Generalized Komar Charges in Gravity Theories

• Generalized Komar charges are on-shell closed (d–2)-forms derived from dynamical fields and symmetries, extending classical definitions of mass and angular momentum.
• They are constructed via the Noether–Wald procedure, incorporating corrections to address obstructions in higher-curvature and matter-coupled theories.
• They play a key role in black hole thermodynamics and Smarr relations by linking geometric invariants with conserved physical quantities across varied gravity models.

A generalized Komar charge is an on-shell closed $(d-2)$-form, constructed from the dynamical fields and symmetries of a gravitational theory, whose surface integral on a closed $(d-2)$-dimensional hypersurface yields a conserved quantity associated to an exact symmetry—typically a Killing vector or its generalization. Generalized Komar charges extend the classical Komar mass and angular momentum definitions of General Relativity to incorporate theories with torsion, higher-curvature interactions, gauge fields, extended objects, and boundary or asymptotic structures. They play a foundational role in deriving Smarr-type relations, black hole thermodynamics, and establishing connections between geometry, topology, and physical invariants across a broad class of classical and quantum gravity models.

1. Classical Komar Charges: Definition and Basic Properties

Let $(M, g_{\mu\nu})$ be a $d$-dimensional Lorentzian spacetime with a Killing vector $\xi^{\mu}$, i.e., $\nabla_{(\mu}\xi_{\nu)}=0$. The standard Komar $(d-2)$-form is

$\omega_\xi = \frac{1}{16\pi G} \star d \xi^{\flat}$

where $\xi^{\flat} = g_{\mu\nu}\xi^{\nu} dx^{\mu}$ and $\star$ denotes the Hodge dual. The Komar charge is then defined by integration over a closed, spacelike $(d-2)$-surface $S$: $$Q[\xi] = \int_S \omega_\xi = \frac{1}{16\pi G} \int_S \epsilon_{\mu_1\cdots\mu_{d-2}\rho\sigma}\nabla^{\rho} \xi^{\sigma}\, dx^{\mu_1}\wedge \cdots \wedge dx^{\mu_{d-2}}$$ In vacuum General Relativity, $d\omega_\xi = 0$ on-shell due to the Einstein equations and Bianchi identity, so $Q[\xi]$ yields a conserved charge—the Komar mass for $\xi$ timelike at infinity, or angular momentum for rotational Killing vectors (Ballesteros et al., 2024).

2. Generalization: Noether–Wald Procedure and Closure Obstructions

For diffeomorphism-invariant theories with generic field content (including higher-curvature, matter, or gauge fields), the construction is systematized by the Noether–Wald formalism. Under an exact symmetry generator $k$, the Noether current $(d-1)$-form

$J[k] = \Theta(\phi, \delta_k \phi) + \iota_k L$

is locally exact: $J[k] = dQ[k]$ (Ortín et al., 2024). For Killing fields, $\delta_k\phi=0$ and $\Theta$ typically vanishes, so $dQ[k]=\iota_k L$. In pure GR, $\iota_k L$ is also exact, and so $Q[k]$ is closed.

In general, however, $\iota_k L$ may not be an exact differential—especially in higher-curvature or matter-coupled theories—introducing an "obstruction" to closedness. When $\iota_k L = d w_k$ on-shell for some $(d-2)$-form $w_k$, the generalized Komar form

$K[k] = -Q[k] + w_k$

is closed on-shell: $dK[k]=0$. The associated charge is $Q_{\text{gen}}[k] = \int_S K[k]$ (Ortín et al., 2024).

This universal construction yields the correct mass, angular momentum, and gauge charges in all standard examples. The explicit algorithm involves computing the symplectic potential $\Theta$, the Noether current/potential, the obstruction $w_k$, and forming $K[k]$ as above (see (Ortín et al., 2024) for detailed procedural steps).

3. Examples of Generalized Komar Charges in Various Theories

Gravity with Torsion (Einstein–Cartan):

For Riemann–Cartan manifolds with a vierbein $e^a$ and independent spin-connection $\omega^{ab}$, torsion $T^a = D e^a$ appears. The dual Komar^* mass,

$$Q_*[\xi] = \frac{1}{16\pi G} \int_{\Sigma} \epsilon_{abcd}\, D_{\mu}T^{\mu ab}\, e^c_{\nu} e^d_{\rho}\, d\Sigma^{\nu\rho}$$

is non-vanishing due to the local violation of the algebraic Bianchi identity. The source current $j^a \sim \star(D T^a)$ acts as a gravitational-magnetic current, generalizing mass to encompass NUT and torsion-induced “magnetic” charges (Kol, 2020).

