4. DERIVATION OF SPECIAL RELATIVITY page 1
4.1. The necessity of deriving special relativistic results Any new theory claiming to be more accurate than the old theory must predict the old theories' verified results. The special relativistic predictions derived here are not new. However, these predictions are derived from a distance-time manifold. Therefore, a different perspective is given to these special relativistic results. In this chapter, I prepare the reader for a photonic distance-time.
4.2. The second postulate of special relativity Einstein's second postulate of special relativity theory is that any particle traveling at speed c has a constant speed relative to any reference frame [1–5]. This postulate is satisfied by the distance-time Euclidean metric function, Eq. (3). According to this function, the ratio of time to distance is always a constant c relative to any reference frame. Therefore, within a prime reference frame, the amount of distance per time an eventon crosses is Eq. (3) with all prime variables. Within a nonprime reference frame, different from the prime frame, the amount of distance per time is Eq. (3) again, but this time all variables are nonprime. Consequently, the rate of the distance per period of time an eventon crosses is a constant speed c relative to the prime and nonprime reference frames. This is Einstein's second postulate of special relativity: that any particle at speed c has a constant speed relative to any reference frame [1–5]. Therefore, the distance-time Euclidean metric function satisfies Einstein's second postulate of special theory of relativity.
4.3. The first postulate of special relativity I next derive Einstein's first postulate of special relativity. Since relative to any reference frame, eventons have a constant speed c, the clocks relative to any reference frame have a constant speed c. This constant clock speed for a reference frame I call the "proper clock speed." Therefore, the amount of time measured by a proper clock measurement of any reference frame is constant. Also, the relationship between rod measurements and clock measurements for an eventon moving in a straight path in and relative to a reference frame is given by Eq. (3). Consequently, a rod measurement relative to a reference frame is also constant. I call this rod measurement the proper rod measurement. If all physical laws relative to any reference frame were defined to be based only upon the proper clock and rod measurements, all physical laws would be constant relative to any reference frame, since clock and rod measurements of any frame are constant. This is equivalent to Einstein's first postulate of special relativity, which states that the laws of physics are constant relative to any reference frame [1–5]. I am assuming that Newton's first two laws of motion hold within the reference frames that I am using to derive Einstein's postulates of special relativity; therefore, the Euclidean distance-time metric function holds within these reference frames. I could put in this article the same type of clock synchronization scenario as found in any book about special relativity. Instead, I state that I synchronize clocks in a similar manner and that there is no need to delineate the method of clock synchronization any further.
4.4. Relativistic kinematics To derive the relativistic equations found in Einstein's special relativity theory, I use Figure 4 and Eqs. (11) through (25). (Eqs. (11) through (25) come from Figure 4.) In Figure 4, relative to the S reference frame, Eq. (18) gives the relationship between the distance-time in the rest speeds of S' and S frames. Converting Eq. (18) to time units, I arrive at ,____________________________________________________________(27) which is Einstein's time-dilation equation [1–5]. However, Eq. (18) [the origin of Eq. (27)] compares the magnitude of distance-times occurring in the S' to that of S. It is not a general transformation of S' coordinates to S coordinates. Also, Eq. (18) was derived from the clock measurements of Figure 4. Since the clocks were only located at the S and S' origins in Figure 4, Eq. (27) transfers only the clock measurement at the S' origin to the clock measurement at the S origin. It is assumed that the clocks within the S and S' frames are synchronized by a traditional method of synchronization. Since v and −v lie parallel to the X and X' axes, there is a difference in the clock measurements between the x and x' coordinates according to Eqs. (22) through (25) because there is nonsimultaneity of the synchronized clocks along the relative to the S frame. To derive the Lorentz transformation equations, I first subtract out the distance-time in –v from t', which gives t' – T". (The distance-time in –v is the difference in distance-time between the x and x' coordinates relative to the primed reference frame.) Since I am transferring from t' to t, I subtract the left side of Eq. (25) from t', and this results in ._______________________________________________________-(28) Secondly, in Eq. (27), I replace t' with Eq. (28). This results in .___________________________________________________________(29) Lorentz's clock transformation equation for transforming clock measurements in S' to S is in Eq. (29) [1–5]. The Y' axis is perpendicular to the X' axis. Therefore, according to Eq. (29), the difference between points on the Y' axis has no effect on time measurement. However, clock measurements on the Y' axis would still be dilated according to Eq. (27). I next derive the equation to transform the y' coordinate to the y coordinate. Eventon A in Figure 4 travels from (0, 0) at t = 0 to (x, y) at t = t in S, and from (0', 0') at t' = 0 to (0', y') at t' = t' in S'. Therefore, according to the distance-time metric, I have Eq. (13) in S, and I have ______________________________________________________________-(30) in S'. According to Eq. (14), eventon A crosses vt distance-time in the positive X axis direction. Using Eq. (14), I substitute vt for x in Eq. (13). Then, I solve for y to get .__________________________________________________________(31) Combining Eqs. (27), (30), and (31), I solve for y and y', which results in .______________________________________________________________-(32) The transformation equation between the y and y' coordinates is given by Eq. (32). The same results can be derived for the z and z' coordinates. I next derive the transformation equation that transforms the x' and t' coordinates to the x coordinate. Again I use eventon B to give the distance-time metric Eqs. (20) and (21), for the X and X' axes. Using Eqs. (20) and (21) to substitute for t', x', and t in Eq. (29), I change Eq. (29) from time units to distance units and derive .____________________________________________________________(33) Lorentz's transformation equation, which transforms the x' and t' coordinates to x is given by Eq. (33) [1–5]. Employing similar methods to those I have already used, I can derive Lorentz's transformation equations, which are ____________________________________________________________(34) and .___________________________________________________________(35)
Finally, I turn to the subject of length contraction. The scalar coordinate is different from the vector coordinates. The scalar coordinate is what the clock measures. I give the following scenario. Relative to an observer, there are two clocks. Relative to this observer, clock R is at rest, and clock S has a speed. To start, they are both at the same time t = 0. Later, clock R has measured more time than clock S. In other words, relative to the observer, clock R is at t = A, and clock S is at t = B, where A > B. However, the S clock is at B at the same time that the R clock is at A relative to the observer. This means that in the same period of time it took the R clock to go from 0 to A, the S clock went from 0 to B. From the observer's perspective, therefore, the S clock measured less time than the R clock. The observer sees the start times of both clocks simultaneously, and he sees the end times of both clocks simultaneously. Consequently, the S clock period of time must span the period of the R clock’s time. In conclusion, the smaller amount of distance-time measured by the S clock must be dilated—not contracted—relative to the observer. However, this is the result for the scalar coordinate t only—not for the vector coordinates. In Figure 4, eventon B travels across less distance-time in the S' frame than it traverses in the S frame. The relationship between the distance eventon B traverses in the two reference frames is given by Eq. (26). Since the eventon is moving along the X and X' axes, its location is given by vector coordinates, not scalar coordinates. Relative to an observer in the S frame, eventon B begins at a coordinate that moves in the S' frame but not in the S frame. Consequently, when the eventon arrives at the end coordinates, the starting coordinates of the S and S' are now at different locations. This causes an observer in the S frame to determine that eventon B traverses less distance-time in the S' frame. Unlike the distance-time in the scalar coordinate, the distance-time in the S' frame does not extend across the distance-time in the S frame. As a result, according to an observer in frame S, the lesser distance-time must be contracted—not dilated—given by Eq. (26), which is the length-contraction equation found in special relativity. |
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