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Black Holes within Asymptotic Safety

Frank Saueressig

Research Institute for Mathematics, Astrophysics and Particle Physics

Radboud University Nijmegen

B. Koch and F. Saueressig, Class. Quant. Grav. 31 (2014) 015006

B. Koch and F. Saueressig, Int. J. Mod. Phys. A29 (2014) 8, 1430011

FFP14, Marseille, July 16, 2014

– p. 1/28

Outline

• Why quantum gravity?

• Asymptotic Safety in a nutshell

• Black holes within Asymptotic Safety

• Summary

– p. 2/28

Motivations for Quantum Gravity

1. internal consistency

Rµν − 12gµν R+Λ gµν

︸︷︷︸

classical

= 8πGN Tµν︸︷︷︸

quantum

– p. 3/28

︸︷︷︸

classical

quantum

2. singularities in solutions of Einstein equations

• black hole singularities

• Big Bang singularity

– p. 3/28

︸︷︷︸

classical

quantum

3. cosmological observations:

• small positive cosmological constant

• initial conditions for structure formation

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︸︷︷︸

classical

quantum

3. cosmological observations:

• small positive cosmological constant

• initial conditions for structure formation

General Relativity is incomplete

Quantum Gravity may give better answers to these puzzles

– p. 3/28

The quantum gravity landscape

a) Treat gravity as effective field theory: [J. Donoghue, gr-qc/9405057]

• compute corrections in E2/M2Pl ≪ 1 (independent of UV-completion)

• breaks down at E2 ≈ M2Pl

– p. 4/28

b) UV-completion requires new physics:

• string theory:

requires: supersymmetry, extra dimensions

• loop quantum gravity:

keeps Einstein-Hilbert action as “fundamental”

new quantization scheme

– p. 4/28

• string theory:

c) Gravity makes sense as Quantum Field Theory:

• UV-completion beyond perturbation theory: Asymptotic Safety

• UV-completion by relaxing symmetries: Horava-Lifshitz

– p. 4/28

• string theory:

c) Gravity makes sense as Quantum Field Theory:

• UV-completion beyond perturbation theory: Asymptotic Safety

• UV-completion by relaxing symmetries: Horava-Lifshitz

– p. 4/28

UV-completion of gravity within QFT

Central ingredient: fixed point of renormalization group flow

β-functions vanish at fixed point g∗i :

• RG flow can “end” at a fixed point keeping limk→∞ gk = g∗ finite!

trajectory has no unphysical UV divergences

well-defined continuum limit

• 2 classes of RG trajectories:

relevant = end at FP in UV

irrelevant = go somewhere else...

• predictive power:

number of relevant directions

= free parameters (determine experimentally)

[scholarpedia ’13]

– p. 5/28

Proposals for UV fixed points

• isotropic Gaussian Fixed Point (GFP)

fundamental theory: Einstein-Hilbert action

perturbation theory in GN

– p. 6/28

fundamental theory: higher-derivative gravity

perturbation theory in higher-derivative coupling

– p. 6/28

• non-Gaussian Fixed Point (NGFP)

fundamental theory: interacting

non-perturbatively renormalizable field theories

– p. 6/28

• anisotropic Gaussian Fixed Point (aGFP)

fundamental theory: Horava-Lifshitz gravity

Lorentz-violating renormalizable field theory

– p. 6/28

• anisotropic Gaussian Fixed Point (aGFP)

fundamental theory: Horava-Lifshitz gravity

Lorentz-violating renormalizable field theory

– p. 6/28

Quantum gravity as quantum field theory

Requirements:

a) ultraviolet fixed point

controls the UV-behavior of the RG-trajectory

ensures the absence of UV-divergences

– p. 7/28

Requirements:

b) finite-dimensional UV-critical surface SUV

fixing the position of a RG-trajectory in SUV

⇐⇒ experimental determination of relevant parameters

guarantees predictive power

– p. 7/28

Requirements:

c) classical limit:

RG-trajectories have part where GR is good approximation

recover gravitational physics captured by General Relativity:

(perihelion shift, gravitational lensing, nucleo-synthesis, . . .)

– p. 7/28

Quantum gravity as quantum field theory: Asymptotic Safety

Requirements:

a) non-Gaussian fixed point (NGFP)

c) classical limit:

RG-trajectories have part where GR is good approximation

recover gravitational physics captured by General Relativity:

(perihelion shift, gravitational lensing, nucleo-synthesis, . . .)

