Generalized Komar Charge

• Generalized Komar charge is an on-shell closed (d-2)-form that extends the standard Komar construction to include additional terms for matter couplings and modified gravity.
• It restores the Gauss-law property by adding precise corrections, ensuring conserved charges can be reliably computed across boundaries such as horizons and asymptotic infinity.
• It plays a key role in deriving Smarr formulas, black-hole thermodynamics, and establishing BPS bounds in supergravity and related gravity theories.

A generalized Komar charge is an on-shell closed $(d-2)$-form that extends the vacuum Komar construction to situations in which the ordinary Komar form is no longer closed. In its standard vacuum form, for a Killing vector $k$, the Komar $(d-2)$-form is

$$\mathbf{K}[k]=-\frac{1}{16\pi G_N^{(d)}}\star d\hat k,$$

with $\hat k=k_\mu dx^\mu$. In pure General Relativity it obeys $d\mathbf K[k]\doteq 0$, but matter couplings, cosmological terms, higher-curvature interactions, topological terms, torsion, or weakened symmetry assumptions generally obstruct this closure. The generalized Komar charge restores an on-shell Gauss-law property by adding precisely those extra terms needed so that the surface integral can still be moved through the bulk and evaluated at infinity, on horizons, or on other homologous surfaces. In modern usage, the concept spans matter-coupled gravity, supergravity, AdS and Kaluza–Klein settings, first-order and torsionful formalisms, and certain non-Killing or conformal generalizations (Ortín, 6 May 2026, Ortín et al., 2024).

1. Vacuum antecedents and the basic Komar mechanism

In vacuum General Relativity, the ordinary Komar construction is tied to a Killing vector. The essential property is on-shell closedness: $d\mathbf K[k]\doteq 0.$ For stationary asymptotically flat solutions, the integral at infinity satisfies

$$\int_{S^{d-2}_\infty}\mathbf K[k]=\frac{d-3}{d-2}M,$$

up to the usual normalization conventions (Ortín, 6 May 2026).

A closely related current formulation treats the Killing vector as a 1-form $\xi$. In that language, the ordinary Komar current is

$J_K=-\delta d\xi,$

and is co-closed because $k$0. This expresses the same conservation mechanism in differential-form language and makes clear that the standard Komar charge is fundamentally a boundary flux derived from an antisymmetric 2-form superpotential $k$1 (Peng et al., 2019).

The reason this construction is so important is geometric rather than merely asymptotic. Since the form is closed on shell, its integral is invariant under deformations of the integration surface through regions where the field equations hold. This is the structural basis of Komar-type Smarr formulas: the same charge evaluated at spatial infinity yields asymptotic conserved quantities, while evaluation on a horizon yields thermodynamic quantities such as $k$2 and $k$3 (Ortín, 6 May 2026, Ballesteros et al., 2024).

2. Matter couplings, Noether–Wald theory, and the general construction

Once matter is present, the ordinary Komar form generally ceases to be closed on shell. In supergravity, gauge fields and scalars contribute to the Einstein equations, so $k$4 is no longer closed, and the natural conserved $k$5-form must include matter potentials or momentum maps (Ortín, 6 May 2026). A central misconception addressed in the recent literature is that the generalized Komar charge should simply be the Noether–Wald charge $k$6 evaluated on a Killing vector. In a generic exactly gauge- and diffeomorphism-invariant theory this is false: on shell,

$k$7

so $k$8 is not generally closed by itself (Ortín et al., 2024).

The corrected object is obtained by finding a $k$9-form $(d-2)$0 such that $(d-2)$1, and then defining

$(d-2)$2

Equivalently, the construction can be written as

$(d-2)$3

This gives a compact algorithm for producing an on-shell closed generalized Komar charge in arbitrary exactly gauge- and diffeomorphism-invariant theories (Ortín et al., 2024).

A complementary result identifies a broad class of theories in which the on-shell Lagrangian itself is a universal total derivative. If a global transformation rescales the Lagrangian with nonzero weight $(d-2)$4, then the variational identity implies

$(d-2)$5

This provides a solution-independent candidate for the exact form needed in generalized Komar constructions and explains why homogeneous field transformations, especially weighted rescalings, are so useful in practice (Cerdeira et al., 16 Jun 2025).

Concrete matter-coupled realizations follow this pattern. In Einstein–Maxwell theory, the generalized charge contains gravitational, electric, and magnetic momentum-map terms; in axion–dilaton gravity it is a symplectic pairing of electric and magnetic potentials with dual field strengths; in Einstein–nonlinear electrodynamics it also contains an additional term associated with a dimensionful coupling promoted to a field in the covariant phase space (Ortín et al., 2024, Mitsios et al., 2021, Barbagallo et al., 4 May 2026).

