11institutetext: Department of Natural and Mathematical Sciences, Özyeğin
University, 34794 İstanbul Turkey

# The variational method, backreactions, and the absorption probability in
Wald type problems

Koray Düztaş

(Received: date / Revised version: date)

###### Abstract

We argue that the variational method in Wald type thought experiments,
involves order of magnitude problems when one imposes the fact that $\delta M$
is inherently a first order quantity itself. One observes that the
contribution of the second order perturbations is actually of the fourth
order. Therefore backreactions have to be explicitly calculated. Here, we re-
consider the overspinning problem for Kerr-Newman black holes interacting with
test fields. We calculate the backreaction effects due to the induced increase
in the angular velocity of the event horizon, which brings a partial solution
to the overspinning problem. To bring an ultimate solution, we argue that the
absorption probability should be taken into account in Wald type problems
where black holes interact with test fields. This fundamentally alters the
course of the analysis of the thought experiments. Due to the fact that a
small fraction of the challenging modes is absorbed by the black holes,
overspinning is prevented for both nearly extremal and extremal cases. Some
extreme cases are easily fixed by backreaction effects. The arguments do not
apply to the generic overspinning by fermionic fields for which the absorption
probability is positive definite.

###### pacs:

04.20.DwSingularities and cosmic censorship

##  1 Introduction

One of the main unsolved problems in classical general relativity is the
validity of the cosmic censorship conjecture due to Penrose ccc . The
conjecture aims to circumvent the problems that could arise if a curvature
singularity is allowed to be in causal contact with distant observers. This is
achieved by forbidding the existence of naked singularities in a physical
universe. The gravitational collapse of a massive object should end up in a
black hole surrounded by an event horizon rather than a naked singularity, as
prescribed by Penrose and Hawking singtheo .

The natural question at this stage is whether the event horizon of a black
hole can be destroyed by test bodies or fields to expose the curvature
singularity lurking at the center. The possibility to destroy an event horizon
was first evaluated in a thought experiment constructed by Wald wald74 . Wald
started with an extremal Kerr-Newman black hole and attempted to increase the
charge and angular momentum beyond the extremal limit by sending in test
bodies from infinity. It turns out that the test bodies that could potentially
overcharge or overspin an extremal Kerr-Newman black hole are not absorbed by
the black hole. The event horizon is stable and the smooth structure of the
space-time is maintained excluding the black hole region inside the event
horizon. Later, Hubeny adapted an alternative approach to Wald type problems
where one starts with a nearly extremal black hole instead of an extremal one
hu . She showed that a nearly extremal Reissner-Nordström black hole can be
overcharged into a naked singularity by a test body. Jacobson and Sotiriou
applied an analogous analysis to show that nearly extremal Kerr black holes
can be overspun by test bodies js . Düztaş and Semiz derived the same result
for nearly extremal Kerr black holes interacting with test fields overspin .
In these works the overspinning and overcharging of nearly extremal black
holes are not quite generic, which suggests that they should be fixed by
employing backreaction effects. It was argued that the self force effects can
prevent the overcharging backhu  and overspinning backjs  of nearly extremal
black holes by test bodies. In a very recent work we showed that the
absorption of the test fields that could overspin nearly extremal black holes
is not allowed due to the increase in the angular velocity of the event
horizon before the absorption of the field kerrmog .

In literature there exist various attempts to overspin or overcharge black
holes with test bodies f1 ; saa ; gao ; siahaan ; magne ; dilat ; higher ; v1
; he ; wang ; gim ; jamil ; shay ; shay2 ; zeng , and fields semiz ; emccc ;
duztas ; toth ; natario ; duztas2 ; mode ; taub-nut ; kerrsen ; gwak5 ; gwak6
; hong ; yang ; bai . The effect of quantum tunnelling and particle creation
has also been incorporated in Wald type problems q1 ; q2 ; q3 ; q4 ; q5 ; q6 ;
q7 . In recent years, the possibility to destroy the event horizon in the
asymptotically anti-de Sitter cases has also become an active field of
research btz ; gwak1 ; gwak2 ; gwak3 ; gwak4 ; chen ; he2 . ( For a recent
review see ong )

In Wald type problems the backreaction effects are difficult to compute and
most of the time the results are restricted to order of magnitude estimates.
Recently Sorce and Wald designed a new type of gedanken experiment by adapting
a variational approach w2 . They derived an explicit expression for the second
order effects, so one does not have to explicitly compute self force or finite
size effects. Currently, the Sorce-Wald method is widely accepted among the
researchers working on Wald type problems. Recently we have also employed
Sorce-Wald method to test the stability of the event horizon for Martinez,
Teitelboim, Zanelli (MTZ) black holes mtz . In our analysis, we have imposed
the fact that $\delta M$ is inherently a first order quantity. We observed
that, imposing this fact causes order of magnitude problems to arise in the
method developed by Sorce and Wald. In section (2), we further scrutinize the
Sorce-Wald method by imposing the fact that $\delta M$ is inherently a first
order quantity itself, for test bodies and fields. Unfortunately, it turns out
that the order of magnitude problems do not pertain to the MTZ case. They are
also manifest in the case of Kerr-Newman black holes, for which the Sorce-Wald
method was developed. We present the details in section (2.1).

The order of magnitude problems in the Sorce-Wald method suggest that the
backreactions should be explicitly calculated. Based on this argument, we re-
visit the overspinning problem for nearly extremal and extremal Kerr-Newman
black holes interacting with test fields, in section (3). In a recent work we
showed that extremal Kerr-Newman black holes which satisfy $J^{2}/M^{4}<(1/3)$
can be overspun by scalar test fields generic . We argued that the
overspinning is not quite generic and it is prone to be fixed by backreaction
effects. In section (3.1), we show that nearly extremal black holes can also
be overspun and employ the backreaction effects based on Will’s argument that
the angular velocity of the event horizon increases before the absorption of
the test field will . The employment of this backreaction effect brings a
partial solution to the problem. The destruction of the event horizon can be
prevented for certain classes of nearly extremal and extremal black holes,
with a sufficiently large magnitude of angular momentum. In section (3.2), we
show that the same argument applies to extremal black holes. We analytically
derive the relevant magnitude of the initial value of angular momentum, for
both nearly extremal and extremal cases.

The interaction of test fields with black holes is actually a scattering
problem. The field is partially absorbed by the black hole and partially
reflected back to infinity. The fact that only a fraction of the incoming
field is absorbed by the black hole has been ignored in all the thought
experiments constructed so far, including the works of this author. In section
(4), we take the absorption probabilities of test fields into account, which
fundamentally changes the course of the analysis of the problem. We show that
a very small fraction of the challenging test fields are absorbed by the black
hole which has no practical effect on the mass and angular momentum parameters
of the space-time. In section (4.1) we evaluate the optimal perturbations with
the lowest possible energy relative to their angular momentum and charge. We
show that the absorption probability is zero for the optimal perturbations. In
that case, the test field is entirely reflected back to infinity. The space-
time parameters remain identically the same after the interaction with the
test field. In sections (4.2) and (4.3) we perturb nearly extremal and
extremal black holes with challenging modes with frequencies slightly larger
than the optimal perturbations. We show that the absorption probability is
very low for the challenging modes. It turns out that most of the energy and
angular momentum carried by the challenging modes is reflected back to
infinity. Still there exists a set of fine-tuned parameters that seem to be
capable of overspinning extremal and nearly extremal Kerr-Newman black holes.
In sections (4.2) and (4.3), we also show that, these anomalies are remedied
by backreaction effects due to the induced increase in the angular velocity of
the event horizon.

##  2 Sorce-Wald method

The Kerr-Newman metric describes a black hole surrounded by an event horizon
provided that the spacetime parameters satisfy the main inequality

$M^{2}\geq a^{2}+Q^{2}$ (1)

where $M$, $a\equiv J/M$, and $Q$ are respectively the mass, angular momentum
and charge parameters of the spacetime. In Wald type problems, one starts with
an extremal or nearly extremal black hole satisfying the main criterion (1)
and attempts to increase the angular momentum and/or charge parameters beyond
the extremal limit, by sending in test bodies or fields from infinity. The
main assumption in these thought experiments is that the interaction of the
black hole with test bodies and fields does not alter the background geometry
of the spacetime, but leads to perturbations in the mass, angular momentum,
and charge parameters. After a sufficiently long time, the spacetime is
supposed to settle down to a new Kerr-Newman solution with modified
parameters. Apparently the energy, angular momentum, and the charge of the
test body or field should be very small compared to the initial parameters of
the spacetime so that the assumption that the background geometry in the final
state is also a Kerr-Newman solution, is justified. Then, one can check if the
final parameters of the spacetime represent a Kerr-Newman black hole
satisfying the inequality (1) or a naked singularity which violates it. For
that purpose we prefer to define

$\delta_{\rm{fin}}\equiv
M_{\rm{fin}}^{2}-Q_{\rm{fin}}^{2}-\frac{J_{\rm{fin}}^{2}}{M_{\rm{fin}}^{2}}$
(2)

If the contribution of the second order terms are taken into account in
calculating $\delta_{\rm{fin}}$, one should also incorporate the effect of
backreactions which bring second order corrections to (2), so that the
calculation can be considered consistent. As we mentioned in the introduction,
the overcharging of Reissner-Nordstro̧m black holes hu , and the overspinning
of Kerr black holes js ; overspin  can be fixed by backreaction effects backhu
; backjs ; kerrmog . The backreaction effects comprise finite size effects,
self interaction, gravitational radiation, the effect of black hole radiation,
induced increase in the angular velocity of the horizon and many more possible
effects pertaining to the specific problem. The Sorce-Wald method has been
developed to bring an ultimate solution to the problem of determining and
calculating the backreactions w2 . Sorce and Wald (SW) attempted obtain an
expression for the full second order correction $\delta^{2}M$ without having
to calculate the backreaction effects explicitly. To check whether the event
horizon can be destroyed SW first derive an expression for the minimum energy
of the incoming test body or field so that it is absorbed by the black hole.

