U Illinois Circle Campus at Chicago
Mathematics Department
October 30, 2001
Roy Lisker
ASSERTION: Under the above set of assumptions, in a non-linear universe U, it is not possible, from the existence of a clock C0 , which measures a duration of length T, to construct, ( save by trial and error), a clock measuring a duration 1/2 T . More generally, it is not possible to construct a clock measuring an interval of time aT, where a is any constant 0
RULERS Definition : A RULER is a mechanical system,an instrument for measuring lengths. It has the property that this distance between its endpoints remains invariant under rotations, translations and reflections.
CONSTRUCTION : To determine the midpoint of a segment S = {P1,P2} of length L, by motions of rulers in one spatial dimension:
Choose a ruler of length R
In this way one builds up a sequence of "remainders", R1, R2, R3..... If R and L are incommensurable, this sequence is infinite, and must converge to zero 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 . Let Hk = (1/2)Nk and locate the point on the segment S at the distance HkRk = Dk . Then the sequence of lengths {Dk} must converge to the point 1/2L .
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 however, to observe simply that the succession of rulers converges to zero. Obviously this is much simpler. One can use the familiar Euclidean construction involving parallel lines to find the midpoint of any segment.
All that is needed is a way of constructing parallel lines. This can be done with marked rulers. Since rulers are postulated to be able to move about freely, they can be employed effectively as compasses on the plane. 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 same construction as normally done, with a compass in 3-space
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. They are treated at some length in the original paper. The following Axiom, which may appear self-evident to some, or merely a definition, is crucial:
This depends of course upon the assumption that the universe U is deterministic, not quantum. Therefore, under the set of assumptions that we are working with, in a non-quantum, non-relativistic universe , all clocks are periodic non-reversible dynamical systems . The postulate of relativity allows one to escape this periodicity requirement precisely because it asserts that a quantum of light is sytem with a constant dynamical state variable, namely c = speed.
By assumption, given C, a clock C1 with period T1 < T can always be constructed. We wind up both clocks and set them going that time t = 0.
Obviously we can tell if T1 exactly divides T, and, in that case, can compute n such that T = nT1. If n is even we are finished. Note however, 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, we can make the simplifying assumption that T1 pulses exactly once in the time interval [ 0, T ], or
0 < 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 I, and I was incommensurable with T, that the convergence of the remainders cannot be guaranteed . This is because the procedure of building clocks to select between Ta = I , and Tb = C1- I , cannot be made without bringing in time reversal.
It is perhaps illegitimate to employ a play on words to say that, even if this selection process were possible, the carrying out of the full Euclidean algorithm process could "never" be accomplished because it would require an infinite amount ot time!
In practical terms, in the normal world in which both quantum and relativistic effects are disregarded, there is no way to subdivide the ticking period of one's clocks without, at some point, treating time as a spatial dimension and 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' when one does this, but that is what is in fact done.
If one then brings back relativity, there are no absolute velocities, and once is forced again to reply on the speed of the light quantum as the only reliable way to get around the limitations of clocks as periodic systems and thereby effect the subdivision of time intervals. CLOCKS VERSUS RULERS
Let L be a pre-assigned distance between end-points P1 and P2 . Since we are dealing with a Hilbert-Euclid universe, we assume that this enables us to locate and draw the entire line segment S connecting P1 and P2 . We look at the methods of construction for the midpoint of S in
If Z is the terminal point of the ruler, R* , and Y is 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 the two and call this R1. Lay off R1 against L . If R1 goes into L exactly, then we can compute a new number N1 such that L = N1R1 , and check to see if N1 is even. If R1 does not exactly divide L , or If N1 is odd, then choose a ruler R1*
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 identifying time with a spatial dimension, will involve a host of new considerartions:
Return to Home Page