# Loop Quantum Gravity motivated multihorizon rotating black holes

Jitendra Kumar a<EMAIL_ADDRESS>Shafqat Ul Islama<EMAIL_ADDRESS>Sushant G. Ghosh${}^{a,\;b}$<EMAIL_ADDRESS><EMAIL_ADDRESS>a Centre for
Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India  b
Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics
and Computer Science, University of KwaZulu-Natal, Private Bag 54001, Durban
4000, South Africa

###### Abstract

With a semiclassical polymerization in the loop quantum gravity (LQG), the
interior of the Schwarzschild black holes provides a captivating single-
horizon regular black hole spacetime. The shortage of rotating black hole
models in loop quantum gravity (LQG) substantially restrains the progress of
testing LQG from observations. Motivated by this, starting with a spherical
LQG black hole as a seed metric, we construct a rotating spacetime using the
revised Newman-Janis algorithm, namely, the LQG-motivated rotating black holes
(LMRBH), which encompasses Kerr ($l=0$) black holes as an exceptional case. We
discover that for any random $l>0$, unlike Kerr black hole, an extremal LMRBH
refers to a black hole with angular momentum $a>M$. The rotating metric, in
parameter space, describes (1) black holes with an event and Cauchy horizon,
(2) black holes with three horizons, (3) black holes with only one horizon or
(4) no horizon spacetime. We also discuss the horizon and global structure of
the LMRBH spacetimes and its dependence on $l/M$ that exhibits rich spacetime
structures in the ($M,\;a,\;l$) parameter space.

##  I Introduction

That the gravitational collapse of an adequately massive star ($\sim
3.5M_{\odot}$) inevitably guides to a spacetime singularity is a reality
demonstrated by an elegant theorem due to Hawking and Penrose [1] (see also
Hawking and Ellis [2]). However, it is a general belief that these
singularities do not live in nature but are an artefact of classical general
relativity. By its very definition, the existence of a singularity conveys
spacetime failing to exist, signalling a breakdown of the laws of physics.
Thus, singularities must be replaced by other objects in a more unified theory
for these laws to exist. The extreme condition at the singularity indicates
that one should count on quantum gravity, developed to fix this singularity
[3]. In particular, recent work indicates that loop quantum gravity (LQG) may
be capable of resolving the singularities [4, 5, 6]. Because of the intrinsic
problem in solving the whole system, the emphasis is on spherically symmetric
black hole spacetimes [7, 8, 9].

LQG is a non-perturbative quantum gravity technique within which the
discretized eigenvalues of a set of geometrical operators handle the core of
spacetime. Remarkably, the non-zero area gap associated with the geometrical
operators is crucial in removing classical singularities [10]. It turns out
that within the framework of LQG and considering the semi-classical regimes,
one can construct various effective but spherical symmetric black hole models
such that singularity occuring in the general relativity is now substituted by
a transition surface that connects a black hole and a white hole region [11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. The most
successful endeavour is the phase space quantization, i.e., the polymerization
techniques formulated in LQG, which settles the big-bang singularity [27, 28,
29]. It is naturalistic to represent the black-hole spacetimes by assuming the
quantum corrections introduced by polymerization schemes. Especially the semi-
classical polymerization technique, that preserves the LQG’s idea of spacetime
discreteness, turns out to be attractive and worthwhile [30, 31, 32, 33, 8].
Most of the regular black holes closely connected to a potential theory of
quantum gravity are not derived from but are _inspired_ by quantum gravity.
They make up applicable phenomenological models but make their physical
justification less straightforward [34].

Peltola and Kunstatter [35], following earlier works [8, 9, 36, 37, 32, 38,
39, 40], used the effective field theory technique to the symmetry reduced
black hole Hamiltonian reported in terms of the two configuration space
variables: the areal radius $r$ and a function $\rho$ obtained from the
conformal mode of the metric. The effective field equations that arise are
readily resolvable and guide to the following black hole exterior metric [33,
35]

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(\sqrt{1-\frac{l^{2}}{y^{2}}}-\frac{2M}{y}\right)dt^{2}+\frac{\left(1-\frac{l^{2}}{y^{2}}\right)^{-1}}{\left(\sqrt{1-\frac{l^{2}}{y^{2}}}-\frac{2M}{y}\right)}dy^{2}$
(1) $\displaystyle+$ $\displaystyle y^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$

Thereby the dynamical field equations in the partially polymerized theory
admit the above static and spherically symmetric black holes [33, 35] with $t$
being the Schwarzschild time, $r$ being the areal radius and $l$ being the
polymer length scale. The solution is asymptotically flat at $y\to\infty$. One
of the most striking features of this four-dimensional quantum-corrected black
hole metric (1) is that it has a single horizon at $y\equiv
y_{+}=\sqrt{4M^{2}+l^{2}}$.

|
---|---
Figure 1: Behaviour of $\rho$, $\rho+P_{\theta}$ for different values of the
parameter $l$.

Thus we have several effective polymerized spherically symmetric black hole
solutions. However, the spherical black hole models can hardly be tested by
astrophysical observations, as the black hole spin, i.e., rotating black
holes, typically found in nature, plays a fundamental role in any
astrophysical process. Further, without rotating black hole models, LQG
substantially hinders the progress of testing LQG theory from observations.
The Kerr metric [41] is the most important black hole solutions of general
relativity that can arise from gravitational collapses of massive star. Also,
it turns out that when applied to other models in LQG, the revised Newman-
Janis works quite well in generating rotating metrics starting with their non-
rotating seed metrics [42, 43, 26, 44, 45, 46, 47, 48, 49]. This encouraged us
to pursue rotating or axisymmetric generalization of the spherical metric (1)
or finding a Kerr-like metric, namely, a LQG motivated rotating black hole
(LMRBH) metric suitable for test with astrophysical observations. This is
achieved by by starting with a spherical LQG black hole (1) as a seed metric,
we construct a rotating spacetime using the revised Newman-Janis algorithm. We
discuss the various black hole properties including the horizon structure, and
construct the Penrose diagrams and the embedding diagrams .

The organization of this paper is as tails. We begin in Sec. II with the
building of the rotating counterpart of the spherical seed metric (2), namely,
the LMRBH metric. Before this, we also discuss generic features of the LQG
polymarized single horizon spherical black hole in the same section. In Sec.
III, we discuss the horizon structure and phase space of the LMRBH spacetimes
and we present its peculiar properties, e.g, the LMRBH spacetimes, depending
on the values of parameters $a$ and $l$, can admit up to three horizons. Other
is the possibility that the black hole spin parameter $a$ may surpass the
black hole mass parameter $M$. An extreme black hole in LMRBH will have $a>M$;
we provide numerical illustrations in Sec. III. In Sec. IV, we investigate the
the global structure of the LMRBH via Penrose diagrams of LMRBH and the
extended spacetime. We also discuss here the embedding diagrams of the
equatorial plane of LMRBH spacetime for the selected regions of the parameter
space. Finally, we outline our main results in Sec. V.

##  II LQG motivated Polymerized Rotating Black Holes

We can do away with the coordinate singularity at $y=l$ for the metric (1), by
using the transformation $y=\sqrt{r^{2}+l^{2}}$; then the metric takes the
form

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(\frac{r-2M}{\sqrt{r^{2}+l^{2}}}\right)dt^{2}+\frac{1}{\left(\frac{r-2M}{\sqrt{r^{2}+l^{2}}}\right)}dr^{2}$
(2) $\displaystyle+$ $\displaystyle(r^{2}+l^{2})(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$

Here, the radial coordinate $r$ assumes the full range $0\leq r\leq\infty$.
This metric irrespective of the value of $l$ always represents a black hole
and has a single horizon, namely event horizon located at fixed radial
coordinate, $r=2M$.

Figure 2: Plot showing behaviour of horizons ($\Delta(r)$ vs $r$) for fixed
$a/M$ and varying $l/M$; the points where $\Delta(r)=0$ are the locations of
horizons.

It is necessary to investigate the behaviour of scalar invariants in order to
assess the regularity of solution (2). For this metric, the invariants Ricci
scalar $R=R_{ab}R^{ab}$, ($R_{ab}$ Ricci tensor) and Kretschmann scalar
$K=R_{abcd}R^{abcd}$ ($R_{abcd}$ Reimann tensor), respectively, assume the
form

$\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{1}{\left(l^{2}+r^{2}\right)^{5/2}}\left[l^{2}(6M-5r)+2\left(l^{2}+r^{2}\right)^{3/2}-2r^{3}\right]$
(3) $\displaystyle K$ $\displaystyle=$
$\displaystyle\frac{1}{\left(l^{2}+r^{2}\right)^{5}}[12r^{4}\left(l^{2}+4M^{2}\right)+r^{2}\left(41l^{4}-48l^{2}M^{2}\right)$
(4)
$\displaystyle+36l^{4}M^{2}+56l^{2}Mr^{3}-60l^{4}Mr+4l^{6}-16Mr^{5}+8r^{6}$
$\displaystyle-8r^{2}\left(l^{2}+r^{2}\right)^{3/2}(r-2M)]$

Hence, these invariants are everywhere finite, and vanish rapidly at far
distances from the black hole. As a result, the polymerized black hole metric
describes a globally regular spacetime.