$\delta M-\Omega_{H}\delta J-\Phi_{H}\delta Q\geq 0$ (3)

where $\Omega_{H}=a/(r_{+}^{2}+a^{2})$, $\Phi_{H}=(Qr_{+})/(r_{+}^{2}+a^{2})$,
and $r_{+}$ is the horizon radius. The condition (3) is well known in black
hole physics. The first derivation without assuming the validity of cosmic
censorship known to this author is by Needham in 1980 needham . The condition
(3) determines the lowest possible energy for a given combination of angular
momentum and charge that would allow the absorption of a test field. The
perturbations with the lowest possible energy are referred to as the optimal
perturbations. The perturbations that do not satisfy the condition (3) are not
absorbed by the black hole. If the absorption of these perturbations was
allowed, they would lead to a generic destruction of the event horizon since
they carry relatively large angular momentum and charge. However, the
condition (3) only applies to the perturbations that satisfy the null energy
condition. For fermionic fields there is no lower bound on the energy that
would prevent the absorption of the challenging modes. In generic  we argued
that the absence of the lower bound for the energy of the fermionic fields
leads to a generic destruction of the event horizon.

Sorce and Wald proceed by parametrizing a nearly extremal black hole as

$M^{2}-Q^{2}-(J/M)^{2}=M^{2}\epsilon^{2}$ (4)

which is common in Wald type problems. The small parameter $\epsilon$
determines the closeness of the black hole to extremality. For $\epsilon\ll 1$
the black hole is very close to extremality in which case the effect of the
interactions with test bodies and fields become relevant. Next, Sorce and Wald
define the function:

$f(\lambda)=M(\lambda)^{2}-Q(\lambda)^{2}-J(\lambda)^{2}/M(\lambda)^{2}$ (5)

If $f(\lambda)<0$ the inequality (1) is violated and the event horizon cannot
exist. Next $f(\lambda)$ is expanded to second order in $\lambda$

$\displaystyle f(\lambda)$ $\displaystyle=$
$\displaystyle\left(M^{2}-Q^{2}-\frac{J^{2}}{M^{2}}\right)$ (6)
$\displaystyle+$ $\displaystyle 2\lambda\left(\frac{M^{4}+J^{2}}{2M^{3}}\delta
M-\frac{J}{M^{2}}\delta J-Q\delta Q\right)$ $\displaystyle+$
$\displaystyle\lambda^{2}\left[\frac{M^{4}+J^{2}}{2M^{3}}\delta^{2}M-\frac{J}{M^{2}}\delta^{2}J-Q\delta^{2}Q+\frac{4J}{M^{3}}\delta
J\delta M\right.$ $\displaystyle-$ $\displaystyle\left.\frac{1}{M^{2}}(\delta
J)^{2}+\left(\frac{M^{4}-3J^{2}}{M^{4}}\right)(\delta M)^{2}-(\delta
Q)^{2}\right]$

To avoid any confusion we refer to $\delta M,\delta J,\delta Q$ as first order
perturbations, and $\delta^{2}M,(\delta M)^{2},...$ terms as second order
perturbations. For the first order perturbations Needham’s condition (3)
implies that

$\displaystyle f(\lambda)$ $\displaystyle\geq$ $\displaystyle
M^{2}\epsilon^{2}+\frac{2}{M^{4}+J^{2}}\left((J^{2}-M^{4})Q\delta
Q-2JM^{2}\delta J)\right)\lambda\epsilon$ (7) $\displaystyle+$ $\displaystyle
O(\lambda^{2},\epsilon^{3},\epsilon^{2}\lambda)$

The equations (6) and (7) above, are the equations (119) and (120) in the
relevant paper of Sorce and Wald w2 . To derive (7), one imposes the condition
(3) that the test body or field is absorbed by the black hole, and expresses
$\delta M$ in terms of $\delta J$ and $\delta Q$. At this point Sorce and Wald
claim that neglecting the terms of order $O(\lambda^{2})$ it is possible to
make $f(\lambda)<0$. This statement aims to convince the readers that the
variational method reproduces the previous results that the nearly extremal
black holes can be overcharged or overspun when the second order terms are
neglected. The main claim of SW is that $f(\lambda)$ becomes positive again by
considering the effect of the terms that are second order in $\lambda$. Here
we show that these two results cannot be obtained by the method developed by
Sorce and Wald, when one does not ignore the fact that $\delta M$ is
inherently a first order quantity, itself.

###  2.1 Sorce-Wald method with the correct test body/field approximation

The small parameter $\lambda$ is introduced in (5) to ensure that the
variation of the function $f$ from its initial value $f(0)$ is small, in
accord with the test body/field approximation. However, in (6) the parameter
$\lambda$ explicitly multiplies the terms $\delta M,\delta J,\delta Q$. Since
$\delta M$ is inherently a first order quantity, the $\lambda\epsilon\delta M$
terms actually contribute to third order to $f(\lambda)$, while the
$\lambda^{2}\delta^{2}M$ terms contribute to fourth order. One cannot ignore
the fact that $\delta M$ is a small quantity itself and proceed as if $\delta
M\sim M$, as it was done in the derivation of Sorce and Wald.

To clarify the fact that $\delta M$ is a small quantity itself, we let

$\delta M=M\zeta$ (8)

where $\zeta$ parametrises the energy of the perturbation and the fact that
$\zeta\ll 1$ ensures that the test body/field approximation is not violated.
In principle the parameters determining the magnitude of the perturbations and
the closeness to extremality need not be equal. However, for numerical
calculations one can let $\epsilon\sim\zeta$. Imposing the fact that the
perturbations are small themselves (7) implies that

$f(\lambda)\sim O(\epsilon^{2})-O(\lambda\epsilon\zeta)$ (9)

which is valid to first order in $\lambda$. (Note that the term
$(J^{2}-M^{4})$ is negative.) It is easy to see that when one imposes the fact
that the first order perturbations are of the “first order” themselves, it is
not possible to make $f(\lambda)$ –defined by SW– negative for the first order
terms. The variational method does not reproduce the previous results due
Hubeny hu , Jacobson-Sotiriou js  and Düztaş-Semiz overspin . For the first
order perturbations, the results of SW contradict with the previous results
when the fact that $\delta M$ is inherently a small quantity is taken into
account.

Though it is manifest in (9) that $f(\lambda)$ defined by SW, cannot be made
negative for the first order terms, it would be appropriate to elaborate on
this subject considering the fact that the SW method is widely accepted in
black hole physics. For simplicity let us consider a neutral body or a field
$(\delta Q=0)$, incident on a nearly extremal black hole. For the optimal
perturbations (3) implies that

$J\delta J=M\delta M(r_{+}^{2}+a^{2})$

Imposing the fact that $\delta M=M\zeta$ by the definition (8)

$\displaystyle J\delta J$ $\displaystyle=$ $\displaystyle
M^{2}\zeta\left[M^{2}(1+\epsilon)^{2}+\frac{J^{2}}{M^{2}}\right]$ (10)
$\displaystyle=$
$\displaystyle(M^{4}+J^{2})\zeta+O(\epsilon\zeta,\epsilon^{2}\zeta)$

where we have substituted $r_{+}=M(1+\epsilon)$ for a nearly extremal black
hole parametrized as (4). Now we substitute the expression for $J\delta J$
derived in (10) to the expression for $f(\lambda)$. For the optimal
perturbations one derives

$f(\lambda)=M^{2}\epsilon^{2}-4M^{2}\lambda\epsilon\zeta-O(\lambda\epsilon^{2}\zeta,\lambda\epsilon^{3}\zeta)$
(11)

Again it is manifest in (11) that $f(\lambda)$ defined by SW, cannot be made
negative for small $\lambda$, $\epsilon$ and $\zeta$. The claim that
$f(\lambda)$ can be made negative by the terms first order in $\lambda$
requires one to assume that $\delta M\sim M$, which apparently contradicts the
test body/field approximation.

