Proof of the Weak Gravity Conjecture from Black Hole Entropy

Transcription

1 Proof of the Weak Gravity Conjecture from Black Hole Entropy Grant N. Remmen Berkeley Center for Theoretical Physics Miller Institute for Basic Research in Science University of California, Berkeley arxiv: with Cliff Cheung & Junyu Liu Gravity, Cosmology & Physics Beyond the Standard Model June 2018, Paris

2 Introduction: The swampland program

3 The swampland program theory space landscape swampland How do we define this boundary? Landscape: Set of EFTs consistent with UV completion in quantum gravity Swampland: Set of EFTs inconsistent with UV completion in quantum gravity String theory provides a large number of consistent vacua, with different sets of low-energy laws of physics. But it does not completely populate the space of all possible EFTs.

4 The swampland program How to find the boundary of the landscape? 1. Observe examples in string theory and make conjectures Example: Weak Gravity Conjecture 2. Prove bounds from infrared physics principles Unitarity Causality Analyticity Examples: Einstein-Maxwell theory Higher-curvature gravity ( R 2, R 4 terms) Massive gravity Cheung, GNR [ ] (@ ) 4 and F 4 couplings Arkani-Hamed et al. [hep-th/ ] Cheung, GNR [ ] Bellazzini, Cheung, GNR [ ]; Cheung, GNR [ ] Adams et al. [hep-th/ ] This paper: Prove the Weak Gravity Conjecture using a new IR argument related to black hole entropy.

5 Introduction: The Weak Gravity Conjecture An ultraviolet consistency condition for quantum gravity. Statement: For any U(1) gauge theory coupled consistently with quantum gravity, there must exist in the spectrum a state with charge q and massm such that Thus, gravity is the weakest force. q, m q m > 1 m Pl Original justification: Arkani-Hamed et al. [hep-th/ ] Q, M q, m q, m q, m q, m A black hole of charge Q and massm can only decay into states satisfying q m > Q M Extremal BH decay =) WGC Why BH decay? BH remnant pathologies

6 Thermodynamics thought experiment

7 Comparing systems Consider two systems: T T same macrostate

8 Comparing systems Consider two systems: System without extra microstates System with extra microstates Entropy: S S System with extra modes has greater entropy: S = S S >0

9 Black hole entropy comparison We can compute the black hole s entropy in two situations: Theory L without Theory L = L + L with higher-derivative terms higher-derivative terms Q, M same Q, M Present in theory in UV: Massive states that generated higher-curvature terms Integrated out to generate EFT Compare entropy in the two theories: S = S S >0

10 Black hole entropy comparison We can compute the black hole s entropy in two situations: Pure Einstein-Maxwell theory IR EFT Q, M same Q, M r Area à dictated by Einstein equation Area A = à + A dictated by higherderivative-corrected Einstein equation Entropy S = Ã/4G Entropy given by Wald s formula: Z L S = 2 µ H R µ

11 Einstein-Maxwell effective action Pure Einstein-Maxwell theory IR EFT el = 1 2apple 2 R 1 4 F µ F µ L = L + L L = c 1 R 2 + c 2 R µ R µ + c 3 R µ R µ + c 4 RF µ F µ + c 5 R µ F µ F + c 6 R µ F µ F + c 7 F µ F µ F F + c 8 F µ F F F µ Other types of terms that we can drop: All terms involving r F µ, since the Bianchi identities allow us to write these in terms of r µ F µ =0and terms already included Dimension-independent total derivatives

12 Einstein-Maxwell effective action Pure Einstein-Maxwell theory IR EFT el = 1 2apple 2 R 1 4 F µ F µ L = L + L L = c 1 R 2 + c 2 R µ R µ + c 3 R µ R µ + c 4 RF µ F µ + c 5 R µ F µ F + c 6 R µ F µ F + c 7 F µ F µ F F + c 8 F µ F F F µ We will prove a positivity bound on a combination of the. We ll then demonstrate, surprisingly, that this bound precisely implies the Weak Gravity Conjecture. c i

13 Proof of S>0

14 Assumptions For the purposes of this proof, we assume: 1. There exist quantum fields at a mass scale m satisfying m, where is the scale at which QFT breaks down. In general, can be much smaller than the Planck scale. E m Pl.. m BH QFT.

15 Assumptions For the purposes of this proof, we assume: 1. There exist quantum fields at a mass scale m satisfying m, where is the scale at which QFT breaks down. In general, can be much smaller than the Planck scale. 2. The fields couple to photons and gravitons so that the higherdimension operators are generated at tree level, e.g., R, F 2 so: c i / 1/m 2 1/ 2 QFT effect Quantum gravity slop Couplings like this are common in string theory: dilaton and moduli are massless in supersymmetric limit, and acquire masses if SUSY is broken. 3. We will consider black holes with charge large enough that the specific heat is positive. As we ll see, this will be necessary for our Euclidean path integral argument.

