Reply to Itin, Obukhov and Hehl paper "An Electric Charge has no Screw Sense - A Comment on the Twist-Free Formulation of Electrodynamics by da Rocha & Rodrigues"
aa r X i v : . [ m a t h - ph ] D ec Reply to Itin, Obukhov and Hehl paper “AnElectric Charge has no Screw Sense - A Commenton the Twist-Free Formulation ofElectrodynamics by da Rocha & Rodrigues”
Rold˜ao da Rocha (1) and Waldyr A. Rodrigues Jr. (2)(1)Centro de Matem´atica, Computa¸c˜ao e Cogni¸c˜aoUniversidade Federal do ABC, 09210-170, Santo Andr´e, SP, Brazil [email protected] (2)Institute of Mathematics, Statistics and Scientific ComputationIMECC-UNICAMP CP 606513083-859 Campinas, SP, Brazil [email protected] or [email protected] Abstract
In this note we briefly comment a paper by Itin, Obukhov and Hehlcriticising our previous paper ([3]). We show that all remarks by ourcritics are ill conceived or irrelevant to our approach and moreover weprovide some pertinent new comments to their critical paper, with theaim to clarify even more our view on the subject.
Authors of [2] said that it is a reaction to [3], a paper of ours which gives aClifford bundle approach to classical electrodynamics. It is our opinion that ourpaper deals appropriately with all their comments (some of them unfortunatelynot appropriate), but here it turns out necessary to repeat at least a crucial remark of [3] and make some additional few comments. The first and moreimportant is that what we show in our paper is that in an oriented
Lorentzianspacetime we can formulate classical electrodynamics using only pair form fields,viewed as sections of an appropriate Clifford bundle (thus dispensing the use ofimpair form fields ) in a coherent way using good (but eventually not so wellknown) Mathematics. This is due to the fact quoted in our paper and firstspelled by de Rham [4] (an author often quoted, often not read) that: We use the the term impair forms (as originally used by de Rham) instead the term twistedforms to avoid any sequence of words that could seem not adequate due to one of the meaningsof the word twist in English. However, we insist here, our formulation of electrodynamics inan oriented spacetime does not need the use of twisted forms, but does not claim that thoseobjects cannot be used. Si la vari´et´e V est orient´e, c’ est-`a-dire si elle est orientable et si l”on a choisi uneorientation ε , `a toute forme impaire α est associ´ee une forme paire εα . Par la suite, dans lecas d’une variet´e orientable, en choissant une foi pour toutes une orientation, il serai possibled’´eviter l’emploi des formes impaires. Mais pour les vari´et´es non orientables, ce concept estr´eellement utile et naturel . ”Using only pair forms of course, does not mean – contrary to what our criticsthink and spell – that the resulting differential equations of our theory are notinvariant under arbitrary coordinate transformations. This is so because alldifferential equations in our approach are writing intrinsically. However, whenusing pair forms the sign of a charge resulting from the evaluation of the integralof a pair current 3-form J depends of course, on the handiness of the coordinatechart using for performing the evaluation. The relevant question is: does itimply any contradiction with observed phenomena ? As clearly shown in ourpaper through a very carefully analysis using good mathematics the answer is no . However, our critics are not happy with our analysis and continue to insist adnauseam that charge does not have a screw sense and as such the electromagneticcurrent must be an impair (twisted) 3-form field J because they “may want toput charge on a (non-orientable) M¨obius strip . . . ”. Well, suppose for a whilethat the M¨obius strip M ¨ o is sitting (embedded) on R (the rest space of aninertial frame). To eventually calculate its charge we need to start with a 2-formsurface charge density J defined on R . Now had our critics read our Remark 13(see also [1]) they could be recalled of the fact that being J a pair or an impair2-form we cannot define its integral over the Mobi¨us strip . So, we concludethat it is only in fiction that someone can think in putting a real physical chargedistribution (made of elementary charge carriers) on a M¨obius strip, and leavingaside this physical impossibility we cannot see any necessity for the use of impairforms. Our critics said that our statement that the Clifford bundle works onlywith pair forms and could not apply to Physics if there is real need for the use ofimpair forms is unsubstantied. They justify their assertion quoting Demers [5]which deals with a non associative ‘Clifford like’ algebra structure involving pairand impair forms. This structure has nothing to do with the Clifford algebraused (as fibers) in our Clifford bundle, which is an associative algebra, a propertythat makes that formalism a very powerful computational tool. We recall alsothat as detailed in our paper our formalism which writes ‘Maxwell equation’(no misprint here) with pair differential forms can be split in two different ways.The first one results in two equations using only pair forms and the second oneresults in an equation using pair forms and another one using impair forms.However to do that it is crucial to understand that there exists two differentHodge star operators, one pair and one impair.