The main claim of SW is that the negativeness of $f(\lambda)$ can be fixed by
the contribution of the terms that are second order in $\lambda$. Though the
fact that $f(\lambda)$ cannot be made negative by the first order terms
renders this claim irrelevant, it is necessary to evaluate the contribution of
the second order terms when the derivation is corrected by imposing $\delta
M=M\zeta$. To second order in $\lambda$ we have

$f(\lambda)\sim
O(\epsilon^{2})-O(\lambda\epsilon\zeta)+O(\lambda^{2}\zeta^{2})$ (12)

It is manifest in (12) that the contribution of the second order perturbations
vanishes in (6), as it becomes fourth order when multiplied by the square of
the small parameter $\lambda$. (Note that the leading term in (12) –which is
zeroth order in $\lambda$– is actually second order in $\epsilon$.) In that
respect it is not possible to incorporate the effect of the second order
perturbations into the analysis using the SW method. Moreover, when the
analysis is corrected by imposing the fact that $\delta M$ is a first order
quantity, even the first order perturbations $(\delta M,\delta J,\delta Q)$ do
not contribute to $f(\lambda)$ as one can observe in (11) and (12).

In the previous works by Hubeny, Jacobson-Sotiriou, and Düztaş-Semiz,
$\delta_{\rm{fin}}$ defined in (2) is made negative for nearly extremal black
holes which corresponds to making $f(\lambda)$ negative in the derivation of
SW hu ; js ; overspin . The numerical value of $\delta_{\rm{fin}}$ turns out
to be of the order $-M^{2}\epsilon^{2}$ which suggests that the destruction of
the event horizon can be fixed by the second order corrections due to the
backreaction effects. Later, these corrections were indeed achieved by
employing backreaction effects backhu ; backjs ; kerrmog . The SW method does
not reproduce any of these results when one imposes the fact that $\delta M$
is a first order quantity for test bodies and fields. One observes that the
terms first order in $\lambda$, contribute to $f(\lambda)$ to third order so
they cannot make $f(\lambda)$ negative. The terms that are second order in
$\lambda$, cannot fix anything since their contribution is of the fourth
order. Therefore the SW method cannot be used to evaluate the effect of second
order perturbations. Backreactions have to be explicitly calculated.

##  3 Re-visiting the over-spinning problem

In this section we re-visit the over-spinning problem for Kerr-Newman black
holes and explicitly calculate the backreaction effects, which supervenes on
the argument that they cannot be calculated using the SW method. We start by
attempting to overspin a nearly extremal Kerr-Newman black hole parametrised
as (4), by a neutral, scalar test field with frequency $\omega$ and azimuthal
wave number $m$. We adapt the parametrization (8) so that the field carries
energy $\delta M=M\zeta$, in accord with the test field approximation. At the
end of the interaction, the final parameters of the space-time satisfy:

$\displaystyle M_{\rm{fin}}$ $\displaystyle=$ $\displaystyle M+M\zeta$
$\displaystyle J_{\rm{fin}}$ $\displaystyle=$ $\displaystyle
J+\frac{m}{\omega}M\zeta$ $\displaystyle Q_{\rm{fin}}$ $\displaystyle=$
$\displaystyle Q$ (13)

where $M,J,Q$ are the initial parameters which satisfy (4). Now, we demand
that the black hole is overspun at the end of the interaction; i.e.
$\delta_{\rm{fin}}<0$

$\delta_{\rm{fin}}=(M+M\zeta)^{2}-Q^{2}-\frac{(J+\frac{m}{\omega}M\zeta)^{2}}{(M+M\zeta)^{2}}<0$
(14)

We can substitute $Q^{2}=M^{2}-J^{2}/M^{2}-M^{2}\epsilon^{2}$ by using (4).
Re-arranging (14), we get

$M^{2}\left(\zeta^{2}+2\zeta+\epsilon^{2}+\frac{J^{2}}{M^{4}}\right)<\frac{J+\frac{m}{\omega}M\zeta)^{2}}{M^{2}(1+\zeta)^{2}}$
(15)

We define the dimensionless parameter

$\alpha\equiv J/M^{2}$ (16)

Note that for a nearly extremal Kerr-Newman black hole parametrised as (4),
the sum $J^{2}/M^{2}+Q^{2}$ has a fixed value which is equal to
$M^{2}(1-\epsilon^{2})$ for a fixed mass $M$. However nearly extremal black
holes satisfying (4) may have different values of angular momentum and charge
keeping the sum $J^{2}/M^{2}+Q^{2}$ fixed. We use the dimensionless parameter
$\alpha$ to identify different Kerr-Newman black holes –with a fixed mass $M$–
that all satisfy (4). Also note that, substituting $\alpha\equiv J/M^{2}$ the
parametrisation (4) can be re-written in terms of the dimensionless variable
$\alpha$.

$1-\alpha^{2}-\frac{Q^{2}}{M^{2}}=\epsilon^{2}$ (17)

We proceed by taking the square root of both sides of (15). The condition
$\delta_{\rm{fin}}<0$ reduces to

$\omega<\omega_{\rm{max}}=\frac{m\zeta}{M\left[(1+\zeta)\sqrt{\zeta^{2}+2\zeta+\epsilon^{2}+\alpha^{2}}-\alpha\right]}$
(18)

We have considered the interaction of a nearly extremal black hole
parametrised as (4) with a test field carrying energy $\delta M=M\zeta$ and
angular momentum $\delta J=(m/\omega)\delta M$. Note that $\delta J$ is
inversely proportional to the frequency $\omega$. The equation (18) implies
that a test field with a frequency $\omega<\omega_{\rm{max}}$, will contribute
to the angular momentum parameter of the black hole with a magnitude
sufficiently larger than its contribution to the energy parameter so that the
final parameters of the black hole describe a naked singularity with
$\delta_{\rm{fin}}<0$. In that case we could conclude that the nearly extremal
Kerr-Newman black hole is overspun into a naked singularity. However, we
should also demand that the test field is absorbed by the black hole, i.e.
$\omega$ is larger than the limiting frequency for superradiance, which we
denote by $\omega_{\rm{sl}}$. For a nearly extremal black hole parametrised as
(4), which is perturbed by a neutral test field $(\delta Q=0)$, the
superradiance limit is given by

$\omega_{\rm{sl}}=\frac{ma}{r_{+}^{2}+a^{2}}=\frac{m}{M\left[\frac{(1+\epsilon)^{2}}{\alpha}+\alpha\right]}$
(19)

Kerr-Newman black holes with different values of $\alpha$ defined in (16),
have different superradiance limits. For lower values of $\alpha$ which
describe black holes with relatively low angular momentum, the superradiance
limit will also be low. In that case the absorption of the modes with
relatively low frequencies will be allowed. The test fields with frequency in
the range $\omega_{\rm{sl}}<\omega<\omega_{\rm{max}}$ simultaneously satisfy
the two conditions that the field is absorbed by the black hole and it
contributes to the angular momentum parameter with a sufficiently large
magnitude to overspin the black hole into a naked singularity. We can conclude
that the test fields with energy $\delta M=M\zeta$ and frequency in the range
$\omega_{\rm{sl}}<\omega<\omega_{\rm{max}}$ can be used to overspin a nearly
extremal Kerr-Newman black hole into a naked singularity, provided that
$\omega_{\rm{sl}}<\omega_{\rm{max}}$ . Comparing (18) and (19) one observes
that the upper limit for the frequency of the incident field
$\omega_{\rm{max}}$ derived in (18), is larger than the superradiance limit
$\omega_{\rm{sl}}$ for any value of $\alpha$, in the relevant range $(0,1)$.
It turns out that every nearly extremal Kerr-Newman black hole satisfying (4)
can be overspun by neutral test fields, regardless of the specific value of
$\alpha$ defined in (16). The validity of this conclusion is limited to the
case, where one ignores the backreaction effects.

###  3.1 Backreactions for nearly extremal black holes

In this derivation we have not ignored the contribution of the second order
terms $(\delta M)^{2}$ and $(\delta J)^{2}$. Therefore we have to employ the
backreaction effects to test whether the destruction of the event horizon can
be fixed. Since, the Sorce-Wald method is invalid we have to explicitly
determine and calculate the backreaction effects. Backreaction effects will
bring second order corrections to $\delta_{\rm{fin}}$ which can in principle
restore the event horizon. The most legitimate type of backreaction effect for
an overspinning problem is the induced increase in the angular velocity of the
event horizon before the absorption of the test body/field occurs, which was
suggested by Will will . The induced increase in the angular velocity of the
event horizon leads to an increase in the superradiance limit. This implies
that the absorption of the challenging modes with relatively low frequencies
can be prevented. In a recent paper we have employed this backreaction effect
for the overspinning problem of Kerr-MOG black holes kerrmog .

