3. DISTANCE-TIME THEORY page 3

3.16. Dilation of distance-time

In Figure 2, I place point P at (x1, 0), point Q at (0, y1), and point O at (0, 0), which is the origin. In this figure, an eventon travels straight from P to Q, crossing

._________________________________________(4)

PQ is the distance-time given in distance units. In order for this eventon to travel from P to Q, it must cross

,________________________________________________(5)

and

.________________________________________________(6)

PO is the distance-time parallel to the X axis, and QO is the distance-time parallel to the Y axis. Therefore, this eventon traveling straight from P to Q crosses PQ distance-time, while crossing PO distance-time parallel to the X axis and QO distance-time parallel to the Y axis. Using Eqs. (4), (5), and (6), I derive

_________________________________________________(7)

and

_________________________________________________(8)

As a result, PO and QO are dilated along the path that the eventon takes between P and Q when PO and QO are smaller than PQ.

PO and QO dilated between P and Q occur slower than speed c. Dividing PQ by c, I change PQ from distance units to time units, T. Since PO distance-time and QO distance-time occur in a period of time T, I divide Eqs. (7) and (8) by T, which gives

_______________________________________________(9)

and

_______________________________________________(10)

Both Eqs. (9) and (10) show that the distance in PO and QO, dilated between P and Q, have a speed smaller than or equal to c. If in Figure 2 I send a different eventon traveling straight between P and O or Q and O, it travels at speed c; therefore, actual distance occurs at speed c between P and O as well as between Q and O. However, the PO and QO distances in Eqs. (5) and (6) are dilated between P and Q, and are only parallel to the X and Y axes in Figure 2. They do not necessarily lie on the X and Y axes. According to Eq. (9), PO has a speed c if PO = PQ. According to Eqs. (4) through (6), this only occurs when QO equals zero distance, making PO the actual distance occurring between P and O. Also, QO has speed c and is the actual distance-time between Q and O when PO equals zero distance.

3.17. Visible space

Visible space is the space an observer sees. Since light particles crossing distance-time give an observer a special view, I derive a three-dimensional view of space by placing an observer in a distance-time manifold.

In this section on visible space, I assume that light is traveling in a vacuum. In Figure 3, an observer stands a distance from the broadside of a wall. Also, in Figure 3, I place points P, Q, and O from Figure 2 such that the observer is at point Q, and on the wall's surface lie points P and O. Light beams travel a straight path in Figure 3 from P to Q and from O to Q. The observer at Q sees that the light that travels from P to Q crosses PO distance parallel to the wall's surface. Because the path from Q to O is perpendicular to the wall, the observer sees that the light that travels from O to Q does not cross any distance parallel to the wall's surface. Therefore, the observer sees P extended out from O a distance of PO. Since P can be any point on the wall's surface, the observer sees the entire wall's surface extended out, a distance in any direction on the wall's surface from O, when the light rays reflected off of the wall's surface reach the observer.

The fraction of distance in PQ parallel to the straight path between Q and O is QO distance. Since P can be any point on the wall's surface, every light ray from the wall's surface reaching the observer must traverse QO distance parallel to the straight path between Q and O. Thus, the observer sees the wall's surface a depth of QO distance away and in the past.

I make the plane represented by the wall's surface infinitely large, and I make the distance QO any possible distance in any direction from the observer and perpendicular to the wall's surface. The point P can now be any point within the three-dimensional distance-time manifold, and the observer sees a three-dimensional space in the past. However, the distances within this visible space do not always travel as fast as actual space at speed c. Visible space is only the space that an observer sees. It is not necessarily the actual space, which has a speed c. According to Eq. (9), the distance of PO that the observer sees on the wall's surface has a speed smaller than or equal to c. The observer only sees the actual distance PO occur at speed c if the observer is located at point O, making QO equal to zero distance and PQ = PO. However, if the observer is at a distance away from O, the distance that the observer sees between P and O is dilated along the path between P and Q, and is slower than speed c. Hence, visible space has a speed smaller than or equal to the speed of actual space.