On 2015-01-22 11:04, William Unruh wrote:
On 2015-01-22, David Malone <dwmal...@walton.maths.tcd.ie> wrote:
William Unruh <un...@invalid.ca> writes:
Note UTC differs from TAI by an interger number of seconds, AND that
integer changes with the leap second. Ie, it cannot be continuous if TAI
is continuous.
That assumes that UTC can be represented as a real number with the
standard topology, which doesn't seem to be what TF.460 says. It
describes each second as labelled, which means that you have to
stitch together all possible unit intervals for each second with
some topology, and then you can ask if the path taken by UTC through
this space is continuous.
General relativity assures us that time exists and is measured by a
metric. Take that metric and define a proper time say at the center of
the earth. Now one can ask whether TAI or UTC is a function of that
time.
Consider some labeling of the time. Jun 30 23:59:00 and Jul 1 00:01 let
us say. Now when we look at TAI, that second one is one second one is
120 seconds ( as measured by that metric) later than the first. For
UTC it is 121 seconds later than the first. As one hunts in toward
midnight, say Jun 30 23:59:58 vs Jul 1 00:00:02 say, that interval is
still 1 second different in the two scales. And for Jun 30
23:59:59.999999999 and Jul 1 00:00:00.000000001
while TAI says that difference is .000000002 sec, UTC says it is
1.000000002 sec different.
That for all purposes is a discontinuous function of the time as defined
by General relativity. Now, it is true that UTC does give a name to that
second that lies between the two times, but giving it a name does not
make the function continous.
UTC is a function which is linear and continuous for all times which are
not the leap second, but is discontinous at the leap second. The limit
of the function as delta t goes to zero of the two scales is not the
same. Limit as delta t goes to zero t_relativity (UTC Jun 30 23:59:59.999...
-Delta t) is not equal to Limit as delta t goes to zero t_relativity
(UTC Jul 1 00:00:00:.00000.... +Delta t) while it is for TAI. The fact
that UTC gives a name ( 23:59:60) to that extra second does not alter
the above fact.
The fact that UTC publishes a list of when those discontinuities occur
is also irrelevant. That one says a function is discontinuous at some
value of x and how much it is discontinous, does not alter the fact that
it is discontinous.
TAI, TT, UTC, UT, UT0, UT1, UT2 are empirical time scales based on
measurements not functions, with some scales having fairly simple
relations, and UTC stepping by leap seconds.
The relative values on these scales are only available accurately
when they are published about a month after the time, with estimates
available later each day from some labs, based only on those individual
labs' standards, which may need corrected later in the month.
So none of these simplified arithmetical approaches are anything more
than working approximations to the nearest jiffy, and they are not really
useful unless you are working in astronomy or physics related fields.
POSIX allows you to do useful calculations on civil times based on mean
solar seconds, but there are no useful sources for synchronizing or
calibrating POSIX time, as all time sources use scales based on SI seconds.
--
Take care. Thanks, Brian Inglis
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