Abstract

Our aim in this paper is a reply to Seshadri’s comments [J. Opt. Soc. Am. A 29, 2532 (2012)] on a previous article of ours, titled “Cherenkov radiation versus X-shaped localized waves” [J. Opt. Soc. Am. A 27, 928 (2010)], as well as to his more extended criticism of the extended special relativity theory, called by him nonrestricted relativity, and in particular of the extended Maxwell equations.

© 2012 Optical Society of America

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References

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  1. S. R. Seshadri, “Cherenkov radiation versus X-shaped localized waves: comment,” J. Opt. Soc. Am. A 29, 2532–2535 (2012).
    [CrossRef]
  2. M. Zamboni-Rached, E. Recami, and I. M. Besieris, “Cherenkov radiation versus X-shaped localized waves,” J. Opt. Soc. Am. A 27, 928–934 (2010).
    [CrossRef]
  3. S. C. Walker and W. A. Kuperman, “Cherenkov-Vavilov formulation of X waves,” Phys. Rev. Lett. 99, 244802 (2007).
    [CrossRef]
  4. H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves, Theory and Applications (Wiley, 2008).
  5. J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992).
    [CrossRef]
  6. P. Saari and K. Reivelt, “Evidence of X-shaped propagation invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138(1997).
    [CrossRef]
  7. E. Recami, M. Zamboni-Rached, and C. A. Dartora, “Localized X-shaped field generated by a superluminal charge,” Phys. Rev. E 69, 027602 (2004) and references therein.
    [CrossRef]
  8. E. Recami and M. Zamboni-Rached, “Localized waves: a review,” Adv. Imaging Electron Phys. 156, 235–355 (2009).
    [CrossRef]
  9. A. Sommerfeld, “Überlichtgeschwindigkeitsteilchen,” Proc. K. Ned. Akad. Wet. 8, 346–367 (1904).
  10. A. Sommerfeld, “Zur Electronentheorie (3 Tiele),” Nach. Kgl. Ges. Wiss. Göttingen, Math. Naturwiss. Klasse 99–130, 363–439 (1904), 201–236 (1905).
  11. P. O. Fröman, “Historical background of the tachyon concept,” Arch. Hist. Exact Sci. 48, 373–380 (1994).
    [CrossRef]
  12. O.-M. Bilaniuk, V. K. Deshpande, and E. C. G. Sudarshan, “‘Meta’ relativity,” Am. J. Phys. 30, 718–723 (1962).
    [CrossRef]
  13. O.-M. Bilaniuk and E. C. G. Sudarshan, “Particles beyond the light barrier,” Phys. Today 22, 331–339 (1969).
    [CrossRef]
  14. E. Recami, “Classical theory of tachyons,” Riv. Nuovo Cimento 9(6), 1–178 (1986).
    [CrossRef]
  15. R. Mignani and E. Recami, “Crossing relations derived from (extended) relativity,” Int. J. Theor. Phys. 12, 299–320 (1975).
    [CrossRef]
  16. M. Pavšič and E. Recami, “Charge conjugation and internal space-time symmetries,” Lett. Nuovo Cimento 34, 357–362 (1982).
    [CrossRef]
  17. E. Recami, “Tachyon mechanics and causality: a systematic thorough analysis of the tachyon causal paradoxes,” Found. Phys. 17, 239–296 (1987).
    [CrossRef]
  18. A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509–533 (1982).
    [CrossRef]
  19. R. Mignani and E. Recami, “Tachyons do not emit Cherenkov radiation in vacuum,” Lett. Nuovo Cimento 7, 388–390 (1973).
    [CrossRef]
  20. A. B. Utkin, “Droplet-shaped waves: causal finite-support analogs of X-shaped waves,” J. Opt. Soc. Am. A 29, 457–462 (2012).
    [CrossRef]
  21. P. M. Morse, Theoretical Acoustics (Princeton University, 1985).
  22. E. Arias, C. H. G. Bessa, and N. F. Svaiter, “An analog fluid model for some tachyonic effects in Field Theory,” Mod. Phys. Lett. A 26, 2335–2344 (2011) and references therein.
    [CrossRef]
  23. E. Recami, “The Tolman antitelephone paradox: its solution by tachyon mechanics,” 1985, reprinted in Electron. J. Theor. Phys. (EJTP) 6, 1–8 (2009).
  24. E. Recami, “Superluminal motions? A bird’s-eye view of the experimental status-of-the-art,” Found. Phys. 31, 1119–1135 (2001).
    [CrossRef]
  25. E. Recami, “Superluminal waves and objects: an up-dated overview of the relevant experiments,” arXiv:0804.1502 [physics].
  26. E. Recami and W. A. Rodrigues, “A model theory for tachyons in two dimensions,” in Gravitational Radiation and Relativity, J. Weber and T. M. Karade, eds., Vol. 3 of Proceedings of the Sir Arthur Eddington Centenary Symposium (World Scientific, 1985), pp. 151–203.
  27. A. O. Barut and H. C. Chandola, “’Localized’ tachyonic wavelet solutions to the wave equation,” Phys. Lett. A 180, 5–8(1993).
    [CrossRef]
  28. E. Recami and R. Mignani, “Magnetic monopoles and tachyons in special relativity,” Phys. Lett. B 62, 41–43 (1976).
    [CrossRef]

