#5.1 The necessity of deriving special relativistic results

#5.2 The second postulate of special relativity

#5.3 The first postulate of special relativity

#5.4 Relativistic kinematics

#5.5 A matter-wave at rest

#5.6 Doppler effect of matter

#5.7 Different here-nows for different reference frames

#5.8 Does matter have a memory?

#5.9 The solution to another causality paradox for faster-than-light quantum tunneling

#5.10 The nondispersive matter-wave

5.1 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. Moreover, there is one possibly testable, new prediction in this part of my paper. When De Broglie designed his matter wave, he had no alternative but to embed it into the four-dimensional space-time continuum of special relativity. Since a matter wave could not have a constant speed in a four-dimensional space-time continuum, it would be defined as a dispersive wave. However, I predict that a matter wave placed in a distance-time manifold has a constant total velocity c and is a nondispersive wave, which contrasts with De Broglie’s prediction of matter being a dispersive wave. (Waves packets are dispersive because the speeds of the waves within the packet vary with wavelength. [7] A wave packet with a constant speed would not possess waves of varying speeds.) (For more discussion on the nondispersive matter wave, see subsections 5.6 and 5.10.) Also, in section 5 of my paper, I prepare the reader for the photonic reference frame and give a solution for another type of causality paradox for faster-than-light quantum tunneling.

5.2 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, 2, 3] 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 primed reference frame, the amount of distance per time an eventon crosses is

Within a nonprimed reference frame, different from the primed frame, the amount of distance per time is

Consequently, the rate of the distance per period of time an eventon crosses is a constant speed c relative to the primed and nonprimed 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, 2, 3] Therefore, the distance-time Euclidean metric function satisfies Einstein’s second postulate of special theory of 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, then 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, 2, 3] 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.

In order to derive the relativistic equations of Einstein’s special relativity theory, I shall use figure 4 and Eqs. (11) through (25), which were derived 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

which is Einstein’s time dilation equation. [1, 2, 3] However, Eq. (18) [the origin of Eq. (28)] 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. (28) transfers only the clock measurement at the S’ origin to the clock measurement at the S origin. 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). Therefore, in order to find a more general transformation equation between S and S’ clock measurements, I first subtract out the distance-time in - v between the x and x’ coordinates from t’. Since I am transferring from t’ to t, I subtract Eq. (25) from t’ and this results in

Secondly, in Eq. (28) I replace t’ with Eq. (29). This results in

Lorentz’s clock transformation equation for transforming clock measurements in S' to S is in Eq. (30). [1, 2, 3]

The Y’ axis is perpendicular to the X’ axis. Therefore, according to Eq. (30), 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. (28).

I next derive the equation to transform the y’ coordinate to the y coordinate. Eventon A in figure 4 travels form (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 in S

                                               and in S’

                                            .                                           Equation 31               

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

                                                                                 Equation 32

Combining Eqs. (28), (31), and (32), I solve for y and y’, which results in

                                                                                         Equation 33

The transformation equation between the y and y’ coordinates is given by Eq. (33). 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. (30), I change Eq. (30) from time units to distance units and derive

Lorentz’s transformation equation, which transforms the x’ and t’ coordinates to x is given by Eq. (34). [1, 2, 3] Using similar methods to those I have already used, I can derive Lorentz’s transformation equations, which are

                                              and

I define matter as a wave with a frequency f and a wavelength w, and I refer to matter as a "matter-wave." I place a matter-wave in the S’ reference frame of figure 4. Relative to S’, the matter-wave cycles at a rest speed c across distance-time. Since the matter-wave cycles across distance-time, I divide the wavelength of the matter-wave by c to change distance units to time units. This results in a corresponding frequency to the wavelength. Therefore, the matter-wave has a frequency-wavelength analogous to distance-time. I treat S’ as the source for the matter-wave. The source emits N cycles of the matter-wave across the proper clock measurement ct. Earlier, I derived the proper clock measurement to be constant relative to both S and S’. The frequency-wavelength of the matter-wave is

