# The extremal Reissner-Nordström black holes: an exact charged scalar
quasiresonance.

David Senjaya<EMAIL_ADDRESS>High Energy Physics Theory Group,
Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok
10330, Thailand  Supakchai Ponglertsakul<EMAIL_ADDRESS>Strong Gravity
Group, Department of Physics, Faculty of Science, Silpakorn University, Nakhon
Pathom 73000, Thailand

###### Abstract

In this letter, we present a novel exact scalar quasibound states solutions in
the extremal Reissner-Norström black hole background. We start with the
construction of the governing covariant relativistic scalar field equation,
the Klein-Gordon equation in the extremal Reissner-Norström black hole
background and applying the separation of variables anzat. The exact
relativistic scalar wave’s angular solution is found in terms of the spherical
harmonics while the two independent radial wave solutions are, for the first
time, exactly found and presented in terms of the double confluent Heun
functions. The solutions are settled in the gravitational potential well and
behave like an ingoing waves approaching black hole’s horizon, vanishing when
approaching infinity. The gravitationally bounded charged massive scalar
fields are found to have quantized complex valued energy levels while
imaginary energy levels are obtained for the charged massless scalar field, of
both cases, indicating decaying states. Further investigation shows that the
extreme Reissner-Nordstöm black hole does not support scalar cloud. And with
the help of the obtained exact radial solutions, the Hawking radiation of the
extremal Reissner-Nordstöm black hole is investigated and we find the zero
temperature of the black hole’s horizon.

##  I Introduction

The efforts to formulate the theory of of quantum systems in a curved space-
time background goes back to the year of 1920s-1930s. The study was carried
out independently by Fock, Shcrödinger and Pauli to obtain the picture of the
quantum mechanics in a curved space [1, 2, 3]. The effect of the space-time
curvature on the Hydrogen atom has also been investigated in 1970s [4]. It was
found that the presence of the gravitational field represented by the space-
time curvature alters the energy levels of the Hydrogen’s electron.

A concrete effort to investigate the analytical solutions of the relativistic
scalar filed in a curved space-time by solving the Klein-Gordon equation for a
massive field was carried out for the first time by Rowan and Stephenson in
1977 [5]. However, due to the complexity of the Klein-Gordon’s radial
equation, the analytical solutions were obtained only for asymptotic regions,
i.e., close to the exterior of the event horizon and far away from the event
horizon. The radial asymptotic solution found by Rowan and Stephenson is given
in terms of the Hypergeometric functions that subjects to a polynomial
condition, from where, the hydrogenic-like energy levels expression is
obtained. Interestingly, scattering and bound states of scalar field on the
Schwarzschild spacetime are obtained analytically in [6]. Quasibound states
are computed explicitly in case of the Reissner-Nordström black holes [7, 8,
9] and the Kerr black hole [10]. In addition, for acoustic black holes, the
quasibound state solutions are obtained in many scenarios [11, 12, 13].
Together with the advances in the research of the family of the Heun
differential equations, we have developed a method to exactly solve the radial
equation of the Klein-Gordon equation in several black hole backgrounds. Exact
solutions of quasibound states of massless and massive neutral scalar field in
various static spherically symmetric backgrounds have successfully been
obtained in terms of either the confluent Heun or the general Heun functions
[6, 11, 12, 13, 14, 8, 15, 16].

On the otherhand, the failure of numerical methods to solve the Klein-Gordon
equation with extremal black holes background is a well known problem [17,
18]. The difficulty forces the numerical investigation stop short at the near-
extremal limit [19, 20, 21, 22, 23, 24, 25, 26]. Thus, in this paper, the
charged scalar field in an extremal Reissner-Norström black hole background
will be investigated analytically.

We will start with the discussion of properties of the Reissner-Norström
space-time in Sec II and the Klein-Gordon equation that is non-minimally
coupled to the electromagnetic vector potential is constructed in Sec III.
Using the ansatz of separation of variables, we solve the temporal and the
angular parts respectively in terms of the harmonic function and the Spherical
Harmonics. The radial part is then treated carefully and is brought to its
normal form. We find that the radial equation of the charged scalar field in
the extremal Reissner-Norström black hole background has exact solutions in
terms of the double confluent Heun functions. By applying the polynomial
conditions of the double confluent Heun function, the complex valued spectrum
of the gravito-electromagnetically bound states are found for both massive and
massless cases. This are discussed in Sec III.3.

We will also examine the analytical energy levels expression in the so-called
small black hole limit, given by the condition,

$M_{\mathrm{black\,hole}}\ll\frac{m_{\mathrm{Plank}}^{2}c^{2}}{E_{0}},$ (1)

where $m_{\mathrm{Plank}}$ is Plank mass, $E_{0}$ is the scalar particle’s
rest energy and $c$ is the speed of light. In this particular condition, the
imaginary part of the complex valued resonance frequencies are suppressed [27,
28], and we recover the expression found in [5] after setting $q=0$,

$\displaystyle\frac{E_{n}}{E_{0}}\approx 1-\frac{\mathcal{E}^{2}}{2n^{2}},$
(2) $\displaystyle\mathcal{E}={\left(\frac{E_{0}r_{s}}{\hbar c}\right)}^{2}.$
(3)

Having the exact energy expression, we investigate the presence of the charged
scalar cloud around the extremal Reissner-Norström black hole in Sec IV by
imposing the $\tilde{\omega}=qQ$ and $\tilde{\omega}_{I}=0$ conditions and
found out that the extremal Reissner-Norström black hole does not support
scalar cloud.

In Sec V, using the Damour-Ruffini method, we make use the obtained radial
exact solutions to calculate the Hawking radiation of the Extremal Reissner-
Norström black hole’s horizon. We find out that the Extremal Reissner-Norström
black hole’s horizon does not radiate and it has zero Hawking temperature. We
summarize our findings in Sec VI.

##  II The Reissner-Nordström Black Hole

It did not take long after Einstein completed his general theory of relativity
in 1915 for Karl Schwarzschild to find the first exact static spherically
symmetric black hole solution [29]. In the same year, Hans J. Reissner
generalized the Schwarzschild’s solution by including an electric charge into
the black hole solution [30]. Two years later, Gunnar Nordström independently
obtained the same solution [31].

The Reissner-Nordström (RN) space-time describes the space-time geometry
outside a static spherically symmetric charged massive body with mass $M$ and
electric charge $Q$. The space-time is described by this following line
element,

$\displaystyle
ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2},$ (4) $\displaystyle f(r)=\frac{(r-r_{-})(r-r_{+})}{r^{2}},$ (5)

where the inner ($r_{-}$) and outer ($r_{+}$) horizons are given by

$r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}.$ (6)

Generally, the RN metric has two event horizons when $M>Q$. The RN metric
develops to naked singularity as the event horizons become complex number
i.e., $M<Q$. Interestingly, the event horizons are degenerate when
gravitational mass $M$ equals electric charge $Q$ yielding the extremal black
hole. Here in this article, we focus particularly on extremal RN black hole
i.e., $r_{-}=r_{+}=M$.

