Ɇɟɬɨɞɨɥɨɝɢɱɟɫɤɢɟ ɚɫɩɟɤɬɵ ɜɵɜɨɞɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ ɩɪɢ ɜɵɛɨɪɟ ɫɬɚɧɞɚɪɬɨɜ ɞɥɢɧɵ ɢ ɜɪɟɦɟɧɢ

ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 2, ʋ1, 1999 ɝ.
ɫɬɪ.119-124
Ɇɟɬɨɞɨɥɨɝɢɱɟɫɤɢɟ ɚɫɩɟɤɬɵ ɜɵɜɨɞɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ
ɩɪɢ ɜɵɛɨɪɟ ɫɬɚɧɞɚɪɬɨɜ ɞɥɢɧɵ ɢ ɜɪɟɦɟɧɢ
ȼ.Ƚ. ɉɢɜɨɜɚɪɨɜ 1, Ɉ.Ⱥ. ɇɢɤɨɧɨɜ 2
1
Ɏɢɧɚɧɫɨɜɵɣ ɮɚɤɭɥɶɬɟɬ ɆȽɌɍ, ɤɚɮɟɞɪɚ ɩɪɢɤɥɚɞɧɨɣ ɦɚɬɟɦɚɬɢɤɢ
ɢ ɟɫɬɟɫɬɜɟɧɧɨ - ɧɚɭɱɧɵɯ ɞɢɫɰɢɩɥɢɧ
2
ɗɥɟɤɬɪɨɦɟɯɚɧɢɱɟɫɤɢɣ ɮɚɤɭɥɶɬɟɬ ɆȽɌɍ, ɤɚɮɟɞɪɚ ɮɢɡɢɤɢ
Ⱥɧɧɨɬɚɰɢɹ. ȼ ɪɚɛɨɬɟ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɜɨɩɪɨɫɵ ɜɵɛɨɪɚ ɫɬɚɧɞɚɪɬɨɜ ɞɥɢɧɵ ɢ ɜɪɟɦɟɧɢ ɜ ɢɧɟɪɰɢɚɥɶɧɵɯ
ɫɢɫɬɟɦɚɯ ɨɬɫɱɟɬɚ (ɂɋɈ) ɢ ɪɟɚɥɶɧɨɫɬɢ ɥɨɪɟɧɰɨɜɫɤɨɝɨ ɫɨɤɪɚɳɟɧɢɹ ɞɥɢɧɵ ɢ ɡɚɦɟɞɥɟɧɢɹ ɯɨɞɚ ɞɜɢɠɭɳɢɯɫɹ
ɱɚɫɨɜ. Ʉɚɤ ɢɡɜɟɫɬɧɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɜɵɜɨɞɹɬɫɹ ɧɚ ɨɫɧɨɜɟ ɞɜɭɯ ɩɨɫɬɭɥɚɬɨɜ ɫɩɟɰɢɚɥɶɧɨɣ ɬɟɨɪɢɢ
ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ (ɋɌɈ). Ʉɪɨɦɟ ɷɬɢɯ ɞɜɭɯ ɩɨɫɬɭɥɚɬɨɜ ɬɪɟɛɭɟɬɫɹ ɬɪɟɬɢɣ, ɤɚɫɚɸɳɢɣɫɹ ɫɜɨɣɫɬɜ
ɩɪɨɫɬɪɚɧɫɬɜɚ ɢ ɜɪɟɦɟɧɢ. ɉɭɫɬɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ ɨɞɧɨɪɨɞɧɨ ɢ ɢɡɨɬɪɨɩɧɨ, ɜɪɟɦɹ ɨɞɧɨɪɨɞɧɨ ɢ
ɨɞɧɨɧɚɩɪɚɜɥɟɧɨ. Ɍɚɤɚɹ ɫɢɫɬɟɦɚ ɩɨɫɬɭɥɚɬɨɜ ɹɜɥɹɟɬɫɹ ɧɟɩɪɨɬɢɜɨɪɟɱɢɜɨɣ ɢ ɞɨɫɬɚɬɨɱɧɨɣ ɞɥɹ ɜɵɜɨɞɚ
ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ. ȼ ɪɚɛɨɬɟ ɢɫɫɥɟɞɭɟɬɫɹ ɜɨɩɪɨɫ ɨɛ ɨɝɪɚɧɢɱɟɧɢɹɯ, ɧɚɤɥɚɞɵɜɚɟɦɵɯ ɧɚ ɜɵɛɨɪ
ɫɬɚɧɞɚɪɬɨɜ ɞɥɢɧɵ ɢ ɜɪɟɦɟɧɢ ɩɨɫɬɭɥɚɬɚɦɢ ɗɣɧɲɬɟɣɧɚ.
Abstract. In the work questions of choosing of length and time standards in inertial systems of counting (ISC)
and reality of Lorentz length reduction and moving clock slowing down are considered. As it is known Lorentz
transformations are deduced on the basis of two postulates of special theory of relativity (STR). Besides, the
third postulate concerning characteristics of time and space is needed. Empty space is homogenic and isotropic,
time is homogenic and one way directional. Such system of postulates is non-contradictory and enough to
conclude Lorentz transformations. So, in this work the question of limitations which apply to choosing of length
and time standards has been examined.
1. ȼɜɟɞɟɧɢɟ
ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɹɜɥɹɸɬɫɹ ɮɭɧɞɚɦɟɧɬɨɦ ɋɌɈ. ɂɡɜɟɫɬɧɨ, ɱɬɨ ɷɬɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɦɨɝɭɬ
ɛɵɬɶ ɜɵɜɟɞɟɧɵ ɤɨɪɪɟɤɬɧɨ ɧɚ ɨɫɧɨɜɟ ɧɟɫɤɨɥɶɤɢɯ ɩɨɫɬɭɥɚɬɨɜ, ɞɜɚ ɢɡ ɤɨɬɨɪɵɯ, ɫɜɹɡɚɧɧɵɯ ɫ ɢɦɟɧɟɦ
ɗɣɧɲɬɟɣɧɚ, ɢɡɜɟɫɬɧɵ ɤɚɤ ɱɚɫɬɧɵɣ ɩɪɢɧɰɢɩ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɢ ɩɨɫɬɭɥɚɬ ɨ ɩɨɫɬɨɹɧɫɬɜɟ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ.
Ʉɪɨɦɟ ɷɬɢɯ ɩɨɫɬɭɥɚɬɨɜ ɬɪɟɛɭɟɬɫɹ ɬɪɟɬɢɣ, ɤɚɫɚɸɳɢɣɫɹ ɫɜɨɣɫɬɜ ɩɪɨɫɬɪɚɧɫɬɜɚ – ɜɪɟɦɟɧɢ; ɩɭɫɬɨɟ
ɩɪɨɫɬɪɚɧɫɬɜɨ ɢɡɨɬɪɨɩɧɨ ɢ ɨɞɧɨɪɨɞɧɨ, ɚ ɬɟɱɟɧɢɟ ɜɪɟɦɟɧɢ ɜ ɧɟɦ ɨɞɧɨɪɨɞɧɨ. ȿɫɥɢ ɦɵ ɩɨɩɵɬɚɟɦɫɹ
ɪɚɡɨɛɪɚɬɶɫɹ ɜ ɞɨɫɬɚɬɨɱɧɨɫɬɢ ɷɬɢɯ ɩɨɫɬɭɥɚɬɨɜ ɞɥɹ ɨɞɧɨɡɧɚɱɧɨɝɨ ɜɵɜɨɞɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ ɩɨ
ɢɦɟɸɳɟɣɫɹ ɭɱɟɛɧɨɣ ɥɢɬɟɪɚɬɭɪɟ (Ⱦɟɬɥɚɮɮ, əɜɨɪɫɤɢɣ, 1989; Ɇɚɬɜɟɟɜ, 1986; ɋɚɜɟɥɶɟɜ, 1989 ɋɢɜɭɯɢɧ,
1980; Ɍɪɨɮɢɦɨɜɚ, 1997), ɬɨ ɫɪɚɡɭ ɠɟ ɜɫɬɪɟɬɢɦɫɹ ɫ ɟɳɟ, ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ ɨɞɧɢɦ, ɩɪɟɞɩɨɥɨɠɟɧɢɟɦ. Ɉɧɨ
ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɟɫɥɢ ɩɪɢ ɫɩɟɰɢɚɥɶɧɨɦ ɜɵɛɨɪɟ ɂɋɈ ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɂɋɈ K ɨɬɧɨɫɢɬɟɥɶɧɨ K’ V, ɬɨ
ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɂɋɈ K ɨɬɧɨɫɢɬɟɥɶɧɨ K’ ɪɚɜɧɚ ɦɢɧɭɫ V. ɉɨɩɵɬɤɢ ɨɛɨɫɧɨɜɚɧɢɹ ɷɬɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ ɫ
ɩɨɦɨɳɶɸ ɫɥɨɜ "ɨɱɟɜɢɞɧɨ" ɢ "ɹɫɧɨ" ɧɟ ɦɨɝɭɬ ɫɥɭɠɢɬɶ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨɦ ɷɬɨɝɨ ɭɬɜɟɪɠɞɟɧɢɹ, ɬ.ɤ.
