphantom{\frac{2\left(68\epsilon^{2}+171\right)g^{5}}{945\pi}}\right]\,,$
(42) $\displaystyle Q_{2}^{\circ}$ $\displaystyle=E_{2}^{\circ}$
$\displaystyle=\epsilon\,\left[-\frac{4g^{2}}{15\pi}+\frac{4\left(4\epsilon^{2}+9\right)g^{4}}{105\pi}\right.$
$\displaystyle\quad-\frac{1}{453600\pi^{3}}\left(\left(36018\epsilon^{4}-26537\epsilon^{2}+84816\right)\pi^{2}\right.$
$\displaystyle\quad\left.\left.-\,107520\epsilon^{4}-68096\epsilon^{2}+175616\right)g^{6}+\mathcal{O}\left(g^{8}\right)\vphantom{\frac{4\left(4\epsilon^{2}+9\right)g^{4}}{105\pi}}\right]\,,$
(43) $\displaystyle\frac{Q_{3}^{\circ}}{\mathrm{i}}$
$\displaystyle=B_{3}^{\circ}$
$\displaystyle=\epsilon\sqrt{1-\epsilon^{2}}\,\left[-\frac{4g^{3}}{35\pi}+\frac{\left(62\epsilon^{2}+51\right)g^{5}}{945\pi}\right.$
$\displaystyle\quad-\frac{1}{9979200\pi^{3}}\left(\left(394522\epsilon^{4}+54107\epsilon^{2}-2568216\right)\pi^{2}\right.$
$\displaystyle\quad\left.\left.-\,1520640\epsilon^{4}-9022464\epsilon^{2}+19058688\right)g^{7}\right.$
$\displaystyle\quad\left.+\,\mathcal{O}\left(g^{9}\right)\vphantom{\frac{\left(62\epsilon^{2}+51\right)g^{5}}{945\pi}}\right]\,.$
(44)

The mass, angular momentum, electric and magnetic moments are alternating
positive and negative, to be more precise,

$\displaystyle\text{for}\,\;l=0,2,4,...:\,$ $\displaystyle M_{2l}\geq
0\,,\quad\\!J_{2l+1}\geq 0\,,$ $\displaystyle E_{2l}\geq 0\,,\quad
B_{2l+1}\geq 0\,,$ (45) $\displaystyle\text{for}\,\;l=1,3,5,...:\,$
$\displaystyle M_{2l}\leq 0\,,\quad\\!J_{2l+1}\leq 0\,,$ $\displaystyle
E_{2l}\leq 0\,,\quad\\!B_{2l+1}\leq 0\,.$ (46)

It should be emphasized that the global prefactors, whereby

$J_{n}\sim\sqrt{1-\epsilon^{2}}\,,\quad E_{n}\sim\epsilon\,,\quad
B_{n}\sim\epsilon\sqrt{1-\epsilon^{2}}\,,$ (47)

ensure that for vanishing charge, i.e. $\epsilon=0$, all electromagnetic
multipole moments $Q_{n}$ become zero and for vanishing rotation, i.e.
$\epsilon=1$, all angular momentum and magnetic moments, $J_{n}$ and $B_{n}$,
respectively.333Additionally, mass and electric moments contain only even
powers of $g$ and angular momentum and magnetic moments only odd, due to a
symmetric and antisymmetric transformation behaviour, respectively, under a
change of sense of rotation. For further details, see [8] or [25].

Note that disc configurations with $\epsilon=0$ rotate with maximal angular
velocity and ones with $\epsilon=1$ have no rotation at all. This becomes
obvious in the Newtonian limit where each dust particle in the disc is in an
equilibrium of gravitational, electric and centrifugal force.

The influence of the global prefactors is also reflected in figs. 1, 2, 3 and
4. There, the dependence of the multipole moments normalized by the proper
radius, $M_{n}^{\circ}$, $J_{n}^{\circ}$, $E_{n}^{\circ}$ and $B_{n}^{\circ}$,
on the specific charge $\epsilon$ is depicted for $n=0,...,7$. The relativity
parameter is set to $g=0.6$.

Figure 1: Normalized mass moments $M_{n}^{\circ}$ for $n=\\{0,2,4,6\\}$ as
functions of the specific charge $\epsilon$ plotted for $g=0.6$. Figure 2:
Normalized angular momentum moments $J_{n}^{\circ}$ for $n=\\{1,3,5,7\\}$ as
functions of the specific charge $\epsilon$ plotted for $g=0.6$. Figure 3:
Normalized electric moments $E_{n}^{\circ}$ for $n=\\{0,2,4,6\\}$ as functions
of the specific charge $\epsilon$ plotted for $g=0.6$. Figure 4: Normalized
magnetic moments $B_{n}^{\circ}$ for $n=\\{1,3,5,7\\}$ as functions of the
specific charge $\epsilon$ plotted for $g=0.6$. Figure 5: Normalized mass
moments $M_{n}^{\circ}$ for $n=\\{0,2,4,6\\}$ as functions of the relativity
parameter $g$ plotted for $\epsilon=0.3$. Figure 6: Normalized angular
momentum moments $J_{n}^{\circ}$ for $n=\\{1,3,5,7\\}$ as functions of the
relativity parameter $g$ plotted for $\epsilon=0.3$.

Figs. 5 and 6 show the dependence of the normalized gravitational multipole
moments on the relativity parameter $g$, where $\epsilon=0.3$. As can be seen
there, in the limit $g\to 0$ all gravitational multipole moments
$P_{n}^{\circ}$ vanish and with increasing $g$ the first ones (to be more
precise, $n\leq 1$ for all $\epsilon$) grow monotonically. The electromagnetic
multipole moments $Q_{n}$ show a completely analogous behaviour.

Remarkably, as can be seen in figs. 1, 2, 3, 4, 5 and 6, all multipole moments
are perfectly ordered:

$\displaystyle\left|M_{n}\right|$ $\displaystyle\geq\left|M_{n+2}\right|\,,$
(48) $\displaystyle\left|J_{n}\right|$
$\displaystyle\geq\left|J_{n+2}\right|\,,$ (49)
$\displaystyle\left|B_{n}\right|$ $\displaystyle\geq\left|B_{n+2}\right|\,,$
(50) $\displaystyle\left|E_{n}\right|$
$\displaystyle\geq\left|E_{n+2}\right|\,.$ (51)

Notice that this ordering as well as the alternating sign can also be observed
for the (exact) gravitational multipole moments of the uncharged disc of dust
[15].

Interesting are also the ratios $\frac{P_{n}^{\circ}}{P_{n-2}^{\circ}}$ and
$\frac{Q_{n}^{\circ}}{Q_{n-2}^{\circ}}$ of the individual moments. According
to figs. 7 and 8, the absolute values of the ratios are greater, the higher
the gravitational multipole moments are. The corresponding plots of the ratios
of the electromagnetic multipole moments look almost identical to those of the
gravitational moments and in fact agree at $g=0$.444This is not a coincidence,
as in the Newtonian limit $\Lambda=\epsilon\,\Xi$ holds. In addition, there is
a pairwise agreement at $g=0$:
$\frac{P_{3}^{\circ}}{P_{1}^{\circ}}=\frac{P_{4}^{\circ}}{P_{2}^{\circ}}=-\frac{3}{7}\left(=\frac{Q_{3}^{\circ}}{Q_{1}^{\circ}}=\frac{Q_{4}^{\circ}}{Q_{2}^{\circ}}\right)$
and
$\frac{P_{5}^{\circ}}{P_{3}^{\circ}}=\frac{P_{6}^{\circ}}{P_{4}^{\circ}}=-\frac{5}{9}\left(=\frac{Q_{5}^{\circ}}{Q_{3}^{\circ}}=\frac{Q_{6}^{\circ}}{Q_{4}^{\circ}}\right)$.

Figure 7: Ratios $\frac{P_{n}^{\circ}}{P_{n-2}^{\circ}}$ for
$n=\\{2,3,...,7\\}$ as functions of the specific charge $\epsilon$ plotted for
$g=0.6$. Figure 8: Ratios $\frac{P_{n}^{\circ}}{P_{n-2}^{\circ}}$ for
$n=\\{2,3,...,7\\}$ as functions of the relativity parameter $g$ plotted for
$\epsilon=0.3$.

