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?


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