V’=dx’/dt’, ɚ V=dx/dt,
ɝɞɟ ɤɚɠɞɵɣ ɢɡ ɞɢɮɮɟɪɟɧɰɢɚɥɨɜ dx’ ɢ dt’ ɢɡɦɟɪɹɟɬɫɹ ɜ ɂɋɈ K’, ɚ dx ɢ dt — ɜ ɂɋɈ K.
ȼɵɡɵɜɚɟɬ, ɬɚɤɠɟ ɧɟɞɨɭɦɟɧɢɟ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨ "ɧɚ ɩɚɥɶɰɚɯ" ɛɟɡ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɨɫɧɨɜɧɵɯ
ɩɨɫɬɭɥɚɬɨɜ ɬɨɝɨ ɮɚɤɬɚ, ɱɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ a ɢ b ɜ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɯ
y’ = a y
z’ = b z
ɪɚɜɧɵ ɟɞɢɧɢɰɟ. ɇɚ ɫɚɦɨɦ ɞɟɥɟ, ɤɚɤ ɷɬɨ ɛɵɥɨ ɩɨɤɚɡɚɧɨ ɜ ɦɨɧɨɝɪɚɮɢɢ (Ɇɚɧɞɟɥɶɲɬɚɦ, 1972), ɤɚɤ ɩɟɪɜɨɟ,
ɬɚɤ ɢ ɜɬɨɪɨɟ ɭɬɜɟɪɠɞɟɧɢɹ ɹɜɥɹɸɬɫɹ ɫɥɟɞɫɬɜɢɹɦɢ ɞɜɭɯ ɩɨɫɬɭɥɚɬɨɜ ɗɣɧɲɬɟɣɧɚ.
ȿɫɬɶ ɟɳɟ ɨɞɢɧ ɜɨɩɪɨɫ, ɫɜɹɡɚɧɧɵɣ ɫ ɜɵɜɨɞɨɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ, ɤɨɬɨɪɵɣ ɧɚɦ
ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜɚɠɧɵɦ ɢ ɤɨɬɨɪɵɣ ɜ (Ɇɚɧɞɟɥɶɲɬɚɦ, 1972; Ɇɚɬɜɟɟɜ, 1986; ɋɢɜɭɯɢɧ, 1980) ɧɟ
ɡɚɬɪɚɝɢɜɚɟɬɫɹ — ɷɬɨ ɜɨɩɪɨɫ ɨ ɜɵɛɨɪɟ ɫɬɚɧɞɚɪɬɨɜ ɞɥɢɧɵ ɢ ɜɪɟɦɟɧɢ ɜ ɂɋɈ K ɢ K’ ɢ ɢɯ ɫɨɝɥɚɫɨɜɚɧɢɢ. Ȼɟɡ
ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ ɷɬɨɝɨ ɜɨɩɪɨɫɚ ɨɫɬɚɟɬɫɹ ɧɟɹɫɧɵɦ ɩɨɫɬɭɥɚɬ ɨ ɩɨɫɬɨɹɧɫɬɜɟ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ ɩɪɢ
119
ɉɢɜɨɜɚɪɨɜ ȼ.Ƚ., ɇɢɤɨɧɨɜ Ɉ.Ⱥ. Ɇɟɬɨɞɨɥɨɝɢɱɟɫɤɢɟ ɚɫɩɟɤɬɵ ɜɵɜɨɞɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ...
ɩɟɪɟɯɨɞɟ ɨɬ ɂɋɈ K ɤ ɂɋɈ K’ ɢ ɨɛɪɚɬɧɨ, ɬɚɤ ɤɚɤ ɫɤɨɪɨɫɬɶ ɫɜɟɬɚ ɢɡɦɟɪɟɧɚ ɜ ɪɚɡɧɵɯ ɧɟɫɨɝɥɚɫɨɜɚɧɧɵɯ ɞɪɭɝ
ɫ ɞɪɭɝɨɦ ɟɞɢɧɢɰɚɯ. ȿɫɥɢ ɧɟɢɡɜɟɫɬɧɵ ɩɟɪɟɯɨɞɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɦɟɠɞɭ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɫɬɚɧɞɚɪɬɚɦɢ,
ɬɨ ɥɸɛɚɹ ɫɜɹɡɶ ɦɟɠɞɭ ɢɡɦɟɪɹɟɦɵɦɢ ɜɟɥɢɱɢɧɚɦɢ a ɢ b, ɜɵɬɟɤɚɸɳɚɹ ɢɡ ɩɨɫɬɭɥɚɬɚ ɨ ɩɨɫɬɨɹɧɫɬɜɟ ɫɤɨɪɨɫɬɢ
ɫɜɟɬɚ, ɬɟɪɹɟɬ ɫɦɵɫɥ. Ɍɚɤɠɟ ɬɟɪɹɸɬ ɫɦɵɫɥ ɢ ɫɚɦɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɩɪɟɠɞɟ ɱɟɦ
ɩɨɥɶɡɨɜɚɬɶɫɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ Ʌɨɪɟɧɰɚ, ɭɫɬɚɧɚɜɥɢɜɚɸɳɢɦɢ ɫɜɹɡɶ ɦɟɠɞɭ ɤɨɨɪɞɢɧɚɬɚɦɢ ɢ ɜɪɟɦɟɧɟɦ
ɨɞɧɨɝɨ ɢ ɬɨɝɨ ɠɟ ɫɨɛɵɬɢɹ ɜ ɂɋɈ K ɢ K’, ɧɟɨɛɯɨɞɢɦɨ ɩɪɟɞɥɨɠɢɬɶ ɪɟɰɟɩɬ ɜɵɛɨɪɚ ɫɬɚɧɞɚɪɬɨɜ ɞɥɢɧɵ ɢ
ɜɪɟɦɟɧɢ ɜ ɨɛɟɢɯ ɂɋɈ ɢ ɭɫɬɚɧɨɜɢɬɶ ɫɜɹɡɶ ɦɟɠɞɭ ɧɢɦɢ. ɂ ɬɨɥɶɤɨ ɩɨɫɥɟ ɷɬɨɝɨ ɨɬɤɪɵɜɚɟɬɫɹ ɜɨɡɦɨɠɧɨɫɬɶ
ɤɨɪɪɟɤɬɧɨɝɨ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɜɬɨɪɨɝɨ ɩɨɫɬɭɥɚɬɚ ɗɣɧɲɬɟɣɧɚ ɩɪɢ ɩɨɥɭɱɟɧɢɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ.