3. Supergravity and supersymmetric vanishing theorems

Supergravity provides one of the sharpest formulations of the generalized Komar idea. In minimal $(d-2)$6 supergravity, the appropriate object is not merely a spacetime 2-form but a closed bosonic 2-superform in superspace, the super-Komar form,

$(d-2)$7

which obeys

$(d-2)$8

Its lowest component is the spacetime generalized Komar 2-form, and the construction requires not only the Lorentz and electric momentum maps but also a magnetic momentum map $(d-2)$9, needed for closure and duality covariance (Bandos et al., 2024).

A stronger result holds for supersymmetric solutions in several supergravity theories. For the Killing vector constructed as a bilinear of the Killing spinors, the generalized Komar $$\mathbf{K}[k]=-\frac{1}{16\pi G_N^{(d)}}\star d\hat k,$$0-form vanishes identically. This has been shown explicitly in ungauged $$\mathbf{K}[k]=-\frac{1}{16\pi G_N^{(d)}}\star d\hat k,$$1 supergravity coupled to vector multiplets, ungauged $$\mathbf{K}[k]=-\frac{1}{16\pi G_N^{(d)}}\star d\hat k,$$2 supergravity coupled to vector multiplets, and pure $$\mathbf{K}[k]=-\frac{1}{16\pi G_N^{(d)}}\star d\hat k,$$3 supergravity (Ortín, 6 May 2026).

This vanishing theorem has a direct physical use. Because the generalized Komar integral at infinity is precisely the mass-plus-charge combination constrained by supersymmetry, its vanishing for the supersymmetric Killing vector provides a coordinate-independent route to BPS bounds (Ortín, 6 May 2026). This suggests that, in supergravity, generalized Komar charges are not only corrected mass functionals but also geometric encodings of Killing-spinor identities.

4. Asymptotics, topology, and modified gravitational sectors

Generalized Komar charges also arise when the obstruction is not ordinary matter but asymptotics, topology, or modifications of the gravitational action.

In Einstein gravity with negative cosmological constant, the naive Komar mass diverges for asymptotically AdS spacetimes. One remedy is an improved Komar potential

$$\mathbf{K}[k]=-\frac{1}{16\pi G_N^{(d)}}\star d\hat k,$$4

whose surface integral yields the correct finite mass and angular momentum of asymptotically AdS black holes in Einstein gravity (Peng et al., 2020). In Lovelock gravity, the generalization is more structural: the ordinary $$\mathbf{K}[k]=-\frac{1}{16\pi G_N^{(d)}}\star d\hat k,$$5 is replaced by a theory-dependent antisymmetric boundary tensor whose divergence reproduces the Lovelock equations of motion, preserving a Komar-type relation on shell. This includes the cosmological-constant case via Killing potentials and gives a finite AdS Komar mass without infinite background subtraction (0804.1832).

In pure five-dimensional gravity with Kaluza–Klein boundary conditions, the naive 5d Komar charge associated with time translations does not give the physical 4d Einstein-frame mass. The correction uses the fact that generalized Komar charges are ambiguous up to the addition of any other on-shell closed $$\mathbf{K}[k]=-\frac{1}{16\pi G_N^{(d)}}\star d\hat k,$$6-form. A higher-form symmetry specific to the Kaluza–Klein topology supplies the needed extra closed 3-form, and the corrected charge also contains the Kaluza–Klein monopole contribution, leading to electric-magnetic duality invariance (Barbagallo et al., 18 Jun 2025).

In first-order gravity with a Holst term, the Noether–Wald derivation gives

$$\mathbf{K}[k]=-\frac{1}{16\pi G_N^{(d)}}\star d\hat k,$$7

The second term is topologically closed by itself, and for time translations in Taub–NUT it shifts the asymptotic mass according to

$$\mathbf{K}[k]=-\frac{1}{16\pi G_N^{(d)}}\star d\hat k,$$8

This is interpreted as a gravitational analogue of the Witten effect, with NUT charge acting as a magnetic source for an induced mass (Cerdeira et al., 18 Jun 2025).

A related but distinct development concerns the dual Komar mass. In ordinary smooth Riemannian General Relativity it vanishes because $$\mathbf{K}[k]=-\frac{1}{16\pi G_N^{(d)}}\star d\hat k,$$9 for the Komar 2-form field strength $\hat k=k_\mu dx^\mu$0. On Riemann–Cartan manifolds, however, torsion modifies the algebraic Bianchi identity and produces a local gravitational-magnetic current, so the dual Komar mass becomes a volume integral over a local torsion-dependent source (Kol, 2020). A plausible implication is that generalized Komar constructions probe not only stress-energy corrections but also the cohomological and parity-odd structure of the gravitational sector.

5. Beyond exact Killing symmetry

A separate line of work generalizes Komar charges by weakening the symmetry assumptions rather than by adding matter-dependent terms.