Quantum Einstein Gravity (QEG)

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Asymptotic Safety

in a nutshell

– p. 9/28

Effective average action Γk for gravityC. Wetterich, Phys. Lett. B301 (1993) 90

M. Reuter, Phys. Rev. D 57 (1998) 971

central idea: integrate out quantum fluctuations shell-by-shell in momentum-space

– p. 10/28

Effective average action Γk for gravityC. Wetterich, Phys. Lett. B301 (1993) 90

M. Reuter, Phys. Rev. D 57 (1998) 971

central idea: integrate out quantum fluctuations shell-by-shell in momentum-space

• scale-dependence governed by functional renormalization group equation

k∂kΓk[h, g] =12STr

Γ(2)k +Rk

)−1k∂kRk

vertices of Γk incorporate quantum-corrections with p2 & k2

– p. 10/28

Approximate solutions of the flow equation

approximate Γk by scale-dependent Einstein-Hilbert action:

Γk ≈ 1

16πG(k)

d4x√g [−R+ 2Λ(k)] + Sgf + Sgh

• two running couplings: G(k),Λ(k)

– p. 11/28

Approximate solutions of the flow equation

approximate Γk by scale-dependent Einstein-Hilbert action:

Γk ≈ 1

16πG(k)

d4x√g [−R+ 2Λ(k)] + Sgf + Sgh

• two running couplings: G(k),Λ(k)

explicit β-functions for dimensionless couplings gk := k2G(k) , λk := Λ(k)k−2

• Particular choice of Rk (Litim cutoff)

k∂kgk =(ηN + 2)gk ,

k∂kλk = − (2− ηN )λk − gk2π

5 11−2λk

− 4− 56

11−2λk

• anomalous dimension of Newton’s constant:

ηN =gB1

1− gB2

B1 = 13π

5 11−2λ

− 9 1(1−2λ)2

− 7]

, B2 = − 112π

5 11−2λ

+ 6 1(1−2λ)2

– p. 11/28

Einstein-Hilbert-truncation: the phase diagramM. Reuter and F. Saueressig, Phys. Rev. D 65 (2002) 065016 [hep-th/0110054]

−0.2 −0.1 0.1 0.2 0.3 0.4 0.5

−0.75

−0.5

−0.25

Type IIIaType Ia

Type IIa

Type Ib

Type IIIb

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Connecting the quantum and classical regimesM. Reuter, H. Weyer, JCAP 0412 (2004) 001, hep-th/0410119

identify RG trajectory realized in Nature by measurement of GN ,Λ

?P1 P2

−7010

• NGFP: quantum regime (G(k) = k−2g∗,Λ(k) = k2λ∗)

• T: flow passes extremely close to GFP

• P1 → P2: classical regime (G(k) = const,Λ(k) = const)

• λ . 1/2: IR fixed point?

– p. 13/28

Charting the RG-flow of the R2-truncationO. Lauscher, M. Reuter, Phys. Rev. D66 (2002) 025026, hep-th/0205062

S. Rechenberger, F.S., Phys. Rev. D86 (2012) 024018, arXiv:1206.0657

Extending Einstein-Hilbert truncation with higher-derivative couplings

Γgravk [g] =

d4x√g

16πGk(−R+ 2Λk) +

-0.5-0.25

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Charting the theory space spanned by Γgrav

R8 . . .

R7 . . .

R6 . . .

R5 . . .

R4 . . .

R3 CµνρσCρσ

κλCκλµν RR + 7 more

R2 CµνρσCµνρσ RµνRµν

Einstein-Hilbert truncation

polynomial f(R)-truncation

R2 + C2-truncation

– p. 15/28

key results: Asymptotic Safety

pure gravity:

• evidence for Asymptotic Safety

⇒ non-Gaussian fixed point provides UV completion of gravity

• low number of relevant parameter:

⇒ dimensionality of UV-critical surface ≃ 3[ R. Percacci and A. Codello, arXiv:0705.1769]

[ P.F. Machado and F. Saueressig, arXiv:0712.0445]

[ D. Benedetti, P.F. Machado and F. Saueressig, arXiv:0901.2984]

– p. 16/28

key results: Asymptotic Safety

pure gravity:

• evidence for Asymptotic Safety

⇒ non-Gaussian fixed point provides UV completion of gravity

• low number of relevant parameter:

⇒ dimensionality of UV-critical surface ≃ 3[ R. Percacci and A. Codello, arXiv:0705.1769]