ɉɨɥɟɡɧɨ, ɤɪɨɦɟ ɬɨɝɨ, ɩɪɨɫɥɟɞɢɬɶ ɜ ɩɪɨɰɟɫɫɟ ɜɵɜɨɞɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ, ɧɟ ɧɚɤɥɚɞɵɜɚɸɬ ɥɢ
ɩɨɫɬɭɥɚɬɵ ɗɣɧɲɬɟɣɧɚ ɤɚɤɢɟ-ɥɢɛɨ ɨɝɪɚɧɢɱɟɧɢɹ ɧɚ ɜɵɛɨɪ ɫɬɚɧɞɚɪɬɨɜ ɞɥɢɧɵ ɢ ɜɪɟɦɟɧɢ.
2. ȼɵɛɨɪ ɫɬɚɧɞɚɪɬɨɜ ɞɥɢɧɵ ɢ ɜɪɟɦɟɧɢ ɜ ɂɋɈ
ɇɚɩɨɦɧɢɦ, ɱɬɨ ɜɫɹɤɚɹ ɂɋɈ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɬɟɥɚɦɢ ɨɬɫɱɟɬɚ, ɫɢɫɬɟɦɨɣ ɤɨɨɪɞɢɧɚɬ, ɦɚɫɲɬɚɛɧɵɦɢ
ɥɢɧɟɣɤɚɦɢ ɢ ɢɞɟɧɬɢɱɧɵɦɢ ɫɢɧɯɪɨɧɢɡɨɜɚɧɧɵɦɢ ɱɚɫɚɦɢ. ɉɨɞ ɱɚɫɚɦɢ ɦɵ ɛɭɞɟɦ ɩɨɧɢɦɚɬɶ ɥɸɛɨɣ
ɩɟɪɢɨɞɢɱɟɫɤɢɣ ɩɪɨɰɟɫɫ, ɚ ɩɨɞ ɜɪɟɦɟɧɟɦ — ɩɨɤɚɡɚɧɢɟ ɱɚɫɨɜ. ɉɪɢ ɜɵɜɨɞɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ
ɢɫɩɨɥɶɡɭɟɬɫɹ ɨɛɵɱɧɨ ɞɟɤɚɪɬɨɜɚ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ. ȼɵɛɟɪɟɦ ɜ ɂɋɈ K ɫɬɚɧɞɚɪɬɵ ɞɥɢɧɵ ɢ ɫɬɚɧɞɚɪɬ
ɜɪɟɦɟɧɢ (ɱɬɨ ɫɜɹɡɚɧɨ ɫ ɜɵɛɨɪɨɦ ɩɟɪɢɨɞɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ). ɋɬɚɧɞɚɪɬɵ ɞɥɢɧɵ ɦɨɝɭɬ ɛɵɬɶ ɪɚɡɥɢɱɧɵ ɩɨ
ɨɫɹɦ x, y, z. Ɉɛɨɡɧɚɱɢɦ ɷɬɢ ɫɬɚɧɞɚɪɬɵ ɱɟɪɟɡ ex, ey, ez ɢ et, Ⱥɧɚɥɨɝɢɱɧɨ ɩɨɫɬɭɩɢɦ ɨɬɧɨɫɢɬɟɥɶɧɨ ɂɋɈ K’;
ɬɚɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɫɬɚɧɞɚɪɬɵ ɛɭɞɭɬ ex’, ey’, ez’, et’. ɂɡ ɨɞɧɨɪɨɞɧɨɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɢ ɜɪɟɦɟɧɢ ɢ
ɫɩɟɰɢɚɥɶɧɨɝɨ ɜɵɛɨɪɚ ɂɋɈ K ɢ K’ ɨɛɳɢɣ ɜɢɞ ɮɭɧɤɰɢɨɧɚɥɶɧɨɣ ɫɜɹɡɢ ɦɟɠɞɭ ɤɨɨɪɞɢɧɚɬɚɦɢ ɢ ɜɪɟɦɟɧɟɦ
ɨɞɧɨɝɨ ɢ ɬɨɝɨ ɠɟ ɫɨɛɵɬɢɹ, ɢɡɦɟɪɟɧɧɵɦɢ ɜ ɂɋɈ K ɢ K’ ɫɜɨɞɢɬɫɹ ɤ ɜɢɞɭ
x’ = a11 x + a14 t
y’ = a22 y
(1)
z’ = a33 z
t’ = a41 x + a44 t
ɝɞɟ a11, a14, a41, a44, a22, a33 — ɩɚɪɚɦɟɬɪɵ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ.
ȼɜɟɞɟɦ ɜɦɟɫɬɨ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ x, y, z, t, x’, y’, z’, t’ ɢɯ ɛɟɡɪɚɡɦɟɪɧɵɟ ɡɧɚɱɟɧɢɹ x*, y*, z*, t*,
x*’, y*’, z*’, t*’, ɬɨɝɞɚ
x*’ex’=a11x*ex+a14t*et
y*’ey’=a22y*ey
(2)
z*’ez’=a33z*ez
t*’et’=a41x*ex+a44t*et.
Ɂɚɦɟɬɢɦ, ɱɬɨ a11, a22, a33, a44 — ɛɟɡɪɚɡɦɟɪɧɵɟ ɜɟɥɢɱɢɧɵ a14 ɢ a41 — ɪɚɡɦɟɪɧɵɟ.
Ɋɚɫɫɦɨɬɪɢɦ ɞɚɥɟɟ ɫɨɛɵɬɢɟ, ɫɨɫɬɨɹɳɟɟ ɜ ɩɪɢɯɨɞɟ ɜ ɬɨɱɤɭ ɫ ɤɨɨɪɞɢɧɚɬɨɣ x* ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t*
ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɂɋɈ K’. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (2) ɢɡ ɩɟɪɜɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫɥɟɞɭɟɬ
0=a11x*ex+a14t*et.
(3)
ɉɨɞɟɥɢɦ ɥɟɜɭɸ ɢ ɩɪɚɜɭɸ ɱɚɫɬɢ ɪɚɜɟɧɫɬɜɚ ɧɚ et t*ɩɨɥɭɱɢɦ
0=a11x*ex/ ett*+a14.
ȿɫɥɢ ɨɛɨɡɧɚɱɢɬɶ x*/ t* ɱɟɪɟɡ V*, ɬɨ ɜɢɞɢɦ, ɱɬɨ
a14 = - V* a11ex/ et.
120
(4)
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 2, ʋ1, 1999 ɝ.
ɫɬɪ.119-124
Ɍɚɤ ɤɚɤ a11 — ɛɟɡɪɚɡɦɟɪɧɨ, ɬɨ a14 — ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ ɫɤɨɪɨɫɬɢ. ɋɚɦɚ ɠɟ ɜɟɥɢɱɢɧɚ V* ex/ et
ɢɦɟɟɬ ɩɪɨɫɬɨɣ ɮɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ — ɷɬɨ ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɂɋɈ K’ ɨɬɧɨɫɢɬɟɥɶɧɨ K.
Ɋɚɫɫɦɨɬɪɢɦ ɞɚɥɟɟ ɜ ɂɋɈ K’ ɫɨɛɵɬɢɟ, ɤɨɬɨɪɨɟ ɫɨɫɬɨɢɬ ɜ ɩɪɢɯɨɞɟ ɧɚɱɚɥɚ ɂɋɈ K ɜ ɦɨɦɟɧɬ t*’ ɜ
ɬɨɱɤɭ x*’. Ɍɨɝɞɚ ɤɨɨɪɞɢɧɚɬɵ ɷɬɨɝɨ ɠɟ ɫɨɛɵɬɢɹ ɜ K: x* = 0, y* = 0, z* = 0, ɚ ɜɪɟɦɹ ɷɬɨɝɨ ɫɨɛɵɬɢɹ t*, ɬɚɤ
ɱɬɨ ɩɟɪɜɨɟ ɢ ɱɟɬɜɟɪɬɨɟ ɭɪɚɜɧɟɧɢɹ ɫɢɫɬɟɦɵ (2) ɩɪɢɧɢɦɚɸɬ ɜɢɞ
x*’ex’=a14t*et,
t*’et’=a44t*et.
