Time in accelerated reference frames

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Luigi Fortunati

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May 31, 2022, 3:54:35 AMMay 31
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In accelerated reference frames, the clocks do not stay synchronized
with each other.

Yet on Earth, which is an accelerated reference frame, all the clocks
that are at the same altitude remain perfectly synchronized with each
other wherever they are, why?

J. J. Lodder

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May 31, 2022, 5:48:51 AMMay 31
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No. They don't.
It is just that the gravitational effects of the sun can be ignored.
(with the available precision)
Remember that the radius of the Earth
is very small on the scale of the AU.

Jan

Tom Roberts

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Jun 1, 2022, 1:33:50 PMJun 1
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On 5/31/22 2:54 AM, Luigi Fortunati wrote:
> In accelerated reference frames, the clocks do not stay synchronized
> with each other.

Hmmm. Clocks that are at the same "altitude" relative to the
acceleration do remain synchronized.

Note also that "accelerated frame" is an oxymoron -- "frame" implies a
set of four mutually-orthogonal coordinate axes, which can occur ONLY
for inertial coordinates.

> Yet on Earth, which is an accelerated reference frame,

No, it is not. On the surface of the earth, a "small" region of
spacetime can be considered to be equivalent to an accelerated system in
flat spacetime, but larger regions on the surface are nowhere close to
an accelerated system in flat spacetime. Here "small" depends on one's
measurement accuracy.

> all the clocks that are at the same altitude remain perfectly
> synchronized with each other wherever they are, why?

Because in weak gravity, "gravitational time dilation" depends on the
gravitational potential, which primarily depends on altitude (as in an
accelerated system in flat spacetime). This is only approximate: when
measured very accurately, the potential at a given altitude depends on
the density of the material below, and on the positions of sun, moon,
and planets above -- at 15,000 feet above earth's geoid, the potential
over Pike's Peak is measurably different from that over Death Valley.

Tom Roberts

J. J. Lodder

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Jun 8, 2022, 4:51:14 PMJun 8
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Certainly, but one should realise that there is no such thing
as an absolute value of the Newtonian potential.
It depends on which masses you consider to be relevant,
at your level of approximation.
Or in other words, where you consider a practical 'at infinity' to be.

Fortunati says correctly that all clocks on a Newtonian equipotential
system, calculated with respect to all masses on the Earth,
will remain synchronised. (for all practical purposes, on Earth)
But they will not remain synchronised when you also consider
the (as yet unobservably small) gravitational effects of the Sun
at diferent places on Earth.

In practical terms, for all 'sub-lunar' calculations
the relativistically corrected TCG timescale will do.
(which takes only terrestrial relativistic corrections into account)
If you want to go further out in the Solar system you need TCB,
which corrects for solar gravitational effects.

BTW, for practical purposes all those relativistic time scales
are computed as corrections to TAI,

Jan

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