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7.3. New law of speeds composition, superlight speeds and causality principle

Let us introduce into consideration a third inertial reference frame G (besides two inertial reference frames A and B from chapter 3), which moves at a speed s in the positive direction of x' axis of the reference frame B, at that let the x'' axis of the reference frame G to coincide with axes x and x' and axes y'' and z'' of the reference frame G let be parallel to the appropriate axes of inertial reference frames A and B.  Let us also assume that at the zero moment of time of all that three inertial referencce frames the origins of all three reference frames coincide with each other. Let us designate by the letter w the speed of the inertial frame G motion relatively the reference frame A. This speed w we must determine using  known speeds u (the speed of motion of the refeerence frame B relatively the reference frame A) and s (the speed of motion of the reference frame G relatively the reference frame B).

Let us write down transformations of coordinates and time for events happening to a body, which is at rest in the reference frame G, from the reference frame G to the reference frame B and from the reference frame G to the reference frame A (having omitted trivial equalities for coordinates y and z)

x' = Г_s (x'' + B_sc_ot'' ), c_st' = Г_s(c_ot'' + B_sx''), (7.24)

x = Г_w (x'' + B_wc_ot'' ), c_wt' = Г_w(c_ot'' + B_wx''), (7.25)

where    Г_s= (1 - B_s²)^-1/2;    B_s= s/c_s;     c_s = c_o(1 + s²/c_o²)^1/2.

Having solved transformations (7.24) with respect to events coordinates in the reference frame G we obtain

x'' = Г_s (x' - B_sc_st' ),  c_ot'' = Г_s(c_st' - B_sx'). (7.26)

Substituting expressions (7.26) into transformations (7.25) we have

x = Г_sГ_w (1 - B_sB_w)[x' + c_st' (B_w - B_s)/( 1 - B_sB_w)],

c_wt = Г_sГ_w (1 - B_sB_w)[c_st'  + x' (B_w - B_s)/( 1 - B_sB_w)]. (7.27)

Transformations (7.27) are transformations of coordinates and time for events happening to a boddy, which is at rest in the reference frame G, from the inertial reference frame B to the inertial reference frame A. Comparing transformations (7.27) with transformations (6.10) we can write down transformations (7.27) in the form

x = Г_u (x' + B_uc_st'),   c_w t = Г_u (c_s t'  + B_u x'),  (7.28)

where

B_u= (B_w - B_s)/( 1 - B_sB_w); (7.29)

Г_u= Г_sГ_w (1 - B_sB_w);  (7.30)

Г_u= (1 - B_u²)^-1/2;    B_u= u/c_u;     c_u = c_o(1 + u²/c_o²)^1/2.

From the expression (7.29) we find

B_w= (B_u + B_s)/( 1 + B_uB_s). (7.31)

From  expressions (7.30) and (7.31) we have

Г_w= Г_uГ_s (1 + B_uB_s). (7.32)

From expressions (7.31) and (7.32) we can also obtain the formula

w = u Г_s+ s Г_u. (7.33)

Expressions  (7.31) and (7.33) are two various forms of the speeds composition law in the new space-time theory. The quantities  u and s enter into the new speeds composition law  (7.31) or (7.33) simmetrically. At that the new speeds composition law is converted into the speeds composition law of the special relativity theory under a condition, if the speeds u, s and w are small as compared with the speed of light c_o. Indeed, the expression (7.31) can be written in the form

...(7.31a)

At small values of quantities  u, s and w  each square root in the expression (7.31а) is approximately equal to unity. Then the expression (7.31a) is converted into the speeds composition law of the special relativity theory

. (7.34)

But it is known, that if we consider transformations as components of some set, a product of two transformations is equal to a transformation resulting from two consequently applied transformations and this product of two transformations can be replaced by one transformation, which is a component of the same set, to which transformations-multipliers belong, then such set of transformations forms a group (see p.p. 262-268 in   [21]. Mandelshtam L. I.   Lectures on optics, theory of relativity and quantum mechanics.  Мoscow: Nauka, 1972. p. 161).