##  III The Klein-Gordon Equation

The quasibound states dynamics is governed by the covariant equation
describing the behaviour of a relativistic charged scalar field in a curved
spacetime is given by the Klein-Gordon equation,

$D_{\nu}D^{\nu}\psi+\mu^{2}\psi=0,$ (7)

where,

$D_{\nu}=\nabla_{\nu}-iqA_{\nu}.$ (8)

Scalar field’s mass and charge are denoted by $\mu$ and $q$ respectively. The
electromagnetic potential is chosen to be,

$\displaystyle A_{\nu}$ $\displaystyle=\left(\frac{Q}{r},0,0,0\right)$ (9)

The scale field ansatz is given by,

$\displaystyle\psi\left(t,r,\theta,\phi\right)$ $\displaystyle=e^{-i\omega
t}R\left(r\right){Y_{l}^{m}}\left(\theta,\phi\right),$ (10)

where $Y_{l}^{m}\left(\theta,\phi\right)$ is the spherical harmonics. Under
extremal RN spacetime, the Klein-Gordon equation can be expressed as,

$\displaystyle
R^{\prime\prime}+\frac{2}{r-r_{+}}R^{\prime}+\left(\frac{r^{2}}{(r-r_{+})^{4}}\left(\omega
r+qQ\right)^{2}-\frac{\mu^{2}r^{2}+\ell(\ell+1)}{(r-r_{+})^{2}}\right)R$
$\displaystyle=0,$ (11)

where $R^{\prime}=dR/dr$ and $\ell(\ell+1)$ is the eigenvalue of spherical
harmonic function. Now, we define a new variable,

$\displaystyle x$ $\displaystyle\equiv\frac{r-r_{+}}{r_{+}}.$ (12)

Thus, the event horizon is at $x=0$ and $x\to\infty$ as one approaches
infinity. The Klien-Gordon equation becomes,

$\displaystyle\frac{d^{2}R}{dx^{2}}+p\left(x\right)\frac{dR}{dx}++q\left(x\right)R=0,$
(13)

where,

$\displaystyle p(x)$ $\displaystyle=\frac{2}{x},$ (14) $\displaystyle q(x)$
$\displaystyle=\left[\tilde{\omega}^{2}-\tilde{\mu}^{2}\right]+2\left[\tilde{\omega}\left(2\tilde{\omega}+qQ\right)-\tilde{\mu}^{2}\right]\frac{1}{x}+\left[6\tilde{\omega}^{2}+6\tilde{\omega}qQ+q^{2}Q^{2}\right.$
$\displaystyle-\ell(\ell+1)-\tilde{\mu}^{2}\left.\right]\frac{1}{x^{2}}+\left[2\left(2\tilde{\omega}+qQ\right)\left(\tilde{\omega}+qQ\right)\right]\frac{1}{x^{3}}+\left[\left(\tilde{\omega}+qQ\right)^{2}\right]\frac{1}{x^{4}}.$
(15)

We define $\tilde{\omega}=\omega r_{+}$ and $\tilde{\mu}=\mu r_{+}$. Then, we
transform the radial differential equation into the normal form by following
the method described in Appendix A. The radial equation (13) in the normal
form reads,

$\displaystyle{\partial}^{{\rm
2}}_{x}Y\left(x\right)+K\left(x\right)Y\left(x\right)$ $\displaystyle=0,$ (16)

where $Y(x)\equiv xR(x)$. Since,
$K(r)=-\frac{1}{2}\frac{dp}{dx}-\frac{1}{4}p^{2}+q$ (see Appendix A), equation
(14) implies that $K(x)=q(x)$. This equation generally describes scalar
perturbation on extremal RN background. In the following subsection, we shall
impose specific boundary condition corresponding to quasibound states.

###  III.1 The Boundary Conditions

The quasibound states are quasilocalized relativistic quantum states bound in
the black hole’s gravitation potential well. They have quasistationary energy
levels and behave like an ingoing wave close to the black hole’s horizon and
vanishing at infinity.

Let us consider the normal form of the radial equation (16). Far away from the
horizon, $x\to\infty$, the differential equation can be approximated in this
following form,

${\partial}^{{\rm
2}}_{x}Y\left(x\right)+\left[\tilde{\omega}^{2}-\tilde{\mu}^{2}\right]Y\left(x\right)=0,$
(17)

we obtain the decaying solution, we obtain a decaying solution,

$Y\left(x\right)=Ae^{-\sqrt{\tilde{\mu}^{2}-\tilde{\omega}^{2}}x},$ (18)

when ${\tilde{\omega}}<{\tilde{\mu}}$ is satisfied.

Near the horizon, i.e. $x\to 0$, the radial equation (16) is then,

${\partial}^{{\rm
2}}_{x}Y\left(x\right)+\frac{\left(\tilde{\omega}+qQ\right)^{2}}{x^{4}}Y\left(x\right)=0,$
(19)

where the solutions are obtained as follows,

$\displaystyle Y\left(x\right)$
$\displaystyle=x\left[C_{1}e^{i\left(\frac{\tilde{\omega}+qQ}{x}\right)}+C_{2}e^{-i\left(\frac{\tilde{\omega}+qQ}{x}\right)}\right],$
(20)

where $C_{1},C_{2}$ are integration constants. Now, we define
$k=\tilde{\omega}+qQ$ and redefining the radial variable
$\tilde{x}=\frac{1}{x}$. We can set $C_{1}=0$ to obtain ingoing wave solution.

###  III.2 The Radial Exact Solution

By comparing (14) with the normal form of double confluent Heun differential
equation (65) in Appendix A. One can identity the following,

$\displaystyle\epsilon$
$\displaystyle=2\sqrt{\tilde{\mu}^{2}-\tilde{\omega}^{2}},$ (21)
$\displaystyle\gamma$ $\displaystyle=2i(\tilde{\omega}+qQ),$ (22)
$\displaystyle\delta$ $\displaystyle=2+2iqQ+4i{\tilde{\omega}},$ (23)
$\displaystyle\alpha$
$\displaystyle=2\left(\frac{\epsilon}{2}-{\tilde{\mu}}^{2}+\left(2{\tilde{\omega}}+qQ\right)\left({\tilde{\omega}}+\frac{i\epsilon}{2}\right)\right),$
(24) $\displaystyle\beta$
$\displaystyle={\tilde{\mu}}^{2}+\ell(\ell+1)-iqQ\left(1+\epsilon-2i{\tilde{\omega}}\right)-2{\tilde{\omega}}\left({\tilde{\omega}}+i\left(1+\frac{\epsilon}{2}\right)\right).$
(25)

Here, we have successfully solved all of the double confluent Heun’s
parameters. Thus, the novel exact solutions of the Klein-Gordon equation in
the extremal Reissner-Nordström background can be written as follows,

$\psi=\psi_{0}e^{-i\omega
t}Y^{m_{l}}_{l}\left(\theta,\phi\right)\left[Ax^{\frac{1}{2}(\delta-2)}e^{\frac{1}{2}\left(\epsilon
x-\frac{\gamma}{x}\right)}\operatorname{HeunD}\left(\beta,\alpha,\gamma,\delta,\epsilon,x\right)\right.\\\
\left.+Bx^{-\frac{1}{2}(\delta-2)}e^{-\frac{1}{2}\left(\epsilon
x-\frac{\gamma}{x}\right)}\operatorname{HeunD}\left(-2+\beta+\delta,\alpha-2\epsilon,-\gamma,4-\delta,-\epsilon,x\right)\right],$
(26)

where HuenD is the double confluent Huen function.

###  III.3 The Scalar’s Energy Quantization

Now, let us apply the polynomial condition of the double confluent Heun as
given in (66). The polynomialization of the radial solution is important since
any infinite sequence of polynomials $p_{n}$, where $p_{n}$ having degree $n$
forms a basis for the infinite dimensional vector space of all polynomials.
The set of the polynomial functions can be constructed as orthogonal basis by
using the Gram-Schmidt orthogonalization. The polynomial condition terminates
the infinite series of the radial solution at $n_{r}^{th}$ order and leads to
quantization of relativistic scalar’s energy as follows,

$\frac{4\tilde{\omega}^{2}-2\tilde{\mu}^{2}+2\tilde{\omega}qQ}{2\sqrt{\tilde{\mu}^{2}-\tilde{\omega}^{2}}}+\left(1-\frac{4\tilde{\omega}^{2}+2q^{2}Q^{2}+6\tilde{\omega}qQ}{2i(\tilde{\omega}+qQ)}\right)=-n_{r},$
(27)

where $n_{r}$ is the principal quantum number that can be connected with the
angular momentum quantum number $\ell$ and the excitation number $N$ as
$n_{r}=N+\ell$.

In addition, we investigate the energy quantization in the small black hole
limit. Taking $\tilde{\omega}=\omega r_{+}\ll 1$, $qQ\ll 1$ and
$\frac{\tilde{\omega}}{\tilde{\mu}}\approx 1$, for massive charged scalar
field, a Hydrogenic energy levels expression is obtained,

$\displaystyle\tilde{\omega}{\rm=}{\tilde{\mu}}\sqrt{1-\frac{(\tilde{\mu}+qQ)^{2}}{{\left(n_{r}+1\right)}^{2}}},$
(28)

.

