Time Dilation at Velocity Explained

Musings on the mnp Model and the Constituent Model

Time dilation as a function of velocity has a clearer explanation due to recent advances in the general Constituent Model and the more specific mnp Model. This explanation joins the explanation of relativistic mass and momentum that emerges from both Models (Constituent Models - Useful Supersets of the mnp Model).

The author recently (2018-07-17) realized that if the electrical/magnetic basis called Axis propagates at c in all directions rather than just along the line of Travel of the basic entities carrying the Axis information, then many explanations in the mnp Model become easier and more consistent with experience and experiment. Improved explanation and understanding of gravitation's lack of charge effects in spite of individual graviton's Axis information, magnetism itself, and the guide waves that lead to diffraction and "interference" by electrons at surfaces or edges or edges of slits are all advantages for the recent understanding/development of the mnp Model. The blog on uniform spread of electrical/magnetic polarity information awaits better pictures of magnetism and the frame. Perhaps the biggest advantage of Axis/polarity information propagating at c is that concomitant explanations in the Constituent Model become easier.

Time in. The words movement and moving refer to particles. Travel and traveling applies to the constituents or basic entities, which are seen as always traveling at c in a flat Minkowski space⁽¹⁾ in these absolute frame models⁽²⁾.

This post will demonstrate a generalized Constituent approach to measured time.⁽¹⁾ If time is measured by the oscillation of electron shells or nuclei, round-trip time for information or constituents themselves is involved. Since charge/polarity information propagates at c and constituents themselves travel at c, the time to cross a nucleus or shell when that nucleus or shell is moving can be examined as a proxy for oscillation. The easiest situation is when the information or constituents are traveling across the particle.

Figures 1 to 3) Oscillation, v = 0

At v=0, with particle width d_t, the time across is d_t/c and the time to return is d_t / c for a round trip of 2d_t / c. If the oscillation is around the ellipsoidal or circular surface of the particle, the round trip will be π d_t / c.

Figures 4 to 6) Transverse Oscillation, v > 0

If the particle is moving at v, information or constituents can be seen as traveling c on a diagonal, v on the longitudinal direction, and at a rate of sqrt(c²-v²) in the lateral/transverse direction. So the time for the constituent(s) or wave information to cross the particle d_t/sqrt(c²-v²), and the ratio of traversal time at rest to traversal time in motion is (2d_t/c) / (2d_t/sqrt(c²-v²)) or c/sqrt(c²-v²). The frequency of the oscillations or the ticks of a "clock" in a moving particle compared to one at rest is sqrt(c²-v²)/c, a familiar time dilation expression. If the path is not across but around the surface of a spheroid or ellipsoidal shell, the distance traveled is π d_t at v=0 and the time to make a circuit is π d/c. When the particle is moving, the time is π d_t / sqrt(c²-v²). The ratios are the same.

Examining the time taken to make a forward and backward circuit in a rest or moving particle is important and appears to involve different expressions. When the particle is at rest, the formulae for oscillation time is the same, 2d / c. When moving, the forward and backward times are different. Forward, d / (c - v), which takes longer. Backwards, since the back end of the particle is catching up with the returning information or constituents, the time is d / (c + v), shorter than the forward traversal. The total round trip is d_l(c - v) + d_l(c+v) / (c² - v²) or 2d_l c / (c² - v²), which looks and is very different from the convenient expression when the information or constituents are traveling transverse to particle movement.

Figures 7 to 9) Longitudinal Oscillation, v > 0

But note that the classic formula for length compression has a factor of sqrt(c²-v²) / c, so if d_l is d sqrt(c² - v²) / c, then the total round trip time is 2d / sqrt(c² - v²) and the ratio of absolute time taken for an oscillation for moving compared to rest particles is exactly the same as the time to make a round trip perpendicular to the direction of movement: c / sqrt(c² - v²). The ratio of oscillations or clock ticks is the inverse or sqrt(c² - v²) / c, also the same ratio as with oscillation transverse to the direction of movement. As expected, oscillations and clock ticks are slower for the moving particle.

Length compression actually saves the discussion of time slowing with movement, just as it saved discussions of the two way speed of light in absolute frame models early in the development of the mnp Model!

IF (and only if) length contraction occurs by the Lorentz or special relativity formulae, then time dilation by the classic formula is seen as real and equal across all diameters and circumferences of the particle.

Status of Length Contraction in the mnp and Constituent Models.

