Loop Quantum Gravity and Planck Stars

Carlo Rovelli

jl t

vn →

r = rin r =2m r

Saclay, march 2014 Loop Quantum Gravity and Planck Starw

jl i. Loop quantum gravity: the theory vn → • Loop cosmology

ii. Planck Stars [F Vidotto, CR, 2014] t

• Observations

r = rin r =2m r

Carlo Rovelli Loop Quantum Gravity and Planck Stars no uncontrollable infinities

existence, not uniqueness

Is there a consistent quantum theory

whose classical limit is general relativity,

in 4 lorentzian dimensions, with its standard matter couplings?

unification not addressed

What are its physical consequences?

Can we observe some specific effects?

Carlo Rovelli Loop Quantum Gravity and Planck Stars Newton: Particles Space Time

Faraday-Maxwell: Particles Fields Space Time

Special relativity: Particles Fields Spacetime

Quantum mechanics: Quantum-Fields Spacetime

General relativity: General-covariant fields

Quantum gravity: General-covariant quantum fields

Carlo Rovelli Loop Quantum Gravity and Planck Stars Loop Quantum Gravity

n0 2 L N = L [SU(2) /SU(2) ] l State space H n Operators: where Kinematics

Graph (modes, links)

Transition amplitudes W (hl)= dhvf (hf ) A(hvf ) C ZSU(2) f v Y Y e v 1 hvf A(hf )= dge (2jf + 1) Trj[hf Y†geg Y ] Vertex amplitude e0 f j SL(2,C) Xf Z Yf 2-complex (vertices, edges,C faces) Dynamics

Simplicity map Y : Hj !Hj,j j; m j, (j + 1); j, m | i 7! | i hf = hvf v Y

With a cosmological constant Λ > 0: Amplitude: SL(2,C) ➞ SL(2,C)_q network evaluation.

Carlo Rovelli Loop Quantum Gravity and Planck Stars Main results

q 1. The amplitudes with positive cosmological constant are UV and IR finite: W <

(Han, Fairbairn-Meusburger, 2011). C 1

2. The classical limit of the vertex amplitude converges (appropriately) to the Regge Hamilton function (with cosmological constant).

(Barrett et al, Conrady-Freidel, Bianchi-Perini-Magliaro, Engle, Han..., 2009-2012).

3. The boundary states represent classical 3d geometries.

(Canonical LQG 1990’, Penrose spin-geometry theorem 1971).

4. Boundary geometry operators have discrete spectra.

(Canonical LQG main results, 1990’).

Carlo Rovelli Loop Quantum Gravity and Planck Stars General relativity : Spacetime is a field (like electric field)

Quantum gravity : Space is formed by “quanta”: it is formed by elementary “bricks”

Discreteness of Geometry.

Fundamental length (like strings).

Quantum theory : All fields are formed by “quanta” (like photons).

Carlo Rovelli Loop Quantum Gravity and Planck Stars Geometry + quantum theory

Geometry: - 6 lengths or, 4 normals (up to rotations) L~ 1

L~ a,a=1...4 i, j =1, 2, 3

Closure constraint: ~ a La =0

Volume: V 2 = 9 L~ P(L~ L~ ) 2 1 · 2 ⇥ 3 A = L~ a | a| Quantum geometry:

i j 2 ij k [La,Lb]=lP ab ✏ k La

Carlo Rovelli Loop Quantum Gravity and Planck Stars Complete set of commuting operators Aa,V

Spectrum is discrete A =8⇥~G jl(jl + 1)

p N Basis that diagonalises A ,V : ja,v ,ja a | i 2 2

A a ,V are only five quantities: geometry is never sharp

Minimal volume

Carlo Rovelli Loop Quantum Gravity and Planck Stars jl

→ vn →

Spin network states j ,v | l ni

Basic states of Loop Quantum Gravity

Carlo Rovelli Loop Quantum Gravity and Planck Stars Boundary geometry: spinfoams

2 L N State space = L [SU(2) /SU(2) ] H Al Operator: triad (metric) Polyhedron l l

Area and volume form a complete set of commuting observables and have discrete spectra

Nodes: discrete quanta of volume (“quanta of space”) Links: discrete quanta of area.

