On 2006-01-11, David Malone wrote: > [A lot of discussion on this list seem to revolve around people > understanding terms in different ways. In an impractical example > of that spirit...]
Anyway: excuse me for repeating some basics of classical mechanics; but I believe it to be necessary. > To say if TAI is a monotone function of UTC, you need to put an > order on the set of possible TAI and UTC values. To say if UTC is > a continuous function of TAI, you need to put a topology on both. Yes: there is an order on the set of values of timescales - it is a basic property of spacetime models that one can distinguish past and present, at least locally. Spacetime is a differentiable 4-dimensional manifold, its coordinate functions are usually two times differentiable or more. In particular, the set of values of timescales does indeed have a topology (which is Hausdorff). > To me, TAI seems to be a union of copies of [0,1) labelled by > YEAR-MM-DD HH:MM:SS where you glue the ends together in the obvious > way and SS runs from 00-59. You then put the obvious order on it > that makes it look like the real numbers. TAI is determined as a weighted mean of the (scaled) proper times measured by an ensemble of clocks close to the geoid - so the values of TAI must belong to the same space as these proper times, which (being line integrals of a 1-form) take their values in the same space as the time coordinates of spacetime (such as TCB and TCG). No gluing is needed. And yes: this space is diffeomorphic to the real line. All of this is completely independent from the choice of a particular calendar or of the time units to be used for expressing timescale values. > OTOH, UTC seems to be a union of copies of [0,1) labelled by > YEAR-MM-DD HH:MM:SS where SS runs from 00-60. You glue both the end > of second 59 and 60 to the start of the next minute, in adition to > the obvious glueing. > I haven't checked all the details, but seems to me that you can put > a reasonable topology and order on the set of UTC values that > will make UTC a continious monotone function of TAI. The topology > is unlikely to be Hausdorf, but you can't have everything. If you subtract a time from a timescale value, you get another timescale value. If you mean to say that UTC takes its values in a different space than TAI then you cannot agree with UTC = TAI - DTAI, as in the official definition of UTC. And if you say that UTC - TAI can be discontinuous (as a function of whatever) with both UTC and TAI continuous then you must have a subtraction that is not continuous. Strange indeed. Where did I misinterpret your post? And can you give some reference for your assertions? Michael