2012 (2)

2011 (1)

E. Arias, C. H. G. Bessa, and N. F. Svaiter, “An analog fluid model for some tachyonic effects in Field Theory,” Mod. Phys. Lett. A 26, 2335–2344 (2011) and references therein.
[CrossRef]

2010 (1)

2009 (2)

E. Recami and M. Zamboni-Rached, “Localized waves: a review,” Adv. Imaging Electron Phys. 156, 235–355 (2009).
[CrossRef]

E. Recami, “The Tolman antitelephone paradox: its solution by tachyon mechanics,” 1985, reprinted in Electron. J. Theor. Phys. (EJTP) 6, 1–8 (2009).

2007 (1)

S. C. Walker and W. A. Kuperman, “Cherenkov-Vavilov formulation of X waves,” Phys. Rev. Lett. 99, 244802 (2007).
[CrossRef]

2004 (1)

E. Recami, M. Zamboni-Rached, and C. A. Dartora, “Localized X-shaped field generated by a superluminal charge,” Phys. Rev. E 69, 027602 (2004) and references therein.
[CrossRef]

2001 (1)

E. Recami, “Superluminal motions? A bird’s-eye view of the experimental status-of-the-art,” Found. Phys. 31, 1119–1135 (2001).
[CrossRef]

1997 (1)

P. Saari and K. Reivelt, “Evidence of X-shaped propagation invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138(1997).
[CrossRef]

1994 (1)

P. O. Fröman, “Historical background of the tachyon concept,” Arch. Hist. Exact Sci. 48, 373–380 (1994).
[CrossRef]

1993 (1)

A. O. Barut and H. C. Chandola, “’Localized’ tachyonic wavelet solutions to the wave equation,” Phys. Lett. A 180, 5–8(1993).
[CrossRef]

1992 (1)

J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992).
[CrossRef]

1987 (1)

E. Recami, “Tachyon mechanics and causality: a systematic thorough analysis of the tachyon causal paradoxes,” Found. Phys. 17, 239–296 (1987).
[CrossRef]

1986 (1)

E. Recami, “Classical theory of tachyons,” Riv. Nuovo Cimento 9(6), 1–178 (1986).
[CrossRef]

1982 (2)

A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509–533 (1982).
[CrossRef]

M. Pavšič and E. Recami, “Charge conjugation and internal space-time symmetries,” Lett. Nuovo Cimento 34, 357–362 (1982).
[CrossRef]

1976 (1)

E. Recami and R. Mignani, “Magnetic monopoles and tachyons in special relativity,” Phys. Lett. B 62, 41–43 (1976).
[CrossRef]

1975 (1)

R. Mignani and E. Recami, “Crossing relations derived from (extended) relativity,” Int. J. Theor. Phys. 12, 299–320 (1975).
[CrossRef]

1973 (1)

R. Mignani and E. Recami, “Tachyons do not emit Cherenkov radiation in vacuum,” Lett. Nuovo Cimento 7, 388–390 (1973).
[CrossRef]

1969 (1)