The decision to restrict our discussion to the first eight multipole moments
$P_{n}$ and $Q_{n}$ and the relativity parameter to $g\leq 0.65$ is based on
the convergence behaviour of the multipole moments. As an example, to estimate
the convergence of the electric moments, we calculate the resulting change by
adding the $K$th order to the post-Newtonian expansion at order $K-1$ relative
to the solution at $K$th order:

$\displaystyle\delta
E_{n\,|\,K}^{\circ}\coloneqq\frac{\left|E_{n\,|\,K}^{\circ}-E_{n\,|\,K-1}^{\circ}\right|}{\left|E_{n\,|\,K}^{\circ}\right|}\,,$
(52) $\displaystyle\text{where}\quad
E_{n\,|\,K}^{\circ}\coloneqq\sum_{k=1}^{K}E_{n2k}^{\circ}g^{2k}\,.$ (53)

Figure 9: Convergence estimate of the normalized multipole moments
$E_{0}^{\circ}$, $E_{2}^{\circ}$, $E_{4}^{\circ}$ and $E_{6}^{\circ}$ at order
$k=K$ plotted for $g=0.65$ and $\epsilon=0$.

According to fig. 9, the relative change by adding the last order is for all
$E_{n}^{\circ}$ ($n=0,..,6$) less than or equal to $2.2\text{\times}{10}^{-4}$
for $g=0.65$. (In this case $\epsilon=0$ shows the worst convergence
behaviour.) Moreover, as can already be guessed from the calculation scheme,
the higher the multipole moments, the less good the convergence. For the mass,
angular momentum and magnetic moments we get an even better, but otherwise
analogous convergence behaviour.555For $\epsilon=0$ and $g=0.65$ one gets:
$\delta M_{6\,|\,10}^{\circ}=$7.5\text{\times}{10}^{-5}$$, $\delta
J_{7\,|\,10}^{\circ}=$1.9\text{\times}{10}^{-4}$$, $\delta
B_{7\,|\,10}^{\circ}=$9.7\text{\times}{10}^{-5}$$.

Table 1: Normalized multipole moments $M_{6}^{\circ}$, $J_{7}^{\circ}$: charged disc in the limit $\epsilon=0$ (post-Newtonian expansion up to tenth order) versus uncharged disc (exact solution). |  | $\text{charged disc}\rvert_{\epsilon=0}$ | uncharged disc
---|---|---|---
$g=0.55$ | $\left|M_{6}^{\circ}\right|$ | $1.628\,655\,0\text{\times}{10}^{-3}$ | $1.628\,655\,1\text{\times}{10}^{-3}$
| $\left|J_{7}^{\circ}\right|$ | $8.066\,392\,3\text{\times}{10}^{-4}$ | $8.066\,395\,2\text{\times}{10}^{-4}$
$g=0.6$ | $\left|M_{6}^{\circ}\right|$ | $1.487\,599\,7\text{\times}{10}^{-3}$ | $1.487\,600\,0\text{\times}{10}^{-3}$
| $\left|J_{7}^{\circ}\right|$ | $7.363\,625\,1\text{\times}{10}^{-4}$ | $7.363\,645\,0\text{\times}{10}^{-4}$
$g=0.65$ | $\left|M_{6}^{\circ}\right|$ | $1.314\,102\,5\text{\times}{10}^{-3}$ | $1.314\,103\,3\text{\times}{10}^{-3}$
| $\left|J_{7}^{\circ}\right|$ | $6.359\,058\,8\text{\times}{10}^{-4}$ | $6.359\,180\,7\text{\times}{10}^{-4}$

A direct comparison of the derived multipole moments (in terms of the post-
Newtonian expansion) in the limit $\epsilon=0$ with the exact solutions of the
uncharged rotating disc of dust [15] shows an excellent agreement. In table 1
we compare the sixth and seventh gravitational multipole moments for different
values of the relativity parameter. In particular, the relative error of the
multipole moments of the charged rotating disc of dust in the limit
$\epsilon=0$ with respect to the exact solutions at $g=0.65$ is
$6.1\text{\times}{10}^{-7}$ for $M_{6}^{\circ}$ and
$1.9\text{\times}{10}^{-5}$ for $J_{7}^{\circ}$.

##  IV Multipole conjecture

Filter and Kleinwächter [16] formulated an interesting conjecture about the
multipole moments of (uncharged) rigidly rotating perfect fluid bodies: The
absolute values of all higher multipole moments $P_{n}$ ($n\geq 2$) of an
axistationary, rigidly rotating, perfect fluid body, surrounded by vacuum, are
always greater than those of the corresponding moments of the Kerr spacetime
with the same mass and angular momentum.

An obvious question now is whether this conjecture can be extended to more
general, particularly charged, bodies.

The multipole moments of the Kerr-Newman spacetime, with mass $M^{\text{KN}}$,
angular momentum $J^{\text{KN}}$ and charge $Q^{\text{KN}}$, are given by [6]:

$\displaystyle
P_{n}^{\text{KN}}=m_{n}^{\text{KN}}=M^{\text{KN}}\left(\mathrm{i}\frac{J^{\text{KN}}}{M^{\text{KN}}}\right)^{n}\,,$
(54) $\displaystyle
Q_{n}^{\text{KN}}=q_{n}^{\text{KN}}=Q^{\text{KN}}\left(\mathrm{i}\frac{J^{\text{KN}}}{M^{\text{KN}}}\right)^{n}\,.$
(55)

With these we state:

######  Generalized multipole conjecture.

For the gravitational and electromagnetic multipole moments, $P_{n}$ and
$Q_{n}$, of an isolated, axisymmetric, stationary, physically well-defined
body of ordinary matter, with mass $M$, angular momentum $J$ and charge $Q$,
holds for all $n\geq 2$:

$\left|P_{n}\right|\geq\left|\frac{J^{n}}{M^{n-1}}\right|\,,\quad\left|Q_{n}\right|\geq\left|\frac{QJ^{n}}{M^{n}}\right|\,.$
(56)

Furthermore, in case of $J\neq 0$, equalities apply if and only if the body
reaches a black hole limit.

The goal of this section is to test the generalized multipole conjecture using
the charged rotating disc of dust as a concrete and physically meaningful
candidate that satisfies the requirements of the conjecture.666Note that the
Kerr-Newman spacetime furnished with the mass $M$, angular momentum $J$ and
charge $Q$ of the charged rotating disc of dust does not describe a black hole
but a hyperextreme solution ($Q^{2}+\frac{J^{2}}{M^{2}}>M^{2}$). To this end
we plot the quantities $X_{n}$ and $Y_{n}$ that have to be less than or equal
to $1$ if the conjecture is valid:

$X_{n}\coloneqq\left|\frac{J^{n}}{M^{n-1}P_{n}}\right|\leq 1\,,\quad
Y_{n}\coloneqq\left|\frac{QJ^{n}}{M^{n}Q_{n}}\right|\leq 1\,.$ (57)

Notice that due to the global prefactors of $P_{n}$ and $Q_{n}$ the quantities
$X_{n}$ and $Y_{n}$ stay regular in the limits $\epsilon\to 1$ and
$\epsilon\to 0$. In fact, $X_{n}$ and $Y_{n}$ vanish for $\epsilon\to 1$ and
take finite values in the limit $\epsilon\to 0$ (see also figs. 12 and 13).

In the limit $\epsilon\to 1$ the angular momentum $J$ vanishes and therefore
also all Kerr-Newman multipole moments, $P_{n}^{\text{KN}}$ and
$Q_{n}^{\text{KN}}$, with $n\geq 1$ . Since the multipole moments of the disc,
$P_{n}$ and $Q_{n}$, become zero only for odd, but not for even $n$, the disc
spacetime clearly does not attain the Reissner-Nordström solution in this
limit.

Figure 10: $X_{n}$ for $n=\\{2,3,...,7\\}$ as functions of the relativity
parameter $g$ plotted for $\epsilon=0$. Figure 11: $Y_{n}$ for
$n=\\{2,3,...,7\\}$ as functions of the relativity parameter $g$ plotted for
$\epsilon=0$. Figure 12: $X_{n}$ for $n=\\{2,3,...,7\\}$ as functions of the
specific charge $\epsilon$ plotted for $g=0.6$. Figure 13: $Y_{n}$ for
$n=\\{2,3,...,7\\}$ as functions of the specific charge $\epsilon$ plotted for
$g=0.6$.

