This essay is awarded 4th Prize in the 2023 Essay Competition of the Gravity
Research Foundation

# Black holes that are too cold to respect cosmic censorship

Shahar Hod  The Ruppin Academic Center, Emeq Hefer 40250, Israel  The Hadassah
Institute, Jerusalem 91010, Israel

###### Abstract

In this essay it is proved that there are black holes that are dangerously
cold. In particular, by analyzing the emission spectra of highly charged black
holes we reveal the fact that near-extremal black holes whose Bekenstein-
Hawking temperatures lie in the regime $T_{\text{BH}}\lesssim m^{6}_{e}/e^{3}$
may turn into horizonless naked singularities, thus violating the cosmic
censorship principle, if they emit a photon with the characteristic thermal
energy $\omega=O(T_{\text{BH}})$ [here $\\{m_{e},e\\}$ are respectively the
proper mass and the electric charge of the electron, the lightest charged
particle]. We therefore raise here the conjecture that, in the yet unknown
quantum theory of gravity, the temperatures of well behaved black-hole
spacetimes are fundamentally bounded from below by the relation
$T_{\text{BH}}\gtrsim m^{6}_{e}/e^{3}$.

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##  .1 Introduction

The mathematically elegant singularity theorems of Hawking and Penrose HawPen
; Pen2  have questioned the utility of Einstein’s theory of general relativity
in describing gravitational phenomena in highly curved spacetimes. In
particular, the Einstein field equations are known to lose their predictive
power in the presence of infinitely curved regions that contain spacetime
singularities.

Following this intriguing observation and in order to guarantee the
deterministic nature of a self-consistent theory of gravity, Penrose
conjectured that a mysterious (and diligent) “cosmic censor” protects far away
observers from being exposed to the pathological properties of spacetime
singularities Pen2 . This physically important principle asserts, in
particular, that spacetime singularities are always hidden inside of black
holes with stable shielding horizons. If true, the cosmic censorship principle
would guarantee that we live in a spacetime region in which general relativity
is a self-consistent theory of gravity Pen2 .

The most studied curved spacetime in the physics literature, the Kerr-Newman
spacetime, describes a black hole of mass $M$, electric charge $Q$, and
angular momentum $J=Ma$ that contains an hidden singularity. The
characteristic inequality (we use natural Planck units in which
$G=c=\hbar=k_{\text{B}}=4\pi\epsilon_{0}=1$) Chan ; NoteQQ

$M^{2}-Q^{2}-a^{2}\geq 0\ $ (1)

provides a necessary condition for the existence of an engulfing event horizon
that protects far away observers from being exposed to this inner spacetime
singularity. Extremal black-hole configurations, which satisfy the critical
relation $M^{2}-Q^{2}-a^{2}=0$, are on the verge of exposing their inner
singularities.

In the present essay we shall explicitly prove that well behaved charged black
holes that respect the condition (1) may turn into horizonless naked
singularities that violate the cosmic censorship principle if they quantum
mechanically emit massless photons whose characteristic energies are of the
same order of magnitude as the thermal energy of the black hole.

##  .2 Hawking evaporation of near-extremal charged black holes

We shall now analyze the physical and mathematical properties of the Hawking
emission spectra of near-extremal charged black holes whose Hawking-Bekenstein
temperatures are characterized by the strong dimensionless inequality Bek1 ;
Haw1

$\epsilon\equiv 2\pi
MT_{\text{BH}}={{M(M^{2}-Q^{2})^{1/2}}\over{[M+(M^{2}-Q^{2})^{1/2}]^{2}}}\ll
1\ .$ (2)

The emission rate of neutral bosonic fields from the black hole is given by
the familiar Hawking relation Haw1

${{dN}\over{dt}}={{1}\over{2\pi}}\sum_{l,m}\int_{0}^{\infty}{{S_{lm}(\omega)}\over{e^{\omega/T_{\text{BH}}}-1}}d\omega\
,$ (3)

where $\\{l,m\\}$ are the spheroidal and azimuthal angular harmonic indices of
the radiated field mode. The partial back-scattering of the emitted fields by
the effective curvature-centrifugal barrier outside the black hole is encoded
in the energy-dependent absorption probability factor $S_{lm}(\omega)$ Haw1 .

