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

Jochen Bern Jochen.Bern at LINworks.de
Mon Jan 26 12:17:53 UTC 2015

```On 01/22/2015 07:04 PM, William Unruh wrote:
> 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.

(Bad choice because relativity says that clocks down the gravity well
run faster, but we've been ignoring that fact so far, anyway, so ...)

> [...] 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.

Actually I agree with that last sentence, but not in the way you expect.
It's the *metric* that UTC defines, along with the representation, that
makes the function continuous. Basically, UTC not only says that
"23:59:60" is valid and shall be ordered between 23:59:59 and 00:00:00,
but that it represents one SI second (as measured with any suitable
instrument) wherever it is officially inserted.

Hence, the difference between "23:59:59.999999999 UTC" and
"00:00:00.000000001 UTC" is 0.000000002 SI seconds wherever a leap
second is *not* inserted, and 1.000000002 SI seconds where it is,
*because that's what you were told how to count it*, and since computing
the difference between the two UTC timestamps (with a list of past and
present leap seconds at hand) correspondingly results in "0.000000002
UTC seconds" or "1.000000002 UTC seconds" unless the timestamps are in
the uncertain future, the two notions still agree down to any resolution
you want -> continuous, linear, derivative with slope = 1.

> The fact that UTC publishes a list of when those discontinuities occur
> is also irrelevant.

UTC says that leap seconds are part of the UTC representation of time
(i.e., on the conversion function's ordinate) and correspond to an
actual SI second of physical time that passes (i.e., it's present on the
abscissa as well). Your refusing both punches a square hole, one square
second large, into the function's graph - that's not a discontinuity in
a strict sense, it's a stretch where the function remains undefined by
your refusal to acknowledge the definition. Anyone willing to re-insert
that square with the diagonal line on it into the graph gets a straight
line from one edge of the paper to the other, and has no reason
whatsoever to see a discontinuity.

Regards,
J. Bern
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