Smarr Formulas for Black Holes

• Smarr formulas express black hole mass as a bilinear function of horizon thermodynamic properties, highlighting scaling symmetries in general relativity and beyond.
• They are derived using methods like Euler’s theorem and Komar charge integrals, linking ADM mass to entropy, angular momentum, and electromagnetic charges.
• Extended formulations incorporate variable coupling constants, ensuring consistency in theories like Lovelock gravity, nonlinear electrodynamics, and scalar hair models.

A Smarr formula is a mass relation for black holes that expresses the total mass (or ADM energy) as a bilinear in the horizon thermodynamic quantities and conserved charges, with coefficients fixed by dimensional analysis and the underlying gravitational action. Originally derived for four-dimensional stationary black holes in general relativity, Smarr formulas have since been generalized to arbitrary dimension, diverse gravity theories, and black objects with more complicated matter content or topology. Their derivation is intimately tied to the scaling symmetries of the underlying field equations or action, Komar–Noether charges, and, in advanced settings, to extended thermodynamic phase spaces including additional couplings or scalar hair.

1. Classical Smarr Relations: General Structure and Dimensional Analysis

In Einstein gravity, the Smarr formula for an asymptotically flat, stationary black hole in $D$ spacetime dimensions takes the form

$(N-2)M = (N-1)\Omega_H J + (N-2)\Phi Q + (N-1)TS$

where $N=D-1$ is the spatial dimension, $M$ is the ADM mass, $\Omega_H$ is the horizon angular velocity, $J$ is the angular momentum, $\Phi$ is the electrostatic potential, $Q$ is the electric charge, $T$ is the Hawking temperature, and $S$ is the Bekenstein–Hawking entropy ($S=A_H/4G$). The coefficients arise from requiring homogeneity under the scaling of all dimensionful quantities: under $r \rightarrow \lambda r$, $M$ scales as $\lambda^{N-2}$, $J$ and $S$ as $\lambda^{N-1}$, $Q$ as $\lambda^{N-2}$, etc. The general derivation relies on either Euler’s theorem for homogeneous functions or on Komar surface integrals and their horizon/infinity matching (Banerjee et al., 2010, Modak, 2012).

Specializing to $D=4$, the standard Smarr relation becomes

$M = 2TS + 2\Omega_H J + \Phi Q$

with the first law encoded in the differential relation

$dM = T dS + \Omega_H dJ + \Phi dQ$

2. Geometric Derivation: Komar Charges and Surface Integrals

A geometric route to the Smarr formula employs Komar charge integrals, which associate conserved quantities to each Killing field $K^\mu$,

$$Q[K] = -\frac{1}{16\pi G}\int_{S^{D-2}} \star dK^\flat$$

In vacuum, $dQ[K] = 0$ on shell, so by Stokes’ theorem,

$\int_{S^{\infty}} Q[K] = \int_{\mathcal{B}} Q[K]$

with $S^\infty$ at spatial infinity and $\mathcal{B}$ the (bifurcation) horizon. Evaluating for the generator $\chi^\mu = \partial_t + \Omega_H \partial_\varphi$ leads directly to

$K_\chi = 2ST$

at the horizon for arbitrary dimension, and thereby to the global Smarr formula for $M$ after tracking Komar normalizations (Banerjee et al., 2010, Modak, 2012, Ballesteros et al., 2024).

In the presence of matter, the Komar charge is generalized to include additional contributions (e.g., electromagnetic potentials, scalar terms), ensuring closure on shell and allowing an extended Smarr relation with work terms for every conserved charge (Ballesteros et al., 2024).

3. Extended Thermodynamics: Variable Coupling Constants

When the gravitational action includes dimensionful couplings—such as a cosmological constant $\Lambda$, Lovelock or higher-derivative couplings $\alpha_k$, or matter couplings—the standard Smarr formula is modified. Extended thermodynamics treats these couplings as thermodynamic charges, promoting, for example, $P=-\Lambda/8\pi$ to a “pressure” and introducing the conjugate thermodynamic volume $V = (\partial M/\partial P)_{S,J,Q,\dots}$,

$$dM = TdS + \Omega_H dJ + \Phi dQ + V dP + \sum_k \Psi^{(k)} d\beta_k + \dots$$

and the Smarr relation acquires additional terms,

$$(D-3)M = (D-2)TS + (D-2)\Omega_H J + (D-3)\Phi Q - 2PV + \sum_k 2(k-1)\Psi^{(k)}\beta_k$$

where each $\Psi^{(k)}$ is the conjugate “potential” to a coupling $\beta_k$, determined by scaling arguments and the extended first law (Kastor et al., 2010, Hajian et al., 27 Nov 2025, Hajian et al., 2021).

