The Relativistic Ellipse

In 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 proportions

The 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 Theories

When 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"