# [ntp:questions] Leap second to be introduced in June

William Unruh unruh at invalid.ca
Thu Jan 22 18:04:00 UTC 2015

```On 2015-01-22, David Malone <dwmalone at walton.maths.tcd.ie> wrote:
> William Unruh <unruh at 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.

```