# Quantum tunneling from high dimensional Gödel black hole

Hui-Ling Lia,b111Electronic address<EMAIL_ADDRESS>Zhong-Wen Fenga, Xiao-
Tao Zua a School of Physical Electronics, University of Electronic Science and
Technology of China, Chengdu, 610054, People’s Republic of China
b College of Physics Science and Technology,
Shenyang Normal University, Shenyang 110034, People’s Republic of China

###### Abstract

Abstract: Considering quantum gravity effect, we investigate the quantum
tunneling from high dimensional Kerr-Gödel black hole using generalized Dirac
equation. As a result, revised tunneling probability is obtained, and the
corrected Hawking temperature is also presented.

Keywords: generalized Dirac equation, Kerr-Gödel black hole, quantum gravity
effect, quantum tunneling

PACS: 04.70.-s, 04.70.Dy, 97.60.Lf

##  I Introduction

Since Hawking made a striking discovery that a black hole can radiate
particles from its event horizon with a black body spectrum and the
temperature is proportional to its surface gravity 1 , a considerable amount
of work has been done relating to static, stationary and non-stationary black
holes’ Hawking radiation 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 ;
14 ; 15 ; 16 ; 17 ; 18 . Recently, incorporating quantum gravity effects into
black hole’s physics, generalized uncertainty principle (GUP) with a minimal
measurable length has attracted a lot of interest 19 ; 20 ; 21 ; 22 ; 23 ; 24
; 25 ; 26 . A minimal observable length arises naturally when incorporating
gravity into quantum field, which also leads to a modification of original
uncertainty principle. This modification now called GUP. GUP produces some
very interesting and novel implications and influences the well know
traditional Hawking radiation and revises a serise of results dramatically.
Consider a GUP with a minimal observable length, applying elegant Parikh-
Wilczek tunneling method, K. Nozari et al. 23  have succeeded in presenting
scalar particle’s quantum tunneling from a schwarzchild black hole. At the
same time, in this framework, this method has been extended to noncommutative
black holes 24 . Latter, considering a GUP with natural cutoffs as a minimal
length, a minimal momentum and a maximal momentum, K. Nozari et al. 27
discussed on quantum tunneling of massless particles’ acrossing black hole
horizon and remarked on the role played by these natural cutoffs on the
tunneling probability. On the other hand, a GUP with a minimal measurable
length also means a correction to Dirac equation in curved spacetime. Very
recently, adopting Kerner and Mann’s Hamilton-Jacobi anatz and modified Dirac
equation based on GUP, Chen et al. 28 ; 29 ; 30  addressed the fermions
tunneling from black holes and black rings, and discussed on the issue about
black hole evaporation and remnants. However, as for higher dimensional black
holes, the issue of quantum corrections to fermion’s Hawking radiation is
comparatively less discussed.

Higher dimensional black holes and their properties have attracted
considerable attention 31 ; 32 ; 33 ; 34 ; 35 , in particular, with the
development of the conformal field theory (CFT) correspondence, allowing
statement to be made about four dimensional quantum field theory using
solutions of the Einstein equation constant in five dimensions, and with the
advent of brane-world theories 36 ; 37 , raising the possibility of direct
observation of Hawking radiation and of as probes of large spatial extra
dimensional in future high energy colliders 38 ; 39 . On the other hand, a
realistic black hole must be localized inside the cosmological background, and
a natural consideration for it is our universe. The Gödel universe is an exact
solution to Einstein’s equation in the presence of a cosmological constant and
homogeneous pressureless matter. In recent years, there has been great
interest in studying Gödel-type solutions to five-dimensional supergravity 40
; 41 ; 42 ; 43 ; 44 ; 45 . These developments motivate us to explore five
dimensional Gödel black hole’s quantum correction to fermion tunneling via
modified Dirac equation based on the GUP.

##  II A brief review of the modified Dirac equation

In a curved background, the Dirac equation can be written as

$\displaystyle
i\gamma^{\mu}\left(\partial_{\mu}+\Omega_{\mu}+\frac{i}{\hbar}eA_{\mu}\right)\psi+\frac{\mu}{\hbar}\psi=0,$
(1)

where

$\displaystyle\Omega_{\mu}=\frac{i}{2}\omega_{\mu}^{\phantom{\mu}ab}\Sigma_{ab},\qquad\Sigma_{ab}=\frac{i}{4}[\gamma^{a},\gamma^{b}],\qquad\left\\{\gamma^{a},\gamma^{b}\right\\}=2\eta^{ab},$
$\displaystyle\omega_{\mu\phantom{a}b}^{\phantom{\mu}a}=e_{\nu}^{\phantom{\nu}a}e^{\lambda}_{\phantom{\lambda}b}\Gamma_{\mu\lambda}^{\nu}-e^{\lambda}_{\phantom{\lambda}b}\partial_{\mu}e_{\lambda}^{\phantom{\lambda}a},\qquad\gamma^{\mu}=e^{\mu}_{\phantom{\mu}a}\gamma^{a},\qquad\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}=2g^{\mu\nu}.$
(2)

The modified Dirac equation is suggested on the basis of a minimal observable
length. In the framework of generalized uncertainty principle (GUP), this
length is obtained. In one dimension, the GUP is given by

$\displaystyle\triangle x\triangle P\geqslant\frac{\hbar}{2}[1+\beta(\triangle
P)^{2}],$ (3)

where the value of $\beta=\beta_{0}l_{P}^{2}/\hbar^{2}$ is very small, the
dimensionless parameter $\beta_{0}$ satisfies $\beta_{0}<10^{-34}$ and $l_{P}$
stands for Planck length. On the basis of the corrected Heisenberg algebra
$[x_{i},P_{j}]=i\hbar\delta_{ij}[1+\beta P^{2}]$, the GUP is presented.
Position operator $x_{i}$ and momentum operator $P_{j}$ are defined as 46

$\displaystyle x_{i}=x_{0i},$ $\displaystyle P_{i}=P_{0i}(1+\beta P^{2}),$ (4)

where $P_{0}^{2}=\Sigma P_{0j}P_{0j}$, the operators $P_{0j}$ and $x_{0i}$
satisfy the usual fundamental commutation relations
$[x_{0i},P_{0j}]=i\hbar\delta_{ij}$. Other modifications may be refereed to
[47-49]. Substituting Eq. (4) into Eq. (1), in a curved background, the
generalized Dirac equation was presented by the revised commutation relations
50 , which can be expressed as follows 28 ; 29 ; 30

$\displaystyle\Big{[}i\gamma^{0}\partial_{0}+i\gamma^{i}\partial_{i}\left(1-\beta\mu^{2}\right)+i\gamma^{i}\beta\hbar^{2}\left(\partial_{j}\partial^{j}\right)\partial_{i}+\frac{\mu}{\hbar}\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta\mu^{2}\right)$
$\displaystyle+i\gamma^{\mu}\frac{i}{\hbar}eA_{\mu}\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta\mu^{2}\right)+i\gamma^{\mu}\Omega_{\mu}\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta\mu^{2}\right)\Big{]}\psi=0.$
(5)

In the following section, using the generalized Dirac equation in Eq.$(5)$, we
will focus on discuss the Kerr-Gödel black hole’s quantum correction to the
fermion tunneling in the five-dimensional gravity background.

##  III Quantum correction to fermion tunneling

The solution of five dimensional Kerr-Gödel black hole is of the
Schwarzschild-Kerr black hole embedded in a Gödel universe. The metric can be
expressed as 45

$\displaystyle{\rm d}s^{2}=$ $\displaystyle-$ $\displaystyle\\!\\!f(r){\rm
d}t^{2}+\frac{1}{V(r)}{\rm d}r^{2}+\frac{r^{2}}{4}{\rm d}\theta^{2}-2a(r){\rm
d}t\,{\rm d}\phi$ (6) $\displaystyle+$
$\displaystyle\\!\\!\left(\frac{r^{2}}{4}\sin^{2}\theta+\frac{r^{2}V(r)-4a^{2}(r)}{4f(r)}\cos^{2}\theta\right){\rm
d}\psi^{2}$ $\displaystyle+$
$\displaystyle\\!\\!\frac{r^{2}V(r)-4a^{2}(r)}{4f(r)}{\rm
d}\phi^{2}-2a(r)\cos\theta{\rm d}t\,{\rm d}\psi$ $\displaystyle+$
$\displaystyle\\!\\!2\left(\frac{r^{2}V(r)-4a^{2}(r)}{4f(r)}\cos\theta\right){\rm
d}\phi\,{\rm d}\psi,$

where

$\displaystyle
f(r)=1-\frac{2m}{r^{2}},\,\,\,\,\,\,a(r)=jr^{2}+\frac{ml}{r^{2}},$
$\displaystyle
V(r)=1-\frac{2m}{r^{2}}+\frac{16j^{2}m^{2}}{r^{2}}+\frac{8jml}{r^{2}}+\frac{2ml^{2}}{r^{4}}.$
(7)

The parameter $j$ stands for the Gödel parameter and is responsible for the
rotation of the spacetime. The parameter $l$ is related to the rotation of the
black hole. When $j=0$ this metric reduces to the 5-dimensional Kerr black
hole with the two possible rotation parameters $(l_{1},l_{2})$ of the general
5-dimensional Kerr spacetime set equal to $l$. When $l=0$ the solution becomes
the Schwarzschild-Gödel black hole. When the parameters $j$ and $l$ are set to
zero the metric simply reduces to the 5-dimensional Schwarzschild black hole,
whose mass is proportional to the parameter $m$. When $m=l=0$, the metric
reduces to that of the five dimensional Gödel universe.