We envisage a test field with angular momentum $\delta J$ incident on a black
hole with mass $M$. According to the estimate in will , the angular velocity
of the event horizon increases by an amount

$\Delta\omega=\frac{\delta J}{4M^{3}}$ (20)

The increase in the angular velocity of the event horizon results in an
increase in the superradiance limit, which will be modified as

$\omega_{\rm{sl}}^{\prime}=\omega_{\rm{sl}}+\Delta\omega$ (21)

In (18) we have derived the maximum value for the frequency of a test field
that could overspin a nearly extremal Kerr-newman black hole parametrised as
(4). We noted that the fields with frequency in the range
$\omega_{\rm{sl}}<\omega<\omega_{\rm{max}}$ can lead to overspinning. However
as the test field is incident on the black hole the angular velocity of the
event horizon will increase, which will lead to a modification in the
superradiance limit given derived in (21). If the modified value of the
superradiance limit exceeds the frequency of the incoming field, the
absorption of the test field is prevented and the event horizon cannot be
destroyed. The test field will be scattered back to infinity with a larger
magnitude. Note that $\delta J$ and $\Delta\omega$ given in (20) are inversely
proportional to the frequency $\omega$. For that reason, if the modified value
of the superradiance limit exceeds the incoming frequency for
$\omega\simeq\omega_{\rm{max}}$, it will exceed the incoming frequency even
further for smaller values in the range
$\omega_{\rm{sl}}<\omega<\omega_{\rm{max}}$, as we have argued in kerrmog .
Therefore it is critical to calculate $\Delta\omega$ for the frequencies
arbitrarily close to $\omega_{\rm{max}}$.

To calculate backreactions, we first envisage a test field with frequency
arbitrarily close to but slightly less than $\omega_{\rm{max}}$, incident on a
nearly extremal Kerr-Newman black hole parametrised as (4). The test field
carries energy $\delta M=M\zeta$ and angular momentum $\delta
J=(m/\omega)\delta M$, where $\omega\lesssim\omega_{\rm{max}}$. According to
the derivation in the previous section, this test field will lead to the
overspinning of the nearly extremal Kerr-Newman black hole. Now we incorporate
the backreaction effects due to the induced increase in the angular velocity
of the event horizon. We check if the modified value of the superradiance
limit derived in (21), exceeds the frequency of the incoming field. For
simplicity we let $\epsilon\simeq\zeta$ and substitute
$\omega=\omega_{\rm{max}}$ in (20) to calculate $\Delta\omega$.

$\Delta\omega=\frac{\left(\frac{m}{\omega}\right)M\epsilon}{4M^{3}}=\frac{\left[(1+\epsilon)\sqrt{2\epsilon^{2}+2\epsilon+\alpha^{2}}-\alpha\right]}{4M}$
(22)

For $\omega\simeq\omega_{\rm{max}}$, the increase in the superradiance limit
is given by $\Delta\omega$ in (22), which leads to the modified value of the
superradiance limit denoted by $\omega_{\rm{sl}}^{\prime}$ derived in (21).
Then, we need to compare $\omega_{\rm{sl}}^{\prime}$ and $\omega_{\rm{max}}$.
If $\omega_{\rm{sl}}^{\prime}$ is larger than $\omega_{\rm{max}}$, no net
absorption of the test field will occur and overspinning will be prevented.

It turns out that $\omega_{\rm{sl}}^{\prime}$ defined in (21) is indeed larger
than $\omega_{\rm{max}}$ provided that

$\alpha\gtrsim 0.50$ (23)

Note that the parameter $\alpha\equiv J/M^{2}$ defined in (16) is used to
distinguish different nearly extremal Kerr-Newman black holes that all satisfy
(4). These black holes may have different angular momentum and charge
parameters keeping the sum $(J^{2}/M^{2}+Q^{2})$ fixed. Without employing
bakreaction effects, one derives that every nearly extremal Kerr-Newman black
hole can be overspun by test fields, regardless of the specific value of
$\alpha$. This includes the Kerr limit $Q\to 0$,
$\alpha^{2}\to(1-\epsilon^{2})$. When one employs backreaction effects due to
the induced increase in the superradiance limit, it turns out that there are
two there are two possibilities depending on the specific value of $\alpha$.
If $\alpha<0.5$, the modified value of the superradiance limit
($\omega^{\prime}_{\rm{sl}}$), will still be smaller than $\omega_{\rm{max}}$.
The absorption of the modes in the range
$\omega^{\prime}_{\rm{sl}}<\omega<\omega_{\rm{max}}$ will lead to
overspinning. However if $\alpha>0.5$, the increase in the superradiance limit
will be sufficient to prevent the absorption of all challenging modes. Since
$\omega^{\prime}_{\rm{sl}}$ is larger than $\omega_{\rm{max}}$ for this class
of nearly extremal Kerr-Newman black holes, all the modes with
$\omega<\omega_{\rm{max}}$ that could potentially overspin the black hole will
be reflected back to infinity without any net absorption. Thus, the
backreaction effects will prevent overspinning. The argument is also valid in
the Kerr limit $Q\to 0$. For nearly extremal Kerr black holes, the parameter
$\alpha$ has a unique value which is equal to $\sqrt{1-\epsilon^{2}}$.
Manifestly, $\alpha>0.5$ for nearly extremal Kerr black holes and the
overspinning problem is fixed by employing backreaction effects.

It would be appropriate to give a numerical example to elucidate the subject.
For that purpose, let us consider a nearly extremal Kerr-Newman black hole
with $\alpha=0.5$ and $\epsilon=0.01$. Initially the parameters of this black
hole satisfy

$M^{2}-(J^{2}/M^{2})-Q^{2}=(0.0001)M^{2}$

or equivalently

$1-\alpha^{2}-(Q^{2}/M^{2})=0.0001$

For $\alpha=0.5$ the initial parameters of the black hole are given by
$J=0.5M^{2}$ and $Q^{2}=0.7499M^{2}$. Now we perturb this black hole with a
test field with energy $\delta M=M\zeta$ and angular momentum $\delta
J=(m/\omega)\delta M$. To choose the frequency of the test field we calculate
the critical values. We choose $m=1$ which is the mode with the highest
probability of absorption. We use equation (19) to calculate the superradiance
limit.

$\omega_{\rm{sl}}=0.39366(1/M)$

Letting $\zeta=\epsilon=0.01$, we can find the maximum value for the frequency
of the test field that could overspin the black hole, which is analytically
derived in (18)

$\omega_{\rm{max}}=0.399908(1/M)$

If we choose the frequency of the incoming field in the range
$\omega_{\rm{sl}}<\omega<\omega_{\rm{max}}$, the test field with energy
$\delta M=0.01M$ will be absorbed by the black hole and it will lead to
overspinning. For example let us choose

$\omega=0.395(1/M)$

Using $\delta M=0.01M$ and we can calculate $\delta J$

$\delta J=(m/\omega)\delta M=0.025316M^{2}$

The final parameters of the black hole satisfy

$\delta_{\rm{fin}}=(M+\delta M)^{2}-\frac{(J+\delta J)^{2}}{(M+\delta
M)^{2}}-Q^{2}=-000319M^{2}$

Note that $Q^{2}=0.7499M^{2}$ for $\alpha=0.5$. The fact that
$\delta_{\rm{fin}}$ is negative imply that the final parameters of the black
hole describe a naked singularity. One can choose any value in the range
$\omega_{\rm{sl}}<\omega<\omega_{\rm{max}}$, and verify that
$\delta_{\rm{fin}}$ is negative.

Now we employ the backreaction effects due to the induced increase in the
angular velocity of the horizon. We have argued that the increase in the
angular velocity of the horizon leads to an increase in the superradiance
limit, which will prevent the absorption of the challenging modes for a class
of nearly extremal Kerr-Newman black holes with $\alpha\gtrsim 0.5$. The
increase in the superradiance limit $(\Delta\omega)$ is analytically derived
in (22). Note that $(\Delta\omega)$ is inversely proportional to the frequency
of the incoming field so the minimum increase occurs for
$\omega\simeq\omega_{\rm{max}}$ considering the challenging modes. Using (22)
we calculate the minimum increase in the superradiance limit for this black
hole

$\Delta\omega=0.00625(1/M)$

With this increase the superradiance limit is modified as

$\omega_{\rm{sl}}^{\prime}=0.39991(1/M)$

which implies that the modes with $\omega<0.39991(1/M)$ will not be absorbed
by the black hole. Since this value is larger than $\omega_{\rm{max}}$, none
of the challenging modes will be absorbed by the black hole. Thus, the
increase in the superradiance limit prevents overspinning as no net absorption
of the challenging modes occur.

However for smaller values of $\alpha$, $\omega_{\rm{sl}}^{\prime}$ will still
be less than $\omega_{\rm{max}}$. Then, the frequencies in the range
$\omega_{\rm{sl}}^{\prime}<\omega<\omega_{\rm{max}}$ can be used to overspin
the nearly extremal black hole. The increase in the angular velocity of the
event horizon does not bring an ultimate solution to the overspinning problem
for Kerr-Newman black holes. However, backreactions is an open problem.
Various different forms of backreactions can be considered which can possibly
fix the overspinning problem.

