To begin the explanation of the motion of dilated frames in terms of contraction, length contraction must be explained in terms of contraction theory. Until now I have used contraction to describe the diminishing of the velocity of light and the size of space relative to the total amount of space within the Universe. It is also possible to use the contraction principle to describe both the reduction in the distance that light travels over a given non-dilated time interval and the reduction in the size of space within a dilated time continuum as measured by someone in a non-dilated time continuum relative to the distance that light travels over a given time interval and the size of space in the non-dilated time continuum.
According to special relativity, a photon considered to be in a dilated reference frame moves through a reduced amount of space during a given non-dilated time interval relative to the distance that a photon in a non-dilated frame moves during that same interval. However, because that time interval is defined as dilated, less time passes due to a slowing in the rate at which time passes, and a constant velocity for light is maintained. Also, in describing special relativity's length contraction, it was pointed out that in directions parallel to the direction of motion of a source the factor by which the velocity of light in the dilated time frame, as measured from the non-dilated time frame and in terms of non-dilated time, is contracted relative to the velocity of light in the non-dilated frame is 1/(u sq.). However, since distances in this direction in the dilated frame are also contracted relative to distances in the non-dilated frame by a factor of 1/u, the velocity of light in the dilated frame, in terms of non-dilated time, is effectively contracted relative to the velocity of light in the non-dilated frame by a factor of only 1/u. When time dilation is taken into account, the velocity remains the same as for a non-dilated frame. In contraction theory, but not in relativity theory, the ratio between the velocity of light and the radius of the Universe, or in other words, the size of the space in which the light exists, is a of fundamental importance. For the past frame which corresponds to the dilation factor u, this is greater by a factor of u relative to the present frame. Also, as explained earlier, according to contraction theory, in directions perpendicular to a source's velocity, the velocity of light in a dilated frame (and non-dilated frame) is contracted by a factor of 1/u relative to the velocity of light in the corresponding past frame represented by the dilation factor u. [In special relativity this has no relevance since past frames don't exist.] There is also a contraction in the size of space in terms of the size of objects, measuring sticks, ect., for the dilated frame relative to the size of space in the past frame. Also, because there is a slowing in the rate at which time passes for this frame, there is a further contraction in the distance that light travels for the duration of one non-dilated second, by a factor of 1/u. This gives a total contraction factor for the distance transgressed by a photon in a dilated frame over this duration relative to the distance transgressed by a photon in the past frame which corresponds with this time dilation factor of
1/(u sq.). In directions parallel to a source's motion, the factor by which the size of space and the velocity of light in the dilated frame is contracted relative to the size of space and the velocity of light in the past frame is 1/u2. With the slowing in the rate of passage in time, there is a total contraction in the distance that light travels in the dilated frame over a non-dilated time interval relative to the past frame by a factor of 1/(u cubed). Consequently, the total amount of space as measured in all directions for the dilated time continuum is contracted relative to the total amount of space as measured in all directions for the past frame corresponding to the time dilation factor u. The main point here is that from the perspective of the non-dilated frame, the affect of time dilation on the size of space, as measured in terms of light moving through it, is the contraction of the distance scale of the dilated frame relative to both the size of the corresponding past frame and also relative to the size of the present non-dilated frame, this being reflected in the reduced distance that light travels over the duration of a non-dilated second and the length contraction of special relativity. In contraction theory this contraction is in fact what causes the motion.
The basic reason for the contraction of the dilated frame is that the larger size of the past frame has a greater rate of contraction, and when this is put into the context of the smaller sized space of the present frame, the result is a contracted frame. The past frame contracts to the size of the present, and the present aspect of the dilated time continuum contracts relative to the present aspect of the non-dilated time continuum.
The dilated frame's contracted size relative to both the past and the present non-dilated frames, plus the fact that dilated frames have a velocity relative to non-dilated frames, indicate how a dilated frame's universal expansion and contraction can be described differently than the expansion and contraction of a non-dilated frame. Time dilation causes an increase in the rate of contraction in the size of space in terms of light moving through it, the degree of this increase determined by the size of the past frame that corresponds with the dilation factor. The direction of the increase in contraction causes a velocity for the frame. Below I illustrate this.