###  II.1 Energy conditions

From the Einstein field equation, we have $G^{\mu}_{\nu}=T^{\mu}_{\nu}$. Here,
$T^{\mu}_{\nu}$ is the energy momentum tensor associated with matter, which
reads

$T^{\mu}_{\nu}=\mbox{diag}(-\rho,P_{r},P_{\theta},P_{\phi}).$ (5)

We use Hawking-Ellis’ approach to assess the status of the various energy
conditions [50, 51, 52]. The weak energy condition (WEC) demands
$T_{\mu\nu}\xi^{\mu}\xi^{\nu}\geq 0$ for any timelike vector $\xi^{\mu}$. The
WEC requires

$\rho\geq 0;~{}~{}~{}\rho+P_{i}\geq 0,~{}~{}~{}(i=r,\;\theta,\;\phi)$ (6)

which guarantees that the energy density as measured by any local observer is
non-negative [50]. The strong energy condition (SEC), in addition to Eq. (6)
requires

$\rho+P_{r}+2P_{\theta}\geq 0.$ (7)

In our case, i.e. for the metric (2), we have

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{4l^{2}M+\left(l^{2}+r^{2}\right)^{3/2}-3l^{2}r-r^{3}}{\left(l^{2}+r^{2}\right)^{5/2}},$
(8) $\displaystyle\rho+P_{r}$ $\displaystyle=$
$\displaystyle-\frac{2l^{2}(r-2M)}{\left(l^{2}+r^{2}\right)^{5/2}},$ (9)
$\displaystyle\rho+P_{\theta}$ $\displaystyle=$
$\displaystyle\rho+P_{\phi}=\frac{l^{2}\left(2\sqrt{l^{2}+r^{2}}+6M-5r\right)}{2\left(l^{2}+r^{2}\right)^{5/2}}$
(10)
$\displaystyle+\frac{2r^{2}\left(\sqrt{l^{2}+r^{2}}-r\right)}{2\left(l^{2}+r^{2}\right)^{5/2}},$
$\displaystyle\rho+P_{r}+2~{}P_{\theta}$ $\displaystyle=$
$\displaystyle-\frac{l^{2}(r-2M)}{\left(l^{2}+r^{2}\right)^{5/2}}.$ (11)

First we note that $\rho+P_{r}>0$ for $r<2M$. The behaviour of $\rho$ and
$\rho+P_{\theta}$ is depicted in Fig. 1 which are positive for most of the
values of $r$. Thus, the WEC is mostly satisfied when $r<2M$. The violation
can be very small, depending on the value of $l$, as seen from Fig. 1.
Similarly, $\rho+P_{r}+2P_{\theta}>0$ for $r<2M$ and SEC is obeyed for most
values of $r<2M$, and violation, as in the case of WEC, is minimal.

The solution (2) will be the spherical _seed_ metric to generate the rotating
LMRBH metric by a non-complexification technique [53, 54].

Figure 3: Plot showing behaviour of horizons ($\Delta(r)$ vs $r$) for fixed
$l/M$ and varying $a/M$; the points where $\Delta(r)=0$ are the locations of
horizons.

###  II.2 Rotating metric

The Newman$-$Janis algorithm (NJA) has been commonly used to generate rotating
black hole solutions from their non-rotating or spherical counterparts [55].
While this algorithm was developed within general relativity, it has been more
recently applied to non-rotating solutions in modified gravity theories [56,
57, 58, 59]. Exercising the Newman$-$Janis algorithm to an arbitrary non-
general relativity spherically symmetric solution can introduce pathologies in
the resulting axially symmetric metric [60]. Hence, we derive the rotating
LMRBH black hole metric, using the modified NJA developed by Azreg-Aïnou’s - a
non-complexification procedure[53, 54]. The researchers have successfully
applied it for generating imperfect fluid rotating solutions in the
Boyer$-$Lindquist coordinates from spherically symmetric static black holes
[61, 62, 63, 64, 47], including some other rotating LQG metrics. We derived
the LQG-motivated rotating black hole (LMRBH) metric, starting from the
spherical metric (2), via the modified Newman$-$Janis algorithm [53, 54],
which in Boyer$-$Lindiquist coordinates reads

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left[1-\frac{2M(r)\sqrt{r^{2}+l^{2}}}{\rho^{2}}\right]dt^{2}+\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}$
(12) $\displaystyle-4aM(r)\sqrt{r^{2}+l^{2}}\sin^{2}\theta
dtd\phi+\frac{\mathcal{A}\sin^{2}\theta~{}}{\rho^{2}}d\phi^{2}$

where

$\displaystyle M(r)$ $\displaystyle=$ $\displaystyle
M-\frac{r-\sqrt{r^{2}+l^{2}}}{2}~{}~{}~{}~{}~{}\rho^{2}=r^{2}+l^{2}+a^{2}\cos^{2}\theta,$
$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle
r^{2}+l^{2}+a^{2}-2M(r)\sqrt{r^{2}+l^{2}},$ $\displaystyle\mathcal{A}$
$\displaystyle=$
$\displaystyle(r^{2}+l^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta.$ (13)

Figure 4: Parameter space ($M,a,l$) for LMRBH metric (12). We spell acronyms
and details in the text. The red transition line (EBH) corresponds to the
extremal black holes with degenerate horizons, where the outer two horizons
merge. The blue transition line also corresponds to the black hole with
degenerate horizons, but unlike the red line, the inner two horizons merge.
For the green line, $\Delta(r)=0$ admits two positive and $r=0$ root while for
dashed green line, we have $r=0$ as the only root.

The LMRBH metric (12) reverts to the standard Kerr black hole metric in the
limit $l\to 0$, and to the spherical metric (2) when $a=0$. The LMRBH
spacetime, like Kerr spacetime, admits two killing vectors $\eta^{\mu}_{(t)}$
and $\eta^{\mu}_{(\phi)}$ being invariant under $t$ and $\phi$
transformations.

The horizons are surface defined by $g^{rr}=0=\Delta$ and the radii of
horizons are zeroes of

$(r-2M)\sqrt{r^{2}+l^{2}}+a^{2}=0.$ (14)

Eq. (14), for special case $l=0$, reduces to $r^{2}-2Mr+a^{2}=0$ which admits
solutions $r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$, which are radii of Kerr black
hole. Analysis of $\Delta(r)$ reveals that $\Delta(r)$ has a local maxima at
$r=\left(M-\sqrt{M^{2}-2l^{2}}\right)/2$ and a local minima at
$r=(M+\sqrt{M^{2}-2l^{2}})/2$ when $l<M/\sqrt{2}$ and is strictly increasing
for $l\geq M/\sqrt{2}$. Also $\Delta(r)\to\pm\infty$ as $r\to\pm\infty$.
Therefore, $\Delta(r)=0$ will necessarily have at least one root and can have
maximum of three roots depending on the values of $a$ and $l$, with
possibility of one of the roots being negative (cf. Figures 2 and 3). We name
the roots $r_{1}$, $r_{2}$, $r_{3}$ with $r_{3}\leq r_{2}\leq r_{1}$. The
largest root, $r_{1}$ is always an event horizon. $r_{2}\leq r_{1}$, if it
exists, is always a Cauchy horizon. While $r_{3}$ is an additional horizon
inside the Cauchy horizon and when $r_{3}<0$, it corresponds to a horizon
inside $r=0$ surface. The parameter space ($M,a,l$) for LMRBH is depicted in
Fig. 4. Note that positive roots of $\Delta(r)=0$ correspond to the horizons
of LMRBH with the larger/smaller root corresponding to the event /Cauchy
horizon.

Lines and Dots in Fig. 4 | Transition | Critical value of $l$ for given $a$ | Critical value of $a$ for given $l$
---|---|---|---
|  |  $a/M$ |  $l_{c}/M$ |  $l/M$ |  $a_{c}/M$
Red line | BH-II to NH | 1.04 | 0.566629 | 0.2 | 1.01005
| 1.07 | 0.60462 | 0.4 | 1.04091
Dashed Red line | BH-III to BH-I | 1.1 | 0.645459 | 0.64 | 1.11044
| 1.12 | 0.674884 | 0.68 | 1.12694
Green line | BH-III to BH-II | 0.5 | 0.125 | 0.3 | 0.774597
| 0.7 | 0.245 | 0.55 | 1.04880
Dashed Green line | BH-I to NH | 1.12 | 0.6272 | 0.65 | 1.14017
| 1.2 | 0.72 | 0.7 | 1.18321
Blue line | BH-III to BH-I | 0.6 | 0.180746 | 0.4 | 0.884923
| 0.8 | 0.32442 | 0.6 | 1.06646
Red Dot | Surrounded by BH-III, BH-I and NH | 1.095 | 0.601 | 0.601 | 1.095
Black Dot | Surrounded by BH-III, BH-I | 1.139 | 0.7076 | 0.7076 | 1.139
Table 1:  Table summarizing the critical values of parameters
($a_{c}/M,l_{c}/M$) for different lines and dots in parameter space in Fig. 4.

It turns out that, for a given $a\;(l)$, there is a critical value of $l(a)$,
$l_{c}(a_{c})$ (on the red line), such that $\Delta(r)=0$ has a double root
which corresponds to an extremal LMRBH with degenerate horizons. Further, for
$a<a_{c}(l>l_{c})$, $\Delta(r)=0$ admit two simple positive zeros
corresponding LMRBH with Cauchy and event horizons (BH-II), while for
$a>a_{c}(l<l_{c})$, $\Delta(r)=0$ admit not positive zeros and correspond to
no-horizon (NH) spacetimes. Similarly, for given $a\;(l)$, we can find
critical value of $l(a)$, $l_{c}(a_{c})$ (see Table 1) on other transition
lines and dots (Fig. 4).

