[In what follows, I will agree to your request and use L rather than l for length symbol, and further use V rather than v for volts]
Really? Go read my quote from your #321 again. Yes it is what you were arguing. And even more clearly so in your #352:
"And taken to its logical conclusion, this would again imply that breakdown should occur *immediately* upon turning on any field source, regardless of its state of motion, since there will always be *some* vp that will sense E > E_crit. That's obviously false, so again something must be wrong with your argument."
Up to you whether to admit that claim, made twice, is mistaken.
Provided that frame, in which E >= Ecrit
, relates to motion wrt to an E source meeting invariant |E|*|L| >= ~ 104
V/cm.cm (E and L being orthogonal). As I have consistently maintained, two criteria must be met.
Only nominally 'volts', as clearly explained in #356: "there is an invariant |E|*|L| product (E and source characteristic length L being orthogonal) that must be exceeded before breakdown is possible in *any* other inertial frame." Further, in #360: "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)" and later; "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." Notice I expressed it variously as either (V/cm)cm, or V/cm.cm, in order to avoid any confusion that the quantity is really 'volts'.
Re-reading #338, where this scenario began, it's more than clear E and L are to be taken as orthogonal in order to make any sense at all. And as quoted above, that criteria was expressly stated in #356. Forget it all somehow?
Clear by now surely that L is normal
to applied E. And btw, just as clearly, relative velocity (re gamma factor) is also taken as normal to E, hence along the same axis as L. That was also explicit and implicit in #338 and later. Otherwise, frame dependent E coupled to frame dependent duration would make no sense.
It's the most natural frame, but the invariant product (let's express it now as |ExL|, even though it's not a cross product) is all that matters here. Choose any frame you like, so long as E and orthogonal L, measured in that same frame, give a scalar product >= ~ 104
V/cm.cm, which is *not* to be taken as volts.
Huh? The 'possible answer' is directly given in what you quoted above! It's the second criteria, as stated, given first criteria is already met.
How much more explicit can I get!?
Actually, that's |ExL| >= ~ 104
V/cm.cm, in any frame whatsoever, and understood that V/cm.cm does not mean volts.
Congrats - got that bit right.
Because, as discussed at length above, you got the first criteria wrong. The two criteria must be simultaneously met. Which amounts to nothing more than saying an E source must transform by the LT's into an applied E >= Ecrit
in any vp's rest frame, for a minimum period of time in that frame given by HUP. For any given E value in say source rest frame, L minimum is thus set there. And vice versa. Minimum gamma factor for any vp's whizzing through in source rest frame is then set solely by E there according to gamma >= Ecrit
/E. A no brainer combo.
The answer should by now be self-evident. And btw, I take it we agree transverse velocity of a created pair is an overall neutral mass flow, effecting the current along E only insofar as 'relativistic mass' of pair will be very high, making motion along E extremely sluggish as seen in source rest frame. In fact, as per discussion in #360, it implies many created pairs simply whiz right through without ever being collected by E source.
And that error of understanding what L must be referring to invalidates the rest of your critique. Still, there is an issue regardless as per #360. While it's not simply a voltage V, but an an variant product |ExL| >= ~ 104
V/cm.cm, it can be easily enough met. So the real task, once all the basic misunderstandings above are finally cleared away, is to figure out what is really going on and why. So far, no order of magnitude estimate of expected pair flux has been made. Could be typically so small as to go unnoticed. If QFT already has a pat answer making that moot, no-one here has so far offered it.
Assuming no pair creation near high-tension power lines etc., one conjecture might be to suppose that applied E somehow suppresses high gamma factor flux of vp's in source rest frame. Immediate problem with that is for a capacitor configuration, many high gammma vp's would be originating outside of appreciable applied E region, so subsequent reduction to gamma < gammacrit
within E region implies weird electrodynamics indeed.