Smarr Formula in Black Hole Thermodynamics

• Smarr Formula is a foundational relation in black hole thermodynamics that connects a black hole's mass with horizon area (entropy), surface gravity (temperature), angular momentum, and charge.
• It is derived using geometric techniques such as Komar integrals, scaling arguments, and Noether charge methods, reinforcing the connection between the formula and the first law of black hole mechanics.
• Modern generalizations extend the formula to nonlinear electrodynamics, higher dimensions, and quantum corrections, incorporating additional thermodynamic variables like the cosmological constant and higher-curvature couplings.

The Smarr formula is a foundational relation in black hole physics expressing the total mass of a black hole in terms of its horizon area (entropy), surface gravity (temperature), angular momentum, charge, and—in extensions—additional quantities such as cosmological constant, coupling constants, or higher curvature terms. Its principal mathematical underpinning is the scaling property (homogeneity) of stationary black hole solutions, and it is tightly linked to the geometric first law of black hole mechanics. Recent developments have shown that its structure persists, though often with important modifications, in a wide range of generalized and quantum-corrected gravity theories, non-linear field couplings, higher dimensions, and entropy models beyond extensivity.

1. Classical Formulation and Physical Content

In four-dimensional, asymptotically flat, stationary, and axisymmetric Einstein–Maxwell gravity, the Smarr formula establishes a relationship among the ADM mass $M$, surface gravity $\kappa$ (encoded as temperature $T=\kappa/2\pi$), event horizon area $\mathcal A$ (entropy $S=\mathcal{A}/4$), angular momentum $J$, horizon angular velocity $\Omega$, electric potential $\Phi$, and charge $Q$ (Gulin et al., 2017): $M = 2 T S + 2 \Omega J + \Phi Q.$ This relation reflects the homogeneity of $\kappa$0 as a function of its extensive variables under a scaling transformation and can be derived by combining Komar integrals for conserved quantities with Einstein’s equations.

The same structure holds (modulo dimension-dependent scaling coefficients) for generalizations to higher spacetime dimensions $\kappa$1, for example, for the charged rotating Myers–Perry black hole (Banerjee et al., 2010, Modak, 2012): $\kappa$2 For asymptotically AdS solutions, the Smarr formula admits a further extension involving the cosmological constant $\kappa$3 interpreted as thermodynamic pressure $\kappa$4 and its conjugate volume $\kappa$5. For the BTZ black hole in three dimensions (Erices et al., 2017, Liang et al., 2017),

$\kappa$6

where $\kappa$7 is interpreted as enthalpy due to the presence of $\kappa$8.

2. Methodologies of Derivation: Geometric, Scaling, and Noether Approaches

The Smarr formula is intimately tied to geometric properties of the spacetime, in particular through Komar integrals associated with Killing fields. The key steps are:

• Evaluation of Komar charges (mass, angular momentum) at infinity and, via the divergence theorem, at the event horizon, equating the two using Einstein’s equations (Modak, 2012, Banerjee et al., 2010, Clément et al., 2017).
• Application of the first law of black hole thermodynamics, $\kappa$9, together with Euler’s theorem for homogeneous functions, leveraging the scaling properties of $T=\kappa/2\pi$0 in its extensive variables (Erices et al., 2017, Liu et al., 2015).
• Noether charge and Wald’s covariant phase space formalism, giving a generalized identity in any diffeomorphism-invariant theory as the balance between surface integrals of the Noether potential and volume integrals of the Lagrangian (Liberati et al., 2015, Kastor et al., 2010).

In Lovelock and higher-curvature gravities, the Smarr formula includes additional conjugate pairs $T=\kappa/2\pi$1 for each higher-curvature coupling $T=\kappa/2\pi$2 (Kastor et al., 2010, Liberati et al., 2015): $T=\kappa/2\pi$3

3. Generalizations: Nonlinear Theories, Matter Fields, and Entropy New Models

Nonlinear Electrodynamics and Additional Coupling Terms

In Einstein gravity coupled to nonlinear electrodynamics (NLE), the Smarr formula acquires a correction due to the nonvanishing trace of the energy-momentum tensor (Gulin et al., 2017, Balart et al., 2017): $T=\kappa/2\pi$4 where

$T=\kappa/2\pi$5

and $T=\kappa/2\pi$6 captures the deviation from Maxwell theory. When the Lagrangian depends homogeneously on a coupling $T=\kappa/2\pi$7, $T=\kappa/2\pi$8 can be written as a product of conjugate variables, $T=\kappa/2\pi$9, with $\mathcal A$0 defined via a horizon integral. In specific models, such as power–Maxwell or Born–Infeld, this extra term can often be related to a suitable thermodynamic conjugate, and in the quantum-corrected Euler–Heisenberg case, it encodes the leading QED vacuum polarization effects.

Inclusion of Magnetic Charge and Dyonic Solutions

Rotating black holes with both electric and magnetic (dyonic) charges obey a Smarr formula where the electromagnetic contributions combine into a complex quantity (Clément et al., 2017, Manko et al., 2015): $\mathcal A$1 with $\mathcal A$2 the electric and magnetic charges, and $\mathcal A$3 their respective horizon potentials. In these cases, the total “asymptotic” mass may differ from the horizon mass by a contribution attributed to the energy of Dirac strings associated with magnetic monopoles.