O.-M. Bilaniuk and E. C. G. Sudarshan, “Particles beyond the light barrier,” Phys. Today 22, 331–339 (1969).
[CrossRef]

1962 (1)

O.-M. Bilaniuk, V. K. Deshpande, and E. C. G. Sudarshan, “‘Meta’ relativity,” Am. J. Phys. 30, 718–723 (1962).
[CrossRef]

1904 (2)

A. Sommerfeld, “Überlichtgeschwindigkeitsteilchen,” Proc. K. Ned. Akad. Wet. 8, 346–367 (1904).

A. Sommerfeld, “Zur Electronentheorie (3 Tiele),” Nach. Kgl. Ges. Wiss. Göttingen, Math. Naturwiss. Klasse 99–130, 363–439 (1904), 201–236 (1905).

Arias, E.

E. Arias, C. H. G. Bessa, and N. F. Svaiter, “An analog fluid model for some tachyonic effects in Field Theory,” Mod. Phys. Lett. A 26, 2335–2344 (2011) and references therein.
[CrossRef]

Barut, A. O.

A. O. Barut and H. C. Chandola, “’Localized’ tachyonic wavelet solutions to the wave equation,” Phys. Lett. A 180, 5–8(1993).
[CrossRef]

A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509–533 (1982).
[CrossRef]

Besieris, I. M.

Bessa, C. H. G.

E. Arias, C. H. G. Bessa, and N. F. Svaiter, “An analog fluid model for some tachyonic effects in Field Theory,” Mod. Phys. Lett. A 26, 2335–2344 (2011) and references therein.
[CrossRef]

Bilaniuk, O.-M.

O.-M. Bilaniuk and E. C. G. Sudarshan, “Particles beyond the light barrier,” Phys. Today 22, 331–339 (1969).
[CrossRef]

O.-M. Bilaniuk, V. K. Deshpande, and E. C. G. Sudarshan, “‘Meta’ relativity,” Am. J. Phys. 30, 718–723 (1962).
[CrossRef]

Chandola, H. C.

A. O. Barut and H. C. Chandola, “’Localized’ tachyonic wavelet solutions to the wave equation,” Phys. Lett. A 180, 5–8(1993).
[CrossRef]

Dartora, C. A.

E. Recami, M. Zamboni-Rached, and C. A. Dartora, “Localized X-shaped field generated by a superluminal charge,” Phys. Rev. E 69, 027602 (2004) and references therein.
[CrossRef]

Deshpande, V. K.

O.-M. Bilaniuk, V. K. Deshpande, and E. C. G. Sudarshan, “‘Meta’ relativity,” Am. J. Phys. 30, 718–723 (1962).
[CrossRef]

Fröman, P. O.

P. O. Fröman, “Historical background of the tachyon concept,” Arch. Hist. Exact Sci. 48, 373–380 (1994).
[CrossRef]

Greenleaf, J. F.

J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992).
[CrossRef]

Kuperman, W. A.

S. C. Walker and W. A. Kuperman, “Cherenkov-Vavilov formulation of X waves,” Phys. Rev. Lett. 99, 244802 (2007).
[CrossRef]

Lu, J.-Y.

J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992).
[CrossRef]

Maccarrone, G. D.

A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509–533 (1982).
[CrossRef]

Mignani, R.

E. Recami and R. Mignani, “Magnetic monopoles and tachyons in special relativity,” Phys. Lett. B 62, 41–43 (1976).
[CrossRef]

R. Mignani and E. Recami, “Crossing relations derived from (extended) relativity,” Int. J. Theor. Phys. 12, 299–320 (1975).
[CrossRef]

R. Mignani and E. Recami, “Tachyons do not emit Cherenkov radiation in vacuum,” Lett. Nuovo Cimento 7, 388–390 (1973).
[CrossRef]

Morse, P. M.

P. M. Morse, Theoretical Acoustics (Princeton University, 1985).

Pavšic, M.

M. Pavšič and E. Recami, “Charge conjugation and internal space-time symmetries,” Lett. Nuovo Cimento 34, 357–362 (1982).
[CrossRef]

Recami, E.

