Yikes, utterly hairy math!

From: Ben Bean (kavs_delethis__at_sysmatrix.net)
Date: 11/02/04 

Date: Tue, 2 Nov 2004 10:34:20 -0500

I am fond of maintaining that, thanks to relativity, a radio signal emitted
halfway between two neighboring equatorial villages arrives at the two
villages simultaneously, in the actual rotating frame ie. coordinate system
to which those villages belong. But I don't know the math. Maybe you can
help, considering your know-how.

This much I know: if you use an inertial frame unmoving WRT Earth's center,
then there is a sizeable computed disparity between eastbound and westbound
EM signal transit times for the scenario cited. Yet if you choose an
inertial frame that is moving eastward, at Earth's tangential velocity WRT
Earth's center, and the chosen frame is momentarily oriented tangential to
Earth at the midpoint of the two villages when the signal emits; then the
calculated disparity between eastbound and westbound EM signal transit times
pretty much disappears altogether. When I did some trial calculations, about
three orders of magnitude of east-vs-west disparity vanished when I employed
the nearest tangential IF instead of employing UTC. The only problem is that
such an inertial frame is merely a real close approximation... those 2
villages (in motion) aren't actual natives of such frame.

I don't know how to cipher the actual rotating metric, but based on what I
have shown above, it seems mighty clear that in the actual rotating
coordinate system of Earth, there must be null or negligible disparity
between eastward and westward transit times of an EM signal, over a suitably
small range, landmark to native landmark.

The actual ciphering is too much for my depth. And to make matters more
confusing, I believe that separated equatorial clocks are ever
unsynchronized, according to their native rotating scheme?

I am perfectly well aware of the Sagnac findings, but I think that when
you're dealing with arcs that are far less than one radian in length,
relativity (eg. light's behavior) still operates as expected, especially for
a slowly-rotating orb such as Earth.

Relevant Pages

  • Yikes, utterly hairy math!
    ... halfway between two neighboring equatorial villages arrives at the two ... villages simultaneously, in the actual rotating frame ie. coordinate system ... if you use an inertial frame unmoving WRT Earth's center, ... calculated disparity between eastbound and westbound EM signal transit times ...
    (sci.physics.relativity)
  • Re: Yikes, utterly hairy math!
    ... >> halfway between two neighboring equatorial villages arrives at the two ...
    (sci.physics.relativity)
  • Re: Yikes, utterly hairy math!
    ... > halfway between two neighboring equatorial villages arrives at the two ... > villages simultaneously, in the actual rotating frame ie. coordinate system ...
    (sci.physics.relativity)