For asymptotically flat dynamical spacetimes, a generalized Komar energy can be built from the normal evolution vector

$\hat k=k_\mu dx^\mu$1

without assuming a Killing or even asymptotically Killing field. Under Weinberg asymptotic flatness, the central relation is

$\hat k=k_\mu dx^\mu$2

and equality $\hat k=k_\mu dx^\mu$3 follows if $\hat k=k_\mu dx^\mu$4. The important point is that this extends ADM–Komar equality to a broad class of dynamical asymptotically flat spacetimes (Wang et al., 7 Sep 2025).

At a more formal level, generalized Komar currents for non-Killing vectors can be organized by the second-order operators $\hat k=k_\mu dx^\mu$5, $\hat k=k_\mu dx^\mu$6, and $\hat k=k_\mu dx^\mu$7. The paper on generalized Komar currents for vector fields shows that the usual Komar current can be rewritten as a particular linear combination of these operators and then generalized to arbitrary vectors satisfying a second-order constraint, summarized by

$\hat k=k_\mu dx^\mu$8

with

$\hat k=k_\mu dx^\mu$9

This unifies almost-Killing, conformal Killing, and related constructions (Peng et al., 2019).

In locally rotationally symmetric spacetimes, the same strategy can be specialized to conformal Killing vectors. The conformal Komar current remains divergence-free, admits both kinematic and matter/curvature forms, and on a conformal Killing horizon the associated Noether charge is proportional to the surface gravity, giving it a thermodynamic interpretation (Sherif, 5 Jun 2025). These developments indicate that “generalized Komar charge” can denote either an on-shell closed matter-corrected $d\mathbf K[k]\doteq 0$0-form or a Komar-type current associated with weakened symmetry generators; the two traditions are related but not identical.

6. Smarr formulas, black-hole thermodynamics, and solitonic constraints

The most persistent use of generalized Komar charges is the derivation of Smarr formulas. In axion–dilaton gravity, the generalized Komar 2-form is

$d\mathbf K[k]\doteq 0$1

and integrating it at infinity and on the horizon yields

$d\mathbf K[k]\doteq 0$2

Here the correction from $d\mathbf K[k]\doteq 0$3 is essential: the bare Wald–Noether 2-form is not $d\mathbf K[k]\doteq 0$4-invariant, whereas the generalized Komar form is (Mitsios et al., 2021).

In Einstein–nonlinear electrodynamics, the generalized Komar charge contains electric, magnetic, and coupling-dependent contributions,

$d\mathbf K[k]\doteq 0$5

and yields a Smarr formula with an explicit $d\mathbf K[k]\doteq 0$6-term. In the Bardeen solution, the standard Komar part tends to zero as $d\mathbf K[k]\doteq 0$7, while the nonlinear electromagnetic contribution carries the entire ADM mass, clarifying how regular black holes can avoid a central singular source (Barbagallo et al., 4 May 2026).

Generalized Komar charges also underpin non-existence theorems. For black holes and boson stars, on-shell closed $d\mathbf K[k]\doteq 0$8-form charges imply that a regular horizonless asymptotically flat solution with only an outer boundary must have vanishing asymptotic generalized Komar integral. This is the basis of no-soliton and no-boson-star statements in several matter-coupled theories (Ballesteros et al., 2024). The same work shows that generalized symmetric fields—configurations invariant under a combination of spacetime isometries and global symmetries—can evade these theorems. This suggests that the generalized Komar framework not only encodes conserved charges but also sharply diagnoses when apparently stationary matter is truly compatible with regular self-gravitating solitons.

Quasi-local and deformed variants fit naturally into this picture. In Kerr–Newman, effective Komar mass and angular momentum can be defined on finite-radius surfaces, and the horizon generator satisfies the local identity $d\mathbf K[k]\doteq 0$9, interpreted as a local precursor of the generalized Smarr formula (Modak et al., 2010). In noncommutative-geometry-inspired charged black holes, the standard Killing integral evaluated on a deformed metric yields a radius-dependent Komar energy,

$d\mathbf K[k]\doteq 0.$0

and a generalized Smarr formula with explicit $d\mathbf K[k]\doteq 0.$1-dependent corrections (Larranaga et al., 2012). In dyonic dihole solutions, magnetic charge does not redefine Komar mass itself, but it does require an enhanced Smarr formula through an additional horizon boundary term (Cabrera-Munguia et al., 2014).

Taken together, these results establish the generalized Komar charge as a flexible but tightly structured concept: a corrected codimension-2 conserved form whose precise definition depends on what obstructs the ordinary Komar closure, but whose role remains constant—turning bulk field equations into boundary identities for mass, angular momentum, gauge charges, coupling contributions, BPS bounds, and the existence or non-existence of regular gravitating configurations.