The horizon of the black hole can be found by setting

$\displaystyle
V(r)=1-\frac{2m}{r^{2}}+\frac{16j^{2}m^{2}}{r^{2}}+\frac{8jml}{r^{2}}+\frac{2ml^{2}}{r^{4}}=0,$
(8)

which yield outer and inter horizons,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!r_{\pm}=\sqrt{m(1-8mj^{2}-4lj)\pm
m\sqrt{(1-8mj^{2}-4lj)^{2}-\frac{2l^{2}}{m}}}.\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$

At the event horizon $r_{+}$, the Hawking temperature of 5-dimensional Kerr-
Gödel black hole can be determined as 14

$\displaystyle T_{BH}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{V^{\prime}(r_{+})\sqrt{-f(r_{+})}r_{+}}{8\pi a(r_{+})}$
(10) $\displaystyle=$
$\displaystyle\\!\frac{m[r_{+}^{2}(1-8j^{2}m-4jl)-2l^{2}]}{\pi
r_{+}^{2}\sqrt{-4j^{2}r_{+}^{6}+(1-8j^{2}m)r_{+}^{4}+2ml^{2}}},$

where

$\displaystyle V^{\prime}(r_{+})=\partial_{r}V(r)|_{r=r_{+}},\qquad
f(r_{+})=1-\frac{2m}{r_{+}^{2}},\qquad
a(r_{+})=jr_{+}^{2}+\frac{ml}{r_{+}^{2}}.$ (11)

The angular velocity at the event horizon is

$\displaystyle\Omega_{+}=-\frac{f(r_{+})}{a(r_{+})}.$ (12)

Now, applying the modified Dirac equation based on GUP and Kernerand Mann’s
fermion tunneling method, we focus on investigating the quantum correction to
fermion tunneling from the Kerr-Gödel black hole. In order to avoid the
dragging effect, we first introduce a dragging coordinate transformation

$\displaystyle{\rm d}\phi=\Omega{\rm
d}t=\frac{4a(r)f(r)}{r^{2}V(r)-4a^{2}(r)}{\rm d}t.$ (13)

Then, the Kerr-Gödel black hole’s metric can be expressed as

$\displaystyle{\rm d}s^{2}=\\!$ $\displaystyle-$
$\displaystyle\\!\frac{4a(r)f(r)}{r^{2}V(r)-4a^{2}(r)}{\rm
d}t^{2}+\frac{1}{V(r)}{\rm d}r^{2}+\frac{r^{2}}{4}{\rm d}\theta^{2}$ (14)
$\displaystyle+$
$\displaystyle\\!\frac{r^{2}f(r)\sin^{2}\theta+(r^{2}V(r)-4a^{2}(r))\cos^{2}\theta}{4f(r)}{\rm
d}\psi^{2}.$

Here, we will only analysis the spin-up case since the final result is the
same as the spin-down case as can be presented by using the methods described
below. As for the spin-up case, we adopt the following ansatz

$\displaystyle\psi_{\uparrow}(t,r,\theta,\psi)=\left(\begin{array}[]{clr}A(t,r,\theta,\psi)\\\
0\\\ B(t,r,\theta,\psi)\\\
0\end{array}\right)\exp\left(\frac{i}{\hbar}I_{\uparrow}(t,r,\theta,\psi)\right),$
(19)

and construct the gamma matrices of the Gödel black hole as following

$\displaystyle\gamma^{\,t}=\sqrt{-g^{tt}}\left(\begin{array}[]{clr}i&\,\,\,0\\\
0&\,\,\,i\end{array}\right),\qquad\gamma^{\,r}=\sqrt{V(r)}\left(\begin{array}[]{clr}0&\sigma^{3}\\\
\sigma^{3}&0\end{array}\right)$ (24)
$\displaystyle\gamma^{\,\theta}=\sqrt{g^{\theta\theta}}\left(\begin{array}[]{clr}0&\sigma^{1}\\\
\sigma^{1}&0\end{array}\right)\qquad\gamma^{\,\psi}=\sqrt{g^{\psi\psi}}\left(\begin{array}[]{clr}0&\sigma^{2}\\\
\sigma^{2}&0\end{array}\right)$ (29)

and $\sigma^{i}(i=1,2,3)$ are Pauli matrices as follows:

$\displaystyle\sigma^{1}=\left(\begin{array}[]{clr}0&\,\,\,1\\\
1&\,\,\,0\end{array}\right),\,\,\,\,\,\,\sigma^{2}=\left(\begin{array}[]{clr}0&-i\\\
i&\,\,\,0\end{array}\right),\,\,\,\,\,\,\sigma^{3}=\left(\begin{array}[]{clr}1&\,\,\,0\\\
0&-1\end{array}\right).$ (36)

Substituting the ansatz (15) and gamma matrices (16) into the generalized
Dirac equation (5), and adopting WKB approximation, after neglecting the high
order of $\hbar$, we get the following four equations:

$\displaystyle-
iA\sqrt{-g^{tt}}\partial_{t}I_{\uparrow}-B\sqrt{g^{rr}}(1-\beta\mu^{2})\partial_{r}I_{\uparrow}-A\mu\beta\left[g^{rr}(\partial_{r}I_{\uparrow})^{2}+g^{\theta\theta}(\partial_{\theta}I_{\uparrow})^{2}+g^{\psi\psi}(\partial_{\psi}I_{\uparrow})^{2}\right]{}$
$\displaystyle+B\beta\sqrt{g^{rr}}\left[g^{rr}(\partial_{r}I_{\uparrow})^{2}+g^{\theta\theta}(\partial_{\theta}I_{\uparrow})^{2}+g^{\psi\psi}(\partial_{\psi}I_{\uparrow})^{2}\right]\partial_{r}I_{\uparrow}+A\mu(1-\beta\mu^{2})=0,$
(37) $\displaystyle
iB\sqrt{-g^{tt}}\partial_{t}I_{\uparrow}-A\sqrt{g^{rr}}(1-\beta\mu^{2})\partial_{r}I_{\uparrow}-B\mu\beta\left[g^{rr}(\partial_{r}I_{\uparrow})^{2}+g^{\theta\theta}(\partial_{\theta}I_{\uparrow})^{2}+g^{\psi\psi}(\partial_{\psi}I_{\uparrow})^{2}\right]{}$
$\displaystyle+A\beta\sqrt{g^{rr}}\left[g^{rr}(\partial_{r}I_{\uparrow})^{2}+g^{\theta\theta}(\partial_{\theta}I_{\uparrow})^{2}+g^{\psi\psi}(\partial_{\psi}I_{\uparrow})^{2}\right]\partial_{r}I_{\uparrow}+B\mu(1-\beta\mu^{2})=0,$
(38) $\displaystyle
A\big{\\{}-(1-\beta\mu^{2})\sqrt{g^{\theta\theta}}\partial_{\theta}I_{\uparrow}+\beta\sqrt{g^{\theta\theta}}\partial_{\theta}I_{\uparrow}\left[g^{rr}(\partial_{r}I_{\uparrow})^{2}+g^{\theta\theta}(\partial_{\theta}I_{\uparrow})^{2}+g^{\psi\psi}(\partial_{\psi}I_{\uparrow})^{2}\right]{}$
$\displaystyle-i(1-\beta\mu^{2})\sqrt{g^{\psi\psi}}\partial_{\psi}I_{\uparrow}+i\beta\sqrt{g^{\psi\psi}}\left[g^{rr}(\partial_{r}I_{\uparrow})^{2}+g^{\theta\theta}(\partial_{\theta}I_{\uparrow})^{2}+g^{\psi\psi}(\partial_{\psi}I_{\uparrow})^{2}\right]\big{\\}}=0,$
(39) $\displaystyle
B\big{\\{}-(1-\beta\mu^{2})\sqrt{g^{\theta\theta}}\partial_{\theta}I_{\uparrow}+\beta\sqrt{g^{\theta\theta}}\partial_{\theta}I_{\uparrow}\left[g^{rr}(\partial_{r}I_{\uparrow})^{2}+g^{\theta\theta}(\partial_{\theta}I_{\uparrow})^{2}+g^{\psi\psi}(\partial_{\psi}I_{\uparrow})^{2}\right]{}$
$\displaystyle-i(1-\beta\mu^{2})\sqrt{g^{\psi\psi}}\partial_{\psi}I_{\uparrow}+i\beta\sqrt{g^{\psi\psi}}\left[g^{rr}(\partial_{r}I_{\uparrow})^{2}+g^{\theta\theta}(\partial_{\theta}I_{\uparrow})^{2}+g^{\psi\psi}(\partial_{\psi}I_{\uparrow})^{2}\right]\big{\\}}=0.$
(40)

