Smarr formula

Smarr formula
Name	Smarr formula
Field	Relativity
Introduced	1973
Notable	Larry Smarr

Smarr formula The Smarr formula is a relation in classical General relativity connecting conserved quantities of stationary black hole solutions such as mass, angular momentum, electric charge, surface gravity, and horizon area. It was originally derived in the context of the Kerr and Reissner–Nordström families and has influenced developments in black hole thermodynamics, ADM mass, and the formulation of the First law of black hole mechanics. The relation appears in analyses of solutions in Einstein–Hilbert action frameworks and in extensions involving cosmological constant and nontrivial matter fields.

Introduction

The Smarr formula was introduced to relate the total Arnowitt–Deser–Misner mass of a stationary asymptotically flat spacetime to geometric and gauge quantities defined on the event horizon, notably the horizon area, angular velocity, electric potential, and charges. It played a role alongside results by Bekenstein, Hawking, and Wald in establishing analogies between black hole mechanics and thermodynamics, and it has been applied in studies involving the Kerr–Newman solution, the Schwarzschild metric, and charged rotating configurations.

Derivation

Derivations of the Smarr formula employ methods from differential geometry and variational principles, often invoking Komar integrals, scaling arguments, and Hamiltonian methods. Early derivations used Komar expressions associated with Killing vectors in stationary spacetimes and relied on identities related to the Einstein tensor and asymptotic boundary terms evaluated at spatial infinity and on the event horizon. Alternative derivations use the Noether charge formalism developed by Iyer and Wald or the Hamiltonian approach tied to the ADM formalism and surface terms introduced by Regge and Teitelboim.

Applications to Black Hole Thermodynamics

The Smarr formula underpins the identification of horizon area with entropy in the framework initiated by Bekenstein and quantified by Hawking's semiclassical calculations, feeding into the thermodynamic interpretation encoded in the First law proved by methods of Bardeen, Carter, and Hawking. It connects intensive quantities like surface gravity and angular velocity with extensive quantities like mass and angular momentum in contexts including the Kerr–Newman family and has been used in analyses of phase behavior in spacetimes with a cosmological constant interpreted as pressure in extended thermodynamics studies by Kubizňák and Mann.

Generalizations and Extensions

The original Smarr relation has been extended to include contributions from a cosmological constant, scalar fields, higher-curvature corrections, and gauge fields arising in supergravity and string theory. Work generalizing the relation appears in contexts of Lovelock gravity, Gauss–Bonnet corrections, and asymptotically anti-de Sitter solutions relevant for the AdS/CFT correspondence studied by Maldacena and others. Extensions also treat nonstationary horizons via isolated and dynamical horizon frameworks developed by Ashtekar, Booth, and Fairhurst, and incorporate magnetic charges and higher-form fields encountered in Type II string theory compactifications.

Examples and Specific Solutions

For the Schwarzschild black hole the Smarr relation reduces to a proportionality between mass and horizon area reflecting the simple static, spherically symmetric case studied in early Oppenheimer–Snyder collapse models. In the rotating charged Kerr–Newman solution the formula explicitly involves mass, angular momentum, charge, surface gravity, angular velocity, and electrostatic potential as in calculations by Carter and later treatments in textbooks by Misner, Thorne, and Wheeler. Charged solutions such as Reissner–Nordström and rotating solutions in anti-de Sitter space show modified Smarr relations when a cosmological constant or gauge-field hair is present, with explicit examples appearing in work by Henneaux, Teitelboim, and researchers studying holographic duals.

Physical Interpretation and Implications

Physically the Smarr formula encodes balance relations among conserved charges and horizon properties, supporting the interpretation of horizon area as an entropy-like quantity and surface gravity as a temperature-like quantity in semiclassical contexts pioneered by Hawking and Bekenstein. It provides constraints used in uniqueness theorems for stationary black holes proven by Israel, Robinson, and Mazur, informs stability analyses by connecting perturbative parameters in studies by Detweiler and Press, and guides investigations into microscopics in approaches such as string-theoretic counting by Strominger and Vafa. The relation continues to serve as a diagnostic tool in numerical relativity simulations by groups at institutions like Caltech and Max Planck Institute exploring mergers and horizon dynamics.

Category:General relativity