16 Euclidean path integral Positively charged black hole, charge Q and mass M, spacetime dimensiond Perturbed metric g µ = eg µ + g µ perturbed Lagrangian L = L e + L computed from Inverse temperature of perturbed BH, M S = e +, defines periodicity in Euclidean time for the Euclidean path integral, Z where e F ( ) = Z( )= I[ĝ,Â] d[ĝ]d[â] e I = I e + I is the Euclidean action (spacetime integral of Wick-rotated Lagrangian) F ( ) is the free energy ĝ, Â are integration variables for the metric and gauge field

17 Euclidean path integral Positively charged black hole, charge Q and mass M, spacetime dimensiond Perturbed metric g µ = eg µ + g µ perturbed Lagrangian L = L e + L computed from Ultraviolet completion: introduce integration variable that are integrated out when we go from UV to IR: Z d[ĝ]d[â]d[ ˆ] e I UV[ĝ,Â, ˆ] = Z ˆ I[ĝ,Â] d[ĝ]d[â] e for the heavy fields We define the vev of ˆ to be zero in flat space. For the on-shell black hole in the L theory, 6= 0, since equations of motion dictate R, F 2

18 Going off shell We can evaluate the Euclidean action at any field configuration we wish, including one that does not satisfy the classical equations of motion. In particular, let s evaluate I UV at ˆ =0, which turns off all the higherdimension operators in L, so we have the simple mathematical fact: I UV [ĝ, Â, 0] = e I[ĝ, Â] where ei is the Euclidean action for pure Einstein-Maxwell theory. This observation will allow us to compare the two black hole entropies in L and L e via an argument that only involves working in a single theory.

19 Free energy inequality Putting our thermodynamic argument together, we have the string of (in)equalities relating the free energies of an Einstein-Maxwell and perturbed Reissner-Nordström black hole at the same temperature: log Z( )=I UV [g,a, ] <I UV [eg, A e, 0] = I[eg e, A e ] = log Z( e ) by saddle-point approximation where g,a, are the solutions to classical EoM in UV theory, with periodicity

20 Free energy inequality Putting our thermodynamic argument together, we have the string of (in)equalities relating the free energies of an Einstein-Maxwell and perturbed Reissner-Nordström black hole at the same temperature: log Z( )=I UV [g,a, ] <I UV [eg, A e, 0] = I[eg e, A e ] = log Z( e ) by saddle-point approximation if the extremum is a local minimum (will discuss shortly)

21 Free energy inequality Putting our thermodynamic argument together, we have the string of (in)equalities relating the free energies of an Einstein-Maxwell and perturbed Reissner-Nordström black hole at the same temperature: log Z( )=I UV [g,a, ] <I UV [eg, A e, 0] = I[eg e, A e ] = log Z( e ) by saddle-point approximation if the extremum is a local minimum by the off-shell relation we found previously, relating and I e I UV

22 Free energy inequality Putting our thermodynamic argument together, we have the string of (in)equalities relating the free energies of an Einstein-Maxwell and perturbed Reissner-Nordström black hole at the same temperature: log Z( )=I UV [g,a, ] <I UV [eg, A e, 0] = I[eg e, A e ] = log Z( e ) by saddle-point approximation if the extremum is a local minimum by the off-shell relation by saddle-point approximation, again Now, log Z( e ) does not correspond to the free energy of a pure Reissner- Nordström black hole of mass M, since is the perturbed inverse temperature ( = e ). To account for this, we have log Z( e ) = log Z( e e e log Z( e e ) = log Z( e e ) M@ M S usingm e log Z( e e ) and M S

23 Free energy inequality log Z( e ) = log Z( e e ) M@ M S By the definition of free energy in the canonical ensemble, log Z( )=S M =(1 M@ M )S log Z( e e )= S e e M =(1 M@M ) S e Using the above expressions and reshuffling terms, our inequality log Z( ) < log Z( e ) i.e., F ( ) < F e ( ), becomes S>0

24 Minimization of the Euclidean action We needed the saddle point, corresponding to the classical solution, to be a local minimum. Equivalently, we needed the Euclidean action to be stable under small off-shell perturbations. What about conformal saddle-point instabilities? These have been shown to be gauge artifacts.gibbons, Hawking, Perry (1978); Gibbons, Perry (1978) The Euclidean Schwarzschild black hole is known to have a bona fide instability. Gross, Perry, Yaffe (1982) However, this instability is always connected with negative specific heat. Prestidge [hep-th/ ]; Reall [hep-th/ ]; Monteiro, Santos [ ] For large enough charge, the specific heat of the black hole is positive. In D =4, this requires q/m > p 3/2 in natural units. Hereafter, we ll focus on black holes where this is satisfied.

