A Simple Derivation of the Lorentz Tranformation for Time Intervals

Universal Field Cosmology assumes that all the electromagnetic energy of the Universe is in a “stationary” universal energy field, and all matter is always moving though this energy field at the speed of light. A part of this motion is assumed to be experienced by us as our movement through time, and part of it as movement through space. These two movements must add up (directly or by a vector addition) to exactly the speed of light, as many experiments have shown that the relative speed of electromagnetic radiaton (EMR) and matter in our universe is always equal to the speed of light. This is another way of stating the constancy of the speed of light. So, let's see what this implies for the speed of passing of time when we are moving through space with a velocity “v” (with respect to a point that is “at rest,” that I’ll define soon).

A thought experiment about frames of reference

Let’s call our velocity of travel through time “T”. As we just stated in words, T + v = c, where “v” is our velocity through space and “c” is the velocity of light. When we are “at rest,” our velocity though space is zero, or v = 0. Our velocity of travel through time when we are “at rest,” “T0” is, according to this theory, as just stated, T0+ 0 = c, so:

T0 = c   .           .           .           .           .           .           .           .           .           (1)

If our direction of travel through time could be specified in the three space dimensions, and we were traveling with a velocity “v” in the same direction as the light from a light source, then the constancy of the speed of light would, as we've stated, require that our speed of travel through time plus our speed of travel through space add up to the speed of light. We have expressed this as: T + v = c or, subtracting v from each side of the equation:

T = c – v          .           .           .           .           .           .           .           .           (2)


This can be expressed in a diagram like this:



From Figure 1 you can see that if “v” increases, “T” must decrease, as “c” always stays the same (you can adjust the lengths of the “T” and “v” arrows but they must always add up to the length of the “c” arrow). In other words, our speed of travel through time would slow down by exactly the same amount as our speed of travel through space increases, since “c” must remain constant. But if we turned around and traveled in the opposite direction at a speed “v” then you can see from Figure 2 that as our speed through space, “v”, increases, our speed of travel through time,“T”, would also have to increase (rather than decrease) to keep “c” constant. We can express this as: T – v = c or, adding v to each side of the equation:

T = c + v          .           .           .           .           .           .           .           .           (3)



So our speed of travel through time, and hence our perception and measurement of how fast time passes, would decrease as we speed up in one direction, but would increase as we speed up when going in the opposite direction! Not only has this never been observed, but it is also a long-standing assumption in scientific theories that the Universe is “isotropic,” meaning that the laws of science operate in the same way regardless of direction in space. Fortunately, though, there is one possible solution that overcomes this problem and is consistent with an isotropic Universe.

If the direction of travel through time were perpendicular to all three space dimensions, then it would have no component in any of the space dimensions to cause the speed of light or the speed of passing of time to depend on the direction in which we are moving. For this to be the case, our experience of time passing must be because of our movement through a fourth dimension perpendicular to all three space dimensions — the “time dimension.” This movement must, for a given observer, pass through a single “point in space,” since there can be no component of the movement in any of the three space dimensions. This is the point that is “at rest” that I mentioned earlier. We will consider later what the nature of these “points in space” is, where this happens. For now, though, I will just observe that these “points in space” form a rest frame that is, in a very real way, absolutely at rest, and that this rest frame is a requirement of this theory.

The Lorentz Transformation for time intervals, the “Time Dilation Equation,” follows from the above thought experiment

In this isotropic case we have just mentioned, T + v = c (Equation 2) must be a vector addition, as shown in Figure 3, where the direction of travel through time at speed “T” is always at right angles to our direction of travel through space at speed “v”, regardless of the direction we are traveling through space.




To determine how the rate at which time passes for us, “T”, varies with our speed of travel through space, “v”, for this isotropic solution, we can apply Pythagoras’ Theorem to Figure 3 to get:

T˛ + v˛ = c˛      .           .           .           .           .           .           .           .           (4)

Subtracting v˛ from each side, we get:

T˛ = c˛ - v˛

Multiplying both numerator and denominator of the terms on the right side by c˛ we get:

T˛ = c˛.c˛/c˛ - c˛.v˛/c˛

Taking the common factor of c˛ out of both terms on the right we get:

T˛ = c˛(1 - v˛/c˛)

And taking the square root of each side (note: x˝ is a way of writing the square root of x), we get:

T = c(1 - v˛/c˛)˝           .           .           .           .           .           .           .           (5)

From Equation 1, though, we know that T0 = c, so:

T = T0(1 - v˛/c˛)˝         .           .           .           .           .           .           .           (6)

This equation shows how, according to this theory, the speed at which an object travels through time, “T”, will slow down, as its speed through space, “v”, increases. (This equation will be used in Article 13 to determine the size of the “flat spots” in our universe containing superclusters of galaxies.) Already it is a form of the Lorentz transformation for time, but it is not yet in its most usual form, which is in terms of the duration of an interval of time “t” (such as a second, or an hour). As the speed of passing of time decreases (as time passes more slowly), the duration of each second (or hour) will increase (each hour will last longer). In other words there is a reciprocal relationship between them, where:


T = k/t            .            .           .            .           .            .            .           (7)


where k is a constant, the value of which depends on the units used to measure time, but which otherwise remains the same. Substituting k/t for T, and k/t0  for T0 into (6) we get:

k/t = (k/t0)(1 - v˛/c˛)˝

 

Dividing each side by 'k' to cancel out the 'k's on each side of the equation we get:

1/t = (1/t0)(1 - v˛/c˛)˝

If two mathematical expressions are equal then their reciprocals are also equal, so we can invert both sides of the equation to get:

t = t0/(1 - v˛/c˛)˝           .           .           .           .           .           .        .        (8)

where t0 is the duration of a given time interval at rest (say one second), and “t” is the duration of that time interval when traveling at a speed of “v” compared to the “point in space” the observer is occupying, referred to earlier. This is the most recognizable form of the Lorentz transformation for a time interval. Sometimes it is also written:

t = γt0   .           .           .           .           .           .           .           .           .   (9)

where γ = 1/(1 - v˛/c˛)˝ , the “Lorentz factor.”



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