##  III Phase Diagram

The range of possibilities with each spacetime structure corresponding to a
particular region of the parameter space are defined as follows. We summarise
the range of cases with a “phase diagram” depicted in Fig. 4. We use the
following terminology for the parameter space under consideration for
different spacetime structures (cf. Fig. 4).

  * •

BH-I: In this region of parameter space (The grey region in Fig. 4)
$\Delta(r)=0$ has only one positive root (cf. Fig. 5) corresponding to the
single event horizon similar to the spherical black hole (1), and we refer to
this as single horizon LMRBH .

  * •

BH-II: $\Delta(r)=0$ has two positive roots $(r_{2}<r_{1})$ in this region and
one negative root ($r_{3}$) (light blue region in Fig. 4). The former case
corresponds to the Cauchy ($r_{2}$) and event horizon ($r_{1}$) (cf. Fig. 5).
This region is the most physically relevant one for considering rotating black
holes and we refer to this as generic LMRBH.

  * •

BH-III: In the yellow region in Fig. 4, $\Delta(r)=0$ admits three positive
roots viz $r_{1}$, $r_{2}$, $r_{3}$ and we address it as LMRBH with additional
interior horizon (cf. Fig. 7).

  * •

NH (peach colour region in Fig. 4): $\Delta(r)=0$ has only one root (cf. Fig.
7) which is negative. This region is not of physical interest being ruled out
by EHT observations of Sgr A* and is referred to as no-horizon (NH) spacetime.

  * •

EBH (Red line): This represents an extremal LMRBH with degenerate horizons
($r_{2}=r_{1}$), where the outer two horizons merge. $\Delta(r)=0$ also admits
one more negative root $r_{3}$ (cf. Fig. 6). At this line, there is a
transition from BH-II - NH or vice-versa.

  * •

Dashed Red line: This is an extreme case of BH-III where, like EBH, the outer
two horizons merge. But, unlike EBH, the third root $r_{3}$ of $\Delta(r)=0$
is positive (cf. Fig. 8). At this line, the transition occurs from BH-III -
BH-I or vice-versa.

  * •

Blue line: It represents a black hole with two degenerate horizons as a
limiting case of BH-III where the inner two horizons merge. $\Delta(r)=0$
admits two equal positive roots $r_{2}=r_{3}$ and one larger root $r_{1}$ (cf.
Fig. 6). At this line, the transition takes place from BH-III - BH-I or vice-
versa.

  * •

Green line: It represents a black hole with two horizons, located at positive
values of $r$, and a null throat surface $r=0$ (cf. Fig. 8). At this line, the
transition happens from BH-III - BH-II or vice-versa.

  * •

Dashed Green line: Like, green line, the surface $r=0$ is null but, unlike it,
$\Delta(r)=0$ has no other roots than $r_{1}=0$ (cf. Fig. 7). At this line,
the transition happens from BH-I - NH or vice-versa.

  * •

Red Dot: The transition lines red and green intersect at this point. It
represents an extremal black hole with a null throat. Here, $\Delta(r)=0$
admits three roots, $r_{3}=0$ and $r_{1}=r_{2}\approx 0.763M$ for ($a,l$)
$\approx$ ($1.095M,0.601M$) (cf. Fig. 8).

  * •

Black Dot: At this point, transition lines red and blue merge. It represents
an extremal black hole with three degenerate horizons. Here, $\Delta(r)=0$
admits three equal roots, $r_{1}=r_{2}=r_{3}=0.5M$ for ($a,l$) $\approx$
($1.139M,0.707M$) and it is referred as ultra extremal black hole.

|
---|---
|
Figure 5: Plot showing $\Delta(r)$ in the parameter space (Fig. 4) (a) BH-I, where $\Delta(r)=0$ admits only one positive root $r_{1}=$ corresponding to event horizon (Top left), (b) BH-II where $\Delta(r)=0$ admits two distinct positive roots $r_{1}$ (event horizon) and $r_{2}$ (Cauchy horizon), and one negative root $r_{3}$ corresponding to additional third horizon (Top right). Penrose diagrams of LMRBH in the parameter space $(M,a,l)$: (a) BH-I (Bottom left) and (b) BH-II (Bottom right). They will identify the surfaces connected by the arrows as the same surfaces. |
---|---
|
Figure 6: Plot showing $\Delta(r)$ in the parameter space (Fig. 4) (a) EBH $l<0.601M$, where $\Delta(r)=0$ admits two positive equal roots $r_{1}=r_{2}$, corresponding to degenerate horizons and one negative root $r_{3}$, corresponding to the additional third horizon (Top left), (b) Blue line, where $\Delta(r)=0$ admits two positive equal roots $r_{2}=r_{3}$ corresponding to degenerate horizons and one larger root $r_{1}$ corresponding to event horizon(Top right). Penrose-Carter diagrams of the parameter space $(M,a,l)$ showing the region (a) EBH $l<0.601M$ (Bottom left) and (b) Blue line (Bottom right). The surfaces connected by the arrows are to be identified as the same surfaces. |  |
---|---|---
Figure 7: Plot showing $\Delta(r)$ in the region of parameter space (Fig. 4) (a) NH, where $\Delta(r)=0$ admits only one negative root $r_{1}$, corresponding to horizon inside $r=0$ surface (Left), (b) Dashed green line, $l>0.601M$ where $\Delta(r)=0$ admits only one root $r_{1}=0$, showing that transition surface $r=0$ is null (Middle), (c) BH-III, where $\Delta(r)=0$ admits three distinct positive roots $r_{1},r_{2},r_{3}$ with $r_{1}$ and $r_{2}$, respectively, corresponding to event and Cauchy horizon while $r_{3}$ is the additional third horizon (Right). The Penrose diagrams for NH and dashed green line, $l>0.601M$ are same as in Fig. 5 (left) except that the surface $r=0$ lies in the regions I and I′ and is timelike for NH; while for the case of dashed green line, $l>0.601M$, $r=r_{1}$ is replaced by the null surface $r=0$. The Penrose diagram for BH-III is the same as in Fig. 5 (right), except that the surface $r=0$ surface lies in the regions IV and IV′ and is spacelike. |  |
---|---|---
Figure 8: Plot showing $\Delta(r)$ in the region of parameter space (Fig. 4)
(a) Green line, $l<0.601M$ where $\Delta(r)=0$ admits two distinct positive
roots $r_{1}$ (event horizon) and $r_{2}$ (Cauchy horizon) and one additional
root at $r_{3}=0$ (null transition surface) (Left), (b) Dashed red line
$l>0.601M$, where $\Delta(r)=0$ admits two positive equal roots $r_{1}=r_{2}$,
corresponding to degenerate horizons and one smaller positive root $r_{3}$
corresponding to the additional third horizon (Middle), (c) Red dot
$a=1.095M,l=0.601M$, where $\Delta(r)=0$ admits two positive equal roots
$r_{1}=r_{2}$, corresponding to degenerate horizons and one root at $r_{3}=0$
(null transition surface) (Right). The Penrose diagram for green line,
$l<0.601M$ is same as in Fig. 5 (right) except that $r=r_{3}$ is replaced by
the null surface $r=0$. The Penrose diagrams for dashed red line and red dot
are the same as in Fig. 6 (left) except that the surface $r=0$ lies in the
regions III and III′ and is spacelike for dashed red line; while for the case
of red dot, $r=r_{3}$ is replaced by the null surface $r=0$.

##  IV Structure of rotating spacetime: Penrose diagrams

Penrose diagrams are the way to present the causal structure of spacetime on
2-D finite sheet of paper. In the diagrams the light cone structure remains
exactly same as in the Minkowski spacetime i.e. the null rays run at
$45\degree$ with respect to the space and time axis. So, from the diagrams we
can easily get the information what spacetime points are causally connected to
what other spacetime points. We have drawn the diagrams for the LMRBH
spacetime, corresponding to all the different regions in the parameter space
(Fig. 4), along the symmetric $z$-axis ($\theta=0$).

Penrose diagrams for the region BH-I are same as for its non-rotating
counterpart given in [33, 35]. Like the non-rotating version, it has two
exterior regions (I and I′), two black hole and white hole interior regions
($r=r_{1}$ to $r=0$), and two quantum corrected interior regions ($r=0$ to
$r=-\infty$) and the surface $r=0$, lying in the regions II and II′, is
spacelike and hidden behind the event horizon. Penrose diagrams for regions NH
and dashed green line, $l>0.601M$ are similar as for BH-I except the fact that
for region NH, the surface $r=0$ lies in the regions I and I′ of Penrose
diagram for BH-I and is time-like while for the dashed green line, $l>0.601M$
the surface $r=0$ is null.

Penrose diagrams for region BH-II look similar to that of Kerr black hole
except for the extra horizon at $r=r_{3}$. In the region beyond $r=r_{3}$, $t$
and $r$ interchange their roles, $r$ behaving as the time-coordinate while $t$
behaving as a space-coordinate. Like Kerr, one has event horizon at $r=r_{1}$
and a Cauchy horizon at $r=r_{2}$, but the classical ring singularity of the
Kerr black hole is replaced by the timelike transition surface $r=0$ inside
the event horizon. Also, in contrast to Kerr black hole, Penrose diagrams for
region BH-II have quantum corrected interior regions ($r_{3}$ to $r=-\infty$),
and the diagrams continue ad infinitum in the vertical as well the horizontal
direction. Penrose diagrams for regions BH-III and green line, $l<0.601M$ are
similar as for BH-II except for the fact that for region BH-III, the surface
$r=0$ lies in the regions IV and IV′ of the Penrose diagram for BH-II and is
spacelike, while for the green line, $l<0.601M$ the surface $r=0$ is null.