In the limit where
$\frac{\left({\tilde{\mu}}+qQ\right)^{2}}{\left(n_{r}+1\right)^{2}}\ll 1$, one
can expand (28) as a Taylor Series as follows,

$\tilde{\omega}=\tilde{\mu}\left[1-\frac{(\tilde{\mu}+qQ)^{2}}{2{\left(n_{r}+1\right)}^{2}}-\frac{(\tilde{\mu}+qQ)^{4}}{8{\left(n_{r}+1\right)}^{4}}-\frac{(\tilde{\mu}+qQ)^{6}}{16{\left(n_{r}+1\right)}^{6}}+\dots\right].$
(29)

If $q$ is set to be zero, we recover the well-known energy levels of the so
called gravitational atom,

$\displaystyle\frac{\omega_{n}}{\mu}\approx
1-\frac{\kappa^{2}}{2(n_{r}+1)^{2}},\kappa={\mu M}^{2},$ (30)

that can be found in [32, 33, 34, 35, 36]. Analogous to the electronic
transition that emits photon, graviton is emitted as level transition occurs
in a gravitational atom [37, 38].

In the case of massless field $({\tilde{\mu}}=0)$, energy level can be solved
as follows,

$\displaystyle\tilde{\omega}=-\frac{qQ}{2}+\frac{\left(n_{r}+1\right)}{4}i.$
(31)

Clearly, the interaction between black hole and field dictates the real part
of quasibound state frequency while the imaginary part solely depends on the
principal quantum number $n_{r}$. The positive real part requires that $qQ<0$,
therefore, there is no superradiance in this case [39]. In addition, it is
shown in [40] that extremal Reissner-Nordström black hole does not suffer from
superradiant instability. In the absence of scalar field charge ($q=0$), this
formula agrees with those obtained in [8].

Figure 1: Quasibound state’s frequencies with ${\tilde{\mu}}=0.1$ where
$n_{r}=0-10$ increasing from bottom to top.

Figure 2: Quasibound state’s frequencies as a function of ${\tilde{\mu}}$ for
fixed $qQ=-0.1$.

Figure 3: Quasibound state’s frequencies as a function of $-qQ$ with
$n_{r}=0$. The colored curves are denoted by the legend in the right plot.

In Fig. 1, we plot quantized quasibound states frequencies in a complex
$\tilde{\omega}$ plane for a scalar field with mass $\tilde{\mu}=0.1$ in the
extremal Reissner-Nordström black hole background where the charge
configurations are varied as $qQ=-0.01,-0.03,-0.05$. We present the quasibound
states frequencies for $n_{r}=0,1,...,10$ where the lowest $n_{r}$ is always
associated with lowest value of $Im(\tilde{\omega})$. The leap of the
imaginary part of the quasibound states frequencies, $\Delta
Im(\tilde{\omega})$, seems steady while the real part of the quasibound states
frequencies significantly increase for lower $n_{r}$ and slightly increase for
higher $n_{r}$, similar to the hydrogenic atom’s electronic level transitions.

In Fig. 2, we separately present the real and imaginary quasibound states
frequencies for a fixed charge configuration $qQ=-0.1$. We solve the
$\tilde{\omega}$ for various value of $\tilde{\mu}$ with $n_{r}=0,1,2,3$. It
is important to mention that the imaginary part of the quasibound states
frequencies runs from positive, zero, and finally entering negative zone. The
zero imaginary part of the frequencies indicates the existence of a scalar
cloud which will be elaborated more in the following section.

In Fig. 3, we present visualizations of the real and imaginary part of the
quasibound states frequencies for various $qQ-\tilde{\mu}$ configuration. The
$Re(\tilde{\omega})~{}\text{vs}~{}-qQ$ graph shows us that larger value of
$-qQ$ causes larger drop of $Re(\tilde{\omega})$. Classically, this is
understood as the adding up of the attractive electrostatic potential to the
gravitational potential enhancing the binding energy of the scalar fields to
the central black hole. The $Im(\tilde{\omega})~{}\text{vs}~{}-qQ$ graph also
shows us that the quasibound states are stable since all values of the
$Im(\tilde{\omega})$ are negative indicating decaying states. Moreover, we
also notice that the value of $Im(\tilde{\omega})$ increases in magnitude
following the increase in magnitude of $qQ$. This indicates that the power of
the extremal charged black hole in absorbing scalar particles is increased
with the help of the attractive electrostatic interaction. Also notice that
for each value of $qQ$, larger $\tilde{\mu}$ leads to more rapid decay.

##  IV The Charged Scalar Cloud

In this section, we will investigate the existence of the scalar cloud around
the extremal Reissner-Nordström black hole. The scalar cloud is an stable
configurations of a charged scalar field stationarily bound in a black hole
background. The scalar cloud have a real valued frequency that matches the
$qQ$ configuration as follows [41, 42, 43],

$\displaystyle\tilde{\omega}=qQ,$ (32) $\displaystyle\tilde{\omega}_{I}=0.$
(33)

The first scalar cloud condition, classically, is understood as the
equilibrium between gravitational-electrostatic force balance. The second
energy condition indicates that there is no ingoing scalar flux impenetrates
the outer horizon. Therefore, they do not grow or decay over time.

Applying the first condition to the exact energy expression (27), we obtain,

$\frac{6\tilde{\omega}^{2}-2\tilde{\mu}^{2}}{2\sqrt{\tilde{\mu}^{2}-\tilde{\omega}^{2}}}+\left(1+3i\tilde{\omega}\right)=-n_{r}.$
(34)

where the analytical solution must be complex valued, thus, violates the
second condition of the scalar cloud. Now, let us check whether there exists a
massless charged scalar cloud around the extremal Reissner-Nordström black
hole by setting $\tilde{\mu}=0$ in the equation (34), here we obtain,

$\tilde{\omega}=i\frac{(n_{r}+1)}{6}.$ (35)

Again, the solution is purely imaginary, violating the second criterion of the
scalar cloud. Therefore, there is no solution for $\tilde{\mu}$ that satisfies
the energy condition (33). Thus, the charge scalar cloud surrounding an
extreme Reissner-Nordström black hole does not exist. The same conclusion is
obtained via asymptotical method and can be found in [41].

However, it is possible to find a condition for a long-lived massive scalar
cloud by taking $\tilde{\omega}=\omega r_{+}\ll 1$ and
$\frac{\tilde{\omega}}{\tilde{\mu}}\approx 1$ limit of the (34) and obtain a
real valued energy levels expression as follows,

$\tilde{\omega}=\tilde{\mu}\sqrt{1-\frac{4\tilde{\mu}^{2}}{(n_{r}+1)^{2}}}.$
(36)

Despite the extremal Reissner-Nordström black hole does not support scalar
cloud. But, it is possible to find scalar configuration that minimizes the
decay.

##  V The Hawking Radiation

In this section, the Hawking radiation coming out of the extremal Reissner-
Nordström black hole’s horizon will be investigated. With the exact solution
of the radial wave function in hand, we can apply the Damour-Ruffini method
[44, 45] in order to derive the radiation distribution function without having
to go through the black hole’s thermodynamics. Let us start with rewriting the
complete radial solution of the Klein-Gordon equation as follows,

$R=Ax^{\frac{1}{2}(\delta-2)}e^{\frac{1}{2}\left(\epsilon
x-\frac{\gamma}{x}\right)}\operatorname{HeunD}\left(\beta,\alpha,\gamma,\delta,\epsilon,x\right)\\\
+Bx^{-\frac{1}{2}(\delta-2)}e^{-\frac{1}{2}\left(\epsilon
x-\frac{\gamma}{x}\right)}\operatorname{HeunD}\left(-2+\beta+\delta,\alpha-2\epsilon,-\gamma,4-\delta,-\epsilon,x\right).$
(37)

Approaching the horizon $r_{+}$ i.e.,$x\to 0$, the wave function can be
expanded up to the first order of $x$, where,

$\displaystyle
e^{\frac{1}{2}\left(-\frac{\gamma}{x}\right)}\operatorname{HeunD}\left(\beta,\alpha,\gamma,\delta,\epsilon,x\to
0\right)$ $\displaystyle\equiv\phi_{1},$ (38) $\displaystyle
e^{\frac{1}{2}\left(\frac{\gamma}{x}\right)}\operatorname{HeunD}\left(-2+\beta+\delta,\alpha-2\epsilon,-\gamma,4-\delta,-\epsilon,x\to
0\right)$ $\displaystyle\equiv\phi_{2}.$ (39)