It is difficult to directly obtain the solution of the above equations. Due to
there are three Killing vectors $(\partial/\partial_{t})^{\mu}$,
$(\partial/\partial_{\phi})^{\mu}$ and $(\partial/\partial_{\psi})^{\mu}$ in
the five dimensional Kerr-Gödel black hole, so the action $I_{\uparrow}$ in
the dragging coordinate frame can be decomposed as

$\displaystyle I_{\uparrow}=-(\omega-J\Omega)t+L\psi+R(r,\theta),$ (41)

here $J$ and $\omega$ are the tunneling particle’s angular momentum and
energy, respectively. From Eqs. (20) and (21), we can get

$\displaystyle(\sqrt{g^{\theta\theta}}\partial_{\theta}I_{\uparrow}+i\sqrt{g^{\psi\psi}}\partial_{\psi}I_{\uparrow})\big{\\{}\beta\left[g^{rr}(\partial_{r}I_{\uparrow})^{2}+g^{\theta\theta}(\partial_{\theta}I_{\uparrow})^{2}+g^{\psi\psi}(\partial_{\psi}I_{\uparrow})^{2}\right]-(1-\beta\mu^{2})\big{\\}}=0.$
(42)

Due to $\beta$ is a very small quantity, the second term in above equation is
not zero. Thus, we can obtain

$\displaystyle\sqrt{g^{\theta\theta}}\partial_{\theta}I_{\uparrow}+i\sqrt{g^{\psi\psi}}\partial_{\psi}I_{\uparrow}=0.$
(43)

That means

$\displaystyle
g^{\theta\theta}(\partial_{\theta}I_{\uparrow})^{2}+g^{\psi\psi}(\partial_{\psi}I_{\uparrow})^{2}=0.$
(44)

Substituting Eqs. (22) and (25) into Eqs. (18) and (19), and canceling $A$ and
$B$, we obtain the differential equation

$\displaystyle\beta^{2}\frac{r^{2}f(r)V^{4}(r)}{r^{2}V(r)-4a^{2}(r)}(\partial_{r}R)^{6}+\beta\frac{r^{2}f(r)V^{3}(r)}{r^{2}V(r)-4a^{2}(r)}(\mu^{2}\beta-2)(\partial_{r}R)^{4}{}$
$\displaystyle+\frac{r^{2}f(r)V^{2}(r)}{r^{2}V(r)-4a^{2}(r)}\left[(1-\beta\mu^{2})^{2}+2\beta\mu^{2}(1-\beta\mu^{2})\right](\partial_{r}R)^{2}{}$
$\displaystyle-\frac{r^{2}f(r)V(r)}{r^{2}V(r)-4a^{2}(r)}\mu^{2}(1-\beta\mu^{2})^{2}-(\omega-J\Omega)^{2}=0.$
(45)

From Eq. (26), we can see that $R(r,\theta)$ has explicit $r$-dependence and
is independent of $\theta$. Thus, $R(r,\theta)=R(r)+\Theta(\theta)$ and
$\partial_{r}R(r,\theta)=\partial_{r}R(r)$. Integrating at the event horizon
$r_{+}$ and keeping the leading order term of $\beta$, we obtain the
expression of the action $R(r)$

$\displaystyle R_{\pm}(r)$ $\displaystyle=$ $\displaystyle\pm\int\frac{{\rm
d}r}{rV(r)}\sqrt{\frac{r^{2}V(r)-4a^{2}(r)}{f(r)}\left[\mu^{2}\frac{r^{2}f(r)V(r)}{r^{2}V(r)-4a^{2}(r)}+(\omega-J\Omega)^{2}\right]}{}$
(46)
$\displaystyle\times\left[1+\beta\mu^{2}+\beta\frac{r^{2}V(r)-4a^{2}(r)}{r^{2}f(r)V(r)}(\omega-J\Omega)^{2}\right]{}$
$\displaystyle=$ $\displaystyle\pm
i\pi\frac{(\omega-J\Omega_{+})r_{+}^{2}a^{2}(r_{+})}{\sqrt{-f(r_{+})}(r_{+}^{2}-r_{-}^{2})}(1+\beta\eta)+\mbox{Real
Part},$

where $\pm$ correspond to the outgoing/ingoing solution, and

$\displaystyle\eta=\frac{3}{2}\mu^{2}-2J\frac{\omega-J\Omega_{+}}{a(r_{+})}+\frac{\omega^{2}-J^{2}\Omega_{+}^{2}}{2f(r_{+})}+\frac{(\omega-J\Omega_{+})^{2}[(r_{+}^{2}-r_{-}^{2})^{2}+4a^{2}(r_{+})(3r_{+}^{2}-r_{-}^{2})]}{f(r_{+})(r_{+}^{2}-r_{-}^{2})^{2}}.$

As Akhmedov E T, Akhmedova V et al. have indicated in Refs. 51 ; 52 ; 53 ; 54
; 55 , the expression of the tunneling rate
$\Gamma\propto\exp\left(-\mbox{Im}\oint P\,{\rm d}r\right)$, instead of
$\Gamma\propto\exp\left(-2\mbox{Im}\int P\,{\rm d}r\right)$ is invariant under
canonical transformations. So the fermion’s tunneling rate of the five-
dimensional Gödel black hole can be written as

$\displaystyle\Gamma\propto$ $\displaystyle\exp\left(-\mbox{Im}\oint P\,{\rm
d}r\right){}$ (48) $\displaystyle=\exp\left[-\mbox{Im}\left(\int
P_{\text{out}}\,{\rm d}r-\int P_{\,\text{in}}\,{\rm d}r\right)\right]{}$
$\displaystyle=\exp\left[\mp 2\mbox{Im}\int P_{\text{out},\,\text{in}}\,{\rm
d}r\right],$

where $P=\partial_{r}R$, $\pm$ signs in front of this integral in last step
correspond to $P_{\text{out}}$ and $P_{\,\text{in}}$, which have the
relationship $P_{\text{out}}=-P_{\text{\,in}}$. So the tunneling probability
of the five-dimensional Kerr-Gödel black hole can be calculated as

$\displaystyle\Gamma\propto$
$\displaystyle\exp\left[-2\pi\frac{(\omega-J\Omega_{+})r_{+}^{2}a(r_{+})}{\sqrt{-f(r_{+})}(r_{+}^{2}-r_{-}^{2})}(1+\beta\eta)\right].$
(49)

As the authors suggested in Refs. [51, 54,55], by calculating the tunneling
rate in this way, the contribution from the time part of the action was
overlooked. In order to work out the correct tunneling rate, we introduce
general Kruskal coordinates $(T,R)$. Order

$\displaystyle{\rm
d}r^{*}=\sqrt{\frac{r^{2}V(r)-4a^{2}(r)}{4a(r)f(r)V(r)}}\,{\rm d}r.$ (50)

As a result, we can write (14) as

$\displaystyle{\rm d}s^{2}=$
$\displaystyle\pm\frac{4a(r)f(r)}{r^{2}V(r)-4a^{2}(r)}\kappa^{-2}e^{-2\kappa
r^{*}}\left(-{\rm d}T^{2}+{\rm d}R^{2}\right)+\frac{r^{2}}{4}{\rm
d}\theta^{2}{}$ (51)
$\displaystyle+\frac{r^{2}f(r)\sin^{2}\theta+[r^{2}V(r)-4a^{2}(r)]\cos^{2}\theta}{4f(r)}{\rm
d}\psi^{2},$

where $+(-)$ sign stands for the Kerr-Gödel metric outside (inside) the event
horizon, and $\kappa=2\pi/T_{BH}$ is the Kerr-Gödel black hole’s surface
gravity. The region exterior of the Kerr-Gödel black hole is