This “universal Smarr framework” applies to higher-curvature gravities (such as Lovelock) and any additional coupling constant with well-defined scaling. The construction is systematically achieved by promoting couplings to parameters via auxiliary fields, dynamically associating each to a global symmetry and conserved charge whose chemical potential is evaluated at the black hole horizon (Hajian et al., 27 Nov 2025).

4. Smarr Formulas in Specific Gravity Theories

4.1 Lovelock Gravity and Higher-Derivative Theories

In Lovelock gravity, black holes are characterized by higher-order curvature terms with their own dimensionful couplings $\tilde{\beta}_k$. The extended first law and Smarr formula are (Kastor et al., 2010, Liberati et al., 2015),

$$\delta M = T\delta S + \sum_k \Psi^{(k)} \delta\tilde{\beta}_k$$

$$(D-3)M = (D-2)TS - \sum_k 2(k-1)\Psi^{(k)}\tilde{\beta}_k$$

The potentials $\Psi^{(k)}$ receive explicit boundary, “volume” (bulk), and horizon contributions, and the mass and entropy receive nontrivial corrections, including possible topological terms (e.g., Euler-Gauss–Bonnet in even $D$) which can be subtracted consistently from the entropy definition for the physically relevant thermodynamics (Liberati et al., 2015).

4.2 Black Holes with Scalar Hair and Other Extensive Charges

In beyond-Horndeski theories or black holes endowed with primary or secondary scalar hair, promoting the scalar sector coupling constants to dynamical charges is necessary for a consistent Smarr formula. The improved first law reads (Myung et al., 5 May 2025),

$dM = T_H dS + \Phi_\lambda d\lambda$

with the corresponding Smarr relation from Euler scaling,

$M = 2T_H S + \Phi_\lambda \lambda$

This analysis demonstrates that a “naive” first law holding only the hair parameter $q$ fixed fails to yield a consistent Smarr formula; promoting all dimensionful couplings to thermodynamic variables restores homogeneity and correct scaling (Myung et al., 5 May 2025, Hajian et al., 27 Nov 2025).

4.3 Nonlinear Electrodynamics and Additional Matter Sectors

For black holes coupled to nonlinear electrodynamics (NLED), the Smarr formula receives further corrections related to the trace of the energy-momentum tensor, responsible for the loss of naive scale-invariance (Balart et al., 2017, Gulin et al., 2017). The generalized Smarr formula can be written as

$M = 2TS + \Phi_H q - \int_V \omega(r) dV$

where $\omega = T^\mu{}_\mu/2$ encodes a vacuum-polarization or “work” density due to nonlinearities. In general NLED, the correction can be written as a product of the nonlinear coupling and its conjugate “vacuum polarization” variable, paralleling the $\beta\mathcal{C}$ structure in master Smarr formulas (Gulin et al., 2017).

5. Special Cases: Lower Dimensions, Exotic Gravity and Distorted Horizons

5.1 Three-Dimensional Gravity and BTZ/Exotic Black Holes

For 3D black holes (BTZ and their extensions), the Smarr relation takes a homogeneous form with zero on the left,

$0 = TS + \Omega J - 2VP$

with $M$ interpreted as enthalpy, and the thermodynamic volume $V$ can differ from its naive geometric value in higher-derivative or Chern–Simons extended models (Liang et al., 2017). In these models, consistency enforces “locking” relations among dimensionless couplings and the AdS length, a constraint absent in pure Einstein gravity.

5.2 Distorted and Multi-Horizon Black Holes

For stationary, axisymmetric black holes with external distortions, the Smarr formulas for both outer and inner horizons remain formally unaltered,

$$M = \pm \frac{\kappa_\pm A_\pm}{4\pi} + 2\Omega_\pm J$$

with the distortion parameters absorbed into modified redshift factors and horizon data (Shoom, 2015). The Smarr relation maps between outer and inner horizons under a characteristic discrete symmetry, illustrating a duality structure.