Notice that, the $\operatorname{HeunD}(x\to 0)\to\infty$ regardless the value
of the parameter. Therefore, $\phi_{1}$ can be considerably finite while
$\phi_{2}$ diverges as one moves closer to the horizon. Therefore, in the near
horizon limit, the radial wave can be rewritten in this following form,

$R=A\phi_{1}x^{\frac{1}{2}(\delta-2)}+B\phi_{2}x^{-\frac{1}{2}(\delta-2)}.$
(40)

The scalar’s radial wave consists of two parts, i.e. $\psi_{+in}$ is the
ingoing wave and $\psi_{+out}$ is the outgoing wave as follows,

$\displaystyle
R=\left\\{\begin{array}[]{cc}\psi_{+in}=B\phi_{2}x^{-\frac{1}{2}(\delta-2)}\\\
\psi_{+out}=A\phi_{1}x^{\frac{1}{2}(\delta-2)}\end{array}\right.,$ (43)

where after defining $\kappa=\frac{1}{2}(\delta-2)$, the combination of the
radial and temporal solution leads to this following simple wave equation,

$\displaystyle
R=\left\\{\begin{array}[]{cc}\psi_{+in}=B\phi_{2}e^{-i\left(\omega
t+\kappa\ln{x}\right)}\\\ \psi_{+out}=A\phi_{1}e^{-i\left(\omega
t-\kappa\ln{x}\right)}\end{array}\right.,$ (46)

Furthermore, we can investigate the ratio $\frac{\phi_{2}}{\phi_{1}}$ as
follows,

$\displaystyle\frac{\phi_{2}}{\phi_{1}}=\frac{\operatorname{HeunD}\left(-2+\beta+\delta,\alpha-2\epsilon,-\gamma,4-\delta,-\epsilon,x\to
0\right)}{\operatorname{HeunD}\left(\beta,\alpha,\gamma,\delta,\epsilon,x\to
0\right)}e^{\left(\frac{\gamma}{x}\right)},$ (47)

and as the $\operatorname{HeunD}$ are series solutions, the behaviour of the
quantity $\frac{\phi_{2}}{\phi_{1}}$ is mostly determind by the exponential
term. So, in the limit $x\to 0$, we obtain,

$\lim_{x\to 0}\frac{\phi_{2}}{\phi_{1}}=\infty.$ (48)

Now, let us apply the Damour-Ruffini method as the following. Suppose we have
an ingoing wave hitting the apparent horizon at $r_{+}$ and inducing a
particle-antiparticle pair where the particle will enhance the reflected wave
and the antiparticle will become the transmitted wave. Analytical continuation
of the wave function $\psi\left(x\right)$ describes unique antiparticle state
for a particular particle state and it is defined as
$\psi_{-out}\equiv\psi_{+out}\left(x\to xe^{-i\pi}\right)$ is obtained simply
by changing $x\to-x=xe^{-i\pi}$ as follows,

$\begin{split}\psi_{-out}&=A\phi_{1}\left(xe^{-i\pi}\right)^{\frac{1}{2}(\delta-2)},\\\
&=\psi_{+out}e^{-\frac{1}{2}i\pi(\delta-2)}.\end{split}$ (49)

We can therefore, consider the following ratio,

$\displaystyle{\left\lvert\frac{\psi_{-out}}{\psi_{+in}}\right\rvert}^{2}={\left\lvert\frac{\psi_{+out}}{\psi_{+in}}\right\rvert}^{2}e^{-2i\pi(\delta-2)},$
(50)
$\displaystyle={\left\lvert\frac{\psi_{+out}}{\psi_{+in}}\right\rvert}^{2}e^{{4\pi}\left[2\tilde{\omega}+qQ\right]},$

where we substitute $\delta$ as in (23). Using the fact that total probability
of the particle wave going out from the horizon and the antiparticle wave
going in must be equal to 1, we obtain this following distribution function,

$\displaystyle\left\langle\frac{\psi_{out}}{\psi_{in}}\mathrel{\left\lvert\vphantom{\frac{\psi_{out}}{\psi_{in}}\frac{\psi_{out}}{\psi_{in}}}\right.\kern-1.2pt}\frac{\psi_{out}}{\psi_{in}}\right\rangle=1={\left\lvert\frac{A\phi_{1}}{B\phi_{2}}\right\rvert}^{2}\left\lvert
1-e^{{4\pi}\left[2\tilde{\omega}+qQ\right]}\right\rvert,$ (51)

and let us write the black hole’s apparent horizon’s radiation distribution
function,

$\left\lvert\frac{B}{A}\right\rvert^{2}=\zeta(T_{H})=\frac{1}{e^{\frac{\hbar\omega}{k_{B}T_{H}}}-1},$
(52)

where $k_{B}$ is the Boltzmann constant. The equation (51) in the near horizon
limit, now can be rewritten as follows [45],

$\zeta(T_{H})=\left(\lim_{x\to
0}\frac{\phi_{1}}{\phi_{2}}\right)\times\left\lvert
1-e^{{4\pi}\left[2\tilde{\omega}+qQ\right]}\right\rvert=0.$ (53)

By comparing (52) and (53), we can conclude that the extremal Reissner-
Nordström black black hole’s horizon has a zero Hawking temperature, thus,
does not radiate. This is in contrast with the non-extremal Reissner-Nordström
black hole where the apparent horizon’s radiation distribution function is
obtained as [8],

$\zeta(T_{H})=\left\lvert\frac{B}{A}\right\rvert^{2}=\frac{1}{e^{\frac{4\pi}{\delta_{r}}\left[\frac{E}{\hbar
c}r^{2}_{+}\right]}-1}=\frac{1}{e^{\frac{\hbar\omega}{k_{B}T_{H}}}-1},$ (54)

leads to a non-zero Hawking temperature,

$\displaystyle T_{H}=\frac{\delta_{r}c\hbar}{4\pi k_{B}r^{2}_{+}},$ (55)

where $\delta_{r}=r_{+}-r_{-}$ is as defined in equation (6). One can also set
$Q=M$ and directly substitute the expression into the Hawking temperature
formula of the non-extremal Reissner-Nordström’s above to get the same
conclusion that the extremal Reissner-Nordström black black hole’s horizon has
a zero Hawking temperature.

##  VI Summary

In this work, we explore the quasibound spectrum of charged relativistic
bosonic quantum states of the massive and massless fields in an extremal
Reissner-Nordström black hole space-time. We start with the Klein-Gordon
equation with extremal Reissner-Nordström black hole background. Due to the
static spherically symmetry, the temporal and the angular parts of the
relativistic wave solution are obtained respectively in terms of the harmonic
function and the Spherical Harmonics. We carefully deal with the radial
equation and successfully find the exact solutions in terms of the double
confluent Heun functions

The quantization of the radial solution is done by applying the polynomial
condition of the double confluent Heun function. The relation between the
double confluent Heun’s parameters turns out to be an energy quantization
rule, presented in (27). We also do a further investigation for the $\omega
r_{+}\to 0$ limit and obtain the Hydrogenic atom’s energy level (28). Similar
expressions are also obtain via asymptotic methods, can be found in [46, 47].
We also obtain exact energy level for for massless charged scalar field around
the extremal Reissner-Nordström black hole. We find that imaginary part of the
quasibound state can either be positive, zero and negative while the real part
can only be positive if $qQ<0$. Increasing scalar mass ${\tilde{\mu}}$ leads
to a more rapid decaying modes where the real part is increasing with
${\tilde{\mu}}$. We can see that larger magnitude of $-qQ$ increases the
combined attractive potential leads to enhancement of both the scalar fields’
binding energy, $Re(\tilde{\omega})$ and the decay, which is represented by
the magnitude of $-Im(\tilde{\omega})$.

Using the exact quantized energy expression (27), we search for a possible
charged scalar cloud around an extremal Reissner-Norström black hole. We found
that there is no possible charged scalar cloud configuration for the extremal
Reissner-Norström black hole. The same conclusion is also found in [41]. We
have also investigated the Hawking radiation of the extremal Reissner-Norström
black hole’s horizon via the Damour-Ruffini method. We find that the extremal
static spherically symmetric charged black hole’s horizon has zero
temperature, thus, does not radiate. The same result is also obtained via
black hole thermodynamics analysis in [48].