Interestingly, and most importantly for our analysis, it has been explicitly
proved in Pagec  that the $l$-dependent centrifugal barrier outside the black
hole mainly affects (blocks) the propagation of high-$l$ modes. Thus, the
Hawking spectra of spherically symmetric large-mass black holes Notelrg  are
dominated by the emission of massless field modes with the smallest known
angular momentum: namely, by electromagnetic field quanta with $l=1$ Notel1 .

The appearance of both a thermal factor $(e^{\omega/T_{\text{BH}}}-1)^{-1}$
and an absorption factor $S_{lm}(\omega)$ in the black-hole emission formula
(3) implies that the radiation spectrum has a well defined peak which is
characterized by the relation (below we shall determine the exact value of the
dimensionless ratio $\omega_{\text{peak}}/T_{\text{BH}}$)

$\omega_{\text{peak}}\sim T_{\text{BH}}\ll 1\ .$ (4)

The energy-dependent absorption probability factor of the dominant
electromagnetic field mode with $l=1$ is known in a closed analytical form in
the characteristic regime (4) Pagec ; Hodec :

$S_{1m}(\nu)={256\over
9}\epsilon^{8}(4\nu^{8}+5\nu^{6}+\nu^{4})\cdot[1+O(\nu\epsilon)]\ ,$ (5)

where we have used here the dimensionless frequency-temperature parameter

$\nu\equiv{{\omega}\over{4\pi T_{\text{BH}}}}\ .$ (6)

Substituting the relation (5) into the integral expression (3) for the black-
hole radiation rate, one obtains the remarkably compact near-extremal relation
Notemd

${{dN_{\gamma}}\over{dt}}={{512\epsilon^{9}}\over{3\pi
M}}\int_{0}^{\infty}{\cal N}(\nu)d\nu\ ,$ (7)

where

${\cal N}(\nu)\equiv{{(1+\nu^{2})(1+4\nu^{2})\nu^{4}}\over{e^{4\pi\nu}-1}}\ .$
(8)

From Eq. (8) one deduces that the emission spectra of black holes in the
dimensionless near-extremal regime (2) are characterized by a well defined
peak at $\nu_{\text{peak}}\simeq 0.399$. Thus, the characteristic energy of an
emitted field quantum is given by the functional relation [see Eq. (6)]

$E=\omega=\nu_{\text{peak}}\cdot 4\pi T_{\text{BH}}\ .$ (9)

The emission of neutral field modes from near-extremal black holes is
dangerous from the point of view of the cosmic censorship principle since it
allows the emitting black hole to reduce its mass without reducing its
electric charge. The emission of neutral field quanta therefore decreases the
magnitude of the expression $M^{2}-Q^{2}-a^{2}$ that appears in the necessary
condition (1) for the existence of a shielding horizon that protects far away
observers from being exposed to the pathological properties of the inner
black-hole singularity.

In particular, the emission of a photon with the characteristic energy (9) and
an azimuthal angular momentum $m$ (with $|m|\in\\{0,1\\}$) produces a new
spacetime configuration whose physical parameters are characterized by the
following relations:

$M_{\text{new}}=M-E\ \ \ \ ;\ \ \ \ Q_{\text{new}}=Q\ \ \ \ ;\ \ \ \
|a|_{\text{new}}={{|m|}\over{M-E}}\ .$ (10)

Intriguingly, taking cognizance of the necessary condition (1) for the
existence of a black-hole horizon that covers the central singularity, one
deduces from (9) and (10) that a near-extremal black hole in the dimensionless
low-temperature regime

$T_{\text{BH}}<T^{\text{critical}}_{\text{BH}}\equiv{{{\cal C}}\over{\pi
M^{3}}}\ $ (11)

with ${\cal C}\equiv\nu_{\text{peak}}+\sqrt{\nu^{2}_{\text{peak}}+{1/4}}$ that
emits a photon with the characteristic energy (9) and angular momentum
$l=|m|=1$ leaves behind it an horizonless naked singularity that violates the
black-hole condition (1).