ɉɨɞɟɥɢɦ ɩɟɪɜɨɟ ɭɪɚɜɧɟɧɢɟ ɧɚ ɜɬɨɪɨɟ ɢ ɩɨɥɶɡɭɹɫɶ (4) ɩɨɥɭɱɢɦ
x*’ex’/( t*’et’)= -( ex/ et )V*( a11 / a44 ).
Ɉɬɫɸɞɚ
( ex/ e’x )(1/( e’t/ et ))V*’= - V* a11 / a44,
ɝɞɟ V*’ ( e’x/ e’t ) - ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ K ɨɬɧɨɫɢɬɟɥɶɧɨ K’. ȼɜɟɞɹ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɟɪɟɯɨɞɚ
ȕx=( e’x/ ex ); ȕt=( e’t/ et ),
ɩɨɥɭɱɢɦ ɟɳɟ ɨɞɧɨ ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ
a44 = - (V*/V*’)( ȕt /ȕx )a11 .
(5)
ɂɫɩɨɥɶɡɭɹ (4) ɢ (5), ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ (2) ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɜ ɜɢɞɟ
x*’ ȕx =a11(x* - V* t*)
y*’ ȕy =a22y*
(6)
z*’ ȕz =a33z*
t*’ ȕt =a41x*(ex /et ) – (V*/V*’)( ȕt /ȕx )a11t*.
Ɋɚɫɫɦɨɬɪɢɦ ɞɚɥɟɟ ɞɜɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɩɪɨɰɟɫɫɚ — ɩɨɫɵɥɤɭ ɢ ɩɪɢɟɦ ɫɜɟɬɨɜɨɝɨ ɫɢɝɧɚɥɚ ɢɡ ɬɨɱɤɢ A ɜ
ɬɨɱɤɭ B. ɉɭɫɬɶ ɬɨɱɤɚ A ɫɨɜɩɚɞɚɟɬ ɫ ɧɚɱɚɥɨɦ ɤɨɨɪɞɢɧɚɬ ɂɋɈ K ɜ ɦɨɦɟɧɬ t* = 0, ɬɨɱɤɚ B — ɫ
ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɨɣ ɩɪɨɫɬɪɚɧɫɬɜɚ, ɢɦɟɸɳɟɣ ɤɨɨɪɞɢɧɚɬɵ x*, y*, z* ɢ ɜɪɟɦɹ t*. Ɍɨɝɞɚ ɜɵɩɨɥɧɹɟɬɫɹ
ɪɚɜɟɧɫɬɜɨ
x2 + y2 + z2 = c2t2
ɢɥɢ
ex2x*2+ ey2y*2+ ez2z*2= c*2t*2 (el2 /et2 )et2 = c*2el2 t*2,
(7)
ɝɞɟ el — ɫɬɚɧɞɚɪɬ ɞɥɢɧɵ ɜɞɨɥɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɫɜɟɬɚ ɜ K. ɗɬɢ ɠɟ ɞɜɚ ɫɨɛɵɬɢɹ ɜ ɂɋɈ K’ ɢɦɟɸɬ
ɤɨɨɪɞɢɧɚɬɵ (0, 0, 0) ɢ t*’ = 0 ɞɥɹ ɬɨɱɤɢ A ɢ (x*’, y*’, z*’) ɜ ɦɨɦɟɧɬ t*’, ɞɥɹ ɬɨɱɤɢ B, ɬɚɤ ɱɬɨ
x’2 + y’2 + z’2 = c’2t’2
ɢɥɢ
ex’2 x*’2 + ey’2 y*’2+ ez’2 z*’2= c*’2 el ’2 t*’2,
(8)
ɝɞɟ el’ — ɫɬɚɧɞɚɪɬ ɞɥɢɧɵ ɜɞɨɥɶ ɧɚɩɪɚɜɥɟɧɢɹ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɫɜɟɬɚ ɜ K’. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɪɚɜɟɧɫɬɜɨ
ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ ɜɨ ɜɫɟɯ ɂɋɈ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
c*=c*’( ȕl /ȕt ),
121
(9)
ɉɢɜɨɜɚɪɨɜ ȼ.Ƚ., ɇɢɤɨɧɨɜ Ɉ.Ⱥ. Ɇɟɬɨɞɨɥɨɝɢɱɟɫɤɢɟ ɚɫɩɟɤɬɵ ɜɵɜɨɞɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ...
ɝɞɟ ȕl=el’/el . ɉɪɟɨɛɪɚɡɭɟɦ ɫ ɩɨɦɨɳɶɸ (8) ɲɬɪɢɯɨɜɚɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɢ ɜɪɟɦɹ ɜ ɧɟɲɬɪɢɯɨɜɚɧɧɵɟ, ɬɨɝɞɚ
ɲɬɪɢɯɨɜɚɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ A ɢ ɲɬɪɢɯɨɜɚɧɧɨɟ ɜɪɟɦɹ ɩɟɪɜɨɝɨ ɫɨɛɵɬɢɹ ɩɟɪɟɣɞɭɬ ɜ
ɧɟɲɬɪɢɯɨɜɚɧɧɵɟ, ɚɧɚɥɨɝɢɱɧɨ ɲɬɪɢɯɨɜɚɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ B ɢ ɲɬɪɢɯɨɜɚɧɧɨɟ ɜɪɟɦɹ ɜɬɨɪɨɝɨ
ɫɨɛɵɬɢɹ ɩɟɪɟɣɞɭɬ ɜ ɧɟɲɬɪɢɯɨɜɚɧɧɵɟ. Ɍɟɦ ɫɚɦɵɦ ɛɭɞɟɬ ɫɨɜɟɪɲɟɧ ɩɟɪɟɯɨɞ ɨɬ ɂɋɈ K’ ɤ ɂɋɈ K, ɩɪɢ ɷɬɨɦ
ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (9) c’ ɞɨɥɠɧɚ ɛɵɬɶ ɡɚɦɟɧɟɧɚ ɧɚ c. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɜɨɡɧɢɤɚɸɳɟɟ ɫɨɨɬɧɨɲɟɧɢɟ ɢ
ɫɨɨɬɧɨɲɟɧɢɟ (7) ɞɨɥɠɧɵ ɛɵɬɶ ɬɨɠɞɟɫɬɜɟɧɧɨ ɪɚɜɧɵ ɞɪɭɝ ɞɪɭɝɭ.
ɉɨɥɭɱɢɦ ɬɟ ɭɫɥɨɜɢɹ, ɩɪɢ ɤɨɬɨɪɵɯ ɷɬɨ ɜɵɩɨɥɧɹɟɬɫɹ:
a112 = 1+ c*2 a412 (el2/et2 ),
a11 = c*2 a41 (el2 ȕt /(et ex’) )(1/ V*’),
(10)
a112 ( V*2/ V*’2)( ȕt2 / ȕx2)(1- ex‘2 V*’2 /(el2 ȕt2 c*2))= 1.
Ʉɪɨɦɟ ɬɨɝɨ,
a222 = 1,
a332 = 1.
ɋɨɨɬɧɨɲɟɧɢɹ (10) ɫɨɜɦɟɫɬɧɵ, ɟɫɥɢ
V*’ = - V*( ȕt /ȕx ),
a11 = ((1 – ex2 V*2/(el2 c*2))-1/2,
(11)
a41 = - et ex V*/[ el2 c*2 ((1 – ex2 V*2/(el2 c*2))1/2].