I next adapt De Broglie’s equations to be used for the matter-wave. [4, 5] In this use of De Broglie’s equations, which are

the rest frequency is multiplied by Planck’s constant and the rest wavelength is divided into Planck’s constant. This results in the rest energy momentum of a matter-wave. Consequently, matter possesses a scalar rest momentum, not a rest mass. Combining Eqs. (37), (36), and (39), the rest momentum-energy equation can be derived:

                                                                                      Equation 40

The rest velocity c multiplied by proportionality factor m is equal to the rest momentum. Therefore,

                                                                                      Equation 41

The proportionality factor m must be determined distinctly for each matter-wave. Combining Eqs. (40) and (41), Einstein’s mass energy equation can be derived:

Eq. 42 is for a body at rest. [1, 2, 3]

In Eq. (16), v represents S’ frame’s velocity relative to S, and r represents the rest speed within S’ relative to S. The relationship between v and r in Eq. (16) equals a constant c. Therefore, relative to S, the matter-wave placed at rest in S’ of figure 4 has a velocity v in the positive direction of the X axis, a rest speed r, and a total constant speed c. Although De Broglie used Eqs. (38) and (39), his description of matter as a wave had only a varying velocity v; however, my description gives a constant total speed c with varying partial speeds of v and r. Since my description of a matter-wave has a constant total velocity, then the matter-wave described in distance-time theory is a nondispersive wave. This is in contrast to De Broglie’s dispersive matter-wave concept.

Using Eq. (19), I create, in figure 6, a mnemonic device which represents the distance-time crossed by the matter-wave relative to S. In this figure, ct is the total distance-time that the matter-wave crosses and is the event line for the matter-wave. D’ is the distance-time in the rest speed, r, of the matter-wave relative to S. D" is the distance-time the source (S’ reference frame) with the matter-wave crosses in the velocity v relative to S. While the source moves across D", relative to S, it emits N cycles of matter-wave across D’. This gives a wavelength for the matter-wave, relative to S, of

                                                                                      Equation 43

I use Eqs. (43) and (37) for substitutions into Eq. (18) to get

                                                                                Equation 44

In Eq. (44) the wavelength w is shorter than the wavelength w_o. This is caused by the source moving in approximately the same direction as the matter-wave, thereby causing a Doppler effect. Dividing Eq. (44) by c, I change Eq. (44) from distance units to time units and derive

Because of the Doppler effect, the frequency of the particle wave is increased to f according to Eq. (45).

In Eqs. (44) and (45), I relate the original frequency-wavelength, f_ow_o=c, relative to S' to the total frequency-wavelength, fw=c, relative to S. I next relate f_ow_o=c to f' w’=c, which is the frequency-wavelength in the rest speed r of the matter-wave within S’, relative to S. Relative to S, the fraction of the distance-time in rest speed r is D’. According to Eq. (18), D’ is dilated across ct, which is the total distance-time the matter-wave crosses relative to S. Therefore, the wavelength w’, in the rest speed r, is also dilated out from w to the same extent that D’ is dilated in Eq. (18). In Eq. (18), I replace ct with w’ and D’ with w. Combining this result with Eqs. (44) and (45), I derive w_o=w’ and f_o=f ’. Therefore, the frequency-wavelength of the matter-wave in speed r relative to S is equal to the original frequency-wavelength of the matter-wave relative to S’.

