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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
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Outline
• Why quantum gravity?
• Asymptotic Safety in a nutshell
• Black holes within Asymptotic Safety
• Summary
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Motivations for Quantum Gravity
1. internal consistency
Rµν − 12gµν R+Λ gµν
︸ ︷︷ ︸
classical
= 8πGN Tµν︸︷︷︸
quantum
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Motivations for Quantum Gravity
1. internal consistency
Rµν − 12gµν R+Λ gµν
︸ ︷︷ ︸
classical
= 8πGN Tµν︸︷︷︸
quantum
2. singularities in solutions of Einstein equations
• black hole singularities
• Big Bang singularity
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Motivations for Quantum Gravity
1. internal consistency
Rµν − 12gµν R+Λ gµν
︸ ︷︷ ︸
classical
= 8πGN Tµν︸︷︷︸
quantum
2. singularities in solutions of Einstein equations
• black hole singularities
• Big Bang singularity
3. cosmological observations:
• small positive cosmological constant
• initial conditions for structure formation
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Motivations for Quantum Gravity
1. internal consistency
Rµν − 12gµν R+Λ gµν
︸ ︷︷ ︸
classical
= 8πGN Tµν︸︷︷︸
quantum
2. singularities in solutions of Einstein equations
• black hole singularities
• Big Bang singularity
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
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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
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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
b) UV-completion requires new physics:
• string theory:
requires: supersymmetry, extra dimensions
• loop quantum gravity:
keeps Einstein-Hilbert action as “fundamental”
new quantization scheme
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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
b) UV-completion requires new physics:
• string theory:
requires: supersymmetry, extra dimensions
• loop quantum gravity:
keeps Einstein-Hilbert action as “fundamental”
new quantization scheme
c) Gravity makes sense as Quantum Field Theory:
• UV-completion beyond perturbation theory: Asymptotic Safety
• UV-completion by relaxing symmetries: Horava-Lifshitz
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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
b) UV-completion requires new physics:
• string theory:
requires: supersymmetry, extra dimensions
• loop quantum gravity:
keeps Einstein-Hilbert action as “fundamental”
new quantization scheme
c) Gravity makes sense as Quantum Field Theory:
• UV-completion beyond perturbation theory: Asymptotic Safety
• UV-completion by relaxing symmetries: Horava-Lifshitz
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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]
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Proposals for UV fixed points
• isotropic Gaussian Fixed Point (GFP)
fundamental theory: Einstein-Hilbert action
perturbation theory in GN
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Proposals for UV fixed points
• isotropic Gaussian Fixed Point (GFP)
fundamental theory: Einstein-Hilbert action
perturbation theory in GN
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Proposals for UV fixed points
• isotropic Gaussian Fixed Point (GFP)
fundamental theory: Einstein-Hilbert action
perturbation theory in GN
• isotropic Gaussian Fixed Point (GFP)
fundamental theory: higher-derivative gravity
perturbation theory in higher-derivative coupling
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Proposals for UV fixed points
• isotropic Gaussian Fixed Point (GFP)
fundamental theory: Einstein-Hilbert action
perturbation theory in GN
• isotropic Gaussian Fixed Point (GFP)
fundamental theory: higher-derivative gravity
perturbation theory in higher-derivative coupling
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Proposals for UV fixed points
• isotropic Gaussian Fixed Point (GFP)
fundamental theory: Einstein-Hilbert action
perturbation theory in GN
• isotropic Gaussian Fixed Point (GFP)
fundamental theory: higher-derivative gravity
perturbation theory in higher-derivative coupling
• non-Gaussian Fixed Point (NGFP)
fundamental theory: interacting
non-perturbatively renormalizable field theories
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Proposals for UV fixed points
• isotropic Gaussian Fixed Point (GFP)
fundamental theory: Einstein-Hilbert action
perturbation theory in GN
• isotropic Gaussian Fixed Point (GFP)
fundamental theory: higher-derivative gravity
perturbation theory in higher-derivative coupling
• non-Gaussian Fixed Point (NGFP)
fundamental theory: interacting
non-perturbatively renormalizable field theories
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Proposals for UV fixed points
• isotropic Gaussian Fixed Point (GFP)
fundamental theory: Einstein-Hilbert action
perturbation theory in GN
• isotropic Gaussian Fixed Point (GFP)
fundamental theory: higher-derivative gravity
perturbation theory in higher-derivative coupling
• non-Gaussian Fixed Point (NGFP)
fundamental theory: interacting
non-perturbatively renormalizable field theories
• anisotropic Gaussian Fixed Point (aGFP)
fundamental theory: Horava-Lifshitz gravity
Lorentz-violating renormalizable field theory
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Proposals for UV fixed points
• isotropic Gaussian Fixed Point (GFP)
fundamental theory: Einstein-Hilbert action
perturbation theory in GN
• isotropic Gaussian Fixed Point (GFP)
fundamental theory: higher-derivative gravity
perturbation theory in higher-derivative coupling
• non-Gaussian Fixed Point (NGFP)
fundamental theory: interacting
non-perturbatively renormalizable field theories
• anisotropic Gaussian Fixed Point (aGFP)
fundamental theory: Horava-Lifshitz gravity
Lorentz-violating renormalizable field theory
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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
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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
b) finite-dimensional UV-critical surface SUV
fixing the position of a RG-trajectory in SUV
⇐⇒ experimental determination of relevant parameters
guarantees predictive power
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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
b) finite-dimensional UV-critical surface SUV
fixing the position of a RG-trajectory in SUV
⇐⇒ experimental determination of relevant parameters
guarantees predictive power
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, . . .)