$\displaystyle T_{\text{out}}=e^{\kappa r^{*}}\sinh(\kappa t),$ $\displaystyle
R_{\text{out}}=e^{\kappa r^{*}}\cosh(\kappa t),$ (52)

where $\kappa=2\pi/T_{BH}$ is the surface gravity. The interior region is
expressed as

$\displaystyle T_{\text{in}}=e^{\kappa r^{*}}\cosh(\kappa t),$ $\displaystyle
R_{\text{in}}=e^{\kappa r^{*}}\sinh(\kappa t).$ (53)

We can refer to [47-49], setting the time $t$ as $t\rightarrow t-\pi
i/2\kappa$ an additional contribution coming from the time part of the action
can be discoved that is $\mbox{Im}[(\omega-J\Omega_{+})\varDelta
t_{\text{out,\,in}}]=-\pi(\omega-J\Omega_{+})/2\kappa$. Eventually,
incorporating the temporal contribution, we turns out to be of the form

$\displaystyle\Gamma\propto$
$\displaystyle\exp\left\\{\left[\mbox{Im}((\omega-J\Omega_{+})\varDelta
t_{\text{out}})+\mbox{Im}((\omega-J\Omega_{+})\varDelta
t_{\text{in}})-\mbox{Im}\oint P\,{\rm d}r\right]\right\\}{}$ (54)
$\displaystyle=\exp\left[-4\pi\frac{r_{+}^{2}a(r_{+})(\omega-J\Omega_{+})}{\sqrt{-f(r_{+})}(r_{+}^{2}-r_{-}^{2})}(1+\frac{1}{2}\beta\eta)\right].$

As a consequence, the temperature including the modification terms of the
Gödel black hole can be determined as

$\displaystyle T=\frac{\sqrt{-f(r_{+})}(r_{+}^{2}-r_{-}^{2})}{4\pi
r_{+}^{2}a(r_{+})(1+\beta\eta/2)}=T_{BH}(1-\frac{1}{2}\beta\eta),$ (55)

where

$\displaystyle T_{BH}=\frac{\sqrt{-f(r_{+})}(r_{+}^{2}-r_{-}^{2})}{4\pi
r_{+}^{2}a(r_{+})}=\frac{r_{+}V^{\prime}(r_{+})\sqrt{-f(r_{+})}}{8\pi
a(r_{+})}$ (56)

is the Kerr-Gödel black hole’s Hawking temperature. According to the above
expressions (35) and (36), we can see that nonlinear terms arise. That is to
say, when the quantum gravity effect is taken into account, the emission
spectrum of the higher dimensional Gödel black hole is no more pure thermal.
This means the information can be supplied. And the revised tunneling
temperature is no longer the traditional Hawking temperature but depends on
the emitted particle’s mass $\mu$, angular momentum $J$ and energy $\omega$.
If gravity corrected term is neglected, the results are consistent with the
previous those in the literatures 14 ; 17 .

##  IV Conclusion

Comparing with three or four dimensional spacetime, five dimensional case
becomes more complicated. As mentioned former, it is significate to
investigate quantum correction to higher dimensional case. In this current
paper, we have studied the fermion tunneling from a five dimensional Kerr
Gödel black hole when incorporating quantum gravity effects. For this purpose,
we have first introduced a dragging coordinate transformation. For a rotating
black hole, due to there exist dragging effect, a reasonable tunneling
radiation should be described in a dragging coordinate framework. Here we have
made the coordinate transformation $\phi=\Omega dt$, departing from one in
Refs. [28-30], where the angular is fixed when integrating the action for the
rotating spacetime. This transformation is more natural and convenient to
derive tunneling radiation, especially for the higher dimensional case, we can
get exact expression of the tunneling probability and tunneling temperature.
After making the dragging coordinate transformation, applying Kerner and
Mann’s fermion tunneling method and the spacetime’s symmetries of the Kerr
Gödel black hole, we have resolved the modified Dirac equation proposed by
authors in Refs. [28-30]. As a result, the revised tunneling probability has
been workout, and the corrected tunneling temperature also discovered. We find
out that tunneling rate and tunneling temperature depend on the tunneling
particle’s mass, energy and angular momentum. Furthermore, there exists
nonlinear terms in the imaginary part of the action. This means tunneling rate
is no longer purely thermal and the information may be supplied in
correlations between tunneling modes.

###### Acknowledgements.

This work is supported in part by the Science and Technology Project
Foundation of the Education Department of Liaoning Province, China (Grant No.
L2011195).

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# Estimating sea ice properties from wave observations in sea ice

J.J. Voermans  Department of Infrastructure Engineering, University of
Melbourne, Melbourne, Australia  J. Rabault  Arctic Technology Department, The
University Centre in Svalbard, Longyearbyen, Norway  A. Marchenko  T. Nose
Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, Chiba,
Japan  T. Waseda  Graduate School of Frontier Sciences, The University of
Tokyo, Kashiwa, Chiba, Japan  A.V. Babanin  Department of Infrastructure
Engineering, University of Melbourne, Melbourne, Australia

###### Abstract

The Marginal Ice Zone is a highly dynamic region where the atmosphere, ocean,
waves and sea ice meet. Waves play a fundamental role in this coupled system,
but our progress in understanding wave-ice interactions is currently hindered
by the lack of observations of sea ice properties in-situ. In this study we
aim to estimate the ice thickness and effective elastic modulus of sea ice
passively using observations of waves in ice obtained from vibrations sensors.
Specifically, we use three low-cost geophones deployed in triangular formation
with sides of about 200 m on fast ice. The focus here is on three major wave
events that were recorded, each consisting of initial high frequency
vibrations with a frequency at around 10 Hz, followed by low frequency
dispersive waves within a frequency range of 0.08–0.28 Hz. Based on the phase
speed of the initial high frequency vibrations, we estimate the purely elastic
effective modulus to be 4–4.5 GPa. By comparing the arrival times of the
dispersive low frequency wave events at the geophones to the dispersion
relationship of waves in a thin elastic ice sheet we obtain estimates of the
effective elastic modulus in the range of 0.4–0.7 GPa. This is close to the
measured effective elastic modulus through cantilever experiments of 0.5 GPa
but considerably smaller than the default value of 5.5 GPa currently in use in
contemporary wave models. We could not, however, obtain explicit estimates of
the ice thickness and effective modulus individually from the measured low
frequency dispersive waves as their impacts on the shape of the dispersion
relationship are similar in this frequency range. That is, information of one
is required to estimate the other. Distinction is only possible for dispersive
wave events larger than about 1 Hz, such as for vibrations generated by
(thermal) cracking, which are frequently observed in our dataset. Using source
triangulation, we find that the major wave events observed in this study
originate from the nearby glacier. We suspect that the high frequency
vibrations are caused by the calving of the glacier. With our results, we show
that low-cost geophones can be used to estimate sea ice properties in fast ice
and substantiate that this to be possible on very large ice floes as well.
Progress in retrieving sea ice properties from wave measurements is expected
to increase the dataset of in-situ sea ice properties observations in number
and geographic coverage and improve our understanding of wave-ice
interactions.

##  1 Introduction

Waves can propagate hundreds of kilometers intro the sea ice before the
majority of their energy is attenuated by wave scattering (Montiel et al.,
2016; Kohout and Meylan, 2008) and wave dissipation processes, including
turbulence (Kohout et al., 2011; Voermans et al., 2019; Herman, 2021),
internal friction (Wang and Shen, 2010), overwash (Nelli et al., 2020) and
ice-floe collisions (Herman et al., 2019; Rabault et al., 2019; Løken et al.,
2022). The rate of attenuation is largely determined by the properties of the
ice where, in general, thicker and larger ice floes result in larger
attenuation of wave energy. When the wave steepness is sufficiently high,
waves can break the ice (Dumont et al., 2011; Voermans et al., 2020), thereby
reducing its attenuation capacity and allowing waves to penetrate further into
the ice cover, ultimately leading to increased break-up activity and an
increase of the Marginal Ice Zone (MIZ) width (Asplin et al., 2012; Collins
III et al., 2015). Wave steepness can be further impacted by the change in
wave group velocity when entering the ice which, in the case of short waves,
tends to decrease the wave height and thus wave steepness (Collins et al.,
2017). Implementation of these complex feedbacks between the waves and the ice
in our forecasting models is thus strongly dependent on our collective
knowledge of wave attenuation, wave dispersion and wave-induced sea ice break-
up in terms of the sea ice properties and, importantly, the availability
thereof (e.g., Kousal et al., 2022; Boutin et al., 2020).

