G. D. Maccarrone

On the shape of tachyons

SummaryWe study some aspects of the experimental behaviour of tachyons, in particular by finding out their « apparent » shape. A Superluminal particle, which in its own rest frame is spherical or ellipsoidal (and with an infinite lifetime), would « appear » to a laboratory frame as occupying the whole region of space bound by a double cone and a twosheeted hyperboloid. Such a structure (the tachyon « shape ») rigidly travels with the speed of the tachyon. However, if the Superluminal particle has a finite lifetimein its rest frame, then in the laboratory frame it gets afinite space extension. As a by-product, we are able to interpret physically the imaginary units entering—as is well known—the transverse co-ordinates in the Superluminal Lorentz transformations. The various particular or limiting cases of the tachyon shape are thoroughly considered. Finally, some brief considerations concerning possible experiments to look for tachyons are added.RiassuntoSi studiano alcuni aspetti del comportamento dei taohioni, in particolare trovando quale « apparirebbe » la loro forma. Una particella Superluminale, che sia sferica o ellissoidale (e con vita di durata finita) nel proprio riferimento a riposo, a un osservatore nel laboratorio sembrerebbe occupare Tintera regione di spazio limitata da un doppio cono e da un iperboloide a due falde. Tale struttura (la « forma » del tachione) viaggerà rigidamente con la velocità del tachione. Si noti però che, se la particella Superluminale ha una vita finita (nel suo riferimento a riposo), allora nel laboratorio essa risulta avere un’estensione spazialefinita. Come conseguenza della precedente analisi, siamo in grado d’interpretare fisicamente le unità immaginarie che entrano — come noto — nelle coordinate trasversali per azione delle trasformazioni di Lorentz Superluminali. Si esaminano dettagliatamente i vari casi particolari o casi limite della forma dei tachioni. Infine, si aggiungono alcune considerazioni circa eventuali esperimenti atti alla ricerca effettiva dei tachioni.РезюмеМы исследуем некоторые аспекты экспериментального поведения тахионов, в частности, посредством нахождения их кажущейся формы. Суперлюминальная частица, которая в своей собственной системе соординат является сферической или эллипсоидальной (и с бесконечным временем жизни), в лабораторной системе координат представляется занимающей всю область пространства, ограниченную двойным конусом и гиперболоидом с двумя слоями. Такая структура (« форма » тахиона) движется со скоростью тахиона. Однако, ески суперлюминальная частица имеет конечное время жизни в своей собственной системе координат, то в лабораторной системе координат эта частица занимает конечное пространство. Как вспомогательный результат, мы можем физически интерпретировать мнимые единицы, входящие, как известно, в попереченые координаты в суперлюминальных преобразованиях Лоренца. Подробно исследуются различные частные и предельные случаи формы тахиона. В заключение, проводится обсуждение возможных экспериментов по наблюдению тахионов.

Considerations about the apparent «superluminal expansions» observed in astrophysics

SummaryThe orthodox models devised to explain the apparent «superluminal expansions» observed in astrophysics—and here briefly summarized and discussed together with the experimental data—do not seem to be too much successful, especially when confronted with the most recent observations, suggesting complicated expansion patterns, even with possible accelerations. At this point it may be, therefore, of some interest to explore the possible alternative models in which actual Superluminal motions take place. To prepare the ground, we start from a variational principle, introduce the elements of a tachyon mechanics within special relativity, and argue about the expected behaviour of tachyonic objects when interacting (gravitationally, for instance) among themselves or with ordinary matter. We then review and develop the simplest «Superluminal models», paying particular attention to theobservations which they would give rise to. We conclude that some of them appear to be physically acceptable and are statistically favoured with respect to the orthodox ones.RiassuntoI modelli ortodossi costruiti per spiegare le apparenti «espansioni superluminali» osservate in astrofisica—e qui brevemente riassunti e discussi, insieme coi dati sperimentali relativi—sembra non abbiamo avuto troppo successo, specialmente se posti a confronto con le piú recenti osservazioni, le quali suggeriscono la presenza di meccanismi di espansione complicati anche con possibili accelerazioni. A questo punto, quindi, può essere di qualche interesse esplorare i possibili modelli alternativi, in cui abbiano luogo effettivi moti Superluminali. Per preparare il terreno partiamo da un principio variazionale, introduciamo le basi di una meccanica dei tachioni nell’ambito della relatività speciale, e individuiamo il comportamento che ci si aspetta da parte dei tachioni quando essi interagiscono (ad esempiogravitazionalmente) tra di loro o con la materia ordinaria. Si esaminano, quindi, e si sviluppano i piú semplici modelli «Superluminali», prestando particolare attenzione alleosservazioni a cui essi darebbero luogo. Si conclude che alcuni di tali modelli appaiono fisicamente accettabili e statisticamente favoriti rispetto a quelli ortodossi.РезюмеВ этой работе обсуждаются ортодоксальные модели, разработанные для объяснения очевидного «сверхсветового расширения», наблюденного в астрофизике, наряду с экспериментальными данными. Отмечаются недостатки этих моделей. Особенно, когда модели сопоставляются с недавними наблюдениями, предполагающими сложную картину расширения, даже с возможными ускорениями. Следовательно, с этой точки зрения представляет интерес исследовать возможные альтернативные модели, в которых могут иметь место действительные суперсветовые движения. Сначала мы исходим из вариационного принципа, вводим элементы тахионной механики в специальнцю теорию относительности и аргументируем ожидаемое поведение тахионных объектов, взаимодействующих (гравитационным образом) друг с другом и с обыкновенным веществом. Затем развиваем простейшие «суперсветовые модели», обращая особое внимание на наблюдаемые величины, которые получаются в этих моделях. Мы отмечаем, что некоторые из рассмотренных моделей являются физически приемлемыми и статистически предпочтительными относительно ортодоксальных моделей.

