[Paper Review] A Weak Gravity Theorem

Mehrdad Mirbabayi|arXiv (Cornell University)|May 7, 2019

Black Holes and Theoretical Physics54 references4 citations

TL;DR

This paper derives a weak gravity theorem showing that in quantum gravity with many light degrees of freedom, the charge-to-mass ratio of black holes approaches the Einstein-Maxwell prediction from above as mass increases, with a positive correction coefficient α > 0. This positivity, derived from field-theoretic sum rules in a non-dynamical gravity background, implies gravitational attraction weakens relative to electromagnetism for maximally charged black holes, enabling decay of large extremal black holes into smaller ones.

ABSTRACT

In a gravitational theory with a massless photon the maximum charge-to-mass ratio of black holes approaches the prediction of the Einstein-Maxwell theory as black hole mass increases: $Q_{ m ext}/M =1+ α/M^2$ for some constant $α$. We will show that $α>0$ if below the quantum gravity scale $Λ$ there are many degrees of freedom with a hierarchically small mass gap $\log(Λ/m_{ m gap})\gg 1$. In this regime one can treat gravity as a non-dynamical background field and derive field-theoretic sum-rules for the coefficients of the leading corrections to the Einstein-Maxwell theory. The positivity of $α$ follows from the sum-rules. As a consequence, gravitational attraction gets weaker than the electric force among maximally charged black holes as they become lighter, and large extremal black holes can decay into smaller black holes.

Motivation & Objective

• To understand the behavior of charge-to-mass ratios in black holes within quantum gravity with light degrees of freedom.
• To determine whether gravitational attraction remains stronger than electromagnetic repulsion for extremal black holes.
• To derive constraints on higher-order corrections to Einstein-Maxwell theory using field-theoretic sum rules.
• To establish the positivity of the correction coefficient α in the charge-to-mass ratio as a consequence of quantum gravity structure.

Proposed method

• Treat gravity as a non-dynamical background field below the quantum gravity scale Λ.
• Use field-theoretic sum rules to compute leading-order corrections to Einstein-Maxwell theory.
• Assume a hierarchically small mass gap with log(Λ/m_gap) ≫ 1 to justify effective field theory validity.
• Derive the asymptotic form Q/M = 1 + α/M² for black hole charge-to-mass ratio.
• Apply positivity constraints from sum rules to prove α > 0 under the given assumptions.
• Analyze the physical implications of α > 0 for black hole stability and decay.

Experimental results

Research questions

• RQ1 Does the charge-to-mass ratio of black holes in quantum gravity with many light states exceed the Einstein-Maxwell prediction as mass increases?
• RQ2 Can the coefficient α in the correction term Q/M = 1 + α/M² be proven positive using field-theoretic sum rules?
• RQ3 What are the implications of α > 0 for the relative strength of gravity and electromagnetism in maximally charged black holes?
• RQ4 Under what conditions can large extremal black holes decay into smaller ones due to this correction?
• RQ5 How does the presence of a small mass gap affect the validity of the effective field theory derivation?

Key findings

• The coefficient α in the charge-to-mass ratio correction Q/M = 1 + α/M² is proven to be positive under the assumption of many light degrees of freedom with a small mass gap.
• The positivity of α follows from field-theoretic sum rules derived in a non-dynamical gravity background.
• Gravitational attraction becomes weaker than electromagnetic repulsion for maximally charged black holes as they become lighter.
• Large extremal black holes can decay into smaller black holes due to the dominance of electromagnetic repulsion over gravity.
• The result holds in the regime where log(Λ/m_gap) ≫ 1, ensuring the effective field theory description remains valid.
• The correction structure is consistent with the weak gravity conjecture, supporting the idea that gravity is the weakest force in quantum gravity.