[ P.F. Machado and F. Saueressig, arXiv:0712.0445]

gravity coupled to matter:

• gravity + scalars: asymptotic safety survives 1-loop counterterm

• non-Gaussian fixed point compatible with standard-model matter

[ R. Percacci and D. Perini, hep-th/0207033]

[ P. Dona, A. Eichhorn and R. Percacci, arXiv:1311.2898]

• prediction of the Higgs mass mH ≃ 126 GeV

[ M. Shaposhnikov and C. Wetterich, arXiv:0912.0208]

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Black holes in Asymptotic Safety

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Classical black hole solutions with cosmological constant

Einstein’s equations in vacuum

Rµν − 12gµν R+ Λ gµν = 0

black holes: spherical symmetric, static solutions

ds2 = −f(r)dt2 + f(r)−1dr2 + r2dΩ22

f(r) = 1− 2GM

r− 1

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Classical black hole solutions with cosmological constant

Einstein’s equations in vacuum

Rµν − 12gµν R+ Λ gµν = 0

black holes: spherical symmetric, static solutions

ds2 = −f(r)dt2 + f(r)−1dr2 + r2dΩ22

f(r) = 1− 2GM

r− 1

horizons

• Λ ≤ 0 : black hole horizon rbh

• Λ > 0,M < (3G√Λ)−1 : black hole + cosmological horizon rbh < rcosmo

Λ > 0,M ≥ (3G√Λ)−1 : naked singularity

horizon temperature:

∂f(r)

∣∣∣∣r=rhorizon

– p. 18/28

Quantum physics from average action Γk

Γk provides effective description of physics at scale k

capture quantum effects by “RG-improvement” scheme:

• transition: classical SEH → average action Γk[g]

one-parameter family of effective actions valid at different scales

k-dependence captures quantum corrections

– p. 19/28

Quantum physics from average action Γk

Γk provides effective description of physics at scale k

capture quantum effects by “RG-improvement” scheme:

• transition: classical SEH → average action Γk[g]

one-parameter family of effective actions valid at different scales

k-dependence captures quantum corrections

extracting physics information from Γk:

• single-scale problem may allow for “cutoff-identification”:

based on physical intuition:

express RG-scale k through physical cutoff ξ

⇒ modification of classical system by quantum effects

– p. 19/28

Practical RG-improvement schemes

given: physically motivated cutoff-identification k = k(ξ)

1. improved classical solutions

solve classical equations of motion

solutions: replace GN −→ G(k(ξ))

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2. improved classical equations of motion

compute equations of motion from classical action

equations of motion: replace GN −→ G(k(ξ))

solve RG-improved equations of motion

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3. improved average action

Γk: replace GN −→ G(k(ξ))

k2 ∝ R −→ Einstein-Hilbert action 7→ f(R)-gravity theory

compute modified equations of motion

solve modified equations of motion

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3. improved average action

Γk: replace GN −→ G(k(ξ))

k2 ∝ R −→ Einstein-Hilbert action 7→ f(R)-gravity theory

compute modified equations of motion

solve modified equations of motion

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Cutoff identification for black holes[A. Bonanno, M. Reuter, gr-qc/9811026]

[A. Bonanno, M. Reuter, hep-th/0002196]

[K. Falls, D. F. Litim, A. Raghuraman, arXiv:1002.0260]

requirements for cutoff-identification k = k(physical scale)

• invariance under coordinate transformations

• respect symmetries of solution

• “reasonable” asymptotic behavior

– p. 21/28

Cutoff identification for black holes[A. Bonanno, M. Reuter, gr-qc/9811026]

[A. Bonanno, M. Reuter, hep-th/0002196]

[K. Falls, D. F. Litim, A. Raghuraman, arXiv:1002.0260]

requirements for cutoff-identification k = k(physical scale)

• invariance under coordinate transformations

• respect symmetries of solution

• “reasonable” asymptotic behavior

proposal

k(P ) =ξ

d(P ), d(P ) =

√|ds2|

• results compatible with improved e.o.m and action scheme

short distance behavior

k(r) =3ξ

√2GM r−3/2 (1 +O(r))

• full function k(r) can be found numerically

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High-energy behavior of RG-improved Schwarzschild black holes

