[[Mod. note -- Please limit your text to fit within 80 columns,
preferably around 70, so that readers don't have to scroll horizontally
to read each line. I have manually reformatted this article. -- jt]]

When Galileo and Isaac Newton talked about time, they were referencing
the rotation of the earth. Modern scientists are talking about
transitions of a cesium isotope atom. I am going back to the concept
of time these earlier scientists used because , if we take the
example of a clock in a flying airplane, modern scientists say that
clock will disagree with a clock on the ground. If the clock on
the ground agrees with the rotation of the earth, then, obviously,
the clock in the airplane does not. But its time can still be
expressed with the Galilean transformation equations used by Galileo
and Newton.
x'=x-vt
y'=y
z'=z
t'=t
The last equation persuaded scientists to switch over to the Lorentz
equations because if t'=t, then x=ct and x'=ct' cannot both be true,
which they are in the Lorentz equations. The results of the
Michelson-Morely experiment can still be shown by Galilean equations.
There have always been faster and slower clocks. Scientists like
Galileo and Newton understood this. How do you show the time of a
faster or slower clock with the Galilean equations? You use another
set of Galilean equations with different variables of velocity and
time. So the inverse equations to the above equations would be,
if describing time shown by a slower clock in an airplane
x = x' - (-vt/n')n'
y = y'
z = z'
n = n'
n' is the time of the slower clock in the airplane. (-vt/n') is the
velocity of the ground relative to the airplane according to the time
of the clock in the airplane. n = n' shows that the time of the
clock in the airplane is being used in both frames of reference.
So then to show that the speed of light is 186,000 miles per second
in both frames of reference, as scientists said the results of the
Michelson-Morley experiment required, all you do is say x=ct and x'=cn'.
x'=x-vt
cn'=ct-vt
n' = t-vt/c
This value for the time of the slower clock is the same as the
numerator of Lorentz's equation for t'.
t-vt/c = t - vct/c^2 = t - vx/c^2
Having x in this expression is unnecessary in the Galilean equations
because there is no length contraction in those equations. The
spatial coordinates are the same as in the original set of equations.
The inverse equations also convert back to the original equations
if you cancel out the (n')'s.
x = x' - (-vt/n')n'
x = x' +vt
t = t'
If we apply these equations to the times of clocks on the planets
of the solar system, we can see an interesting relationship. Mercury
is the fastest moving planet, having a speed of 30 mi./sec in its
orbit. A clock on Mercury would be faster than a clock on earth
because earth is moving slower, 20 mi/sec. But the t from which
n' is derived to show the time of the clock on Mercury is not derived
from the time of a clock on earth. The time of the clock on earth
has an n' derived from the same t and is a faster clock than the
one on Mercury because earth is farther from the sun and is moving
slower. As we consider the outer planets each succeeding planet
has a slower speed of orbit and a faster clock. The time t in the
equation is the time of a clock, say halfway between the sun and
the nearest star, where the rate of a clock would be faster than
the rate of a clock on any planet. So the time of this interstellar
clock would be unaffected by gravitation and would not be moving
relative to the sun. The speed of Neptune in its orbit would be v
in the equation
x = x' - (-vt/n')n'
t would be the time of the interstellar clock, and n' would be the
time of the clock on Neptune. Then the time of a clock on each
planet could be obtained the same way until you come to Mercury,
which would have the slowest clock. Scientists of today calculate
all of this using the time of a clock on earth as the basis for
their calculations, which is much more difficult than this method.

[[Mod. note --

Scientists doing planetary ephemerises and celestial mechanics often
use the time of a clock at the solar system barycenter (= center of mass),
so as to avoid the complications of the Earth's changing speed and
changing distance from the Sun (and hence changing depth in the Sun's
gravitational potential).

In this context, let me (once again) put in a plug for a beautiful
paper
Carroll O. Alley,
"Proper Time Experiments in Gravitational Fields with Atomic Clocks,
Aircraft, and Laser Light Pulses",
pages 363-427 in
"Quantum Optics, Experimental Gravity, and Measurement Theory",
eds. Pierre Meystre and Marlan O. Scully,
Proceedings Conf. Bad Windsheim 1981,
1983 Plenum Press New York, ISBN 0-306-41354-X.
Alley's paper describes a set of experiments which directly compare
stationary and moving atomic clocks. In these experiments the moving
clock was in an airplane flown in a "racetrack" pattern over Chesapeake
Bay, illuminated by a ground-based pulsed laser so that the stationary
and moving clocks could be compared *in real time* while the airplane was
in flight. The results (particularly figures 44-47) clearly show both
special- and general-relativistic (gravitational) time effects.

Unfortunately I don't know of any open-access copy of Alley's paper
online, but I have a pdf of it and can send it to anyone who's
interested; email me privately at (remove -color)
<dr.j.thornburg@gmail-pink.com> if you'd like a copy. -- jt]]

On Tuesday, February 13, 2024 at 1:55:21=E2=80=AFAM UTC-5, Robert Winn wrote:
[[Mod. note -- quoted text snipped here. -- jt]]

> When Galileo and Isaac Newton talked about time, they were referencing
> the rotation of the earth. Modern scientists are talking about
> transitions of a cesium isotope atom. I am going back to the concept
> of time these earlier scientists used because , if we take the
> example of a clock in a flying airplane, modern scientists say that
> clock will disagree with a clock on the ground. If the clock on
> the ground agrees with the rotation of the earth, then, obviously,
> the clock in the airplane does not. [[...]]

[[Mod. note -- quoted text snipped here. -- jt]]

I am not sure, but I heard that ring light gyroscopes may be interesting

[[Mod. note -- Yes, they're a fascinating technology. See
https://en.wikipedia.org/wiki/Ring_laser_gyroscope
for a nice introduction. And here
https://arxiv.org/abs/1007.1861
is a proposal for a very sensitive ring laser gyroscope to try to detect
the Earth's "spin dragging" gravitomagnetic field (previously measured by
Gravity Probe B
https://en.wikipedia.org/wiki/Gravity_Probe_B
and by Lageos I and II
https://en.wikipedia.org/wiki/LAGEOS
-- jt]]