Special Relativity and Conic Sections - Interpretation of the Experiment

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In order to understand what the experiment means, we must write down the equation of the curve defined by projecting the wavefront curve at to the (standard) coordinates of the plane of . I do this by introducing certain coordinates for the plane of . (We saw this in a Euclidean context in Note 3 of Planes Intersecting Cones.) The "speed" of (as measured by ) is defined to be

I assume for this experiment that -- otherwise, there is nothing to do.

We saw that the wavefront curve was an ellipse in the (stationary) frame of by using the basis in coordinates, given by

,

,

,

where these vectors satisfy the hyperbolic orthogonality relations,

Note 10. Coordinates adapted to the projected ellipse

We see that there are orthogonal unit vectors U and V in the plane of so that the projected ellipse is given by the parametric equation

From this it follows that the projected curve is an ellipse with major axis on the line generated by , as illustrated at the right.

Here is the interpretation: The projected ellipse is the projection to the plane of the observer of his measured spatial positions of the wavefront at the observer's time . It is not a circle because this wavefront does not consist of simultaneous events for (since ).

The major axis of this ellipse contains, between the focus at the origin and the second focus, the line of spatial positions of the world-line of between his time . In particular, he moves from the origin (first focus) to the center in 1 unit of his time, which is units of the observer's time. His projection must therefore move to the second focus at his time .

The rays of light, emitted at various angles from the emission event, project to segments connecting the first to the second focus via a point on the boundary of the projected ellipse. The questions, of course, are:

• Why should the sum of the lengths be constant?
• What does this constant sum represent?

And the answers will now be fairly easy to see.

Consider a ray of light that bounces from the emission to the reception event. Describe its itinerary in the reference frame of observer . It starts from the origin event

and arrives at the mirror at event

Next, it reflects to the reception event,

In the first leg of its itinerary, it lies in the light cone, so

Since the square roots of both sides of this equation are equal, we conclude that the spatial projection of that path has length , which is , since .

In the second leg of its itinerary, it also lies in a light cone, so

For the same reason, the spatial projection of that path has length .

Therefore, the sum of the spatial lengths is . This is equal to the time that observer measures from the emission event to the reception event. That is, of course, the time coordinate in the frame of of the event . But from the definition of in the previous section,

we see that this is

And this, of course, is the length of the major axis of the projected ellipse. That is a physical interpretation of why the sum of the lengths of the paths from a point on the ellipse to its foci is constant. It is the time measures between the emission and reception event, which is the same for all rays of light.

Next, I consider the focus-directrix property of the ellipse. I showed in Planes Intersecting Cones that the intersection of the plane of simultaneity  for  with the plane of simultaneity  for is a line that I call :

.

I represented the tangent plane, the plane of simultaneity  for ,  as the graph of a function of x and y:

In the present context, the function has the following interpretation. With each point there is a unique point on the plane of simultaneity   for that projects to it. is the time coordinate that standard observer ascribes to it. The line is the set of events with and .

Note 11. Interpretation of the eccentricity

The calculation above shows that the value of the eccentricity is the speed of as measured by ,

= ,

and it gives us the focus-directrix property of the projected ellipse.

Now, we might ask how general this construction is. It is clear that, for any particular direction of motion of with respect to , we will obtain among the projected ellipses one representative of the similarity class of ellipses for each speed . And every similarity class will be represented, simply by choosing the representative with semi-minor axis of length 1, and semi-major axis of length

Therefore this physical argument is completely general.

What happens if is stationary with respect to ? In that case, this construction breaks down when we attempt to build the observer's orthonormal basis. But in that case, there is no need to build such a basis. The foci collapse to the center, the ellipse is a circle, and everything happens as it does in the frame of .