That's because you are not saying the same thing as me at all. I accept you think we are just talking past each other, but not so. Your criteria is simply E >= Ecrit
in source rest frame - period. The scalar invariant you express later in #357 follows from that imposed condition. Whether source of E there is a micron or a million miles long is irrelevant in such a view. Your statement further down "This includes any effect of "minimum duration", as I said before." is true only in the limited sense that, with the source rest frame E >= Ecrit
imposed, LT's naturally determine a frame dependent vp 'duration' that will be
greater seen in another frame, as will E there - offset by a higher vp energy (thus inertia) seen there. That invariance recipe seems right because there is no 'moving observer causes physics' paradox. But it fails to consider the pov from a vp pair that cares only that E >= E crit
for a minimum HUP time span in it's own rest frame, and can't care less what the source rest frame sees. [And btw, it's still ok to accept my request to apply your position to the rotating hoops (or annulus pair) capacitor scenario I gave in #318 (oh yeah, that Xmas present can still be yours).]
I am claiming, based on example in #338 and codified in #356 as |E|*|l| >= 'volts'min
(even though it is not really volts per se), something quite different. That perspective that matters is that of vp's in their rest frame. Which in turn means that a source must have a minimal product of apllied E *and* length normal to E, measured in any given frame, before, seen in any
other inertial frame, a 'rest' vp pair will receive a minimal impulse capable of boosting to real status. And then only if E >= Ecrit
in that vp 'rest' frame, or alternately where gamma factor of any vp pair passing through in source rest frame yields the equivalent. This criteria is very different from demanding E >=Ecrit
in source rest frame. On that basis it remains the case your reductio ad absurdum argument that immediately upon switching on any source of E breakdown should occur is wrong. Putting it more concretely, from #338 example, a capacitor must roughly have M = |E|*|l|>= 104
(v/cm)cm before a vp pair passing through at any relative gamma factor whatsoever can be elevated to real pair status.
Our criteria are thus fundamentally different - if nothing else on an elementary dimensional analysis level. For you, the source E 'does all the energy pumping'. For me, it is often the inherent KE of relative lateral motion (vacuum vp spectrum alone can provide that) that mostly 'does the pumping' - applied E has more the role of catalyst. Yes the E source discharges when breakdown current flows, but the source power drain per vp-pair->rp-pair creation is vastly different depending on relative motion of source. From my outlook there is a transverse energy budget that cannot be ignored. In past entries I have argued that transverse energy is a result of KE energy pumped into gross motion of the source making it move relative to the underlying local LET rest frame. But some more thought and it becomes obvious there are relevant transverse motions omnipresent and inherent in vacuum as vp 'sea'.
But this leads up to my new outlook. A logical conclusion from M = |E|*|l|>= ~ 104
(v/cm)cm is that there must be some breakdown occurring just having a high voltage structure sitting around with such a perfectly achievable parameter mix. Why? Courtesy of the vacuum. We all agree vp spectrum is frame invariant. This immediately requires that in any frame there must be a finite ultra-relativistic random flux component of vp's, and some fraction will be exceeding breakdown criteria merely by passing between the plates of a suitably dimensioned and charged capacitor in the right directions. Some will strike the plates, some will simply pass through as newly created ultra-relativistic real pairs - subsequently smashing into say air molecules, or recombining to produce a presumably faint but finite ultra-high energy gamma-ray flux. The latter in particular seems disturbing because there is no evident energy drain from the E source involved at all. The vacuum itself seems to provide all the 'oomph' needed there. My own 'magnetic suppression' counter-argument in #356 cammot be germaine I think because seen in source rest frame, a high gamma factor vp passing through and elevated to real status simply executes an ever so slightly parabolic path; that's all.
Bizarre as that seems, imo above is the logical conclusion if
the vacuum vp picture we have been working from is true. Somehow I doubt gamma rays emanating from certain electrostatically charged structures would have gone undetected till now, so not rushing to patent a 'free-energy-from-vacuum' device. After all the vacuum is generally believed to have either precisely or next to zero real energy density - despite the notorious ~ 10120
order of magnitude problem.
Upshot is my new perspective renders prior arguing for LET over SR on vacuum breakdown criteria sort of moot. If there is a top notch QFT expert here that can knock the above on it's head - please step up now. Otherwise, maybe someone can pick the above matter up, wrap it a bit differently, and run it under their own banner on another thread. So be it. I'd rather end this windy-twisty saga on that note.