Many theories and models currently exist, with varying degree of complexity
and representation of the underlying wave-ice physical processes (e.g., see
Shen, 2019; Squire, 2020; Rogers et al., 2021, for an overview). All these
models rely in one way or another on the properties of the ice. Notable sea
properties are the ice thickness and elastic modulus, but others include the
flexural strength, roughness (both above and below the waterline) and
horizontal dimensions of the ice floes. Validation of the theories and models
against field experimental observations is however very limited, largely due
to the difficulty of obtaining observations of waves and sea ice in the field.

Measuring waves and sea ice properties in the MIZ is hindered by the extreme
complexity of this harsh environment which poses logistical and technological
challenges. Despite this, significant progress has been made in estimating
wave properties in the MIZ due to developments and advancements in low-cost
instrumentation (e.g., Rabault et al., 2022) and satellite derived products
(Ardhuin et al., 2015; Horvat et al., 2020), which already has led to a
substantial increase of the availability of in-situ wave observations (Rabault
et al., 2023). In contrast, access to observations of the physical and
mechanical properties of sea ice remain restricted, with the exception perhaps
of sea ice thickness which may be estimated from satellite derived products
(e.g., Paţilea et al., 2019), although at relatively coarse spatial resolution
and primarily for thin ice. Importantly, the elastic modulus remains a
sparingly documented property of sea ice in relation to wave-ice interaction
studies.

The elastic modulus $E$ is equal to the ratio of the stress and strain in the
ice sheet during elastic behavior (Timco and Weeks, 2010). When considering
most engineering applications, however, the ice does not behave as purely
elastic and delayed elasticity may become important during small loading
rates. We may define the total recoverable strain as the sum of the purely
elastic strain and the delayed strain, denoted by an effective elastic modulus
$E^{*}$ (Williams et al., 2013). Timco and Weeks (2010) suggest typical values
of the effective elastic modulus between 1–5 GPa, where Williams et al. (2013)
opt for a somewhat higher value of $4-7$ GPa, and Karulina et al. (2019)
slightly lower with observations between 0.3–4 GPa. Due to the absence of
observational data of $E^{*}$, the default value in contemporary wave models
is currently 5.5 GPa (WW3DG, 2019) which, by lack of better alternatives, is
commonly adopted in wave related studies in the MIZ (e.g., Boutin et al.,
2018; Williams et al., 2017; Li et al., 2021). This poses two problems.
Firstly, it provides an additional tuning parameter to fit model output to the
observations, thereby potentially masking the mechanistic output of the models
or, in other words, it obscures our qualitative understanding of wave-ice
interactions (e.g., Voermans et al., 2021). Secondly, an unknown elastic
modulus leads to quantitative uncertainty in the model output including
critical dynamical variables, such as the MIZ width. For example, choosing
wave scattering as an attenuation model, Williams et al. (2017) showed in
their numerical experiments that a doubling of the effective elastic modulus
may double the MIZ width during swell events. We note that an uncertainty of
$E^{*}$ by a factor of two (or even more) is far from uncommon as in most
wave-ice studies $E^{*}$ is not measured nor inferred. The sensitivity of the
wave-ice interaction models to the properties of the ice, and thus their
collective outcome and feedback into the dynamics of the MIZ as a whole,
highlights the need for more frequent and accurate observations of sea ice
properties.

Traditional methods of measuring sea ice properties tend to be laborious,
whether it being sea ice coring to obtain sea ice samples and measuring sea
ice thickness, or in-situ cantilever experiments to retrieve effective
mechanical properties of the ice (Marchenko et al., 2020; Karulina et al.,
2019; Timco and Weeks, 2010). Ideally, sea ice properties are inferred from
long-term in-situ sensor deployments without the requirement of continuous
human presence, which would then allow for higher temporal and/or spatial
resolution of such observations. Examples include floe-size distribution from
stereo-cameras (e.g., Alberello et al., 2019) and ice thickness from ice-mass-
balance buoys (e.g., Richter-Menge et al., 2006). Measuring the mechanical
properties of ice without physical activity seems less straightforward. The
elastic modulus of sea ice, for instance, can be estimated based on
observations of the propagation speed of waves in the ice (Stein et al., 1998;
Yang and Giellis, 1994; Moreau et al., 2020). This can be inferred from the
phase speeds of flexural waves, compressive and/or shear waves, but typically
involves the deployment of a series of seismometers or geophones to track the
waves in the spatial domain (Moreau et al., 2020; Yang and Giellis, 1994;
Oliver et al., 1954). This can be an intensive and expensive exercise with
some studies using ten to hundreds of vibration sensors to measure the phase
speed of waves in ice created by hammer blows, explosions or jumping people.
However, the use of active sources is, to some extent, a restriction of the
method’s generalization as it still requires human presence to generate these
active sources. Examples of studies using passive sources do exist, such as
Voermans et al. (2021); Sutherland and Rabault (2016); Marsan et al. (2012),
who used accelerometers and seismometers to estimate the ice thickness and/or
the elastic modulus from the phase speed of gravity waves, but those will
require the presence of gravity waves in the first place and may fail once the
ice is broken due to the restrictions in the horizontal length scales of the
ice relative to the wave length. In this study, we aim to retrieve sea ice
properties from passive wave observations. Specifically, we focus here on the
retrieval of sea ice thickness and effective elastic modulus of sea ice from
observations of waves in landfast sea ice using three geophones.

##  2 Methods

###  2.1 Experimental Setup

Figure 1: Experimental setup in Tempelfjorden, Svalbard. Instrument deployment
consists of three geophone logger (also see inset), a hydrophone and four
wave-ice buoys. In-situ cantilever experiments were performed near the ice
edge, marked by the ‘+’ sign.

To measure waves in sea ice, various instruments were deployed on landfast sea
ice from 11–28 February 2022 in Tempelfjorden, Svalbard. The experimental
setup consist of multiple ice buoys (Rabault et al., 2020), one hydrophone
(Audiomoth Dev v1.0.0. with an Aquarian Audio h2a hydrophone) and three
geophone loggers. The ice buoys focus on the frequency range between 0.05–1 Hz
(i.e., ocean waves), whereas the geophones target sea ice vibrations with
frequencies up to 240 Hz. The hydrophone logger was deployed to record the
acoustic radiation of ice events underwater, such as cracking and break-up,
with a sampling rate of 48kHz. The ice motion loggers were deployed along a
line parallel to the main axis of the fjord and the geophones were deployed
closest to the glacier-side of the fjord in triangular formation with sides of
about 200 m. The distance between the geophones needs to be large enough to
record the fastest waves, such as compressive and/or shear waves which have
phase speed of a few thousand meters per second, but small enough to limit
significant wave transformation and wave attenuation effects. The hydrophone
was deployed next to the geophone closest to the open water (see Fig. 1). In
this study no data from the ice buoys were used as the measured wave energy
during the measurement campaign was very small.

Ice thickness was measured at the sites of the ice buoys at the time of
deployment. The thickness was 35, 40, 47 and 52 cm thick in the direction from
the open water towards the glacier, respectively. The thickness did not change
noticeable over the duration of the experiment. Air temperatures measured at
Svalbard Lufthavn (45 km from the deployment site) varied between -27∘ and
-5∘C. The local water depth is roughly 40 m at the site (Marchenko et al.,
2013).

###  2.2 Geophone logger

A custom geophone logger was designed for this project to record sea ice
vibrations. While various low-cost and open-source designs are available to
record the analog signals of the geophones, such as the Geophonino of Soler-
Llorens et al. (2016), we developed a custom logger instead for general
flexibility and to suit our experimental design. Most notably, the logger was
designed to record five analog channels continuously at a sampling rate of
1000 Hz to an SD-card, including timestamps at high temporal accuracy (the
order of microseconds) by using GPS time synchronization. The latter allows
cross correlation between instruments without the need of instruments to
communicate with each other or be connected to a centralized system.

The hardware of the instrument consists of five main components: (1) a
microcontroller, Arduino Due; (2) SD-card breakout, used to store the recorded
signals; (3) GPS breakout, used for time synchronization and recording of its
location; (4) signal amplifiers, to amplify the voltage signals generated by
(5) the geophone. Both firmware and design of the hardware are made open-
source and can be found at http://github.com/jvoermans/Geophone_Logger.

