Weak Gravity Conjecture from Black Hole Entropy Clifford Cheung - - PowerPoint PPT Presentation

Weak Gravity Conjecture from Black Hole Entropy

Clifford Cheung

CC, Liu, Remmen (1801.08546) CC, Remmen (1407.7865) CC, Remmen (1402.2287)

q ≥ m/mPl

weak gravity conjecture (WGC)

A U(1) gauge theory consistently coupled to gravity requires a charged state with “Gravity is the weakest force.” paraphrased by the slogan

Arkani-Hamed, Motl, Nicolis, Vafa (2006)

mPl ≥ v

g ≥ m/mPl

The WGC is natural in QFT. For example, in gauge theory with SU(2) → U(1) breaking.

circumstantial evidence #1

The WGC is automatically satisfied in many examples in string theory.

(mW = gv)

circumstantial evidence #2

Exact global symmetries are suspect due to black hole no-hair theorems. Without the WGC, the limit induces an exact global symmetry. The WGC obstructs the global limit by placing a lower bound on . q

q → 0

q, m q, m q, m q, m

circumstantial evidence #3

Consider the decay of a charged black hole. Q, M

black hole

For an extremal black hole, , so number of particles in final state Q/q mQ/q < M = total rest mass in final state =

conservation

• f charge

conservation

• f energy

Q = M/mPl q > m/mPl

• thermodynamic issues???
• tension with holography???

Violation of WGC implies that extremal black holes are stable remnants. These might be pathological: Satisfaction of WGC implies that extremal black holes can decay.

• in the swampland
• merely pathological?

Many properties of EFTs that arise from consistent UV completion are in actuality dictated by IR constraints: anomaly cancellation, charge quantization, positivity.

(diagnose in UV) (diagnose in IR)

OR Question: is a theory that violates the WGC

Can WGC violation yield IR pathologies?

energy light neutral states heavy charged state that satisfies WGC

mPl

Studying the IR spectrum is not enough!

∆L =

8

X

I=1

ciOi

O1 = R2

O2 = RµνRµν

O3 = RµνρσRµνρσ

O4 = RFµνF µν

O5 = RµνF µρF ν

ρ

O6 = RµνρσF µνF ρσ

O7 = FµνF µνFρσF ρσ

O8 = FµνF νρFρσF σµ

We consider Einstein-Maxwell theory plus higher dimension operator corrections.

A physical bound must be invariant under a change of field basis.

gµν → gµν + r1gµνR + r2Rµν + r3gµνF 2 + r4F 2

µν

A physical bound must be invariant under a change of field basis.

gµν → gµν + r1gµνR + r2Rµν + r3gµνF 2 + r4F 2

µν

This shifts the operator coefficients by

c1 → c1 − r2 − r2/2

c2 → c2 + r2

c3 → c3

c4 → c4 + r2/4 − r3 − r4/2

c5 → c5 − r2 + r4

c6 → c6

c7 → c7 + r4/4

c8 → c8 − r4

c3

c6

c0 ≡ c2 + 4c3 + c5 + c6 + 4c7 + 2c8

c9 ≡ c2 + c5 + c8

There are only four field basis invariants :

Any physical bound should be a function of these combinations of coefficients, e.g.

c3

c6

c0 ≡ c2 + 4c3 + c5 + c6 + 4c7 + 2c8

c9 ≡ c2 + c5 + c8

There are only four field basis invariants :

α0c0 + α3c3 + α6c6 + α9c9 > 0

can black holes ensure WGC?

It is possible that the WGC might be satisfied automatically by small black holes. The higher-dimension operator corrections decouple for large black holes. Q M = 1 + 128π2c0 5M 2 · · ·

Planck scale classical GR

c0 > 0

c0 < 0

M

Q M

Compute first order perturbed metric.

perturbed black hole system

Study a black hole of mass and charge perturbed by higher-dimension operators. L = ˜ L + ∆L

M

Q

gµν = ˜ gµν + ∆gµν

Einstein- Maxwell

8

X

i=1

ciOi

Reissner- Nordstrom ∝ ci

S = −2⇡ Z

Σ

L Rµνρσ ✏µν✏ρσ

correction from shift

• f horizon

= −2⇡ ˜ A ˜ L Rµνρσ + ˜ A ∆L Rµνρσ + ∆A ˜ L Rµνρσ + · · · ! ✏µν✏ρσ

Bekenstein- Hawking entropy correction from modified interactions

˜ S

∆S

+

=

Now compute the black hole Wald entropy. The entropy is corrected by two effects.

