Horace Heffner wrote:
It seems to me, to understand the "planes around the earth problem"
quantitative data is needed. In the absence of such data, it seems
reasonable to me that the differences in time can be ascribed in part
to the fact the earth clock is rotating. If the earth were not
rotating, and the earth had no mass, then the two traveling clocks
should show the same final time upon return, and that time should be
less than the earth's clock, due to the acceleration the traveling
clocks experience.
Indeed, that's what SR predicts.
If all you want is the answer, and if you're willing to assume a
"Styrofoam Earth" (no gravity), then there's an easy trick to find all
the time rates and final clock readings: Take the point of view of a
hypothetical _stationary_ observer at the center of the Earth. Then the
clock rates of all three revolving clocks can be computed trivially:
they're just 1/gamma times the base rate, where "gamma" is determined by
looking at their velocity as observed by the "man in the middle".
In this case, gravity just adds a fudge factor which causes "lower"
clocks to run slower; for aircraft flying in the atmosphere versus
someone standing on the surface, a net altitude difference of about
0.1%, I suspect the "gravity correction" is dwarfed by the other effects.
The fact that all three clocks differ must be due in part to the fact
the clock going against the earth's rotation must experience much
greater accelerations to make the trip, assuming the trips are made in
about the same length of time, or assuming that they are made as
orbital flights.
>
It takes much more energy and acceleration to perform
an east-to-west launch than a west-to-east launch, both on take-off and
landing.
In a steady state orbital situation, a coasting situation, it is
interesting as to which clock might be advancing more rapidly, the
earth clock, the east-to-west satellite, or the west-to-east
satellite. From the satellite's perspectives, they should experience
no acceleration during this interval, while the earth based clock, if
in an enclosed box or not, would be, by Einstein's equivalence
principle, experiencing acceleration.
>
It thus seems that the longer
the flights last, the more time the orbiting clocks should gain over
the earth clock. There should not be any *final* difference between
the two orbiting clocks due to this coasting part of the journey.
Final here means when all three clocks are brought back together.
Bang-on. Right again. No prob.
But wait -- your counter-orbiting clocks are _in_ _orbit_ which means
they're traveling at the same speed RELATIVE TO A FIXED OBSERVER! On
the other hand, the two aircraft in the Sagnac experiment went around
the Earth at fixed speed RELATIVE TO THE SURFACE OF THE EARTH. From the
POV of a fixed observer, the one going east was going faster than the
one going west, by a non-trivial amount. If the planes had nominal
airspeeds of 500 mph, then the man in the middle saw the eastbound plane
going east at 1500 MPH, while the "westbound" plane was actually _also_
going EAST (flying backwards!) at about 500 MPH. Since the eastbound
plane was going 3 times as fast as the westbound plane, of course its
clock ran slower.
Again, figure the problem from the point of view of a non-moving
observer at the center of the earth, and the difference is obvious. The
confusion sets in when we look at it from a frame of reference attached
to the Earth's surface.
I have to wonder what the data actually looks like.
Me, too.
Horace Heffner