As can be seen in figs. 10 and 11 and figs. 12 and 13, indeed all $X_{n}$ and
$Y_{n}$ are smaller than $1$ in the region $g\in\left[0,0.65\right]$.
Furthermore, there is a similar ordering as for the individual moments:

$X_{n}\geq X_{n+1}\,,\quad Y_{n}\geq Y_{n+1}\,.$ (58)

Therefore, most critical are $X_{2}$ and $Y_{2}$ and for increasing $n$ the
conjecture is even better fulfilled.

On the other hand, the first $X_{n}$ and $Y_{n}$ are also the quantities with
the best convergence behaviour. As $n$ becomes larger, the convergence of
$X_{n}$ and $Y_{n}$ (as it contains $P_{n}$ and $Q_{n}$, respectively) becomes
less and less good. The relative change (as defined in eq. 52) by adding the
last order is $\delta X_{7\,|10}=$1.9\text{\times}{10}^{-2}$$ and $\delta
Y_{7\,|10}=$9.7\text{\times}{10}^{-3}$$, where $g=0.65$ and
$\epsilon=0$.777Specifically for $X_{7}$ and $Y_{7}$, the convergence gets
slightly worse for higher $\epsilon$: $\delta
X_{7\,|10}=$4.1\text{\times}{10}^{-2}$$ and $\delta
Y_{7\,|10}=$3.4\text{\times}{10}^{-2}$$, for $\epsilon=1$ and $g=0.65$.
However, as $X_{n}$ and $Y_{n}$ for high $n$ are not decisive for the
verification of the conjecture, this is not problematic.

The good convergence of $X_{2}$ and $Y_{2}$ allows verification of the
conjecture for slightly higher values of the relativity parameter until
$g\approx 0.8$ and $g\approx 0.75$, respectively.888For $\epsilon=0$: $\delta
X_{2\,|10}=$1.9\text{\times}{10}^{-2}$$ at $g=0.8$ and $\delta
Y_{2\,|10}=$2.8\text{\times}{10}^{-2}$$ at $g=0.75$. The conjecture remains
true there. For the limit $g\to 1$, Breithaupt et al. [9] provided strong
evidence that the multipole moments of the charged rotating disc of dust
converge to those of the (extreme) Kerr-Newman spacetime.

Additionally, one can observe from figs. 12 and 13 that all values of $X_{n}$
and $Y_{n}$ decrease equally with increasing $\epsilon$. This fact is very
convenient for our purposes, since for the most critical case, $\epsilon=0$,
the exact multipole moments [15] of the uncharged disc of dust [12, 13, 14]
are available. The relative error of $X_{2}|_{\epsilon=0}$ with respect to the
corresponding exact value of the uncharged disc of dust is
$1.3\text{\times}{10}^{-2}$ for $g=0.8$. For the gravitational multipole
moments of the uncharged disc of dust Filter and Kleinwächter showed that the
conjecture is fulfilled [16].

In summary, the conjecture holds at least up to $g\approx 0.8$ for the
gravitational and up to $g\approx 0.75$ for the electromagnetic multipole
moments. In the limit $g\to 1$ the quantities $X_{n}$ and $Y_{n}$ converge to
$1$ according to Breithaupt et al. and for the most critical case,
$\epsilon=0$, by means of the exact solution of the uncharged disc of dust it
was proven that $X_{n}\leq 1$. All this taken together, we conclude that
(within the scope of accuracy of the post-Newtonian expansion) the generalized
multipole conjecture stated above is fulfilled for the multipole moments of
the charged rotating disc of dust.

##  V Conclusions and outlook

The individual mass, angular momentum, electric and magnetic moments of the
charged rotating disc of dust, in agreement with the uncharged disc [15],
follow an ordering in which the higher they are, the less they contribute to
the gravitational and electromagnetic field. This effect is in addition to the
suppression of higher orders in the asymptotic expansions. Therefore,
primarily the first multipole moments are relevant for the far-field behaviour
of the gravitational and electromagnetic field of the charged rotating disc of
dust.

Of course, it would be desirable to have a proof of the generalized multipole
conjecture. This would first require a precise notion of physical well-
definedness.999Perhaps one should consider a stronger condition, like
uniformly rotating, charged perfect fluid bodies with convective electric
currents only, see [26]. In case of non-vanishing angular momentum the
conjecture could serve as a powerful tool to distinguish ordinary physical
bodies from black holes by measuring their multipole moments in the far-field.
In fact, if the necessary condition $Q^{2}+\frac{J^{2}}{M^{2}}\leq M^{2}$ for
a black hole is satisfied, the measurement of the multipole moments for $n=2$
would already make the difference.

###### Acknowledgements.

This work has been funded by the Deutsche Forschungsgemeinschaft (DFG) under
Grant No. 406116891 within the Research Training Group RTG 2522/1. The authors
would like to thank Andreas Kleinwächter for the computed multipole moment
values of the uncharged disc of dust and Martin Breithaupt for the provided
results of the post-Newtonian expansion up to tenth order.

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# TRIVIALITY OF A SURFACE-LINK WITH MERIDIAN-BASED FREE FUNDAMENTAL GROUP

Akio KAWAUCHI
Osaka City University Advanced Mathematical Institute
Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
<EMAIL_ADDRESS>
( )

###### ABSTRACT

It is proved that every disconnected surface-link with meridian-based free
fundamental group is a trivial (i.e., an unknotted-unlinked) surface-link.
This result is a surface-link version of the author’s recent announcement
result on smooth unknotting of a surface-knot.

x

Keywords: Boundary surface-link, Stably trivial Surface-link, Trivial surface-
link, Unknotted-unlinked surface-link.

Mathematics Subject Classification 2010: Primary 57Q45; Secondary 57N13

1\. Introduction

A surface-link is a closed oriented (possibly disconnected) surface $F$
embedded in the 4-space ${\mathbf{R}}^{4}$ by a smooth (or a piecewise-linear
locally flat) embedding. When $F$ is connected, it is also called a surface-
knot. When the surface $F$ is fixed, it is also called an $F$-link. Two
surface-links $F$ and $F^{\prime}$ are equivalent by an equivalence $f$ if $F$
is sent to $F^{\prime}$ orientation-preservingly by an orientation-preserving
diffeomorphism $f:S^{4}\to S^{4}$. A trivial surface-link is a surface-link
$F$ which is the boundary of the union of mutually disjoint handlebodies
smoothly embedded in $S^{4}$, where a handlebody is a 3-manifold which is a
3-ball or a disk sum of some number of solid tori. A trivial surface-knot is
also called an unknotted  surface-knot. A trivial disconnected surface-link is
also called an unknotted-unlinked surface-link. For every given closed
oriented (possibly disconnected) surface $F$, a trivial $F$-link exists
uniquely up to equivalences [3]).

In the previous paper [8] following the arguments of [5, 6, 7], the author
claimed that a surface-knot $F$ in ${\mathbf{R}}^{4}$ is a trivial surface-
knot if and only if it has the infinite cyclic fundamental group.

In this paper, this result is generalized to a surface-link with meridian-
based free fundamental group. For our argument here, a surface-link in
${\mathbf{R}}^{4}$ is considered as a surface-link in the one-point
compactification ${\mathbf{R}}^{4}\cup\\{\infty\\}=S^{4}$. Then the exterior
of a surface-link $F$ is the compact 4-manifold $E=\mbox{cl}(S^{4}\setminus
N(F))$ for a tubular neighborhood $N(F)$ of $F$ in $S^{4}$. Let $q_{0}$ be a
fixed base point in the interior of $E$. A surface-link with meridian-based
free fundamental group is a surface-link $F$ in $S^{4}$ such that the
fundamental group $\pi_{1}(E,q_{0})$ is a free group with a meridian basis. A
trivial surface-link is a surface-link with meridian-based free fundamental
group. The following unknotting-unlinking result is our main theorem.

x

Theorem 1.1. A surface-link is a trivial surface-link if and only if it is a
surface-link with meridian-based free fundamental group.

x

This result answers positively the problem [9, Problem 1.55 (B)] for a
$cS^{2}$-link (i.e., a 2-link with $c$ components) for any $c$. In this paper,
surface-links are discussed in DIFF-category and the same argument goes well
in PL-category. Even in TOP-category, the author believes that the same
argument goes well by using a TOP-transversality in [1], so that in the case
of a surface-knot, the method of proof is different from the proof given by
[2, 4].