## Acknowledgments

David Senjaya acknowledges this research project is supported by the Second
Century Fund (C2F), Chulalongkorn University. SP acknowledges funding support
from the NSRF via the Program Management Unit for Human Resources &
Institutional Development, Research and Innovation [grant number B39G670016].

##  Appendix A The Normal Form and double confluent Heun differential equation

In this appendix, we discuss a useful transformation of a linear second order
differential equation [49]. Let us first consider a general second order
differential equation in the following form

$\displaystyle\frac{d^{2}y}{dx^{2}}+{\tilde{p}}(x)\frac{dy}{dx}+{\tilde{q}}(x)y=0.$
(56)

Now, we assume that the solution can be written as

$\displaystyle y=Y(x)e^{-\frac{1}{2}\int{{\tilde{p}}(x)}dx}.$ (57)

If one assumes that $e^{-\frac{1}{2}\int{{\tilde{p}}(x)}dx}\neq 0$, then (56)
can be recast into the normal form

$\displaystyle\frac{d^{2}Y}{dx^{2}}+\left(-\frac{1}{2}\frac{d{\tilde{p}}}{dx}-\frac{1}{4}{\tilde{p}}^{2}+{\tilde{q}}\right)Y=0.$
(58)

Now, Let us consider the double confluent Heun differential equation [50],

$\displaystyle\frac{d^{2}y}{dx^{2}}+\left(\frac{\gamma}{x^{2}}+\frac{\delta}{x}+\epsilon\right)\frac{dy}{dx}+\left(\frac{\alpha}{x}-\frac{\beta}{x^{2}}\right)y$
$\displaystyle=0.$ (59)

This equation has two irregular singular points at $x=0$ and $x=\infty$. A
general solution of this equation is [50]

$\displaystyle y$
$\displaystyle=A\operatorname{HeunD}\left(\beta,\alpha,\gamma,\delta,\epsilon,x\right)+Be^{\frac{\gamma}{x}-\epsilon
x}x^{2-\delta}\operatorname{HeunD}\left(-2+\beta+\delta,\alpha-2\epsilon,-\gamma,4-\delta,-\epsilon,x\right)$
$\displaystyle\equiv\mathcal{H}(x),$ (60)

where $\operatorname{HeunD}$ is doubly confluent Heun function. One puts the
double confluent Heun differential equation in its normal form by recognizing

$\displaystyle{\tilde{p}}$
$\displaystyle=\frac{\gamma}{x^{2}}+\frac{\delta}{x}+\epsilon,$ (61)
$\displaystyle{\tilde{q}}$
$\displaystyle=\frac{\alpha}{x}-\frac{\beta}{x^{2}},$ (62) $\displaystyle y$
$\displaystyle=Y(x)e^{-\frac{1}{2}\left(\epsilon
x-\frac{\gamma}{x}\right)}x^{-\frac{\delta}{2}}\equiv\mathcal{H}(x),$ (63)

Therefore, the normal form of (59) is obtained

$\displaystyle\frac{d^{2}Y}{dx^{2}}+K(x)Y$ $\displaystyle=0,$ (64)

where

$\displaystyle K(x)$
$\displaystyle=-\frac{1}{2}\frac{d{\tilde{p}}}{dx}-\frac{1}{4}{\tilde{p}}^{2}+{\tilde{q}},$
$\displaystyle=-\frac{\epsilon^{2}}{4}+\frac{1}{x}\left(-\frac{\epsilon\delta}{2}+\alpha\right)+\frac{1}{x^{2}}\left(\frac{\delta}{2}-\frac{\delta^{2}+2\epsilon\gamma}{4}-\beta\right)$
$\displaystyle~{}~{}~{}~{}~{}+\frac{1}{x^{3}}\left(\gamma-\frac{\delta\gamma}{2}\right)+\frac{1}{x^{4}}\left(-\frac{\gamma^{2}}{4}\right).$
(65)

Remark that the double confluent Heun function becomes polynomial when the
following condition is met

$\displaystyle\frac{\alpha}{\epsilon}$ $\displaystyle=-n_{r},\quad
n_{r}=0,1,2,\dots.$ (66)

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# Human-Computer Interaction Glow Up: Examining Operational Trust and
Intention Towards Mars Autonomous Systems

Thomas Chan111Assistant Professor, Positive Augmented Research and
Development, California State University Northridge Autonomy Research Center
for STEAHM, California State University Northridge  Jeremy Argueta  Autonomy
Research Center for STEAHM, California State University Northridge  Jazlyn
Armendariz  Autonomy Research Center for STEAHM, California State University
Northridge  Allison Graham  Autonomy Research Center for STEAHM, California
State University Northridge  Sarah Hwang  Autonomy Research Center for STEAHM,
California State University Northridge  Basak Ramaswamy222Product Owner,
Mission Planning, Sequencing and Analysis, Jet Propulsion Laboratory,
California Institute of Technology Jet Propulsion Laboratory, California
Institute of Technology  So Young Kim333Manager, Human Systems Integration,
Jet Propulsion Laboratory, California Institute of Technology Jet Propulsion
Laboratory, California Institute of Technology  & Scott Davidoff444Manager,
Human-Centered Design, Jet Propulsion Laboratory, California Institute of
Technology Jet Propulsion Laboratory, California Institute of Technology

###### Abstract

Tactful coordination on earth between hundreds of operators from diverse
disciplines and backgrounds is needed to ensure that Martian rovers have a
high likelihood of achieving their science goals while enduring the harsh
environment of the red planet. The operations team includes many individuals,
each with independent and overlapping objectives, working to decide what to
execute on the Mars surface during the next planning period. The team must
work together to understand each other’s objectives and constraints within a
fixed time period, often requiring frequent revision.

This study examines the challenges faced during Mars surface operations, from
high-level science objectives to formulating a valid, safe, and optimal
activity plan that is ready to be radiated to the rover. Through this
examination, we aim to illuminate how planning intent can be formulated and
effectively communicated to future spacecrafts that will become more and more
autonomous.

Our findings reveal the intricate nature of human-to-human interactions that
require a large array of soft skills and core competencies to communicate
concurrently with science and engineering teams during plan formulation.
Additionally, our findings exposed significant challenges in eliciting
planning intent from operators, which will intensify in the future, as
operators on the ground asynchronously co-operate the rover with the on board
autonomy. Building a marvellous robot and landing it onto the Mars surface are
remarkable feats – however, ensuring that scientists can get the best out of
the mission is an ongoing challenge and will not cease to be a difficult task
with increased autonomy.

##  1 Introduction

Operating a rover on Mars to advance our knowledge about the universe’s
formation and future requires deep expertise in many scientific and
engineering domains. Scientists who study the geology and atmosphere of Mars
must negotiate with scientists seeking traces of ancient life to decide the
most valuable data we can collect at any point[1, 2].

In addition, the scientists have to work with instrument engineers who know
how to utilize onboard instruments best to collect the data as efficiently and
accurately as possible. All of these instruments have to negotiate the
resource usage checking in with vehicle system engineers. Vehicle system
engineers, which are further divided into sub-systems, are responsible for
enabling observations, returning data to earth, and maintaining the overall
safety of the rover. In the end, a fully detailed and validated activity plan
is uplinked to the rover to execute. Under normal conditions, execution only
minimally deviates from the plan[1, 2].

Future of space operations will involve more autonomous spacecraft where some
of the decisions currently made on the ground will be delegated to the flight
system. The autonomous flight system has to work with the operators on the
ground to achieve their intent, which requires operators to communicate their
intent to the flight system. The key motivation for this study was to observe
– how human operators communicate with each other during the planning process
to inform how planning intent could be formulated for the more autonomous
spacecrafts of the future.

We conducted a series of semi-structured interviews with science and
engineering operations representatives from two Mars surface missions to
answer this question. We focused on how operators exchange information
regarding available resources, constraints, and planning intent, how input
from one team was translated into actions across the other team. We analyzed
the interview recordings by delving into how decisions were made, what kind of
language was used, and what range of soft skills were crucial. An atomic
research model [3] a common technique in user experience research, was
utilized to generate and test emergent mechanistic themes linked to the
planning and negotiation process. Results found that three latent themes were
most commonly linked to foster mission coordination:

1\. Corporate Knowledge Gluers – people who fill in any mission-relevant
knowledge gaps to update rotating or missing personnel; 2\. Bridge Builders –
people who are conduits to foster collective understanding between science and
engineering teams; 3\. Efficiency Optimizers – people who help streamline the
tasks and attention of teams to meet mission deadlines.