After some tedious calculation, we obtain

B = (1 − ξ)2c0 + 20ξc3 − 5ξ(1 − ξ)(2c3 + c6)

where the charge-to-mass parameter is

ξ = r 1 − Q2 M 2

so

ξ = 0

ξ = 1

(extremal) (neutral)

∆S ∝ ˜ S × B M 2

invariant under field redefinitions!

positive entropy shift → WGC

For highly charged black holes, the entropy shift is proportional to the WGC bound. Entropy and WGC are intimately connected! ∆S ∝ c0

∆S > 0

c0 > 0

positive entropy shift

implies

black holes ensure WGC

(ξ ⌧ 1)

positive entropy shift → WGC

The WGC/entropy linkage is true in arbitrary dimensions and for multiple U(1) factors.

• z2

− z2 − z1

• z1

inconsistent with WGC ing charge-to-mass ratios for two

• z2

− z2 − z1

• z1

consistent with WGC Figure 1. Vectors representing c

• 10
• 5

5 10

• 4
• 2

2 4 6 8 10 c0/c3 c6/c3

• 1.0
• 0.5

0.0 0.5 1.0

• 4
• 2

2 4 c3/c0 c6/c0

ξ = 0 ξ = 1

• ther implications

Which entropy?

S ∼ M 2 + log M + 1 + · · ·

Local higher-dimension operators comprise the last term, which naively never dominates. However, the actual parametric size of the dimensionless prefactors is crucial. From dimensional analysis, we know that the entropy receives contributions of the form

In a weakly coupled completion of gravity, the higher-dimensional operators are enhanced. E.g. the Virasoro-Shapiro amplitude implies