A stabilization of a surface-link $F$ is a connected sum
$\bar{F}=F\\#_{k=1}^{s}T_{k}$ of $F$ and a trivial torus-link
$T=\cup_{k=1}^{s}T_{k}$. By granting $T=\emptyset$, we understand that a
surface-link $F$ itself is a stabilization of $F$. The trivial torus-link $T$
is called the stabilizer with $T_{k}\,(k=1,2,...,s)$ the stabilizer components
on the stabilization $\bar{F}$.

A stably trivial surface-link is a surface-link $F$ such that a stabilization
$\bar{F}$ of $F$ is a trivial surface-link. In [8, Corollary 1.2], it is shown
that every stably trivial surface-link is a trivial surface-link. Therefore,
the proof of Theorem 1.1 is completed by combining [8, Corollary 1.2] with the
following lemma, which generalizes the result of [3, Theorem 2.10] to a
surface-link.

x

Lemma 1.2. Every surface-link with meridian-based free fundamental group is a
stably trivial surface-link.

x

2\. Proof of Lemma 1.2.

Throughout this section, the proof of Lemma 1.2 will be given.

Let $F$ be a surface-link in $S^{4}$ with components $F_{i}\,(i=1,2,\dots,r)$.
Let $N(F)=\cup_{i=1}^{r}N(F_{i})$ be a tubular neighborhood of
$F=\cup_{i=1}^{r}F_{i}$ in $S^{4}$ which is a trivial disk bundle over $F$,
and $E=\mbox{cl}(S^{4}\setminus N(F))$ the exterior of a surface-link $F$.
Then the boundary $\partial E=\partial N(F)=\cup_{i=1}^{r}\partial N(F_{i})$
is a trivial circle bundle over $F=\cup_{i=1}^{r}F_{i}$. Identify $\partial
N(F_{i})=F_{i}\times S^{1}$ such that the composite inclusion

$F_{i}\times 1\to\partial N(F_{i})\to\mbox{cl}(S^{4}\setminus N(F_{i})$

induces the zero-map in the integral first homology, where $S^{1}$ denotes the
set of complex numbers of norm one. Let $q_{i}\times 1$ be a point in
$F_{i}\times S^{1}$ for every $i\,(i=1,2,\dots,r)$.

Let

$K=(\cup_{i=1}^{r}a_{i})\bigcup(\cup_{i=1}^{r}S_{i})$

be a connected graph in $E$ such that

x

(1) $a_{i}$ is an edge embedded in $E$ joining $q_{0}$ and $q_{i}\times 1$
such that the interiors of $a_{i}\,(i=1,2,\dots,r)$ are mutually disjoint,

(2) $S_{i}=q_{i}\times S^{1}\,(i=1,2,\dots,r)$, and

(3) the inclusion $K\to E$ induces an isomorphism
$\pi_{1}(K,q_{0})\to\pi_{1}(E,q_{0})$ such that the element $t_{i}=[a_{i}\cup
S_{i}]\in\pi_{1}(E,q_{0})$ is the $i$th meridian generator.

x

By the assumption that $\pi_{1}(E,q_{0})$ is a free group with a meridian
basis, there is a graph $K$ with properties (1), (2) and (3). Further, we have
the following lemma.

x

Lemma 2.1. The composite inclusion $F_{i}\times 1\to\partial N(F_{i})\to E$ is
null-homotopic for all $i$.

x

Proof of Lemma 2.1. Since $\partial N(F_{i})=F_{i}\times S^{1}$, the
fundamental group elements between the factors $F_{i}\times 1$ and
$q_{i}\times S^{1}$ are commutive. On the other hand, since $\pi_{1}(E,q_{0})$
is a free group, the image of the homomorphism $\pi_{1}(a_{i}\cup F_{i}\times
1,q_{0})\to\pi_{1}(E,q_{0})$ is in the infinite cyclic group $<t_{i}>$
generated by $t_{i}$. The surface $F_{i}\times 1$ in $\partial
N(F_{i})=F_{i}\times S^{1}$ is chosen so that the inclusion $F_{i}\times
1\to\mbox{cl}(S^{4}\setminus N(F_{i}))$ induces the zero-map in the integral
first homology. This implies that the inclusion $F_{i}\times 1\to E$ is null-
homotopic. ∎

x

Let

$K(\partial N)=(\cup_{i=1}^{r}a_{i})\cup(\cup_{i=1}^{r}\partial N(F_{i}))$

be a polyhedron in $E$. Let $p:K(\partial N)\to K$ be the map defined by the
projection $F_{i}\times S^{1}\to q_{i}\times S^{1}$ sending $F_{i}$ to $q_{i}$
for all $i$.

x

Lemma 2.2. The map $p:K(\partial N)\to K$ extends to a map $g:E\to K$.

x

Proof of Lemma 2.2. Since $K$ is a $K(\pi,1)$-space, there is a map $f:E\to K$
inducing the inverse isomorphism $\pi_{1}(E,q_{0})\to\pi_{1}(K,p_{0})$ of the
isomorphism $\pi_{1}(K,p_{0})\to\pi_{1}(E,q_{0})$. Then the composite map
$fj:K(\partial N)\to K$ of $f$ with the inclusion $j:K(\partial N)\to E$ and
the map $p:K(\partial N)\to K$ induces the same homomorphism

$(fj)_{\\#}=p_{\\#}:\pi_{1}(K(\partial N),q_{0})\to\pi_{1}(K,q_{0}).$

Since $K$ is a $K(\pi,1)$-space, the map $fj$ is homotopic to $p$. By the
homotpy extension property in [11], there is a map $g:E\to K$ homotopic to the
map $f$ such that $gj=p$. ∎

x

Replacing $g$ by a piecewise smooth approximation keeping the map $p$ fixed,
we can use a transverse regularity argument to obtain a regular point
$q_{i}\times t_{i}\in q_{i}\times S^{1}$ for each $i\,(i=1,2,\dots,r)$ such
that the preimage $V_{i}=g^{-1}(q_{i}\times t_{i})$ is a compact oriented
smooth 3-manifold with boundary $\partial V_{i}=p^{-1}(q_{i}\times
t_{i})=F_{i}\times t_{i}$. By discarding a closed component from $V_{i}$, we
assume that $V_{i}$ is connected. Let $\alpha_{ik}\,(k=1,2,\dots,n_{i})$ be
mutually disjoint proper arcs in $V_{i}$ such that the exterior
$V_{i}^{*}=\mbox{cl}(V_{i}\setminus N_{i}))$ for a regular neighborhood
$N_{i}=N(\cup_{k=1}^{n_{i}}\alpha_{ik})$ of the arcs
$\alpha_{ik}\,(k=1,2,\dots,n_{i})$ in $V_{i}$ is a handlebody. Let $V(C)_{i}$
be a compact connected oriented smooth 3-manifold with boundary $\partial
V(C)_{i}=F_{i}$ obtained from $V_{i}$ by adding a collar $C_{i}$ joining
$F_{i}\times t_{i}$ with $F_{i}$ in the trivial disk bundle $N(F_{i})$ over
$F_{i}$. Let $\alpha(C)_{ik}\,(k=1,2,\dots,n_{i})$ be mutually disjoint proper
arcs in $V(C)_{i}$ obtained from the arcs $\alpha_{ik}\,(k=1,2,\dots,n_{i})$
by extending them until they reach $F_{i}$ through the collar $C_{i}$. Then
the exterior $V(C)_{i}^{*}=\mbox{cl}(V(C)_{i}\setminus N(C)_{i}))$ for a
regular neighborhood $N(C)_{i}=N(\cup_{k=1}^{n_{i}}\alpha(C)_{ik})$ of the
arcs $\alpha(C)_{ik}\,(k=1,2,\dots,n_{i})$ in $V(C)_{i}$ is a handlebody for
all $i$. Let $\alpha(C)^{0}_{ik}$ be an arc obtained from a simple arc
$\beta_{ik}$ in $F_{i}$ with $\partial\beta_{ik}=\partial\alpha(C)_{ik}$ by
pushing the interior of $\beta_{ik}$ into the collar $C_{i}$. Note that every
loop in $V_{i}$ is null-homotopic in $E$ since the map $g$ induces an
isomorphism $g_{\\#}:\pi_{1}(E,q_{0})\cong\pi_{1}(K,q_{0})$. This means that
the arc $\alpha(C)_{ik}$ is homotopic to an arc $\alpha(C)^{0}_{ik}$ by a
homotopy in $(S^{4}\setminus F)\cup F_{i}$ relative to $F_{i}$. By an argument
of HoK, we see that the $1$-handle $h_{ik}$ on $F$ thickening the arc
$\alpha(C)_{ik}$ is a trivial 1-handle whose surgery of $F$ is the connected
sum of $F$ and a trivial torus-knot on the component $F_{i}$. Since the
surface-link obtained from $F$ by the surgery along the 1-handles $h_{ik}$ for
all $i$ and $k$ bounds a system of handlebodies
$V(C)_{i}^{*}\,(i=1,2,\dots,r)$, the surface-link $F$ is a stably trivial
surface-link. This completes the proof of Lemma 1.2. ∎

