The leading term in time difference with radius is the g_{tt} component of
the Schwarzschild metric
g_{00}(r) = 1 - 2GM/rc^2
We have 2GM/c^2 = 0.0088m which is the Schwarzschild radius. From R to R'
we can integrate this
∫g_{00}(r) dr = R' - R - 2GM/c^2 ln(R'/R).
For R' = R + δr we can approximate this as
∫g_{00}(r) dr = δr - 2GM/c^2 δr/R = g_{00}(r)δr
We take R the radius of the Earth 6.4×10^6m and so we have
2GM/c^2 δr/R = 1.38×10^{-9}δr
as the difference from δr in a flat space and the curved spacetime result
from ∫g_{00}(r) dr. For a centimeter this is then 1.38×10^{-11}m. So the
deviation from a Euclidean space result between R and R' is by this factor.
Dividing by the speed of light gives a time deviation of 4.6×10^{-18}sec.
The universe started 13.8 billion years ago and a year is 3.15×10^7sec so
the universe has been around 4.35×10^{17}sec and so one second deviation
since the start of the universe would be 2.3×10^{-18}, which is only half
the result above.
LC
On Thursday, November 29, 2018 at 8:29:12 AM UTC-6, John Clark wrote:
>
> In yesterday's issue of the journal Nature Scientists at the National
> Institute of Standards and Technology (NIST) reported they have made a new
> type of clock that is the most accurate ever, it's called a Ytterbium
> Lattice Clock. It's about 100 times better than any previous clock, if set
> at the time of the Big Bang 13.8 billion years ago today it would be off by
> less than one second.
>
> https://www.nature.com/articles/s41586-018-0738-2
>
> It's so good the main source of error is due to General Relativity, if you
> lift the clock up by just one centimeter the Earth's gravitational field is
> slightly weaker and so the clock runs noticeably faster, that may be why
> NIST is now working on a portable version of their Ytterbium Lattice Clock.
> If GPS satellites had clocks this good they'd know where they were relative
> to the Earth to within a centimeter and so could tell users on the ground
> where they were within a centimeter; and that would be more than good
> enough for jet fighters to automatically land on aircraft carriers without
> a pilot, even at night in a heavy fog in a bad storm with the deck tossing
> up and down. It would be by far the best instrument ever made to detect
> tiny changes in the gravitational field, and that would make it much easier
> to find things buried deep underground. The Earth just became more
> transparent. It might even be used to detect Gravitational Waves and Dark
> Matter.
>
> John K Clark
>
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