On the energy balance of Newtonian Gravitation

Technical Summary: On the Energy Balance of Newtonian Gravitation

Problem Statement
The paper addresses the long-standing ambiguity in defining the energy density and energy flux of the Newtonian gravitational field. While several expressions have been proposed—most notably those by Maxwell, Bondi, and Ohanian—they yield different local distributions of energy and energy current, despite yielding the same total gravitational energy when integrated over all space under standard boundary conditions at infinity. The authors argue that the traditional reliance on boundary conditions at spatial infinity to resolve this ambiguity is physically unsatisfactory. Instead, the problem requires a local analysis using differential energy balances and boundary conditions on energy fluxes, treating the gravitational field within a continuum mechanical framework.

Methodology
The authors employ a continuum-mechanical approach, drawing on concepts from fluid mechanics and non-equilibrium thermodynamics. Key methodological elements include:

• Reference Frames: A rigorous distinction is made between local (Eulerian) balances, observed in a fixed laboratory frame, and substantial (Lagrangian) balances, observed in a frame comoving with the fluid mass.
• Classical Holography: The analysis utilizes the "classical holographic property" of the gravitational field, where the bulk gravitational force density ($\rho \mathbf{g}$) can be expressed as the divergence of a gravitational pressure tensor ($\nabla \cdot \mathbf{P}_g$). This allows the conversion of bulk force terms into surface traction terms.
• Thermodynamic Constraints: The study applies the Second Law of Thermodynamics to ideal, dissipation-free continua. By analyzing the entropy production and using divergence separation techniques, the authors derive constraints on the energy balance equations.
• Off-Shell vs. On-Shell Analysis: The paper distinguishes between "off-shell" relations (mathematical identities valid without assuming the field equations) and "on-shell" relations (where the Poisson equation $\Delta \phi = \rho$ holds).

Key Contributions and Results

1. Equivalence of Energy Balances:
   The paper demonstrates that the energy balances associated with the Maxwell ($a=1$), Ohanian ($a=0$), and Bondi ($a=1/2$) energy densities are mathematically equivalent. They differ only by the addition of a sourceless conservation law, termed the Green balance. This balance is derived from the time derivative of the Green identity ($\nabla \cdot (\phi \nabla \phi)$) and represents a "gauge-like" freedom in the definition of the energy balance. Consequently, the differences between these formulations are not physical but rather a redistribution of energy between density and current terms without affecting the source term (power density).

2. The Green Balance:
   The authors identify a specific, observer-dependent, sourceless balance (the Green balance) that governs the transformation between the different energy formulations. They show that any arbitrary energy balance parameterized by $a$ can be expressed as a linear combination of the Ohanian balance and the Green balance. This clarifies that the ambiguity in energy density is a consequence of adding a conserved quantity that vanishes upon integration over space.

3. Thermodynamic Distinction of the Ohanian Form:
   While the balances are equivalent on-shell, the paper argues that the Ohanian energy density ($\varepsilon_0 = \rho \phi + \frac{1}{2}g^2$) is physically distinguished from a thermodynamic perspective.

   • When the Ohanian density is used as the starting point for a thermodynamic derivation (specifically, separating the divergence of the energy current from the source term), the Second Law of Thermodynamics naturally forces the system "on-shell," deriving the Poisson equation ($\Delta \phi = \rho$) as a condition for zero dissipation.
   • In contrast, the Maxwell and Bondi densities do not yield the field equations directly through this thermodynamic procedure; they require the field equations to be assumed a priori.

4. Classical Holography and Source Terms:
   The study confirms that the source terms in the energy balances—whether expressed as the power of the bulk force ($\mathbf{f} \cdot \mathbf{v}$) or the power of the surface traction (via the gravitational pressure tensor)—are equivalent due to the classical holographic property. This equivalence holds locally and is independent of specific boundary conditions at infinity.

5. Lagrangian Field Theory Connection:
   The paper notes that the Ohanian energy density is the negative of the standard gravitational Lagrangian density used in classical field theory. The authors show that only the Ohanian form, when used as a Lagrangian, correctly reproduces the Poisson equation via the Euler-Lagrange equations without requiring ad-hoc assumptions about the matter density being fixed.

Significance and Claims
The paper claims that the ambiguity in Newtonian gravitational energy is not a fundamental physical indeterminacy but a mathematical redundancy governed by the Green balance. The primary significance of the work lies in:

• Localizing the Problem: Shifting the analysis from global boundary conditions to local differential balances and flux conditions.
• Selecting a Preferred Form: Identifying the Ohanian energy density as the unique form that is consistent with both the Second Law of Thermodynamics (acting as a gauge-fixing principle) and the variational principles of classical field theory.
• Unifying Framework: Providing a unified continuum-mechanical framework that reconciles the different historical formulations (Maxwell, Bondi, Ohanian) by showing they are merely different representations of the same underlying physical reality, connected by a sourceless conservation law.

The authors conclude that while the different energy balances are physically equivalent in their integrated results, the Ohanian formulation is "distinguished" because it naturally generates the field equations and satisfies thermodynamic requirements without additional constraints, whereas the other forms rely on the field equations being imposed externally.