x

A surface link $F$ with $r(\geq 2)$ components is a boundary surface-link if
it bounds mutually disjoint $r$ bounded connected oriented smooth 3-manifolds
in $S^{4}$. The following corollary is contained in the proof of Lemma 1.2
whose technique is known in classical link theory (see [10]).

x

Corollary 2.3. A disconnected surface link $F$ is a boundary surface-link if
there is an epimorphism from the fundamental group $\pi_{1}(E,q_{0})$ to a
meridian-based free group.

x

Acknowledgements. This work was in part supported by Osaka City University
Advanced Mathematical Institute (MEXT Joint Usage/Research Center on
Mathematics and Theoretical Physics).

x

## References

  * [1] M. H. Freedman and F. Quinn, Topology of 4-manifolds, Princeton Univ. Press (1990).
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  * [4] A. Kawauchi, Splitting a 4-manifold with infinite cyclic fundamental group, revised, Journal of Knot Theory and Its Ramifications 22 (2013) 1350081 (9 pages).
  * [5] A. Kawauchi, A chord diagram of a ribbon surface-link, Journal of Knot Theory and Its Ramifications 24 (2015), 1540002 (24 pages).
  * [6] A. Kawauchi, Supplement to a chord diagram of a ribbon surface-link, Journal of Knot Theory and Its Ramifications 26 (2017), 1750033 (5 pages).
  * [7] A. Kawauchi, Faithful equivalence of equivalent ribbon surface-links, Journal of Knot Theory and Its Ramifications 27 (2018),1843003 (23 pages).
  * [8] A. Kawauchi, Ribbonness of a stable-ribbon surface-link, I. A stably trivial surface-link, revised preprint. http://www.sci.osaka-cu.ac.jp/ kawauchi/SRibbonSLinkI(STrivialCase).pdf
  * [9] R. Kirby (e.d.), Problems in low-dimensional topology, Algebraic and geometric topology (Stanford, 1978) (1978), 273-312. Up-dated version: http://www.math.berkeley.edu/ kirby/.
  * [10] N. Smythe, Boundary links, Topology seminar Wisconsin 1965, Ann. of Math. Studies 60 (1966), 69-72.
  * [11] E. Spanier, Algebraic topology, MacGraw Hill (1966).


# Variable-delay Polarization Modulators for Cryogenic Millimeter-wave
Applications

D.T. Chuss<EMAIL_ADDRESS>NASA Goddard Space Flight Center,
Greenbelt, MD 20771  J.R. Eimer  Department of Physics and Astronomy, The
Johns Hopkins University, Baltimore, MD  D.J. Fixsen  NASA Goddard Space
Flight Center, Greenbelt, MD 20771  J. Hinderks  NASA Goddard Space Flight
Center, Greenbelt, MD 20771  A.J. Kogut  NASA Goddard Space Flight Center,
Greenbelt, MD 20771  J. Lazear  Department of Physics and Astronomy, The Johns
Hopkins University, Baltimore, MD  P. Mirel  NASA Goddard Space Flight Center,
Greenbelt, MD 20771  E. Switzer  NASA Goddard Space Flight Center, Greenbelt,
MD 20771  G.M. Voellmer  NASA Goddard Space Flight Center, Greenbelt, MD 20771
E.J. Wollack  NASA Goddard Space Flight Center, Greenbelt, MD 20771

###### Abstract

We describe the design, construction, and initial validation of the variable-
delay polarization modulator (VPM) designed for the PIPER cosmic microwave
background polarimeter. The VPM modulates between linear and circular
polarization by introducing a variable phase delay between orthogonal linear
polarizations. Each VPM has a diameter of 39 cm and is engineered to operate
in a cryogenic environment (1.5 K). We describe the mechanical design and
performance of the kinematic double-blade flexure and drive mechanism along
with the construction of the high precision wire grid polarizers.

astronomical polarimetry, instrumentation, cosmic microwave background

###### pacs:

42,95,98

††preprint: AIP/123-QED

> ##  I Introduction

A Variable-delay Polarization Modulator (VPM) changes the state of
polarization of electromagnetic radiation via the introduction of a variable
phase delay between two orthogonal polarization components.Chuss  _et al._
(2006) This leads to a transfer function in which an output Stokes parameter
$U^{\prime}$, which is defined at a $45^{\circ}$ angle with respect to the
polarization separation basis (defined by the VPM grid), is modulated
according to

$\displaystyle U^{\prime}=U\cos{\phi}+V\sin{\phi}.$ (1)

Here, $U$ is the linear Stokes parameter that is the difference between the
linear polarization components oriented at $\pm 45^{\circ}$ with respect to
the wires, $V$ is the Stokes parameter corresponding to circular polarization,
and $\phi$ is the introduced phase delay between the two orthogonal
polarizations.

In recent work, VPMs have been realized by the arrangement of a wire grid
polarizer in front of and parallel to a moving mirror Krejny  _et al._ (2008)
(See Fig. 1). This phase delay is a function of the incidence angle, $\theta$,
and the grid-mirror separation, $d$. In the limit where the wavelength is much
larger than the wire, the phase delay can be approximated using the geometric
path difference,

$\phi\approx\frac{4\pi\lambda}{d}\cos{\theta}.$ (2)

Figure 1: Overview of the VPM concept adapted from Chuss et al.Chuss  _et al._
(2012a) The geometric path difference between the two polarizations is
highlighted by the thick solid lines.

For greater fidelity, a transmission line model has been used to connect the
grid-mirror separation with the introduced phase delay as a function of the
wire grid geometry.Chuss  _et al._ (2012a) In addition, metrology techniques
for realizing high precision measurements of the VPM response have been
developed.Eimer  _et al._ (2011)

The Primordial Inflationary Polarization ExploreR (PIPER)Chuss  _et al._
(2010); Kogut  _et al._ (2012) and the Cosmology Large Angular Scale Surveyor
(CLASS)Eimer  _et al._ (2012) will both employ VPMs as the first element of
their optics in their measurement of the polarization of the cosmic microwave
background on large angular scales. The motivation for using the VPM is
twofold. First, the VPM can be made large enough to be placed at the primary
aperture of a relevant telescope. This allows sky polarization to be modulated
while leaving any subsequent instrumental contribution to the polarization
unmodulated, thereby mitigating potential mixing between temperature and
polarization. In addition, the small linear distances (fraction of a
wavelength) required for the grid-mirror separation variation allow a rapid
($\sim$few Hz) modulation in polarization that moves the signal out of the
$1/f$ noise of the environment.

Devices employing small linear motions have potential advantages in both
reliability and power dissipation over those that utilize large angular
motions for cryogenic applications. The PIPER flexures were designed for more
than 3$\times 10^{6}$ cycles, sufficient to survive 8 flights at 3 Hz
operation. It is anticipated that similar designs could be employed having
much longer lifetimes. Superconducting bearings for wave plates Klein  _et
al._ (2011) are a good solution for low-friction rotational operation;
however, parasitic heat removal is a challenge for such non-contacting
solutions.