ɉɨɞɫɬɚɜɥɹɹ (11) ɜ (6) ɩɨɫɥɟ ɩɪɨɫɬɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɩɨɥɭɱɢɦ
x*’ ȕx =(x* - V* t*)/((1 – ex2 V*2/(el2 c*2))1/2
y*’ ȕy =y*
(12)
z*’ ȕz =z*
t*’ ȕt =(t*- x* ex2 V*2/(el2 c*2))/((1 – ex2 V*2/(el2 c*2))1/2.
ɂɡ ɜɵɩɨɥɧɟɧɧɨɝɨ ɢɫɫɥɟɞɨɜɚɧɢɹ ɦɨɠɧɨ ɫɞɟɥɚɬɶ ɜɵɜɨɞ ɨ ɬɨɦ, ɱɬɨ ɩɨɫɬɭɥɚɬɵ ɗɣɧɲɬɟɣɧɚ ɧɟ
ɧɚɤɥɚɞɵɜɚɸɬ ɧɢɤɚɤɢɯ ɨɝɪɚɧɢɱɟɧɢɣ ɧɚ ɜɵɛɨɪ ɫɬɚɧɞɚɪɬɨɜ ɞɥɢɧɵ ɢ ɜɪɟɦɟɧɢ ɤɚɤ ɜɧɭɬɪɢ ɤɚɠɞɨɣ ɢɡ ɂɋɈ
ɬɚɤ ɢ ɧɚ ɨɞɧɨɜɪɟɦɟɧɧɵɣ ɢɯ ɜɵɛɨɪ. Ʉɪɨɦɟ ɬɨɝɨ, ɫɨɨɬɧɨɲɟɧɢɟ (12) ɧɟ ɢɦɟɟɬ ɮɢɡɢɱɟɫɤɨɝɨ ɫɦɵɫɥɚ ɞɨ ɬɟɯ
ɩɨɪ, ɩɨɤɚ ɨɫɬɚɸɬɫɹ ɧɟɢɡɜɟɫɬɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɟɪɟɯɨɞɚ Ex, Ey, Ez, Et. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɢɫɩɨɥɶɡɨɜɚɧɢɟ
ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ ɬɪɟɛɭɟɬ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɝɨ ɫɨɝɥɚɫɨɜɚɧɧɨɝɨ ɜɵɛɨɪɚ ɫɬɚɧɞɚɪɬɨɜ ɞɥɢɧɵ ɢ ɜɪɟɦɟɧɢ
ɜ ɂɋɈ K ɢ K’.
3. ɋɨɝɥɚɫɨɜɚɧɢɟ ɜɵɛɨɪɚ ɫɬɚɧɞɚɪɬɨɜ ɞɥɢɧɵ ɢ ɜɪɟɦɟɧɢ
Ⱦɥɹ ɫɨɝɥɚɫɨɜɚɧɧɨɝɨ ɜɵɛɨɪɚ ɫɬɚɧɞɚɪɬɨɜ ɞɥɢɧɵ ɢ ɜɪɟɦɟɧɢ ɜ ɂɋɈ K ɢ K’ ɜɵɛɟɪɟɦ ɩɪɨɢɡɜɨɥɶɧɵɦ
ɨɛɪɚɡɨɦ ɫɬɚɧɞɚɪɬ ɞɥɢɧɵ ɜ ɂɋɈ K. ɉɭɫɬɶ ɷɬɨ ɛɭɞɟɬ ɧɚɛɨɪ ɫɬɟɪɠɧɟɣ ɨɞɢɧɚɤɨɜɨɣ ɞɥɢɧɵ. ȼ ɂɋɈ K ɦɵ
ɪɚɫɩɨɥɨɠɢɦ ɷɬɢ ɫɬɟɪɠɧɢ ɜɞɨɥɶ ɨɫɟɣ Ox, Oy, Oz , ɬɟɦ ɫɚɦɵɦ ɡɚɞɚɜ ɨɞɢɧ ɢ ɬɨɬ ɠɟ ɫɬɚɧɞɚɪɬ ɞɥɢɧɵ ɩɨ ɜɫɟɦ
ɬɪɟɦ ɨɫɹɦ. Ɉɛɨɡɧɚɱɢɦ ɟɝɨ ɱɟɪɟɡ e. ȼ ɫɢɥɭ ɢɡɨɬɪɨɩɧɨɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɫɬɟɪɠɧɢ ɦɨɝɭɬ ɛɵɬɶ ɩɨɜɟɪɧɭɬɵ
ɩɨɞ ɥɸɛɵɦ ɭɝɥɨɦ ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɜɧɭɬɪɢ ɂɋɈ ɨɞɧɢɦ ɢ ɬɟɦ ɠɟ ɫɬɚɧɞɚɪɬɨɦ ɞɥɢɧɵ ɦɨɠɧɨ ɢɡɦɟɪɹɬɶ
ɪɚɫɫɬɨɹɧɢɹ ɦɟɠɞɭ ɞɜɭɦɹ ɥɸɛɵɦɢ ɬɨɱɤɚɦɢ. Ɂɚɫɬɚɜɢɦ ɞɚɥɟɟ ɷɬɢ ɫɬɟɪɠɧɢ, ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɪɚɫɩɨɥɨɠɢɜ ɢɯ
ɜɞɨɥɶ ɨɫɟɣ Oy ɢɥɢ Oz, ɞɜɢɝɚɬɶɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ V. Ʉɚɤ ɫɥɟɞɭɟɬ ɢɡ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ, ɞɥɢɧɵ
ɫɬɟɪɠɧɟɣ ɧɟ ɢɡɦɟɧɹɸɬɫɹ ɩɪɢ ɬɚɤɨɦ ɢɯ ɪɚɫɩɨɥɨɠɟɧɢɢ. ɋɜɹɠɟɦ ɫ ɧɢɦɢ ɂɋɈ K’. ȼ ɂɋɈ K’ ɷɬɢ ɫɬɟɪɠɧɢ
ɩɨɤɨɹɬɫɹ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɯ ɦɨɠɧɨ ɪɚɫɩɨɥɚɝɚɬɶ ɬɚɤ ɤɚɤ ɧɚɦ ɧɟɨɛɯɨɞɢɦɨ. Ɋɚɫɩɨɥɨɠɢɦ ɨɞɢɧ ɢɡ
ɫɬɟɪɠɧɟɣ ɜɞɨɥɶ ɨɫɢ O’x’, ɨɫɬɚɜɢɜ ɨɫɬɚɥɶɧɵɟ ɩɨɤɨɢɬɶɫɹ ɜɞɨɥɶ ɨɫɟɣ O’y’, O’z’. Ɍɟɦ ɫɚɦɵɦ ɦɵ ɡɚɞɚɞɢɦ
ɫɬɚɧɞɚɪɬ ɞɥɢɧɵ ɜ ɂɋɈ K’ , ɨɞɢɧ ɢ ɬɨɬ ɠɟ ɜɞɨɥɶ ɬɪɟɯ ɨɫɟɣ O’x’, O’y’, O’z’. ɂɡɨɬɪɨɩɢɹ ɩɪɨɫɬɪɚɧɫɬɜɚ
ɩɨɡɜɨɥɹɟɬ ɩɪɨɜɨɞɢɬɶ ɢɡɦɟɪɟɧɢɹ ɷɬɢɦɢ ɫɬɟɪɠɧɹɦɢ ɜɞɨɥɶ ɥɸɛɨɝɨ ɧɚɩɪɚɜɥɟɧɢɹ. Ɉɛɨɡɧɚɱɢɦ ɜɜɟɞɟɧɧɵɣ
ɫɬɚɧɞɚɪɬ ɱɟɪɟɡ e’.