25 Example UV completion Let s see how this works in a particular example: Higher-dimension operators completed by a massive scalar field Euclidean action for the UV completion: Z I UV [g, A, ]= d D x p apple 1 g 2apple 2 R F µ F µ + a apple R + b applef µ F µ r µ r µ m2 2 Equation of motion for : = r 2 1 m 2 a apple R + b applef µ F µ Low-energy effective theory: Z " # I[g, A] = d D x p 1 g 2apple 2 R F µ F µ 1 a 2m 2 apple R + b applef µ F µ 2

26 Example UV completion Euclidean action for the UV completion: Z I UV [g, A, ]= d D x p apple 1 g 2apple 2 R F µ F µ + a apple R + b applef µ F µ r µ r µ m2 2 Low-energy effective theory: Z " # I[g, A] = d D x p 1 g 2apple 2 R F µ F µ 1 a 2m 2 apple R + b applef µ F µ 2 Off-shell UV action: I UV [g, A, 0] = Z d D x p g apple 1 2apple 2 R F µ F µ = e I[g, A] Thus, we have I[g, A] <I[eg, e A] < e I[eg, e A]

27 Unitarity and monotonicity In the explicit example, signs of the couplings depended on the absence of ghosts or tachyons in the UV completion, so S>0is connected with unitarity. Connections with other IR consistency bounds on couplings derived from unitarity and analyticity? Connection with monotonicity theorems for RG flows: If spectrum is hierarchical, we can apply our logic for S>0 to each mass threshold, one at a time. Then: S = Z IR UV ds where ds > 0 for each state. Conjecture: Differential entropy shift may continue to be positive at the quantum (i.e., loop) level, giving us a monotonic function along RG flow.

28 Classical vs. quantum

29 Leading contributions Let s define some rescaled couplings for convenience: d 1,2,3 = apple 2 c 1,2,3, d 4,5,6 = c 4,5,6, d 7,8 = apple 2 c 7,8 Example tree-level completion: Scalar couples to curvature and gauge field as R/apple, apple F 2 Contributions to higher dimension operators: Tree level: (d i ) 1 m 2 from the propagator Loop level: Renormalization of Newton s constant: (apple 2 ) m D 2 Loop-level completions of the gravitational higher-dimension operators: (d i ) apple 2 m D 4

30 Leading contributions Let s define some rescaled couplings for convenience: d 1,2,3 = apple 2 c 1,2,3, d 4,5,6 = c 4,5,6, d 7,8 = apple 2 c 7,8 Example tree-level completion: Scalar couples to curvature and gauge field as R/apple, apple F 2 Contributions to higher dimension operators: Tree level: (d i ) 1 m 2 from the propagator Loop level: Gauge interactions contribute similarly, but enhanced by the chargeto-mass ratio of the fundamental charged particles. If these particles satisfy the WGC, we re already done, so let s conservatively assume the particles fail or marginally satisfy the WGC.

31 Region of interest Estimating the sizes of the entropy corrections for a black hole: q m S D 2 apple 2 + D 2 m D 2 + D 4 m D 4 + D 4 apple 2 m 2 + Bekenstein-Hawking entropy Loop correction tog Loop contribution to Tree contribution to L L

32 Region of interest Estimating the sizes of the entropy corrections for a black hole: q m S D 2 apple 2 + D 2 m D 2 + D 4 m D 4 + D 4 apple 2 m 2 + Tree contribution to L (4th term) dominates over all quantum (i.e., loop) corrections (2nd and 3rd terms), provided: 1 applem D/2 This is consistent with the regime of validity of the EFT, 1/m, since we take m m Pl. We will therefore consider black holes in this size range.