This paper describes the construction and initial validation of the PIPER
VPMs. PIPER is a balloon-borne cosmic microwave background polarimeter that
will operate at 4 frequencies between 200 and 600 GHz in separate flights. The
instrument is enclosed in a large bucket dewar, and each of the elements of
the two telescopes is cooled to 1.5 K by a combination of evaporating liquid
helium and superfluid pumps.Singal  _et al._ (2011) Cooling the telescope
reduces background radiation and mitigates the coupling to variable grid
emission. Because of this, the VPM must be engineered to work at 1.5 K.

The details of the optical design Eimer  _et al._ (2010) and detectors Benford
_et al._ (2010) for PIPER are described in other papers. This work focuses on
the design, construction and initial validation of the VPMs.

##  II Design of the Mechanical Assembly

A schematic of the VPM is shown in Figure 2. The 1.5 K mirror is actuated by a
warm (ambient temperature) motor that couples to the cold mechanism through a
driveshaft. All parts of the mechanism that operate at 1.5 K are constructed
from stainless steel to ensure thermomechanical stability and matching of the
coefficient of thermal expansion. This includes the mirror, which is
constructed from stainless steel honeycomb. The mirror faceplate is lapped
flat, and then gold is deposited onto the mirror to increase the conductivity
of the surface. A photo of the VPM drive mechanism is shown in Figure 3.

Figure 2: The operation of the VPM drive mechanism is shown. A warm motor
turns a cam shaft that is attached to a push rod via an eccentric. This push
rod turns a cold rotary flexure that in turn pushes and pulls on one side of a
kinematic double-blade flexure that maintains the parallelism of the mirror
over the throw. A symmetric pair of rotary flexures (not shown) couples to an
additional push rod that is moved in the opposite direction of the first.
These rotary flexures move counterweights to compensate the motion of the
mirror. Figure 3: Photograph of the PIPER VPM assembly.

To ensure that transport of the mirror is parallel, we have utilized a
kinematic double-blade flexure similar to those used in the prototype Hertz
VPM. Voellmer  _et al._ (2006) This mechanism constrains all degrees of
freedom except for the direction of motion. The PIPER flexure is constructed
out of stainless steel using wire-EDM and is larger than the Hertz flexure to
permit the maximum throw required, 1 mm. This is set by the longest wavelength
modulated, $\lambda=1.5$ mm (200 GHz). The parallelism requirement of the
flexure is 5 arc seconds over the 1 mm throw to ensure that the phase is well-
defined at the shortest operating wavelength, 0.5 mm (600 GHz). In practice,
the flexure is mechanically biased so that the entirety of the 1 mm throw is
done on one side of the neutral (un-stressed) position. This prevents the
flexure from operating through its neutral point where its motion is
potentially not as well controlled.

The drive is powered by a motor operating at ambient temperature. The motor
drives a cam shaft. The cam shaft has two eccentrics that are 180∘ out of
phase. The primary eccentric drives a push rod that causes the central rotary
flexure to rotate. This moves the mirror in a direction perpendicular to the
grid; parallelism is maintained by the double blade flexure. The secondary
eccentric drives a similar mechanism that in turn drives a counterweight to
compensate the mirror motion. This mitigates potential vibrations that can
induce microphonic response in the system.

Because the eccentric determines the limits of the mirror movement, the
operational mode of the VPM can be changed simply by replacing the cam shaft
to achieve the desired throw. This allows a straightforward means to operate
in different observational frequency bands. For PIPER, the throw will be
reconfigured between flights to accommodate the range of planned observing
frequency bands.

##  III Grid Fabrication

One of the most challenging aspects of the VPM is the construction of the
polarizing grids. The grids must hold a flatness tolerance of $<$10 $\mu$m RMS
over a clear aperture of 39 cm.Eimer  _et al._ (2010) In addition, to achieve
high polarization diplexing efficiency, it is important to maintain high
uniformity of the spacing of the wires that comprise the grid.

One historical approach for fabricating free-standing wire grid polarizers is
to wind them directly onto a frame.Chambers, Parker, and Costly (1986);
Lahtinen and Hallikainen (1999) In this technique, wire spacing control is
difficult because the wires are unconstrained along the frame. They can roll
or slide out of position under minimal displacement forces because the wire
tension, and thus the static friction holding them in place on the frame, is
low during winding to avoid wire breakage. In addition, this technique
requires a single spool of wire to be used during fabrication with no
tolerance to wire breakage. An alternative to this technique that mitigates
these issues is to wrap the wires on a cylinder that has been precut with
regularly spaced alignment teeth, and then transfer the wires to their final
frame.Novak, Sundwall, and Pernic (1989) The latter technique has been
extended to make prototype 50 cm diameter polarizers.Voellmer  _et al._ (2008)

Here we describe significant enhancements to this technique that have led to
improved wire uniformity and survivability at cryogenic temperatures. Free-
standing wire grid polarizers have an advantage over comparable deposited
membrane structures in that they do not use dielectric material as a
substrate. The use of dielectrics causes a degradation of the capacitive mode
(polarization perpendicular to wires) and decreases the maximum operating
frequency for a given feature size.Whitborn and Compton (1985) However,
historically, the performance of free-standing wire grid polarizers has been
limited due to wire spacing variability. The grid manufacturing process
presented here greatly improves the uniformity of grid wires and can therefore
greatly enhance the capability of polarizers operating at wavelengths between
1 cm and 60 $\mu$m.

The process of grid fabrication is illustrated in Figure 4, and additional
diagrams are included in the appendix for clarity. The 40 $\mu$m copper-plated
tungsten wire (36 $\mu$m diameter W wire; 0.13 $\mu$m Ni strike; 1.3-2.5
$\mu$m Cu plating)CFW  is wrapped onto a mandrel using a 5-axis CNC mill. In
contrast to previous versions of this technique,Voellmer  _et al._ (2008);
Novak, Sundwall, and Pernic (1989) the mandrel itself is smooth, with grooves
cut only in two wire retainer bars that are recessed into the mandrel. These
grooves serve as guides to set the wire spacing. The precision of the milling
machine used for the wrapping is sufficient that the entire mandrel need not
be grooved to maintain the 1 wire per groove criterion on the wire retainer
bars. This realization reduces risk to the wire coatings that can become
damaged by metal burrs during the winding and unwrapping processes.

To cut the grooves for the wires, we used a rotating end mill oriented at 45∘
with respect to the rotation axis of the mandrel. We use a straight flute
tungsten carbide tipped 1/2${}^{{}^{\prime\prime}}$ diameter end mill in a
climb cut at 2,500 RPM on the spindle, with the mandrel rotating at 2.7 RPM.
This combination of tool speeds and feeds leaves a burr-free groove in the
soft copper retaining bar.

Figure 4:  The process for constructing the large cryogenic wire grid. (top)
The wire is wrapped around a mandrel using a CNC mill. (middle) The wires are
soldered to a pair of grooved wire retainer bars and cut. Heat is supplied to
the wire retainer bars via an integrated cartridge heater during this process.
The wire retainer bars are thermally isolated from the rest of the mandrel.
(bottom) The wires are then transferred to the grid frame, attached via screws
with springs.

Once the wire is wrapped, the wire is fastened to the wire retainer bars.
Previous efforts have used epoxy; however, because of the large size and
cryogenic operating environment of the PIPER grids, we have developed a
technique for soldering the wires to the bars. The wire retainer bars are
attached to the mandrel using thermally insulating stand-offs. This enables
adequate thermal isolation to allow the solder to flow on the wire retainer
bars without heating the mandrel. A cartridge heater within the fixture plate
heats the plate and the wire retainer bars to soldering temperature. No
additional heat is added. The flux core solder used was manufactured by
Amerway, Inc. (62% Sn, 36% Pb, 2% Ag). Amerway  The grooves in the wire
retainer bars that constrain the wire spacing are divided into two parts by a
channel cut perpendicular to the wire direction. This “solder gap” keeps the
solder away from the clear aperture and keeps the aperture side of the grooves
free of solder. This allows the wires to seat in each groove when tensioned as
is explained below.

The wires are then cut at the top-dead-center location between the retainer
bars and then removed from the mandrel. The retaining bars, with wires
attached, are mounted to the stainless steel grid frame using screw-adjustable
helical spring tensioners (See bottom of Fig. 4.) The springs maintain the
tension on the wires upon cooling despite the mismatch in thermal contraction
due to the dissimilar materials of the grid and grid frame. In addition, upon
tensioning, the wires are pulled deeper into the unsoldered part of the
retainer bar grooves. This ensures maximum control over wire spacing.
Additional diagrams of the mandrel and the frame are shown in Figures 9 and 10
in the Appendix.

