# [LEAPSECS] Nit-pick: SI second

Magnus Danielson magnus at rubidium.dyndns.org
Thu Feb 10 17:58:29 EST 2011

On 10/02/11 16:18, Tom Van Baak wrote:

>> Since the velocity of the atomic clock causes relativistic dilation,

>> surely it is not the altitude-above-sea-level, but the radial distance

>> from the earths axis that we are talking about???

>

> 1)

> This seems to be a common misunderstanding. Realize that

> the relativistic dilation we're talking about would occur even if

> the earth were not rotating: it's due to the mass of the earth.

Actually, this shift would change if the earth was not rotating as it
would have a different mass distribution.

> To a first approximation, the magnitude of the blue shift is gh/c².

> This is about 1.1e-16/meter or 1.1e-13/km, which means that a

> clock 1 km above sea level runs faster than a clock at sea level

> by about 10 ns/day. You can actually measure this.

The height over sea level is an approximation at best. The gravity
strength at sea level depends on where on earth you are. Infact, the sea
level itself reflect this. The strength of gravity can shift for many
different reasons.

It is the difference in gravitational level which causes an observed
shift in frequency between two reference frames.

>> I.e. surely both latitude and altitude affect the ceasium? I mean the

>> velocity of the atomic clock as it stands in the lab is dictated by the

>> earth rotation times the radial distance from the earths rotational

>> axis?

>

> I get this question a lot. Let me try several more answers.

>

> 2)

> Yes, the radius of the earth is slightly dependent on latitude,

> perhaps that's what you're thinking about. But a clock at sea

> level at lat 0 will run the same as and a clock at sea level at

> lat 45 or lat 90 because that's what MSL is - the equipotential

> surface of the geoid.

No. You are confusing two concepts, that of equal gravitational strength
and that of mean sea level.
http://en.wikipedia.org/wiki/Sea_level#Difficulties_in_utilization.

The MSL is not a very good reference for this discussion. It just isn't
quite what we believe it to be, so if we are going nit-picking galore,
it's good for ball-parking but not for nitty gritty.

Rather, there is formulas for approximation of gravitational
acceleration as a combination of latitude and height. A more
fine-grained variant would be to include a more detailed gravitational
bump-map, but actually measuring it would be needed.

> So this is why it's not correct to think that time dilation seen by

> clocks at elevation is due to radial velocity. It's the gravitational

> effect that makes clocks run faster at higher elevation.

Then there is the Sagnac effect... which is a relative phase error due
to the earths rotation.

Cheers,
Magnus