Re: Confirmation of Relativity

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

I need to take a break from some late night end-of-semester grading.
So I think I'll make a comment about Leigh's comment:

> ...  Time runs more slowly on Earth than it does on the Moon, and
> more slowly on an artificial satellite than on Earth's surface.

The first claim here is correct, but the second needs to be qualified.
*If* the artificial satellite orbits the Earth in a circular orbit whose
radius is less than twice the radius of the Earth then Leigh's claim is
correct.  But if the orbital radius is greater than twice the radius of
the Earth then it is incorrect.  If the orbit is wildly ellipical then it
is a little trickier to decide on the claim's validity one way or the
other, since one would need to know both the perigee and apogee, and one
would also need to know if the claim was being made for all parts of the
orbit, or for just for the average time dilation per orbit.  Note that
since geostationary satellites orbit farther out than twice the Earth's
radius, time for them is slowed *less* than for observers fixed on the
Earth's surface.

Also it should be noted that time is differently dilated for different
observers fixed to different locations on the Earth's surface itself
where the dominant effect is a latitude-dependent contribution to the
time dilation comes from different surface rotational speeds about the
Earth's spin axis and from different gravitational potentials coming from
the Earth's oblate shape.  But these geographical differences in time
dilation on the Earth are much smaller than those typically coming from a
change in orbital radius for a satellite when the radius change is a
nontrivial fraction of the Earth's radius.

> As a timely example, since selective availability has just been turned
> off (hooray!), there are two effects observable on GPS satellites:
> their clocks run more slowly due to an SR effect, and faster due to
> a GR effect. Which effect do you suppose is dominant?

The Virial Theorem ties both effects to each other where the
gravitational effect is stuck being twice the kinetic effect for a
fixed-radius orbit.  (For an ellipitical orbit this is only true for the
*average* effect, though, where the average is taken over a complete
orbit.)

BTW, an interesting way to get the relative correction for the
relativistic time dilation to leading order in 1/c^2 (relative to an
observer at rest at spatial infinity) for a Newtonian particle of mass m
moving under the influence of only Newtonian gravity is to take the
Newtonian Lagrangian for the particle and divide it by its rest energy,
i.e. [relative time dilation correction] = L/(m*c^2).  (This assumes, of
course, that the zero level of the Newtonian gravitational potential is
taken to be zero at spatial infinity.)

> GPS is the only
> consumer product I know that depends upon GR for its function.

I, too, can't think of any other example of a commercial technology
that needs to account for GR effects to properly function (other
than this GPS example).

David Bowman
David_Bowman@georgetowncollege.edu