Physical stress-energy interpretation of the adopted EED flux

Determine whether the field-only Poynting flux used for scalar-longitudinal radiation is the flux associated with a physical stress-energy tensor for extended electrodynamics, and reconcile it with the canonical and Hilbert stress-energy currents, including the sign of the radiated energy.

Background

The paper derives an extended Poynting balance with a positive scalar-field energy density and an outward scalar-longitudinal flux. However, the extended electrodynamics Lagrangian also yields canonical and potentially Hilbert stress-energy constructions, and these need not agree with the adopted field-only balance.

The paper notes that a tensor treatment can predict an opposite sign for the radiated flux of the same currentless anomalous dipole. Consequently, the multipole power formula is established only as the outgoing flux of the selected current, not as a proof that the radiation carries positive physical energy.

References

What is not established. Four things remain open.

— The Bondi Dipole in Full Numerical Relativity: a Self-Accelerating Positive-Negative Mass Binary  (2608.24577 - Shirokov, 25 Aug 2026) in Section VIII, paragraph beginning “What is not established”

The relation among eq:uS, the canonical current, and a Hilbert stress tensor remains unresolved. The tensor treatment of Ref. can give a different radiated-flux sign.

— Scalar-Longitudinal Radiation in Extended Electrodynamics with Multipole Theory and a Compensated Source Model  (2608.19405 - Rentzber, 19 Aug 2026) in Section 7, subsection “Poynting Functional and the Unresolved Energy Sign”

A detailed physical interpretation of the nonsymmetric mixed sector of the energy--momentum tensor, including the relation between energy flux and momentum density in combined Maxwell and $S$-field configurations, requires a separate study and is left for future work.

— Covariant Electrodynamics with a Scalar Degree of Freedom  (2608.26839 - Sautbekov, 27 Aug 2026) in Section Discussion