33 Black hole spacetime

34 The black hole system Macrostate: Charged black hole in D =4spacetime dimensions with charge Q and mass M measured at spatial infinity Komar formalism: Q = D 3 D 2 apple2 M = Z Z d D 2 p D 2 nµ r F µ i 0 i 0 d D 2 D 2 p nµ r µ K Convenient units: Charge-to-mass parameter: m = apple2 M 8 q = appleq 4 p 2 apple 2 =8 G r q = 1 2 =0=) extremal m 2 =1=) uncharged =1/2 =) q/m = p 3/2

35 Unperturbed solution The Reissner-Nordström black hole: inner horizon outer horizon Static, spherically-symmetric metric: ds 2 = eg µ dx µ dx = f(r)dt e eg(r) dr2 + r 2 d 2 Unperturbed components ( D =4): 2m ef(r) =eg(r) =1 r + q2 r 2 Field strength: ef = Q q dt ^ dr 4 r2 Outer (event) horizon: r = e e = m + p m 2 q 2 = m(1 + ) Extremality condition: m apple 1

36 Perturbed charged black hole metric Need to calculate the change in area of the black hole of fixed due to the higher-dimension operators Q, M From definition of Ricci tensor and spherically-symmetric metric: apple 2 Z M 1 +1 R g(r) =1 dr r 2 t t R r r R i i 4 r r 2 f(r) =g(r)exp r applez +1 r dr Kats, Motl, Padi [hep-th/ ] r g(r) (Rt t R r r) Inputting Einstein equation, R µ 1 2 Rg µ = apple 2 T µ, T µ = 2 ( p gl mat ) p g g µ can rewrite as apple 2 r Z +1 apple 2 M g(r) =1 dr r 2 T t t 4 r r r Z +1 f(r) =g(r)exp appleapple 2 dr r g(r) (T t t T r r)

37 The corrected energy-momentum tensor For now, focus on computing the radial metric component g Need to find the corrected energy T t t Background: et µ = 2 ( p gl p e mat ) g g µ = F µ F 1 4 g µ F F Treat higher-dimension operators as perturbation to background energy-momentum tensor T µ = T µ (g) + T µ (F ) Contribution from g µ Contribution from A µ equation of motion equation of motion (correction to Einstein s equations) (correction to Maxwell s equations)

38 The corrected energy-momentum tensor T (g) µ = Metric part of corrected energy-momentum: 2 ( p g L) p g g µ = c 1 g µ R 2 4RR µ +4r r µ R 4g µ R + c 2 g µ R R +4r r R µ 2 R µ g µ R 4R µ R + c 3 g µ R R 4R µ R 8 R µ +4r r µ R +8Rµ R 8R R µ + c 4 gµ RF F 4RFµ F 2F F R µ +2r µ r F F 2g µ (F F ) + c 5 hg µ R F F 4R F µ F 2R F µ F g µ r r (F F ) +2r r (F µ F ) (F µ F ) + c 6 gµ R F F 6F F R µ 4r r (F µf ) + c 7 gµ F F F F 8F F Fµ F + c 8 g µ F F F F 8F µ F F F To linear order in c i, input background Reissner-Nordström solution

39 The corrected energy-momentum tensor Corrected Maxwell s equations: r F µ = 4c 4 r (RF µ ) +2c 5 r (R µ F R F µ ) +4c 6 r (R µ F ) +8c 7 r (F µ F F ) +8c 8 r (F µ F F ) = r ( F µ ) Gauge field part of corrected energy-momentum: T µ (F ) = F µ F + F F µ 1 2 g µ F F To linear order in c i, input background Reissner-Nordström solution

40 Perturbed solution General form of the metric: Putting everything together, we can compute the correction to the g(r) =1 2m r ds 2 = g µ dx µ dx = f(r)dt g(r) dr2 + r 2 d 2 + q2 q 2 r 2 r 6 8 >< >: rr component: 4 5 (d 2 +4d 3 )(6q 2 15mr + 10r 2 ) +8d 4 (3q 2 7mr +4r 2 )+ 4 5 d 5(11q 2 25mr + 15r 2 ) d 6(16q 2 35mr + 20r 2 )+ 8 5 (2d 7 + d 8 )q 2 9 >= >; f and g are required to have the same zeros, since otherwise there would be a non-lorentzian spacetime region. Can confirm this via direct calculation.

41 Wald entropy formula Wald entropy for black hole in IR EFT, for a spherically symmetric spacetime: S = 2 A L R µ µ g µ,r H Binormal to the horizon: µ = s f(r) g(r) ( t µ r r µ t ) Expand in perturbations: Horizon radius: = + Area: A =4 2 = A e + A Lagrangian: L = e L + L

42 Wald entropy formula Wald entropy for black hole in IR EFT, for a spherically symmetric spacetime: S = 2 A L R µ µ g µ,r H Expand the entropy: S = 2 e A e L R µ + e A L R µ + A! el + µ R µ g µ,

43 Wald entropy formula Wald entropy for black hole in IR EFT, for a spherically symmetric spacetime: S = 2 A L R µ µ g µ,r H Expand the entropy: S = 2 e A e L R µ + e A L R µ + A! el + µ R µ g µ, el R µ = 1 2apple 2 gµ g (symmetrization implied) Since µ µ = 2, yields background e 2 S = apple e 2 A = e A 4G

