Generalized Komar Energy

• Generalized Komar energy is a covariantly defined, quasi-local gravitational charge that extends classical Komar integrals to dynamic and matter-coupled spacetimes.
• It is constructed using generalized methods, including almost-Killing vectors and the Noether–Wald formalism, to derive conserved currents in non-stationary and higher-dimensional settings.
• The framework bridges spacetime geometry, thermodynamics, and holographic principles, offering insights into black hole thermodynamics and universal energy properties.

Generalized Komar energy is a covariantly defined, quasi-local gravitational charge constructed via a generalization of the Komar integral, which, in its classical form, associates conserved charges to spacetime symmetries through Killing vectors. Its modern extensions accommodate dynamical, non-stationary, and matter-coupled settings, and serve as a fundamental tool linking spacetime geometry, thermodynamics, and covariant notions of energy in general relativity and beyond.

1. Foundational Definition and Classical Context

Generalized Komar energy extends the original Komar formulation—where the mass or angular momentum is encoded as a surface integral involving antisymmetrized derivatives of a Killing vector—to non-static settings and more general (possibly matter-coupled) gravitational theories. For a stationary, asymptotically flat spacetime with a Killing vector $\xi^\mu$, the Komar mass is classically given by

$$M_K = -\frac{1}{8\pi} \int_{S_\infty} \nabla^\mu \xi^\nu dS_{\mu\nu}.$$

Generalization to dynamical and non-Killing settings replaces $\xi^\mu$ with an arbitrary evolution vector field, preserving the covariant structure of the integral. For such cases, the antisymmetrized derivative

$$S^{\mu\nu}(\xi) = \frac{1}{2} (\xi^{\nu;\mu} - \xi^{\mu;\nu})$$

defines a conserved current $J^\mu(\xi) = S^{\mu\nu}_{;\nu}$ which, via Stokes’ theorem, yields a boundary charge even in the absence of a genuine symmetry (Wang et al., 7 Sep 2025, Feng, 2018, Peng et al., 2019).

The generalized construction allows $\xi^\mu$ to be, for example, the lapse function times the future-directed unit normal to the constant-$t$ hypersurfaces, applicable to fully dynamical spacetimes, including those involved in numerical relativity (Wang et al., 7 Sep 2025). This covariant approach recovers the ADM mass under mild asymptotic conditions: $$M^{\text{ADM}} = E(\xi) \text{ whenever } {}^{(3)}G_{ij} = o(r^{-3}),$$ where ${}^{(3)}G_{ij}$ is the spatial Einstein tensor, N is the outward normal, and $E(\xi)$ is the generalized Komar energy (Wang et al., 7 Sep 2025).

2. Generalizations in the Presence of Matter and Beyond Stationarity

In the presence of matter or for Lagrangians beyond vacuum gravity, the naive Komar charge loses its on-shell closure property due to additional terms in the action involving non-trivial matter dynamics. The Noether–Wald formalism provides a systematic route to constructing charges in such precisely gauge- and diffeomorphism-invariant theories, leading to the generalized charge (Ortín et al., 15 Nov 2024, Ballesteros et al., 12 Sep 2024): $K[k] = - Q[k] + \omega_k$ where $Q[k]$ is the standard Noether–Wald charge 2-form and $\omega_k$ is a compensating boundary term constructed such that $dK[k] = 0$ holds on-shell (i.e., enforcing the field equations). The addition of $\omega_k$ accounts for the failure of strict invariance under the symmetry generated by $k$, especially in models with non-trivial matter couplings, higher-derivative corrections, or scalar/gauge fields (Ballesteros et al., 12 Sep 2024, Cerdeira et al., 16 Jun 2025).

In Kaluza–Klein compactifications, Komar charges require further modification by inclusion of conserved forms associated with higher-form symmetries. For example, the presence of a harmonic one-form $\mathfrak{h}$ associated with the compact dimension necessitates augmenting the Komar charge by a term restoring electric-magnetic duality and eliminating spurious scalar charges: $$K_{1/2}[\hat{l}]_\mathfrak{h} = \frac{k_\infty}{16\pi G_5}\left\{-\star_E d\ell_E + \frac{1}{2} P_{E, \ell} k^3 \star_E F_E - \frac{1}{2} \tilde{P}_{E, \ell} F_E\right\} \wedge \mathfrak{h},$$ yielding a surface integral that matches the 4D physical mass (Barbagallo et al., 18 Jun 2025).

3. Thermodynamic Identities and Smarr Formulas

Generalized Komar energy underpins horizon thermodynamics via precise geometric-thermodynamic identities. For the Kerr–Newman black hole, the Komar charge evaluated on the null generator ($\chi^\mu = \xi^{(t)} + \Omega_H \xi^{(\phi)}$) yields

$K_\chi = 2 S T,$

where $S$ is the Bekenstein–Hawking entropy and $T$ is the Hawking temperature (Modak et al., 2010). This relation is the local manifestation of the nonlocal generalized Smarr formula,

$$M - 2\Omega_H J - Q\Phi = \frac{\kappa A_{\text{Horizon}}}{4\pi},$$

demonstrating the dual role of Komar energy as both a local horizon property and a bridge to asymptotic conserved charges.

