Definition: A linear universe U is one in which there is some way translating the temporal dimension into a spatial dimension. That is to say, the structure of U allows the measurement of duration by clocks to be replaced by a measurement of distance by rulers. An obvious example of a linear universe is the universe of Special Relativity, or any universe in which the Postulate of Relativity ( constancy of the speed of light) applies.
Let W be a non-linear universe, one in which no isomorphism can be established between time measurement and length measurement. We make the following assumptions:
However, in either a 1-dimensional, 2-dimensional or 3-dimensional Euclidean space, it is possible , using a ruler, to determine,from a given length L, the midpoint l= (1/2) L.
Corollary: if I have a finite collection of clocks ticking off durations t1 < t2 < t3 ....< tn , then I cannot, save by accident or trial and error, construct a clock which ticks in any predetermined interval less than the minimum t1.
(A) RULERS
Fix a moment in time. Let L be a pre-assigned distance in W between end-points p1 and p2. The axioms of a Euclid-Hilbert universe allow one to construct the entire line segment S connecting p1 and p2 . We look at methods for constructing the midpoint of S in
THE ONE DIMENSIONAL CONSTRUCTION:
To determine the midpoint of a segment S = [p1,p2] of length L, by the motions of rulers in one spatial dimension.
Our assumptions allow us to make a ruler of length R < L. Lay off integral lengths of R along S, starting from p1. If R goes into L an integral number of times n , then L = nR . If n is even our work is finished.
If n > 1 is odd, or if R does not divide L exactly, then make a ruler of length R*< R , and compare the following lengths: (1) Ra = R* ; (2)Rb = R-R*. It is important to note that the property of free translation in space has made it possible to construct the length Rb .
Place the leftmost endpoints of rulers R and R* next to each other. If y is the terminal point of R*, z the terminal point of R, then Rb = Length[y,z ].
Either Ra or Rb must be less than (1/2)R in length.
Choose the shorter of these two lengths, and label it R1. Next lay off R1 against L . If R1 goes into L exactly, then we can compute a new number n1 such that L = n1R1. If n1 is even we are finished. If r1 does not exactly divide L , or if n1 is odd, then make a shorter ruler R1*
This process, known as the Euclidean algorithm, can be continued indefinitely
One thereby builds up a sequence of remainder lengths, R1, R2, R3..... If R and L are incommensurable, this sequence is infinite. It must converge to zero however since each remainder is less than or equal to 1/2 of the previous one.
Each Rk goes into L a certain number of times, say nk =[L/Rk] . Let hk = [(1/2)nk] and locate the point on the segment S at the distance dk= hkRk. Then the sequence of lengths {dk} must converge to the point (1/2)L .
Since there are no temporal restrictions on the measuring process, one can get around the Zeno Paradox by positing that each operation takes half the length of time of the previous. It is sufficient for our purposes to observe that the succession of rulers converges to zero.
THE TWO DIMENSIONAL CONSTRUCTION:Here One can find the midpoint of any segment by using the familiar construction from Euclidean Geometry involving parallel lines. All that is needed is a way of constructing parallel lines. This can be done with marked rulers, which are certainly permitted from our initial assumptions. One can restate this as follows: since rulers are postulated to be able to move about freely they can be employed effectively on the plane as compasses . The issues surrounding the use of compasses or marked rulers have nothing to do with the mechanical laws governing the space of the plane in which the construction takes place.
THE THREE DIMENSIONAL CONSTRUCTION. The compass as a mechanical system is allowed by the assumptions governing the universe W.
Lots of assumptions about Causation, and the relationship of the values of a state variable S to the determination of the behavior of a system M are involved here, but we will not go into them. I refer the interested reader to my paper On the Algebraic Structure of Causation The following Axiom is crucial:
One may in fact take this as the definition of what is meant by an equal interval of time. Note that it does not depend on the measurement of time, nor on the assigning of a numerical metric to the instants t1 and t2.
This axiom is itself dependant on numerous conditions discussed in the papers cited above. It is also assumed of course that W is deterministic, not quantum.
By assumption, given C with period T a clock C1 with period T1 < T can always be constructed. We wind up both clocks and set them going simultaneously at time t = 0.
Obviously T1 and T tick together in the first period of C , we can compute n such that T = nT1. If n is even we are finished. Note that C1 was a lucky accident.
If n is odd, we select a new clock, call it C1 and start again.
If T1 does not exactly divide T there is an integer m >1 such that
0 < (m-1)T1 Any such construction must involve some way of "pushing" the initial point of the second cycle of C back to the terminal point of C1 , a mechanical action that is easily achieved with a ruler.
This statement, which is clear yet informal here, is given a more rigorous treatment in the original paper.
Note that, even if it were possible to construct a clock with period J, and J were incommensurable with T, the convergence of remainders cannot be guaranteed . This is because the procedure of building clocks to select between Ta = J , and Tb = C1- J , cannot be made without bringing in time reversal.
It is perhaps a supercilious play on words to say that the carrying out of the full Euclidean algorithm process could "never" be accomplished because it would require an infinite amount ot time! In fact it isn't even possible to set it up.
In the world of daily life in which both quantum and relativistic effects are disregarded there is no way to subdivide the periods of a clock without treating time as a spatial dimension by making the assumption of constant velocity in some mechanical system that effectively replaces time duration with spatial length. Since length is reversible while duration is not, one is in some sense 'cheating' by doing so. But this is what is in fact done.
Bringing back relativity there are no absolute velocities, and one is obliged to rely on the speed of the light quantum as the only reliable way to get around the limitations of clocks as periodic systems, and subdivide arbitrary intervals of time.
(B)CLOCKS
It is clear that the mechanical process of finding the mid-point of a temporal duration T, in the absence of the postulate of relativity,( or some other unambiguous way of mechanically setting up an isomorphism between temporal duration and spatial length. ) involves a host of new difficulties
We will now show that under our set of assumptions for a non-linear W that, given a clock C which pulses in periods of duration T, there can be no procedure (other than lucky accident) for constructing a clock C* of period d= (1/2)T . Any isolated and closed system M with an identical complete set of state variables V at two distinct moments in time t1 and t2 , will pulse forever from and to this state in durations of equal length T = [t1, t2]
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