###  3.2 Backreactions for extremal black holes

By definition, the initial parameters of extremal Kerr-Newman black holes
satisfy:

$M^{2}-Q^{2}-(J^{2})/(M^{2})=0$ (24)

or equivalently

$1-\alpha^{2}-(Q^{2}/M^{2})=0$ (25)

If an extremal Kerr-Newman black hole is perturbed by a neutral test field,
the limiting frequency for superradiance is

$\omega_{\rm{sl-
ex}}=\frac{ma}{r_{+}^{2}+a^{2}}=\frac{m}{M\left(\frac{1}{\alpha}+\alpha\right)}$
(26)

In a recent paper we have shown that $\delta_{\rm{fin}}$ becomes negative for
an extremal Kerr-Newman black hole if the frequency of the test field is less
than the maximum value generic

$\omega<\omega_{\rm{max-
ex}}=\frac{m\zeta}{M\left[(1+\zeta)\sqrt{\zeta^{2}+2\zeta+\alpha^{2}}-\alpha\right]}$
(27)

which is the $\epsilon\to 0$ limit of the value derived for nearly extremal
black holes in (18). Again the two conditions should be satisfied
simultaneously for overspinning to occur. The test field should be absorbed by
the black hole and its contribution to the angular momentum parameter should
be larger that the contribution to the mass parameter. In generic  we have
shown that $\omega_{\rm{sl-ex}}$ is less than $\omega_{\rm{max-ex}}$ for a
class of extremal Kerr-Newman black holes which satisfy

$\alpha^{2}\equiv\frac{J^{2}}{M^{4}}<\frac{1}{3}\Rightarrow\alpha\lesssim
0.577$ (28)

This implies that the test fields with frequency in the range $\omega_{\rm{sl-
ex}}<\omega<\omega_{\rm{max-ex}}$ can be used to overspin the extremal Kerr-
Newman black holes which satisfy (28). The derivation is incomplete as it
ignores the contribution of the backreaction effects. As in the case of nearly
extremal black holes we calculate the induced increase in the angular velocity
of the event horizon before the absorption of the test field.

$\Delta\omega=\frac{\delta J}{4M^{3}}=\frac{(m/\omega)M\zeta}{4M^{3}}$ (29)

By substituting $\omega=\omega_{\rm{max-ex}}$ in (29) we derive that

$\Delta\omega=\frac{1}{4M}\left[(1+\zeta)\sqrt{\zeta^{2}+2\zeta+\alpha^{2}}-\alpha\right]$
(30)

We should add the induced increase in the angular momentum of the extremal
Kerr-Newman black hole derived in (30) to the superradiance limit (26). This
gives us the modified value of the superradiance limit. As we have argued in
the previous section, we demand that the modified value of the superradiance
limit is larger than $\omega_{\rm{max-ex}}$ derived in (27), so that the
absorption of all the challenging modes with $\omega<\omega_{\rm{max-ex}}$ is
prevented. For that purpose we let $\zeta=0.01$, and demand that

$\omega_{\rm{sl-ex}}+\Delta\omega>\omega_{\rm{max-ex}}$ (31)

One derives that (31) is satisfied, i.e. the modified value of superradiance
will exceed the maximum value of the frequency of the test field
$\omega_{\rm{max-ex}}$, provided that

$\alpha\gtrsim 0.31$ (32)

The dimensionless parameter $\alpha$ was introduced to distinguish extremal
Kerr-Newman black holes with different angular momentum and charge parameters
that all satisfy (24) and its dimensionless equivalent (25). In a recent paper
we derived that extremal Kerr-Newman black holes which satisfy $\alpha\lesssim
0.57$ can be overspun by test fields generic . The employment of backreaction
effects bring a further restriction to the class of extremal Kerr-Newman black
holes that can be overspun by test fields. The result (32) implies that the
absorption of all the challenging test fields will be prevented due to the
induced increase in the superradiance limit, provided that
$\alpha\equiv(J)/(M^{2})\gtrsim 0.31$. Let us elucidate the subject with a
numerical example. For that purpose let us consider an extremal Kerr-Newman
black hole with $\alpha=0.32$. The initial parameters of this black hole
satisfy $J=0.32M^{2}$ and $Q^{2}=0.8976M^{2}$ so that the black hole is
extremal. Our first claim is that the black hole can be overspun by test
fields if one ignores the backreaction effects. To verify this claim let us
first calculate the critical values $\omega_{\rm{sl-ex}}$ and
$\omega_{\rm{max-ex}}$ which were analytically derived in (26) and (27).

$\displaystyle\omega_{\rm{max-ex}}=0.298507(m/M)$
$\displaystyle\omega_{\rm{sl-ex}}=0.290276(m/M)$ (33)

We claim that if we choose a test field with frequency in the range
$\omega_{\rm{sl-ex}}<\omega<\omega_{\rm{max-ex}}$ and energy $\delta
M=M\zeta$, it will be absorbed by the extremal black hole and overspin it into
a naked singularity. Let us choose a test field with

$\omega=0.295(m/M);\quad\delta M=M\zeta=0.01M$ $\delta
J=\frac{m}{\omega}\delta M=0.033898M^{2}$

By definition $\delta_{\rm{in}}=0$ for an extremal black hole. Let us
calculate $\delta_{\rm{fin}}$.

$\delta_{\rm{fin}}=(M+\delta M)^{2}-\frac{(J+\delta J)^{2}}{(M+\delta
M)^{2}}-Q^{2}=-0.000276M^{2}$

The negative value of $\delta_{\rm{fin}}$ implies that the black hole is
overspun. However the fact that $\delta_{\rm{fin}}\sim\zeta^{2}$ indicates
that the overspinning can be fixed by backreaction effects.

Our second claim is that the overspinning of this extremal Kerr-Newman black
hole should be fixed by the induced increase in the superradiance limit since
$\alpha>0.31$. Using the analytical result (30), we calculate that, the
limiting frequency for superradiance to occur will increase by an amount

$\Delta\omega=0.008375(1/M)$ (34)

Then, for $m=1$ the modified value of the superradiance limit will be

$\omega^{\prime}_{\rm{sl-ex}}=\omega_{\rm{sl-ex}}+\Delta\omega=0.298651(1/M)$
(35)

Since the modified value of the superradiant limit exceeds $\omega_{\rm{max-
ex}}$, the test fields which could overspin the black hole will not be
absorbed by the black hole. In particular the test field that we have chosen
for our numerical example will not be absorbed, since
$\omega=0.295(1/M)<0.298651(1/M)$. However for smaller values of $\alpha$ the
modified value of the superradiance limit will still be less than
$\omega_{\rm{max-ex}}$. The induced increase in the angular velocity of the
event horizon brings further restrictions to the class of extremal Kerr-Newman
black holes which can be overspun by test fields, though it does not
completely fix the problem. As in the case of nearly extremal black holes, we
note that different type of backreaction effects can be employed to fix the
problem.

##  4 Absorption Probabilities in Wald type problems

As we stated in the introduction, the interaction of black holes with test
fields is actually a scattering problem. The test fields are partially
absorbed by the black hole, and partially reflected back to infinity. For
classical fields, t he transmission and reflection coefficients represent the
ratios of energies that are respectively, absorbed by the black hole and
scattered back to infinity. The conservation of energy implies that the sum of
the coefficients should be unity.

$\frac{\Phi_{\rm{reflected}}}{\Phi_{\rm{incident}}}+\frac{\Phi_{\rm{transmitted}}}{\Phi_{\rm{incident}}}=1$
(36)

Conventionally the relative flux
$(\Phi_{\rm{transmitted}})/(\Phi_{\rm{incident}})$ is interpreted as the
absorption probability of the incident field. This notion would be improper
for the cases of superradiant scattering. If the frequency of the incoming
field is lower than the superradiance limit $(\omega<\omega_{\rm{sl}})$, the
wave carries energy out of the black hole and the absorption probability will
be negative. The conservation of energy described by the equation (36)
continues to hold. Though the notion of a negative probability can be
improper, we shall continue adapt the conventional term “absorption
probability” for the relative flux
$(\Phi_{\rm{transmitted}})/(\Phi_{\rm{incident}})$, throughout this paper.

In all the previous Wald type problems to test the stability of event
horizons, the effect of the absorption probability of the test fields has been
ignored. The works of this author are not exceptions to this general attitude.
Ignoring the absorption probability corresponds to assuming that the
probability is of the order of unity if the field is absorbed by the black
hole. However the test fields are partially absorbed by the black hole and
partially reflected back to infinity. Only the transmitted part of the test
field contributes to the mass, angular momentum, and charge parameters of the
black hole. In this sense, the magnitude of the contribution is directly
proportional to the absorption probability
$(\Phi_{\rm{transmitted}})/(\Phi_{\rm{incident}})$. For the challenging modes,
the absorption probability approaches zero as the frequency becomes close to
the superradiance limit. This fundamentally alters the course of the analysis
for the interaction of test fields with extremal and nearly extremal black
holes.