122
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 2, ʋ1, 1999 ɝ.
ɫɬɪ.119-124
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɬɚɧɞɚɪɬ ɞɥɢɧɵ ɜɞɨɥɶ ɨɫɢ O’x’ ɫɨɜɩɚɞɚɟɬ ɫɨ ɫɬɚɧɞɚɪɬɚɦɢ ɞɥɢɧɵ ɜɞɨɥɶ ɨɫɟɣ O’y’
ɢ O’z’, ɤɨɬɨɪɵɟ ɜ ɫɜɨɸ ɨɱɟɪɟɞɶ ɫɨɜɩɚɞɚɸɬ ɫɨ ɫɬɚɧɞɚɪɬɚɦɢ ɞɥɢɧɵ ɜɞɨɥɶ Oy ɢ Oz. ɇɨ ɬɚɤ ɤɚɤ ɷɬɢ
ɫɬɚɧɞɚɪɬɵ ɪɚɜɧɵ (ɬɨɠɞɟɫɬɜɟɧɧɨ) ɫɬɚɧɞɚɪɬɭ ɜɞɨɥɶ ɨɫɢ Ox, ɬɨ ɫɬɚɧɞɚɪɬɵ ɜɞɨɥɶ Ox ɢ ɜɞɨɥɶ O’x’ ɬɚɤɠɟ
ɪɚɜɧɵ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, e = e’.
ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɢɡɦɟɪɢɦ ɞɥɢɧɭ ɫɬɟɪɠɧɹ, ɩɨɤɨɹɳɟɝɨɫɹ ɜ K’ ɢ ɪɚɫɩɨɥɨɠɟɧɧɨɝɨ ɜɞɨɥɶ ɨɫɢ O’x’
ɜ ɂɋɈ K. ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ ɩɪɢɜɨɞɢɬ ɤ ɞɥɢɧɟ ɷɬɨɝɨ ɫɬɟɪɠɧɹ
l = e(1 – V2/ c2)1/2,
ɝɞɟ e — ɞɥɢɧɚ ɩɨɤɨɹɳɟɝɨɫɹ ɜ K’ ɫɬɟɪɠɧɹ, ɚ V — ɫɤɨɪɨɫɬɶ ɟɝɨ ɩɟɪɟɦɟɳɟɧɢɹ ɜɞɨɥɶ ɨɫɢ Ox ɨɬɧɨɫɢɬɟɥɶɧɨ
ɂɋɈ K. Ɉɬɫɸɞɚ ɜɢɞɧɨ, ɱɬɨ l < e. ɇɨ ɷɬɨ ɧɚɯɨɞɢɬɫɹ ɜ ɨɱɟɜɢɞɧɨɦ ɩɪɨɬɢɜɨɪɟɱɢɢ ɫ ɬɟɦ ɨɛɫɬɨɹɬɟɥɶɫɬɜɨɦ,
ɱɬɨ ɩɪɟɞɥɨɠɟɧɧɚɹ ɧɚɦɢ ɩɪɨɰɟɞɭɪɚ ɫɨɝɥɚɫɨɜɚɧɢɹ ɫɬɚɧɞɚɪɬɨɜ ɨɛɟɫɩɟɱɢɜɚɟɬ ɪɚɜɟɧɫɬɜɨ ɞɥɢɧ ɫɬɟɪɠɧɟɣ ɜ K
ɢ K’. ɋɧɹɬɢɟ ɷɬɨɝɨ ɩɪɨɬɢɜɨɪɟɱɢɹ ɫɜɹɡɚɧɨ ɫ ɩɪɟɞɩɨɥɨɠɟɧɢɟɦ, ɱɬɨ ɮɢɡɢɱɟɫɤɚɹ ɞɥɢɧɚ ɫɬɟɪɠɧɟɣ ɩɪɢ
ɩɟɪɟɯɨɞɟ ɨɬ ɨɞɧɨɣ ɂɋɈ ɤ ɞɪɭɝɨɣ ɨɫɬɚɟɬɫɹ ɧɟɢɡɦɟɧɧɨɣ, ɚ ɦɟɧɹɟɬɫɹ ɥɢɲɶ ɪɟɡɭɥɶɬɚɬ ɢɡɦɟɪɟɧɢɹ ɞɥɢɧɵ,
ɤɨɬɨɪɵɣ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɪɨɰɟɞɭɪɨɣ ɢɡɦɟɪɟɧɢɹ ɞɥɢɧɵ ɞɜɢɠɭɳɟɝɨɫɹ ɫɬɟɪɠɧɹ. ɋɚɦɚ ɩɪɨɰɟɞɭɪɚ ɢɡɦɟɪɟɧɢɹ
ɩɪɨɞɢɤɬɨɜɚɧɚ ɬɪɟɛɨɜɚɧɢɟɦ ɩɨɫɬɨɹɧɫɬɜɚ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ ɢ ɟɟ ɤɨɧɟɱɧɨɫɬɶɸ. Ⱦɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɥɭɱɲɟ ɩɨɧɹɬɶ
ɫɤɚɡɚɧɧɨɟ, ɪɚɫɫɦɨɬɪɢɦ ɫɥɟɞɭɸɳɭɸ ɩɪɨɰɟɞɭɪɭ ɢɡɦɟɪɟɧɢɹ ɞɥɢɧɵ ɫɬɟɪɠɧɹ. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɡɚ ɞɥɢɧɭ
ɫɬɟɪɠɧɹ AB ɜ ɩɪɨɢɡɜɨɥɶɧɨɣ ɂɋɈ K ɛɭɞɟɦ ɩɪɢɧɢɦɚɬɶ ɞɥɢɧɭ ɩɪɨɟɤɰɢɢ ɫɬɟɪɠɧɹ ɧɚ ɩɟɪɩɟɧɞɢɤɭɥɹɪ ɤ ɥɭɱɭ,
ɫɨɟɞɢɧɹɸɳɟɦɭ ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɫ ɫɟɪɟɞɢɧɨɣ ɫɬɟɪɠɧɹ (ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 1)
Ɋɢɫ.1. Ⱦɥɢɧɚ ɫɬɟɪɠɧɹ ɜ ɂɋɈ K1, K2, K3
Ɍɨɝɞɚ ɜ ɂɋɈ K1 ɫ ɧɚɱɚɥɨɦ ɜ ɬɨɱɤɟ O1 ɞɥɢɧɚ ɫɬɟɪɠɧɹ A1B1, ɚ ɜ ɂɋɈ K2 ɫ ɧɚɱɚɥɨɦ O2 ɞɥɢɧɚ ɷɬɨɝɨ ɠɟ
ɫɬɟɪɠɧɹ A2B2. Ɇɵ ɜɢɞɢɦ, ɱɬɨ ɩɪɟɞɥɨɠɟɧɧɚɹ ɩɪɨɰɟɞɭɪɚ ɢɡɦɟɪɟɧɢɹ ɞɥɢɧɵ ɫɬɟɪɠɧɹ ɞɟɥɚɟɬ ɷɬɭ ɜɟɥɢɱɢɧɭ
ɨɬɧɨɫɢɬɟɥɶɧɨɣ, ɯɨɬɹ ɫɚɦ ɫɬɟɪɠɟɧɶ AB ɩɪɢ ɷɬɨɦ ɧɟ ɡɚɬɪɚɝɢɜɚɟɬɫɹ. ɉɨɞɱɟɪɤɧɟɦ, ɱɬɨ ɷɬɨ ɧɟ ɨɛɴɹɫɧɟɧɢɟ
ɥɨɪɟɧɰɨɜɫɤɨɝɨ ɫɨɤɪɚɳɟɧɢɹ ɞɥɢɧɵ, ɚ ɥɢɲɶ ɢɥɥɸɫɬɪɚɰɢɹ ɤ ɮɪɚɡɟ ɨ ɜɨɡɦɨɠɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɢɡɦɟɪɹɟɦɨɝɨ
ɩɚɪɚɦɟɬɪɚ ɨɬ ɩɪɟɞɥɨɠɟɧɧɨɣ ɩɪɨɰɟɞɭɪɵ.