I next derive the relationship between wf=c and w"f "=c, which is the frequency-wavelength of the matter-wave in v relative to S. According to Eq. (17), the fraction of ct in v, D", is dilated across ct. Therefore, w" is also dilated compared to w according to Eq. (17). Replacing ct with w" and D" with w in Eq. (17), I derive

Changing the distance units in w" and w to time units by dividing by c, I change Eq. (46) to frequency units, resulting in

Using Eqs. (38), (39), (41), and (42), along with Eqs. (44) and (45), I derive the following:

                                         and

Both Eqs. (48) and (49) give the relationship between the total momentum-energy, E=Pc, of the matter-wave relative to S and the energy-momentum of the matter-wave at rest, E_o=P_oc, relative to S’. Einstein’s relativistic equation between the rest energy E_o and the total energy E of a body of matter is given by Eq. (49). [1, 2, 3] Both Eqs. (50) and (48) are not found in Einstein’s relativistic equations because there is no total momentum, P, in relativity. However, Eq. (51) is found in relativity and it gives the total energy, E. Dividing Eq. (50) by c and equating the result to M, I produce the result

This is Einstein’s relativistic equation, which relates the rest mass, m, to total mass, M, of a body of matter. [1, 2, 3] However, in the theory of distance-time, m and M are merely proportionality factors. Instead, in distance-time theory, the scalar rest momentum, P_o, is compared to the total momentum, P, in Eq. (48).

Next, I multiply Planck's constant, h, with Eq. (47). This results in

Changing the time units to distance units in Eq. (53), I divide both sides of this equation by c to get

                                         .                                            Equation 54

E"=P"c is the momentum-energy of the matter-wave in the velocity v. Using Eqs. (53) and (54) to substitute for P and E in Eqs. (48) through (51), I derive the following:

                                          and

Within Einstein’s relativity theory, Eq. (57) is found. [1, 2, 3] However, Eqs. (55), (56), and (58) are not found in relativity because P_o, the scalar rest momentum, and E", the vector energy, are not a part of relativity. Since E"=hf" and P"=h/w", Eqs. (57) and (58) can be altered to

                                        and

The wavelength in the velocity v of the matter-wave is given by Eq. (59), and it agrees with Einstein’s special relativity theory and De Broglie’s wave theory for matter. [4, 5] Eq. (60) only occurs in distance-time theory because the matter-wave is a nondispersive wave in this theory.

As I previously discussed, all events at every coordinate are located here-now for a single clock point, t, relative to a reference frame. However, different reference frames experience different sets of events here-now. I use Eq. (35), derived from figure 4, to examine the relationship of here-nows between the different reference frames of S and S’.

In Eq. (35) I define v to be constant while examining the three variables x, t, and t’. As I mentioned earlier, each point of t and t' represents distinct sets of events located here-now relative to the S and S’ frames respectively. (See subsection 3.6.) Also, x(v/c²) in Eq. (35) is the fraction of distance-time that is in the velocity v of the S’ frame, relative to the S frame of figure 4. Eq. (35) subtracts the events in x(v/c²) out of the set of events located here-now in the S frame, represented by t, to get the set of events located here-now in the S’ frame, represented by t’. To see this more clearly, I hold t’ constant in Eq. (35) and vary x, which then varies x(v/c²). Since t’ is constant, t must vary with x(v/c²). Since the single point t' represents one here-now in the S’ frame, all the events in x(v/c²) are happening here-now at t’ relative to the S’ frame. However, x(v/c²) are the events in the velocity v relative to S and are not happening here-now relative to S. Therefore, the difference between the here-nows of the S and S’ frames are the events in x(v/c²), located here-now relative to the S’ frame but not to the S frame. Consequently, the S’ frame cannot measure its own velocity v. The velocity -v of S relative to S’ is a different velocity which the S’ frame can measure. Summing it up, the events in the distance-time of the velocity, v, of a particle are located here-now, relative to that particle, and the particle cannot measure its own velocity. However, relative to an observer who sees the particle moving with the velocity v, these events do not occur here-now but are in the distance-time line of velocity v.