The author does not consider the length contraction formula entirely proven within the mnp Model to the degree that relativistic mass increase has been proven to be essential for motion. Length contraction has been shown as likely in the mnp Model, given that coils must flatten for movement to take place. Since the basic charge entities that are constituents of the mnp Model are traveling in coils, on average each coil must flatten so that the axis is no more than cos^-1(sqrt(c²-v²) / c) from the longitudinal axis. The average longitudinal dimension of those coils must not exceed sqrt(c²-v²)/c. In general, the author feels the coils should have a fairly even distribution from axis parallel to movement to the "maximum" orientation to minimize internal changes of direction, internal rotations, and internal or surface variations in density. This suggests but does not yet prove that the longitudinal dimension of the particle is reduced by a similar factor.

Length compression seems to be real in Constituent Models, but the author feels that the exact value for length compression needs at some point to be proved. A few embryonic thoughts toward proving length compression in the Constituent Model follow. The constituents traveling at c that make up particles must be curving in some fashion, so that the particle remains more or less a unit. The "surface" of the particle does not disintegrate with movement.

Embryonic Thoughts Regarding Particle Shape and Movement

The forward surface will remain intact, so that the range of velocities/directions in a dV portion near the front surface will have a limit on the directions available. In like manner, a dV portion of the back surface will limit the directions to NOT backward from that surface. Integration across that surface may show that the surface itself needs to change from a perfect sphere?? The particle will remain intact, perhaps the surface will remain smooth, though this may be an over simplification.

Developmental thoughts: At the perimeter transverse to movement, constituents cannot be traveling forward at c in direction of moving, but must be skewed to side so that forward progress is less than or equal to [remember rest condition, where it can move forward at c as long as the curvature keeps the constituent within the particle. Actually, must be tangent?? and curving in. Can be a maximum of v forward or less. Or actually exactly v forward if I allow no oscillation of the shape at all. No, just have to travel forward and away from the edge faster than being caught up with, since v 0 can be any direction as long as curvature radius less than or equal to radius of particle. At high speeds, must pull back from the surface if going backward-ish faster than particle surface is catching up.. Back from the edge, can be different. Could this lead to a curvature that represents an ellipse?? A diagram will clarify, for the author as well:

Figure 10) Constituent Travel at "Surface" of Particle

This diagram shows four extreme points - the center front, center back, and two of the lateral midpoints. At the front, forward progress for the constituent must be strictly less than v, at the rear strictly more than v, and at the lateral midpoints strictly more than 0 and less than c. Within, perhaps, some variation that does not affect the integrity of the particle. On further thought, if the particle is not moving at all, the lateral midpoints may be moving at c in any direction, as long as they are curving inward. So the limits on forward or backward progress at the lateral circumference may be more like sqrt(c²-v²) and curving inward.

The particle should remain symmetrical around the axis of movement (since it continues in that direction), the momentum will be consistent radially around that axis.

Acceleration must average 0 at all times for a stationary or non-accelerating particle, so the third derivative and above will also be 0. That acceleration averaging 0 would applies in all planes and axes of symmetry.

The more plausible rules the better, to limit the freedom internally. Acceleration may be limited compared to momentum/mass, which would limit the radius of turning.

At a given point within the particle, the direction amount is a 3d tensor, the magnitude in any direction representing how much constituent is going each way. At the surface, there are of course limits. If we assume no effect on each other, the change through time can be whatever it is. The integral over 3 dimensions of mv will give total mv. Change over time of the tensors should be radially symmetrical about the line of movement. Change over time at any point on the surface should be the velocity of movement. Conditions of continuity might apply, for example, a minimum radius of change might prove useful or necessary for the modeling.

Rotation within the particle around the axis of movement is not expected, but "no rotation" would be better emerging from the math than needing to be imposed.

Consistent density throughout the particle or across the surface would be better emerging from the math than needing to be imposed. Would it be provable??

Conclusion

Time dilation is explained in the Constituent and mnp Models. Length contraction with movement is demonstrated in both Models. However, since a fully convincing proof of time dilation relies on length contraction, the formula for time dilation remains unproved until the formula for length contraction is proved.

Footnotes:

(1) Both mnp and Constituent Models are absolute frame models, so some readers may need to suspend disbelief for a while. The lack of time dilation with physical acceleration, as shown by the muon storage experiment, and the absence of Twin Paradox from the GPS satellite system may make such suspension a little easier.

(2) Insuring that travel applies to basic entities and constituents and that movement and moving applies to particles, so that clarity of reference is maintained, has proved surprisingly difficult. The words were almost randomly distributed in the early drafts of this post. So information should spread or propagate, to maintain clarity.