A(GGGG) ∼

G

G

G

G

φ

∆L ∼ m2

Pl

m2

φ

× R2

∼ s2 m2

Plm2 φ

+ · · ·

Modulo signs, the entropy contributions are

S ∼ M 2 m2

Pl

1 + m2

φ

m2

Pl

! + log ✓ M mPl ◆ + m2

Pl

m2

φ

+ · · ·

Modulo signs, the entropy contributions are

quantum corrections to gravitational constant quantum corrections to higher-dim

• perators
• <latexit sha1_base64="HwkDmf7SZz6qEtnI4Wpt8Sh2Bs=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBI8V7Qe0oWy2k3TpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O6W19Y3NrfJ2ZWd3b/+genjU1kmGLZYIhLVDahGwSW2DcCu6lCGgcCO8H4ZuZ3nlBpnshHM0nRj2kecgZNVZ6EfRoFpz6+4cZJV4BalBgeag+tUfJiyLURomqNY9z02Nn1NlOBM4rfQzjSlYxphz1JY9R+Pj91Ss6sMiRhomxJQ+bq74mcxlpP4sB2xtSM9LI3E/zepkJr/2cyzQzKNliUZgJYhIy+5sMuUJmxMQSyhS3txI2oyY9Op2BC85ZdXSfui7rl17/6y1rgt4ijDCZzCOXhwBQ24gya0gEz/AKb45wXpx352PRWnKmWP4A+fzBz/vjcU=</latexit><latexit sha1_base64="HwkDmf7SZz6qEtnI4Wpt8Sh2Bs=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBI8V7Qe0oWy2k3TpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O6W19Y3NrfJ2ZWd3b/+genjU1kmGLZYIhLVDahGwSW2DcCu6lCGgcCO8H4ZuZ3nlBpnshHM0nRj2kecgZNVZ6EfRoFpz6+4cZJV4BalBgeag+tUfJiyLURomqNY9z02Nn1NlOBM4rfQzjSlYxphz1JY9R+Pj91Ss6sMiRhomxJQ+bq74mcxlpP4sB2xtSM9LI3E/zepkJr/2cyzQzKNliUZgJYhIy+5sMuUJmxMQSyhS3txI2oyY9Op2BC85ZdXSfui7rl17/6y1rgt4ijDCZzCOXhwBQ24gya0gEz/AKb45wXpx352PRWnKmWP4A+fzBz/vjcU=</latexit><latexit sha1_base64="HwkDmf7SZz6qEtnI4Wpt8Sh2Bs=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBI8V7Qe0oWy2k3TpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O6W19Y3NrfJ2ZWd3b/+genjU1kmGLZYIhLVDahGwSW2DcCu6lCGgcCO8H4ZuZ3nlBpnshHM0nRj2kecgZNVZ6EfRoFpz6+4cZJV4BalBgeag+tUfJiyLURomqNY9z02Nn1NlOBM4rfQzjSlYxphz1JY9R+Pj91Ss6sMiRhomxJQ+bq74mcxlpP4sB2xtSM9LI3E/zepkJr/2cyzQzKNliUZgJYhIy+5sMuUJmxMQSyhS3txI2oyY9Op2BC85ZdXSfui7rl17/6y1rgt4ijDCZzCOXhwBQ24gya0gEz/AKb45wXpx352PRWnKmWP4A+fzBz/vjcU=</latexit><latexit sha1_base64="HwkDmf7SZz6qEtnI4Wpt8Sh2Bs=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBI8V7Qe0oWy2k3TpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O6W19Y3NrfJ2ZWd3b/+genjU1kmGLZYIhLVDahGwSW2DcCu6lCGgcCO8H4ZuZ3nlBpnshHM0nRj2kecgZNVZ6EfRoFpz6+4cZJV4BalBgeag+tUfJiyLURomqNY9z02Nn1NlOBM4rfQzjSlYxphz1JY9R+Pj91Ss6sMiRhomxJQ+bq74mcxlpP4sB2xtSM9LI3E/zepkJr/2cyzQzKNliUZgJYhIy+5sMuUJmxMQSyhS3txI2oyY9Op2BC85ZdXSfui7rl17/6y1rgt4ijDCZzCOXhwBQ24gya0gEz/AKb45wXpx352PRWnKmWP4A+fzBz/vjcU=</latexit>

(always)