##  .3 Huge, cold black holes endanger the cosmic censorship principle

The intriguing conclusion that black holes in the regime (11) are too cold to
respect cosmic censorship is based on our assumption that the radiation
spectra of near-extremal black holes are dominated by the emission of neutral
massless field modes (mainly by photons with $l=1$). In particular, since the
physical parameters of the positron, the lightest charged particle of the
Standard Model, are characterized by the strong inequality $e\gg m_{e}$, even
a single positron emission would push a near-extremal black hole away from the
dangerous extremal limit by increasing its temperature (2).

As we shall now prove explicitly, there exists a critical black-hole mass,
$M=M_{\text{min}}$, above which the radiation spectra of near-extremal black
holes are dominated by the emission of neutral massless field modes with the
smallest known angular momentum (that is, by photons with $l=1$ Pagec ). Black
holes in the near-extremal regime (11) with $M>M_{\text{min}}$ may evaporate
into horizonless naked singularities that violate the black-hole condition
(1), thus violating the cosmic censorship principle.

In order to determine the value of the critical black-hole mass
$M_{\text{min}}$, one may use the relation [see Eqs. (2), (7), and (8)]

${{dN_{\gamma}}\over{dt}}={{256\xi}\over{\pi}}\cdot M^{8}T^{9}_{\text{BH}}\ $
(12)

with $\xi\equiv 8\pi^{4}\zeta(5)+75\pi^{2}\zeta(7)+210\zeta(9)\simeq 1764.9$
for the emission rate of neutral massless photons with $l=1$ from near-
extremal black holes in the dimensionless regime (11). In addition, one may
use the fact that, in the regime $1\ll Mm_{e}\ll Qe\ll(Mm_{e})^{2}$, the
emission rate of charged field quanta (positrons) by near-extremal black holes
is well approximated by the Schwinger pair-production formula Pagec ; Sch1 ;
Sch2

${{dN_{e^{+}}}\over{dt}}={{e^{3}}\over{2\pi^{3}m^{2}_{e}}}\cdot\exp({-E_{\text{c}}/E_{+}})\
,$ (13)

where $E_{+}=Q/r^{2}_{+}\simeq 1/Q$ is the electric field strength of the
near-extremal black hole and $E_{\text{c}}=\pi m^{2}_{e}/e$ is the critical
electric field for quantum production of electron-positron pairs.

Our assumption that the emission spectrum of the near-extremal black hole with
the critical temperature (11) is dominated by neutral massless photons with
$l=1$ corresponds to the relation

${{dN_{e^{+}}}\over{dt}}<{{dN_{\gamma}}\over{dt}}\ .$ (14)

Taking cognizance of Eqs. (11), (12), and (13), one can express the inequality
(14) in the form

$\Big{(}{{\pi m^{2}_{e}}\over{e}}M\Big{)}^{19}\cdot\exp\Big{(}-{{\pi
m^{2}_{e}}\over{e}}M\Big{)}<{{512\xi{\cal
C}^{9}}\over{\pi^{8}e^{2}}}\cdot\Big{(}{{\pi m^{2}_{e}}\over{e}}\Big{)}^{20}\
,$ (15)

which yields the critical relation (note that, in natural Planck units, the
physical parameters of the positron are given by $e\simeq 1/137.036^{1/2}$ and
$m_{e}\simeq 4.19\cdot 10^{-23}$)

$M>M_{\text{min}}\equiv{{e}\over{\pi m^{2}_{e}}}\cdot x_{\text{min}}\simeq
1.55\times 10^{43}\cdot x_{\text{min}}\ $ (16)

with $x_{\text{min}}\simeq 2124.7$ Notems .