Formal and Physical Properties of the Generalized (Subluminal and Superluminal) Lorentz Transformations

SummaryWe investigate the mathematical and physical properties of the generalized Lorentz transformations (both subluminal and Superluminal). The form here adopted for the Superluminal Lorentz transformations is the one—recently introduced by us—which satisfies the requested group-theoretical properties. We clarify the role of the interpretation procedure of the imaginary quantities also from the formal point of view, for both the «longitudinal» and the «transverse» co-ordinates. Careful attention is devoted to define four-momentum and three-velocity for tachyons. At last, the shape of a tachyon—obtained by applying to an ordinary particle ageneric Superluminal Lorentz transformation (without rotations)—is studied. As a simplifying tool, we make recourse also to the «light-cone co-ordinates» and to «dilation-invariant» co-ordinates.RiassuntoSi esaminano le proprietà matematiche e fisiche delle trasformazioni generalizzate di Lorentz (sia subluminali, sia Superluminali). La forma qui adottata per le trasformazioni Superluminali di Lorentz è quella—da noi recentemente introdotta—che soddisfa le richieste proprietà gruppali. Quale mezzo di semplificazione, si fa uso anche delle «coordinate sul cono di luce» e di coordinate «invarianti per dilatazioni». Si chiarisce il ruolo della procedura di reinterpretazione anche dal punto di vista formale, e ciò per le coordinate tanto «longitudinali» quanto «trasversali». Si dedica opportuna attenzione a definire quadrimomento e trivelocità per i tachioni. Infine, si studia laforma dei tachioni ottenuti applicando a una particella ordinaria una generica trasformazione Superluminale di Lorentz (senza rotazioni).

Two-body interactions through tachyon exchange

SummaryDue to its relevance for the possible applications to particle physics and for causality problems, we thoroughly analyse in this paper the kinematics of (classical) tachyon exchange between two bodiesA, B, for all possible relative velocities. In particular, the two casesu·V≶c2 are carefully investigated, whereu, V are the bodyB and tachyon speeds relative toA, respectively.RiassuntoA causa della sua possibile rilevanza per le applicazioni alla fisica delle particelle e ai problemi causali, in questo articolo si analizza dettagliatamente la cinematica dello scambio di un tachione tra due corpiA, B, per tutte le velocità relative possibili. In particolare si studiano accuratamente i due casiu·V≶c2, doveu, V sono rispettivamente le velocità del corpoB e del tachione rispetto adA.РезюмеИсходя из возможных применений к физике частиц и проблемам причинности, мы проводим анализ кинематики (классического) обмена тахионами между двумя теламиA, B, для всех возможных относительных скоростей.B частности, исследуются два случаяu·V≶c2, гдеu, V скорости телаB и тахиона относительноA.

The introduction of Superluminal Lorentz transformations: A revisitation

We revisit the introduction of the Superluminal Lorentz transformations which carry from “bradyonic” inertial frames to “tachyonic” inertial frames, i.e., which transform time-like objects into space-like objects, andvice versa. It has long been known that special relativity can be extended to Superluminal observers only by increasing the number of dimensions of the space-time or—which is in a sense equivalent—by releasing the reality condition (i.e., introducing also imaginary quantities). In the past we always adopted the latter procedure. Here we show the connection between that procedure and the former one. In other words, in order to clarify the physical meaning of the imaginary units entering the classical theory of tachyons, we have temporarily to call into play anauxiliary six-dimensional space-time M(3, 3); however, we are eventually able to go back to the four-dimensional Minkowski space-time. We revisit the introduction of the Superluminal Lorentz transformations also under another aspect. In fact, the generalized Lorentz transformations had been previously written down in a form suited only for the simple case of collinear boosts (e.g., they formed a group just for collinear boosts). We express now the Superluminal Lorentz transformations in a more general form, so that they constitute a group together with the ordinary—orthochronousand antichronous—Lorentz transformations, and reduce to the previous form in the case of collinear boosts. Our approach introduces either real or imaginary quantities, with exclusion of (generic) complex quantities. In the present context, a procedure—in two steps—for interpreting the imaginary quantities is put forth and discussed. In the case of a chain of generalized Lorentz transformations, such a procedure (when necessary) is to be applied only at the end of the chain. Finally, we justify why we call “transformations” also the Superluminal ones.

Revisiting the superluminal Lorentz transformations and their group-theoretical properties

SummaryWe briefly revisit the introduction of generalized Lorentz transformations (GLT) in four dimensions, by writing them in a new form which satisfies the requested group-theoretical properties. The new GLTs, in the particular case of boosts, reduce to the ones by Mignani and Recami (the latter, in fact, forming a group only in the case of collinear boosts). Some other aspects are briefly clarified.