Introduction
Black hole thermodynamics and light deflectionin effective loop quantum gravity
Ken Matsuno
Osaka Metropolitan University

1: homogeneous pressureless star: modified Friedmann equation

These spacetimes admit extremal minimal-sized configurationsof quantum gravitational nature characterized by vanishingtemperature and entropy.

Several aspects of quantum black holes have been discussed:
Thermodynamics, Hawking radiation, Quasinormal modes,
Stabilities, Accretion disk, Periapsis shift, Gravitational time delay,
Light deflection, Gravitational lensing, Black hole shadow,

Quantum Oppenheimer-Snyder model [Lewandowski et al.]Ashtekar-Pawlowski-Singh model:

Thermodynamics of quantum corrected polymer black holeand asymptotically safe gravity black hole have been discussed[Mele et al., Mandal, Gangopadhyay].

Some different models of quantum gravity are effective ways tounderstand gravity behaviors at a sufficiently small scale:
String theory, Loop quantum gravity, Asymptotically safe gravity,
Noncommutative gravity, Quadratic gravity, Rainbow gravity,: critical density: uniform density,Quantum gravity effects on black hole shadows in loop
quantum Oppenheimer-Snyder gravitational collapse model
have been discussed [Yang et al.].

M: mass of dust ball with radius

We study thermodynamics and light deflection of static black
holes in loop quantum Oppenheimer-Snyder model as one ofquantum gravity effects in Hawking radiation and gravitational
lensing, respectively.

c : quantum regimec : classical regime

2,

Metric with0 :nowhere and neversingular: Barbero-Immirzi parameter

usual Friedmann equationa( ) can be extend
to interval

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Maximal extension of black hole spacetime

Loop quantum corrected black holes

Bounce universe in loop quantum cosmology

Loop quantum cosmology

Introduction

Though debates on quantum nature of black holes are crucial
and long standing, there is no systematic study by quantum
gravity so far to lay a solid theoretical foundation for arguments.

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Equation of motion for scalar particles

Loop quantum corrected black holes

Gluing interior geometry ds 2APS with exterior spherically symmetric one:
Klein-Gordon equation for uncharged massive scalar particles:

Metric is determined foralong radial freefall geodesics by identificationsuch that induced metric and extrinsic curvature are equal on gluingsurfaces that become a single C1 surface of dusty part of spacetime.

Dust surface radiusruns over [rb, )Wave function of Klein-Gordon equation:
minimum surface

Introducing parameter 0 < < 1 by
0 < < 1/2 :A quantum deformation of Schwarzschild spacetime
as derived in effective loop quantum gravity [Kelly et al.]

1/2 < < 1 :: No horizon

WKB approximation to leading order insingularity

According to Killing vector fields, the action ansatz:( J : scalarenergy and angular momentum ): two Killing horizons

Equation of motion for scalar particles:

dust surface geodesic

A Killing observer perceives energy density:: dark matter candidate
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Inner horizon r is unstable with respect to scalar perturbations
and will probably turn into a null singularity [Cao et al.].

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Modified thermodynamics

Tunneling of scalar particles
Action for outgoing and ingoing modes for classically forbidden trajectory:

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Modified thermodynamics

Modified thermodynamics

Heat capacity:

Temperature:in black hole geometry:

Function W can be written as

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Evaporation of loop quantum corrected BHs

Entropy:

Stefan-Boltzmann law:

At minimum mass Mmin, since temperature, heat capacity andentropy vanish, black hole may not exchange its energy withsurrounding environment.

If =0, static black hole is described by Schwarzschild metric andevaporates completely with dM/dt
near end of evaporation.

Tunneling probability amplitude of scalar particles:

Comparing tunneling probability amplitude with Boltzmann factor
at temperature T, we obtain modified Hawking
temperature of loop quantum corrected black hole:11

Sparsity of Hawking radiation during BH evaporation[Gray et al.]Sparsity: average time gap between two successive particle
emissions over characteristic timescale ofindividual particle emission

As black hole mass decreases, temperature with 0 reaches
local maximum value and then decreases to zero at minimum
value of mass

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At local maximum temperature, system with
0 undergoes
transition from unstable negative heat capacity phase to stable
positive heat capacity cooling down towards cold extremalconfiguration with mass Mmin.
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Large black hole mass limit

Modified sparsity of Hawking radiation

Loop quantum correction prevents black hole to completely
evaporate and results in thermodynamic stable remnant.
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Photon motions around loop quantum corrected BHs

Evaporation of black hole with 0 halts at a minimum mass Mmin,leaving a remnant of the order of Planck mass that no longer radiate.
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Effective potential and photonsphere radius: Temperature
at M = Mmin.

: thermal wavelength of Hawking particle: universal cross section at high frequenciesSparsity is enhancedby quantum gravity
effect.

1 : Hawking radiation is a typical blackbody radiation where its
thermal wavelength is much shorter than emitting body size.
1 : Hawking radiation is not a continuous emission of particles
but a sparse radiation, i.e., most particles are randomly
emitted in a discrete manner with pauses in between.
ex) 4D Sch.= 64 3/27

Similar to black hole evaporations in noncommutative model andasymptotically safe gravity, loop quantum corrected black hole
would take an infinite amount of time to radiate a particle atfinal stage of evaporation, and then turn into remnant when
black hole mass M approaches mass Mmin.

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Light deflection by loop quantum corrected BHs

Hamiltonian for photon:

diverges: Heat capacity: constants of motion: Sparsity

We can concentrate on orbits withEntropy ::equations
Presence of quantum gravity effect decreases photonsphere radiussimilar to four-dimensional Reissner-Nordström black hole case.

Energy conservation equation:

Weak-field limit:
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Weak gravitational deflection angle of photon

Logarithmic term

18Weak gravitational deflection angle of photon

19Summary
We consider thermodynamics and light deflection of 4D static
spherically symmetric black holes in quantum Oppenheimer-Snyder
model in loop quantum gravity.

Loop quantum corrected black hole would be difficult to distinguishby weak deflection gravitational lensing.
cf. Light deflection by 5D squashed Kaluza-Klein BHs in a plasma medium

Deflection angle of photon:[K. Matsuno]

We derive modified Hawking temperature, heat capacity, entropy,
evaporation time, radiation sparsity and weak deflection angle ofphoton in loop quantum corrected black hole spacetime.
Loop quantum correction may slow down increase of Hawking
temperature due to radiation and result in thermodynamic stableremnant.
Modified sparsity of Hawking radiation may become infinite whenmass of loop quantum corrected black hole approaches its
remnant mass.

Deflection angle in weak-field limit:
Deflection angle decreases with increasing /M2 for fixed b/M.
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It would be challenging to detect the correction by extra dimension forprimordial black holes in future observations of gravitational lensing.

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Loop quantum corrected black hole would be difficult todistinguish by weak deflection gravitational lensing.
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20Discussion
If Barbero-Immirzi parameter is0.24, mass of black holeremnant is Mmin 10 8 kg, which is of order Planck mass likeblack hole remnants in minimally geometric deformation
model, quadratic gravity and asymptotically safe gravity.

Since such a black hole remnant would not radiate and its
gravitational interaction would be very weak, it would bedifficult to observe remnants in our Universe directly.
It would be expected that one possible indirect signature of
black hole remnant might be associated with cosmic
gravitational wave background [Chen, Adler].

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