Just as the velocity of light and the radius of the universe can be described in terms of a ratio, the size of the distance scales of dilated reference frames, measured in terms of the velocity of light, can be described relative to the size of the distance scale for the velocity of light in a non-dilated frame. Just as the size of the universe defines the context of the non-dilated time continuum, the size of the non-dilated frame defines the context of dilated frames. Since a frame dilated by a factor of u corresponds to a past time frame in which size is larger than the present by a factor of u, the distance scale for the velocity light in a dilated frame, in it's expanded state (it will be shown that this is also contracted to the size of the non-dilated frame), can be considered to be the same as that of the past frame which corresponds to it. Thus, the ratio of the distance scale of the expanded state of the dilated frame relative to the distance scale of the non-dilated frame is u:1. However, as just explained, from the perspective of the non-dilated frame, in directions perpendicular to motion, the distance scale for the velocity of light in a time continuum dilated by a factor of u is contracted by a factor of 1/u relative to the distance scale of the corresponding past frame, while, in directions parallel to motion, it is contracted by a factor of 1/(u sq.). Thus, from the perspective of the non-dilated frame, in directions perpendicular to motion, it is equal in size to the distance scale of the non-dilated frame, and in directions parallel to motion, it is contracted relative to the present non-dilated frame, by a factor 1/u. Consequently, the ratio between the distance scale of the dilated frame relative to the distance scale of the non-dilated frame appears, from the perspective of the non-dilated frame, to be 1:u. This relationship can be put into terms of integral space by describing the difference between the non-dilated and dilated frames as equal to the ln(u). Also, a ratio of 1:u sq., which will be used soon, defines an integral space equal to ln(u sq). The above description of the relative sizes of the distance scales of the non-dilated frame and dilated frames applies to space in terms of photons moving through it. Matter's motion through space, which, as mentioned earlier, special relativity theory shows to be different in nature than that of a photon, can also be described in terms of the relative size of distance scales of different frames. [It should be noted here that this also holds true for a dilated frame's motion through space, since matter's motion is an aspect of the dilated frame's motion.]
As pointed out earlier, when one describes the motion of matter through space in terms of dilated time, the distance scale that applies to the motion can be considered to be expanded by a factor of u relative to the distance scale that is described by the non-dilated frame, since the rate at which time passes slows by a factor of 1/u for the dilated frame, consequently increasing the rate of velocity by a factor of u when measured in terms of dilated time. Thus, there is an inverse relationship between the size of the distance scale of a dilated frame in terms of matter's motion through space and the size of the distance scale of a dilated frame in terms of a photon's motion through space. Also, each dilated time frame is associated with a relative velocity, with the velocity of the non-dilated frame equal to zero, and greater degrees of dilation associated with greater velocities, with the limit of velocity equal to c. In contraction theory, because of the expanding and contracting nature of space, this can be turned around so that the velocity of a photon can be considered to be zero and it's position constant, while matter can be considered to be moving away from the photon at a velocity of c, this creating the space between the photon and matter. [This is simply a case of considering the motion of matter from it's expanded position.] When matter is in motion relative to the non-dilated frame, thus in a dilated state, it is actually moving away from the photon at a slower rate, thus less space is created between the matter and the photon.
The space creation between the matter and the photon can be put into terms of integral space by considering the integral space created by matter moving at close to the speed of light as being almost zero (little separation between it and the photon) and the integral space of matter moving at slower rates to be equal to the ln of the ratio between c and the velocity (in conventional terms) of matter. The ratio between the velocity of light and the velocity of a non-dilated frame is c to almost zero. This seems meaningless unless one is considering the space of a contracting universe. In such a universe, what would seem to be a non-dilated frame is in actuality a frame that is dilated by a minimum degree, this degree being determined by a minimum velocity that results from the fact that space and the velocity of light itself is contracting with the passage of time. The rate at which the minimum size of space contracts determines the minimum velocity and the minimum degree of time dilation.
The minimum velocity can be described in terms of differences in universal expansion and contraction. (This will eventually lead to a description of gravity in terms of contraction, though for now an explanation of the principle will suffice.) For the frame with the minimum time dilation, universal expansion shifts a point in space outward at a rate of c, and universal contraction shifts this same point back at slightly smaller rate resulting in a reduced rate of separation between a photon and that point in space. However, the whole frame, both matter and the photon, is shifted back toward the original position of the matter, and at the same rate, though in the opposite direction, as the difference in universal expansion and contraction. (This being the result of a minimum gravity.) Consequently, minimum velocity and time dilation simply reflect the contraction of the velocity of light and the size of space with the passage of time, and in fact, are not even manifested in space as velocity. Since minimum size is so small, it's change in size with the passage of time, on the order of Planck?s length, is extremely small.