S ∼ M 2 m2

Pl

1 + m2

φ

m2

Pl

! + log ✓ M mPl ◆ + m2

Pl

m2

φ

+ · · ·

Modulo signs, the entropy contributions are

quantum corrections to gravitational constant quantum corrections to higher-dim

• perators
• <latexit sha1_base64="HwkDmf7SZz6qEtnI4Wpt8Sh2Bs=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBI8V7Qe0oWy2k3TpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O6W19Y3NrfJ2ZWd3b/+genjU1kmGLZYIhLVDahGwSW2DcCu6lCGgcCO8H4ZuZ3nlBpnshHM0nRj2kecgZNVZ6EfRoFpz6+4cZJV4BalBgeag+tUfJiyLURomqNY9z02Nn1NlOBM4rfQzjSlYxphz1JY9R+Pj91Ss6sMiRhomxJQ+bq74mcxlpP4sB2xtSM9LI3E/zepkJr/2cyzQzKNliUZgJYhIy+5sMuUJmxMQSyhS3txI2oyY9Op2BC85ZdXSfui7rl17/6y1rgt4ijDCZzCOXhwBQ24gya0gEz/AKb45wXpx352PRWnKmWP4A+fzBz/vjcU=</latexit><latexit sha1_base64="HwkDmf7SZz6qEtnI4Wpt8Sh2Bs=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBI8V7Qe0oWy2k3TpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O6W19Y3NrfJ2ZWd3b/+genjU1kmGLZYIhLVDahGwSW2DcCu6lCGgcCO8H4ZuZ3nlBpnshHM0nRj2kecgZNVZ6EfRoFpz6+4cZJV4BalBgeag+tUfJiyLURomqNY9z02Nn1NlOBM4rfQzjSlYxphz1JY9R+Pj91Ss6sMiRhomxJQ+bq74mcxlpP4sB2xtSM9LI3E/zepkJr/2cyzQzKNliUZgJYhIy+5sMuUJmxMQSyhS3txI2oyY9Op2BC85ZdXSfui7rl17/6y1rgt4ijDCZzCOXhwBQ24gya0gEz/AKb45wXpx352PRWnKmWP4A+fzBz/vjcU=</latexit><latexit sha1_base64="HwkDmf7SZz6qEtnI4Wpt8Sh2Bs=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBI8V7Qe0oWy2k3TpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O6W19Y3NrfJ2ZWd3b/+genjU1kmGLZYIhLVDahGwSW2DcCu6lCGgcCO8H4ZuZ3nlBpnshHM0nRj2kecgZNVZ6EfRoFpz6+4cZJV4BalBgeag+tUfJiyLURomqNY9z02Nn1NlOBM4rfQzjSlYxphz1JY9R+Pj91Ss6sMiRhomxJQ+bq74mcxlpP4sB2xtSM9LI3E/zepkJr/2cyzQzKNliUZgJYhIy+5sMuUJmxMQSyhS3txI2oyY9Op2BC85ZdXSfui7rl17/6y1rgt4ijDCZzCOXhwBQ24gya0gEz/AKb45wXpx352PRWnKmWP4A+fzBz/vjcU=</latexit><latexit sha1_base64="HwkDmf7SZz6qEtnI4Wpt8Sh2Bs=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBI8V7Qe0oWy2k3TpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O6W19Y3NrfJ2ZWd3b/+genjU1kmGLZYIhLVDahGwSW2DcCu6lCGgcCO8H4ZuZ3nlBpnshHM0nRj2kecgZNVZ6EfRoFpz6+4cZJV4BalBgeag+tUfJiyLURomqNY9z02Nn1NlOBM4rfQzjSlYxphz1JY9R+Pj91Ss6sMiRhomxJQ+bq74mcxlpP4sB2xtSM9LI3E/zepkJr/2cyzQzKNliUZgJYhIy+5sMuUJmxMQSyhS3txI2oyY9Op2BC85ZdXSfui7rl17/6y1rgt4ijDCZzCOXhwBQ24gya0gEz/AKb45wXpx352PRWnKmWP4A+fzBz/vjcU=</latexit>

(always)

• <latexit sha1_base64="HwkDmf7SZz6qEtnI4Wpt8Sh2Bs=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBI8V7Qe0oWy2k3TpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O6W19Y3NrfJ2ZWd3b/+genjU1kmGLZYIhLVDahGwSW2DcCu6lCGgcCO8H4ZuZ3nlBpnshHM0nRj2kecgZNVZ6EfRoFpz6+4cZJV4BalBgeag+tUfJiyLURomqNY9z02Nn1NlOBM4rfQzjSlYxphz1JY9R+Pj91Ss6sMiRhomxJQ+bq74mcxlpP4sB2xtSM9LI3E/zepkJr/2cyzQzKNliUZgJYhIy+5sMuUJmxMQSyhS3txI2oyY9Op2BC85ZdXSfui7rl17/6y1rgt4ijDCZzCOXhwBQ24gya0gEz/AKb45wXpx352PRWnKmWP4A+fzBz/vjcU=</latexit><latexit sha1_base64="HwkDmf7SZz6qEtnI4Wpt8Sh2Bs=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBI8V7Qe0oWy2k3TpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O6W19Y3NrfJ2ZWd3b/+genjU1kmGLZYIhLVDahGwSW2DcCu6lCGgcCO8H4ZuZ3nlBpnshHM0nRj2kecgZNVZ6EfRoFpz6+4cZJV4BalBgeag+tUfJiyLURomqNY9z02Nn1NlOBM4rfQzjSlYxphz1JY9R+Pj91Ss6sMiRhomxJQ+bq74mcxlpP4sB2xtSM9LI3E/zepkJr/2cyzQzKNliUZgJYhIy+5sMuUJmxMQSyhS3txI2oyY9Op2BC85ZdXSfui7rl17/6y1rgt4ijDCZzCOXhwBQ24gya0gEz/AKb45wXpx352PRWnKmWP4A+fzBz/vjcU=</latexit><latexit sha1_base64="HwkDmf7SZz6qEtnI4Wpt8Sh2Bs=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBI8V7Qe0oWy2k3TpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O6W19Y3NrfJ2ZWd3b/+genjU1kmGLZYIhLVDahGwSW2DcCu6lCGgcCO8H4ZuZ3nlBpnshHM0nRj2kecgZNVZ6EfRoFpz6+4cZJV4BalBgeag+tUfJiyLURomqNY9z02Nn1NlOBM4rfQzjSlYxphz1JY9R+Pj91Ss6sMiRhomxJQ+bq74mcxlpP4sB2xtSM9LI3E/zepkJr/2cyzQzKNliUZgJYhIy+5sMuUJmxMQSyhS3txI2oyY9Op2BC85ZdXSfui7rl17/6y1rgt4ijDCZzCOXhwBQ24gya0gEz/AKb45wXpx352PRWnKmWP4A+fzBz/vjcU=</latexit><latexit sha1_base64="HwkDmf7SZz6qEtnI4Wpt8Sh2Bs=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBI8V7Qe0oWy2k3TpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O6W19Y3NrfJ2ZWd3b/+genjU1kmGLZYIhLVDahGwSW2DcCu6lCGgcCO8H4ZuZ3nlBpnshHM0nRj2kecgZNVZ6EfRoFpz6+4cZJV4BalBgeag+tUfJiyLURomqNY9z02Nn1NlOBM4rfQzjSlYxphz1JY9R+Pj91Ss6sMiRhomxJQ+bq74mcxlpP4sB2xtSM9LI3E/zepkJr/2cyzQzKNliUZgJYhIy+5sMuUJmxMQSyhS3txI2oyY9Op2BC85ZdXSfui7rl17/6y1rgt4ijDCZzCOXhwBQ24gya0gEz/AKb45wXpx352PRWnKmWP4A+fzBz/vjcU=</latexit>

