Summary
A new groupG of Lorentz transformations (LT) in four dimensions, generalized also for Superluminal frames, is introduced and particularly studied in its physical implications. With the help of a « principle of duality »—implied byG—between subluminal and Superluminal frames, the meanings of « inertial frame », « equivalence », « principle of relativity », « covariance » may be correspondingly extended. A biunivocal correspondence exists between bradyonic and tachyonic velocities, which we find to be a particular conformal mapping (inversion). Since the groupG consists of generic rotations in space-time, it includes,e.g., also the total-inversion operation (PT). Moreover (for a non « charge »-free universe), it is shown that our generalized special relativity requires covariance underCPT. A « tachyonization principle » is formulated, on the basis of which relativistic physical laws (of mechanics and electrodynamics, at least) can be easily extended to tachyons. Many simple applications are performed of the generalized LT’s (velocity composition law, comparison of the length and time units, Doppler effect, refraction index, …), either useful to clarify our problem or interesting in astrophysics.
Riassunto
Un nuovo gruppoG di trasformazioni di Lorentz (LT) in quattro dimensioni, generalizzato anche per sistemi di riferimento Superluminali, è introdotto e studiato particolarmente nelle sue implicazioni fisiche. Con l’aiuto di un « principio di dualità » — implicato daG — tra sistemi subluminali e Superluminali, è possibile estendere il significato di « riferimento inerziale », « equivalenza », « principio di relatività », « covarianza ». Tra velocità bradioniche e tachioniche esiste una corrispondenza biunivoca, che risulta essere una particolare corrispondenza conforme (inversione). Poiché il gruppoG consiste di rotazioni generiche nello spazio-tempo, esso include per esempio anche l’operazione di inversione totale (PT). Inoltre (per un universo con « cariche »), si mostra che la nostra relatività ristretta generalizzata richiede la covarianza perCPT. Si formula un « principio di tachionizzazione », in base al quale le leggi fisiche relativistiche (quelle almeno della meccanica e dell’elettrodinamica) possono essere facilmente estese al caso dei tachioni. Si applicano le LT generalizzate ad alcuni semplici casi (legge di composizione delle velocità, confronto di unità di tempo e di lunghezza, effetto Doppler, indice di rifrazione, …) utili per chiarire il nostro problema o di interesse in astrofisica.
Реэюме
Вводится новая группаG преобраэований Лорентца в четырех иэмерениях, обобшенная также для сверхсветяшихся систем отсчета. Исследуются фиэические применения новой группыG. Испольэуя «принцип дуальности» между субсветяшимися и сверхсветяшимися системами отсчета, может быть расщирен фиэический смысл понятий « инерциальной системы отсчета», «зквивалентности», « принципа относительности » и « ковариантности ». Сушествует соответствие между скоростями брадионов и тахионов, которое получается как реэультат конкретного конформного отображения (инверсии). Так как группаG содержит врашения в пространстве и времени, то она включает, например, также операцию полной инверсии(РТ). В случае эарядовой инверсии наща обобшенная специальная теория относительности требует ковариантности относительноОРТ. Формулируется «принцип тахиониэации», на основе которого релятивистские фиэические эаконы (по крайней мере, механики и злектродинамики) могут быть легко обобшены для тахионов. Рассмотрено много простых применений обобшенных преобраэований Лорентца (эакон сложения скоростей, сравнение единиц длины и времени, зффект Допплера, козффициент преломления и т.д.), полеэных либо для прояснения нащей проблемы, либо интересных в астрофиэике.
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References
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After the completion of paper I, we became aware of the existence of papers (2–4), which approached our problem too. Criticizing ref. (2) is the content of ref. (3). Paper (4), byAntippa, puts forward a theory yielding the same results as Parker’s (7), both works being confined tobidimensional space-time. The original features of ref. (4),i.e. introducing unidirectionality in time for bradyons and in space for tachyons, seems extraneous to our own philosophy.
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D. Weingarten: preprint NBI-HE-71-3 (N. Bohr Institute, Copenhagen, Dec. 1971).
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An erratum to this article is available at http://dx.doi.org/10.1007/BF02785527.
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Mignani, R., Recami, E. Generalized Lorentz transformations in four dimensions and superluminal objects. Nuov Cim A 14, 169–189 (1973). https://doi.org/10.1007/BF02734611
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DOI: https://doi.org/10.1007/BF02734611