In this work we incorporate the effect of absorption probabilities into Wald
type problems. For that purpose, we modify the contributions of a test field
to the mass, angular momentum, and charge parameters of the black hole taking
the absorption probabilities into consideration. If a test field carries
energy $M\zeta$ and the absorption probability of the field is $\Gamma$, the
energy absorbed by the black hole will be

$E_{\rm{abs}}=\Gamma(M\zeta)$ (37)

while the energy reflected back to infinity is

$E_{\rm{ref}}=(1-\Gamma)(M\zeta)$ (38)

The expressions (37) and (38) are direct consequences of the fact that only
the transmitted part of the test field contribute to the parameters of the
black hole. As we have mentioned above, in all the previous problems, the
field is assumed to be entirely absorbed by the black hole (the absorption
probability of the field is $\Gamma$ is assumed to be of the order of unity)
so that $\delta M=M\zeta$. After a sufficiently long time, the mass and
angular momentum parameters of the black hole will be modified as:

$\displaystyle M_{\rm{final}}$ $\displaystyle=$ $\displaystyle
M+E_{\rm{abs}}=M+\Gamma(M\zeta)$ $\displaystyle J_{\rm{final}}$
$\displaystyle=$ $\displaystyle
J+J_{\rm{abs}}=J+\frac{m}{\omega}\Gamma(M\zeta)$ (39)

where $E_{\rm{abs}}$ and $J_{\rm{abs}}$ are the energy and the angular
momentum absorbed by the black hole. The absorption probability $\Gamma$ can
be positive, negative or zero. The absorption probability $\Gamma$ appearing
in (39) should not be confused with the small parameter $\lambda$ introduced
by Sorce and Wald. $\Gamma$ is not necessarily a small parameter. It can be of
the order of unity for test fields with frequency $\omega\gg\omega_{\rm{sl}}$.
It is identically zero for the optimal perturbations with
$\omega=\omega_{\rm{sl}}$, and it is negative for the test fields in the
superradiant range $\omega<\omega_{\rm{sl}}$

The absorption probabilities $\Gamma_{s\omega lm}$ for the wave modes with
spin $s$, frequency $\omega$, spheroidal harmonic $l$ and azimuthal wave
number $m$, were first calculated by Page page . The absorption probability of
the wave depends on the parameters of the black hole such as the mass $M$, the
charge $Q$, the angular momentum $J$, the area of the event horizon $A$, the
surface gravity $\kappa$, the angular velocity of the event horizon $\Omega$
and the electrostatic potential of the event horizon $\Phi$. The parameters of
the black hole are not independent. In particular The area and the surface
gravity satisfy

$\kappa A=4\pi(r_{+}-M)$

where $(r_{+}-M)=\sqrt{M^{2}-a^{2}-Q^{2}}$ for a Kerr-Newman black hole. In
this work we are interested in the absorption probability of scalar fields
($s=0$) incident on Kerr-Newman black holes. The modes with $m=0$ do not
contribute to angular momentum, therefore we should consider the modes with
$m\geq 1$. For $l=1$ Page’s results imply that

$\Gamma_{0\omega
1m}=\frac{1}{9}\frac{A}{\pi}[M^{2}-(m^{2}-1)a^{2}-Q^{2}](\omega-m\Omega)\omega^{3}$
(40)

The factor $(\omega-m\Omega)$ implies that the absorption probability will be
negative if the incident field is in a superradiant mode
$(\omega<\omega_{\rm{sl}})$. In that case the mass of the black hole will
decrease. However, the angular momentum will decrease by a much larger
magnitude and $\delta_{\rm{fin}}$ will be positive. Therefore the modes in the
superradiant range $(\omega<\omega_{\rm{sl}})$ do not lead to overspinning.

###  4.1 Optimal perturbations

Optimal perturbations satisfy the inequality (3) at the lower limit so that

$\delta M=\Omega\delta J+\phi\delta Q$ (41)

This constitutes the lower limit to allow the absorption of a test body or
field. For neutral test fields with energy $\delta M$ and angular momentum
$\delta J=(m/\omega)\delta M$, (41) implies that the frequency of the test
field is equal to the superradiance limit $(\omega=\omega_{\rm{sl}}=m\Omega)$
for the optimal perturbations. Since the test bodies and fields with lower
energies than the optimal perturbations are not absorbed by the black holes,
they need not be considered for the overspinning and overcharging problems.
The optimal perturbations carry the lowest possible energy relative to their
angular momentum and charge. Thus, when one ignores absorption probabilities,
these modes appear to be more likely to lead to overspinning or overcharging
than any other mode that is absorbed by the black hole.

However when one incorporates the absorption probabilities into the problem,
the course of the analysis is fundamentally altered. It is manifest in (40)
that the absorption probability is zero for the optimal perturbations with
$\omega=\omega_{\rm{sl}}=m\Omega$. In that case the field is entirely
reflected back to infinity, with the same amplitude. No net absorption of the
field occurs. The final parameters of the spacetime given by (39) are
identically equal to the initial parameters.

$\displaystyle M_{\rm{final}}=M+\Gamma(M\zeta)=M$ $\displaystyle
J_{\rm{final}}=J+\frac{m}{\omega}\Gamma(M\zeta)=J$ (42)

Therefore the optimal perturbations do not challenge the stability of the
event horizon. Whether we start with an extremal or a nearly extremal black
hole, we end up with the same black hole surrounded by an event horizon. The
parameters of the black hole remain invariant after the interaction. The black
hole maintains its initial state. When absorption probability is taken into
account, the optimal perturbations become irrelevant for the overspinning and
overcharging problems.

###  4.2 Nearly extremal black holes and challenging modes

The modes that could potentially overspin black holes have frequencies close
to the superradiance limit. The absorption probabilities of these modes are
approach zero as one approaches the superradiance limit. By definition, a test
field carries a small amount of energy and angular momentum and only a small
fraction of its energy and angular momentum will be absorbed by the black hole
if its frequency is close to the superradiance limit. Therefore it seems to be
very difficult for a test field to drive a nearly extremal black hole to
extremality and beyond, when the absorption probability is taken into account.
To evaluate this quantitatively, let us consider a scalar field with frequency

$\omega=m\Omega(1+\xi)$ (43)

where the small parameter $\xi\ll 1$ assures that the frequency of the
incoming field is close to the superradiance limit $\omega_{\rm{sl}}=m\Omega$.
The scalar field is incident on a nearly extremal Kerr-Newman black hole
parametrised as (4), where $\epsilon\ll 1$ determines the closeness of the
black hole to extremality. The highest absorption probability occurs for
$m=1$. Substituting $\omega=\Omega(1+\xi)$ in (40)

$\displaystyle\Gamma_{0\omega 11}$ $\displaystyle=$
$\displaystyle\frac{8}{9}M^{2}(1+\epsilon)[M^{2}-Q^{2}](\Omega\xi)[\Omega(1+\xi)]^{3}$
(44) $\displaystyle\sim$ $\displaystyle\xi+O(\epsilon\xi)$

where we have used $A=8\pi M^{2}(1+\epsilon)$ for a nearly extremal black
hole. The leading term in the absorption probability of a challenging mode is
of the order of $\xi$. Now, we can re-evaluate the overspinning problem taking
the absorption probability into consideration. We start with a Kerr-Newman
black hole satisfying

$M^{2}-Q^{2}-\frac{J^{2}}{M^{2}}=M^{2}\epsilon^{2}$

which is perturbed by a test field with

$\displaystyle m=1;\quad\omega=m\Omega(1+\xi)$ $\displaystyle\delta
M=M\zeta;\quad\delta J=\frac{1}{\omega}\delta M;\quad\delta Q=0$
$\displaystyle\Gamma\sim\xi$ (45)

We should note that we choose the energy, frequency, and the azimuthal wave
number of the test field to challenge the stability of the event horizon. The
absorption probability of this test field is not a choice; it follows from
(40). After the interaction of the test field with the Kerr-Newman black hole,
the final parameters of the space-time will take the form

$\displaystyle M_{\rm{final}}$ $\displaystyle=$ $\displaystyle M+M\zeta\xi$
$\displaystyle J_{\rm{final}}$ $\displaystyle=$ $\displaystyle
J+\frac{1}{\omega}M\zeta\xi$ $\displaystyle Q_{\rm{final}}$ $\displaystyle=$
$\displaystyle Q$ (46)

We demand that the final parameters of the space-time describe a naked
singularity.

