Virial Identity: Constraining Energy and Stability

• Virial identity is a fundamental integral constraint that links spatially averaged kinetic and potential energies through scaling symmetries in various physical systems.
• It generalizes the classical virial theorem by incorporating stress tensors, boundary terms, and field contributions, with applications in solitons, black holes, and quantum gases.
• The identity provides a practical tool for diagnosing energy balance and numerical convergence in simulations of multi-scale and non-linear systems.

The virial identity is a fundamental integral constraint derived from scaling or dilation symmetries, appearing broadly in particle mechanics, classical and quantum field theory, statistical mechanics, continuum physics, and general relativity. Its core purpose is to relate spatially averaged kinetic and potential contributions, providing a diagnostic for energy balance, imposing necessary conditions for the existence of nontrivial solutions, and serving as a stringent numerical accuracy check in non-linear systems. The concept generalizes the classical virial theorem of Clausius by incorporating stress tensors, fields, and boundary terms, with numerous applications to solitons, black holes, quantum gases, and astrophysical systems.

1. Historical Genesis and the Classical Virial Theorem

Clausius formulated the original virial theorem for N-particle systems, introducing the virial function $I(t)$ and its second time derivative: $$I(t) = \sum_{i=1}^N m_i |x_i|^2, \quad \frac{d^2 I}{dt^2} = 4T + 2\sum_{i} x_i \cdot F_i,$$ where $T$ is the total kinetic energy and $F_i$ are the forces (Pommaret, 2015). The time-average, under the bounded $I(t)$, yields $\langle 2T + x \cdot F \rangle = 0$, encoding the mean energy balance. In the continuum limit, the theorem relates the trace of the stress tensor and external body forces, generalizing to media with internal stresses and isotropic pressure: $$\frac{d^2I}{dt^2} = 2K + 2\int_\Omega x \cdot f \; dV + 6P V$$ for fluids or gases.

2. Scaling, Dilatation, and the Derrick Argument

Field-theoretic virial identities are often obtained via scaling arguments (the Derrick theorem). For an effective action $$S[q_j(r)] = \int \mathcal{L}(q_j, q_j', q_j'', r) dr$$, a radial scaling $r \to \tilde r = r_i + \lambda (r - r_i)$ induces a variation in the action ($\lambda$ parameter), whose stationarity leads to the general virial identity (Oliveira et al., 2022, Herdeiro et al., 2022, Herdeiro et al., 2021, Pombo et al., 2024): $$\int_{r_i}^\infty \left[ \sum_j \frac{\partial \mathcal{L}}{\partial q_j'} q_j' - \mathcal{L} - \frac{\partial \mathcal{L}}{\partial r}(r - r_i) \right] dr = [\text{boundary contributions}]$$ When second derivatives enter, as in gravitational actions, total derivatives must be separated, and boundary terms—most critically the Gibbons-Hawking-York (GHY) term in GR—must be included for a well-posed identity (Herdeiro et al., 2021).

3. Virial Identities in General Relativity and Gauge Theory

In GR, virial identities play a foundational role in constraining black hole and soliton solutions. The presence of second derivatives in the Einstein-Hilbert action and necessary boundary terms complicate the calculation, but judicious gauge choices—such as the "sigma-m" gauge for spherical symmetry and newly proposed gauges for axial symmetry—can trivialize the gravitational sector so that only matter contributions remain (Oliveira et al., 2022, Pombo et al., 2024, Herdeiro et al., 2022).

For axisymmetric systems, the identity takes the form: $$0 = \int d^3x \sqrt{-g} \left[ 2(T^t_{\;t} - T^r_{\;r} - T^\theta_{\;\theta}) + (T^t_{\;t} - T^\varphi_{\;\varphi}) \mathcal{F}(\omega, m, F_W) \right]$$ where stress-energy components encode energy density, stresses, and angular momentum, and $\mathcal{F}$ quantifies time-rotation interplay. Applications include verifying the existence of hairy Kerr black holes and providing stringent numerical convergence checks (Oliveira et al., 2022).

The formal adjoint of the Spencer operator for the conformal group underpins the virial theorem in gauge field theory, establishing its link with Maxwell, Weyl, and Cosserat equations (Pommaret, 2015). The tracelessness of the stress-energy tensor is a geometric manifestation of the virial identity under conformal symmetry.

4. Generalized Virial Identities: Radial Families and Physical Diagnostics

Generalized virial identities parameterized by a radial exponent $\alpha$ (the "α-family" identity) enable the decomposition of the global constraint into core-vs-tail diagnostics (Lozano-Mayo, 29 Dec 2025). For O(n)-symmetric field configurations: $$\alpha \int_0^\infty d\rho \rho^{\alpha-1} \mathcal{C}_\rho = \int_0^\infty d\rho \rho^\alpha (-\partial_\rho \mathcal{G}_\rho)$$ Choosing $\alpha=1$ recovers Derrick's scaling law, while negative (core) or large positive (tail) $\alpha$ weight different regions radially, distinguish errors or mechanisms (e.g., core singularity vs asymptotic decay). BPS configurations satisfy all virial identities pointwise; for generic solitons and bounces, the error as a function of $\alpha$ diagnoses numerical convergence, and in multi-scale models (Skyrmions, sphalerons) isolates dominant stabilizing forces (Lozano-Mayo, 29 Dec 2025).