Lorentz-Violating and Æther/Hořava Theories

In Einstein–Æther and IR Hořava gravity, the Smarr formula must be modified to accommodate the presence of the æther vector field $\mathcal A$4 and its couplings. In four dimensions, for static, spherically symmetric, charged black holes, the formula at a universal or Killing horizon is (Ding et al., 2015, Ho et al., 2017): $\mathcal A$5 where the horizon temperature $\mathcal A$6 is not simply $\mathcal A$7, but encodes contributions from the æther field. The additional term arises from the flux of the æther stress and can be expressed as an explicit surface term involving the projection of acceleration or twist at the horizon. The distinction between Killing and universal horizons is crucial, as the latter encodes the causal boundary for arbitrarily fast signals in these non-Lorentz-invariant theories (Pacilio et al., 2017, Ho et al., 2017).

4. Modern Generalizations: Couplings as Thermodynamic Variables

Dimensionful couplings, including the cosmological constant, higher-derivative couplings, and scalar field hair parameters, must be included as bona fide thermodynamic variables to achieve a universal and consistent Smarr relation in modified gravity (Hajian et al., 27 Nov 2025, Myung et al., 5 May 2025). The differential and integrated first law then take the forms: $\mathcal A$8

$\mathcal A$9

where each dimensionful coupling $S=\mathcal{A}/4$0 appears as a charge, with associated conjugate potential $S=\mathcal{A}/4$1 constructed—by a gauge extension—as a Noether charge at the horizon. This approach yields a universal formulation that is necessary for the thermodynamic consistency of black holes in gravitational theories with multiple couplings, as in Lovelock, Horndeski, and massive gravity (Kastor et al., 2010, Hajian et al., 27 Nov 2025).

In scalar-tensor or hairy black holes, the Smarr formula must account for the scalar charge or coupling in an appropriately extended phase space. In shift- and $S=\mathcal{A}/4$2-symmetric beyond-Horndeski gravity, one obtains, after promoting the coupling to a thermodynamic variable (Myung et al., 5 May 2025): $S=\mathcal{A}/4$3 where $S=\mathcal{A}/4$4 plays the role of a “secondary hair,” and $S=\mathcal{A}/4$5 is the conjugate potential.

5. Quantum, Statistical, and Non-Additive Generalizations

Generalized Entropy Models

When the Bekenstein–Hawking area law $S=\mathcal{A}/4$6 is replaced by a generalized monotonic function $S=\mathcal{A}/4$7, the Smarr relation must reflect this new scaling (Zafar et al., 1 May 2026): $S=\mathcal{A}/4$8 where $S=\mathcal{A}/4$9 recovers the standard result when $J$0, but yields fundamentally different scaling for, e.g., Tsallis, R\'enyi, Barrow, or Sharma–Mittal entropy. The coefficient in front of the $J$1 term is determined by the effective degree of homogeneity of the entropy function, and the relation is consistent with the corresponding generalized first law.

Deformations from Quantum Gravity or Noncommutative Corrections

In black holes with noncommutative geometry-inspired metrics (Banerjee et al., 2010, Larranaga et al., 2012), the Komar energy or mass at the horizon is found to be

$J$2

where $J$3 is a nonperturbative, exponentially suppressed correction in the noncommutative parameter (e.g., $J$4). This term leads to a breakdown of the area law and a nonvanishing Komar energy at extremality ($J$5). Similar correction structures are observed in regular or deSitter-core black holes.

6. Smarr Formula and Differential Geometry

The Smarr relation can be formulated using the Geroch-Held-Penrose (GHP) formalism, expressing thermodynamic variables as geometric invariants of the horizon (Guilabert et al., 2024). For horizons of arbitrary topology,

$J$6

where the internal energy $J$7 is defined as an integral of the Penrose $J$8-curvature over the horizon cross-section, $J$9 is identified with an expansion scalar, $\Omega$0 with area, $\Omega$1 with the effective pressure from matter and cosmological constant, and $\Omega$2 as a geometric volume. Generalizations to extended gravity theories such as $\Omega$3 involve a rescaling of $\Omega$4, $\Omega$5, and $\Omega$6 by $\Omega$7. This geometric approach links horizon thermodynamics to the bundle of null directions and curvature, suggesting deep connections with the microphysical interpretation of black hole entropy and the constraints imposed by horizon topology.

7. Summary Table: Smarr Formula in Key Contexts

Context	Representative Smarr Formula
4D Kerr–Newman	$\Omega$8
D-dim Myers–Perry	$\Omega$9
AdS/BTZ with $\Phi$0	$\Phi$1 or $\Phi$2
NLED	$\Phi$3 (with $\Phi$4 from trace of $\Phi$5)
Generalized Entropy	$\Phi$6
Lovelock/Higher-Curvature	$\Phi$7
All Couplings Included	$\Phi$8
Lorentz Violation	$\Phi$9

In every setting, the Smarr formula encodes the scaling structure of black hole thermodynamics, provides a stringent test for the consistency of thermodynamic extensions, and acts as a bridge connecting classical, quantum, geometric, and statistical gravitational frameworks.