From the speeds composition law (7.33) it follows, that in the new space-time theory there is no upper limit for bodies motion speeds. For example, if  u/c_o = 0.8 and s/c_o = 0.6, we have Г_u = 1.28; Г_s = 1.17; w = 1.7 c_o, i. e. the total speed is 1.7 times greater than the light speed c_o. This means, that if the square-law dependence c_u = c_o(1 + u²/c_o²)^1/2 of light speed upon light source speed exists in the nature, then the statements of the special relativity theory about impossibility of motion at superlight speed  and about impossibility of   existence of such interaction, which propagates faster than the light in vacuum, are erronous. If the square-law dependence c_u = c_o(1 + u²/c_o²)^1/2 of light speed upon light source speed exists in the nature, then superlight speeds of particles motion should also be real. The more espetially as   the superlight speeds do not result in violation of the causality principle in the new space-time theory.

7.4. Superlight speeds and the causality principle in case of new transformations of coordinates and time

Let a body be at rest in the inertial reference frame G, which we have considered in subsection 7.3, and two events happen to this body. The first event happens to this body at a moment, when this body is in point x₁ of the reference frame A and when a chronometer being at rest in point x₁ of the reference frame A indicates time t₁. Let the second event to happen  to this body at a moment t₂, when this body is in point x₂ of the reference frame A.

Expressions (7.24), (7.25) and (7.28) are transformations of coordinates and time for these events from one of the three inertial reference frames A, B and G to any other one from A, B, G reference frames.

Having determined the primed quantities from transformations (7.28) we have

x' = Г_u (x - B_uc_wt),   c_s t' = Г_u (c_w t  - B_u x). (7.35)

From the second equation of transformations (7.35) it follows, that in the inertial reference frame B a time interval between this two events happening to a body, which is at rest in the reference frame G, is determined by coordinates of the same two events in the reference frame A using the expression

c_s (t₂' - t₁') = Г_u [c_w (t₂ - t₁)  - B_u (x₂ - x₁)]. (7.36)

But in the inertial reference frame A the body, to which the two events under consideration happen, moves at a speed of w. That is why the coordinates of this two events in the reference frame A are connected with each other by the expression

(x₂ - x₁) = w (t₂ - t₁). (7.37)

Then, substituting the expression (7.37) into the expression (7.36), we have

c_s (t₂' - t₁') = Г_u c_w (1 - B_u B_w)(t₂ - t₁)]. (7.38)

From the expression (7.38) it follows, that at any values of the speeds u and w and if (t₂ - t₁) > 0, then always we shall have (t₂' - t₁') > 0.

Indeed, in the expression (7.38)

Г_u= (1 + u²/c_o²)^1/2;     c_w = c_o(1 + w²/c_o²)^1/2;     c_s = c_o(1 + s²/c_o²)^1/2;      t₂ - t₁ = L_o/(u Г_u).

That is why at any values of speeds u, s and w, including at u < c_o and   w > c_o, we shall have

B_u < 1_,    B_w < 1,   (1 - B_uB_w ) > 0. (7.39)

The inequalities (7.39) are right always.

In the special relativity theory instead of the expression (7.38) we have [54]. Terletsky Ya. P. Paradoxes of relativity theory. Moscow, Nauka, 1966, p. 74.

(7.40)

From the formula (7.40) it follows, that in the special relativity theory at w < c_o and u < c_o, if (t₂ - t₁) > 0, then (t₂' - t₁') > 0, but at w > c_o we can match such speed u, at which (1 - u w / c_o²) < 0 and, consequently, at (t₂ - t₁) > 0 we shall have (t₂' - t₁') < 0. And this means,  that according to the special relativity theory the assumption about existence of superlight speeds results in the violation of the causality principle.

In order to clarify, under what conditions in the new space-time theory the superlight speeds are detected, we have at first to obtain the equations, which follow from the new space-time theory, for connection between parameters of electromagnetic field   in two inertial reference frames, which are moving one wwith respect to the other.

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