The Hawking radiation spectrum of the near-extremal black hole with the
critical temperature $T^{\text{critical}}_{\text{BH}}={{{\cal C}}/{\pi
M^{3}}}$ [see Eq. (11)] is dominated, in the large-mass regime (16), by the
emission of neutral massless photons that endanger the integrity of the black-
hole horizon.

##  .4 Summary and discussion

The Penrose cosmic censorship principle asserts that general relativity is a
deterministic theory of gravity and that pathological spacetime singularities
are always hidden inside of black holes with stable shielding horizons HawPen
; Pen2 .

In the present essay we have explicitly proved that near-extremal (cold and
large) black holes may evaporate into naked singularities that violate the
cosmic censorship principle. In particular, taking cognizance of the
analytically derived relations (11) and (16), one deduces that the threat to
the validity of the principle is limited to the extreme physical regime

$T_{\text{BH}}<T^{\text{critical}}_{\text{BH}}={{{\cal C}}\over{\pi
x^{3}_{\text{min}}}}\cdot\Big{(}{{\pi m^{2}_{e}}\over{e}}\Big{)}^{3}\simeq
10^{-140}\simeq 10^{-108}{\ {}^{\circ}K}\ $ (17)

of ultra-cold black holes.

Since we believe that cosmic censorship should be one of the cornerstones of a
self-consistent theory of gravity in curved spacetimes, we here raise the
conjecture that, in the yet unknown quantum theory of gravity, the
temperatures of well behaved black-hole spacetimes are fundamentally bounded
from below by a relation of the form

$T_{\text{BH}}\gtrsim{{m^{6}_{e}}\over{e^{3}}}\ .$ (18)

If the lower bound (18) is indeed respected, then the emission of
characteristic quanta from near-extremal black holes would not endanger the
validity of cosmic censorship, a principle which is fundamentally important
for a self-consistent formulation of the microscopic quantum theory of
gravity.

ACKNOWLEDGMENTS

This research is supported by the Carmel Science Foundation. I thank Don Page
for interesting correspondence. I would also like to thank Yael Oren, Arbel M.
Ongo, Ayelet B. Lata, and Alona B. Tea for stimulating discussions.

## References

  * (1) S. W. Hawking and R. Penrose, Proc. R. Soc. Lond. A314, 529 (1970).
  * (2) R. Penrose in General Relativity, an Einstein Centenary Survey, eds. S.W. Hawking and W. Israel (Cambridge University Press, 1979).
  * (3) S. Chandrasekhar, The Mathematical Theory of Black Holes, (Oxford University Press, New York, 1983).
  * (4) We shall henceforth assume, without loss of generality, the relation $Q>0$ for the electric charge of the black hole.
  * (5) J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973).
  * (6) S. W. Hawking, Commun. Math. Phys. 43, 199 (1975).
  * (7) D. Page, arXiv:hep-th/0012020.
  * (8) In Eq. (16) below we shall provide a precise definition for the concept of “large-mass black holes” in the context of black-hole evaporation.
  * (9) As nicely discussed in Pagec , the presence of the $l$-dependent centrifugal barrier around the radiating black hole implies that, as compared to the emission rate of photons with the minimally allowed value $l=1$ of the spheroidal angular index, the emission rates of massless fields with $l>1$ to infinity are suppressed by several factors of the large dimensionless quantity $1/(MT_{\text{BH}})$.
  * (10) S. Hod, The Euro. Phys. Jour. C (Letter) 78, 634 (2018).
  * (11) Here we have taken into account the three possible modes $m=-1,0,1$ for the electromagnetic field with $l=1$ and the two possible polarizations for each mode.
  * (12) J. Schwinger, Phys. Rev. 82, 664 (1951).
  * (13) S. Hod, Phys. Rev. D 59, 024014 (1999).
  * (14) Note that the critical black-hole mass (16) corresponds to $M_{\text{min}}\simeq 3.6\times 10^{8}M_{\odot}$.