Here, we used a triaxis GS-One 10 Hz geophone, which has a spurious frequency
$>240$ Hz and sensitivity of 85.5 V/m/s. Based on the datasheet, the geophone
response is linear for $f>10$ Hz, and decays exponentially for smaller
frequencies. The geophones used in this study were not calibrated, meaning
that the recorded voltages cannot be converted to surface elevation with
sufficient accuracy. However, phase information of the recorded vibrations are
unaffected. The $x$ and $y$-components of the geophones were amplified by a
gain of 50, whereas the signal of the $z$-component was split and amplified by
gains of 5, 50 and 1000 to increase the range of sensitivity of the logger. We
note that the z-component of geophone 1 did not work for unknown reasons.

###  2.3 Elastic modulus

Figure 2: Photographs of vertical (a) and horizontal (b) thin sections of sea
ice in polarized light. Thin sections were made from a specimen of sea ice
from the Templefjorden, March 2022. Yellow strips scale 5 cm length

To provide comparison of our estimates of the effective elastic modulus, in-
situ cantilever experiments were performed on the ice to obtain estimates of
the effective elastic modulus $E^{*}$. Eigen frequency of bending oscillations
of floating fixed-ends beam depends on the elastic modulus of the beam along
the beam axis. The eigen frequency depends also on the added mass of the beam
determined by the beam geometry. The fixed-ends beam was cut from sea ice near
the ice edge of the Templefjorden on March 9, 2022 (see Fig. 1). The beam
length was $2L_{b}=3$ m, the beam width was $2b=0.27$ m, and the ice thickness
was $H=0.3$ m. The ice temperature was measured between -1.9∘C and -2.5∘C, the
mean ice salinity was 5 ppt. The ice had columnar structure with the diameter
of columns up to 2 cm. Boundaries of ice columns had complicated geometry
(Fig. 2). The length of some columns exceeded 7 cm. The water temperature
below sea ice was measured as -1.7 C.

The accelerometer Bruel & Kjær DeltaTron Type 8344 designed for the
measurements of vibrations in the frequency range 0.2 Hz – 3 kHz was used to
measure the eigen frequency of the fixed-ends beam in the vertical direction.
It was placed in the middle of the beam surface. The oscillations were
initiated by jumping on the ice near the beam. An example of the records after
five tests is shown in Fig. 3(a). One can see that period of oscillations are
very similar. The eigen frequency of the beam was found as $\omega=104$ rad/s.
After the test, the beam was cut near the ends and the frequency of free heave
oscillations of the beam was measured with the same accelerometer. The eigen
frequency of the free-floating beam was measured at $\omega_{b}=4$ rad/s (Fig.
3b).

Figure 3: Natural oscillations of the beam with fixed ends (a), and the free-
floating beam (b).

The elastic modulus of a fixed-ends beam is determined by:

$\begin{array}[]{l}E^{*}=\frac{12(\omega^{2}-\omega_{b}^{2})(\rho_{i}H+m_{add})}{H^{3}}\frac{L_{b}}{2.365}\\\
\omega_{b}^{2}=\frac{\rho_{w}g}{\rho_{i}H+m_{add}}\end{array}$ (1)

where $\omega_{b}$ is the frequency of heave oscillations of the free-floating
beam, $m_{add}$ is the added mass per unit surface of the beam, $g$ is the
gravitational acceleration, and $\rho_{i}$ and $\rho_{w}$ are the densities of
the ice and water, respectively. Based on the observations, the effective
elastic modulus was estimated at $E^{*}=488$ MPa.

###  2.4 Data Analysis

Source triangulation was used to determine where the vibrations originate from
and at what speed the vibrations were propagating. The arrival times of
vibration events at the location of the geophones may be determined by:

$\begin{array}[]{l}t_{1}=t_{0}+\sqrt{(x_{1}-x_{0})^{2}+(y_{1}-y_{0})^{2}}/c\\\
t_{2}=t_{0}+\sqrt{(x_{2}-x_{0})^{2}+(y_{2}-y_{0})^{2}}/c\\\
t_{3}=t_{0}+\sqrt{(x_{3}-x_{0})^{2}+(y_{3}-y_{0})^{2}}/c\end{array}$ (2)

where $t$ refers to time and $(x,y)$ the coordinates in a relative frame of
reference and $c$ is the propagation speed of the waves. Subscripts refer to
the geophone (1–3) or the source (0), i.e., $t_{1}$ is the instant at which a
vibration event arrives at the location of geophone 1 at $(x_{1},y_{1})$. Eq.
2 contain four unknowns and thus would normally require four geophones to
identify where the vibrations are coming from. However, if information is
known about the speed at which these vibrations propagate, one more equation
is available and the source of the vibration event can be determined with
three geophones. In the case of compressive and shear waves, the wave speed is
given, respectively, by (Stein et al., 1998):

$\begin{array}[]{l}c_{c}=\sqrt{\frac{E}{\rho_{i}(1-\theta^{2})}}\\\
c_{s}=\sqrt{\frac{E}{2\rho_{i}(1+\theta)}}\end{array}$ (3)

where $\theta$ is the Poisson ratio, here taken as 0.3. For flexural waves in
sea ice we assume the ice to be a thin elastic plate such that the dispersion
relation of flexural waves in ice can be modeled as (Fox and Squire, 1991;
Collins et al., 2017):

$\frac{\omega^{2}}{Lk^{4}/\rho_{w}-M\omega^{2}+g}=k\tanh(kd)$ (4)

with $L=E^{*}H^{3}/(12[1-\theta^{2}])$, $M=H\rho_{i}/\rho_{w}$, where
$\omega=2\pi/T$ is the radian frequency with wave period $T$, the wave number
$k$, and water depth $d$. From Eq. 4 the propagation speed of wave energy of
flexural waves (the group velocity $c_{g}$) can be determined as
$c_{g}=\partial\omega/\partial k$. We note that Eq. 4 introduces two
additional unknowns, the ice thickness $H$ and effective elastic modulus
$E^{*}$. However, as flexural waves are dispersive, observations of $t_{1}$,
$t_{2}$ and $t_{3}$ across a wide range of wave frequencies may provide enough
information to determine $x_{0}$, $y_{0}$, $H$ and $E^{*}$.

We use here two different approaches in retrieving information of the ice
properties and the source of vibration events. We focus for this on flexural
waves. The first approach considers the difference in arrival times of
vibrations with frequency $f$ between pairs of geophones to estimate the group
velocity as
$c_{g,ij}(f)=(\sqrt{(x_{i}-x_{0})^{2}+(y_{i}-y_{0})^{2}}-\sqrt{(x_{j}-x_{0})^{2}+(y_{j}-y_{0})^{2}})/(t_{i}-t_{j})$,
where $i$ and $j$ refer to the geophone numbers. For this, arbitrary
$(x_{0},y_{0})$ are taken to estimate $c_{g}(f)$ for each pair, where
solutions for $c_{g}(f)$ and $(x_{0},y_{0})$ are found when
$c_{g,12}=c_{g,13}=c_{g,23}$. For different $f$, this provides observations of
$c_{g}$ across a range of $f$ to which the dispersion relationship Eq. 4 can
be fitted. For the second approach, Eqs. 2 and 4 are solved iteratively to
obtain the lowest root-mean-square error (RMSE) in $t_{1}$, $t_{2}$ and
$t_{3}$ with variable $(x_{0},y_{0},t_{0})$, $H$ and $E^{*}$. The first
approach may be seen as local as the phase speed of the vibrations are
estimated locally at the site of the geophones, whilst the second approach may
be viewed as non-local as the sea ice properties between the vibration source
and the geophones will influence the results.

##  3 Results

In this study we focus on three major events that were recorded on the 18th,
21st and 22nd of February by all three geophones. We refer to these events as
event I–III, respectively. In Fig. 4a,b the timeseries of the $y$ and $z$
components of geophone 3 during event III are shown. We note that events I and
II have similar characteristics as event III, except that the magnitude of the
vibrations recorded during event III were the largest. At around $t=170$ s,
high frequency vibrations are observed, followed by low frequency waves over
the next few hundred seconds. The high frequency vibrations are strongest in
the horizontal direction and relatively weak in the vertical component,
suggesting this may be a compressive wave. The low frequency waves induces
surface velocities similar in magnitude in both horizontal and vertical
direction. While magnitude of the voltage amplitudes from $t=400$ to 550 s as
observed in Fig. 4b is increasing, we would like to reiterate that this is not
directly translatable to an increased amplitude of the surface elevation of
the ice as the geophones response to low frequencies is not linear. In Fig. 4c
the corresponding timeseries of sound pressure recorded by the hydrophone is
shown. We note that we have shifted the time series of the hydrophone
recordings by 65 seconds to match the records of the geophone loggers. This
shift is caused by clock drift as the hydrophone logger, unlike the geophone
loggers, is not synchronized by GPS. Based on the audio records, we suspect
the event to originate from the glacier and caused by calving (a snapshot of
the audio record during this initial event can be found at
http://doi.org/10.5281/zenodo.7750699).