classical corrections to higher-dim

• perators

(possibly)

S ∼ M 2 m2

Pl

1 + m2

φ

m2

Pl

! + log ✓ M mPl ◆ + m2

Pl

m2

φ

+ · · ·

Modulo signs, the entropy contributions are

quantum corrections to gravitational constant quantum corrections to higher-dim

• perators
• <latexit sha1_base64="HwkDmf7SZz6qEtnI4Wpt8Sh2Bs=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBI8V7Qe0oWy2k3TpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O6W19Y3NrfJ2ZWd3b/+genjU1kmGLZYIhLVDahGwSW2DcCu6lCGgcCO8H4ZuZ3nlBpnshHM0nRj2kecgZNVZ6EfRoFpz6+4cZJV4BalBgeag+tUfJiyLURomqNY9z02Nn1NlOBM4rfQzjSlYxphz1JY9R+Pj91Ss6sMiRhomxJQ+bq74mcxlpP4sB2xtSM9LI3E/zepkJr/2cyzQzKNliUZgJYhIy+5sMuUJmxMQSyhS3txI2oyY9Op2BC85ZdXSfui7rl17/6y1rgt4ijDCZzCOXhwBQ24gya0gEz/AKb45wXpx352PRWnKmWP4A+fzBz/vjcU=</latexit><latexit sha1_base64="HwkDmf7SZz6qEtnI4Wpt8Sh2Bs=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBI8V7Qe0oWy2k3TpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O6W19Y3NrfJ2ZWd3b/+genjU1kmGLZYIhLVDahGwSW2DcCu6lCGgcCO8H4ZuZ3nlBpnshHM0nRj2kecgZNVZ6EfRoFpz6+4cZJV4BalBgeag+tUfJiyLURomqNY9z02Nn1NlOBM4rfQzjSlYxphz1JY9R+Pj91Ss6sMiRhomxJQ+bq74mcxlpP4sB2xtSM9LI3E/zepkJr/2cyzQzKNliUZgJYhIy+5sMuUJmxMQSyhS3txI2oyY9Op2BC85ZdXSfui7rl17/6y1rgt4ijDCZzCOXhwBQ24gya0gEz/AKb45wXpx352PRWnKmWP4A+fzBz/vjcU=</latexit><latexit sha1_base64="HwkDmf7SZz6qEtnI4Wpt8Sh2Bs=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBI8V7Qe0oWy2k3TpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O6W19Y3NrfJ2ZWd3b/+genjU1kmGLZYIhLVDahGwSW2DcCu6lCGgcCO8H4ZuZ3nlBpnshHM0nRj2kecgZNVZ6EfRoFpz6+4cZJV4BalBgeag+tUfJiyLURomqNY9z02Nn1NlOBM4rfQzjSlYxphz1JY9R+Pj91Ss6sMiRhomxJQ+bq74mcxlpP4sB2xtSM9LI3E/zepkJr/2cyzQzKNliUZgJYhIy+5sMuUJmxMQSyhS3txI2oyY9Op2BC85ZdXSfui7rl17/6y1rgt4ijDCZzCOXhwBQ24gya0gEz/AKb45wXpx352PRWnKmWP4A+fzBz/vjcU=</latexit><latexit sha1_base64="HwkDmf7SZz6qEtnI4Wpt8Sh2Bs=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBI8V7Qe0oWy2k3TpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O6W19Y3NrfJ2ZWd3b/+genjU1kmGLZYIhLVDahGwSW2DcCu6lCGgcCO8H4ZuZ3nlBpnshHM0nRj2kecgZNVZ6EfRoFpz6+4cZJV4BalBgeag+tUfJiyLURomqNY9z02Nn1NlOBM4rfQzjSlYxphz1JY9R+Pj91Ss6sMiRhomxJQ+bq74mcxlpP4sB2xtSM9LI3E/zepkJr/2cyzQzKNliUZgJYhIy+5sMuUJmxMQSyhS3txI2oyY9Op2BC85ZdXSfui7rl17/6y1rgt4ijDCZzCOXhwBQ24gya0gEz/AKb45wXpx352PRWnKmWP4A+fzBz/vjcU=</latexit>