5.3 BPS Multicenter and Composite Black Holes

For multicenter BPS black holes, the ADM mass depends on the mutual separation and individual and total charges. The Smarr-type mass formula involves explicit moduli dependence,

$$M^2 = A\left(1+\alpha J^2\left(1+\frac{2M}{r}+\frac{A}{r^2}\right)\right)$$

and leads to a first law including variations in the inter-center distance (“force” terms) as well as electromagnetic and angular degrees of freedom (Torrente-Lujan, 2019).

6. Smarr Formulas under Lorentz Symmetry Violation

Lorentz-breaking theories, with dynamical aether fields or preferred time slicing (Einstein–aether, IR Hořava gravity), admit generalized Smarr formulas where new work terms and aether charges arise: $$\beta_M M = \beta_T T S + \beta_\Omega \Omega_H J + \sum_i \beta_i \mathcal{Q}_i + \beta_p p V$$ The coefficients $\beta_j$ are fixed by scaling dimensions, and the “æther work” terms reflect conserved quantities associated with asymptotic misalignment of the preferred frame or foliation (Pacilio et al., 2017, Ding et al., 2015, Ding et al., 2016). The structure of the Smarr relation, including its compatibility with universal horizons and generalized thermodynamic variables, indicates robustness of black-hole thermodynamics even under strong Lorentz symmetry breaking.

7. Universalization and Consistency of Smarr Relations

A key insight is that the naive Smarr formula is only consistent if all dimensionful coupling constants—cosmological, higher-derivative, matter, or scalar—are promoted to variables within the black hole thermodynamic phase space. This “universal Smarr formula” ensures closure of the scaling argument and resolves pathologies in diverse settings, such as higher-curvature gravity, Horndeski theory, black branes, and 3D models. The prescription, now systematized and proven for a broad class of gravitational theories, forms the foundation for black-hole chemistry and extended thermodynamic frameworks (Hajian et al., 27 Nov 2025).

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References:

• (Banerjee et al., 2010) Banerjee, Modak, Samanta, "Killing Symmetries and Smarr Formula for Black Holes in Arbitrary Dimensions"
• (Modak, 2012) Modak, "Generalized Smarr formula as a local identity for arbitrary dimensional black holes"
• (Liang et al., 2017) Hansen, "Smarr formula for BTZ black holes in general three-dimensional gravity models"
• (Kastor et al., 2010) Kastor, Ray, Traschen, "Smarr Formula and an Extended First Law for Lovelock Gravity"
• (Liberati et al., 2015) Anastasiou, Olea, "Smarr Formula for Lovelock Black Holes: a Lagrangian approach"
• (Myung et al., 5 May 2025) Anabalón et al., "Smarr formula for black holes with primary and secondary scalar hair"
• (Hajian et al., 27 Nov 2025) Hajian, Tekin, "A Universal Smarr Formula via Coupling Constants"
• (Balart et al., 2017) Balart, Fernando, "A Smarr formula for charged black holes in nonlinear electrodynamics"
• (Gulin et al., 2017) Gulin, Smolić, "Generalizations of the Smarr formula for black holes with nonlinear electromagnetic fields"
• (Hajian et al., 2021) Hajian, Özşahin, Tekin, "First law of black hole thermodynamics and Smarr formula with a cosmological constant"
• (Shoom, 2015) Shoom, "Distorted stationary rotating black holes"
• (Torrente-Lujan, 2019) Torrente-Lujan, "Smarr Mass formulas for BPS multicenter Black Holes"
• (Pacilio et al., 2017) Berglund, Bhattacharyya, Mattingly, "An improved derivation of the Smarr Formula for Lorentz-breaking gravity"
• (Ding et al., 2015) Ding, Wang, Wang, "Charged Einstein-aether black holes and Smarr formula"
• (Ding et al., 2016) Lin, Wang, Wu, Yang, "Three-dimensional charged Einstein-aether black holes and Smarr formula"
• (Ballesteros et al., 2024) Murata, "Generalized Komar charges and Smarr formulas for black holes and boson stars"
• (Erices et al., 2017) Erices, Fuentealba, Riquelme, "Electrically charged black hole on AdS$_3$: scale invariance and the Smarr formula"