44 Wald entropy formula Wald entropy for black hole in IR EFT, for a spherically symmetric spacetime: S = 2 A L R µ µ g µ,r H Expand the entropy: S = 2 e A e L R µ + e A L R µ + A! el + µ R µ g µ, Interaction contribution: S I = 2 e A L R µ µ eg µ, e S = S e S = SI + S H

45 Wald entropy formula Wald entropy for black hole in IR EFT, for a spherically symmetric spacetime: S = 2 A L R µ µ g µ,r H Expand the entropy: S = 2 e A e L R µ + e A L R µ + A! el + µ R µ g µ, Horizon contribution: S H = 2 A el R µ µ eg µ, e = 2 apple 2 A S = S e S = SI + S H

46 Interaction contribution Variation of the action with respect to the Riemann tensor: L = 2c 1 Rg µ g +2c 2 R µ g +2c 3 R µ R µ + c 4 F F g µ g + c 5 F µ F g + c 6 F µ F (anti)symmetrization implied Inputting our unperturbed background to compute to O(c i ), we have: S I S I = e S 2 m 2 (1 + ) 3 [8d 3 2(1 )(d 2 +6d 3 +2d 4 + d 5 +2d 6 )] written in terms of the rescaled coefficients

47 Horizon contribution Expand metric as g(r) =eg(r)+ g(r) and horizon radius = + Enforce horizon condition to compute horizon shift: 0=g( ) =eg(e )+ g(e )+ Shift in the horizon eg(e ) =) = g(e eg(e ) A = A e A =8 e = 8 e g(e eg(e ) Inputting our unperturbed background to compute S H to O(c i ), we have: S H = e S 4(1 ) 5m 2 (1 + ) 3 [(1 + 4 )(d 2 +4d 3 + d 5 + d 6 ) + 10 d 4 + 2(1 )(2d 7 + d 8 )]

48 Near-extremal limit Note that S H diverges in the strict! 0 limit Physical origin: inner and outer horizons degenerate, so@eg(e )/@ e =0 How small can we consistently take? Demanding S e S =) Can make this bound arbitrarily small by making BH arbitrarily large But recall that for wave function renormalization to be subdominant, we required: 1 Since d i 1 applem 2, the bound on becomes m 2 d i m 2 apple 2 m 2 which can be made parametrically small for weakly coupled theories ( m m Pl )

49 Near-extremal limit Note that S H diverges in the strict! 0 limit Physical origin: inner and outer horizons degenerate, so@eg(e )/@ e =0 Further test: What about the temperature? We ve checked that M S for the perturbed black hole agrees with the surface gravity computed from the metric. For near-extremal black holes, Demanding Again imposing e =) 1 applem 2 implies e m/ and d i /m 3 d i 1/2 m applem which we can still take parametrically small, since m m Pl

50 New positivity bounds

51 General bounds Total black hole entropy shift: S = e S 4 5m 2 (1 + ) 3 (1 ) 2 (d 2 + d 5 ) + 2( )d 3 +(1 )(1 6 )d 6 + 2(1 ) 2 (2d 7 + d 8 ) Entropy bound S>0 implies (1 ) 2 d d 3 5 (1 )(2d 3 + d 6 ) > 0 where d 0 = d 2 +4d 3 + d 5 + d 6 +4d 7 +2d 8 Coefficients are required to satisfy this bound for all values of 2 (0, 1/2) Each value of gives a linearly independent bound

52 General bounds Allowed region in d 0 -d 3 -d 6 space:

53 General bounds Another visualization of the excluded regions: d 6 /d 3 0 -d 6 /d d 0 /d 3 -d 0 /d 3 d 3 > 0 d 3 < 0

54 The Weak Gravity Conjecture In 1 (near-extremal) limit, the bound becomes d 0 > 0 How is this connected to the Weak Gravity Conjecture? In Einstein-Maxwell theory + higher-curvature terms, the extra operators modify the allowed black hole charges Original, unperturbed extremality condition is ez = q m =1 New extremality value is z =1+ z Compute by imposing horizon condition: 0=g(,z)=eg(e, ez)+ g(e, eg(e, ez)+ z eg(e, ez) z = g(e, z eg(e, ez)

55 The Weak Gravity Conjecture In 1 (near-extremal) limit, the bound becomes d 0 > 0 How is this connected to the Weak Gravity Conjecture? Direct computation: Same combination of coefficients z = 2d 0 5m 2 > 0 Consistency of black hole entropy proves the Weak Gravity Conjecture. Since z grows as the BH gets smaller, extremal BHs can keep on decaying to yet lighter extremal black holes until they reach the scale of the UV completion.