Geometry is quantized: (i) eigenvalues are discrete (ii) operators do not commute → coherent states theory → semiclassical spacetime

States describe quantum geometries: not quantum states in spacetime but rather quantum states of spacetime

Carlo Rovelli Loop Quantum Gravity and Planck Stars Quantum field theory

→

The quanta of a field are particles (Dirac) (Discreteness of the spectrum of the energy of each mode)

p ...p F 3 | 1 ni ( , ,W) a(k),a†(k) F A A 3 W F eynman rules !

Carlo Rovelli Loop Quantum Gravity and Planck Stars Quantum gravity

→

Quantum granularity of spacetime (1995) (Discreteness of the spectrum of geometrical operators such as volume)

,j ,v F 3 | l ni ( , ,W) L~ F A A 3 l W T ransition amplitudes !

Carlo Rovelli Loop Quantum Gravity and Planck Stars Large distance limit

A “spinfoam”: a two-complex colored with spins v on faces and intertwiners on edges. e

jf

iSRegge iSRegge Theorem : For a 5-valent vertex A(jf ,ve) e + e j 1 [Barrett, Pereira, Hellmann, ⇠ Gomes, Dowdall, Fairbairn 2010]

[Freidel Conrady 2008, W eiS iS[g] Bianchi, Satz 2006, Z Dg e C j !1 C C ! Magliaro Perini, 2011] !1 Z

⇤ ⇤ Theorem : For a 5-valent vertex q iSRegge iSRegge ⇤~G A (jf ,ve) e + e q = e [Han 2012] j 1,q 1 ⇠ ⇠

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↵ 2 and (2) SU eomor- nodes N ↵ ~ 2 and (2) eomor- nodes ! ~ ! Results: I. Theory

1. The amplitudes (with positive cosmological constant) are UV and IR finite: (Han, Fairbairn-Moesburger, 2011).

W q < C 1

3. The classical limit of the vertex amplitude converges to the Regge Hamilton function. (Barrett et al, Conrady-Freidel, Bianchi-Perini-Magliaro, Engle, Han..., 2009-2013).

⇤ ⇤ iSRegge iSRegge ⇤~G Av e + e ,q= e j 1,q 1 ⇠ ⇠

Carlo Rovelli Loop Quantum Gravity and Planck Stars Results: II. Applications

1. Scattering - n-point functions (CR, Alesci, Bianchi, Perini, Magliaro, Ding, Zhang, Han)

2. Black hole thermodynamics - microscopic derivation of Bekenstein-Hawking entropy. (Krasnov, CR, Ashtekar, et al, Bianchi 2012).

Ac3 S = k 4~G 3. Quantum cosmology

- Singularity resolution (Bojowald, Ashtekar, Lewandowski, Singh) - Bounce? - Spinfoam cosmology (Vidotto)

a˙ 2 8⇡G ⇢ = ⇢ 1 a2 3 ⇢ ✓ Pl ◆

Carlo Rovelli Loop Quantum Gravity and Planck Stars A process and its amplitude

hl Boundary state Boundary state = in ⌦ out Boundary

“Spacetime region” Amplitude of the process A = W ( )

Loop Quantum Gravity gives a mathematical definition of the state of space, the boundary observables, and the amplitude of the process.

Carlo Rovelli Loop Quantum Gravity and Planck Stars Boundary values of the gravitational field = geometry of box surface = distance and time separation of measurements

Particle detectors = field measurements

Spacetime region Distance and time measurements = gravitational field measurments

In GR, distance and time measurements are field measurements like the other ones: they are part of the boundary data of the problem.