M. Zamboni-Rached, E. Recami, and I. M. Besieris, “Cherenkov radiation versus X-shaped localized waves,” J. Opt. Soc. Am. A 27, 928–934 (2010).
[CrossRef]

E. Recami and M. Zamboni-Rached, “Localized waves: a review,” Adv. Imaging Electron Phys. 156, 235–355 (2009).
[CrossRef]

E. Recami, “The Tolman antitelephone paradox: its solution by tachyon mechanics,” 1985, reprinted in Electron. J. Theor. Phys. (EJTP) 6, 1–8 (2009).

E. Recami, M. Zamboni-Rached, and C. A. Dartora, “Localized X-shaped field generated by a superluminal charge,” Phys. Rev. E 69, 027602 (2004) and references therein.
[CrossRef]

E. Recami, “Superluminal motions? A bird’s-eye view of the experimental status-of-the-art,” Found. Phys. 31, 1119–1135 (2001).
[CrossRef]

E. Recami, “Tachyon mechanics and causality: a systematic thorough analysis of the tachyon causal paradoxes,” Found. Phys. 17, 239–296 (1987).
[CrossRef]

E. Recami, “Classical theory of tachyons,” Riv. Nuovo Cimento 9(6), 1–178 (1986).
[CrossRef]

M. Pavšič and E. Recami, “Charge conjugation and internal space-time symmetries,” Lett. Nuovo Cimento 34, 357–362 (1982).
[CrossRef]

A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509–533 (1982).
[CrossRef]

E. Recami and R. Mignani, “Magnetic monopoles and tachyons in special relativity,” Phys. Lett. B 62, 41–43 (1976).
[CrossRef]

R. Mignani and E. Recami, “Crossing relations derived from (extended) relativity,” Int. J. Theor. Phys. 12, 299–320 (1975).
[CrossRef]

R. Mignani and E. Recami, “Tachyons do not emit Cherenkov radiation in vacuum,” Lett. Nuovo Cimento 7, 388–390 (1973).
[CrossRef]

E. Recami, “Superluminal waves and objects: an up-dated overview of the relevant experiments,” arXiv:0804.1502 [physics].

E. Recami and W. A. Rodrigues, “A model theory for tachyons in two dimensions,” in Gravitational Radiation and Relativity, J. Weber and T. M. Karade, eds., Vol. 3 of Proceedings of the Sir Arthur Eddington Centenary Symposium (World Scientific, 1985), pp. 151–203.

Reivelt, K.

P. Saari and K. Reivelt, “Evidence of X-shaped propagation invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138(1997).
[CrossRef]

Rodrigues, W. A.

E. Recami and W. A. Rodrigues, “A model theory for tachyons in two dimensions,” in Gravitational Radiation and Relativity, J. Weber and T. M. Karade, eds., Vol. 3 of Proceedings of the Sir Arthur Eddington Centenary Symposium (World Scientific, 1985), pp. 151–203.

Saari, P.

P. Saari and K. Reivelt, “Evidence of X-shaped propagation invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138(1997).
[CrossRef]

Seshadri, S. R.

Sommerfeld, A.

A. Sommerfeld, “Überlichtgeschwindigkeitsteilchen,” Proc. K. Ned. Akad. Wet. 8, 346–367 (1904).

A. Sommerfeld, “Zur Electronentheorie (3 Tiele),” Nach. Kgl. Ges. Wiss. Göttingen, Math. Naturwiss. Klasse 99–130, 363–439 (1904), 201–236 (1905).

Sudarshan, E. C. G.

O.-M. Bilaniuk and E. C. G. Sudarshan, “Particles beyond the light barrier,” Phys. Today 22, 331–339 (1969).
[CrossRef]

O.-M. Bilaniuk, V. K. Deshpande, and E. C. G. Sudarshan, “‘Meta’ relativity,” Am. J. Phys. 30, 718–723 (1962).
[CrossRef]

Svaiter, N. F.

E. Arias, C. H. G. Bessa, and N. F. Svaiter, “An analog fluid model for some tachyonic effects in Field Theory,” Mod. Phys. Lett. A 26, 2335–2344 (2011) and references therein.
[CrossRef]

Utkin, A. B.

Walker, S. C.

