Black HolesA window into Strong (Quantum?) Gravity

Roberto CasadioDIFA - University of Bologna

INFN-FLAG

Vulcano Workshop 2016

1.Gravitational collapse of quantum matter

2.Gravity: Status Quo

3.Horizon Quantum Mechanics*: Schwarzschild radius of quantum source

• Single particle: horizon wave-function and the GUP

• BEC black holes: fuzzy horizon and semiclassical limit

4.Summary and outlook

Plan of the talk

*Papers:Entropy 17 (2015) 6893IJMP D 25 (2016) 1630006 JCAP 09 (2015) 002JHEP 05 (2015) 096PLB 732 (2014) 105; 747 (2015) 68PRD 90 (2014) 084040; 91 (2015) 124069EPJC 74 (2014) 2685; 75 (2015) 160; 75 (2015) 445

*Collaborators:R. Cavalcanti (ABC Fed)A. Giugno (LMU) A. Giusti (Bologna)O. Micu (ISS)J. Mureika (Loyola)F. Scardigli (Kuwait)D. Stojkovic (Buffalo)

1) Gravitational collapse

Standard CLASSICAL picture = General Relativity

CLASSICAL matter and GEOMETRICAL space-time*

ds2= −

1−2M

r

dt2+

1−2M

r

−1

dr2+ r

2dΩ

2*Prototype background:

But matter is QUANTUM…!

Central Singularity

|0; t = −1i

|0; t = +1i

e−

i

~

RH dt

|0; t = +1i =X

excitations = Hawking radiation

Standard SEMICLASSICAL picture = QFT on curved space-time

Background: CLASSICAL matter and GEOMETRICAL space-time Foreground: QUANTUM particles

But world is QUANTUM…!(?)

1) Gravitational collapse

p

~/GN = Mp ' 1019

GeV

p

~GN = `p ' 10−35

m

~

Mp

' GN Mp

Q(FT) scale

(Compton length)

(Strong) Gravity scale

(Schwarzschild radius)

Quantum Gravity(black holes)

Numerology or…?

Perhaps we need Quantum Gravity … but what might it really be?

GN =

`p

mp

1) Gravitational collapse

MPQG

E

Effective QFT of (linearised) Gravity

GR or QG (?)

Black Holes

Standard model (QFT) General Relativity+

L

LP

E

L'

MP

LP

QG collapse

GR collapse

Singularity Th

ρ

ρP

1

ρ

ρP

& 1

ρ

ρP'

E

L3

L3

P

MP

ρ

ρP

1

ρ

ρP

& 1

2) Gravity: Status Quo

BH (Quantum Gravity) vs Hadron (QCD)

Confines everythingin non-perturbative regime

(within causal horizon)

Asymptotically classical M ! Mp

(Almost) no data!

Expectations:

2) Gravity: Status Quo

Confines quarksin non-perturbative regime

(below ) ΛQCD ' 220MeV

Asymptotically free E ! ΛQCD

A lot of data!

Facts:

Employ effective Quantum (Field) Theory (~ chiral theory)

What is the Schwarzschild radius of QM states?

Many attempts at quantising “pure” black hole/horizon degrees of freedom(independently of source)

But GR is non-linear:(non-perturbatively) quantise black hole/horizon and matter source!

“Background field approach”

3) Horizon Quantum Mechanics

Classical spherically symmetric system:

Sphere is a trapping surface (“escape velocity = c”) if

4π R2

H= 4π r

2

Areal radiusSchwarzschild radius

RH = 2 `pE

mp

E

Misner-Sharp mass

GN =

`p

mp

ds2 = gtt dt2+ grr dr

2+ r2

(

dθ2 + sin2 θ dφ2

)

grr = 1−2GN E(t, r)

r

Einstein equation (“Hamiltonian constraint”):

E =4π

3

Zr

0

ρ(t, r0) r2 dr0

3) Horizon Quantum Mechanics

A) “Hamiltonian constraint” relates ADM mass to gravitational radius

|Ψi =X

α,β

C(Eα, RHβ) |Eαi|RH,βi

0 =

H −mp

2 `pRH

|Ψi =X

α,β

Eα −mp

2 `pRHβ

C(Eα, RHβ) |Eαi|RHβi

C(Eα, RHβ) = C(Eα, 2 `p Eα/mp) δαβ

H |Eαi = Eα |Eαi

RH |RHαi = RHα |RHαi

3) Horizon Quantum Mechanics

B) trace out gravitational d.o.f. = source wave-function:

C) trace out source d.o.f. = horizon wave-function:

| Hi =X

γ

hEγ |X

α

CS(Eα) |Eαi|RHαi

=X

γ

CS(mpRHγ/2 `p) |RHγi

| Si =X

γ

hRHγ |X

α

C(Eα, 2 `pEα/mp) |Eαi|RHαi

=X

γ

CS(Eγ) |Eγi (spectral decomposition)

3) Horizon Quantum Mechanics

D) local construction also available, see arXiv:1605.tomorrow

E) extensions to modified gravity theories (under way…)

Localised particle at rest: Gaussian wave-function:

Energy spectrum:

Horizon wave-function:

RH = 2 `pE

mp

|ψSi =X

E

C(E) |Ei

E2= p2 +m2

∆ =~

`∼ m

(flat space)

ψS(r) ' e−

r2

2 `2

ψS(p) ' e−

p2

2∆2

ψH(rH) ' e−

`2

r2H

8 `4p

3) Single Particle

1 2 3 4

0.2

0.4

0.6

0.8

1.0

`/`p ∼ mp/mMBH & mp

“Fuzzy” minimum mass

[ArXiv:1305.3195]

P<(r < rH) = PS(r < rH)PH(rH) PS(r < rH) = 4π

ZrH

0

|ψS(r)|2 r2 dr

PH(rH) = 4π r2H|ψH(rH)|

2

PBH =

Z∞

0

P<(r < rH) drH

Probability density particle is inside its own Schwarzschild radius = horizon:

Probability particle is a Black Hole:

3) Single Particle

Two uncertainties:

N.B. Uncertainty derived with standard canonical commutators: (gravity is more than kinematics...?)

[q, p] = i ~

Minimum measurable length

1 2 3 4

2

4

6

8

γ = 1

h∆r2i `

2 h∆r2Hi

`2p

`2

h∆p2i `2p

`2

∆r =p

h∆r2i+ γ

q

h∆r2Hi

= `pmp

∆p+ 2 γ `p

∆p

mp

3) Single Particle

Horizon uncertainty:

Semiclassical black holes cannot be made of a single particle!

∆RH =

q

hR2Hi − hRHi2 ' `2p/`

hRHi ' `2p/`

∆RH

hRHi' 1

3) Single Particle

BH = self-sustained gravitons = BEC

RH

M ' N m

RH 'pN `p

m 'mppN

N>>1

E ' m+ U ' 0

∆E = m

∆U = m

Hawking quantum

U ' −m`p

mp

M N m

r λm `p mp

m

' −mN

↵

m2

m2p

N α ' 1

M ' mNpN

−mΓ

pN

−m3

p/`p

M2

Γ ∼1

N2N

2 1√N `p

3) BEC Black Hole [Dvali & Gomez]

BH with Planckian hair:

-4 -2 2 4

0.01

0.02

0.03

1-mode discrete spectrum(BEC ground state)

Continuous spectrum(excited states)

|ψSi =1

N !

NX

σi

"

NO

i=1

|ψ(i)S i

#

N-particle state:

RHr

∆rH

hrHi

γ

N

|ψ(i)S i ' |mi+ γ

Z

∞

m

dωi

m3/2

(ωi −m)

exp(

ωi−mm

)

− 1|ωii

C(E > M) ∼γ

m3/2

(E −M)

exp(

E−Mm

)

− 1

3) BEC Black Hole

1. Horizon Quantum Mechanics so far:

• GUP for single particle

• Semiclassical behaviour for radiating BEC black holes (end of evaporation)

• Quantum Hoop conjecture (for 2 particles in 1+1D)

• Minimum mass quantum black holes from dressed graviton propagator

• Charged sources and Quantum Cosmic Censorship

• Black holes in higher and lower dimensions

• Horizon wave-function in cosmology

2. Modified Time Evolution

3. Generalise to spinning systems (~ Kerr)

4. Analyse (2-)particle collisions with angular momentum+spin

5. (Hope for) quantum description of gravitational collapse

4) Summary and outlook

Thank you!