Not only were these skills necessary for smooth and efficient planning
operations, these three main latent themes were associated with specific
operations roles which were created months after the operations began, as the
need for better coordination and communication became apparent. Furthermore,
we observed that planning intent is gradually formulated through many
iterations and interactions between science and engineering teams.

The end goal of this work is to inform the design of autonomous planning
systems which can allow users to express their intent seamlessly and co-
operate a rover with trust. Our study provides a human centered approach to
this collaboration problem, in contrast with research in autonomy domain,
which focuses on how to ensure spacecraft can achieve intent once it is
perfectly formulated.

In the next section we give an overview of Mars surface operations to provide
readers some context about our study. In section III, we describe the study
protocol and data analysis methods employed. Results section provides a more
granular description of socio mechanical themes summarized above. In the
following section we discuss additional observations and several areas of
further investigation to enable a happy collaboration between humans and
autonomy.

##  2 Mars Surface Operations in a Nutshell

In this section we provide a high level overview of Mars surface operations.
Note that operations are very intricate, covers a long duration, includes
multiple processes with different life cycles and involves many roles.
Operators need to train for months and shadow senior operators to be able to
partake. Our description below is a brief summary to help readers interpret
our study results in context.

The goal of any space mission is to achieve high level science goals that are
conceptualized years before operations begin. Scientists start with strategic
plans development by setting deadlines for milestones to meet in the next 2 to
3 month time frame. The strategic plans are produced taking into account
overall capabilities of the rover, such as how far it can drive in a single
day, or how many observations can be conducted in average in a day.

A long term plan is derived from the strategic plan, covering the next 3 to 5
sols, looking ahead from the current sol. When the long term plan is
developed, scientists have more context to decide on specific science
observations, and a more accurate estimate of resources that will be available
[4].

Only during tactical planning an exact activity plan is formulated. The
process starts with the n+1 portion of the long term plan and the latest data
received from the spacecraft describing its most recent status. At the end of
the process, every activity has to be defined precisely and translated into
commands to be executed in the next plan cycle on Mars, which is often covers
one sol (Mars solar day) [5].

Broadly speaking, during the tactical process, scientists have to decide which
exact observations to perform, while engineering teams inform the scientists
about which observations are feasible and how many of them can be accommodated
by running various simulations. High level abstractions of activities (e.g.
"Take a picture of that rock from this angle") and simulation results are
displayed on visual timelines [6] for the team to evaluate options. The
operations team need to come to a consensus of what science activities are of
priority for the next day given various constraints - environmental,
engineering, and science at the end of the process.

Once an activity plan that is safe for the rover, and one that can achieve
optimal science is produced, engineering teams translate the plan into
commands that the spacecraft understands. The command products are then
uplinked to the spacecraft to execute onboard for the next planning cycle.

##  3 Method

###  3.1 Procedures

A total of 10.0019 hours of semi-structured interviews were conducted to
address our targeted inquiries about the Mars surface missions’ planning and
negotiation processes. The interview questions aimed to elicit both broad and
detailed responses regarding planning and negotiations while offering the
researchers opportunities to get clarification and gain context for each
response. Questions were piloted with operations personnel and tailored to
match commensurate Mars missions experience. Specifically, the decision tree
of questions corresponded with the interviewees’ experience on Mars Science
Laboratory-Curiosity Rover (MSL; for details visit
https://mars.nasa.gov/msl/home/) and/or Mars2020-Perseverance Rover (M2020;
for details visit https://mars.nasa.gov/mars2020/). Recruitment was conducted
through email invitations. Invitations were sent to identified vital personnel
currently active on either MSL or/and M2020 missions. Interviews were
performed virtually using WebEx (Cisco Systems, Milpitas, CA) and lasted
between 45-90 minutes, and were transcribed.

###  3.2 Participants

A total of ten semi-structured interviews were conducted with Mars surface
mission operators. The roles of participants spanned multiple scientific and
engineering capacities. Transparently, although official roles (i.e., titles)
were initially targeted to gain the spectrum of planning and negotiating
experiences – most participants noted that they performed multiple roles at
different time frames or in parallel. Additionally, their roles were modified,
expanded, or promoted to meet emergent mission needs. For instance, Project
Scientists also stepped in as Science Operations Coordinators/Science
Engineering Liaisons when needed. Similarly, few Science Planners that we
interviewed later performed the Tactical Uplink Lead role. Consequently, with
much overlap, seven out of ten participants had coordination roles, five out
of ten participants focused on science objectives, and five out of ten
participants focused on engineering tasks. Nine interviewees were from the JPL
organization, while one was outside of JPL. Four participants worked only on
the MSL mission, while six were on both MSL and M2020 missions. Seven of the
ten participants were classified as "veterans" on Mars missions due to their
extensive work history. The other three participants were considered in the
"early stages" of their mission operations career.

###  3.3 Analysis Plan

Interviews were transcribed, reviewed by each team member, and then moments
extracted and coded by team members into an Atomic Research[7, 3] repository.
The coding of key observations was guided by grounded theory—an inductive
approach that utilizes constant comparative methods to establish analytic
distinctions across multiple iterations [8]. Emergent themes were developed
from interrelated categories to better characterize and connect multiple
observations. Following the process of atomic user research, facts from the
observations were anonymized and codified from video records in an Airtable
repository [9]. Emergent patterns were rigorously tested for their
generalizability (i.e., applied to other interviewees’ experiences). If
generalizability was achieved, more than three common experiences, then they
were labeled as insights and aggregated into the higher-level categories that
we report on in our findings section below.

##  4 Results

Our analyses of semi-structured interviews yielded three emergent latent
variable themes with 14 specific functions, as shown in Fig. 1. Latent themes
are the overarching [unobserved] constructs derived from a composite of the
observed measures – in our case, the actual functions performed [10].
Functions were the observed units generated from systematic analyses and honed
operationalizations by the current study’s researchers. Again, latent themes
and functions were only identified if they were applied across at least three
negotiation and planning mission contexts.

Figure 1: Overview of the Latent Themes

###  4.1 Latent Theme I - Corporate Knowledge Gluer - 28.09% of all total
quotes

This first latent theme illustrates the critical role of Mars mission people
who fill in any mission-relevant knowledge gaps to update rotating or missing
personnel. Corporate Knowledge Gluers perform functions that focus on
maintaining the continuity of mission awareness – reducing gaps in critical
knowledge, and provide updates to lessen learning curves. An exemplar quote
that highlights this high-level role is "…the main part of that role is just
coordinating and keeping everybody up to date about what everybody else is
doing…each role can get very focused on their little piece of the
puzzle…basically that person putting all the pieces together and making sure
they all fit." -P2Q19. The continuity Corporate Knowledge Gluers provide are
highlighted even more during off-nominal conditions. They "get everyone up to
speed," and need to do it quickly to ensure that contingency plans are
negotiated in time to make adjustments. The specific functions that make up
the Corporate Knowledge Gluer composite:

  1. 1.

Information Clarifier - Strengthens understanding of objectives and clears up
any misunderstandings during negotiations. This function performed by
Corporate Knowledge Gluers targets ensuring that current or changing
objectives are understood by science, engineering, and operations teams. The
people from this function draws upon what has happened during the mission and
explains the current course. An exemplary quote that illustrates this
function, "You might hear two requests that sounded slightly
contradictory…bringing those forward to the rest of the team…and clarifying
anything that was unclear" -P1Q2.

  2. 2.

Cocktail Host - Acts as the day to day point of contact to maintain mission
structure. This function targets the stable presence of the "go-to person" who
knows Sol’s current course and objectives. Cocktail Hosts also direct any
person who needs more detailed updates to connect. An exemplary quote that
showcases this function, "“That person who’s…listening to every
conversation…efficiently make sure that the right people are hearing the right
information and understand the constraints and the intent of why we’re doing
things" -P5Q4.

  3. 3.