Gauge and Scalar Fields:

For Einstein–Maxwell(-scalar) systems, additional closed $(d-2)$-forms appear, built from electromagnetic or scalar momentum maps (potentials arising from gauge or global symmetries): $$\Omega_{\xi} = \omega_{\xi} + \text{(matter Noethers)} - w_{\xi}, \quad d\Omega_{\xi}=0$$ This covers dyonic black holes, axion–dilaton theories, and general models with electric-magnetic and scalar charges (Mitsios et al., 2021, Ballesteros et al., 2024, Cabrera-Munguia et al., 2014, Manko et al., 2015).

Higher Derivative (Lovelock/Higher-Order) Theories:

For Lovelock actions, the Komar potential generalizes to

$Q^{ab}[k] = -2 P^{abcd} \nabla_c k_d,$

with $$P^{abcd} = \frac{\partial \mathcal{L}}{\partial R_{abcd}}$$ (possibly after background subtraction) (0804.1832, Peng et al., 2019). This construction remains valid in $f(R)$, Gauss–Bonnet, or arbitrary higher-derivative pure-gravity Lagrangians.

Theories with Chern–Simons Terms and p-form Gauge Fields:

For actions including Chern–Simons interactions ($G \wedge G \wedge V$) or higher-form kinetic terms, generalized on-shell closed forms $\mathbf{K}[k]$ incorporate “electric” and “magnetic” momentum maps for the associated gauge symmetries. This structure is essential in 5D minimal supergravity, its reductions, and 11D M-theory (Barbagallo et al., 27 Dec 2025).

Supersymmetric Extensions (SUGRA):

In supergravity, the generalized charge is promoted to a superform in superspace, involving both Killing vectors and their spinorial superpartners. Closure is achieved via the Killing supervector equations and their associated “momentum maps” for all gauge and Lorentz symmetries (Bandos et al., 2024, Bandos et al., 2024).

4. Dual Komar Charges and Magnetic Mass

A significant feature of generalized Komar charges is the possibility to define “dual” charges, notably the dual Komar mass: $Q_*[\xi] = \frac{1}{16\pi G} \oint_{S^2_\infty} F,$ where $F = dA = 2\nabla_{[\mu} \xi_{\nu]}$ plays the role of the “magnetic” potential. In torsion-free GR, $Q_*$ vanishes unless string-like singularities (e.g. Misner strings in Taub–NUT spacetime) are present. However, in Riemann–Cartan geometry or with explicit torsional sources, $Q_*$ becomes a local, physical bulk charge. The dual mass coincides with the dual supertranslation charge at infinity in asymptotically flat spacetimes (Kol, 2020, Bordo et al., 2019, Cerdeira et al., 18 Jun 2025).

5. Structural and Algorithmic Properties

The class of generalized Komar charges possesses several key properties:

• Universality: Any exactly gauge- and diffeomorphism-invariant theory admits a closed $(d-2)$-form charge for every exact symmetry, constructed from the Lagrangian and symmetry data (Ortín et al., 2024, Cerdeira et al., 16 Jun 2025).
• On-shell Conservation: $dK[k]=0$ holds by virtue of the equations of motion.
• Freedom in Charge Basis: The set of generalized charges is defined up to the addition of any other on-shell closed $(d-2)$-form; in certain situations (notably Kaluza–Klein reductions and higher-form symmetries) this redundancy is crucial for duality invariance and correct mass-matching (Barbagallo et al., 18 Jun 2025, Barbagallo et al., 27 Dec 2025).
• Solution Independence from Homogeneity: If the Lagrangian is homogeneous under a global rescaling, the on-shell Lagrangian becomes a total derivative via a functional version of Euler's theorem. Thus, generalized Komar charges are solution-independent constructs in such theories (Cerdeira et al., 16 Jun 2025).
• Explicit Algorithm: A 7-step method exists for producing the closed form of the Komar charge in any such theory (variation→Noether current→potential→obstruction→correction→integration), requiring only the Lagrangian data and the symmetry (Ortín et al., 2024).