To define the plane of the wires, we implement a “grid flattener,” a polished
ring pressed lightly against the wires. This technique of separating the
tensioning component from the plane definition has been implemented in
previous work.Krejny  _et al._ (2008); Voellmer  _et al._ (2008)

The tension is set to be 0.7 N (130 MPa) on each wire, or approximately 25% of
the breaking strength of the tungsten wire. This is implemented by measuring
the spring constants and setting the compression distance. This leads to a
resonant frequency of 190 Hz, before application of a grid flattener.
Application of the grid flattener increases the resonant frequencies of the
individual wires. An image of the final grid is shown in Figure 5 before
application of the flattener.

To summarize, we highlight key innovations in this fabrication process that
separate it from previous work.Novak, Sundwall, and Pernic (1989); Voellmer
_et al._ (2008) (1) The grooves have been cut on the retainer bar only. (2)
The wire-guide grooves are cut using an end mill oriented at 45∘ to the
mandrel axis to produce highly regular and burr-free cuts. (3) The wires are
soldered to the retainer bars for cryogenic compatibility using an _in situ_
soldering technique. (4) The interface between the retainer bars and the grid
frame is designed to pull the wires deeper into the grooves that ultimately
define the spacing uniformity. (5) The entire wire retainer bar is attached to
the grid frame, so the wires are put into their position-defining grooves once
and never removed.

Figure 5:  The final cryogenically-compatible polarizing grid is shown. The
dimensions of the grid are 41.7$\times$40 cm2.

##  IV Validation

Figure 6:  A micrograph of a section of the wires is shown with a millimeter
scale for comparison. The spacing of the wires is $117.0\pm 5.7\,\mu$m.

###  IV.1 Uniformity of Wire Spacing

The wire spacing has been measured using a STIL S.A. CHR 150 confocal
spectrometer.STI  In this configuration, the grid was placed on the X-Y stage
of a milling machine, and the vertical distance to the grid wires was measured
as a function of horizontal position. The average spacing of the wires was
found to be $117.0\pm 5.7\,\mu$m. The error is determined by the standard
deviation of many measurements taken using the confocal spectrometer. The
reported error is partially due to the finite beam size of the confocal
spectrometer beam that is not well characterized. Thus, this measurement
represents an upper limit on the 1-$\sigma$ deviation. A micrograph of the
wires in Figure 6 shows the wire uniformity.

###  IV.2 Flatness and Parallelism

The grid-mirror separation is measured at three places near the edge of the
mirror, evenly spaced in angle 120∘ from each other. Capacitance sensorsKLA
are mounted to the frame and measure the distance between the sensor and the
back of the mirror with better than 0.5 $\mu$m precision using a standard
calibration curve. Figure 7 shows the tilt of the mirror as a function of time
for a few modulation cycles (top) and mirror displacement for 10 minutes of
data (bottom). The tilt was calculated using the displacement data from the
three capacitance sensors. The deviation from parallelism is $\sim\pm 2.5$ arc
seconds. This corresponds to a maximum tilt of about 7 $\mu$m across the 30 cm
illuminated aperture of PIPER. At 270 $\mu$m, this is 0.5% of the wavelength,
and so the induced phase error is small.

Figure 7:  (Top) The mirror tilt is shown as a function of time. Here the cam
is driven at a frequency of 1.4 Hz using a test motor. Several periods of the
motion are shown. (Bottom) The tilt is shown as a function of the mirror
displacement a for 10 minutes of data. At each point, the mirror displacement
is determined by averaging the readings from the three capacitance sensors.

The grid flatness with the current grid flattener is $\pm 8.7\mu$m
(1-$\sigma$) as measured with the confocal spectrometer.

###  IV.3 Cryogenic Compatibility

A test grid was constructed by wrapping the central 15 cm of the PIPER grid
frame at the full 41.7 cm length. This sample was cycled to 77 K ten times and
did not fail. In addition, the flexure mechanism and drive train were cooled
to 4 K in a liquid helium dewar and remained fully operable.

The VPM mirrors are constructed of stainless steel honeycomb that is
internally held together using a brazing process developed by Bodycote.Bod
The face sheet used for the mirror was 1.5 mm material that will be lapped to
a flatness of $<$5 $\mu$m and vapor deposited with 2.5 $\mu$m of gold. We
tested this construction for thermal survivability by thermally cycling the
mirror 10 times to liquid nitrogen (77 K) temperature. We found no change in
the flatness; however, the precision surface (lapped and vapor deposited) was
not applied.

###  IV.4 Grid Polarization Efficiency

We have made initial measurements of the grid efficiency using a quasioptical
testbed coupled to a Vector Network Analyzer (VNA). We have transferred a
sample of a test grid to a 15 cm diameter circular frame and mounted it to a
small VPM. Measurements of the polarization transfer function were made from
76-117 GHz. The data analysis is identical to that for previously published
results.Chuss  _et al._ (2012b) Figure 8 shows the single frequency
polarization transfer function from 76-117 GHz. For each grid-mirror
separation, a polarization spectrum is obtained. Each point in Figure 8 is the
mean of 100 data points (1.36 GHz bandwidth.) Data from 9 grid-mirror
separations are included in the plot.

For reference, we have superposed the transmission line model for a VPM from
Chuss et al.Chuss  _et al._ (2012a) evaluated using the mean wavelength over
the band (3.15 mm) and the PIPER wire diameter (40 $\mu$m.) The resistive
circuit element has been calculated using the conductivity of copper. The
measured grid efficiency is greater than 99%. This measurement is limited by
the uncertainty in the VNA absolute calibration.

Figure 8:  The polarization transfer function of a sample grid in a test VPM
is shown. The data points are polarization measurements as a function of phase
from 76-117 GHz. The measurement is consistent with unit efficiency to the
accuracy of the measurement. The red curve is a transmission line modelChuss
_et al._ (2012a) evaluated for the measured grid parameters.

##  V Summary

We have designed, constructed, and validated a cryogenic variable-delay
polarization modulator (VPM) for the PIPER suborbital cosmic microwave
background polarimeter. The achieved specifications for the VPM are shown in
Table 1.

Property | Value | Units
---|---|---
Maximum mirror throw | 1.0 | mm
Mirror tilt at maximum throw | 5 | arc seconds
Clear aperture | 39 | cm
Wire diameter | 40 |  $\mu$m
Wire separation | 117.0 |  $\mu$m
Wire separation error | 5.7 |  $\mu$m
Grid flatness | 8.7 |  $\mu$m
Min. wire resonance | 190 | Hz
Polarization Efficiency | $>99$ | %
Table 1: The parameters of the VPM for PIPER.

## Acknowledgements

We thank Alyssa Barlis, Adam Blake, Matheus Teixeira, and Vien Ha for work in
the initial testing of the fabrication process and Paul Cursey for machining
support. We thank Mackenzie Turvey for assistance with the electromagnetic
testing. In addition, we would like to thank Mike Jackson for modeling and
finite element analysis for the flexure and drive system. This work was funded
by a NASA APRA suborbital grant.

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## Appendix: Diagrams of Mandrel and Grid Frames

Figure 9:  The mandrel for winding the grids is shown. The wire is wrapped
around a mandrel using a CNC mill. The wires are soldered to a pair of grooved
wire retainer bars and cut. The wires are then transferred to the grid frame,
attached via screws with springs.  Figure 10:  Two views of the grid are shown
(top and middle) along with a detail drawing of the attachment of the wire
retainer bars (bottom).


# Poster Abstract: Activity-based Profiling for Energy Harvesting Estimation

Jiajie Li, Amani Abusafia, Abdallah Lakhdari, and Athman Bouguettaya  The
University of Sydney, AustraliaAustralia jiajie.li, amani.abusafia,
abdallah.lakhdari<EMAIL_ADDRESS>
(2023)

###### Abstract.

We propose a novel activity-based profiling framework to estimate IoT users’
harvested energy based on their daily activities. Energy is harvested from
natural sources such as the kinetic movement of IoT users. The profiling
framework captures the users’ physical activity data to define activity-based
profiles. These profiles are utilized to estimate the harvested energy by IoT
users. We train and evaluate our framework based on a real Fitbit dataset.

