Re: Relativity

[David Bowman <dbowman@tiger.gtc.georgetown.ky.us>]

David Abineri asked for help answering two questions asked by his student.

> 1. If an ellipse is travelling at relativistic speed in a direction parallel
> to its major axis, could it appear to a staionary observer as a perfect
> circle? It would seem that the major axis could contract to equal the minor
> axis but if the distance between the foci is also shrunk proportionally is it
> not still an ellipse because they are not coincident?

If an ellipse with eccentricity epsilon is observed in a frame moving along
the ellipse's major axis then in that frame the ellipse *will be* a circle if
the speed of motion is v/c = epsilon.  How the ellipse will *look* as it
passes by a localised observer's eyes is a different matter.

Besides the *real* length contraction of the ellipse there will be effects due
to differential Doppler shifting of light reflected off of different parts of
the ellipse due to different instantaneous angles from different parts of the
ellipse to the observer's eyes (because there are different components of
radial and transverse motion along and perpendicular to the line of sight
connecting the observer's eyes and a given spot on the ellipse).  This effect
can be minimized by just having the observer view the ellipse from a great
distance from the ellipse's path. 

There is another optical effect which is related to the above effect due to
the differences in the light travel time from the various parts of the ellipse
to the observer's eyes.  This means that the image seen by the observer is
*not* of the ellipse *as it was* at some time in the past but different parts
of the image are formed by light of different ages so the image of different
parts of the ellipse are from times when the ellipse was in different
positions.  Thus the ellipse will not *appear* to the observer as a circle as
it passes by *even though* it is going at the proper speed for it to actually
*be* a circle in that frame.  Actually, the shape of the ellipse *will appear*
to change shape in a way synchronized with the Doppler changes in color as it
passes by--even though the ellipse remains a circle throughout its motion in
the frame with v/c = epsilon.

> 2. If a disk moves past me at relativistic speed while it rotates about its
> center at a constant angular velocity, what shape will I see? Since the top of
> the disk is moving at a different speed relative to me than the bottom of the
> disk, hwo does the different amount of contraction distort the disk from my
> point of view.

The actual Lorentz-contracted shape of the disk depends only on the
translational velocity of the disk and not on its angular velocity.  The shape
is the same as if the disk was not rotating.  Thus the disk would become an
ellipse with an eccentricity of v/c where the original rest-radius is the
semimajor axis along the direction transverse to the motion, and the semiminor
axis is (1 - (v/c)^2)^(1/2) times the rest-radius oriented along the motion.

Again the *apparent* shape seen (as opposed to the actual Lorentz-contracted
shape) by a localized observer's eyes will also include the effects of the
kinds of optical distortions mentioned above.  Since different parts of the
disk are moving with different velocites the effects would be more complicated
than those listed above for a nonrotating object.  To get the apparent shape
at some fixed time for the observer, one needs to subtract off the
displacement that each element of the disk has undergone during the light-
travel time for the light leaving that location to get to the observer's eyes.
This calculation is more difficult than the case of a nonrotating object
above.

David Bowman
dbowman@gtc.georgetown.ky.us