56 The Weak Gravity Conjecture small black holes large black holes q m Planck scale classical GR m Since z grows as the BH gets smaller, extremal BHs can keep on decaying to yet lighter extremal black holes until they reach the scale of the UV completion.

57 Entropy, area, and extremity Consistency of black hole entropy proves the Weak Gravity Conjecture. Why did the same combination of coefficients appear? 0=g(,z)=eg(e, ez)+ g(e, eg(e, ez)+ z eg(e, ez) For near-extremal BH with q, m fixed, z =0, so = g/@ e eg d 0 For exactly extremal BH with free charge and mass, z = g/@ ez eg Since radial component of metric is e eg>0 Metric dictates gravitational potential, which decreases with so and z have the same sign m, ez eg>0 Near-extremal entropy shift is dominated by horizon shift, so S z>0

58 Generalized Weak Gravity Conjecture This logic generalizes to theories with multiple Abelian gauge fields: Define vector z in charge-to-mass ratio space All possible large BH states = unit ball Generalized WGC: unit ball convex hull of lighter states Cheung, GNR [ ] consistent with WGC inconsistent with WGC ~z 1 ~z 1 ~z 2 ~z 2 ~z 2 ~z 2 ~z 1 ~z 1

59 Generalized Weak Gravity Conjecture This logic generalizes to theories with multiple Abelian gauge fields: Define vector z in charge-to-mass ratio space All possible large BH states = unit ball Generalized WGC: unit ball convex hull of lighter states Metric only depends on ez = ez, so earlier argument applies, using z = z ez/ ez, and implying > 0 () z ez > 0 Thus, for finite-mass, charged BH, the unit ball expands in all directions.

60 Generalized Weak Gravity Conjecture This logic generalizes to theories with multiple Abelian gauge fields: Define vector z in charge-to-mass ratio space All possible large BH states = unit ball Generalized WGC: unit ball convex hull of lighter states Metric only depends on ez = ez, so earlier argument applies, using z = z ez/ ez, and implying > 0 () z ez > 0 Thus, for finite-mass, charged BH, the unit ball expands in all directions. Consistency of black hole entropy proves the generalized Weak Gravity Conjecture.

61 Generalization to arbitrary dimension

62 Unperturbed solution The Reissner-Nordström black hole: inner horizon outer horizon Static, spherically-symmetric metric: ds 2 = eg µ dx µ dx = f(r)dt e eg(r) dr2 + r 2 d 2 D 2 Unperturbed components: 2apple ef(r) M =eg(r) =1 (D 2) D 2 r D 3 + Q 2 apple 2 (D 2)(D 3) 2 D Field strength: 2r2(D 3) Q ef = D 2 rd dt ^ dr 2

63 Unperturbed solution The Reissner-Nordström black hole: Convenient units: inner horizon m = q = apple 2 M (D 2) D 2 appleq p (D 2)(D 3) D 2 outer horizon Redefined radial coordinate: Outer (event) horizon: x = r D 3 x = e Unperturbed components: 2m ef(r) =eg(r) =1 x + q2 x 2 Field strength: Q ef = dt ^ dr D 2 rd 2 e = m + p m 2 q 2 = m(1 + ) Extremality condition: q m apple 1

64 Unperturbed solution The Reissner-Nordström black hole: Convenient units: inner horizon m = q = apple 2 M (D 2) D 2 appleq p (D 2)(D 3) D 2 outer horizon Redefined radial coordinate: x = r D 3 Unperturbed components: 2m ef(r) =eg(r) =1 x + q2 x 2 Field strength: Q ef = dt ^ dr D 2 rd 2 Thermodynamic stability (positive specific heat) requires: q m > p 2D 5 D 2 =) < D 3 D 2

65 Perturbed charged black hole metric Inversion of the Ricci tensor works the same as before: 2apple 2 M 2apple 2 g(r) =1 (D 2) D 2 r D 3 (D 2)r D 3 apple 2apple 2 Z +1 f(r) =g(r)exp dr r D 2 g(r) (T t t T r r) r Z +1 r dr r D 2 T t t Again, compute corrections to metric by treating higher-order terms as perturbations T µ We find: g(r) =1 2m x + q2 q 2 x 2 x 2(2D 5) D 3 8X i (x)c i i=1