$(M+M\zeta\xi)^{2}-Q^{2}-\frac{\left(J+\frac{1}{\omega}M\zeta\xi\right)^{2}}{(M+M\zeta\xi)^{2}}<0$
(47)

As in the previous sections, we eliminate $Q$ from (47), and the define the
dimensionless variable $\alpha=J/M^{2}$. After some algebra one derives that
the condition (47) is equivalent to

$\omega<\omega_{\rm{max}}=\frac{\zeta\xi}{M\left[(1+\zeta\xi)\sqrt{2\zeta\xi+\epsilon^{2}+\alpha^{2}}-\alpha\right]}$
(48)

We have assumed that the frequency of the incoming field is slightly larger
than the superradiance limit by imposing $\Gamma=\xi$. Remember that the
superradiance limit for a neutral test field is

$\omega_{\rm{sl}}=\frac{m}{M\left[\frac{(1+\epsilon)^{2}}{\alpha}+\alpha\right]}$

The maximum value of the frequency of the incident field derived in (48), has
to be larger than the superradiance limit. Letting $\epsilon=\zeta=\xi=0.01$,
one derives that this will only be possible if

$\alpha\lesssim 0.01041$ (49)

We started with a nearly extremal Kerr-Newman black hole parametrised as (4).
We perturbed this black hole with a test field with energy $\delta M=M\zeta$.
For overspinning to occur the frequency of the test should be as small as
possible since the contribution to the angular momentum is inversely
proportional to the frequency. The test field should also be absorbed by the
black hole which entails that the frequency should be larger than the
superradiance limit. Therefore we the frequency should be slightly larger than
the superradiance limit ($\omega=m\Omega(1+\xi)$) for the test field. Using
(40) which follows from the seminal results by Page page , one can show that
the absorption probability for this field is of the order of $\xi$. The final
parameters of the black hole are given by (39), which implies that only the
fraction of the test field that is absorbed by the black hole contributes to
the mass, angular momentum, (and charge) parameters. We demand that the final
parameters of the black hole describe a naked singularity; i.e.
$\delta_{\rm{fin}}<0$. We derive that this demand is satisfied if the
frequency of the field is smaller than the maximum value derived in (48). This
value is larger than the superradiance limit for a class of Kerr-Newman black
holes with $\alpha=J/M^{2}\lesssim 0.01041$. To clarify the reader, we note
that for a nearly extremal Kerr-Newman black hole with mass $M=1$ and
$\alpha=0.01$, the angular momentum and charge parameters satisfy
$J^{2}=0.0001$ and $Q^{2}=0.9998$ with $\epsilon=0.01$. (See equation (17). )
For this class of nearly extremal Kerr-Newman black holes there exist modes
with positive absorption probability, that could increase the angular momentum
parameter beyond the extremal limit. This stems from the fact that the modes
with very low frequencies can have positive absorption probabilities for such
low values of $\alpha$. For these modes, the contribution to angular momentum
will be very large as it is inversely proportional to the frequency. However,
the induced increase in the angular velocity of the event horizon will also be
very large for these fields. The modified value of the superradiance limit,
which was analytically derived in (21), will considerably exceed the frequency
of the test field and its absorption will be prevented.

Let us clarify this argument with a quantitative examle. For that purpose we
consider a nearly extremal Kerr-Newman black hole with $\alpha\equiv
J/M^{2}=0.01$. For this black hole, we can use (48) and the general expression
for the superradiance limit to find that

$\displaystyle\omega_{\rm{max}}=0.009998\frac{1}{M}$
$\displaystyle\omega_{\rm{sl}}=0.009802\frac{1}{M}$ (50)

Let us consider a test field with

$\displaystyle m=1;\quad\omega=0.0099\frac{1}{M}\sim m\Omega(1+\xi)$
$\displaystyle\delta M=M\zeta=0.01M$ $\displaystyle\delta
J=\frac{m}{\omega}\delta M=1.0101M^{2}$ $\displaystyle\Gamma\sim\xi$ (51)

Without considering backreaction effects, one derives that the final
parameters of the black hole given in (46) describe a naked singularity rather
than a black hole. However, one can notice that $\delta J$ is too large for
this field; in particular it is even larger than $M^{2}$. Let us calculate the
modified value of the limiting frequency due to the induced increase in the
angular momentum of the horizon, for this mode. Using (20) and (21),

$\displaystyle\Delta\omega=\frac{\delta
J}{4M^{3}}=0.2525\left(\frac{1}{M}\right)$
$\displaystyle\omega_{\rm{sl}}^{\prime}=\omega_{\rm{sl}}+\Delta\omega=0.262302\left(\frac{1}{M}\right)$
(52)

The modified value of the superradiance limit is far larger than the frequency
of the incoming field, which assures that the test field will not be absorbed
by the black hole. Therefore the backreaction effects fix the overspinning
problem for nearly extremal Kerr-Newman black holes with $\alpha\equiv
J/M^{2}\lesssim 0.01041$. Moreover, one can argue that the challenging modes
for this class of Kerr-Newman black holes cannot be treated in the test field
approximation, since $\delta J\gtrsim M^{2}$. In this case these modes can be
excluded, and the overspinning problem becomes irrelevant. In either case, the
incorporation of the absorption probability brings an ultimate solution to the
overspinning problem for nearly extremal Kerr-Newman black holes.

###  4.3 Extremal black holes and challenging modes

We mentioned that the absorption probability is very low for the challenging
modes. Most of the energy and angular momentum carried by the test field is
reflected back to infinity. This leads one to conclude that it should be very
difficult for a test field to drive a nearly extremal black hole to
extremality and beyond. The situation is different for extremal black holes. A
tiny excess amount of angular momentum can lead to overspinning, since
$\delta_{\rm{in}}$ is equal to zero. In this section, we calculate the
possibility to overspin an extremal Kerr-Newman black hole by neutral test
fields by taking the absorption probabilities into consideration. By
definition, an extremal black hole satisfies (24) and its dimensionless
equivalent (25). The dimensionless parameter $\alpha\equiv J/M^{2}$ allows us
to distinguish extremal black holes with different parameters of charge and
angular momentum. As in the case of nearly extremal black holes we attempt to
destroy the horizon by sending in a test field with energy $\delta M=M\zeta$,
and frequency close to the superradiance limit, such that the absorption
probability takes the form $\Gamma\sim\xi$. Proceeding the same way as in the
previous section, we choose $m=1$ for the test field which maximizes the
absorption probability. We find that the event horizon will be destroyed if
the frequency of the test field satisfies

$\omega<\omega_{\rm{max-
ex}}=\frac{\zeta\xi}{M\left[(1+\zeta\xi)\sqrt{2\zeta\xi+\alpha^{2}}-\alpha\right]}$
(53)

which is the $\epsilon\to 0$ limit of the result found in (48). Remember that
the superradiance limit for a neutral test field incident on an extremal black
hole is

$\omega_{\rm{sl}}=\frac{1}{M\left[\frac{1}{\alpha}+\alpha\right]}$

where $m=1$. Letting $\zeta=\xi=0.01$, one observes that the upper limit
$\omega_{\rm{max-ex}}$ derived in (53), and the lower limit $\omega_{\rm{sl}}$
for the range of the frequencies that can lead to overspinning, almost
coincide in the range $0<\alpha<1$. The upper limit derived in (53) is
slightly larger than the superradiance limit provided that

$\alpha\lesssim 0.70707$ (54)

where the dimensionless parameter $\alpha$ defined in (16), distinguishes
extremal Kerr-Newman black holes with different angular momentum and charge
parameters, keeping the sum $(Q^{2}+J^{2}/M^{2})$ fixed. For extremal Kerr-
Newman black holes satisfying (54), there exists a narrow range of frequencies
$\omega_{\rm{sl}}<\omega<\omega_{\rm{max-ex}}$ that can lead to overspinning.
However, this can easily be fixed by employing backreaction effects. The
induced increase in the angular momentum of the event horizon can be
calculated as

$\Delta\omega=\frac{\delta
J}{4M^{3}}=\frac{1}{M}\frac{(1+\zeta\xi)\sqrt{2\zeta\xi+\alpha^{2}}-\alpha}{4\xi}$
(55)

where we have used

$\delta J=\frac{m}{\omega}\delta M=\left(\frac{1}{\omega_{\rm{max-
ex}}}\right)M\zeta$

As we have argued previously it is critical to calculate $\Delta\omega$ for
the upper limit $\omega_{\rm{max-ex}}$. Substituting the superradiance limit
for extremal black holes and the induced increase in the superradiance limit
derived in (55), the modified value of the superradiance limit takes the form:

$\displaystyle\omega^{\prime}_{\rm{sl}}$ $\displaystyle=$
$\displaystyle\omega_{\rm{sl}}+\Delta\omega$ (56) $\displaystyle=$
$\displaystyle\frac{1}{M\left[\frac{1}{\alpha}+\alpha\right]}+\frac{1}{M}\frac{(1+\zeta\xi)\sqrt{2\zeta\xi+\alpha^{2}}-\alpha}{4\xi}$

To overspin an extremal Kerr-Newman black hole, the frequency of a test field
should be less than the maximum value derived in (53). A test field with such
a low frequency can have a positive absorption probability only for a class of
extremal black holes satisfying (54). We checked if the overspinning can be
fixed by the induced increase in the angular momentum of the black hole, which
modifies the superradiance limit. Due to the induced increase in the angular
momentum, the superradiance limit increases. The absorption of the modes with
frequencies lower than the modified value of the superradiance limit is
prevented, though their absorption probability appears to be positive when one
ignores backreaction effects based on the induced increase in the angular
momentum. If this modified value exceeds the maximum value derived in (53),
the absorption of all the challenging modes with low frequencies and positive
absorption probabilities will be prevented. One observes that the modified
value of the superradiance limit exceeds the maximum value of the frequency
that can lead to overspinning, for any value of $\alpha$ in the range
$0<\alpha<1$. To clarify this, we have plotted $\omega_{\rm{max-ex}}$,
$\omega_{\rm{sl}}$, and $\omega_{\rm{sl}}^{\prime}$ as a function of $\alpha$
in figure (1).