Similar structures arise in black holes with noncommutative geometry (Banerjee et al., 2010, Larranaga et al., 2012, Gangopadhyay, 2012): Komar energy relations and Smarr formulas are deformed by quantum-corrective, non-polynomial terms, leading to phenomena such as nonvanishing Komar energy at extremality ($T_H = 0$). The deformation generically appears as

$$E = 2 S T_H + \mathcal{O}(\sqrt{\theta} e^{-M^2/\theta}),$$

with $\theta$ the noncommutative parameter (Banerjee et al., 2010).

For solutions parameterized by additional charges (dipole, quadrupole moments, or in non-linear electrodynamics), the Komar energy and its products reflect intricate dependencies on horizon parameters, and mass-independent (universal) combinations can sometimes be found (Pradhan, 2016).

4. Covariant Charges in Non-Symmetric and Higher-Dimensional Theories

Generalized Komar charges do not require global Killing symmetries. Using the formalism of conserved currents built from almost-Killing vectors, conformal Killing vectors, or more general vector fields satisfying second-order differential constraints, a spectrum of globally defined, divergence-free currents and corresponding charges is available (Feng, 2018, Peng et al., 2019). Such constructions allow for charge definitions that probe departures from symmetry—quantifying energy, angular momentum, or more general invariants even in radiating or time-dependent scenarios.

In higher dimensions, the Komar integral and ADM mass can be related via Hamiltonian and curvature-based surface integrals. However, for Kaluza–Klein (KK) compactifications, care is required: Hamiltonian energy, ADM mass, and Komar integrals can yield different values unless augmented to account for compactification-induced charges and boundary conditions (Barzegar et al., 2017). Duality invariance and regularity conditions are essential for proper account of lower-dimensional physical charges (Barbagallo et al., 18 Jun 2025).

In supersymmetric extensions, generalized Komar energy incorporates not only bosonic but also fermionic contributions, as encapsulated in superforms built from Killing supervectors and their spinor partners. This construction ensures closure of the conserved form and its applicability within the first law for supergravity black holes (Bandos et al., 24 Dec 2024).

5. Physical and Thermodynamic Fluctuations, and Holographic Relations

In cosmological contexts, generalized Komar energy quantifies the total active gravitational energy enclosed by a horizon. The statistical fluctuations of both horizon and volume (Komar) energies are calculated thermodynamically, leading to the result

$$\frac{\sigma_H^2}{\langle E_H\rangle^2} = 1/N_\text{surf} = 1/N_\text{bulk} = \frac{\sigma_V^2}{\langle E_V\rangle^2},$$

where $N_\text{surf}$ is the number of degrees of freedom on the horizon and $N_\text{bulk}$ those within the volume. This equality, particularly realized in the de Sitter limit, expresses a holographic connection: bulk and boundary energy fluctuations are equated at equilibrium (Namboothiri et al., 19 Aug 2024). The time evolution of fluctuations generally shows a pronounced spike during transitions to accelerated expansion, before stabilizing.

6. Algorithmic Construction and Universal Properties

A universal algorithm for constructing generalized Komar charges in arbitrary exactly gauge- and diffeomorphism-invariant theories is established (Ortín et al., 15 Nov 2024, Cerdeira et al., 16 Jun 2025). The key steps include:

• Varying the action under the relevant symmetry to extract the Noether–Wald charge $Q[k]$.
• Identifying compensating terms $\omega_k$ such that the resulting (d–2)-form $K[k] = - Q[k] + \omega_k$ is closed on-shell.
• Using scaling (e.g., global rescaling) symmetries and the field-theoretic Euler theorem to show that the on-shell Lagrangian can always be written as a total derivative, yielding a unique, solution-independent potential whose contraction gives $\omega_k$ (Cerdeira et al., 16 Jun 2025).

The addition of on-shell closed forms is physically required in scenarios such as Kaluza–Klein reductions, ensuring that the charges computed from higher-dimensional theories match the correct four-dimensional physical interpretations and respect duality invariances (Barbagallo et al., 18 Jun 2025).

7. Interpretations, Applications, and Special Cases

Generalized Komar energy provides a unifying framework for defining gravitational energy in a spectrum of physically relevant scenarios:

• In black hole spacetimes, it links geometric quantities at the horizon to thermodynamic quantities, supplying local versions of global Smarr-type relations (Modak et al., 2010, Cabrera-Munguia et al., 2014).
• It enables charge definitions in regular black holes and spacetimes with non-linear electrodynamics, diagnosing mass-independent product universality or lack thereof (Pradhan, 2016).
• For dynamical, possibly radiating spacetimes, it ensures consistency with Hamiltonian (ADM) definitions of mass under weak asymptotic decay conditions, hence aligning local and global descriptions of gravitational energy (Wang et al., 7 Sep 2025).
• In theories with nontrivial matter couplings or gauge fields, it manages the failure of strict invariance by including necessary compensating charges and terms, extending the notion of energy conservation and enabling proofs or circumventions of nonexistence theorems for gravitational solitons and boson stars (Ballesteros et al., 12 Sep 2024).

In summary, the generalized Komar energy framework systematically extends the concept of gravitational energy from stationary, vacuum contexts to encompass covariant, local, and thermodynamic properties of dynamical, matter-coupled, higher-dimensional, and quantum-corrected spacetimes. It serves as the geometric backbone for black hole thermodynamics, bridges local and global charges, and accommodates the rich structure of modern gravitational theories.