(always)

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classical corrections to higher-dim

• perators

(possibly)

S ∼ M 2 m2

Pl

1 + m2

φ

m2

Pl

! + log ✓ M mPl ◆ + m2

Pl

m2

φ

+ · · ·

mPl m2

φ

M m2

Pl

⇠ R

Possible for small black holes still within the EFT:

Why ?

∆S > 0

#1) Two QFTs that differ by one state.

Einstein-Maxwell

mPl

Einstein-Maxwell +

mPl

φ

mφ

#1) Two QFTs that differ by one state.

Einstein-Maxwell

mPl

Einstein-Maxwell +

mPl

φ

˜ L

˜ L + ∆L

ci ∝ 1 m2

φ

mφ

low-energy EFTs are defined here

#1) Two QFTs that differ by one state.

Einstein-Maxwell

mPl

Einstein-Maxwell +

mPl

φ

mφ

Black holes are common macrostates. #1) Two QFTs that differ by one state.

Einstein-Maxwell

mPl

Einstein-Maxwell +

mPl

φ

mφ

Q, M

Q, M

Black holes are common macrostates.

˜ S

˜ S + ∆S

#1) Two QFTs that differ by one state.

Einstein-Maxwell

mPl

Einstein-Maxwell +

mPl

φ

mφ

Q, M

Q, M

More microstate degeneracy, more entropy.

˜ S

˜ S + ∆S

<

#1) Two QFTs that differ by one state.

#2) A one-parameter family of QFTs.

mPl

mPl

mφ

mφ

mφ

heavy φ

light φ

#2) A one-parameter family of QFTs.

mPl

mPl

mφ

mφ

mφ

Lighter states, more entropy (at fixed Q,M).

heavy φ

light φ

#2) A one-parameter family of QFTs.

Einstein-Maxwell +

φ

#3) One QFT analyzed at two scales.

mPl

mφ

Einstein-Maxwell +

φ

Compute the same entropy in two ways.

mPl

mφ

Q, M

#3) One QFT analyzed at two scales.

Einstein-Maxwell +

φ

Compute the same entropy in two ways.

˜ S + ∆S

compute entropy in low energy EFT

mPl

mφ

#3) One QFT analyzed at two scales.

compute entropy in full theory

Einstein-Maxwell +

φ

Compute the same entropy in two ways.

˜ S + ∆S

compute entropy in low energy EFT

mPl

mφ

#3) One QFT analyzed at two scales.

˜ S + Sφ

compute entropy in full theory

Einstein-Maxwell +

φ

Compute the same entropy in two ways.

