In the article titled “Relativistic Localizing Processes Bespeak an Inevitable Projective Geometry of Spacetime” [1], there are minor errors in three formulas, a somehow inappropriate but not very embarrassing notation, and some extremely important missing information that are corrected and indicated as follows.
Throughout the paper, a better notation could be used as follows for the emission coordinates of the localized events . More precisely, the emission coordinates of events define points in the localization grids , e.g., , unlike all the other emission coordinates intervening in the formulas of the paper and which define points in the positioning grids . For more clarity, another notation could have been used for the points in like, for example, taking the letter instead of and thus rather noting all along . Therefore, especially according to Definition 2 in Section titled “Consistency between the Positioning and Localizing Protocols: Identification,” the consistency property is satisfied iff for all , , , , or and where .
In Section 5.4 titled “The Projective Frame, the Time Stamps Correspondence, and the Consistency,” formulas (18), (19), and (22) must be corrected as follows. Formula (18) should be corrected to readwhere , and formula (19) should be corrected to readwhere and . Formula defines a system of equations giving univocally and as fractions (ratios) of two affine functions in and . We get similar formulas associated with the other three primary events , , and . Also, formula (22) should be corrected in a similar manner to formula .
At the end of Section titled “The Local Projective Structure,” Theorem 22 on consistency (‘Consistency Theorem’ for RLSs in (3+1)-dimensional spacetime) must absolutely be moved and, actually, inserted just before formula (20) which is then justified as indicated below. And, moreover, its proof must be completed as follows for greater clarity because its presentation is far too concise. More precisely, at the end of the third paragraph of Section 5.4 titled “The Projective Frame, the Time Stamps Correspondence, and the Consistency,” and just before the sentence “Now, besides, we have necessarily the relations:” which precedes formula (20), the following text must be inserted (and then, following numbering, Theorem 22 becomes Theorem 19):
Theorem 19 (‘-dimensional consistency’). The localization and the positioning protocols or systems in a -dimensional spacetime are consistent.
Proof. It is obvious, because (1) the RLS in the (3+1)-dimensional case has a causal structure which can be decomposed in four causal substructures, each equivalent to the one given for the RLS in the (2+1)-dimensional case, and (2) we need only the “angles” to localize in each of these subsystems of localization. More precisely, each event has only four degrees of freedom. Hence, only four angles in the set of angles and are independent. We can therefore, in particular, consider the four angles as entirely and univocally defined by the four angles only. Besides, the RLS in the (3+1)-dimensional case has a causal structure which can be decomposed in four causal substructures, each equivalent to the one given for the RLS in the (2+1)-dimensional case. And we need only the angles to localize in each of these subsystems of localization. More precisely, for instance, at , we have the following projection of projective frames from to : , , , and . And we have similar projections at primary events , , and . Then, considering the (2+1)-dimensional RLS case with the emitters , , , and we deduce, from the consistency Theorem 11 in the (2+1)-dimensional case, the consistency for the first three emission coordinates . And the same can be deduced for the (2+1)-dimensional RLS with the emitters , , , and , hence the consistency for . From these two cases, we deduce the consistency for the four emission coordinates .”
Then, just after formula (20) the following text must be inserted:
“Formula (20) is well justified by the previous -dimensional consistency Theorem 19 in the (3+1)-dimensional case. Indeed, in the same way as for the eight angles and , only four of the eight emission coordinates involved in formula (20) are independent. In addition, there are at most only two attribution possibilities for the emission coordinates of event : or . And so we can say that is entirely and univocally determined by . We therefore generally have two points assigned to in the localization grid . However, according to the -dimensional consistency Theorem 19 we can claim that the worldlines of the four main emitters and the ancillary emitter are not duplicated in the localization grid; i.e., to an event of one of these five worldlines corresponds one and only one point in the localization grid. Therefore, in the localization grid, the two points and are in the intersection of the four past null cones of the four primary events. However, this intersection contains only one point at most and therefore consistency implies necessarily that and therefore equalities (20) above. ”
The author apologizes to the readers for these errors and lack of information and explanation.