ɉɟɪɟɣɞɟɦ ɬɟɩɟɪɶ ɤ ɫɨɝɥɚɫɨɜɚɧɢɸ ɜɵɛɨɪɚ ɫɬɚɧɞɚɪɬɨɜ ɜɪɟɦɟɧɢ ɜ ɂɋɈ K ɢ K’. ȼ ɂɋɈ K
ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɤɨɥɟɛɚɬɟɥɶɧɵɦ ɩɪɨɰɟɫɫɨɦ, ɜɨɡɧɢɤɚɸɳɢɦ ɩɪɢ ɞɜɢɠɟɧɢɢ ɜɞɨɥɶ ɨɫɢ Y ɫɜɟɬɨɜɨɝɨ ɢɦɩɭɥɶɫɚ
ɦɟɠɞɭ ɞɜɭɦɹ ɡɟɪɤɚɥɚɦɢ, ɩɥɨɫɤɨɫɬɢ ɤɨɬɨɪɵɯ ɩɚɪɚɥɥɟɥɶɧɵ ɤɨɨɪɞɢɧɚɬɧɨɣ ɩɥɨɫɤɨɫɬɢ XOZ. ȿɫɥɢ L –
ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɡɟɪɤɚɥɚɦɢ, ɬɨ
T = 2 L/c.
Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɩɨɫɬɭɩɢɬɶ ɫ ɜɵɛɨɪɨɦ ɫɬɚɧɞɚɪɬɚ ɜɪɟɦɟɧɢ ɜ ɂɋɈ K’. ȼɵɛɢɪɚɹ ɪɚɫɫɬɨɹɧɢɟ
ɦɟɠɞɭ ɡɟɪɤɚɥɚɦɢ L’, ɪɚɜɧɨɟ L, ɱɬɨ ɦɨɠɧɨ ɫɞɟɥɚɬɶ ɜɫɟɝɞɚ, ɤɚɤ ɨɛ ɷɬɨɦ ɝɨɜɨɪɢɥɨɫɶ ɜɵɲɟ, ɢ ɢɫɩɨɥɶɡɭɹ
ɩɨɫɬɭɥɚɬ ɨ ɩɨɫɬɨɹɧɫɬɜɟ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ ɜ K ɢ K ’, ɩɨɥɭɱɢɦ
T ’ = T = 2 L/c.
ɇɨ ɷɬɨ ɭɬɜɟɪɠɞɟɧɢɟ ɧɚɯɨɞɢɬɫɹ ɜ ɩɪɨɬɢɜɨɪɟɱɢɢ ɫ ɪɟɡɭɥɶɬɚɬɨɦ ɢɡɦɟɪɟɧɢɹ ɯɨɞɚ ɞɜɢɠɭɳɢɯɫɹ ɱɚɫɨɜ
ɧɚ ɨɫɧɨɜɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ, ɩɨɡɜɨɥɹɸɳɢɯ ɫɞɟɥɚɬɶ ɜɵɜɨɞ ɨ ɡɚɦɟɞɥɟɧɢɢ ɯɨɞɚ ɞɜɢɠɭɳɢɯɫɹ ɱɚɫɨɜ.
ɗɬɨ ɩɪɨɬɢɜɨɪɟɱɢɟ ɦɨɠɧɨ ɫɧɹɬɶ, ɟɫɥɢ ɩɪɟɞɩɨɥɨɠɢɬɶ, ɱɬɨ ɩɪɨɰɟɞɭɪɚ ɢɡɦɟɪɟɧɢɹ ɩɟɪɢɨɞɚ ɤɨɥɟɛɚɧɢɹ ɜ
ɂɋɈ ɞɜɢɠɭɳɢɯɫɹ ɜɦɟɫɬɟ ɫ K’ ɱɚɫɨɜ ɫɚɦ ɩɪɨɰɟɫɫ ɤɨɥɟɛɚɧɢɹ ɧɟ ɡɚɬɪɚɝɢɜɚɟɬ ɢ ɧɟ ɜɥɢɹɟɬ ɧɚ ɮɢɡɢɱɟɫɤɭɸ
123
ɉɢɜɨɜɚɪɨɜ ȼ.Ƚ., ɇɢɤɨɧɨɜ Ɉ.Ⱥ. Ɇɟɬɨɞɨɥɨɝɢɱɟɫɤɢɟ ɚɫɩɟɤɬɵ ɜɵɜɨɞɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ...
ɜɟɥɢɱɢɧɭ ɩɪɨɦɟɠɭɬɤɚ ɜɪɟɦɟɧɢ ɦɟɠɞɭ ɞɜɭɦɹ ɤɨɥɟɛɚɧɢɹɦɢ ɱɚɫɨɜ, ɚ ɞɚɟɬ ɥɢɲɶ ɪɟɡɭɥɶɬɚɬ ɢɡɦɟɪɟɧɢɹ
ɩɟɪɢɨɞɚ, ɚɧɚɥɨɝɢɱɧɨ ɬɨɦɭ, ɱɬɨ ɦɵ ɢɦɟɥɢ ɩɪɢ ɢɡɦɟɪɟɧɢɢ ɞɥɢɧɵ, ɢ ɨɬɪɚɠɚɟɬ ɩɪɨɰɟɞɭɪɭ ɢɡɦɟɪɟɧɢɹ
ɜɪɟɦɟɧɢ.
ɉɨɞɨɛɧɨ ɬɨɦɭ, ɤɚɤ ɦɵ ɢɥɥɸɫɬɪɢɪɨɜɚɥɢ ɡɚɜɢɫɢɦɨɫɬɶ ɢɡɦɟɪɟɧɢɹ ɞɥɢɧɵ ɬɟɥɚ ɨɬ ɩɪɟɞɥɚɝɚɟɦɨɣ
ɩɪɨɰɟɞɭɪɵ ɢɡɦɟɪɟɧɢɹ, ɦɨɠɧɨ ɩɪɨɢɥɥɸɫɬɪɢɪɨɜɚɬɶ ɢ ɢɡɦɟɪɟɧɢɟ ɜɪɟɦɟɧɧɵɯ ɢɧɬɟɪɜɚɥɨɜ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ
ɪɟɰɟɩɬɚ ɢɡɦɟɪɟɧɢɹ ɜɪɟɦɟɧɢ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɪɢɦɟɦ ɡɚ ɜɪɟɦɟɧɧɨɣ ɢɧɬɟɪɜɚɥ ɦɟɠɞɭ ɞɜɭɦɹ ɫɨɛɵɬɢɹɦɢ,
ɩɪɨɢɫɯɨɞɹɳɢɦɢ, ɧɚɩɪɢɦɟɪ, ɧɚ ɭɞɚɥɹɸɳɢɯɫɹ (ɢɥɢ ɩɪɢɛɥɢɠɚɸɳɢɯɫɹ) ɷɥɟɤɬɪɢɱɤɚɯ, ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ
ɦɟɠɞɭ ɞɜɭɦɹ ɡɜɭɤɨɜɵɦɢ ɝɭɞɤɚɦɢ (ɢɦɩɭɥɶɫɚɦɢ ɫɜɟɬɚ), ɢɫɩɭɳɟɧɧɵɦɢ ɜ ɦɨɦɟɧɬ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ
ɫɨɛɵɬɢɣ. ȿɫɥɢ ɷɥɟɤɬɪɢɱɤɚ ɛɭɞɟɬ ɩɪɢɛɥɢɠɚɬɶɫɹ ɤ ɧɚɦ, ɬɨ ɜɪɟɦɟɧɧɵɟ ɢɧɬɟɪɜɚɥɵ, ɩɪɢɧɢɦɚɟɦɵɟ ɧɚɦɢ,
ɛɭɞɭɬ ɦɟɧɶɲɟ ɪɟɚɥɶɧɵɯ, ɚ ɟɫɥɢ ɭɞɚɥɹɬɶɫɹ, ɬɨ ɛɨɥɶɲɟ. ɇɨ ɜɪɹɞ ɥɢ ɤɬɨ-ɧɢɛɭɞɶ ɜɨɡɶɦɟɬɫɹ ɭɬɜɟɪɠɞɚɬɶ, ɱɬɨ
ɪɟɚɥɶɧɵɣ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ ɬɨɬ, ɤɨɬɨɪɵɣ ɦɵ ɜɨɫɩɪɢɧɢɦɚɟɦ, ɚ ɧɟ ɬɨɬ, ɤɨɬɨɪɵɣ
ɢɡɦɟɪɹɟɬɫɹ ɧɚ ɷɥɟɤɬɪɢɱɤɟ ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ.