˜ S + ∆S

compute entropy in low energy EFT

˜ S + Sφ

=

mPl

mφ

#3) One QFT analyzed at two scales.

proof ofXXXX

∆S > 0

proof ofXXXX

∆S > 0

classical entropy shift quantum field theoretic

∆S > 0

}

QFT applies

c0 > 0

}

states

mPl

general QFT

Assume QFT applies in some energy range. Einstein-Maxwell is natural “baseline” theory.

∆S > 0

Einstein-Maxwell is natural “baseline” theory.

+ O(1/mPl)

+ O(1/mPl)

}

QFT applies

c0 > 0

}

states

mPl

general QFT

Assume QFT applies in some energy range.

+ O(1/Λ)

∆S > 0

c0 > 0

Λ

}

QFT applies

+ O(1/Λ)

}

states

mPl

general QFT

Note we can make a weaker assumption. Assume QFT applies in some energy range.

Free energy from the Euclidean path integral: I[ˆ g] = ˜ I[ˆ g] + ∆I[ˆ g]

Z(β) = Z

β

[dˆ g]e−I[ˆ

g]

Free energy from the Euclidean path integral: Black hole mass, temperature, entropy are: I[ˆ g] = ˜ I[ˆ g] + ∆I[ˆ g]

Z(β) = Z

β

[dˆ g]e−I[ˆ

g]

S = βM + log Z(β)

M = −∂β log Z(β)

β = ∂MS

Z(β) = Z

β

[dˆ g][dˆ φ]e−IUV[ˆ

g, ˆ φ] =

Z

β

[dˆ g]e−I[ˆ

g]

Classically, higher-dimensional operators are generated at tree-level by heavy states.

Dropping heavy states reduces the theory to the “baseline”, e.g. Einstein-Maxwell. IUV[ˆ g, 0] = ˜ I[ˆ g]

Z(β) = Z

β

[dˆ g][dˆ φ]e−IUV[ˆ

g, ˆ φ] =

Z

β

[dˆ g]e−I[ˆ

g]

Classically, higher-dimensional operators are generated at tree-level by heavy states.

perturbed Reissner- Nordstrom insert classical solutions

log Z(β) ∼ −IUV[g, φ]

Compute the free energy via the saddle point approximation.

>

perturbed Reissner- Nordstrom insert classical solutions

log Z(β) ∼ −IUV[g, φ]

−IUV[˜ g, 0]

definition of extremization

Compute the free energy via the saddle point approximation.

Compute the free energy via the saddle point approximation. >

perturbed Reissner- Nordstrom insert classical solutions

log Z(β) ∼ −IUV[g, φ]

log ˜ Z(β) ∼ −˜ I[˜ g]

−IUV[˜ g, 0]

=

We derive an inequality at fixed temperature.

pure Reissner- Nordstrom definition of extremization truncating heavy states yields Einstein-Maxwell

The free energy inequality can be written as: log Z(β) > log ˜ Z(β) = (1 + ∆β∂ ˜

β) log ˜

Z(˜ β)

The free energy inequality can be written as: log Z(β) > log ˜ Z(β) = (1 + ∆β∂ ˜

β) log ˜

Z(˜ β)

= log ˜ Z(˜ β) − M∆β

M∂M∆S

The free energy inequality can be written as: log Z(β) > log ˜ Z(β) = (1 + ∆β∂ ˜

β) log ˜

Z(˜ β)

= log ˜ Z(˜ β) − M∆β

(1 − M∂M)S

(1 − M∂M) ˜ S

M∂M∆S

The free energy inequality can be written as: log Z(β) > log ˜ Z(β) = (1 + ∆β∂ ˜

β) log ˜

Z(˜ β)

= log ˜ Z(˜ β) − M∆β

This implies that classical black hole entropy from heavy states at fixed and satisfies:

(1 − M∂M)S

(1 − M∂M) ˜ S

∆S > 0

M

Q

Corollary: each correction from a heavy field is positive, yielding a differential bound: Integrating out states sequentially translates to extremizing over each field sequentially: Z µ

Λ

dc0 = c0(µ) > 0

dc0 > 0

conclusions

• WGC violation might be visible in low

energy pathologies of the effective action for photons and gravitons.

• Small black holes automatically fulfill the

WGC if the entropy shift from higher dimension operators is positive.

• The classical entropy shift from tree-

level heavy particle exchange is positive via a simple thermodynamic proof.

thank you!