Quantum Black Holes and Global Symmetries Daniel Klaewer Max-Planck-Institute for Physics, Munich Young Scientist Workshop 2017, Schloss Ringberg
Outline 1) Quantum fields in curved spacetime 2) The Unruh effect 3) Quantum black holes and Hawking radiation 4) Problems with global symmetries 2
Quantum Gravity? ‣ Collider physics described by (perturbative) quantum field theory ‣ Prototype: scalar field theory � 1 � � 2 η µ ν ∂ µ φ∂ ν φ − 1 d 4 x 2 m 2 φ 2 − λφ 4 S = ∼ λ � � L free L int ‣ Dictated by unification of quantum mechanics i � ∂ t | ψ � = ˆ H | ψ � 1 0 0 0 ‣ With special relativity 0 − 1 0 0 η µ ν = 0 0 − 1 0 d � 2 = � µ ν x µ x ν = ( ct ) 2 − � x 2 0 0 0 − 1 3
Quantum Gravity? ‣ General relativity described by highly non-linear, complicated Einstein field equations for the metric (spacetime geometry) g µ ν 1 0 0 0 g 00 g 01 g 02 g 03 0 − 1 0 0 g 10 g 11 g 12 g 13 η µ ν = → g µ ν = 0 0 − 1 0 g 20 g 21 g 22 g 23 0 0 0 − 1 g 30 g 31 g 32 g 33 2 Rg µ ν = 8 π G R µ ν − 1 R ∼ d ( g − 1 dg ) + ( g − 1 dg ) 2 c 4 T µ ν ‣ Naive treatment as perturbative QFT runs into big problems (non- renormalizability) Gravity has to be quantized - we just don’t know for sure how! For this talk we will remain ignorant! 4
Necessity of Quantization 2 Rg µ ν = 8 π G R µ ν − 1 c 4 T µ ν � � gravity matter 2 Rg µ ν = 8 π G c 4 ⟨ ˆ R µ ν − 1 ‣ Why not just quantize RHS? T µ ν ⟩ ‣ Superposition states: gravitational field is “average” over possible outcomes. Upon measurement we have collapse and discontinuous change of the gravitational field. ‣ Violates locality, Lorentz invariance, also , which is ∇ µ ⟨ ˆ T µ ν ⟩ ̸ = 0 inconsistent with the proposed equation. 5
Quantum Fields in Curved Spacetime! ‣ Study quantum fields in classical gravity background (e.g. black hole) � 1 � � 2 η µ ν ∂ µ φ∂ ν φ − 1 d 4 x 2 m 2 φ 2 − λφ 4 � 1 2 g µ ν ∂ µ φ∂ ν φ − 1 � � � d 4 x 2 m 2 φ 2 − λφ 4 det( g ) ‣ Surprisingly, leads to non-trivial, robust insights about quantum gravity ‣ Works as long as curvature is not too strong (black hole singularity) 6
The Vacuum State ‣ QFT vacuum: state of lowest energy ˆ H | Ω ⟩ = E min | Ω ⟩ ‣ Equivalently killed by annihilation operators for every particle a I | Ω ⟩ = 0 ˆ ‣ Lorentz symmetry: inertial observers agree on vacuum [ H, Λ µ ν ] = 0 ‣ In fact only true for inertial observers ‣ In general relativity: no privileged class of observers! The definition of “vacuum” or “particle” in GR is inherently ambiguous ‣ Mathematically: creation/annihilation operators of two observers related by Bogolyubov transformation � � a † a † ˆ � ˆ b I = � b I | Ω A � = J | Ω A � � = 0 α IJ ˆ a J + β IJ ˆ β IJ ˆ J J J 7
The Unruh Effect ‣ Equivalence Principle: gravity is locally equivalent to accelerated frame of reference ‣ For a qualitative picture, we thus consider constantly accelerated observer (constant proper acceleration) ct = c 2 α sinh( ατ /c ) ‣ Calculate expectation value of number density operator of the accelerated observer A in Minkowski vacuum 1 ⟨ Ω M | ˆ n A ( E ) | Ω M ⟩ = � 2 π cE � exp − 1 � α ‣ This is precisely Planck’s law! A sees radiation! T = � k B α Unruh Effect x = c 2 α cosh( ατ /c ) c 2 π 8