ɋɤɚɡɚɧɧɨɟ ɩɨɡɜɨɥɹɟɬ ɭɬɜɟɪɠɞɚɬɶ, ɱɬɨ ɥɨɪɟɧɰɨɜɫɤɨɟ ɫɨɤɪɚɳɟɧɢɟ ɞɥɢɧ ɢ ɡɚɦɟɞɥɟɧɢɟ ɯɨɞɚ
ɞɜɢɠɭɳɢɯɫɹ ɱɚɫɨɜ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɨɬɪɚɠɟɧɢɟ ɜɜɟɞɟɧɧɵɯ ɪɟɰɟɩɬɨɜ ɢɡɦɟɪɟɧɢɹ ɜɪɟɦɟɧɢ ɢ
ɞɥɢɧɵ ɞɜɢɠɭɳɢɯɫɹ ɨɛɴɟɤɬɨɜ.
3. Ɂɚɤɥɸɱɟɧɢɟ
ȼɵɩɨɥɧɟɧɧɵɣ ɚɧɚɥɢɡ ɜɨɡɦɨɠɧɵɯ ɨɝɪɚɧɢɱɟɧɢɣ ɧɚ ɜɵɛɨɪ ɫɬɚɧɞɚɪɬɨɜ ɞɥɢɧɵ ɢ ɜɪɟɦɟɧɢ ɩɪɢ
ɜɵɛɨɪɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ ɧɚ ɨɫɧɨɜɟ ɩɨɫɬɭɥɚɬɨɜ ɗɣɧɲɬɟɣɧɚ ɭɤɚɡɵɜɚɟɬ ɧɚ ɧɟɨɛɯɨɞɢɦɨɫɬɶ
ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɝɨ ɫɨɝɥɚɫɨɜɚɧɢɹ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɷɬɢɯ ɫɬɚɧɞɚɪɬɨɜ ɜ ɢɫɩɨɥɶɡɭɟɦɵɯ ɂɋɈ K ɢ K ’. Ȼɟɡ ɬɚɤɨɝɨ
ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɝɨ ɫɨɝɥɚɫɨɜɚɧɢɹ ɜɬɨɪɨɣ ɩɨɫɬɭɥɚɬ ɗɣɧɲɬɟɣɧɚ ɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ, ɧɚ ɧɚɲ ɜɡɝɥɹɞ,
ɥɢɲɚɸɬɫɹ ɮɢɡɢɱɟɫɤɨɝɨ ɫɨɞɟɪɠɚɧɢɹ. ɉɪɟɞɥɨɠɟɧɧɚɹ ɩɪɨɰɟɞɭɪɚ ɫɨɝɥɚɫɨɜɚɧɢɹ ɩɨɡɜɨɥɹɟɬ ɫɞɟɥɚɬɶ ɜɵɜɨɞ ɨ
ɬɨɦ, ɱɬɨ ɥɨɪɟɧɰɨɜɫɤɨɟ ɫɨɤɪɚɳɟɧɢɟ ɞɥɢɧɵ ɢ ɡɚɦɟɞɥɟɧɢɟ ɯɨɞɚ ɞɜɢɠɭɳɢɯɫɹ ɱɚɫɨɜ ɧɟ ɡɚɬɪɚɝɢɜɚɟɬ ɫɚɦɢ
ɮɢɡɢɱɟɫɤɢɟ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɟ ɢ ɜɪɟɦɟɧɧɵɟ ɢɧɬɟɪɜɚɥɵ, ɚ ɥɢɲɶ ɨɬɪɚɠɚɸɬ ɱɟɪɟɡ ɩɪɟɞɥɚɝɚɟɦɵɟ ɩɪɨɰɟɞɭɪɵ
ɢɡɦɟɪɟɧɢɹ ɞɥɢɧɵ ɞɜɢɠɭɳɟɝɨɫɹ ɬɟɥɚ ɢ ɫɤɨɪɨɫɬɢ ɯɨɞɚ ɞɜɢɠɭɳɟɝɨɫɹ ɩɟɪɢɨɞɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ ɬɪɟɛɨɜɚɧɢɟ
ɩɨɫɬɨɹɧɫɬɜɚ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ ɩɪɢ ɩɟɪɟɯɨɞɟ ɨɬ ɨɞɧɨɣ ɂɋɈ ɤ ɞɪɭɝɨɣ.
Ʌɢɬɟɪɚɬɭɪɚ
Ⱦɟɬɥɚɮ Ⱥ.Ⱥ., əɜɨɪɫɤɢɣ Ȼ.Ɇ. Ʉɭɪɫ ɮɢɡɢɤɢ. ɍɱɟɛ. ɩɨɫɨɛɢɟ ɞɥɹ ɜɬɭɡɨɜ. Ɇ., ȼɵɫɲ. ɲɤ., 608ɫ., 1989.
Ɇɚɧɞɟɥɶɲɬɚɦ Ʌ.ɂ. Ʌɟɤɰɢɢ ɩɨ ɨɩɬɢɤɟ, ɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɢ ɤɜɚɧɬɨɜɨɣ ɦɟɯɚɧɢɤɟ. Ɇ., ɇɚɭɤɚ, 440ɫ.,
1972.
Ɇɚɬɜɟɟɜ Ⱥ.ɇ. Ɇɟɯɚɧɢɤɚ ɢ ɬɟɨɪɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ. ɍɱɟɛ. ɩɨɫɨɛɢɟ ɞɥɹ ɮɢɡ. ɫɩɟɰ. ɜɭɡɨɜ. Ɇ., ȼɵɫɲ. ɲɤ.,
320ɫ., 1986.
ɋɚɜɟɥɶɟɜ ɂ.ȼ. Ʉɭɪɫ ɮɢɡɢɤɢ. ɍɱɟɛ. ȼ 3 ɬ. Ɍ.1. Ɇɟɯɚɧɢɤɚ. Ɇɨɥɟɤɭɥɹɪɧɚɹ ɮɢɡɢɤɚ. Ɇ., ɇɚɭɤɚ. Ƚɥ. ɪɟɞ. ɮɢɡ.ɦɚɬ. ɥɢɬ., 352ɫ., 1989.
ɋɢɜɭɯɢɧ Ⱦ.ȼ. Ɉɛɳɢɣ ɤɭɪɫ ɮɢɡɢɤɢ. Ɉɩɬɢɤɚ. Ɇ., ɇɚɭɤɚ, 752ɫ., 1980.
Ɍɪɨɮɢɦɨɜɚ Ɍ.ɂ. Ʉɭɪɫ ɮɢɡɢɤɢ. ɍɱɟɛ. ɩɨɫɨɛɢɟ ɞɥɹ ɫɬɭɞɟɧɬɨɜ. Ɇ., ȼɵɫɲ. ɲɤ., 542ɫ., 1997.
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