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Quantum gravity as quantum field theory: Asymptotic Safety
Requirements:
a) non-Gaussian fixed point (NGFP)
controls the UV-behavior of the RG-trajectory
ensures the absence of UV-divergences
b) finite-dimensional UV-critical surface SUV
fixing the position of a RG-trajectory in SUV
⇐⇒ experimental determination of relevant parameters
guarantees predictive power
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
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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
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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
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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)
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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
ηN
]
• 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
]
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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
0.25
0.5
0.75
1
λ
g
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
0.5
g
λ
<<1
−7010
T
• 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?
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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
[1
16πGk(−R+ 2Λk) +
1
bkR2
]
-0.5-0.25
00.25
0.5Λ
0
0.2
0.4
0.6
0.8
1
g
0200
400
b
B A
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Charting the theory space spanned by Γgrav
k[g]
...
R8 . . .
R7 . . .
R6 . . .
R5 . . .
R4 . . .
R3 CµνρσCρσ
κλCκλµν RR + 7 more
R2 CµνρσCµνρσ RµνRµν
R
1
Einstein-Hilbert truncation
polynomial f(R)-truncation
R2 + C2-truncation
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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]
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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]
gravity coupled to matter:
• gravity + scalars: asymptotic safety survives 1-loop counterterm
[ D. Benedetti, P.F. Machado and F. Saueressig, arXiv:0902.4630]
• 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
3Λr2
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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
3Λr2
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:
T =1
4π
∂f(r)
∂r
∣∣∣∣r=rhorizon
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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
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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
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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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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))
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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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))
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
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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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))
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
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
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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
proposal
k(P ) =ξ
d(P ), d(P ) =
∫
Cr
√|ds2|
• results compatible with improved e.o.m and action scheme
short distance behavior
k(r) =3ξ
2
√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
r
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High-energy behavior of RG-improved Schwarzschild black holes
• classical line element
f(r) = 1− 2G0 M
r
• RG-improvement: couplings become scale-dependent
f(r) = 1− 2G(k)M
r
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High-energy behavior of RG-improved Schwarzschild black holes
• classical line element
f(r) = 1− 2G0 M
r
• RG-improvement: couplings become scale-dependent
f(r) = 1− 2G(k)M
r
• substitute UV-scaling: G(k) = g∗ k−2
f∗(r) = 1− 2 g∗ k−2 M
r
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High-energy behavior of RG-improved Schwarzschild black holes
• classical line element
f(r) = 1− 2G0 M
r
• RG-improvement: couplings become scale-dependent
f(r) = 1− 2G(k)M
r
• substitute UV-scaling: G(k) = g∗ k−2
f∗(r) = 1− 2 g∗ k−2 M
r
• substitute the cutoff-identification k2 ∝ r−3:
f∗(r) = 1− 1
3
(4g∗
3G0ξ2
)
r2
RG improvement resolves black hole singularity
– p. 22/28
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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
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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
• classical line element
f(r) = 1− 2G0 M
r− 1
3Λ0 r
2
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RG-improved black holes including a cosmological constant
• classical line element
f(r) = 1− 2G0 M
r− 1
3Λ0 r
2
• RG-improvement: couplings become scale-dependent
f(r) = 1− 2G(k)M
r− 1
3Λ(k) r2
– p. 24/28
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RG-improved black holes including a cosmological constant
• classical line element
f(r) = 1− 2G0 M
r− 1
3Λ0 r
2
• RG-improvement: couplings become scale-dependent
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
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RG-improved black holes including a cosmological constant
• classical line element
f(r) = 1− 2G0 M
r− 1
3Λ0 r
2
• RG-improvement: couplings become scale-dependent
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
r
(3
4G0λ∗ξ
2
)
− 1
3
(4g∗
3G0ξ2
)
r2
Microscopic black hole is classical Schwarzschild de Sitter solution
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Temperature of RG-improved Schwarzschild black holes
0 2 4 6 8 10m
0.01
0.02
0.03
0.04
0.05T
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
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Temperature of asymptotic (Anti-) de Sitter black holes
0 2 4 6 8 10m
0.005
0.010
0.015
0.020
0.025
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
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Summary
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Asymptotic Safety Program
Gravitational RG flows:
• strong evidence for a non-Gaussian fixed point:
predictive: finite number of relevant parameters
connected to classical gravity
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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?
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