Breaking News Reporter - Broadcasts updates when plans go off-nominal. This
function performed focuses on gather and disseminating critical updates as
mission objectives go off course. Breaking News Reporters have their eyes and
ears on all aspects of the day’s current plans and use their daily connections
with all teams to get and transmit updates in a speedy matter. An exemplary
quote that highlights this function, "If there’s some late-breaking knowledge
of vehicle state…late-breaking news is, "Oh, hey, we can’t do that…we don’t
have enough time…let’s just go on to the thing that we want to do
strategically" -P10Q17.

  4. 4.

Risk-Reward Assessor - Explains the costs associated with a given option. This
function performed targets transmitting mission knowledge of the risks
associated with given decisions. Risk-Reward Assessors do not make specific
decisions, but they process the costs and benefits of each potential decision
at the highest levels. An exemplary quote that illustrates this function, "We
think about what’s really most important… what’s worth the cost that the
engineering folks have now placed on the site from a resource or time
perspective" -P4Q8.

###  4.2 Latent Theme II - Bridge Builder - 40.45% of all total quotes

This second latent theme illustrates the vital role of Mars mission persons in
being conduits to foster collective understanding between science and
engineering. Bridge Builders use "soft skills" such as translating both verbal
and nonverbal languages to fill in gaps for a diverse set of science and
engineering teams. The exemplary quote that of this high-level theme is,"In
the morning, I work with the scientists and understand what their goals for
the day are…I then transfer and keep communicating with the various
engineering roles…of facilitating for that to happen" -P2Q17. Science and
engineers, in general, have different priorities, terminologies, and focuses –
these Bridge Builders offer ago-between to ensure that information amongst the
teams does not "get lost in translation." Especially involving the expression
of scientific intentions and protection of the equipment – this function
becomes critical in maintaining and improving rapport between people who may
have different directions during mission planning and negotiations. The
specific functions that make up the Bridge Builder composite:

  1. 1.

Intent Translator - Communicates an improved comprehension of planned goals
between scientists and engineers. This function performed uses "social
intelligence" to match the amount of contextual information needed to link to
what is behind the scientific intentions. Once the context is given, another
level of interpreting the necessary steps to engineers is given to best direct
them to fulfill the intended activities. An exemplary quote that highlights
this function, "We struggle with the science to engineering communication…I
put a lot of thought into…how to translate the science into intent and then
ask that the engineers understand" -P4Q27.

  2. 2.

Room Reader- Senses the overall tone of the team as situations unfold. This
function enables Mars personnel’s social intelligence systems to gauge the
present mission milieu. Room Readers are especially attuned to gauge the non-
verbal cues during off-nominal conditions when team members may become
overwhelmed – and this is conveyed to other members to match tasks
accordingly. An exemplary quote of this function, “We often talk about getting
the temperature of the room…if everybody’s stressed out…I better find out more
about that because this can affect planning today" -P3Q5.

  3. 3.

People Reader - Gauges specific personalities and working styles of
individuals. This function also uses people’s social intelligence systems to
target conveying the apt interactions style and corresponding information
based on individuals’ characteristics. An exemplary quote of this function, “A
lot of my role is…reading the person I’m talking to…Jane really likes making
the decisions on her own. So I just give Jane the facts…John, you can be like,
“I think that this is the right thing to do" -P5Q37.

  4. 4.

Room Phantom - Toggles between rooms of different teams and listens for
important updates. This function employs their executive set-shifting
abilities as Room Phantoms are engaged with multiple conversations and appear
when relevant updates occur. These people vacillate between being passive or
active in conversations – depending on the priority of the information. An
exemplary quote of this function, "I run between the rooms during
operations…during remote world, I’ve got four WebExs at once… I’m talking to
four groups at once and I’m listening" -P5Q36.

  5. 5.

Day Trader - Facilitates compromises from the negotiations to achieve
consensus. This function enables people to become a broker of mission
objectives exchanges to ensure planning decisions converge with maximizing
buy-in from personnel. An exemplary quote of this function, "Anticipate the
arguments that people are going to have and propose compromises early…everyone
felt like they were getting a little something" -P9Q1.

  6. 6.

Opinion Validator - Acknowledges individuals’ viewpoints and voices to
increase their acceptability with consensus decisions. This function ensures
people’s contributions to mission planning and negotiations are heard. Mainly
during periods under time crunches, not everyone’s opinions could be
thoroughly discussed. Opinion Validators become the buffer to prevent people
from being disgruntled or disengaged in the future in the planning and
negotiation process. An exemplary quote of this function, "The purpose of
planning is to make it so that everybody has had a voice, everybody’s been
able to state their opinion…so that the plan that comes out, everybody’s
like…I’m good with that" -P8Q11.

###  4.3 Latent Theme III - Efficiency Optimizer - 31.46% of all total quotes

This third latent theme illustrates the essential role of Mars mission persons
to streamline the tasks and attention of teams to meet mission deadlines.
Efficiency Optimizers process all inputs at the meta-level and direct focus on
the top priorities, and ensure those priorities do not get lost – especially
under mission time constraints to communicate with Martian rovers. An
exemplary quote that illustrates this high-level theme is, “Show people
clearly what the constraints are and…how best we can address their desires
within the constraints. The sooner you do that, the sooner you avoid a
discussion that is unproductive” -P9Q9. These "make it happen" people go
beyond just stating the apparent objectives by identifying what is pressing
both in the short- and long-term objectives, and process the planning and
barriers to make those objectives successful. The specific functions of the
Efficiency Optimizer composite:

  1. 1.

Constraints Highlighter - Emphasizes the restrictions associated with a given
planning activity. This function spotlights the limitations of the resources
being used to conduct particular science activities. Constraints Highlighters
constantly communicate with engineering and science teams to ensure that plans
are possible given the available resources. Exemplary quote for this function,
“If we’re using too much power, then I let the [anonymized] lead know that
something has to go" -P6Q8.

  2. 2.

Forest-Tree Juggler - Zooms in and out between long-term blueprints and
current situations. This function uses people’s ability to go between big
picture scientific objectives with the current plan being implemented. Forest-
Tree Jugglers keep track of the everyday decisions and whether they have
implications or potentially jeopardize significant mission objectives.
Exemplary quote for this function, “How should we prioritize the mission time,
over months and years? Where should we drive next?…that’s kind of the
strategic planning work that I help facilitate” -P3Q18.

  3. 3.

Urgency Umpire - Facilitates teams to focus on immediate mission needs. This
function brings forth the planning and decisions that need to be made rapidly
to achieve mission objectives. Urgency Umpires push teams to focus on
particular objectives and relegate other tasks to the background until
completion. Exemplary quote for this function, “We’ll say…this observation is
really high priority…because we’re going to need this to make a decision
tomorrow morning" -P7Q10.

  4. 4.

Options Funneler - Narrows down decisions for science and engineering teams to
pick from during planning and negotiations. This function presents a truncated
set of options for teams to make their decisions on. Options Funnelers use
their informed judgment to ensure that the most viable decisions are filtered
and presented. Exemplary quote for this function, “You are a facilitator, you
are not…the decision maker…you provide people with a set of choices" -P2Q8.

