In
order to develop the physical interpretation of the conic construction that
I made on the previous page, I will now replace the 3-dimensional Euclidean
geometry of 
with 2+1-dimensional Hyperbolic geometry. That geometry is determined by the
"Hyperbolic metric" for 
, as opposed to the Euclidean metric.
Like
the Euclidean metric, it is defined by a non-degenerate inner product, but unlike
that metric, the inner product is not positive definite, as I shall explain
below. In fact, there is a 2-dimensional stratified set of vectors called the
"light cone" with the property that their "lengths" are
zero. This lends the geometry a distinctive and interesting character. Still,
many of the familiar properties of Euclidean geometry have their analog here.
And in particular, as I shall show on this page, the ellipses that I constructed
earlier by slicing the standard cone with a plane, may also be realized by forming
the intersection of two light cones.
This
is of course a preliminary for our physical interpretation of the focus-locus
and focus-directrix properties. In the Thought Experiment and Interpretation
of the Experiment sections, I will take the third step, and interpret that
property using special relativity restricted to a 2+1-dimensional spacetime.
The
geometric structure of 
that I will use from now on is determined by an inner product. I continue to
use the 
Cartesian coordinates to specify that inner product, though it is understood
that the inner product itself is an underlying structure that is independent
of any particular choice of coordinates. That structure remains covariant under
a wide class of linear transformations (Lorentz transformations) that preserve
the inner product, just as the Euclidean geometry remains covariant under all
rotations and inversions across planes. The inner product is defined, then,
as follows:
Suppose
that 
. I will use the words "points" and "events" interchangeably
in anticipation of the discussion to come later.
Then
say that the inner product of W with Z, which I shall denote 
is
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Note
that this implies that the light cone at the origin 
is defined as the set of vectors 
such that 
.
If we denote by 
the vector 
, by 
the vector 
and by 
the vector 
, then in this hyperbolic metric:
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(4.2) |
I
use the suggestive name 
for the third vector because in the physical interpretation, in which t
represents the time, the name 
will do double duty as the vector 
which is the event of the first clock tick on the world-line of a special
(stationary) observer, and as the name of the stationary observer itself.
The vectors 
are "orthonormal" in this metric in the above sense.
This
"hyperbolic" metric defines, for each pair of events: 
in 
a number: 
. This number may be positive, negative, or zero. I'll call that number 
the "hyperbolic interval" between events 
. Obviously this is equal to 
. The number is analogous to the "squared distance" between points
in the Euclidean metric.
I
take the liberty of coloring my prose a little (and anticipating the physics
somewhat) by using terms like "light rays", "signals", "clocks"
and "observers". I will say more about these ideas on the Clocks, Light Rays and Rulers
page. If the hyperbolic interval is zero, it means that a ray of light connects
.
If negative, it means that 
are causally connected, in the sense that some inertial observer may
go from one of these to the other: each lies in the interior of the light cone
of the other. And this means that a slower-than-light signal may pass from one
to the other, so that one definitely precedes the other.
If
the hyperbolic interval is positive, it means that each event lies in the exterior
of the other's light cone, and the events are not causally connected. No signal
may pass from one to the other, and for some inertial observers, 
,
while for others, 
,
and for yet others, events 
are simultaneous.
The
following physical aside is the basic physical postulate that Einstein set down
for special relativity (3+1 Hyperbolic geometry), defining, in a sense, the
class of allowable geometric transformations from one inertial observer to another:
The hyperbolic interval separating two events is the same (number) no matter
which coordinate system of an inertial observer is used to measure the coordinates
of the events. This is true as long as all inertial observers choose compatible
units of measure, and use those units for all measurements. This means that
it must be possible to synchronize their clocks when they are pairwise stationary
with respect to one another, and that each measures the speed of light to be
one unit distance per unit time. When they are in uniform motion with respect
to each other, each uses his own system of coordinates to describe events, but
they still measure the same hyperbolic interval between any two events. In fact,
it is possible for an observer to measure this interval using clocks
and light rays alone. An experiment on the next page: Clocks, Light Rays and Rulers,
will allow you to see that for yourself.
In
order to discuss the plane-slicing-cone construction in a physically unified
way, I introduce some geometric lemmas. These lemmas generalize some obvious
facts about ordinary 
with its Euclidean metric. They are rather trivial, but they point the way to
the physical interpretation of this geometric operation.
Lemma 1: (Hyperbolic orthogonal bisector)
Suppose
we are given two distinct events, 
, in 2+1 space with its hyperbolic metric: 
. Let 
be the midpoint of the segment they determine. The set of vectors 
with the property that 
is a plane. This plane is the orthogonal bisector of the segment.
Of
course, the hyperbolic orthogonal bisector of a segment does not appear
perpendicular to the segment, as it would be in the Euclidean metric. For example,
the "light ray" 
is orthogonal to itself! The orthogonal bisector of the "light segment"
connecting
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is the set of events
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This
contains the segment. Generally, though, the picture might look something like:
Lemma 2:
Suppose
we are given two distinct events, 
, in 2+1 space with its hyperbolic metric: 
.
Let 
be the midpoint of the segment they determine. The set of points 
with the property that the interval from 
to 
is equal to the interval from 
to 
,
that is, such that
,
is a plane, and is in fact
equal to the orthogonal bisector of the segment determined by 
.
Lemma 3: (Conic Intersections)
Suppose
we are given two distinct events, 
, in 2+1 space with its hyperbolic
metric: 
. Let 
be the midpoint of the segment they determine, and let plane 
be the orthogonal bisector of the segment 
passing through 
. Let 
be the light cone with vertex at 
,
and let 
be the light cone with vertex at 
. Let 
be the hyperbolic interval from 
:
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Then if 
, then
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and this intersection is either an ellipse or an hyperbola. There are two cases:
a.
If
, the hyperbolic interval is "time-like". In this case, the common
intersection is an ellipse.
An Elliptic Conic Intersection
b.
If
,
the hyperbolic interval is "space-like". In this case, the common
intersection is a hyperbola.
A Hyperbolic Conic Intersection
I
discuss the idiom of Special Relativity in the next section. And in A Thought
Experiment, I will interpret the focus-locus definition of conic sections
in terms of light cones.
