4. DERIVATION OF SPECIAL RELATIVITY page 2

 

4.5. A matter-wave at rest

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

.___________________________________________________________(36)

I next adapt De Broglie's equations to be used for the matter-wave [6–9]. In this use of De Broglie's equations, which are

______________________________________________________________(37)

and

,_____________________________________________________________-(38)

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. Combining Eqs. (36), (35), and (38), I find that the rest momentum-energy equation can be derived:

._____________________________________________________________(39)

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

._____________________________________________________________-(40)

The proportionality factor m must be determined distinctly for each matter-wave. Combining Eqs. (39) and (40), I find that the Einstein's mass-energy equation can be derived:

.____________________________________________________________-(41)

Eq. 41 is for a body at rest [1–5].

The concept of a rest momentum is very similar to the idea of a mass. The only difference is that, in distance-time theory, mass always has a speed. In other words, even if a mass is at rest relative to an observer, it still has a rest speed. Therefore, it is not only a mass; instead, it is a mass with a scalar rest speed, and it has a scalar rest momentum. In distance-time theory, time does not exist without distance. In other words, matter cannot exist without a scalar rest or vectorial relative speed. I have not eliminated the idea of a mass from physics because mass can still be used in this theory. However, I contend that I have made a slightly different concept other than mass with the idea of a rest momentum. Classical theory gave matter a speed across the time axis, which was similar but not the same, because this speed was also not across distance.

 

4.6. Doppler effect of matter

In Eq. (16), v represents S' frame's velocity relative to S, and u represents the rest speed within S' relative to S. The relationship between v and u 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 u, and a total constant speed c. Although De Broglie used Eqs. (37) and (38), 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 u. Using Eq. (19), I create, in Figure 5, 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, u, 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

.______________________________________________________________(42)

I use Eqs. (42) and (36) for substitutions into Eq. (18) to get

.__________________________________________________________-(43)

In Eq. (43) the wavelength w is shorter than the wavelength wo. This is caused by the source moving in approximately the same direction as the matter-wave, thereby causing a Doppler effect. Dividing Eq. (43) by c, I change Eq. (43) from distance units to time units and derive

._________________________________________________________(44)

Because of the Doppler effect, the frequency of the particle wave is increased to f according to Eq. (44).

In Eqs. (43) and (44), I relate the original frequency-wavelength, fowo = c, relative to S' to the total frequency-wavelength, fw = c, relative to S. I next relate fowo = c to f' w' = c, which is the frequency-wavelength in the rest speed u of the matter-wave within S', relative to S. Relative to S, the fraction of the distance-time in rest speed u is D'. According to Eq. (18), D' is dilated across ct, which is the total distance-time the matter-wave crosses relative to S. (Equation 18 is being used as a fraction found within the total distance-time, which is ct. Hence, across ct, D' is dilated—not contracted.) Therefore, the wavelength w', in the rest speed u, 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. (43) and (44), I derive wo = w' and fo = f'. Therefore, the frequency-wavelength of the matter-wave in the rest speed u 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

._____________________________________________________________(45)

Changing the distance units in w" and w to time units by dividing by c, I change Eq. (45) to frequency units, resulting in

______________________________________________________________(46)

Using Eqs. (37), (38), (40), and (41), along with Eqs. (43) and (44), I derive the following:

,___________________________________________________________(47)

,___________________________________________________________(48)

,___________________________________________________________(49)

and

.___________________________________________________________-(50)

Both Eqs. (47) and (48) 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, Eo = Poc, relative to S'. Einstein's relativistic equation between the rest energy Eo and the total energy E of a body of matter is given by Eq. (48) [1–5]. Eq. (50) gives the total energy, E. Dividing Eq. (49) by c and equating the result to M, I produce the following result:

.___________________________________________________________(51)

This is Einstein's relativistic equation, which relates the rest mass, m, to total mass, M, of a body of matter [1–5]. In distance-time theory, the scalar rest momentum, Po, is compared to the total momentum, P, in Eq. (47). Next, I multiply Planck's constant, h, by Eq. (46). This results in

._____________________________________________________________(52)

Changing the time units to distance units in Eq. (52), I divide both sides of this equation by c to get

._____________________________________________________________(53)

E" = P"c is the momentum-energy of the matter-wave in the velocity v. Using Eqs. (52) and (53) to substitute for P and E in Eqs. (47) through (50), I derive the following:

,________________________________________________________(54)

,________________________________________________________(55)

,___________________________________________________________(56)

and

.___________________________________________________________(57)

Within Einstein's relativity theory, Eq. (56) is found [1–5]. However, Eqs. (54), (55), and (57) are not found in relativity because Po, the scalar rest momentum, and E" are not a part of relativity. Since E" = hf" and P" = h/w", Eqs. (56) and (57) can be altered to

__________________________________________________________(58)

and

.__________________________________________________________-(59)

The wavelength in the velocity v of the matter-wave is given by Eq. (58), and it agrees with Einstein's special relativity theory and De Broglie's wave theory for matter [6–9].

 

4.7. Different here-nows for different reference frames

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. (34), derived from Figure 4, to examine the relationship of here-nows between the different reference frames of S and S'.

In Eq. (34), I define v to be constant while examining the three variables x, t, and t'. As I pointed out earlier, each point of t and t' represents distinct sets of events located here-now relative to the S and S' frames, respectively. (See section 3.6.) Also, x(v/c2) in Eq. (34) 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. (34) 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. (34) 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, we see that 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.

 

4.8. Does matter have a memory?

In the preceding section, 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 their 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, 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.

 

4.9. What about a light cone?

Distance-time theory is theoretically very different from special relativity theory. There is no light cone separating timelike intervals from spacelike intervals. I could discuss how vector coordinates, in distance-time theory, have a direction and act similarly to spacelike intervals in special relativity; however, vector coordinates and spacelike intervals are not the same because one cannot separate distance from time in distance-time theory. Hence, there can be no spacelike intervals different from timelike intervals in distance-time theory. I could also discuss how scalar coordinates seem to behave similarly to timelike intervals; however, scalar coordinates also cannot represent time without distance. As a consequence, the idea of a light cone separating timelike from spacelike intervals does not apply to distance-time theory. At best, I may derive an analogue to the light cone. These differences plus other differences previously discussed show that distance-time theory is theoretically very different from special relativity. It is a mistake to think these two theories are not very different ideas.