Penrose diagrams for the region EBH, $l<0.601M$ are similar to the Kerr
extremal black hole except for the extra horizon at $r=r_{3}$. The surface
$r=0$, lying in the regions II and II′ of the diagram, is timelike. Like Kerr
extremal black hole, the diagram for EBH continues ad infinitum in the
vertical direction. Penrose diagrams for the region represented by dashed red
line, $l>0.601M$ and red dot, $a=1.095M,l=0.601M$ are similar as for EBH,
$l<0.601M$ except the fact that for dashed red line, $l>0.601M$, the surface
$r=0$ lies in the regions III and III′ of Penrose diagram for EBH, $l<0.601M$
and is spacelike while for red dot, $a=1.095M,l=0.601M$ the surface $r=0$ is
null.

Penrose diagrams for the region represented by the blue line in the parameter
space look very different from the Penrose diagrams for the Kerr extremal
black hole or the Kerr black hole having two distinct horizons. Here, the
inner two horizons merge at positive $r=r_{2}=r_{3}$ with $t$ and $r$
interchanging their roles beyond it, such that the $r=0$ surface is spacelike.
The diagram continues ad infinitum in the horizontal direction.

Penrose diagram for the black dot ($a,l$) $\approx$ ($1.139M,0.707M$), which
represents an ultra extremal black hole, is exactly same as for the region
BH-I.

###  IV.1 Embedding diagrams

We embed the equatorial plane of the LQG motivated rotating spacetime in 3-D
Euclidean space. At $t=const$ and equatorial plane ($\theta=\pi/2$), the
metric (12) reduces to the form

$\displaystyle dl^{2}$ $\displaystyle=$
$\displaystyle\left[\frac{\rho^{2}}{\Delta}dr^{2}+\frac{\mathcal{A}}{\rho^{2}}d\phi^{2}\right]_{\theta=\pi/2}.$
(15)

The metric of 3-D Euclidean space in cylindrical coordinates can be written as

$\displaystyle dL^{2}$ $\displaystyle=$ $\displaystyle
dZ^{2}+dR^{2}+R^{2}d\phi^{2}.$ (16)

The embedded surface will be axially symmetric and thus can be described by
the function $Z=Z(R)$. For convenience we parametrize the embedding surface as
$Z(r)$, $R(r)$ where $r$ is the radial coordinate of the metric (15). The
metric induced on the embedding surface from the Euclidean space is

$\displaystyle dl^{2}$ $\displaystyle=$
$\displaystyle\left[\left(\frac{dZ}{dr}\right)^{2}+\left(\frac{dR}{dr}\right)^{2}\right]dr^{2}+R^{2}d\phi^{2}.$
(17)

We require that the metric (17) should be equal to metric (15). So, comparing
them we obtain

$\displaystyle R(r)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{A}{\rho^{2}}}|_{\theta=\pi/2},$ (18)
$\displaystyle\left(\frac{dZ}{dr}\right)^{2}+\left(\frac{dR}{dr}\right)^{2}$
$\displaystyle=$ $\displaystyle\frac{\rho^{2}}{\Delta}|_{\theta=\pi/2},$ (19)

which gives

$\displaystyle Z(r)$ $\displaystyle=$
$\displaystyle\int\,dr\sqrt{\frac{\rho^{2}}{\Delta}-\left(\frac{dR}{dr}\right)^{2}}|_{\theta=\pi/2}~{}~{}+~{}const$
(20)

|  |
---|---|---
|  |
Figure 9: Embedding diagrams, from event horizon to some finite radius $r$, in
the equatorial plane ($\theta=\pi/2$) for selected regions of the parameter
space ($M,a,l$). They are embeddable up to $r\to\infty$.

which can be solved numerically and, along with $R(r)$, gives the shape of the
embedding surface provided ($\rho^{2}/\Delta-(dR/dr)^{2})|_{\theta=\pi/2}\geq
0$. The embedding diagrams for the $\theta=\pi/2$ plane, corresponding to
different regions in Fig. 4, are depicted in Fig. 9. We can see from Eq. (20)
that the $\theta=\pi/2$ plane is not embeddable when $\Delta$ is negative as
the quantity under the square root becomes negative. Therefore, the upper
three diagrams in Fig. 9 are the embedding diagrams for the region outside the
event horizon. The boundary at the throat corresponds to the event horizon.
There is no upper limit on $r$, and the diagrams are embeddable up to
$r=\infty$. When we approach the EBH region, the throat in the embedding
diagram gets longer and longer, and for the EBH region, it becomes infinitely
long. For the embedding diagram EBH, we have chosen the values $a=1.02M$ and
$l=0.281M\approx l_{c}$, which are close to but not exactly in the EBH region
and lie in the region BH-II. For ($a=1.2M$, $l=0.54M$) and ($a=1.2M$,
$l=0.72M$), which respectively belong to the NH region and dashed green line,
$l>0.601M$, the $\theta=\pi/2$ plane is embeddable for all positive values of
$r$. The boundaries at the throats of these two diagrams correspond to the
$r=0$ surface.

##  V Discussion and Conclusion

The Kerr metric well describes astrophysical black holes. It is the only
stationary, vacuum axisymmetric metric that satisfies Einstein field equations
and does not have pathologies outside the event horizon. The Event Horizon
Telescope (EHT) observation unveiled the images of supermassive black holes
Sgr A* and M87*; [65, 66, 67, 68] The shadow size is within $~{}10\%$ of the
Kerr predictions, providing us with another tool to investigate the nature of
strong-field gravity. One can put constraints on potential deviations from the
Kerr, i.e., Kerr-like black holes arising in modified gravity from EHT
observations. Hence, it could test fundamental quantum gravity theories, such
as LQG, as the quantum effects cannot be ignored in the strong-field regime. A
rotating black hole plays a vital role in testing the quantum effects of
gravity in this regime. The dearth of rotating black hole models in LQG
substantially hampers the advancement of testing LQG from observations.

Here, we have handled this problem, we have used revised Newman-Janis
algorithm - a viable solution-generating method to generate a rotating
solution from a nonrotating seed metric (2). The resulting LMRBH spacetime
(12) possesses Kerr-like mathematical and several other interesting properties
and exhibits rich spacetime structure. Unlike Kerr black hole, the $a=M$ case
does not yield an extremal black hole in the LMRBH spacetime. We quantified
this by explicitly solving the $\Delta(r)=0$ for real roots and demonstrating
that the black hole spin yields around $a\approx 1.082M,\,1.09M$,
respectively, for $l=0.56M,\,0.6M$. Interestingly, the $\Delta(r)=0$ of LMRBH
spacetime can admit up to three real roots, viz., $r_{3},\,r_{2},\;r_{1}$ with
the possibility of the smallest root $r_{3}$ being negative. When all roots
are positive, $r_{2},\;r_{1}$, like Kerr black hole, are Cauchy and event
horizons, and $r_{3}$ is an additional horizon inside Cauchy horizon. The case
of two positive roots is what we call the generic black hole. As usual,
$r_{2}=r_{1}$ gives an extremal black hole (cf. Fig. 4 \- red line). For
dashed red line also black hole admits $r_{2}=r_{1}$ but on one side of the
line the black hole admits three horizons while on the other side has one
horizon (cf. Fig. 4). At the blue line (cf. Fig. 4), in contrast to the red
line, inner two horizons merge with $r_{3}=r_{2}$ and it is a transition line
between regions BH-III and BH-I. The red and green lines intersect at red dot
where again $r_{2}=r_{1}$, but $r_{3}=0$. Also the black dot in Fig. 4 refers
to the ultra extremal black hole where $r_{3}=r_{2}=r_{1}$. On the green line
the $\Delta(r)=0$ admits two positive roots, viz., $r_{2},r_{1}$ and
$r_{3}=0$, while on dashed green line $r_{1}=0$ is the only root. As another
possibility, $\Delta(r)=0$ has only one positive root, which is like spherical
counterpart of LMRBH with single horizon. However, the region of spacetime
where $\Delta(r)$ has negative root is not relevant for our discussion as that
region is ruled out by EHT observations [69].

As our result exhibits, the LMRBH has exciting properties, e.g., a transition
surface substitutes the classical ring singularity in Kerr black hole at
($r=0,~{}\theta=\pi/2$) and therby it provides sigularity resolution of Kerr
black hole. Thus, we can expect that LMRBH captures some description of LQG.

##  VI Acknowledgments

S.U.I and S.G.G. are supported by SERB-DST through project No.
CRG/2021/005771. J.K. would like to thank CSIR for providing SRF.

