Zeno's Paradox

star field

4 Length/Time Changes

Although transfer time slots replace intrinsic time slots when particles move with at a constant speed, we cannot simply subtract the transfer time slots from the intrinsic time slots to find out how many intrinsic time slots we have left. Because we now recognize that space and time form a unity and do not stand independent of each other.  In figuring the change in time slots, we must account for the companying change in space. Fortunately, the process is simple. As in finding an interval between two events in spacetime, we replace normal time intervals with the squares of those intervals, then take the square root of the result.

  Total Time Squared = Intrinsic Time Squared + Transfer Time Squared.

The times are measured in quantum time intervals. Experience shows that quantized spacetimes based on different minimum dimensions act the same way with regard to time slowing. So, the reduction of aging by speed can be demonstrated in Zenovian spacetime. Back to  the stadium.

The upper bound of speed, c, in stadium spacetime is 100 fps. Achilles runs the first 30 ft in one second, moving at 0.30 c. That one second is the Total Time, for the spectators. Divide that second by the quantum time interval, and you find it contains 100,000 quanta. Achilles travels 30 feet. Divide that distance by the quantum length interval and you find he makes use of 30,000 quanta. To move through them, he needs 30,000 transfer time intervals. Calculate intrinsic time remaining from the sum of the squares.

100,000 x 100,000           =  Intrinsic Time Squared + 30,000 x 30,000

Intrinsic Time Squared   =  9,100,000 quanta time-intervals squared

Intrinsic Time                    =  95394 quanta time intervals

To get time in seconds, divide by the 100,000 time intervals in a second,

                                            =  0.954 second

Patroclus, watching the race, aged one second while Achilles aged only 0.954 second. Patroclus concludes that Achilles' watch is running slow, but our hero's biological processes are running just as slowly.

Suppose a virtual Achilles appears in Universal Coordinated Spacetime and hitches a ride on a 0.3 c interstellar spacecraft. Redo the calculation for the minimum length and minimum time in our current universe. (If you do not want to do this, I can tell you you will get the same answer.)

The upper bound for speed, c, for the interstellar is the speed of light. Moving at 0.3 c requires transfer quanta amounting to 0.3 of the quanta required for 1c.

Max quanta at c = 1/ 1.616229 x 10-35  (reciprocal of minimum length)

                               =  6.18724 x 1034 quanta for 1 sec when c is the speed of light

        0.3 of this     =  1.85617 x 1034 quanta for Achilles at 0.3 c


Max quanta squared           =    38.28194  x 1068

Transfer quanta squared    =   3.445367 x 1068

Difference squared              =   34.836573 x 1068

Square root                           =   5.902252 quanta x 1034

Fractional Shrinkage            =  5.902252 divided by  6.18724

                                                  =  0.954  sec

As well as being the same value as for Zenovian spacetime, it is also the value calculated by Einstein’s special relativity formula. The generality of time dilation among different spacetimes is a property of quantized spacetime, as is the ability to achieve the same result with time steps or space steps, as long as the same type of step is used throughout the calculation.

A similar calculation can be made using the quantized spatial dimension to find the shrinkage in space. In one dimension this is

Total Length Squared = Intrinsic Length Squared + Transfer Length Squared

This affects ensembles of individual particles separated by fixed spaces, as in a measuring rule. When stationary, any pair of these particles has a specific number of quanta between them. This is intrinsic space. When the particles move at a specific velocity, some intrinsic quanta become transfer quanta, decreasing the intrinsic space between particles. The object shrinks. The digital solution for shrinking parallels that for time slowing. In all quantized spacetimes.

Quantization provides a physical picture for the origin of relativistic time dilation and provides a simple calculation of its value. But there remain interesting questions about the nature of the two time components and how their pattern for a particular speed can sustain the momentum of a particle across the universe.

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