6. Physical Applications and Black Hole Thermodynamics

Generalized Komar charges are foundational in:

• Smarr Relations: The equality of integrals of the generalized Komar charge at infinity and on a black hole horizon yields generalized Smarr mass relations encoding temperature, entropy, angular momentum, electric/magnetic/scalar charges, and “work terms” from coupling constants or external potentials (Ballesteros et al., 2024, Mitsios et al., 2021, Cabrera-Munguia et al., 2014, Manko et al., 2015, Barbagallo et al., 27 Dec 2025).
• Asymptotic Charges and Null Infinity: Through the Wald–Zoupas prescription, Komar-type integrals are covariantly extended to non-stationary, radiative spacetimes at null infinity, capturing Bondi, supertranslation, and angular momentum charges (Grant et al., 2021).
• Higher-Derivative and Supergravity Theories: Generalized charges are central in black-hole thermodynamics, entropy laws, mass formulas, and duality invariance for Lovelock, higher-curvature, gauged, and supergravity models (0804.1832, Peng et al., 2019, Bandos et al., 2024, Bandos et al., 2024).
• Topologically Nontrivial or Twisted Solutions: In the presence of nontrivial topology (e.g., NUT charge, magnetic monopoles, Misner strings, higher-form charges), the generalized Komar framework encompasses both “electric” and “magnetic” contributions, including parity-violating effects (e.g., gravitational Witten effect with the Holst term) (Kol, 2020, Cerdeira et al., 18 Jun 2025).
• Global Symmetries and Higher-Form Currents: In Kaluza–Klein reductions and Chern–Simons-enhanced models, extra on-shell closed forms correspond to generalized higher-form symmetries and must be added to the naive Komar basis to restore gauge and duality invariance (Barbagallo et al., 18 Jun 2025, Barbagallo et al., 27 Dec 2025).

7. Extensions, Limitations, and Outlook

The generalized Komar charge formalism is not limited to stationary spacetimes or field theories with conventional symmetry structure. Notable aspects include:

• Non-Stationary or Radiative Spacetimes: Extensions exist for charges associated to Bondi–Metzner–Sachs (BMS) symmetries at null infinity, using the Wald–Zoupas formalism, which reduces to Komar in stationary regions (Grant et al., 2021).
• Higher-Derivative and Nonlocal Theories: The framework accommodates $f(R)$, Lovelock, Gauss–Bonnet, and generic higher-order models, with the Komar form determined by functional derivatives of the Lagrangian (0804.1832, Peng et al., 2019).
• Twisted Symmetries and Nontrivial Matter Sector: Allowing matter fields to be invariant under combination of spacetime and internal/global symmetries (“twisted” or “generalized symmetric” ansatz) permits nontrivial solitonic or hairy black-hole solutions, linked to modified Komar charges (Ballesteros et al., 2024).
• Quasi-locality and Boundary Conditions: In quasi-local Hamiltonian approaches (e.g., Brown–York or FGP boundary prescriptions), generalized Komar charges depend on the chosen boundary conditions, interpolating between Komar, Brown–York, and Neumann-type Hamiltonians (Odak et al., 2021).
• Ambiguities and Redundancies: The non-uniqueness arising from addition of on-shell closed forms is a feature, not a bug, especially in the presence of higher-form symmetries, background subtractions, or duality transformations (Barbagallo et al., 18 Jun 2025, Peng et al., 2019).

A continuing direction is the systematic classification of all possible generalized Komar charges in a given field theory, their algebraic structure, and their role in quantum-gravity microstates, string dualities, and holographic correspondences.

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Key references:

• (Kol, 2020) for dual Komar mass and torsion
• (Ortín et al., 2024, Cerdeira et al., 16 Jun 2025) for systematic construction in general theories
• (Barbagallo et al., 18 Jun 2025, Barbagallo et al., 27 Dec 2025) for higher-form symmetries and Kaluza–Klein reductions
• (0804.1832, Peng et al., 2019) for Lovelock/higher-derivative gravities
• (Bandos et al., 2024, Bandos et al., 2024) for supersymmetric generalizations
• (Ballesteros et al., 2024, Mitsios et al., 2021, Cabrera-Munguia et al., 2014, Manko et al., 2015) for matter and gauge extensions
• (Bordo et al., 2019, Cerdeira et al., 18 Jun 2025) for topological terms and NUT charge
• (Grant et al., 2021, Odak et al., 2021) for null infinity and boundary/ensemble dependence

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Summary Table: (Selection of Core Variants)

Theory Class	Generalized Komar Form	Key Features
Vacuum GR	$\star d\xi$	Mass/ang. mom. via Killing $\xi$
Einstein–Cartan	$\star d\xi$ (electric), $F$ (magnetic), $T^a$	Dual mass from torsion source
Lovelock/Higher Curvature	$-2P^{abcd}\nabla_c\xi_d$	$P=\partial L/\partial R_{abcd}$
Gauge + Scalar Fields	$+$ momentum maps, Noether terms	Electric/magnetic/scalar charges
Chern–Simons/p-form	Closure via momentum maps for $G, \star G$	Higher-form, duality structure
Supergravity	Superform in (super)superspace	Fermionic contributions

This organization provides an operational and conceptual unification of geometric charges in classical and quantum field theories of gravity and their extensions.