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# Searching for high-frequency gravitational waves by ground high field
magnetic resonant sweepings

H. Zheng  School of Physics, State Key Laboratory of Optoelectronic Materials
and Technologies, Sun Yat-sen University, Guangzhou 510275, China  L. F.
Wei111E-mail<EMAIL_ADDRESS>School of Physics, State Key Laboratory of
Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou
510275, China  Information Quantum Technology Laboratory, School of
Information Science and Technology, Southwest Jiaotong University, Chengdu
610031, China  H. Wen  Institute of Gravitational Physics and Department of
Physics, Chongqing University, Chongqing 400044, China  F. Y. Li  Institute of
Gravitational Physics and Department of Physics, Chongqing University,
Chongqing 400044, China

###### Abstract

With laser interferometers, LIGO-Virgo collaboration has recently realized the
direct detections of the intermediate-frequency (i.e., from dozens to hundreds
of Hertz) gravitational waves (GWs) by probing their mechanically-tidal
responses. Alternatively, in this letter we propose a feasible approach to
actively search for the high-frequency GWs by probing their electromagnetic
responses (EMRs) in a high alternating magnetic field. Differing from the
original Gertsenshtein-Zeldovich configuration (in which the EMRs are
proportional to the square of the amplitudes of the GWs, and consequently are
too weak to be detected experimentally), the EMRs of the GWs passing through
the present configuration are linearly related to the amplitudes of the GWs
and thus the relevant signals are detectable with the current weak-light
detection technique. As the wave impedances of the GWs-induced electromagnetic
signals (EMSs) are very different from those the EM radiations in flat space-
time, i.e, ($\cong 377\Omega$), the stronger background noises (without any
GWs information) could be effectively filtered out by using wave-matching
technique. Given the frequency of the applied alternating magnetic field is
conveniently adjustable, the configuration proposed here could be utilized to
actively search for the GWs (if they really exist) in a sufficiently-wide
frequency band (e.g., could be $10^{7}-10^{12}$Hz), once the scale of the
cavity and the sweeping frequency of the applied alternating magnetic field
are experimentally achievable.

PACS number(s): 03.65.Aa, 03.65.Ta, 03.67.Mn

Introduction.–It is well-known that, the existence of gravitational waves
(GWs), i.e., the ripples of the curved space-time book , is one of the most
important predictions in general relativity. Therefore, the direct detections
of the GWs not only provided new evidences to verify the Einstein’s
gravitational theory, but also begun the era of gravitational wave astronomy.
Yet, it is very difficult to verified such a prediction, due to the
significantly-minute amplitudes of the GWs. Interestingly, with the precise
laser interferometer the LIGO-Virgo collaboration Gw150914 ; Gw151226 ;
Gw170104 ; Gw170814  has realized recently the direct detections of the GWs by
probing their minute mechanical displacements with the peak strains being at
the order of $10^{-21}$ and the frequencies being in the low- and intermediate
frequency band (i.e., dozens and hundreds of Hertz). Still, probing the GWs in
wider bands, such as the high frequency ones expected by a series of models
for cosmology and high-energy astrophysical processes Giovannini a ;
Giovannini b ; relic a ; relic b , is a challenge.

Historically, many approaches have been tried to directly probe the GWs or
observe their certain indirect effects. In fact, before the recent LIGO-
Virgo’s direct detections, an indirect experimental evidence of the existence
of the GWs was obtained by observing the change in the orbital period of the
PSR B1913+16 binary PSR B 1913+16 a ; PSR B 1913+16 b , i.e., a pair of stars
with one of them being a pulsar. Also, observing the $B$-mode polarizations in
the cosmic microwave background (CBM) is regarded as another approach to
indirect verify the existence of the GWs in very low frequency band B-mode .
Since 1980s the laser interferometers, including the typical LIGO-Virgo setup
LIGO ; aLIGO ; Virgo  and the proposed Laser Interferometer Space Antenna
(LISA) LISA1 ; LISA2 , and Einstein telescope Einstein telescope 1 ; Einstein
telescope 2 ; Einstein telescope 3 , etc. have been demonstrated to implement
the detections of the GWs-induced mechanical displacements. Similarly, the
pulsar timings and pulsar timing arrays are believed as another astronomic
candidate to probe the GWs in low-frequency band Detweiler1979 .

Besides the mechanically-tidal effects used in the above GW-detectors, it is
believed that the electromagnetic responses (EMRs) of the GWs could be also
observed. This is based on the so-called Gertsenshtein-Zeldovich (GZ) effects
G-Z effect , i.e., the mutual conversions between EMWs and GWs in a high
stationary magnetic field. Unfortunately, the amplitude of such a GW-induced
EMW is too minute to be experimentally detected too minute , as the relevant
detectable quantities are proportional to the square of the perturbative
matric of the passing GWs second-order 1 ; second-order 2 . To overcome such a
difficulty, a series of modified schemes have been proposed by introducing
certain ground auxiliary EMWs; the plane EMWs auxiliary EMWs 1  and the
Gaussian beams auxiliary EMWs 2 ; auxiliary EMWs 3 , to enhance the observable
effects of the perturbative EMW signals.

Continuously, in this letter we propose a feasible approach to actively search
for the desired first-order EM perturbative signals generated by the high-
frequency GWs passing through a high magnetic field. Besides a high stationary
magnetic field utilized in the original GZ configuration, a relatively-weak
alternating magnetic field is additionally applied. By exactly solving the
relevant Einstein-Maxwell equation in a curved space-time, we show that the
GWs-induced EMRs in the present setup are linearly related to the amplitudes
of the passing GWs. These signals could be detected with the current weak-
signal probing technique d1 ; d2 ; d3 ; d4 . As the amplitude and the
frequency of the applied weak alternating magnetic field are locally
controllable, the scheme proposed here could be applied to actively search for
the probably-existing high-frequency GWs (predicted by a series of theoretical
models) in a sufficiently wide frequency band. Also, due to the wave
impedances of the GWs-induced EMW signals, satisfying the Einstein-Maxwell
equation in the curved space-time, are very different from those of the
background EM noises (which obey the flat space-time Maxwell equation and thus
are always take the value of $377\Omega$ in vacuum), the well-developed wave
impedance matching technique Wave impendance 1 ; Wave impendance 2  could be
utilized to safely filter out the background EM noises. As a consequence, only
the GWs-induced EM signals, i.e., the signal photon fluxes, are left to be
conducted into the detectors for detections.

Model and solutions.— A simplified configuration of our setup to actively
search for the GWs is shown in Fig. 1,

Figure 1: A simplified configuration of a ground setup to search for the high
frequency GWs along the $z$ axis: a single-side transmission cavity, i.e.,
region II $(0\leq z\leq l)$, is biased by an alternating magnetic field
$B(t)$, and only the desired GWs-induced EM signals (with the frequencies
$\omega_{g}=\omega_{B},2\omega_{B}$) can pass through the wave-impedance
matchers for being detected.

wherein a background magnetic ${\bf
B}(t)=(0,B^{(0)}_{y}+B^{\prime}_{y}\sin(\omega_{B}t),0)$ is confined in region
II. $B^{(0)}_{y}$ is the high static magnetic field and
$B^{\prime}_{y}(t)=B^{\prime}_{y}\sin\omega_{B}t$ with
$B^{\prime}_{y}(<B^{(0)}_{y})$ being the alternating one of the frequency
$\omega_{B}$. For the simplicity, the transmission of the cavity photons along
the $-z$ direction is neglected. If a monochromatic circular polarized plane
GW, i.e., a perturbed ripple:

$\displaystyle\left\\{\begin{array}[]{cc}h_{\oplus}=h_{xx}=-h_{yy}=A_{\oplus}\Lambda_{\omega_{g}}(z,t),\\\
\\\
h_{\otimes}=h_{xy}=h_{yx}=iA_{\otimes}\Lambda_{\omega_{g}}(z,t),\end{array}\right.$
(4)

with $\Lambda_{\omega_{g}}(z,t)=\exp[i\omega_{g}(z/c-t)]$, passes through the
cavity, then the background flat Minkowski space-time will be perturbed as

$\displaystyle
g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}=\begin{bmatrix}1\\\
&-1+h_{\oplus}&h_{\otimes}\\\ &h_{\otimes}&-1-h_{\oplus}\\\
&&&-1\end{bmatrix}.$ (5)

The electromagnetic field described by the tensor (2) obeys the relevant
Einstein-Maxwell equation

$\displaystyle\left\\{\begin{array}[]{cc}\partial_{\nu}\sqrt{{-g}}F^{\mu\nu}=\partial_{\nu}(\sqrt{{-g}}\
g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta})=0,\\\ \\\
\partial_{[\alpha}F_{\beta\gamma]}=0.\end{array}\right.$ (9)

Above, as the perturbations of the ${\oplus}$\- and ${\otimes}$ polarizations
of the GW are individual, we just discuss the ${\oplus}$-polarization
modulations specifically (the ${\otimes}$-polarization modulations can also be
treated similarly). It is well-known that, in a curved space-time only local
measurements made by the observer traveling in the world-line are physical and
all the observable quantities are just the projections of the relevant tensors
on the tetrads of the observer’s world-line. Obviously, in the present system
the observer should be rest in the static magnetic field, i.e., only the
zeroth component of the four-velocity is non-vanish. Furthermore, with the
simplified boundary conditions for the proposed configuration shown in Fig. 1,
the solution of the Eq. (3) can be conveniently divided into three parts;
i.e.,

$\displaystyle\left\\{\begin{array}[]{ll}&\widetilde{E}_{x}(z,t)=\widetilde{E}^{(0)}_{x}(z,t)+\widetilde{E}^{(1)}_{x}(z,t)+\widetilde{E^{\prime}}^{(1)}_{x}(z,t),\\\
&\widetilde{B}_{y}(z,t)=\widetilde{B}^{(0)}_{y}(z,t)+\widetilde{B}^{(1)}_{y}(z,t)+\widetilde{B^{\prime}}^{(1)}_{y}(z,t),\end{array}\right.$
(12)

with the zero-order solution:

$\displaystyle\widetilde{E}^{(0)}_{x}(z,t)=-\dfrac{1}{2}B^{\prime}_{y}c\sin[\omega_{B}(z/c-t)],\,\widetilde{B}^{(0)}_{y}(z,t)=\frac{\widetilde{E}^{(0)}_{x}(z,t)}{c},$
(13)

describing the usual electro-magnetic induction without the GWs perturbation.
The perturbative electromagnetic field, up to the first-order of the amplitude
$A_{\oplus}$, can be expressed as:

(a) In regin I $(z<0)$:

$\displaystyle\widetilde{E}^{(1)}_{(x)}(z,t)=\widetilde{B}^{(1)}_{(y)}(z,t)=\widetilde{E}^{\prime(1)}_{(x)}(z,t)=\widetilde{B}^{\prime(1)}_{(y)}(z,t)=0;$
(14)

(b) in regin II $(0<z<l)$:

$\displaystyle\widetilde{E}^{(1)}_{(x)}(z,t)$
$\displaystyle=\dfrac{i}{2}A_{\oplus}B^{(0)}_{y}\omega_{g}z\Lambda_{\omega_{g}}(z,t),\,\,\widetilde{B}^{(1)}_{(y)}(z,t)$
$\displaystyle=\frac{\widetilde{E}^{(1)}_{(x)}}{c},$ (15)

and

$\displaystyle\left\\{\begin{array}[]{cc}\widetilde{E}^{\prime(1)}_{(x)}(z,t)=i\sum_{\alpha}\alpha\Phi_{\alpha}[\exp(-i\alpha\omega_{B}z)-1]\Lambda_{\Delta_{\alpha}}(z,t),\\\
\\\
\widetilde{B}^{\prime(1)}_{(y)}(z,t)=i\sum_{\alpha}\alpha\Psi_{\alpha}[\exp(-i\alpha\omega_{B}z)-1]\Lambda_{\Delta_{\alpha}}(z,t),\end{array}\right.$
(19)

with
$\Phi_{\alpha}=A_{\oplus}B^{\prime}_{y}\omega_{g}\Delta_{\alpha}c/[2(\omega^{2}_{g}-\Delta^{2}_{\alpha})],\,\Psi_{\alpha}=A_{\oplus}B^{\prime}_{y}[\omega^{2}_{g}+\Delta^{2}_{\alpha}]/[4(\omega^{2}_{g}-\Delta^{2}_{\alpha})]$,
$\Delta_{\alpha}=\omega_{g}+\alpha\omega_{B},\,\,\alpha=+,-$; and

(c) in regin III $(z>l)$:

$\displaystyle\widetilde{E}^{(1)}_{(x)}(z,t)=\dfrac{i}{2}A_{\oplus}B^{(0)}_{y}\omega_{g}l\Lambda_{\omega_{g}}(z,t),\,\widetilde{B}^{(1)}_{(y)}(z,t)=\frac{\widetilde{E}^{(1)}_{(x)}}{c},$
(20)

with $l=(2m+1)\pi c/\omega_{B},m=0,1,2,...$ and

$\displaystyle\left\\{\begin{array}[]{cc}\widetilde{E}^{\prime(1)}_{(x)}(z,t)=-2i\sum_{\alpha}\alpha\Phi_{\alpha}\Lambda_{\Delta_{\alpha}}(z,t),\\\
\\\
\widetilde{B}^{\prime(1)}_{(y)}(z,t)=-2i\sum_{\alpha}\alpha\Psi_{\alpha}\Lambda_{\Delta_{\alpha}}(z,t).\end{array}\right.$
(24)

It is seen that, besides the usual electro-magnetic inductions in the local
flat space-time, the perturbed electromagnetic field induced by the GWs
passing through the alternating magnetic field includes three frequencies;
$\omega_{g}$ and $\omega_{g}\pm\omega_{B}$. Distinguishing these signals from
ones without any GW information, rather than the mechanical one utilized
usually in the most GW detecting setups (such as the LIGO-Virgo
installations), the GWs could be detected electromagnetically.

Observable effects of GWs-induced perturbed electromagnetic signals.—
Physically, the averaged power densities: $\langle
S\rangle=T^{-1}\int^{T}_{0}Sdt,\,\vec{S}=\vec{E}\times\vec{B}/\mu_{0}$, with
$T$ being the period of the EMWs, could be used to describe the strengths of
the electromagnetic signals. For a detector at the plane $z=z_{0}$ with the
receiving surface $x=[x_{1},x_{2}],\,y=[y_{1},y_{2}]$, the averaged number of
the photons with the frequency $\omega$ being detected per-second can be
calculated as

$n=\frac{1}{\hbar\omega}\int^{x_{2}}_{x_{1}}\int^{y_{2}}_{y_{1}}\langle
S(z_{0})\rangle dxdy.$ (25)

Here, $\hbar$ is Planck constant, and
$S(z_{0})=\widetilde{E}(z_{0},t)\widetilde{B}_{y}(z_{0},t)/\mu_{0}\simeq
S^{(0)}(z_{0})+S^{(1)}_{1}(z_{0})+S^{(1)}_{2}(z_{0})$. The zero-order
perturbative energy flow density, generated by the usual electro-magnetic
induction of the alternating magnetic field, reads
$S^{(0)}(z_{0})=\widetilde{E}^{(0)}_{x}(z_{0})\widetilde{B}^{(0)}_{y}(z_{0})/\mu_{0}$.
The corresponding averaged photon number

$n^{(0)}=\frac{\langle S^{(0)}\rangle_{\omega_{B}}}{\hbar\omega_{B}}=\frac{\Xi
B^{\prime
2}_{y}c}{8\mu_{0}\hbar\omega_{B}},\,\Xi=|x_{2}-x_{1}|\times|y_{2}-y_{1}|.$
(26)

is certainly very large. The energy flow densities of the first-order
perturbative field (which is lineary relative to the amplitude of GWs) are
$S^{(1)}_{1}(z_{0})=[\widetilde{E}^{(0)}_{x}(z_{0})\widetilde{B}^{(1)}_{(y)}(z_{0})+\widetilde{E}^{(1)}_{(x)}(z_{0})\widetilde{B}^{(0)}_{y}(z_{0})]/\mu_{0}$,
and
$S^{(1)}_{2}(z_{0})=[\widetilde{E}^{(0)}_{x}(z_{0})\widetilde{B}^{\prime(1)}_{(y)}(z_{0})+\widetilde{E}^{\prime(1)}_{x}(z_{0})\widetilde{B}^{(0)}_{(y)}(z_{0})]/\mu_{0}$,
respectively. They correspond to two kinds of perturbed EM signals with the
photon fluxes:

$\displaystyle n^{(1)}_{1}=\frac{\langle
S^{(1)}_{1}(z_{0})\rangle}{\hbar\omega_{g}}=\dfrac{\Xi
B^{(0)}_{y}B^{\prime}_{y}l}{4\mu_{0}\hbar}A_{\oplus},$ (27)

generated by the perturbations of the GWs with the frequency
$\omega_{B}=\omega_{g}$, and:

$\displaystyle n^{(1)}_{2}=\frac{\langle
S^{(1)}_{2}(z_{0})\rangle}{\hbar\omega_{g}}=\dfrac{3\Xi B^{\prime
2}_{y}c}{4\mu_{0}\hbar\omega_{g}}A_{\oplus},$ (28)

produced by the perturbations of the GWs with the frequency
$\omega_{g}=2\omega_{B}$, respectively. Specifically, Fig. 2 shows how the
detectable number of the photons $n^{(1)}_{1}$ and $n^{(1)}_{2}$ versus the
amplitude of the local alternating magnetic field $B^{\prime}_{y}$ for the
selected GWs signals.

Figure 2: The detectable photon number of the one-order perturbative signals
$n^{(1)}_{1}$ and $n^{(1)}_{2}$ versus the amplitude $B^{\prime}_{y}$ of the
locally applied alternating magnetic field. The relevant parameters are chosen
as: $B^{(0)}_{y}=10T$.

It is seen clearly that, without the alternating magnetic field (i.e.,
$B^{{}^{\prime}}_{y}=0$) the first-order perturbative photon vanish (i.e.,
$n^{(1)}_{1}=n^{(1)}_{2}=0$). The lower frequency of the Gws, which
corresponds to the higher amplitude of the GWs, and thus induces the stronger
signal photon fluxes. Additionally, the higher amplitude of the alternating
magnetic field, applied locally to search for the GWs, yields the stronger
electromagnetic responses of the GWs and consequently larger signal photon
fluxes. Anyway, with the single-photon detectors (see, e.g., d1 ; d2 ; d3 ; d4
) developed well in recent years, the induced photon fluxes could be detected,
at least theoretically.

