---
title: Komar Charges in Gravitational Theories
url: https://www.emergentmind.com/topics/komar-charges
type: topic
---

# Komar Charges in Gravitational Theories

Komar charges are conserved quantities in diffeomorphism-invariant field theories—most notably in general relativity and its extensions—which are associated to the symmetries (Killing vectors or reducibility parameters) of a given solution. These charges, constructed as surface integrals of closed differential forms, play a central role in the derivation of black-hole mass, angular momentum, and in establishing Smarr formulas and first laws of black hole mechanics. The modern formulation generalizes Komar’s original approach to encompass generic gauge- and diffeomorphism-invariant theories with higher curvature, matter fields, and nontrivial gauge symmetry content [2411.10420].

## 1. Classical Komar Charges in Vacuum General Relativity

Given a $d$-dimensional metric $g_{\mu\nu}$ and a Killing vector $\xi$ of a vacuum solution (i.e., $\nabla_{(\mu}\xi_{\nu)}=0$ and $R_{\mu\nu}=0$), the Komar $(d-2)$–form is
\[
K[\xi] = \frac{1}{16\pi G}\,\star d\xi
\]
where $d\xi$ is the exterior derivative of the Killing 1-form and $\star$ denotes the Hodge dual. On-shell, $dK[\xi]\approx0$, making $K[\xi]$ closed. The conserved charge is obtained by integrating $K[\xi]$ over a closed $(d-2)$–surface—typically a sphere at spatial infinity:
\[
Q_K[\xi] = \frac{1}{16\pi G}\oiint_{S^{d-2}_\infty}\star d\xi
\]
This reproduces, for appropriate choices of $\xi$, the ADM mass, angular momentum, or linear momentum in asymptotically flat spacetimes [2105.05919, 1910.00339].

## 2. Noether–Wald Formalism and the Need for Generalization

In general diffeomorphism-invariant theories, conserved currents and charges are derived using the Noether–Wald formalism. Let $\mathbf{L}[\phi]$ be a local Lagrangian $d$-form. Its variation decomposes as
\[
\delta \mathbf{L} = \mathbf{E}\,\delta\phi + d\mathbf{\Theta}(\phi,\delta\phi)
\]
where $\mathbf{E}$ are the equations of motion and $\mathbf{\Theta}$ is the presymplectic potential. For an exact symmetry parameterized by a reducibility parameter $\xi$ (e.g., a Killing vector or gauge symmetry), the Noether current is
\[
\mathbf{J}[\xi] = \mathbf{\Theta}(\phi, \pounds_\xi\phi) - \iota_\xi \mathbf{L}
\]
which is off-shell closed, $d\mathbf{J}[\xi]=0$. Locally, one writes
\[
\mathbf{J}[\xi]= d\mathbf{Q}[\xi]
\]
introducing the Noether–Wald charge $(d-2)$–form $\mathbf{Q}[\xi]$ [2411.10420, 2105.05919].

In pure gravity, $\mathbf{Q}[\xi]$ coincides with Komar’s form. However, in the presence of matter, higher-curvature corrections, or nontrivial gauge symmetries, $\mathbf{Q}[\xi]$ can fail to be closed on-shell:
\[
d\mathbf{Q}[\xi]|_{\text{on-shell}} = \iota_\xi \mathbf{L}|_{\text{on-shell}} \neq 0
\]
Thus the naive Noether–Wald charge is not generally conserved in the presence of matter/gauge fields or when higher curvature is present [2411.10420, 2409.08268].

## 3. Generalized Komar Charge: Algorithmic Construction

The generalized Komar prescription produces a unique on-shell closed $(d-2)$–form satisfying:
\[
d\mathbf{K}[\xi] = 0\quad \mathrm{(on\, shell)}
\]
even in the presence of generic matter or gauge couplings. The key steps are:

1. Compute the Noether–Wald $(d-2)$–form $\mathbf{Q}[\xi]$ associated to the symmetry $\xi$.
2. Evaluate its on-shell non-closure, which is given by $\iota_\xi \mathbf{L}|_{\text{on-shell}}$.
3. If $\iota_\xi \mathbf{L}|_{\text{on-shell}}=d\mathbf{w}_\xi$ for some $(d-2)$–form $\mathbf{w}_\xi$ (the "momentum map" or boundary piece), define
   \[
   \boxed{
   \mathbf{K}[\xi] = -\,\mathcal{O}_s(\mathbf{Q}[\xi]) + \mathbf{w}_\xi
   }
   \]
   where $\mathcal{O}_s$ denotes evaluation on-shell.
   The result is a closed $(d-2)$–form:
   \[
   d\mathbf{K}[\xi] = -\,\mathcal{O}_s(d\mathbf{Q}[\xi]) + d\mathbf{w}_\xi = 0
   \]
This construction applies precisely when the Lagrangian is exactly diffeomorphism and gauge invariant, i.e., in the absence of Chern–Simons or pseudo-invariance terms [2411.10420].

## 4. Central Definitions and Prototypical Examples

**Noether Current and Charge:**
\[
J^\mu[\xi] = \Theta^\mu(\phi, \pounds_\xi\phi) - \xi^\mu L,\qquad \nabla_\mu J^\mu[\xi]=0
\]
**Standard Komar Charge in GR:**
\[
Q_K[\xi] = \frac{1}{16\pi G}\oiint_{S^{d-2}}\star d\xi
\]
**Generalized Komar Charge (form version):**
\[
\boxed{
\mathbf{K}[\xi] = -\,\mathcal{O}_s(\mathbf{Q}[\xi]) + \mathbf{w}_\xi,\qquad d\mathbf{K}[\xi]\approx 0
}
\]
where $\iota_\xi \mathbf{L}|_{\text{on-shell}} = d\mathbf{w}_\xi$.