→

Carlo Rovelli Loop Quantum Gravity and Planck Stars Loop quantum cosmology (Covariant)

rr

u

rr

i i f f i i 1 f f 1 W (zi,zf )= dG1 G. 2 dG1 G. 2 Pt(H l(zi) ,G1G2 ) Pt(H l(zf ) ,G1 G2 ) 4 SO(4) i f Z Yl Yl

+ 2t~j(j+1) (j) (j ,j) Pt(H, G)= (2j+1)e tr D (H)Y †D (G)Y . j X h i

i 2 3 3 i p (a0 a ) S(a,a0) e ~ 3 3 = e ~

The expanding Friedmann dynamics is [Bianchi, Vidotto, Krajewski CR 2010] recovered

Carlo Rovelli Loop Quantum Gravity and Planck Stars Bounce Loop quantum cosmology (Canonical)

a˙ 2 8⇡G ⇢ 2 = ⇢ 1 a 3 ⇢Pl Result: ✓ ◆ ⇢ z” v” 1 2 2v v =0 ⇢ r z ✓ Pl◆

1 [Bojowald, Ashtekar...,] 8⇡ 2 Generic prediction of a bounce, ⇢Pl = ~G [Barrau,Cailleteau, Grain, 3 followed by an inflationary phase Vidotto, 2011] ✓ ◆

Corrections to the CMB [Agullo, Ashtekar, Nelson, 2012]

Carlo Rovelli Loop Quantum Gravity and Planck Stars Main physical lesson from loop quantum cosmology: When the matter reaches the Planck density, a strong repulsive force of quantum gravitation origin develops. Can we use this to understand what happens in the interior of a black hole?

Planck Stars

Carlo Rovelli Loop Quantum Gravity and Planck Stars Black hole in Eddington-Finkelstein coordinates ds2 = r2d!2 +2dv dr F (r, t)du2. t

Conventional static black hole F (r)=1 2m/r

r = rin r =2m r

Carlo Rovelli Loop Quantum Gravity and Planck Stars Black hole in Eddington-Finkelstein coordinates ds2 = r2d!2 +2dv dr F (r, t)du2. t

Static black hole with Planck star [Haywards] 2mr2 F (r)=1 r3 +2↵2m

r = rin r =2m r

Carlo Rovelli Loop Quantum Gravity and Planck Stars Black hole in Eddington-Finkelstein coordinates ds2 = r2d!2 +2dv dr F (r, t)du2. t

Evaporating black hole and bouncing Planck star 2m(t)r2 F (r)=1 r3 +2↵(t)2m

r = rin r =2m r

Carlo Rovelli Loop Quantum Gravity and Planck Stars Collapsing star (Penrose diagram)

Non-evaporating Evaporating black hole black hole Planck star

Carlo Rovelli Loop Quantum Gravity and Planck Stars Long proper time ~ m3

Short proper time ~ m

A black hole is a bouncing star!

seen in slow motion

Carlo Rovelli Loop Quantum Gravity and Planck Stars n Final stage of the evaporation can be at a m radius larger than LPlanck by a factor m ✓ Planck ◆

Avoiding firewalls gives n=1 This gives a final size for a primordial black hole with a size 10-14 cm.

3 tH 14 r = l 10 cm 348⇡ t P ⇠ r P

A signal at this wavelength ( ~ GeV) ?

Carlo Rovelli Loop Quantum Gravity and Planck Stars Summary

jl

vn → 1. Loop gravity provides a tentative theory of quantum gravity, which is finite and has the correct classical limit. 2. Space is discrete and quantised at the Planck scale. 3. A strong repulsive force develops when the matter energy density reaches the Planck scale.

t 4. Bounce at the Big Bang? Effects on the CMB? 5. Planck Stars? Observable in the gamma rays?

r = rin r =2m r

Carlo Rovelli Loop Quantum Gravity and Planck Stars