Механика математическое моделирование (стр. 4 )

t = a[t - vx’/(V 2- v 2)], (13)

where a = j (v) is yet unknown function, and for brevity it is taken that at the origin of the moving frame (k) if t = 0, so also t = 0". (Einstein’s notations, see [2, p. 14–15]).

Remark 3.1. The transition from (5) to (6) contains an implicit assumption that there exists a unique time surface t (t, x’, v, V) such as (13) containing the time curves corresponding to intervals t0 ,t2 ,t1 in (4), (5). This supposition is emphasized in italics and explicitly written in *(8) to *(10). This surface must be totally optimal with respect to neighboring surfaces in the sense of [7, pp. 1342–1343], and its existence for V = const is not automatic. When a ray of light propagates in the air with different densities at different heights, continuous refraction takes place, thus V ¹ const in which case such surface does not exist and Einstein’s PDEs (6), (7) are invalid as well as resulting solution (13) and all relativistic transformations for V = const that follow in the sequel. Classical theory of relativity applies only to motions in vacuum or in media without refraction. Extension to cases V ¹ const and/or v ¹ const requires major modifications.

Remark 3.2. If we want to include the time of information transmittal into this relativistic framework, we should use relation (3) instead of (1), thus considering the real time of light registration instead of abstract time of light arrivals in (1). Since the clocks are at rest with respect to the system (K) or (k) where they belong, so all deltas in (3) and the time of reflection are constants, though unknown, and assembling all those constants at the right-hand side of (4), we shall get those constants at right of (7), (11), (12), instead of zeros, yielding, instead of (7) or (12), the equation:

¶ t* /¶ x’ + [v/(V 2- v 2)]¶ t* /¶ t = d ,

d = dA - dB - t 0 , (14)

where t 0 ³ 0 is the time of mirror reflection at B. Using the function t (.) of (13), the solution of equation (14) can be written in the form t* = t + pt + qx’, where p, q are constants satisfying the relation p v/(V 2- v 2) + q = d , and having the same sign as d :

t* = t + pt + qx’ = (p + a) t + [q - a v/(V 2- v 2)] x’, p v/(V 2- v 2) + q = d , (15)

which for d = 0 yields p = q = 0 with the solution in (13), and two parameter family of other solutions for some d ¹ 0. We see that Einstein’s PDE (7) yields not only the relation (13) for abstract times of arrival in (1) but, with the modification in (14), it also provides the relations (15) for different real time observations that include delays due to finite speed of information transmittal in the same framework as considered by Einstein. This, of course, should have been expected. In fact, all experiments that confirmed the theory of special relativity carried time delays due to finite speed of information transmittal whereas the theory itself did not contain those time delays which may be comparable with time intervals in (1), thus, with the values t0 ,t1 ,t2 in (4) for smaller lengths AB in (2). Abstract times of arrival were considered instead. The confirmation was, thus, "by default", based on some real data in neighborhoods of the times of arrival.

Remark 3.3. At this point we need to emphasize the meaning of the principle of constancy of the speed of light cited in Section 4 from [1, § 1] that "every ray of light propagates in a still system of coordinates with certain speed V irrespective of whether the ray of light is issued by a resting or moving source". This statement represents the results of experiments made on Earth considered as a still system (K) in which the speed of light issued by a moving source was measured by and with respect to a still measuring device. However, the same measured value of the speed V of the front of that same ray, if measured with respect to the origin of a moving system (k) by a still measuring device in (K), would be affected by the velocity v of the origin of (k) which results in the appearance of terms (V- v), (V+ v) in (5), (6), (9) to (11) and of the term (V 2- v 2) in (7), (12), (13) to (15). According to the above formulation, Einstein argues about measured values of the speed V of a certain physical process (light propagation from a source of light) in a particular setting of a still coordinate frame, produced by real life physical experiments on Earth considered as a still frame (which it is not, but one can define it as still frame with all surrounding space moving around it, and neglect, for the moment, the effects of the accelerated surrounding space, in a thought experiment). Tacit assigning of some absolute mystical sense to the speed of light would be an error. According to Einstein’s words after (2), the value V "is a universal constant (the speed of light in vacuum)", thus, the same for a resting or a moving source in that vacuum which means that light in a moving system (k) propagates with the same speed V, as in the resting system (K), if measured by a device resting or moving in either system with a constant (cf. the principle of relativity) speed v < V (if v ³ V, the synchronization of clocks in (k) and (K) is impossible since a ray of light issued from A, the origin of (k), cannot reach x’Î (K), point B; also, a ray from B cannot reach AÎ (k) moving with v ³ V, so that equation (4) becomes void). We see that the universal constant V should not be confused as something absolute, being the same when measured from anywhere with respect to anything, cf. Einstein’s answer to Ehrenfest cited in Section 2. Furthermore, in this argument based on real life experiments, it is quite reasonable to consider and include into experimental readings the time delays due to finite speed of information transmittal which delays are present there anyway, this leading to relations (3), (14), (15).

3.1. Calibration factor for transformations without time delays

Let us consider first relativistic transformations based on (13), that is, for abstract time (time of arrivals), without time delays due to finite speed of information transmittal. Given, according to experiments, that light in a moving system (k) propagates with the same speed V, Einstein writes [2, p. 15]: "For a ray of light issued at the moment t = 0 in direction of increasing x, we have x = Vt , or x = a V [t – x’ v/(V 2– v 2)]. However, with respect to the origin of system (k), the ray of light, if observed in the still system (K), propagates with the speed V – v, so it follows

x’/(V- v) = t. (16)

Substituting this t into equation for x, we get x = a x’ V 2/(V 2– v 2). "Now, with x’ = x – v t in the expressions for x and t (13), it yields

t = a[t - vx’/(V 2- v 2)] = aa 2(t - vx/ V 2),

a 2 = V 2/(V 2– v 2), (17)

x = Vt = a x’ V 2 / (V 2– v 2) = aa 2(x – vt). (18)

Further, Einstein writes [2, p. 15]: "Considering rays propagating along two other axes, we find

h = V t = a V[t – x’ v/(V 2– v 2)] , whereby

t = y/(V 2 – v 2)0.5 , x’ = 0 ; (19)

hence (with our notation in (17) for a 2)

h = aVy /(V 2 – v 2)0.5 = aa y,

z = aVz /(V 2 – v 2)0.5 = aa z". (20)

To determine the function a(v, V) in (17), (18), (20), Einstein writes in [2, pp. 16–17]: "For this purpose, we introduce one more, the third coordinate system (K’), which with respect to system (k) is in translational motion parallel to x – axis in such a way that its origin moves with velocity - v along x - ppose that at the moment t = 0 all three axes coincide and for t = x = y = z = 0 the time t’ in (K’) is 0. Suppose that x’, y’, z’ are coordinates measured in system (K’). After applying twice our transformation formulae (17), (18), (20), we obtain"

t’ = aa 2(t + vx / V 2) =

=a 2a 4[t- vx/ V 2 + v(x- vt)/V2] = a2a 2 t, (21)

x’ = a a 2(x + vt ) = a 2a 4(x – vt + vt - v 2x/ V 2) = a2a 2 x, (22)

y’ = aa h = a 2a 2 y, z’ = aa z = a 2a 2 z (23)

"Since relations (22)–(23) between x’, y’, z’ and x, y, z do not contain time, the systems (K) and (K’) are at rest with respect to each other, so it is clear that transformation from (K) into (K’) must be the identity transformation". [2, p. 17] Hence, a2a 2 = 1 and also aa = 1 since the axes h, y and z , z have the same directions. Now, using the value a 2 from (17), we get