5. Virial Theorems in Quantum and Statistical Systems

In quantum statistical mechanics, the virial coefficients $b_n$ of Bose and Fermi gases are related via "virial identities" depending on the trapping potential and dimensionality (Faruk et al., 2016): $$b_n^{\rm Bose} = \frac{1}{n^{\chi+1}}, \quad b_n^{\rm Fermi} = \frac{(-1)^{n-1}}{n^{\chi+1}}$$ where $\chi+1 = 1 + d/2 + d/p$. For 1D harmonic traps ($d=1, p=2$) and 2D free gases, all higher virial coefficients (save the second) match in magnitude and alternate in sign, revealing equivalence in low-dimensional quantum statistics.

In MOND and modified gravity, virial identities provide universal constraints (Milgrom, 2013): $$\sum_p \mathbf{r}_p \cdot \mathbf{F}_p = -\frac{2}{3}(G a_0)^{1/2} \left[(\sum_p m_p)^{3/2} - \sum_p m_p^{3/2}\right]$$ This result is theory-independent within the class of modified-gravity MOND models and underlies the mass-velocity-dispersion relation for galaxies.

6. Virial Identities in Relativistic, Fluid, and Dispersive Systems

The virial identity appears in the context of radiating accretion disks, capturing the balance among potential, kinetic, thermal, and radiation energies (Mach, 2011): $$E_{\rm pot} + 2 E_{\rm kin} + 2 E_{\rm th} + \tilde E_{\rm rad} = 0$$ Similar integral identities arise in water waves, where exact virial theorems express time-averaged equipartition of modified kinetic and potential energy, allowing nonlinear energy balances to be tracked precisely (Alazard et al., 2023).

Virial identities for dispersive PDEs (e.g., Dirac equations) translate dilation symmetry into integral constraints connecting solution regularity and energy flux, essential for deriving smoothing and Strichartz estimates (Cacciafesta, 2011).

Relativistic extensions via Boltzmann moment equations provide general virial relations incorporating cosmic expansion, pressure, redshift, and gravitational potential, unifying self-gravity and cosmological background effects (Javadinezhad et al., 2015).

7. Applications, Physical Interpretation, and Numerical Verification

Virial identities serve as:

• Theoretical constraints for existence and uniqueness (no-go theorems, no-hair theorems in GR)
• Numerical diagnostics (violation quantifies error in soliton and black hole solutions)
• Physical balance laws connecting energy density, pressure, stress, and external fields
• Universal scaling relations in statistical mechanics and galaxy dynamics

In GR, by selecting a suitable gauge, one can isolate matter content, greatly simplifying proofs. In numerical practice, violation of the virial identity provides a robust measure of solver accuracy, with typical residuals at the $10^{-3}$ level for highly converged configurations (Oliveira et al., 2022). The α-family identities allow for spatially resolved error analysis, distinguishing inaccuracies localized in the core or tail.

Summary Table: Representative Forms and Applications

Field/System	Representative Virial Identity	Notable Features
Classical Mechanics	$2T + \sum x_i \cdot F_i = 0$ (time-avg)	Clausius theorem
Scalar Soliton (flat)	$\int [(\nabla \phi)^2 / 3 + U(\phi)] d^3x = 0$	Derrick no-go
General Relativity	Complicated gauge-dependent integral; reduces to matter in special gauge	No-hair, energy-momentum balance
Quantum Gases	$b_n^{\rm Bose} = (-1)^{n-1} b_n^{\rm Fermi}, n\ge3$	Equivalence, sign alternation
MOND Gravity	$$\sum \mathbf{r}_p \cdot \mathbf{F}_p = -\frac{2}{3}(G a_0)^{1/2} [...]$$	DM-independent constraints
Water Waves	$\frac{dI}{dt} = 2V(t) - 2\tilde T(t)$	Equipartition, nonlinear exactness
Dirac Equation	Energy-multiplier: commutator-based identity	Smoothing, dispersive bounds

• "A convenient gauge for virial identities in axial symmetry" (Oliveira et al., 2022)
• "Deconstructing scaling virial identities in General Relativity: spherical symmetry and beyond" (Herdeiro et al., 2022)
• "Generalized Virial Identities: Radial Constraints for Solitons, Instantons, and Bounces" (Lozano-Mayo, 29 Dec 2025)
• "General virial theorem for modified-gravity MOND" (Milgrom, 2013)
• "Virial coefficients from unified statistical thermodynamics..." (Faruk et al., 2016)
• "Virial theorem for radiating accretion discs" (Mach, 2011)
• "Virial identities in relativistic gravity: 1D effective actions and the role of boundary terms" (Herdeiro et al., 2021)
• "Clausius/Cosserat/Maxwell/Weyl Equations: The Virial Theorem Revisited" (Pommaret, 2015)
• "Virial identities across the spacetime" (Pombo et al., 2024)
• "Relativistic virial relation for cosmological structures" (Javadinezhad et al., 2015)
• "Virial identity and dispersive estimates for the n-dimensional Dirac equation" (Cacciafesta, 2011)
• "Virial theorems and equipartition of energy for water-waves" (Alazard et al., 2023)