A C-Function For Non-Supersymmetric Attractors

TL;DR

The paper introduces a c-function for static, spherically symmetric, asymptotically flat spacetimes in four-dimensional gravity coupled to gauge fields and moduli, valid for both extremal and non-extremal black holes. The c-function, defined as c(r) = A(r)/4 with A(r) the area of the 2-sphere, monotonically decreases toward the horizon under the null energy condition, attaining the black hole entropy at the horizon; the construction also extends to higher dimensions with p-form gauge fields, where a related c-function decreases toward the near-horizon AdS_{p+1} x S^q region and connects to the boundary conformal anomaly for even p. The work clarifies the relationship between c, V_eff, and central charges, showing that while they coincide at the horizon in supersymmetric cases, they generally differ away from it. The analysis uses the Raychaudhuri equation and energy conditions to establish monotonicity and discusses broader applicability to AdS boundaries and more general matter content.

Abstract

We present a c-function for spherically symmetric, static and asymptotically flat solutions in theories of four-dimensional gravity coupled to gauge fields and moduli. The c-function is valid for both extremal and non-extremal black holes. It monotonically decreases from infinity and in the static region acquires its minimum value at the horizon, where it equals the entropy of the black hole. Higher dimensional cases, involving $p$-form gauge fields, and other generalisations are also discussed.

Figures (1)

• Figure 1: The effective potential $V_{eff}$ as a function of $\phi$