S. C. Walker and W. A. Kuperman, “Cherenkov-Vavilov formulation of X waves,” Phys. Rev. Lett. 99, 244802 (2007).
[CrossRef]

Zamboni-Rached, M.

M. Zamboni-Rached, E. Recami, and I. M. Besieris, “Cherenkov radiation versus X-shaped localized waves,” J. Opt. Soc. Am. A 27, 928–934 (2010).
[CrossRef]

E. Recami and M. Zamboni-Rached, “Localized waves: a review,” Adv. Imaging Electron Phys. 156, 235–355 (2009).
[CrossRef]

E. Recami, M. Zamboni-Rached, and C. A. Dartora, “Localized X-shaped field generated by a superluminal charge,” Phys. Rev. E 69, 027602 (2004) and references therein.
[CrossRef]

Adv. Imaging Electron Phys. (1)

E. Recami and M. Zamboni-Rached, “Localized waves: a review,” Adv. Imaging Electron Phys. 156, 235–355 (2009).
[CrossRef]

Am. J. Phys. (1)

O.-M. Bilaniuk, V. K. Deshpande, and E. C. G. Sudarshan, “‘Meta’ relativity,” Am. J. Phys. 30, 718–723 (1962).
[CrossRef]

Arch. Hist. Exact Sci. (1)

P. O. Fröman, “Historical background of the tachyon concept,” Arch. Hist. Exact Sci. 48, 373–380 (1994).
[CrossRef]

Electron. J. Theor. Phys. (EJTP) (1)

E. Recami, “The Tolman antitelephone paradox: its solution by tachyon mechanics,” 1985, reprinted in Electron. J. Theor. Phys. (EJTP) 6, 1–8 (2009).

Found. Phys. (2)

E. Recami, “Superluminal motions? A bird’s-eye view of the experimental status-of-the-art,” Found. Phys. 31, 1119–1135 (2001).
[CrossRef]

E. Recami, “Tachyon mechanics and causality: a systematic thorough analysis of the tachyon causal paradoxes,” Found. Phys. 17, 239–296 (1987).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992).
[CrossRef]

Int. J. Theor. Phys. (1)

R. Mignani and E. Recami, “Crossing relations derived from (extended) relativity,” Int. J. Theor. Phys. 12, 299–320 (1975).
[CrossRef]

J. Opt. Soc. Am. A (3)

Lett. Nuovo Cimento (2)

R. Mignani and E. Recami, “Tachyons do not emit Cherenkov radiation in vacuum,” Lett. Nuovo Cimento 7, 388–390 (1973).
[CrossRef]

M. Pavšič and E. Recami, “Charge conjugation and internal space-time symmetries,” Lett. Nuovo Cimento 34, 357–362 (1982).
[CrossRef]

Mod. Phys. Lett. A (1)

E. Arias, C. H. G. Bessa, and N. F. Svaiter, “An analog fluid model for some tachyonic effects in Field Theory,” Mod. Phys. Lett. A 26, 2335–2344 (2011) and references therein.
[CrossRef]

Nach. Kgl. Ges. Wiss. Göttingen, Math. Naturwiss. Klasse (1)

A. Sommerfeld, “Zur Electronentheorie (3 Tiele),” Nach. Kgl. Ges. Wiss. Göttingen, Math. Naturwiss. Klasse 99–130, 363–439 (1904), 201–236 (1905).

Nuovo Cimento A (1)

A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509–533 (1982).
[CrossRef]

Phys. Lett. A (1)

A. O. Barut and H. C. Chandola, “’Localized’ tachyonic wavelet solutions to the wave equation,” Phys. Lett. A 180, 5–8(1993).
[CrossRef]

Phys. Lett. B (1)

E. Recami and R. Mignani, “Magnetic monopoles and tachyons in special relativity,” Phys. Lett. B 62, 41–43 (1976).
[CrossRef]

Phys. Rev. E (1)

E. Recami, M. Zamboni-Rached, and C. A. Dartora, “Localized X-shaped field generated by a superluminal charge,” Phys. Rev. E 69, 027602 (2004) and references therein.
[CrossRef]

Phys. Rev. Lett. (2)

S. C. Walker and W. A. Kuperman, “Cherenkov-Vavilov formulation of X waves,” Phys. Rev. Lett. 99, 244802 (2007).
[CrossRef]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138(1997).
[CrossRef]

