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. > > David.
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 23:59:59.999999999999... -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. _______________________________________________ questions mailing list questions@lists.ntp.org http://lists.ntp.org/listinfo/questions