Quantum Black Holes ‣ Classical black hole: nothing can escape from within the horizon ‣ Hawking showed: vacuum of collapse that of free falling observer ‣ Observer at infinity has relative acceleration and sees Hawking radiation ‣ Black holes lose mass and evaporate after all! ‣ Black hole thermodynamics: � c 3 4 π G A 1 � c 3 M 2 = S = k B T = dM = TdS 4 ℓ 2 8 π Gk B M P 9
Some Ballpark Figures ‣ Both Unruh and Hawking radiation are hard to measure ‣ Measuring Hawking radiation requires getting close to small black holes - only option: micro black holes at accelerators ‣ For a 1K black hole we are looking at M ≃ 10 − 8 M � ‣ Unruh radiation is in principle easy to measure but the amount of acceleration is huge, again for 1K we need α ≃ 10 20 m s 2 10
Global Symmetries and Black Holes ‣ Imagine a world with gravity, matter and a continuous global symmetry (no gauge symmetry! no associated force!) ‣ Noether’s theorem guarantees associated charge, e.g. baryon number B = 1 | q ⟩ → e � / 3 | q ⟩ q ⟩ → e − � / 3 | ¯ | ¯ q ⟩ 3 ( N q − N ¯ q ) ‣ To avoid subtleties, assume single particle species interacting only | p ⟩ through gravity and with charge under such symmetry U (1) p | p ⟩ → e � p | p ⟩ ‣ By collapsing of these to a black hole with mass and waiting until N N · m it evaporates to mass we get black holes with arbitrary charge, all of m the same mass/energy 11
No Global Symmetries + � � � ( N · m − ∆ E, N · q ) ( m, N · q ) N · ( m, q ) ‣ Important: Hawking radiation contains same number of + and - charged particles, so black hole cannot lose charge ‣ If we let the black hole completely evaporate, charge is gone! ‣ We have created a process Q = Nq Q = 0 which explicitly violates charge conservation 12
No Global Symmetries: Loophole + � � � ( N · m − ∆ E, N · q ) ( m, N · q ) N · ( m, q ) ‣ Hawking calculation only valid until M BH ≃ M p ‣ What if evaporation stops and remnant forms? ‣ No-Hair theorem: black holes with different but same are Q M indistinguishable from outside ‣ Since we can construct BH with arbitrary for a fixed and thus Q M energy, we see that black holes in the theory have infinite microcanonical entropy! ‣ Leads to various inconsistencies, violates entropy bounds! 13
Bonus: The Stringy Version ‣ We believe string theory is a consistent theory of quantum gravity ‣ Should rather forbid global symmetries then ‣ Explicit mechanism: perturbative string theory is described by two dimensional field theory on the string world sheet � η MN g ab ∂ a X M ∂ b X N + · · · � � S = dtdl X M ( t, l ) string position ‣ Introducing a global symmetry on the world-sheet magically gives rise to a gauge symmetry with associated gauge bosons in the spacetime! 14
Bonus: The AdS/CFT version ‣ Reminder: AdS/CFT is an isomorphism of two very different theories 1) Quantum gravity in Anti-de-Sitter (AdS) space 2) Conformal field theory (non-gravitational) on the AdS boundary CFT AdS global symmetry in CFT gauge symmetry in AdS Contradiction! global symmetry in AdS 15
Conclusion ‣ Even if ignorant about the details of quantum gravity, we can gain non- trivial insights by using usual QFT techniques in curved backgrounds ‣ Accelerated observers experience Unruh radiation ‣ A different manifestation of this is Hawking radiation of black holes ‣ Combining global symmetries with these expectations leads to paradoxes ‣ Hence global symmetries are not allowed in quantum gravity and thus nature! ‣ String theory seems to obey this! 16
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