Robert Winn

2024-02-13 06:55:16 UTC

[[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)

<***@gmail-pink.com> if you'd like a copy. -- jt]]

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)

<***@gmail-pink.com> if you'd like a copy. -- jt]]