A Thought Experiment
I
am now ready to describe the experiment that displays the similarity class properties:
the focus-locus property and the focus-directrix property of the ellipses 
constructed in the Planes Intersecting Cones section. I give a sketch
first, and then discuss the details. You might like to keep the calculation
of that page in mind, where the point 
of the hyperboloid
|
|
represents the first tick
of the moving observer's ( 
)
clock in standard observer's ( 
) coordinates. The speed that 
attributes to 
is in that case
|
|
Sketch of the experiment:
Suppose
a "standard" inertial observer:
is situated at the origin 
of the Euclidean 
plane. This observer watches a "moving"
inertial observer 
pass through the origin at a time
that they both measure to be 0. 
considers 
to be moving at constant velocity
where 
is a unit vector and 
is the speed. I assume for this experiment
that 
.
Both
observers measure the speed of light to be 1. At the event where 
arrives at the origin it emits a circular
pulse of light. From the point of view of 
, after one unit of time as measured
by 
the wavefront forms a curve of events
whose spatial coordinates 
simply form a circle centered at his
origin. The events in the wavefront all have time coordinate 
for 
.
However,
from the point of view of 
, this curve of events is the intersection
of a plane with a cone in his 
coordinatization of spacetime. I called
the intersections obtained physically this way, boost intersections: 
. The time coordinates that he measures
for points in that wavefront are not constant, but, as we saw, the curve forms,
in the Euclidean structure that the plane inherits from the hyperbolic geometry,
an ellipse. That ellipse is the intersection (in 2+1 spacetime) of a certain
plane with a light cone. Observer 
projects this ellipse into his (Euclidean)
plane t = 0, and discovers another
ellipse: 
, this time with one focus at the origin
. The ellipse has semi-minor axis
1, and semi-major axis 
directed along 
.
If
we imagine that each ray of light emitted by 
is reflected back to 
at 
then they all converge again at 
at a time that 
measures to be 
.
For
, each of those rays of light projects
to a pair of intervals, the first, starting at the origin, a focus of the projected
wavefront ellipse, and arriving at the ellipse, and the second reflecting to
the other focus, which will be the spatial 
position that 
ascribes to 
at 
time 
. We will see in Interpretation
of the Experiment that the sum of the two lengths of these intervals
is, for each ray of light, the time coordinate that 
ascribes to the arrival back at 
of the reflected ray. These time coordinates
are the same for all the rays, because their convergence is at a single event,
clearly simultaneous both for
and .While 
measures that time to be 2, 
measures it to be the time coordinate
of the event: " 
at 
time 
." That number is easy to calculate. It is the length of the major axis:
|
|
Since 2 < 
,
this is another example of "time dilation." Observer
thinks that more time has passed than the 2 units that observer
measures at the event: " at
time
."
Obviously,
every ellipse in the Euclidean plane can be obtained up to similarity
in this way since these ellipses have semi-minor axis 1, and semi-major axis
. This will establish what I call the focus-locus property for all ellipses,
and will give a physical interpretation for the constant sum of distances from
the foci to the points on the ellipse.
Now
to interpret the focus-directrix property, I first identify the directrix of
ellipse
in this experiment. This will be the intersection of the plane
with the
plane t = 0. On our assumption
that
this intersection is a line in the plane of .
This will give an interpretation of the directrix, and then of the eccentricity
of
as the tangent of the dihedral angle thus formed in the Euclidean metric. We
will see the details in the Interpretation
of the Experiment section on the next page.
Further,
it is clear that every intersection of a plane with the light cone that
is the result of the 
for arbitrary 
in the interior of the cone can be associated with a unique boost intersection
with the physical interpretation just described. These intersections are geometrically
similar, and are obtained by simple scalar expansion or contraction from the
origin. Therefore, their projections are also geometrically similar. In the
experiment, I work only with the
that derive from the boost intersections, that is, from planes tangent to the
hyperboloid:
That
is the sketch. What makes this work, as we will see below, is the fact that
every ellipse of the form 
is a conic intersection: the intersection of two light cones. I will first establish
some notation to set the experiment up.
Let
us suppose that our standard inertial observer establishes the basis of coordinates:
for this 2+1 dimensional space-time. As before, I will let names like 
stand for the observer and the event of his first clock tick. So let us call
this observer .
And suppose now that there is another inertial observer, say 
,
whose world line intersects the world line of 
at the origin.
In particular,
I assume that both 
and 
have clocks that tick time 0 for each at the origin event.
