ner_tags: O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O B-Model I-Model O O O B-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O
tokens: strengths where Proca stresses can be significant. As emphasized above, the model considered in this report is not that of dark matter; it may, however, account for the conjectured dark-matter effects in astrophysical settings even beyond galactic rotation curves. It may also be tempting to explore the influence of the negative pressure typical for the Proca electrodynamics on the cosmological-scale phenomena with a possible impact on the dark-energy scenarios. To have a significant effect, one would need the presence of a tangled magnetic field, with a correlation length exceeding the photon Compton length, filling the entire space. The field would have to be nearly force-free to survive on the cosmological timescales. Appendix A Quasi-steady-state Maxwell–Proca Equations A set of Maxwell–Proca equations reads as (see Jackson 1975 ; Goldhaber Nieto 1971 ) Here, and φ are the vector and scalar potentials, ρ and and j are the space charge and the current density satisfying the standard continuity equationNote that here we use the same notation for charge density as we do in the main text for the mass density. This, however, should not cause confusion because the charge density does not appear in any equations in the main text. The electric field is related to and φ byThe potentials satisfy the Lorentz gaugeThis (Lorentz) gauge now becomes the only possible one: Equation ( 54 ) is a direct consequence of Equations ( 50 )–( 52 ). The scalar and vector potentials become measurable quantities. Consider a system where the temporal variation of the “sources” (currents and charges) is slow in the sense that the characteristic time of this variation, , is long compared to the light-propagation time over the spatial scale of the system, L ,(we use the hat overscript to distinguish the characteristic time from temperature). Then, according to Equation ( 48 ), one has the following order-of-magnitude relation:If one associates with L and some velocity that characterizes the motions in the system, then one can say that E is first order in v/ c . Substituting the estimate ( 56 ) into Equation ( 50 ), one sees that the last term on the right is small compared to the first term on the left by a factor of and constitutes a correction of the second order. Neglecting this second-order term, one obtains the quasistatic version of Equation ( 50 ):Likewise, estimating A from Equation ( 49 ) as B × L and using Equation ( 56 ), we find from Equation ( 53 ) an order-of-magnitude estimate . Then, comparing the first and the second terms in Equation ( 54 ), we find that the second term is second-order small and can be dropped, so that Equation ( 54 ) is reduced toTaking the divergence of Equation ( 57 ), one finds that, to the same (first) order, the charge-continuity equation is reduced toThereby we obtain a set of equations describing quasi-steady-state processes in the Maxwell–Proca electrodynamics. The set consists of Equations ( 48 ) and ( 49 ) and Equations ( 58