• classical line element

f(r) = 1− 2G0 M

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f(r) = 1− 2G0 M

• RG-improvement: couplings become scale-dependent

f(r) = 1− 2G(k)M

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f(r) = 1− 2G0 M

f(r) = 1− 2G(k)M

• substitute UV-scaling: G(k) = g∗ k−2

f∗(r) = 1− 2 g∗ k−2 M

– p. 22/28

f(r) = 1− 2G0 M

f(r) = 1− 2G(k)M

• substitute UV-scaling: G(k) = g∗ k−2

f∗(r) = 1− 2 g∗ k−2 M

• substitute the cutoff-identification k2 ∝ r−3:

f∗(r) = 1− 1

(4g∗

3G0ξ2

RG improvement resolves black hole singularity

– p. 22/28

Asymptotically Safe black holes and Planck starsS. Hayward, gr-qc/0506126

C. Rovelli and F. Vidotto, arXiv:1401.6562

Loop quantum gravity: modifications of f(r) due to quantum gravitational repulsion:

f(r) = 1− 2mr2

r3 + 2α2m

• α: constant determined from fundamental theory

Asymptotics of solution:

f(r) =

1− α−2r2 , r ≪ 2α2m

1− 2mr

+ . . . r ≫ 2α2m

• quantum gravitational repulsion resolves black hole singularity

• asymptotics agree with classical Schwarzschild solution

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Asymptotically Safe black holes and Planck starsS. Hayward, gr-qc/0506126

C. Rovelli and F. Vidotto, arXiv:1401.6562

Loop quantum gravity: modifications of f(r) due to quantum gravitational repulsion:

f(r) = 1− 2mr2

r3 + 2α2m

• α: constant determined from fundamental theory

Asymptotics of solution:

f(r) =

1− α−2r2 , r ≪ 2α2m

1− 2mr

+ . . . r ≫ 2α2m

• quantum gravitational repulsion resolves black hole singularity

• asymptotics agree with classical Schwarzschild solution

Same behavior has RG improved black hole!

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RG-improved black holes including a cosmological constant

f(r) = 1− 2G0 M

r− 1

3Λ0 r

– p. 24/28

f(r) = 1− 2G0 M

r− 1

3Λ0 r

f(r) = 1− 2G(k)M

r− 1

3Λ(k) r2

– p. 24/28

f(r) = 1− 2G0 M

r− 1

3Λ0 r

f(r) = 1− 2G(k)M

r− 1

3Λ(k) r2

• substitute UV-scaling: G(k) = g∗ k−2,Λ(k) = λ∗ k2

f∗(r) = 1− 2 g∗ k−2 M

r− 1

3λ∗ k2 r2

– p. 24/28

f(r) = 1− 2G0 M

r− 1

3Λ0 r

f(r) = 1− 2G(k)M

r− 1

3Λ(k) r2

• substitute UV-scaling: G(k) = g∗ k−2,Λ(k) = λ∗ k2

f∗(r) = 1− 2 g∗ k−2 M

r− 1

3λ∗ k2 r2

• substitute the cutoff-identification k2 ∝ r−3:

f∗(r) = 1− 2M

4G0λ∗ξ

(4g∗

3G0ξ2

Microscopic black hole is classical Schwarzschild de Sitter solution

– p. 24/28

Temperature of RG-improved Schwarzschild black holes

0 2 4 6 8 10m

classical Schwarzschild black hole

RG-improved without cosmological constant[A. Bonanno, M. Reuter, hep-th/0002196]

RG-improved including Λk with Λ0 = 0

• Λk crucially influences structure of light black holes

Inclusion of Λk prevents remnant formation

– p. 25/28

Temperature of asymptotic (Anti-) de Sitter black holes

0 2 4 6 8 10m

0.030T

AdS black hole Λ0 = −0.001

Schwarzschild black hole Λ0 = 0

dS black hole Λ0 = 0.001

• black holes evaporate completely

• non-Gaussian fixed point controls universal short-distance properties

– p. 26/28

Summary

– p. 27/28

Asymptotic Safety Program

Gravitational RG flows:

• strong evidence for a non-Gaussian fixed point:

predictive: finite number of relevant parameters

connected to classical gravity

– p. 28/28

Asymptotic Safety Program

Gravitational RG flows:

• strong evidence for a non-Gaussian fixed point:

predictive: finite number of relevant parameters

connected to classical gravity

Asymptotically Safe black holes:

• RG improved Schwarzschild black holes

black hole singularity replaced by de Sitter patch

formation of black hole remnants

• RG improved black holes including cosmological constant

microscopic structure: Schwarzschild-de Sitter black hole

no formation of black hole remnants

quantum singularity related to dynamical dimensional reduction?

– p. 28/28

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