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Special Relativity and Conic Sections - Hyperbolic Geometry of 2+1 Spacetime

Author(s): 
James E. White

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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

 

(6.1)

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:

 

(6.2)

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.

Note 5. Proof of Lemma 1

 

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

is the set of events

,

which contains the segment. Generally, though, the picture might look something like the figure at the right.

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 equals the interval from to , that is, such that

,

is a plane, and this plane is in fact the orthogonal bisector of the segment determined by .

Note 6. Proof of Lemma 2

 

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 be the light cone with vertex at . Let be the hyperbolic interval from :

 

.

 

If , then

 

,

 

and this intersection is either an ellipse or an hyperbola.

There are two cases (see the figures below):

  1. If , the hyperbolic interval is "time-like", and the common intersection is an ellipse.
  2. If , the hyperbolic interval is "space-like", and the common intersection is a hyperbola.
An Elliptic Conic Intersection A Hyperbolic Conic Intersection

Note 7. Proof of Lemma 3

 

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.

James E. White, "Special Relativity and Conic Sections - Hyperbolic Geometry of 2+1 Spacetime," Loci (October 2004)

JOMA

Journal of Online Mathematics and its Applications

Dummy View - NOT TO BE DELETED