They are very distinct objects,often confused (as we explained in detail in our paper). We recall that to havethat fact in mind is important because without the explicit introduction of theimpair Hodge dual operator the claim of our critics (that do not even mentionthat object) that Maxwell equation in the Clifford bundle splits in an equation This could be done only if M ¨ o is sitting on M ¨ o × R , which is not the case in the realphysical world. F . Well, thisis simply not true. In our approach it is clear that F is taken as a physicalfield represented by a 2-form field living in Minkowski spacetime and satisfyingMaxwell equation, where a current 1-form J (formed from the charged mattercarriers) acts as source of F . We next argued that F carries energy-momentumand that the total energy-momentum tensor of the F field plus the chargedmatter field is conserved. Under those well defined conditions we proved thatthe coupling of F with J must be given by the Lorentz force law, which mustthen establish the operational way in which those objects must be used whenone is doing Physics. It is in this sense that we said that such law need notbe postulated in classical electrodynamics, and we are sure that any attentivereader of our paper will understand what we said and what we proved. Acomment is also needed, concerning the formulation of the (interesting) metric-free approach to electrodynamics in ‘spacetime’ defended by our critics. Weleave clear in our paper that the spacetime splitting used in their approach makestheir spacetime manifold structure closely to the Newtonian spacetime structure.Here we remind our readers of a journey of our critics to a strange land (whichwe did not visit yet and hope not to visit ever). Indeed, in briefly reviewingthe metric free approach they said that “we make minimal assumptions aboutspacetime, just a 4-dimensional manifold that we decompose into 1+3 by meansof an arbitrary normalized 4d vector n .” Well, normalized with respect to whichmetric if there is no one in the metric-free approach? Finally, we recall that ourapproach to the vector calculus description to Maxwell theory leaves clear onceagain that we can provide a meaningful mathematical and physical descriptionof facts using only appropriate pair Clifford fields doing the role of (polar) vectorfields. We say even more here, to those people that are satisfied with the Gibbsand Heaviside approach to vector calculus with their polar and axial vectors weleave the following issue (that obviously did not exist in our approach). Usually i , j , k are taken as a (Euclidean) orthonormal basis of polar vectors in R (viewedas a vector space). Next, in vector calculus it is introduced the vector productof two polar vectors a and b denoted a × b which is said to be an axial vector.Next we see printed everywhere the equations i × j = k , j × k = i , k × i = j .Now, do i , j , k become also axial by virtue of those equations? References [1] T. Frankel,
The Geometry of Physics , p.116, Cambridge Univ. Press, Cam-bridge, 1997.[2] Y. Itin., Y. N. Obukhov and F. W. Hehl,
An Electric Charge hasno Screw Sense — A Comment on the Twist-free Formulation of Elec- rodynamics by da Rocha & Rodrigues (submitted to Ann. der Phys.
Berlin). [arXiv:0911.5175] [3] R. da Rocha and W. A. Rodrigues Jr.,
Pair and Impair, Even and OddForm Fields, and Electromagnetism, (to appear in
Ann. Phys .(Berlin)) [arXiv:0811.1713] [4] G. de Rham,
Vari´et´es Differentiables. Formes, Courants, Formes Har-moniques , Actualit´es Sci. Ind.1222