66 Perturbed charged black hole metric g(r) =1 where 1 = (D 3)(D 4) D 2 " 2 =2 D 3 D 2 2m x + q2 q 2 x 2 x 2(2D 5) D 3 8X i (x)c i i=1 apple 2 13D2 47D + 40 q 2 8(3D 5)mx + 16(D 2)x 2 3D 7 8D 3 55D D 76 3D 7 # + 2(3D 10)(D 2)x 2 q 2 4(2D 2 10D + 11)mx " 3 =4 D 3 8D 3 48D D 44 q 2 D 2 3D 7 2(4D 2 17D + 16)mx # + 8(D 2)(D 3)x 2 2(D 2)(D 4) m2 x 2 apple (7D 13)(D 2) 4 = 4(D 3) q 2 2(3D 5)mx + 4(D 2)x 2 3D 7 apple q 2

67 " Perturbed charged black hole metric g(r) =1 2m x + apple q2 q 2 x 2 x where apple (5D 9)(D 2) 5 = 2(D 3) q 2 2(2D 3)mx + 3(D 2)x 2 3D 7 (D 2)2 6 = 4(D 3) apple4 3D 7 q2 (3D 5)mx + 2(D 2)x 2 (D 2)(D 3)2 7 =8 3D 7 (D 2)(D 3)2 8 =4 3D 7 2(2D 5) D 3 q 2 q 2 8X i (x)c i i=1 #

68 Calculation of entropy As in D =4, split entropy shift into contributions from interactions in Wald formula and from the shift in the horizon location: S = S S e = SI + S H S I = 2 e A L R µ µ eg µ, e Direct calculation yields: S I = e S 2(D 3) m 2 D 1 D 3 (1 + ) D 3 2(1 ) n 4(D 2)d 3 apple (D 4)d 1 +(D 3)d 2 + 2(2D 5)d 3 +(D 2) d d 5 + d 6 o

69 Calculation of entropy As in D =4, split entropy shift into contributions from interactions in Wald formula and from the shift in the horizon location: S = S S e = SI + S H S H = 2 A el R µ µ eg µ, e = 2 apple 2 A where A = A e A =(D 2) D 2 e D 3 = (D 2) D 2 e g(e )/@e

70 Calculation of entropy As in D =4, split entropy shift into contributions from interactions in Wald formula and from the shift in the horizon location: S = S S e = SI + S H Direct calculation yields: S H = e S 1 (3D 7)m 2 D 1 D 3 (1 + ) D 3 d 1 (1 )(D 3)(D 4)[(11D 24) + D 4] + d 2 (1 )(D 3)[(10D 2 53D + 68) +2D 2 11D + 16] +2d 3 [ (16D 3 128D D 292)(1 ) 2 + 2(3D 7)(4D 2 23D + 32)(1 ) 2(D 2)(D 4)(3D 7)] +2d 4 (1 )(D 2)(D 3)[5(D 2) + D 4] + 2(d 5 + d 6 )(D 2)(D 3)(1 )[2(D 2) + D 3] + 2(2d 7 + d 8 )(D 2) 2 (D 3)(1 ) 2

71 Near-extremal limit Note that S H diverges in the strict! 0 limit Physical origin: inner and outer horizons degenerate, so@eg(e )/@ e =0 How small can we consistently take? Demanding S e S =) Can make this bound arbitrarily small by making BH arbitrarily large But recall that for wave function renormalization to be subdominant, we required: 1 Since d i 1 applem D/2, the bound on becomes m 2 d i m 2 D 3 apple 2 m D 2 which can be made parametrically small for weakly coupled theories ( m m Pl )

72 Near-extremal limit Note that S H diverges in the strict! 0 limit Physical origin: inner and outer horizons degenerate, so@eg(e )/@ e =0 Further test: What about the temperature? We ve checked that M S for the perturbed black hole agrees with the surface gravity computed from the metric. Background inverse temperature: e 2 m 1 D 2 D 3 (1 + ) D 3 = (D 3) Inverse temperature shift for near-extremal BH: Demanding Again imposing e =) 1 d i 1/2 m 1 D 3 implies applem D/2 (D 2)/2 applem d i / 3 1/(D 3) m which we can still take parametrically small, since m m Pl

73 New positivity bounds Total black hole entropy shift: S = e S 1 (3D 7)m 2 D 1 D 3 (1 + ) D 3 d 1 (D 3)(D 4) 2 (1 ) 2 + d 2 (D 3)(2D 2 11D + 16)(1 ) 2 +2d 3 [(8D 3 60D D 128)(1 ) 2 2(D 2)(2D 5)(3D 7)(1 ) + 2(D 2) 2 (3D 7)] +2d 4 (D 2)(D 3)(D 4)(1 ) 2 +2d 5 (D 2)(D 3) 2 (1 ) 2 +2d 6 (D 2)(D 3)(1 )[ 2(2D 5) + D 3] +4d 7 (D 2) 2 (D 3)(1 ) 2 +2d 8 (D 2) 2 (D 3)(1 ) 2

