On 11/23/21 7:51 PM, Mark-T wrote:
> Reviewing the Hafele-Keating experiment, there's something I don't
> get. Why does elapsed time depend on direction?
Because at base it is the plane's speed relative to the ECI that
matters, not relative to the (rotating) earth surface.
[The ECI frame is the locally inertial frame in
which the center of the earth is at rest. It
does not rotate relative to the "fixed stars".
It is locally inertial in the sense of GR, but
is not an inertial frame in the sense of SR;
for discussions like this that is sufficient.]
In a simplified H-K experiment in which gravity can be ignored, the
elapsed time of each trip is just
\integral sqrt(1-v(t)^2/c^2) dt
where v(t) is the speed of the plane relative to the ECI coordinates,
and t is the ECI time coordinate. Due to the rotation of the earth,
v(westbound) < v(eastbound), which accounts for the difference.
NOTE: That equation holds ONLY for inertial coordinates, such as the ECI
coordinates used by the GPS.
In the real H-K experiment, the effect of gravity is small compared to
the above effect of SR. Gravity must be included to obtain numerical
consistency, but it can be ignored for the basic idea.
> The planes flew identical speeds and altitudes.
Relative to the earth's surface. But the their speeds are quite
different relative to the ECI, and that's what matters.
[They were not precisely equal relative to the earth
surface, but were close enough for an elementary
discussion like this.]
> Why should an east-west discrepancy occur? Isn't the earth an
> arbitrary co-ordinate system?
I suppose just about anything could be an "arbitrary coordinate system".
But in physics like this what matters is an INERTIAL coordinate system.
The rotating earth is not such a coordinate system, but the ECI is.
[Again, close enough for a discussion like this.]
> Was this a surprising result, or standard relativity?
It is unsurprising and simply standard relativity.
> Did Einstein predict this?
Not specifically, because he was long dead. But he did discuss similar
situations, and his theory of General Relativity certainly does
accurately predict the H-K result.
> I should add, somebody made a major effort compiling the faq for this
> group. It's a real public service.
It was an effort by many people. I wrote the FAQ entry on the
experimental basis of SR.
Tom Roberts