Phys. Today (1)

O.-M. Bilaniuk and E. C. G. Sudarshan, “Particles beyond the light barrier,” Phys. Today 22, 331–339 (1969).
[CrossRef]

Proc. K. Ned. Akad. Wet. (1)

A. Sommerfeld, “Überlichtgeschwindigkeitsteilchen,” Proc. K. Ned. Akad. Wet. 8, 346–367 (1904).

Riv. Nuovo Cimento (1)

E. Recami, “Classical theory of tachyons,” Riv. Nuovo Cimento 9(6), 1–178 (1986).
[CrossRef]

Other (4)

H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves, Theory and Applications (Wiley, 2008).

P. M. Morse, Theoretical Acoustics (Princeton University, 1985).

E. Recami, “Superluminal waves and objects: an up-dated overview of the relevant experiments,” arXiv:0804.1502 [physics].

E. Recami and W. A. Rodrigues, “A model theory for tachyons in two dimensions,” in Gravitational Radiation and Relativity, J. Weber and T. M. Karade, eds., Vol. 3 of Proceedings of the Sir Arthur Eddington Centenary Symposium (World Scientific, 1985), pp. 151–203.

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Figures (5)

Fig. 1.
Fig. 1.

Pictures (in 3D only) of the surfaces p2E2p2=±m02: In (a) for bradyons, when p2>0; in (b) for luxons, when p2=0; and in (c) for tachyons, when p2<0. The symmetry matter/antimatter [14] is the symmetry w.r.t. the plane E=0. See the text. Reprinted with kind permission of Società Italiana di Fisica.

Fig. 2.
Fig. 2.

Intrinsically spherical (or pointlike, at the limit) object appears in the vacuum as an ellipsoid contracted along the motion direction when endowed with a speed v<c. By contrast, if endowed with a speed V>c (even if the c-speed barrier cannot be crossed, either from the left or from the right), it would appear [14,18] no longer as a particle, but as occupying the region delimited by a double cone and a two-sheeted hyperboloid—or as a double cone, at the limit—and moving with Ssuperluminal speed V [the cotangent square of the cone semiangle, with c=1, being V21. For simplicity, a space axis is skipped. This figure is taken from [14,18]. Reprinted with kind permission of Società Italiana di Fisica.

Fig. 3.
Fig. 3.

Spherical equipotential surfaces of the electrostatic field created by a charge at rest are transformed into two-sheeted rotation hyperboloids, contained inside an unlimited double cone, when the charge travels at superluminal speed [14]. This figures shows, among the others, that a superluminal charge traveling at constant speed, in a homogeneous medium like the vacuum, does not lose energy. See the text. Reprinted with kind permission of Società Italiana di Fisica.

Fig. 4.
Fig. 4.

Let us consider in a system at rest a purely electric uniform field E parallel, e.g., to y. When moving along x with respect to that system, one also observes a magnetic field along z: (a) depicts the ordinary subluminal case with vvx<c, in which case (in Heaviside–Lorentz units) it is Hz<Ey. When moving superluminally with VVx>c, the magnetic field Hz becomes [14] (in the same units) larger than the electric field Ey; see (b). And when V we are left with a purely magnetic field [14]. See the text. Reprinted with kind permission of Società Italiana di Fisica.

Fig. 5.
Fig. 5.

Sketch of Fig. 2 showing that, if the initial subluminal object has sizes Δx, Δy, and Δz (determined by its intersections with the space axes), the corresponding superluminal object moving along x with speed V has along x the size Δx=ΔxV21, regularly given by the generalized Lorentz contraction formula [14]. But it does not have real intersections with the transverse Cartesian axes, so that [14] (forgetting here for the double sign) it is characterized by Δy=iΔy, and Δz=iΔz. See the text. Reprinted with kind permission of Società Italiana di Fisica.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

ds2=±ds2,
L+=L+Z(2);Z(2){+1}{+1,1}.
G=L+Z(4).
ds2={+ds2forv2<c2;ds2forv2>c2.
SLT(U)=±S·LT(u),
±x=ix+γ1U2U(U·x)γUt;±t=γ(tU·x),

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