Special relativity from the dynamical viewpoint

These authors also proposed the label “dynamical” for this viewpoint, which we have adopted Ref. 10.

See, e.g., Ref. 14 or Ref. 15.

Of course there is also no reason a priori to expect that the effects could fit together to produce the Lorentz transformation. Ultimately, it is simply a mathematical fact that some field theories are Lorentz invariant, while none (to the author's knowledge) exist in which effects conspire to cancel rate changes.

Another possibility for the coupling involves one time derivative, e.g., $ϕ(x)ξξ̇$⁠, which is similar to the coupling of an electric potential. A coupling with two time derivatives would be analogous to a gravitational potential.

Derivatives convert the cosines of Eq. (5) into sines, however, the packet built on sines is approximately equal to the packet built on cosines and shifted by $Δx=−π/2k$⁠.

This can be shown using conservation of angular momentum, but this is of course one aspect of Lorentz invariance, since the Lorentz group contains rotations.

Ideas for calculating explicit shape change can be found in Refs. 13 and 8 (the latter proposes numeric calculations of orbiting systems undergoing acceleration). Calculations to lowest order in v/c are also feasible.

The other part of the mass change is actually an increase, because the electron in a lower energy state moves faster and hence has a larger relativistic mass. The virial theorem guarantees that this increase is more than offset by the decrease in field energy.

One can question whether slow carrying is a valid method of synchronization, but if it is not, then there is no clear reason to believe that observers have any useful way to define synchronization (a real possibility in non-Lorentz-invariant scenarios, cf. Sec. IV).

For further discussion of these “perspectival” changes between observers, see Ref. 13.

The fact that formal quantities, especially the metric, may not have their expected operational meaning within the theory, has also been discussed by Brown (Ref. 4) and Brown and Pooley (Ref. 9).

This helps understand the question of conventionality of the definition of simultaneity (see, e.g., Ref. 11). If Lorentz invariance holds then there is a preferred definition of simultaneity and little reason to consider any other; otherwise one will need to consider conventions, but probably no really ideal convention will exist. In no case can a field/wave world be converted back to Newtonian behavior by means of conventions.