In the preceding subsection, I stated that the events, in the velocity v of a body of matter, happen here-now relative to the body. Nevertheless, relative to a body the events in its rest velocity never happen here-now. This means that, events happening between objects in space and time occur within the events of a body’s rest velocity and not within its velocity v. To further illustrate, I give the following example. Imagine a ball having a velocity v relative to an observer standing next to a wall. The observer and the wall would also have a velocity -v relative to the ball. At a point in time, the ball hits the wall and bounces back. Since the ball striking the wall can occur in the future, while not in the now relative to the ball, then the ball striking the wall is not an event occurring in the velocity v of the ball. Thus, it must be occurring in the rest velocity of the ball. (In other words, striking the wall does not occur here-now relative to the ball.) Consequently, the ball would not know ahead of time that it will strike the wall, and the ball will not have a memory of hitting the wall. From this we can conclude that, matter does not have a memory of events.

This subsection is best understood after one has read section 4 of my paper which deals with quantum tunneling and subsection 5.7 which delineates different here-nows for different reference frames.

From figure 4, I derive figure 7 by placing an impenetrable barrier in the S frame. I locate the barrier so that one end is at the S origin and the other end is located on the positive side of the X axis, a distance, d, away. I next send a particle wave in the negative x direction so that it would have a chance of tunneling through the barrier in the negative x direction. If this particle wave tunnels through this barrier within the here-now of the S’ frame, it would travel across a negative period of time, - T", relative to an observer in the S reference frame. As a result, the observer would be able to detect the particle wave finish tunneling a period - T" before it begins to tunnel. However, an observer in the S’ reference frame would see no difference in time between the times the particle wave begins and ends its tunneling. The reason for this is that relative to the observer in the S’ reference frame, the distance-time in - T" occurs here-now. Therefore, this observer sees the tunneling effect occurring infinitely quickly.

Next, the observer in the S frame wishes to produce a causality paradox. (A causality paradox occurs when an effect happens without the cause happening.) The observer, therefore, aims a rifle at the barrier. He wishes to shoot the barrier out of the way of the particle wave during the time period that the particle wave is tunneling through the barrier into the past. The observer, therefore, wishes to knock the barrier away after the particle wave has finished tunneling, but before it can start tunneling. He could then see the particle wave finish tunneling without its starting. This is a causality paradox. However, when he attempts to do this, the paradox never occurs. At this point, the observer may wonder what went wrong. Since the particle wave is traveling via the here-now in the S’ reference frame, the distance-time in - T" occurs here-now relative to the S’ observer and the particle wave. (- T is not a part of the particle wave’s rest velocity, and the particle wave experiences the tunneling instantaneously.) This alone would not affect the behavior of the particle wave. However, T" is located within the event line of the particle wave. Thus, all the events within - T", including the barrier being removed, are experienced ahead of time by the particle wave, and the particle wave acts accordingly. Relative to the observer in the S frame, the particle wave will behave as if it knew ahead of time that the barrier would be removed. Therefore, the particle wave would never tunnel through the barrier back in time; rather, it would move normally across distance-time through the region where the barrier once was located. No tunneling would occur relative to the S’ observer. Relative to the S’ observer, the bullet from the rifle would knock the barrier out of the way before the tunneling can begin or end. Consequently, relative to this observer, the particle wave would not behave as if it knew ahead of time that the barrier would be removed. Both perspectives of the S and S’ frames would be correct; that is, they do not contradict each other, and no causality paradox would occur.

I give the following thought experiment to further illustrate my views on the nondispersive matter wave. Since forces may affect the shape of a matter wave, I place a matter wave inside a jar with a vacuum and place the jar in spaceship A outside of any gravitational fields. Furthermore, there are no other forces acting on the particle so the particle is floating in free space. I place one observer inside spaceship A with the jar and another observer in a different spaceship B, which has a large velocity relative to the jar. The observer in spaceship A is essentially at rest with the jar and the matter wave. If the matter wave were dispersive, it would spread out overtime relative to the observer within the spaceship B. Therefore, relative to the observer in spaceship A, the matter wave would also spread out. However, if the matter wave does not spread out relative to the observer in spaceship A, then it does not spread out relative to the observe in spaceship B. Thus, the matter wave is nondispersive. The latter scenario is more plausible and agrees with distance-time theory, but it does not agree with traditional views. [4]