**Example: Einstein–Maxwell in $d$ dimensions:**
\[
S = \int\bigl(\star (e^a\wedge e^b)\wedge R_{ab} - \tfrac12 F\wedge\star F\bigr)
\]
On-shell, $\iota_k\mathbf{L} = d\mathbf{w}_k$ with
\[
\mathbf{w}_k = (-1)^{d-1}\bigl[-\star(e^a\wedge e^b)P_{kab} - (d-3)P_k\wedge\star F + \tfrac1{d-2}e^a\wedge e^b\wedge P_{k\,ab}\bigr]
\]
where $P_{k\,ab}$ and $P_k$ are Lorentz and Maxwell momentum maps ($P_{k\,ab} = D_a k_b,\, dP_k = -\iota_kF$) [2411.10420].

## 5. Generalizations, Special Cases, and Limiting Regimes

- **Pure gravity or minimally coupled scalars:** $\iota_\xi \mathbf{L}|_{\text{on-shell}}=0$, so $\mathbf{w}_\xi=0$ and the generalized Komar charge reduces to the Komar–Wald–Noether form.
- **Matter, higher-curvature corrections:** $\iota_\xi \mathbf{L}|_{\text{on-shell}}\neq0$; $\mathbf{w}_\xi$ exactly cancels the non-closure, ensuring a genuinely closed $\mathbf{K}[\xi]$.
- **Nontrivial gauge symmetry:** the algorithm relies on the existence of momentum maps for every gauge symmetry, an assumption satisfied in theories without Chern–Simons or pseudo-invariant terms.
- **The presence of Chern–Simons terms or anomalous gauge invariance:** requires additional boundary counterterms and careful handling of total-derivative ambiguities.

This procedure systematically upgrades the naive Wald–Noether charge to a closed, physically meaningful generalization across broad gravitational models, thereby ensuring a consistent definition of mass, angular momentum, and related charges even in the presence of complex gauge/matter interactions [2411.10420, 2409.08268].

## 6. Applications and Physical Significance

Komar charges, both in their original and generalized formulations, provide the linchpin for several key results:

- **Black hole thermodynamics:** The equality of Komar integrals at infinity and at a horizon underpins the Smarr relation, expressing the mass in terms of entropy, surface gravity, angular velocities, charges, and corresponding potentials. In generalized settings, the correction term $\mathbf{w}_\xi$ enables the appearance of, e.g., nontrivial matter potentials, scalar charges, and conjugate variables associated to dimensionful couplings [2409.08268, 2605.02813].
- **Quasi-local energy and angular momentum:** The closedness of $\mathbf{K}[\xi]$ ensures that, in stationary spacetimes, integrals can be evaluated on any homologous $(d-2)$–surface, facilitating quasi-local definitions and analysis of solitonic and black hole solutions.
- **No-go theorems:** For soliton/boson star solutions without horizons, integrating the generalized Komar form shows the necessity of matter configuration ("generalized symmetry ansatz") for non-trivial mass [2409.08268].
- **Supergravity:** In supersymmetric theories, the Komar construction can be extended by incorporating the relevant momentum maps, Killing spinors, and superspace structures, ensuring closure and manifest gauge/supersymmetry invariance. For supersymmetric Killing vectors arising as spinor bilinears, the generalized Komar charge vanishes identically, yielding BPS bounds and central charge saturation [2605.05178, 2411.01020, 2412.18510].
- **Extensions to asymptotically AdS, dS, and Kaluza–Klein spaces:** Modified Komar forms incorporating higher-derivative or boundary-covariant corrections yield the correct normalization and finiteness properties (e.g., Abbott–Deser–Tekin, KBL superpotentials) [2008.06733, 2506.15615].
- **Dual charges and magnetic mass:** The "dual" Komar mass arises via the flux of $d\xi$ rather than $\star d\xi$, capturing NUT/Misner–string charges or, more generally, gravitational magnetic charges. In Riemann–Cartan geometries, local torsion produces genuine magnetic Komar charges [2010.07887].

## 7. Summary of Key Properties and Limitations

| Aspect                       | Classical Komar | Generalized Komar (Ortín–Zatti)      | Limitations/Notes                                        |
|------------------------------|:--------------:|:-------------------------------------:|----------------------------------------------------------|
| **Theory**                   | GR (vacuum)    | Generic diffeo/gauge-invariant theory | Excludes Chern–Simons and pseudo-invariant terms         |
| **Form degree**              | $(d-2)$        | $(d-2)$                               | —                                                        |
| **On–shell closedness**      | Yes            | Yes                                   | Non-closure always compensated by $\mathbf{w}_\xi$       |
| **Surface integral/charge**  | Mass, $J$, etc.| Mass, $J$, matter/duality charges     | —                                                        |
| **Supergravity/SUSY**        | Not included   | Extension with supermultiplet data    | Requires Killing supervector formalism, momentum maps     |
| **Gauge invariance**         | Manifest       | Manifest (for exact invariance)       | Nontrivial for Chern–Simons-like couplings               |
| **Physical use**             | Smarr, 1st law | Smarr, 1st law, BPS, dualities        | Black hole mechanics, soliton no-go, duality covariance   |

Generalized Komar charges thus provide a universal, covariant tool for the rigorous extraction of conserved quantities and the elucidation of integrated structural laws (Smarr, first law) for a wide range of gravitational theories, including matter-coupled, higher-curvature, and supersymmetric models. Their systematic construction via the Noether–Wald approach—augmented by the boundary term $\mathbf{w}_\xi$—guarantees on-shell closure and robust applicability so long as the underlying action is diffeomorphism and gauge invariant in the strict sense [2411.10420, 2409.08268].

Source: https://www.emergentmind.com/topics/komar-charges