As before, I will picture this by using the coordinates of 
to describe the event of the next tick of the clock of 
on the world-line of 
. This event must occur in the future light cone of 
as we saw earlier. Since the hyperbolic interval connecting
the first and second clock tick of 
along the world-line of 
is measured by 
to be -1, it must be -1 in the coordinates of 
. This means that in the coordinates of 
, it lies on the hyperboloid: 
as depicted below:
For
the next step in setting up the experiment, I will choose the orthonormal system
of vectors: 
(in the hyperbolic metric) in a special way. We are free to choose any orthonormal
system we like because of the covariance principle, so I select one adapted
to the motion of
as follows.
|
|
|
Now
let us describe the experiment in these terms. For observer 
moving with respect to an observer 
in such a way that their origins coincide, I will construct the ellipse formed
by intersecting the light cone at the common origin with the hyperbolic orthogonal
bisector of the segment from the origin to the event of the second
clock tick of 
. All
of this is easily formulated as a "thought experiment." This experiment
will lead us, in this and the next section, to give the "physical"
interpretation to the focus-locus and focus-directrix properties of an ellipse.
Following
is the experiment performed by observer 
.
At his time 0, 
emits
a circular pulse of light. In his spatial coordinate system, he has a circular
reflecting mirror of radius 1 with center at his origin. (Imagine a cylinder
surrounding his world-line.) At 
,
the wavefront arrives at the mirror, and is reflected back. It arrives back
at the origin at 
.
See Figures 3 and 4.
In
the coordinate system of 
,
nothing interesting happens. The picture looks like this:
In
2+1 space-time, the process is depicted below, with emission phase the lower
(green) part of the cone, and reception phase the upper (red) part of the picture.
Figure 4
Light
simply moves from the center of his circle to the boundary after 1 second, then
returns to the origin after 2 seconds. Now, let us examine the same process
from the viewpoint of 
. In his coordinates, 
will be "moving" if the velocity 
.
The following pictures illustrate the two halves of the experiment
in his coordinates. In the first picture of Figure 5, the emission cone from
the origin of 
coordinates
is shaded gray. The reception cone - the path of the reflected wavefront is
shaded white.
Figure 5
The wavefront spreads from the origin along the gray section
of the light cone. At 
time 
, it forms
a curve obtained by intersecting the light cone at the origin with the plane
that is the orthogonal bisector of the segment connecting the events on the
world-line of 
:
and 
. We saw in an earlier section Geometry
of 2+1 Spacetime, that the points on the curve of arrival events must have
equal hyperbolic interval (that is, 0) with the emission and reception events,
hence must lie on this orthogonal bisector plane: .
In the second picture of Figure 5, I show three light "rays"
emitted in three directions.
The 3 dimensional picture below shows this orthogonal bisector
plane in grey, and it also depicts the x-y plane t = 0 in yellow. These planes
intersect in the line that I have designated the directrix in the x-y plane that contains the projected ellipse (not visible
in this picture).
Figure 6
In
Figure 7, I show the projection of the picture onto the x
- y plane
of observer 
.
I project the light rays also. We will see later that these light rays travel
from one focus to the other of this projected ellipse. This indicates that the
bridge from the plane-slicing-cone picture to the focus-locus picture is obtained
by projecting the slice ellipse into the spatial ( 
) plane of observer 
. We will see in the next section:
Interpretation of the Experiment exactly why this is so.
Figure 7
We saw in Planes
Intersecting Cones that this plane is also tangent to the hyperboloid at
vector 
|
|
by observing
that in the Euclidean metric, the vectors 
above are perpendicular to the gradient of the hyperboloid at that point. This
Euclidean calculation still gives the result in the present hyperbolic context
since
generate the plane
at .
Returning to Figure 5, the wavefront then follows the white
inverted cone back to the reception event at 
. The skewed and unsymmetrical aspect of the picture is the result of the fact
that observer 
is
actually moving with respect to observer 
.
From the plane-slicing-cone definition of a conic, we see that
the intersection of the two light cones is, in this case, an ellipse. That is,
the events in the light cone along the wavefront at 
form an ellipse (
where =
) in the frame of 
. But since 
is using
a hyperbolic and not a Euclidean metric, let us examine the equation for this
intersection curve in 
coordinates.
The points on this curve are not "simultaneous" in
the frame of 
(although, of course, they are simultaneous in the frame of 
). That curve is, however, obtained by intersecting the light cone with the
plane through vector 
that is parallel to vectors 
,
since these vectors generate the orthogonal bisector according to the relations
derived above:
|
|
If we let 
define coordinates for the plane in the obvious way, then they will give Euclidean
coordinates for the plane satisfying (even in the hyperbolic metric),
|
|
and the points of the form:
|
|
are in the curve, since, as is easy to see, in the hyperbolic metric,
|
|
In the standard Euclidean metric for 
,
using 
coordinates, are still perpendicular, but while
=1, in general, 
. So the intersection is an ellipse
in the plane if we use the induced Euclidean structure from 
.
Emission Phase
Reception Phase
Both Phases
The light cones
Having
developed these pictures, we now ask what the interpretation for the focus-locus
and focus-directrix properties of the
ellipse is. In the next section: Interpretation of the Experiment, we
will give what may be a surprising answer to this question.
