## The
Relativistic EllipseIn
the section on relativistic aberration, we found that if an observer
bounces light off a mirror, and sees behaviour consistent with their
being stationary, then other observers are forced to conclude that
light from a system must be seen to be deflected forwards, if that
system is seen to be moving. If we want to
generalise the result for rays emitted in other directions (and not
just at 90 degrees), we arrive at
the **Relativistic Ellipse**. This gives us the
apparent
angle-change for light emitted in any direction. It can also be used to
calculate a relativistic theory's Doppler shift
predictions, for any angle.
We
can again use the example of an observer who sees light bouncing off a
mirror in such a way as to indicate that their system is stationary,
but this time we'll use a spherical mirror surrounding the observer.
The mirror catches light sent from the central source in any
direction and returns it to its origin. However, viewed from a
system in which the experiment is seen to be *moving*,
the position from which the light-rays originate, and the position at
which they converge, are not the same. The finiteness of the speed of
light means that our second observer reckons that the system has moved
on by a certain distance between the rays being generated and
converging back to a point. According to our second
observer,
the rays are now originating at one position, and being
refocused
somewhere else, and since having two focii is the property of
an ellipse, our second observer reckons
that the individual reflection-events no longer mark out a set of *spherical*
spatial coordinates, but instead describe the surface of an elongated
ellipsoid.
The
ellipse shape has two focal points, one representing the
position
at which light appears to be generated, and the other representing the
point at which is appears to be refocused. The distance between the
two focal points, divided by the ellipse major diameter (its largest
cross-section) tells us the velocity of the "moving" system as a
fraction of the speed of light.
### Ellipse proportionsThe
ellipse's length divided by its width give us the
Lorentz Factor. At
this point we haven't made many assumptions about
the speed
of light. We've only really assumed that the original observer is
entitled to believe that they are "really" stationary, and have
calculated how someone else should see the same situation. If
we want to be more specific about the way that we think that light
propagates, we can arrive at different theories that can use
different scalings for the ellipse ... but this normally
doesn't
affect the angles involved.
### Scalings and TheoriesWhen
we draw our ellipses with a constant cross-section, with their lengths
elongated by the Lorentz factor, we have diagrams representing Einstein's
special theory of relativity.
This not only includes all of special relativity's angle-changes for
light emitted or received in any direction, it also lets us read off
the change in length of each ray, and treat this as the light's
altered wavelength due to
relative motion. Measuring
the distances between one focus and the front and back of the
ellipse, we find that the distances are changed by the ratio **freq'/freq
= SQRT[ (c-v) /
(c+v) ] **
,
which is the relativistic Doppler
relationship for special relativity [note].
If we measure the ray aimed at 90 degrees (which appears as a diagonal
in our ellipse), it's wavelength is increased by the ratio **1 :
SQRT[ 1 - v²/c² ]**
, which
special relativity would refer to as a transverse
Doppler redshift. This version of the
ellipse generates all the key relationships of special relativity.
### Other models The
ellipse method isn't limited to special relativity. If we
wanted
to do the same exercise using old-fashioned emission theory, we'd get
exactly the same proportions and angles, but the wavelengths would be
enlarged by an additional Lorentz factor. We can
imagine a whole
series of potential theories of relativity, which differ from each
other by the scalings of their ellipses, and whose physical predictions
diverge by "Lorentz-like" factors.
Constructing theories in
this way guarantees that we keep many of the "good" relationships of
special relativity (like E=mc²).
Of
this family of potential theories, special relativity is the
only
one that allows us to describe spacetime as being flat in the presence
of bodies with significant relative motion. With special relativity, we
can turn the ellipse outline back into its original
circular
shape and size simply by contracting the shape in the direction of
motion by the Lorentz factor. If we try
to use theories
that don't use these same SR relationships, the conversion is more
difficult: The "SR" ellipses can be normalised by simply tilting the
diagram off the page by an amount that depends on relative velocity,
and we can then create a mesh of interlocking ellipses that generate
the lightcone geometry
and causal relationships of Minkowski spacetime.
Minkowski spacetime allows the worldview of one simply-moving
observer to be "tilted and skewed" to turn it into the worldview of
another. Minkowski spacetime doesn't assume the physical bending of
spacetime, and instead uses a clever combination of the principle of
relativity with projective geometry in four
dimensions. However, if we take the larger
Newtonian ellipse, and try to cram *that*
back into its original circular outline, there's no polite way to do
this without warping spacetime and forcing the internal
distances
to protrude off the page to form a shape that looks like a tilted
gravitational well. The
"ellipse" exercise tells us that special
relativity is the unique relativistic solution for perfectly flat
spacetime. Any relativistic theories that generates "larger" ellipses
than SR must relate increased velocity between bodies with increased
curvature in the region. In these models, the speed of light can't be globally
constant as it is under special relativity, but has to
be only locally constant, and regulated by gravitomagnetic
effects. The
exercise also gives us a new perspective on the origins of special
relativity. Where Newtonian mechanics was thought to be incompatible
with wave theory, it might be more accurate to say that if we forced
wave arguments onto it, the result was incompatible with flat
spacetime. With hindsight, the choice that we were presented with at
the start of the Twentieth Century was to either keep the optical
relationships of Newtonian mechanics and re-implement them in a curved
spacetime model, or to make the simpler assumption that spacetime was
entirely flat, and modify change the Newtonian relationships
(wavelengths, etc) to fit. This second
route gave us special relativity.
##### References*The
relativistic ellipse is introduced in:*- William
Moreau, "Wave
front relativity" American Journal
of Physics
**62** 426-639 (1994) ...
Moreau's
paper explains how a moving sphere can produce elliptical
reflection coordinates, by pointing out that (under special
relativity),
the relativity of simultaneity allows
the nominal timing of events to be offset by an amount that depends on
their separation along the x-axis. For an observer who sees
the sphere moving, SR can describe the the surface as not
being illuminated at the same moment, but sequentially, with
the rear of the
sphere illuminating first, and with the illuminated
band moving
forwards over the sphere's surface as the sphere advances. The result
is that the reflection-events for the sphere mark out an elongated
ellipsoid.
*The
relativistic ellipse also makes a significant appearance in:* - E. Baird,
Relativity in Curved Spacetime (2007), chapter
8, "Aberration of angles"
and chapter 13, "Horrible nasty mathematics"
all original material
copyright © Eric Baird 2007/2008 |