Next, with the usual frequency matching filtering technique, all the
electromagnetic noises, except the ones with the frequency $\omega_{B}$, could
be filtered out. However, the left signals with the same frequency
$\omega_{B}$ still composite of three components; the photon fluxes $n^{(0)}$
without any GWs information, and the desired first-order perturbative ones,
carrying the information of the GWs, $n^{(1)}_{1}$ and $n^{(1)}_{2}$.
Fortunately, the wave impedances of the zero-order perturbative signals with
the energy flow dengsity $S^{(0)}$ generated by the local electro-magnetic
induction, satisfying the Maxwell equation in the flat space-time, always
reads

$Z_{0}=\frac{\mu_{0}E^{(0)}_{x}}{B^{(0)}_{y}}=\sqrt{\frac{\mu_{0}}{\varepsilon_{0}}}\cong
377\Omega.$ (29)

Here, $\varepsilon_{0}$ is the permittivity constant and $\mu_{0}$ the
permeability constant in free space. While, the wave impedances of the first-
order perturbed signals: $n^{(1)}_{1}$ and $n^{(1)}_{2}$, are

$Z_{1}^{(1)}=\frac{\mu_{0}\widetilde{E}_{x}^{(0)}}{\widetilde{B}_{x}^{(1)}}=\frac{\mu_{0}B^{{}^{\prime}}_{y}c^{2}}{B^{(0)}\omega_{g}lA_{\oplus}},\,Z_{1}^{{}^{\prime}(1)}=\frac{\mu_{0}\widetilde{E}_{x}^{(1)}}{\widetilde{B}_{x}^{(0)}}=\frac{\mu_{0}B^{(0)}\omega_{g}lA_{\oplus}}{B^{{}^{\prime}}_{y}},$
(30)

and

$\displaystyle
Z^{(1)}_{2}=\dfrac{\mu_{0}\widetilde{E}^{(0)}_{x}}{\widetilde{B}^{\prime(1)}_{(y)}}=\dfrac{6\mu_{0}c}{5A_{\oplus}},\,Z^{\prime(1)}_{2}=\dfrac{\mu_{0}\widetilde{E}^{\prime(1)}_{(x)}}{\widetilde{B}^{(0)}_{y}}=\dfrac{2\mu_{0}A_{\oplus}c}{3},$
(31)

respectively. Fig. 3 shows how the wave impedances of the first-order
perturbative electromagnetic signals versus the amplitude of the local
alternating magnetic field $B^{\prime}_{y}$. It is seen that
$Z^{(1)}_{1},Z^{(1)}_{2}\gg Z_{0}$ and
$Z^{{}^{\prime}(1)}_{1},Z^{{}^{\prime}(1)}_{2}\ll Z_{0}$. Therefore, with the
wave impedance matching filtering technique, see, e.g., Refs. Wave impendance
1 ; Wave impendance 2 , the electromagnetic signal $n^{(0)}$ without any GWs’
information could be robustly filtering out, and consequently only the
signals; $n^{(1)}_{1}$ and $n^{(1)}_{2}$, carrying the information of the GWs,
could be effectively conducted into the weak light detectors to implement the
desired detections.

Figure 3: The wave impedances $Z^{(1)}_{1}$ and $Z^{{}^{\prime}(1)}_{1}$ of
the first-order perturbative signals $n^{(1)}_{1}$ versus the amplitude of the
applied alternating magnetic field $B^{{}^{\prime}}_{y}$ in a high magnetic
field $B^{(0)}_{y}=10T$ for typical frequenies and amplitudes of the GWs.

Conclusions and Discussions.— In summary, a theoretical proposal to search for
the high-frequency GWs with semi-open side cavity biased locally by an
alternating high magnetic field is proposed. By analytically solving the
Einstein-maxwell equation in the curved space-time, the electromagnetic
responses of the GWs passing through the cavity are obtained. With the present
setup the significantly stronger first-order perturbations of the GWs are
generated and could be detected experimentally with the current developed-well
weak-light detectors. Technically, the GWs-induced first-order perturbative EM
signals could be selected out for detections, as the noises with the different
frequencies and wave impedances could be filtering out by using the usual
frequency matching and wave impedance matching filtering wave techniques.

It is emphasized that the proposed ground setup to search for the GWs works in
a sufficiently wide band, as the frequency $\omega_{B}$ of the applied
alternating magnetic field is adjustable within a very large regime, once the
size of the cavity is experimentally achievable. Table I lists the typical
frequency band of the detectable GWs with the proposed configuration. Here, a
generic argument Thorne : $A_{\oplus}\propto\omega^{-1}$, on the relation
between the amplitude and frequency of the GWs is used. One can see that the
proposal should work for the GWs with the frequency from $10^{7}$Hz to
$10^{12}$Hz, within the size of the cavity being experimentally reachable.

Certainly, the setup proposed here is still very imperfect for the realistic
experiments. Its sensitivity should be further analyzed, in detail, in the
presence of various noises and imperfect factors. Anyway, the present proposal
provides an potential approach to locally deliver the observable first-order
perturbative effects of the GWs passing through a high magnetic field.
Probably, it can serve as an effective complementary of the very successful
detections of the GWs based on their mechanically-tidal effects with the usual
laser interferometers.

Table 1: Various parameters for probing the electromagnetic responses of GWs: amplitude and frequency of the GWs, the size of the high magnetic field regime, and the number of signal photons at unit time. Here, $B^{\prime}_{y}=0.005T$. $A_{\oplus}$ |  | $\omega_{B}(Hz)$ |  | $l(m)$ |  | $n^{(1)}_{1}$ |  | $n^{(1)}_{2}$
---|---|---|---|---|---|---|---|---
$10^{-26}$ |  | $2\pi\times 10^{7}$ |  | $30$ |  | $2.83\times 10^{11}$ |  | $6.76\times 10^{7}$
$10^{-27}$ |  | $2\pi\times 10^{8}$ |  | $3$ |  | $2.83\times 10^{9}$ |  | $6.76\times 10^{5}$
$10^{-28}$ |  | $2\pi\times 10^{9}$ |  | $0.3$ |  | $2.83\times 10^{7}$ |  | $6.76\times 10^{3}$
$10^{-29}$ |  | $2\pi\times 10^{10}$ |  | $0.03$ |  | $2.83\times 10^{5}$ |  | $67.6$
$10^{-30}$ |  | $2\pi\times 10^{11}$ |  | $3\times 10^{-3}$ |  | $2.83\times 10^{3}$ |  | $0.68$
$10^{-31}$ |  | $2\pi\times 10^{12}$ |  | $3\times 10^{-4}$ |  | $28.3$ |  | $0.07$

Acknowledgements.— We thank Profs. Z. B. Li, X. G. Wu and Z. Y. Fang for
useful discussions. This work is partly supported by the National Natural
Science Foundation of China (NSFC) under Grant Nos. U1330201.

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redacted<EMAIL_ADDRESS>
#  UnUnlearning: Unlearning is not sufficient for content regulation in
advanced generative AI

Ilia Shumailov  Google DeepMind  Jamie Hayes  Google DeepMind  Eleni
Triantafillou  Google DeepMind  Guillermo Ortiz-Jimenez  Google DeepMind
Nicolas Papernot  Google DeepMind  Matthew Jagielski  Google DeepMind  Itay
Yona  Google DeepMind  Heidi Howard  Google DeepMind  Eugene Bagdasaryan
Google DeepMind

###### Abstract

Exact unlearning was first introduced as a privacy mechanism that allowed a
user to retract their data from machine learning models on request. Shortly
after, inexact schemes were proposed to mitigate the impractical costs
associated with exact unlearning. More recently unlearning is often discussed
as an approach for removal of impermissible knowledge i.e. knowledge that the
model should not possess such as unlicensed copyrighted, inaccurate, or
malicious information. The promise is that if the model does not have a
certain malicious capability, then it cannot be used for the associated
malicious purpose. In this paper we revisit the paradigm in which unlearning
is used for in Large Language Models (LLMs) and highlight an underlying
inconsistency arising from in-context learning. Unlearning can be an effective
control mechanism for the training phase, yet it does not prevent the model
from performing an impermissible act during inference. We introduce a concept
of ununlearning, where unlearned knowledge gets reintroduced in-context,
effectively rendering the model capable of behaving as if it knows the
forgotten knowledge. As a result, we argue that content filtering for
impermissible knowledge will be required and even exact unlearning schemes are
not enough for effective content regulation. We discuss feasibility of
ununlearning for modern LLMs and examine broader implications.

##  1 Introduction

Recent advancements in Large Language Models (LLMs) raise concerns about their
use for undesirable purposes. Unlearning emerged as a promising solution for
knowledge control, originally developed for removal of privacy-sensitive
information (Bourtoule et al., 2021). Since then, several works have attempted
to utilize unlearning for a host of applications relating to the removal of
undesired knowledge or behaviours: removing harmful capabilities (Lynch et
al., 2024) or harmful responses (Yao et al., 2023; Liu et al., 2024), erasing
backdoors (Liu et al., 2022) or specific information or knowledge pertaining
to a particular topic (Eldan and Russinovich, 2023; Li et al., 2024), erasing
copyrighted content (Yao et al., 2023) and even reducing hallucinations (Yao
et al., 2023). Such applications have been studied in the context of diffusion
models too, with various attempts to use unlearning to remove unsafe concepts
(Zhang et al., 2023; Fan et al., 2023).

This paper discusses the application of unlearning to LLMs for removal of
broadly impermissible knowledge, the use-case often discussed in policy
circles e.g. for removal of biological and nuclear knowledge (Li et al.,
2024). In fact, we uncover a fundamental inconsistency of the unlearning
paradigm for this application. While unlearning aims to erase knowledge, the
inherent in-context learning (ICL) (Brown et al., 2020; Kossen et al., 2024;
Agarwal et al., 2024) capabilities of LLMs introduce a major challenge. We
introduce the concept of ununlearning, where successfully unlearned knowledge
can resurface through contextual interactions. This raises a critical
question: if unlearned information can be readily reintroduced, is unlearning
a truly effective approach for making sure that the model does not exhibit
impermissible behaviours? We discuss the ramifications of ununlearning,
particularly the need for effective content regulation mechanisms to prevent
the resurgence of undesirable knowledge. Ultimately, we question the long-term
viability of unlearning as a primary tool for content regulation.

Note, this paper explicitly only considers the case when unlearning is used
for purposes of content regulation i.e. problems formulated as as a model
developer I do not want my model to be able to perform X, where X can be e.g.
bioweapons development (Li et al., 2024). Entities deploying models operate
under the expectation that those models don’t pose a risk of being exploited
for dangerous applications like weapons development. Importantly, it does not
cover the original use-case of unlearning for the privacy purposes.