74 General bounds Entropy bound S>0 implies (1 ) 2 d 0 +(D 2) 2 (3D 7) d (D 2)(D 3)(3D 7) (1 )(2d 3 + d 6 ) > 0 where d 0 = 1 4 (D 3)(D 4)2 d (D 3)(2D2 11D + 16)d (2D3 16D D 44)d (D 2)(D 3)(D 4)d (D 2)(D 3)2 (d 5 + d 6 )+(D 2) 2 (D 3) d d 8 Coefficients are required to satisfy this bound for all values of 2 0, D 3 D 2 Each value of gives a linearly independent bound

75 New positivity bounds As before, taking the near-extremal ( 1 ) limit implies d 0 > 0 The shift in extremality condition of the black hole in D dimensions is z = 4(D 3) (3D 7)(D 2)m 2 D 3 d 0 Again, we find: same combination of coefficients Consistency of black hole entropy proves the Weak Gravity Conjecture.

76 Examples and consistency checks

77 Field redefinition invariance Any physical observable should be invariant under a reparameterization of the field variables, e.g., g µ! g µ + g µ = g µ + r 1 R µ + r 2 g µ R + r 3 apple 2 F µ F + r 4 apple 2 g µ F F This has the effect of shifting the action, L = 1 2apple 2 which has the net effect of shifting the higher-dimension operator coefficients: gµ R µ 1 2 Rg µ apple 2 T µ d 1! d r 1 D 2 4 r 2 d 5! d r r 3 d 2! d r 1 d 6! d 6 d 3! d 3 d 7! d r 3 + D 4 8 d 4! d r 1 + D 4 1 r r D r 4 d 8! d r 3 r 4

78 Field redefinition invariance There are four combinations of higher-dimension operator coefficients that are invariant under this transformation: d 0 = 1 4 (D 3)(D 4)2 d (D 3)(2D2 11D + 16)d (2D3 16D D 44)d (D 2)(D 3)(D 4)d (D 2)(D 3)2 (d 5 + d 6 )+(D 2) 2 (D 3) d d 8 d 3 d 6 d 9 = d 2 + d 5 + d 8 The total entropy shift S, and hence our bounds, are built out of d 0,d 3,d 6, and hence are field redefinition invariant.

79 Concrete examples 1. Only photon self-interactions ( d 7,8 ). Our bound becomes simply 2d 7 + d 8 > 0. When we compute the four-photon scattering amplitude and apply the analyticity arguments of Adams et al. [hep-th/ ], we find that different choices of photon polarizations give 2d 7 + d 8 > 0 and d 8 > 0, so this is consistent. 2. Scalar completion: L = 1 2apple 2 R 1 4 F µ F µ + generates so a apple R + b applef µ F µ 1 2 r µ r µ 1 2 m2 2 d i = 1 2m 2 a2, 0, 0, 2a b, 0, 0,b 2, 0 d 0 = D 3 8m 2 [(D 4)a + 2(D 2)b ]2 > 0 3. Low-energy description of the heterotic string: d i = 0 {4, 16, 4, 0, 0, 0, 3, 12} 64 Our bound then becomes (6D 2 30D + 37) 2 + 2(D 2) +2D 5 > 0, which is satisfied for all 2 (0, 1) and D>3. Kats, Motl, Padi [hep-th/ ]; Gross, Sloan (1987)

80 Discussion and conclusions

81 Discussion In this work, we relied on a universal notion of thermodynamic entropy: S>0when more microstates are added to a system of a given macrostate, which we proved for tree-level completions in QFT Applying this logic to the system of charged black holes, we can compare the Wald and Bekenstein-Hawking entropy in the Einstein-Maxwell EFT Imposing the entropy bound requires positivity of various combinations of higher- dimension operator couplings R 2, RF 2, and F 4, producing a family of bounds labeled by For a near-extremal BH, these bounds imply positivity of the same combination of coefficients that also guarantees a positive correction to the extremality bound for BHs in the EFT Thus, consistency of BH entropy proves the WGC Generalizes to multiple gauge fields and arbitrary dimension

82 Future directions Can other swampland program bounds be derived using black hole entropy? Broader class of theories, e.g., Einstein-dilaton gravity Other metrics: (A)dS-black hole, non-spherical metrics, etc. More broadly, understand the relationship between entropy bounds and bounds from analyticity, unitarity, and causality Positivity of entropy shifts comes from UV state-counting, reminiscent of bounds from dispersion relations and spectral representations Extended versions of the WGC? Much work remains in separating the swampland from the landscape and new tools continue to be discovered