Wireless energy, IoT, Energy Harvesting, activity profiling, wireless power
transfer, energy services

††journalyear: 2023††copyright: rightsretained††conference: The 22nd
International Conference on Information Processing in Sensor Networks; May
9–12, 2023; San Antonio, TX, USA††booktitle: The 22nd International Conference
on Information Processing in Sensor Networks (IPSN ’23), May 9–12, 2023, San
Antonio, TX, USA††doi: 10.1145/3583120.3589833††isbn:
979-8-4007-0118-4/23/05††ccs: Human-centered computing Ubiquitous
computing††ccs: Human-centered computing Ubiquitous and mobile computing
systems and tools

##  1\. Introduction

_Energy services_ refers to the wireless transfer of energy from an IoT device
(e.g. energy provider) to a nearby device (e.g., energy consumer)
(lakhdari2020Vision, )(abusafia2022services, ) Energy providers, such as smart
shoes, may harvest energy from natural resources (e.g., physical activity)
(sandhu2020phd, )(gorlatova2015movers, ) For example, PowerWalk kinetic energy
harvester produces 10-12 watts of on-the-move power(lakhdari2020Vision, ). The
harvested energy may be used to charge nearby IoT devices wirelessly using new
technologies known as “Over-the-Air wireless charging”
(abusafia2022maximizing, )(Amani2022QoE, )(yang2023Monitoring, ). For example,
Energous may charge devices up to a distance of 4.5 meters (yao2022wireless,
)(yang2022towards, ).

Enabling a sustainable energy services IoT ecosystem will offer ubiquitous
access to energy (abusafia2022services, ). In this environment, IoT devices
can be charged automatically without human intervention. To achieve that, the
system must know when, where, and how much energy an IoT device may
request/offer as a service from/to nearby IoT devices (lakhdari2020Vision, ).
This paper focuses on estimating the energy services that a provider may
offer, specifically on estimating how much energy a provider may harvest to
offer as a service.

Studies have proposed to estimate the energy from daily activities by
measuring the output from kinetic energy harvesters (sandhu2020phd, ).
Harvested energy may be estimated by capturing the kinetic movement of the
device wearer and their body temperature. Moreover, other studies used kinetic
energy harvesting data to recognize daily activities (khalifa2017harke,
)(khalifa2015energy, ) In principle, it is possible to estimate the amount of
harvested energy based on the activity and the worn harvester’s type
(khalifa2017harke, ) Such activity may be collected by wearables such as
smartwatches. In this paper, we envision a novel framework to profile users
based on their daily physical activities, consisting of an activity-based
profiling component and an energy-harvesting estimation model (See Fig. 1)
(lakhdari2020Vision, ). The activity-based profiling component will aim to
classify users based on their daily activities in order to associate them with
the corresponding energy harvesting estimation model. The model will use the
activities profiles and the type of worn harvester to estimate the harvested
energy. As energy harvesting technologies are still developing,we focus on the
activity-based profiling.

Figure 1. Energy Harvesting Estimation System Table 1. Evaluation Metrics of
clustering algorithms

| K-means Clustering | Robust Clustering
---|---|---
Metrics $\backslash$ Feature Selection | Original | PCA | Correlation | Original | PCA | Correlation
Silhouette Score | 0.47574 | 0.51564 | 0.48323 | 0.73409 | 0.68170 | 0.72274
Davies Bouldin Index | 0.55996 | 0.49664 | 0.54936 | 0.38857 | 0.42842 | 0.40737
Calinski Habrasz Index | 72.97150 | 75.68820 | 75.72814 | 81.62642 | 78.73628 | 79.16408

Several studies have used fitness data to model its wearer behaviour for
fitness purposes (ni2019modeling, ). Hence, we propose to use fitness data to
recognize users’ activities for energy harvesting purposes. Even though people
may do the same activity, e.g., walking, they may differ in activity intensity
level, which may harvest different amounts of energy (khalifa2017harke, ).
Therefore, we propose an activity-based profiling framework to identify macro
profiles that represent the major activity behaviour of IoT users. These
profiles will be used to build an energy harvesting model to estimate the
harvested energy based on the users’ activity and the used energy harvester.

##  2\. Profiling Framework

The activity-based profiling framework aims to cluster users according to
their daily activity records. The framework consists of four phases: Data
Acquisition and Pre-processing, Feature Selection, Data Fusion and Activity-
based Profiling (See Fig. 2). In what follows we discuss each phase:

Figure 2. Activity-Based Profiling System Architecture

  1. 1\.

Data Acquisition and Pre-processing: During this phase, users’ health-related
information and daily activity data are obtained from their fitness tracker.
We used a public Fitbit dataset consisting of 940 records of 33 users’ daily
activities over a 2-month period in 2016111kaggle.com/arashnic/fitbit. As the
dataset does not have bio-related data for all users, only activity-related
data is used. Also, three features are removed from the dataset:
ActivityDate’, TrackerDistance’, and LoggedActivityDistance’ because they are
exact duplicates for other attributes.

  2. 2\.

Feature Selection: Feature selection identifies the most representative
features of an IoT user’s daily activity. The following three approaches were
applied separately to the original cleaned dataset, resulting in three
processed datasets:

    1. 2.1

Original module: This module contains all the features of the pre-processed
dataset.

    2. 2.2

PCA module: We apply the Principal Component Analysis (PCA) technique on the
original dataset to transform all the attributes into only three components.

    3. 2.3

Correlation module: This module uses a feature-based heat map to select
features based on their correlation, removing highly positively or negatively
correlated features.

  3. 3\.

Data Fusion Clustering users into groups is not applicable to our fitness
dataset as it has multiple activity records for each user. Therefore, data
fusion is needed to aggregate the records of each user with minimal
information loss. Specifically, we used the statistical values (maximum,
minimum, range, standard deviation, mean, and median) to capture user
characteristics. These statistics are computed for each user in the three
processed datasets from step (2).

  4. 4\.

Activity-based Profiling This phase includes applying K-means clustering to
users’ aggregated records, followed by using the resulting clusters in a
robust clustering approach.

    1. 4.1

K-means Clustering K-means clustering is applied to each of the datasets
obtained earlier. We used the elbow method to determine the optimal number of
clusters for each dataset. As a result, each dataset produced four clusters.

    2. 4.2

Robust Clustering As aforementioned, the outcome of the clustering was four
clusters for each dataset, but with varying cluster sizes (number of users).
To address this, we define robust clustering by identifying common users
across the same clusters in all three sets. Users who are clustered together
in all three K-means sets are defined as one robust cluster. Users with
different clustering neighbours in the original three K-means sets are defined
as separate robust clusters. This produces seven robust clusters/profiles,
each representing an activity-based macro profile for a group of IoT users
with common behaviour, i.e., activity-based behaviour. A harvested-energy
estimation machine-learning model may be used to associate each profile with
an estimated energy amount based on its characteristics, using a user’s daily
records, profile, and type of harvester worn. The energy harvesting estimation
model will be investigated in the future.

##  3\. Evaluation Metrics

The performance of our robust clustering algorithm is evaluated using three
metrics: Silhouette Score ($SS$), Davies-Bouldin Index ($DBI$), and Calinski-
Habrasz Index ($CHI$)(baarsch2012investigation, ). We compare the performance
of the robust clustering algorithm with the K-means clustering applied to each
feature selection module(See step4). To calculate the metrics, each set of
features is used with the robust clustering. The results of the metrics are
shown in Table 1. The $SS$ ranges from -1 to 1, with a score closer to 1
indicating denser and more distinct clusters. The robust clustering approach
outperforms K-means clustering. The original module achieved the highest $SS$.
In contrast, a lower $DBI$ represents a better separation among clusters. The
robust clustering with all modules has a $DBI$ than k-means clustering, with
the original module having the lowest index. A higher $CHI$ indicates better
clustering performance, and the robust clustering approach with all modules
outperforms K-means clustering, with the original module achieving the highest
index.

###### Acknowledgements.

This research was partly made possible by E180100158 grant from the Australian
Research Council. The statements made herein are solely the responsibility of
the authors.

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