# On the Entropy Function and the Attractor Mechanism for Spherically Symmetric Extremal Black Holes

## TL;DR

This paper clarifies the relationship between Wald's entropy formula and Sen's entropy function for extremal black holes, deriving the entropy without rescaling near horizon geometry and exploring the attractor mechanism with scalar fields.

## Contribution

It demonstrates that Sen's entropy function can be derived from Wald's entropy without rescaling the near horizon geometry, and introduces an off-shell entropy function for analyzing the attractor mechanism.

## Key findings

- Derived Sen's entropy from Wald's formula without rescaling
- Defined an off-shell entropy function for attractor analysis
- Clarified the Legendre transformation relation in entropy calculation

## Abstract

In this paper we elaborate on the relation between the entropy formula of Wald and the "entropy function" method proposed by A. Sen. For spherically symmetric extremal black holes, it is shown that the expression of extremal black hole entropy given by A. Sen can be derived from the general entropy definition of Wald, without help of the treatment of rescaling the AdS_2 part of near horizon geometry of extremal black holes. In our procedure, we only require that the surface gravity approaches to zero, and it is easy to understand the Legendre transformation of f, the integration of Lagrangian density on the horizon, with respect to the electric charges. Since the Noether charge form can be defined in an "off-shell" form, we define a corresponding entropy function, with which one can discuss the attractor mechanism for extremal black holes with scalar fields.