Table 1: Codes generated to describe identified functions. Abbreviations are used for three latent categories Corporate Knowledge Gluer (CKG), Bridge Builder (BB), Efficiency Optimizer (EP) Functions |  Percentage of Latent Composition |  Description of theme |  Examples from the open ended responses
---|---|---|---
CGK: Information Clarifier |  20.83% |  Strengthens understanding of objectives and clears up any misunderstandings during negotiations |  “You might hear two requests that sounded slightly contradictory…bringing those forward…and clarifying anything that was unclear."
CGK: Cocktail Host |  20.83% |  Acts as the day to day point of contact to maintain mission structure |  “That person who’s…listening to every conversation…efficiently make sure that the right people are hearing the right information and understand the.constraints and the intent of why we’re doing things.”
CGK: Breaking News Reporter |  16.67% |  Broadcasts updates when plans go off-nominal |  ‘ “…if there’s some late breaking knowledge of vehicle state…late breaking news is, "Oh, hey, we can’t do that…we don’t have enough time…let’s just go on to the thing that we want to do strategically”.’
CGK: Risk-Reward Assessor |  20.83% |  Explains the costs associated with a given option |  “We think about what’s really most important…what’s worth the cost that the engineering folks have now placed on the site from a kind of resource or time perspective.”
BB: Intent Translator |  16.67% |  Communicates an improved comprehension of planned goals between scientists and engineers |  “We struggle with the science to engineering communication…I put a lot of thought into…how to translate the science into intent and then ask that the engineers understand.”
BB: Room Reader |  13.89% |  Senses the overall tone of the team as situations unfold |  “…we often talk about getting the temperature of the room…if everybody’s stressed out…I better find out more about that because this can affect planning today.”
BB: People Reader |  16.67% |  Gauges specific personalities and working styles of individuals |  ‘ “A lot of my role is…reading the person I’m talking to…Jane really likes making the decisions on her own. So I just give Jane the facts…John, you can be like, “I think that this is the right thing to do”.’
BB: Room Phantom |  11.12% |  Toggles between rooms of different teams and listens for important updates |  “I run between the rooms during operations…during remote world, I’ve got four WebExs at once…I’m talking to four groups at once and I’m listening."
BB: Day Trader |  13.89% |  Facilitates compromises from the negotiations to achieve consensus |  “Anticipate the arguments that people are going to have and propose compromises early…everyone felt like they were getting a little something."
BB: Opinion Validator |  13.89% |  Validates individuals’ opinions to increase their acceptability with consensus decisions |  “…the purpose of [anonymized] planning is to make it so that everybody has had a voice, everybody’s been able to state their opinion…so that the plan that comes out, everybody’s like…I’m good with that.”
EP: Constraint Highlighter |  28.57% |  Emphasizes the constraints of a given planning activity |  “…if we’re using too much power, then I let the [anonymized] lead know that something has to go.”
EP: Forest-Tree Juggler |  17.86% |  Zooms in and out between long-term blueprints and current situations |  “How should we prioritize the mission time, over months and years? Where should we drive next?…that’s kind of the strategic planning work that I help facilitate.”
EP: Urgency Umpire |  10.71% |  Facilitates teams to focus on immediate mission needs |  “…we’ll say…this observation is really high priority…because we’re going to need this to make a decision tomorrow morning."
EP: Options Funneler |  21.43% |  Narrows down decisions for science and engineering teams to pick from |  “You are a facilitator, you are not…the decision maker…you provide people with a set of choices."

##  5 Discussion and Future Work

Using our observations, we observe that activity planning for a spacecraft can
be summarized as optimizing the return of high-value and -volume science data,
with resource availability (e.g. memory, energy), and spacecraft health and
safety. We observed many individuals from multiple disciplines participating
in this optimization calculus in a kind of scientific congress. The
coordination and communication challenge amongst the operators gave rise to
the creation of key "linchpin" roles to ensure a smooth and efficient planning
process. We argue that these roles serve two primary purposes.

First, people occupying these roles need to be able to speak the language of
different domains, such that they can succinctly express the constraints that
must be adhered to and how these constraints affect the option space. They
also have to explain the long- and short-term implications of picking one
option over the other to the various science team members. And second, because
the calculus of negotiation can create lead to friction that undermines team
morale, our findings call attention to the fact that constraints are
actualized through the expert practice of "soft skills" functions, We
highlight the key Bridge Builder functions that facilitate the human-
relational factors linked to mission success – and overall longevity of
organizational and team trust forging [11]. Specifically, Bridge Builders were
the most cited functions identified when discussing the planning and
negotiation process. Bridge Builders were commonly cited as the medium through
which scientific intent and communications between engineering and science
teams were honed – exemplified by the quote, "Why did we do the thing that we
did yesterday? from…an engineering perspective, but also from the science
perspective…I really do walk that line, I have to understand both the
engineering side and the science rationale." Beyond these Bridge Builders
carrying the corporate knowledge – they ensure the creation, maintenance, and
evolution of the social milieu [12] – i.e., MSL as a "Well Oiled Machine." The
Well Oiled Machine that is the MSL mission is primarily due to the scaffolding
of science and engineering team members. That is, the processes and workflows
for the MSL mission were built from the ground up with science and engineering
teams co-creating its inputs, functions, and evolution (e.g., creation of
roles). Future promising work may want to focus on the intensive shadowing
process beyond what any training manual or algorithm may be able to specify to
understand how added new operators or machines enter, learn, and evolve the
co-creation of mission-related cultures.

These findings suggest significant considerations for the design of autonomous
systems. Note that when we have more autonomy onboard the spacecraft, we will
likely need more sophisticated software tools that analyze likely outcomes and
constraints, along with specialized roles to coordinate between science and
autonomy, analogous to Science Operations Coordinator who currently
coordinates between science and engineering domains. Operators’ trust will
increase if they interact with a system that can analyze risks accurately,
communicate them clearly for the information consumer, and recognize the
larger mission context. Designers of autonomous systems should consider how
corporate knowledge should be maintained, how bridges could be built across
domains, and how operators can be guided towards optimal solutions during
operations. Even though we may rely on a human versed in the workings of
autonomy, the software we use should simplify their job to avoid rendering
them as a single point of failure.

Additionally, we observed that planning intent is elaborated gradually and
through many iterations. Scientists really begin the planning process with at
best rough estimates of possible science opportunities and resources. As more
data is received, and more evaluations have been performed on the ground, more
precise constraints and opportunities emerge, and a more complete set of the
space of achievable plans comes into view. These often lead to negotiations
among the operators and result in adjustments to the observation plan.

This notion of planning as emergent countervenes somewhat a common assumption
we find in the literature surrounding autonomous planning and scheduling
systems, which posit that intent can be perfectly formulated a priori.
Alternatively, at the very least, we argue that the comprehensive and coherent
expression of intent does not entirely lie within the research domain of AI or
autonomy developers. Our observations instead reveal challenges faced when
eliciting intent from planners while lacking definitive data to guide their
decision-making. In the current operations paradigm that we observed,
decisions required to more completely specify science intent are elaborated on
the ground by enacting iterative comparison of possible futures. In this way,
scientists come to incrementally understand their own science intent through
conversation with other scientists, that get more and more precise over time.
Towards that end, the idea that AI researchers might rely on the perfect
expression of science intent might inject an insincere or even impossible
precondition on the ultimate infusion of their own research systems to teams
in the wild. We would, instead, advocate for an approach where autonomous
systems might instead ask for a science intent that is more realistically the
outcome of an iterative process.

As operators partially delegate decision-making to the spacecraft, there will
be less information on the state of spacecraft during plan execution.
Operators will need to consider multiple scenarios where resources and other
conditions can vary — eliciting such information from operators
comprehensively is not a trivial task — where many permutations must be
considered. We advocate that, to more accurately design operations concepts,
multi-disciplinary research teams are needed to investigate how software
support tools can guide operators to formulate intent coherently,
comprehensively so that optimal science outcomes can be achieved. We see this
as holding the key to the successful operations of the autonomous missions of
the future.

##  6 Conclusion

Infusing autonomy to space missions and their operations requires a great deal
of information exchange between the domain experts and the autonomous system.
In addition, this information exchange has to be asynchronous by virtue of
space operations, where getting real-time confirmation from the earth is
impossible due to communication limitations. Such use of autonomy is more
ambitious than how standard autonomous systems work today where either humans
can take control over anytime (e.g., autonomous cars), or humans are expected
to provide confirmation (e.g., medical diagnostic application), or penalty for
a mistake is low (e.g., Roomba).

By studying how this information is exchanged among human operators, we
revealed vital issues that need to be addressed while exchanging similar
information between autonomy and human operators. We also demonstrated how a
user-centered approach could be taken while developing autonomous systems that
collaborate with humans. Our investigation signals significant challenges that
will be faced and point out essential investigation areas for human-machine
teaming to glow up to its full potential.

## Acknowledgments

This research was carried out in part at the Jet Propulsion Laboratory,
California Institute of Technology, under a contract with the National
Aeronautics and Space Administration (80NM0018D0004) and Cooperative Agreement
(80NSSC19M0200). Funding from National Institutes of Health Bridges to the
Doctorate Research Training Program (1T32GM137863) was also provided to
support student work on this project. The authors wish to thank the Mars
Science Laboratory and Perseverance Rover Mission Operations teams for their
generous support.

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