Figure 1: The superradiance limit $\omega_{\rm{sl}}$ and the upper limit
$\omega_{\rm{max-ex}}$ almost coincide in the range $0<\alpha<1$. The modified
value of superradiance $\omega_{\rm{sl}}^{\prime}$ exceeds the upper limit
$\omega_{\rm{max-ex}}$. (Here we let $M=1$)

The fact that the modified value of the superradiance limit exceeds the upper
limit of frequency $\omega_{\rm{max-ex}}$, indicates that the absorption of
the fine-tuned frequencies in the narrow range
$\omega_{\rm{sl}}<\omega<\omega_{\rm{max-ex}}$ will be prevented; i.e. the
event horizon cannot be destroyed. Taking absorption probabilities into
consideration fixes the overspinning problem for the extremal case as well as
the nearly extremal case.

##  5 Summary and conclusions

In this work we have re-considered the overspinning problem for Kerr-Newman
black holes. First, we have scrutinized the recent analysis by Sorce and Wald
where they employ a variational method and expand the field configurations to
second order in the small parameter $\lambda$. Sorce and Wald claim that they
have obtained an expression for the full second order correction $\delta^{2}M$
without having to calculate the backreaction effects explicitly. In a recent
paper we have also employed the method developed by Sorce and Wald for MTZ
black holes. In that work we have imposed that $\delta M$ is a first order
quantity itself, for test bodies and fields. We have noticed that imposing
this non-controversial fact leads to order of magnitude problems in the SW
method. Therefore we have concluded that it would be better to calculate the
backreaction effects explicitly. Here we argued that the order of magnitude
problems do not pertain to the case of MTZ black holes, they also appear in
the case of Kerr-Newman black holes. The argument is simple. Since $\delta M$
is inherently a first order quantity:

$\lambda\delta M\Rightarrow\mbox{second order, not first}$
$\lambda\epsilon\delta M\Rightarrow\mbox{third order, not second}$
$\lambda^{2}(\delta M)^{2}\Rightarrow\mbox{fourth order, not second}$
$\lambda^{2}\delta^{2}M\Rightarrow\mbox{fourth order, not second}$

Sorce and Wald claim that the function $f(\lambda)$ in the equation (6) –which
is the equation (119) in w2 – can be made negative for the terms first order
in $\lambda$ and the contribution of the terms second order in $\lambda$ makes
$f(\lambda)$ positive again. This implies that the previous results due to
Hubeny hu , Jacobson-Sotiriou js , and Düztaş-Semiz overspin  are reproduced.
Nearly extremal black holes can be destroyed by test bodies and fields and the
destruction of the event horizon is fixed by employing backreaction effects.
We showed that these correct results cannot be reproduced by SW method. The
leading term in $f(\lambda)$ is of the second order $(M^{2}\epsilon^{2})$,
whereas the contribution of the second order perturbations are of the fourth
order $(\lambda^{2}\delta^{2}M)$. Therefore the effect of second order
corrections cannot be incorporated using SW method, unless one fallaciously
imposes $\delta M\sim M$. This assumption apparently contradicts the test
body/field approximation. For that reason, backreactions should be identified
and explicitly calculated for every specific problem.

Based on the argument that the SW method is invalid, we re-visited the
overspinning problem for extremal and nearly extremal Kerr-Newman black holes.
In a recent paper we had shown that there exists a class of extremal Kerr-
Newman black holes which can be overspun by neutral test fields generic . The
overspinning is possible if the extremal Kerr-Newman black hole satisfies
$\alpha^{2}\equiv J^{2}/M^{4}<1/3$. (The dimensionless parameter $\alpha$ is
introduced to distinguish black holes with different angular momentum and
charge parameters keeping the sum $(Q^{2}+J^{2}/M^{2})$ fixed. See equations
(17) and (25). ) There, we gave numerical examples and compared our results
with previous claims. Here we showed that nearly extremal Kerr-Newman black
holes can also be overspun independent of the value of $\alpha$. In our
analysis we do not ignore the contribution of the second order terms $(\delta
M)^{2}$ and $(\delta J)^{2}$ which drastically changes the results. Therefore
we need to calculate the backreaction effects to complete our analysis. In
this work, we employed the backreaction effects due to the increase in the
angular velocity of the horizon which was first suggested by Will will . The
induced increase in the angular velocity of the event horizon leads to an
increase in the superradiance limit. This prevents the absorption of the modes
that could potentially overspin the black hole. We showed that this
backreaction brings further restrictions to the classes of extremal and nearly
extremal black holes that can be overspun by test fields. Overspinning is
prevented for nearly extremal black holes with $\alpha\gtrsim 0.50$ and for
extremal black holes with $\alpha\gtrsim 0.31$. We noted that the effect of
backreactions is an open problem and there could be different sorts of
backreaction effects that could potentially bring a full solution to the
overspinning problem.

The results derived in this work can also be exploited to evaluate the
possibility to overspin Kerr black holes. In the limit $Q\to 0$, extremal Kerr
black holes are identified with $\alpha\equiv J/M^{2}=1$ whereas nearly
extremal ones are parametrised as $\alpha^{2}=1-\epsilon^{2}$. There is no
overspinning problem for extremal Kerr black holes even if one ignores the
backreaction effects, since $\alpha^{2}=1>(1/3)$. The backreaction effects due
to the increase in the superradiance limit fixes the overspinning problem for
nearly extremal Kerr black holes, since $\alpha=1-\epsilon^{2}>(0.50)$. These
findings are in accord with previous results on Kerr black holes.

In all the previous thought experiments –including the works of this author–
the absorption probability of test fields was ignored. Ignoring the
probability corresponds to assuming that it is of the order of unity. However,
the interaction of black holes with test fields is a scattering problem. A
fraction of the test field is absorbed by the black hole, while part of it is
reflected back to infinity. In section (4), we have incorporated the
absorption probabilities in the thought experiments to test whether the event
horizon can be destroyed. The fact that only a small fraction of the
challenging modes is absorbed by the black holes, fundamentally changes the
course of the analysis in favour of the cosmic censorship conjecture. We
calculated the absorption probability for test fields with frequency close to
the superradiance limit, using the seminal results by Page page . The
absorption probability of a test field with frequency $\omega=m\Omega(1+\xi)$
turns out to be of the order $O(\xi)$. The probability approaches zero for
optimal perturbations with frequency $\omega=m\Omega$, which implies that
these fields are entirely reflected back to infinity. The parameters of the
space-time remain invariant after the interaction with these test fields.
Hence, the event horizon cannot be destroyed. For the nearly extremal case, we
derived that there exists a class of Kerr-Newman black holes identified by
$\alpha\lesssim 0.01401$, which can be destroyed by test fields. Overspinning
occurs due to the fact that test fields with very low frequencies can be
absorbed by these black holes. The contribution of these test fields to the
angular momentum parameter will be large, since it is inversely proportional
to the frequency. However the induced increase in the angular momentum of the
event horizon is also large for these perturbations. Therefore the
overspinning is easily fixed by employing the backreaction effects. We noted
that one can also argue that these fields cannot be treated in the test field
approximation due to the large magnitude of $\delta J$. This argument would
render the overspinning problem irrelevant. For the case of extremal black
holes we derived that there exists a set of fine-tuned parameters that can
overspin Kerr-Newman black holes with $\alpha\lesssim 0.70707$. However, the
range of frequencies is very narrow and the overspinning problem is fixed by
employing backreaction effects. Both for extremal and nearly extremal Kerr-
Newman black holes, the ultimate solution to the overspinning problem follows
by the incorporation of the absorption probabilities into the analysis.

In a recent paper we argued that fermionic fields lead to a generic
destruction of the event horizon in the classical picture generic . Since the
fermionic fields do not obey the weak energy condition, one cannot find a
lower bound for the energy of a fermionic field similar to the condition (3).
The absorption probability is positive definite and it approaches zero only as
$\omega$ approaches zero, as confirmed by Page page . For that reason, the
arguments about the absorption probability of scalar fields and its effect on
the overspinning problem developed in this paper, do not apply to fermionic
fields.

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