Figure 4: Timeseries of (a) horizontal and (b) vertical vibrations recorded by
geophone 3 during event III. A high frequency event around $t=200$ s is
closely followed by low frequency waves between $t=300$ s and 600 s. Note that
the vertical axes are scaled differently. Hydrophone recording during is shown
in (c) shifted by 65 s, a shift originating from clock drift of the hydrophone
logger, to match the time of the initial vibrations recorded by the geophone
loggers. Figure 5: Continuous wavelet transform of the timeseries shown in
Fig. 4a. Colour is indicative of the energy, with blue indicating high energy
and yellow low energy. Figure 6: Wavelet synchrosqueezed transform of the
y-component of geophone 3 during events (a) I, (b) II and (c) III,
respectively. Colour is indicative of the energy, with blue indicating high
energy and yellow low energy.

In Fig. 5 the continuous wavelet transform is shown of the timeseries
presented in Fig. 4a. The initial vibrations around $t=200$ s are within a
frequency range of 0.4–18 Hz and do not appear to be dispersive. To obtain
information on the times at which these high frequency vibrations arrived at
the geophones, we cross-correlate between the horizontal components of
geophone pairs to estimate the transit time of the initial high frequency
vibrations between these pairs, i.e., this will provide estimates of
$t_{1}-t_{2}$, $t_{1}-t_{3}$ and $t_{2}-t_{3}$. Following the first approach
(see Data Analysis section), we estimate the propagation speed of these
initial vibrations to be $2178\pm 40$, $2184\pm 20$ and $2321\pm 20$ m/s for
events I–III, respectively, and are coming from the direction of the glacier.
We suspect that these vibrations are compressive waves and thus do not satisfy
the frequency dispersion relation of Eq. 4.

As the low frequency waves that follow the initial high frequency vibrations
appear to be dispersive (see Fig. 5), and the waves are unlikely to have been
generated in the open water, we can obtain the $f$-$t$-relationship from the
wavelet spectrum. For this, we use the wavelet synchrosqueezed transform using
Matlab’s wsst-function and bump wavelet to identify the times of maximum
energy at each frequency (Fig. 6). We note that minima in $c_{g}$ (or maxima
in $t$) can be observed around 0.14 Hz. From the transit times between each
geophone pair and for each wave frequency we can then estimate $c_{g}$ as a
function of $f$. In Fig. 7a estimates of $c_{g}$ are compared against the wave
dispersion model in sea ice (Eq. 4) for various values of the ice thickness
$H$ and effective elastic modulus $E^{*}$. For waves with $f\lesssim 0.1$ Hz
the propagation speed of waves is largely unaffected by the ice. Notable
deviations start to occur for $f\gtrsim 0.11$ where thicker ice and a larger
effective elastic modulus lead to increases in $c_{g}$. Knowing that the ice
thickness near the geophones was measured at about 0.5 m, our initial estimate
of $E^{*}$ is 1.4 GPa. This is, however, strongly based on the observations
during event I in the frequency range 0.16–0.2 Hz. From the transit times
between geophone pairs, i.e., $t_{1}-t_{2}$, $t_{1}-t_{3}$ and $t_{2}-t_{3}$,
we can also identify the direction of the source of the waves relative to the
geophones (Fig. 7b). Assuming that all waves are supposed to come from the
same source for each event, and thus same direction for each event, it
suggests that uncertainty in estimates of $c_{g}$ is largest for the higher
frequencies.

Figure 7: (a) Estimates of group velocity as determined by the transit time
between the three geophones for the three wave events. In colour are given
various estimates of the group velocity based on different effective elastic
modulus $E^{*}$ and sea ice thickness $H$. (b) Estimates of the direction of
the wave source relative to the geophones, where North is taken as 0∘.

In Fig. 8 the dispersion relationship is fitted directly to the observed
$f$-$t$-curves following the second approach (see Data Analysis section) to
obtain best estimates of $H$ and $E^{*}$. The observations match the
dispersion relationship of waves in ice well, although the minima in
$c_{g}(f)$ (or maxima in $t(f)$) tend to be overpredicted (underpredicted).
This may, however, also be an artifact of the wavelet synchrosqueezed
transform.

As impacts of $E^{*}$ and $H$ on $c_{g}$ are similar for frequencies smaller
than about $f\lesssim 1$ Hz, multiple combinations of $E^{*}$ and $H$ can
approximate the observations well (Fig. 9). Noting that the ice thickness
around the geophones was measured at $H\approx 0.5$ m thick, and ice thickness
near the glacier may be estimated at about $H\approx 0.75$ m based on
extrapolation of the ice thickness measurements (a value similar to those
measured near the glacier in 2011, e.g., Marchenko et al. (2011)), the average
ice thickness through which the vibrations traveled to reach the geophones is
roughly 0.63 m. This would result in best estimates of the elastic modulus as
0.67, 0.64 and 0.42 GPa for events I–III, respectively. We note that this is
consistent with the measured values of $E\approx 0.5$ GPa at the ice edge from
the cantilever experiments (see section Elastic Modulus).

Figure 8: Best fits of the dispersion relationship in sea ice (solid line)
against observed arrival times at the three geophones for the three events
(markers). Note, multiple solutions exist for $H$-$E^{*}$ that can replicate
the best fit, see Fig. 9. Figure 9: Best fit solutions to the wave arrival
times as measured by the three geophones for the effective elastic modulus
$E^{*}$ and sea ice thickness $H$.

Aside from estimates of the ice thickness and effective elastic modulus, the
source of the vibrations can be approximated by finding the lowest RMSE fit of
the dispersion relation against the $f$-$t$-curve. Specifically, we find a
global minimum for $(x_{0},y_{0})$ for all three dispersive wave events.
Contours of the RMSE for event III are shown in Fig. 10 with a best estimate
location of the vibration source located near the glacier wall, confirming
that the events are originating from the glacier. In Fig. 11 the best-estimate
source locations of all three events are shown, including the direction of the
high-frequency vibrations. We note that the direction of the high frequency
vibration is based on a single data point and is thus expected to be less
accurate than the estimate based on the low-frequency waves. They are,
nevertheless, consistently directed towards the glacier.

Figure 10: Estimates of the source of the dispersive waves during event III
(plus sign) and associated contours of the RMSE of the best fits. Location of
the geophones are identified by cross-markers. Sentinel-1 image taken on
24-02-2022 is given in colour. Figure 11: Estimates of the source of the
dispersive waves during events I–III (plus markers) and direction of the high
frequency events (solid lines). Location of the geophones are identified by
cross-markers. Sentinel-1 image taken on 24-02-2022 is given in colour.

##  4 Discussion

In this study we estimated the effective elastic modulus and ice thickness
from passive wave observations. We find that the effective elastic modulus
derived from fitting the observed wave arrival times to the modeled arrival
times retrieves estimates of 0.4–0.7 GPa, whereas estimates based on
observations of the group velocity leads to a value of approximately 1.4 GPa.
We believe the former is more accurate given that it uses considerably more
data to perform the fitting. Particularly, we see scatter in the direction
estimates of the vibration sources relative to the instruments for the higher
frequency range (i.e., Fig. 7b) which is a frequency range fundamental in the
fit against the dispersion relationship in ice. Moreover, the measured
effective elastic modulus of 0.5 GPa from cantilever experiments is consistent
with the former as well. Nevertheless, the difference may also be attributed
to the variability of sea ice conditions, in particular the sea ice
temperature. The sea ice temperature near the ice edge was measured between
-2.5∘C and -1.5∘C, and expected to be much lower near the glacier between
-15∘C and -10∘C. Considering either of these estimates, our results suggests
that the usage of a default value of $E^{*}=5.5$ GPa in wave-ice interaction
models may lead to significant errors and/or uncertainties in model
simulations as those measured in this study are significantly lower. Further
field experiments need to be performed to provide more information on the
variability of $E^{*}$ in the field and to provide further recommendations on
how to parameterize $E^{*}$.

Figure 12: Variability of the group velocity $\Delta c_{g}=c_{g}-c_{g,ref}$
for combination solutions of $H$-$E^{*}$ with wave frequency. Here,
$E^{*}=0.7$ GPa and $H=0.54$ m are taken as for the reference velocity
$c_{g,ref}$. Impact of $E^{*}$ and $H$ to the shape of the dispersion
relationship become significant for frequencies above 1 Hz.