A frame's motion can also be described in terms of integral space. Whereas previously the term integral space referred to the size of space relative to the size of the universe in terms of photons moving through it, here the integral space, which I'll refer to as integral m-space (integral space in terms of a photon's motion will henceforth be referred to as integral p-space), is defined by the velocity of matter relative to the constant velocity of light, c, and indicates the amount of space created between matter and photons in terms of matter's motion away from the photon. The integral m-space of any frame equals ln(c/v). As time dilation increases, integral m-space is reduced, this reflecting the reduction in the rate of separation between a photon and matter in a dilated frame.
As explained earlier, the position expansion of non-dilated matter should be described as being equal to 2c rather than c. This means that the positions of matter and photons expand outward at a rate of 2c, and matter's position contracts back toward it's original position at a rate of -2c, while the photon's position moves back at a rate of -c. It was also pointed out earlier that rates of expansion and contraction can be described as the sum of a number of different types of expansions and contractions. When there is a total expansion rate of 2c, there are two types of expansion and two types of contraction that affect the position of matter. One type of expansion is the universal expansion, at a rate of c, that was explained earlier. The second type of expansion, which I'll call frame expansion, is at a rate of c, giving a total expansion of 2c. This expansion is the same as that which I've earlier described for a photon, but this now also applies to the position of matter since we are now considering the separation of a photon from matter in terms of matter moving away from the photon. For a non-dilated frame, one type of contraction, which I'll call frame contraction, brings the matter back at a rate of approximately -c from the photon. The second contraction, the universal contraction described earlier, also brings it back at a rate of approximately -c, resulting in a net velocity for the matter of approximately zero. Universal contraction also brings the photon back at a rate of approximately -c, giving it a net velocity relative to matter of approximately c. Below I illustrate this.
When analyzing this aspect of the universal expansion and contraction, initially the distance scale of the dilated frame should be considered to be expanded relative to the non-dilated frame by a factor of u sq.. The reason for this is as follows. It was pointed out earlier that the creation of integral space in a squared universe occurs at a rate which is equivalent to the rate of creation of integral space in the "actual" universe, which has twice the radius of the "apparent" universe. By considering the distance scale of the dilated frames of the apparent universe to be equal to (u sq.)c, the creation of integral space at the rate of the actual universe is taken into account.
The larger distance scale of the dilated frame is not manifested in the non-dilated frame as such because it is also contracted (the ex-con factor). Again, this is the nature of a dilated frame. It is expanded relative to the non-dilated frame in one regard, but also contracted in other regards. By first considering a dilated frame in it's expanded state, the relationship between velocity and time dilation can be more fully understood.
For reasons that I'll soon explain, this third expansion rate applies only to frame expansion, so initially I'll ignore universal expansion and consider the non-dilated frame's expansion to be equal to c in all directions. Consequently, the expanded distance scale of a dilated frame, which I'll call "sub-frame expansion", results in an increase in expansion for the frame to (u sq.)c. This means that matter's frame position expansion in the dilated frame also increases by a factor of (u sq.), to (u sq.c). I'll call this increase "matter's sub-frame position expansion". Thus, there is an increase in motion due to sub-frame position expansion for matter in a dilated state of (u sq.-1)c over matter's non-dilated frame position expansion. This increase can be described in terms of a ratio, ((u sq).-1):1.
There will also be a contraction because of the larger distance scale of the dilated frame. This type of contraction, which I'll call "sub-frame contraction", occurs at a rate of -((u sq.)-1)c. Consequently, when both sub-frame expansion and sub-frame contraction are included, the result is a net expansion rate for all frames, dilated and non-dilated, equal to u2c-(u2-1)c, which equals c. This is the frame expansion we began with. Below I illustrate this.
Frame contraction counters the net expansion just described, and, in a non-dilated frame, results in a net creation of space between a photon and matter at a rate of c. This is the same separation that is caused by the spatial contraction mentioned earlier, though this is now described as matter moving away from the photon, as opposed to the photon moving away from the matter. In a dilated frame this contraction is reduced by a factor of 1/u sq. (Here is an example of how time dilation affects the different types of expansion and contraction in different ways.) Consequently the change in this contraction motion due to time dilation creates space between a dilated and non-dilated frame at a rate of c-(c/(u sq),), which equals [((u sq.)-1)/(u sq).]c. Frame contraction is illustrated below
For reasons discussed earlier, we should assume that matter has an initial total position expansion rate of 2c. Frame expansion accounts for one-half of this total. The other half of the position expansion is accounted for by universal expansion, at a rate of c. This expansion factor is different in nature from sub-frame expansion. It affects all frames, non-dilated and dilated, at the same rate. This because it defines the space of the apparent universe. Frame expansion and contraction define the space of the actual universe. This is "added on" as an extended aspect of space. This is one reason that usually only the apparent universe is perceived from the non-dilated frame. This will be more fully discussed later.
