[LEAPSECS] Nit-pick: SI second
Tom Van Baak
tvb at LeapSecond.com
Thu Feb 10 10:18:53 EST 2011
> 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???
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.
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.
> 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
I get this question a lot. Let me try several more answers.
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.
Well, it depends on which reference frame you work in. If you
choose a non-rotating earth centered frame, then no, the radial
velocity of a clock at sea level is the same as the radial velocity
of a clock 1 km above sea level -- because they are both zero.
Remember you're comparing two clocks at different elevation
but on the same planet; there's no need to bring in extraneous
factors like rotation of earth about its axis, or its revolution
around the sun, or its trip inside the milky way.
The acceleration of gravity (9.8 m/s²) would be about 0.3%
greater if the earth were not rotating.
There's a slight change in g as a function of latitude. At the
extremes it's 0.5% -- 9.780 at the equator, 9.832 at the poles.
There is also a slight change in g as a function of altitude:
A Seattle sea level g = 9.8079 becomes 9.8049 at 1 km
above MSL (0.03% less) or 9.7771 (0.3% less) at 10 km.
If you want to use a rotating frame then you could do the
following calculations for two clocks separated by 1 km at
Clock A, 1 km above sea level:
distance from earth center: 6378 + 1 = 6379 km
radial velocity: 465.164 m/s
v/c: 155.162 ppm
SR (red) shift: -1.2037e-012, or -104.005 ns/day
Clock B, at sea level:
distance from center of earth: 6378 km
radial velocity: 465.091 m/s
v/c: 155.138 ppm
SR (red) shift: -1.2033e-012, or -103.972 ns/day
So as expected, and as you conjectured, clock A has greater
radial velocity and thus a greater SR time dilation than clock B.
But do you see little the difference is? Although the velocity is
high (over 1000 mph, mach 1.4) the velocity difference is only
0.073 m/s; the SR time dilation difference is -0.033 ns/day.
Compare this with the gh/c² GR calculation. With h = 1km you
get +9.44 ns/day, about 300x greater in magnitude.
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.
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