While our observations were able to provide estimates of $H$-$E^{*}$
combinations, details on the ice thickness were required before $E^{*}$ could
be estimated explicitly. This is largely because the impacts of $E^{*}$ and
$H$ on the shape of the dispersion relationship are too small for the
frequency range of our observations from 0.08–0.28 Hz. To illustrate, in Fig.
12 we compare $\Delta c_{g}=c_{g}-c_{g,ref}$ for different points on the
$H$-$E^{*}$ curve for event III (see Fig. 9), with $E^{*}=0.7$ GPa and
$H=0.54$ m taken as the reference group velocity $c_{g,ref}$. Up to $f\approx
1$ Hz the different combinations of $H$-$E^{*}$ have very limited impact on
the shape of $c_{g}(f)$, meaning that the absolute differences in $c_{g}$ for
different $H$-$E^{*}$ combinations are well within the measurement uncertainty
of $c_{g}$. However, if observations are available for $f>1$ Hz, the range of
$H$-$E^{*}$ solutions may be reduced significantly as differences in the group
velocities between different solutions may exceed 1 m/s, whilst for
observations of $f>10$ Hz we expect that explicit estimates of $E^{*}$ and $H$
become feasible. Although the dispersive waves presented in this study (i.e.,
Fig. 4) does not contain significant or identifiable energy at $f>0.3$ Hz, we
note that measurements of dispersive waves at a higher frequency range are
frequently encountered in our dataset. For instance, in Fig. 13 an example of
vibrations caused by a sequence of suspected (thermal) cracks is shown with
identifiable phase-time information extending beyond 10 Hz. Unfortunately, no
further information of $H$-$E^{*}$ could be obtained from this data here as
these cracking signals were only identifiable in the z-components of the
geophones whilst only two geophones had this component working. Nevertheless,
from the continuous wavelet transform we can see that the events arrived first
at geophone 2 and we can infer from the data that, taking $E^{*}=0.6$ GPa and
$H=0.52$ m, the source of these cracking events was about 110 m from geophone
2 and 260 m from geophone 3 (see dashed lines in Fig. 13).

Figure 13: Example wavelet spectrum of the z-component of the geophone loggers
during suspected cracking events. Black dashed line indicates best fit. Colour
is indicative of the energy, with blue indicating high energy and yellow low
energy.

Observations of the elastic modulus thus far referred to the effective elastic
modulus. However, from the propagation speed of compressive and shear waves in
the ice, one may estimate the purely elastic modulus as well (i.e., Eq. 3).
From the direction of the $x$-$y$ velocity vectors of the geophone signals we
may approximate the direction of the initial high frequency vibration events
(Fig. 14). We note that the directions as observed by geophone 2 and 3 are
well aligned with the direction from the geophones to the glacier for all
three events which supports the hypothesis that the recorded high frequency
vibrations are indeed compressive waves. However, the direction of the vectors
for geophone 1 during these events is consistently shifted by 50∘–60∘ for
unknown reasons, but we suspect that this is because an error in its alignment
during deployment. From Eq. 3 the purely elastic modulus $E$ is then estimated
at $3.98\pm 0.14$, $4.00\pm 0.07$ and $4.52\pm 0.08$ GPa for events I–III,
respectively. We note that the purely elastic modulus is always larger than
the effective elastic modulus, except for very high loading rates where
$E^{*}$ will approach the value of $E$. We thus find here that $E\approx
6E^{*}$ under the conditions experienced during the field experiment, which is
similar to the results of Karulina et al. (2019) who observed a difference by
a factor of 4-5 between measurements of the elastic modulus obtained through
cantilever experiments and direct acoustic observations.

Figure 14: Probability density of the vector direction of initial high
frequency vibrations during event I (solid line), event II (dashed line) and
event 3 (dash-dot line). North and east correspond to 0∘ and 90∘,
respectively.

Based on the geophone and hydrophone records, we suspect events I–III to
originate from calving events at the glacier wall. The number of reported ice
failure events in Spitsbergen glaciers is estimated to be in the range of
50-100 in February based on observations in the years 2010-2012 (Fedorov et
al., 2016), with most events not related to glacier wall calving and occur in
the accumulation zone of glaciers. This seems to be consistent with our
observations where only three significant events were recorded over a period
of about two weeks. From this, we may also conclude that ice failure events at
a distance from glacier front do not bring energy to the sea ice near the
wall.

Although our experiments provide estimates of the effective elastic modulus
similar to that measured in-situ through cantilever experiments, there are
limitations to the methodology used here. Most importantly, we assume here
that the wave dispersion model of Fox and Squire (1991), i.e., Eq. 4, is
valid. Secondly, our observations are most likely tsunami events caused by
glacier calving, in analogy to landslide induced tsunamis. The generation of
such long waves at such proximity of the instruments are restricted to very
limited sites across the polar regions. When waves are generated in open water
and cross various types of sea ice, the cumulative impact of the inhomogeneous
ice on the group velocity of low frequency waves makes it problematic to
estimate the effective elastic modulus and ice thickness based on the arrival
times of waves (i.e., second approach, see Data Analysis section). In such a
case, $E^{*}$ can only be measured by estimating $c_{g}$ from the transit
times between the instruments (i.e., first approach, see Data Analysis
section) or from high frequency vibrations. Noting that a 1 s period wave in
sea ice has a wave length of roughly 20 m, the main length scale restriction
for an ice floe then seems to be the minimum distance required between the
geophones to measure the travel times of vibrations between the geophones.
Based on our observations here we expect this minimum distance to be about 100
m and thus a minimum diameter of an ice floe of a few hundred meters. While we
used three geophones in this study, we recommend to increase this to 4–5
geophones in future experiments, which is expected to increase the accuracy of
estimates of sea ice properties and safeguards project outcomes in case of
instrument malfunctioning. Further development of the geophone logger is
required for automatic detection of significant vibration events and the
corresponding dispersion curves when instruments cannot be retrieved. This
would require developments in on-board data processing capabilities and
changes in hardware to accommodate for this (such as the addition of a
microprocessor like the Raspberry Pi). While the most promising route for
automatic signal detection seems to be through wavelet spectrum analysis
(e.g., see Figs. 6 and 13), further study is required in establishing
threshold criteria for the fitting of $f$-$t$-curves and measures of quality
control when there is no access to the raw data.

##  5 Conclusions

We showed that the elastic modulus and sea ice thickness of fast ice can be
estimated from measurements of waves in sea ice. By determining the arrival
time of wave events at the geophones and their phase speed, we obtain
estimates of the purely elastic modulus during three different major wave
events of 4–4.5 GPa, and the effective elastic modulus of 0.4-0.7 GPa.
Estimates of the effective elastic modulus correspond well to the measured
effective elastic modulus of 0.5 GPa from in-situ cantilever experiments.
Unfortunately, no explicit estimates of the elastic modulus and ice thickness
were possible with our current dataset as their respective impacts on the
shape of the dispersion relationship is similar for frequencies below about 1
Hz. Thus, in-situ measurements of ice-thickness were required to obtain
explicit estimates of the effective elastic modulus. However, explicit
estimates of both ice thickness and elastic modulus will be possible with
observations of dispersive waves with frequencies larger than 1 Hz. We note
that such signals are frequently observed in our dataset, but due to
experimental limitations could not be used here. Nevertheless, our results
confirm that low-cost geophones can be used to estimate sea ice properties in
fast ice passively. Based on the frequent occurrence of high frequency
dispersive wave events in our dataset beyond 10 Hz, we expect this method to
be possible on large ice floes as well. Progress in retrieving sea ice
properties from wave measurements is expected to increase the dataset of in-
situ sea ice properties observations and improve our understanding of wave-ice
interactions.

## Acknowledgements

Alexander V. Babanin acknowledge support from the US Office of Naval Research
grant N62909-20-1-2080. Joey J. Voermans, Alexander V. Babanin were supported
by the Australian Antarctic Program under project 4593. Joey J. Voermans, Jean
Rabault, Aleksey Marchenko, Takuji Waseda, and Alexander V. Babanin
acknowledge the support of the Research Council of Norway and Svalbard Science
Forum under project 311266. Authors would like to acknowledge that this work
contains modified Copernicus Sentinel data 2022 processed by Sentinel Hub. The
acoustic recording of the hydrophone during event III can be found at
http://doi.org/10.5281/zenodo.7750699. Firmware and hardware design is made
open-source and can be found at http://github.com/jvoermans/Geophone_Logger.
All data is made available in an online repository, please visit the github
repository for further directions on accessing the data.

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