With universal and frame expansion, total position expansion occurs at a rate of 2c. When frame contraction is included, the net rate of position expansion is equal to c+[((u sq.)-1)c/(u sq.)]. Below I illustrate this.
The third type of contraction is the universal contraction that was introduced earlier. This brings non-dilated matter back to it's original position, and brings the dilated matter back at a rate such that it's net velocity will equal v/c. The details of this contraction motion will be dealt with later.
When considering expansion and contraction in the opposite direction of motion, frame expansion and contraction occur exactly the same as they do in the direction of motion, even in terms of direction. However, they should be considered relative to a universal expansion rate of -c, that is, in the opposite direction of matter's motion. Again, this reflects the fact that universal expansion and contraction define the space of the apparent universe. Since in this direction universal expansion and frame expansion are equal in magnitude but opposite in direction, they cancel each other out. With frame contraction, the net result is an expansion rate in this direction of -c/(u sq.). Universal contraction will resolve these two expansions in opposite directions to the same rate of velocity, v/c, which is associated with the time dilation factor, u. This will be discussed later.
Expansions and contractions of space can also be described in terms of integral space. The integral m-space created by spatial contraction, equal to ln(c/v), is a result of the separation of matter's position from a photon's position. This separation is caused by various types of expansion and contraction. As just explained, the combination of two types of position expansion, universal and frame, result in an expansion rate of 2c. From this point the return motion of the matter is caused by two types of contraction, frame and universal. The frame contraction causes the separation between matter and a photon to occur at a rate of c/(u sq.). Integral m-space that is created by frame contraction is measured relative the frame expansion rate of c. Frame contraction creates integral space at a rate of ln (u sq.)/((u sq.)-1). The second contraction is universal contraction, which will be described shortly.
In the opposite direction, integral m-space is created relative to the position expansion of matter as determined by the universal expansion in that direction plus frame expansion, which occurs in the direction of motion. Since these cancel each other out, matter's position before frame contraction is the same as it's initial position. Frame contraction changes this position at a rate of -c/(u sq.), in the opposite direction of motion. Thus, the integral m-space creation in this direction equals the integral m-space creation in the direction of motion. Below I illustrate this.
Frame contraction causes integral m-space creation at a rate of ln [(u sq.)/((u sq.)-1)]. The minimum rate of integral m-space creation occurs when u approaches infinity. This would mean that there is almost no separation between matter and a photon moving in the same direction. The maximum rate of integral m-space creation occurs when u approaches one, thus, when velocity approaches zero. As explained earlier, universal contraction shifts the frame (with the matter) back towards it's original position at a rate of c. While universal contraction affects all frames this way, it also affects the integral m-space created by frame contraction, reducing it by one-half, and this results in a shift in the it's position relative to a non-dilated frame. Universal contraction, (the same as matter's position contraction) gives a frame a velocity of -{[1+((u sq.)-1)/(u sq.)]c-{((u sq.)-1)}sq. root}/u]c}. The differences in the velocity between a non-dilated and dilated frame exists because universal contraction reduces the integral space created by frame contraction by one half. Since (1/2)ln[u sq./(u sq.-1)] equals Len[u sq,/(u sq.-1)]sq. root, the result is a rate of velocity for the dilated frame relative to the non-dilated frame equal to c/{dx/dy[u sq./(u sq-1)]sq.root}, which equals [{[(u sq.-1)]/u sq}sq. root]c. This is illustrate below.
The reasons for the reduction of integral space by one half are as follows. As pointed out earlier, frame expansion and contraction define the extended aspect of the Universe, or the actual universe. This extended aspect of the Universe increases the integral space of the Universe by a factor of two. The combination of universal expansion and contraction reduces this by a factor of 1/2. The way this occurs is; starting with the actual universe, spatial contraction creates integral at a rate of 2[ln(t)], this because the initial radius is 2R. Universal expansion can be considered to "push" this created space outwards at a rate of c, thus it is moving towards the edge of the actual universe. The integral space is not, however, diminished in size, because as this outward expansion occurs, the fundamental units of space contract in such a way that integral space is created at rate of ln (t sq.). With universal contraction, the created space is shifted back, the fundamental units of space are expanded, and integral space is reduced by 1/2. This is illustrated below.