##  2 Nomenclature

In what follows, we rely on six main terms:

(Informal) Definition 1. Knowledge refers to information available to the
model. This information can take up different forms and includes e.g. in-
context provided inputs, information stored in the parameters of the model, or
evidence available for retrieval.

(Informal) Definition 2. Content filtering refers to the process of filtering
out queries to and responses from a given model. Filtering can both be a part
of the model, as well as, be external to it (Glukhov et al., 2023).

(Informal) Definition 3. Unlearning refers to a process in which knowledge is
removed from a given model. This is a broad description that can encompasses
different application scenarios. Below, we provide two informal definitions of
unlearning.

(Informal) Definition 4. Unlearning for privacy seeks to remove knowledge that
is defined as a particular subset of the model’s original training dataset,
referred to as the “forget set”. Formal definitions (Ginart et al., 2019;
Sekhari et al., 2021) require the (distribution of the) unlearned model to be
indistinguishable from (the distribution of the) model retrained excluding the
forget set. According to this notion, an unlearning method can either be exact
(Bourtoule et al., 2021; Muresanu et al., 2024), guaranteeing
indistinguishability between the aforementioned distributions, or inexact
(Golatkar et al., 2020; Thudi et al., 2022; Kurmanji et al., 2024), instead
offering an approximation to that goal in exchange for greater efficiency or
better model utility.

(Informal) Definition 5. Unlearning for content regulation seeks to remove
knowledge that is (believed to be) associated with producing impermissible
content. Note that the specification of this knowledge in this case does not
necessarily take the form of identifying a subset of the training dataset.
Instead, it captures more broadly information that we want to remove from the
model (regardless of which subset of training data led to acquiring it), in
order to prevent it from generating impermissible content. This notion aligns
with the recent related notion of Goel et al. (2024) for “corrective
unlearning”.

(Informal) Definition 6. In-Context Learning refers to an emergent capability
of language models to generalise to tasks from task descriptions without these
tasks present in the training data (Brown et al., 2020; Milios et al., 2023;
Kossen et al., 2024). At the time of writing, such a model capability is
ubiquitous, albeit some tasks are not perfectly solvable from descriptions
alone.

For example, it may seem natural to remove any mention of what term bomb means
from the training set of the model, expecting that when asked to make bombs
the model would fail – simply because it possess no knowledge of what the word
bomb means. Yet, if a term bomb, for example under a different name as we show
in Figure 2, is defined as the a substance with certain properties and the
model possess reasoning skills, a recipe for a bomb can be derived by the
model, provided there is enough chemical knowledge still available to the
model.

Importantly, we highlight that even exact unlearning would not stop the model
from performing an impermissible act in the example above. That is because
even though the model never saw the bomb-defining data, it still possesses all
of the knowledge required to construct one. Consequently, no unlearning method
can lead to ‘better removal’ of the impermissible knowledge, compared to the
model that never trained on this knowledge in the first place. We therefore
argue that if ununlearning poses an issue even in this ‘idealized unlearning’
setting, this problem can only be exacerbated under imperfect unlearning.

Given it is hard to reason about knowledge compositionality, i.e. how
knowledge interacts with other knowledge available to the model, it is not
always obvious what logical inference given (benign) knowledge will enable. As
such, attribution of model behaviour to specific knowledge that a user
introduced is not always trivial.

##  3 Types of Knowledge

To best understand the intuition behind ununlearning, we first discuss types
of knowledge available to our models. We broadly classify knowledge into two
main categories: axioms and theorems to represent facts and assumptions, and
the derivative knowledge.

Consider the example depicted in Figure 1, where we have three concepts
defined out of 6 axioms. Here we define an axioms of Ear, Eye, Tail, Big,
Striped, and Gallops as the smallest units of knowledge. We then use these to
define three main theorems. If it has Ear, Eye, and Tail then we say its a
Cat. If it is a Cat, but also is Big and Striped then it is a Tiger. Yet, if
it is Big, Striped and Gallops then its a Zebra.

Figure 1: We broadly separate the knowledge into two main types: axioms and
theorems to represent given facts and derived knowledge respectively. In the
example above we assume that all theorems are defined in terms of underlying
axioms, where some axioms are shared by different theorems. While unlearning
of Cat may let the model forget what Cat means, it is relatively easy to
redefine it provided that the underlying axioms are preserved.  Figure 2: The
figure demonstrates the concept of ununlearning. Here, the model that at first
possess impermissible bomb making knowledge. The defender uses exact
unlearning to remove all instances of usage of the term bomb, making the model
incapable of providing bomb recipes, since it does not possess the knowledge
of what that term describes. The adversary uses the knowledge still available
to the model to describe the concept and as a result the model provides a
response.

Now lets consider the case where concept of Tiger is impermissible and any
queries regarding the Tiger are not allowed. In other words, the model
developer explicitly expects that the model would not be used to reason about
Tigers. We unlearn the concept of Tiger perfectly, using either exact
unlearning or a very strong approximate unlearning scheme. Yet, even with the
concept of Tiger gone, all of the underlying axiomatic knowledge is still
retained inside of the model, since its features are used for other theorems
available in the model, namely Zebra and Big in the example. If this knowledge
were to also be unlearned from the model, it would start impacting the utility
on other permissible tasks e.g. for reasoning about Zebra.

In the same way, many other conflicting objectives will leave the model
possessing impermissible knowledge. For example, it may be desirable for the
model to respond to high-school chemistry essay questions, yet not to have
bomb-making knowledge. Figure 2 presents one such example.

##  4 UnUnlearning

Setting: Assume a model $M$ that takes in $x\in\mathcal{X}$ and outputs
$\mathcal{Y}$. Note that here $\mathcal{X}$ includes both training and
inference-time data. We assume we have an unlearning method $u$ that takes in
a model and points $\hat{\mathcal{X}}\subseteq\mathcal{X}$ and outputs a model
$\hat{M}$ that unlearned the said points $u(M,\hat{X})=\hat{M}$. In this paper
we assume that $\hat{\mathcal{X}}$ is the impermissible knowledge that is
defined and identified by the model developer. We argue that by relying on in-
context learning (ICL) an adversary can bring back the knowledge such that
$\hat{M}$ when prompted with special context outputs the same results are $M$
over $\hat{\mathcal{X}}$. In other words,
$M(\hat{X})\approx\hat{M}(prompt+\hat{X})$.

(Informal) Definition 7. UnUnlearning refers to a process where previously
removed or never learned knowledge is instructed into the model using in-
context learning.

In this paper we argue that ununlearning is a concern that should be taken
into account when unlearning schemes are used and designed for removing
impermissible functionality; as well as, models openly released. Note that
ununlearning applies even to exact unlearning.

##  5 Discussion

There is a number of ramification of ununlearning.

Need for Effective Filtering Mechanisms UnUnlearning implies that for
unlearning to remain effective, one needs to not only remove impermissible
knowledge, but also perform a continuous, active process of suppressing in-
context attempts to reintroduce that knowledge. Fundamental computational
bounds suggest that such filtering is likely limited too, for example because
of dependence on the context in which queries appear (Glukhov et al., 2023).

Definitions and Mechanisms for Unlearning Given the above, is traditional
unlearning by itself even a worthwhile pursuit for controlling the use of
impermissible knowledge? Consider the Tiger–Zebra setting we discussed
earlier. If we accept classic definitions for unlearning, a model that never
trained on any Tiger knowledge represents perfect unlearning by construction.
In other words, no unlearning method would perform better than a model that
never saw any of the knowledge. Yet, since the model with in-context
introduced concept of Tiger can successfully reason about the Tiger it is
clearly not confirming to the objective of not reasoning at all about it.
Here, further exploration into the precise definitions and mechanisms for
ununlearning is essential – we need to find a way to explicitly limit the
reasoning capability for our models that is invariant to prompting and other
types of learning.

Attributing knowledge If a small benign axiom forces the attacker to discover
a malicious theorem, who should be attributed for it? This dilemma mirrors the
age-old philosophical debate – should an act be attributed to an individual
who directly executes the act, the person who gives the order, the
manufacturer of the tool, or the original tool designer (Owen, 1992; Fischer
and Ravizza, 1998)? We argue that given that knowledge can be introduced by
different parties, unlearning should not be assumed to be the sole mechanism
for content policy enforcement.

Forbidding knowledge Instead of filtering out data, it may be better to teach
the model explicitly that some knowledge is off limits (Henderson et al.,
2023). Do note however that this is not a bullet proof solution and will
unlikely to be robust against, for example, mosaic attacks (Glukhov et al.,
2023). Furthermore, the approach of forbidding knowledge requires foreseeing
the types of harmful tasks or use-cases ahead of time and does not guarantee
preventing harmful use more generally, beyond a given set of identified use-
cases. Finally, it is not clear how a model should react to violations of its
content regulations. In privacy literature it is a known fact that existence
of the privacy mechanism can lead to an increased privacy leakage (Wang et
al., 2022), and a similar effect was previously observed in unlearning (Hayes
et al., 2024). In other words, a model rejecting a request to synthesise a
given chemical recipe can also help the malicious user understand what recipes
can be used for malice.

##  6 Conclusion

In this paper we argue that unlearning offers an incomplete solution for
impermissible knowledge removal in LLMs with strong ICL capability.
UnUnlearning forces us to rethink unlearning as a one-size-fits-all solution
and places emphasis on content filtering.

*

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