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. 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.
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. Active 8 years, 8 months ago.
Viewed 1k times. Improve this question. Brian Rushton Brian Rushton 3, 8 8 gold badges 33 33 silver badges 62 62 bronze badges. The hyperbolic space carries all the information of the isometry, well, provided the isometry preserves the sense of time. It's the same idea as how one can study euclidean vector space symmetries by studying the action on the sphere. It comes up naturally in many situations, especially via the geometrization of 3-manifolds.
In practice it seems to be the right object for succinctly describing certain types of "teeming" complexity. To a relativist, a globally hyperbolic spacetime is essentially one in which solutions to Cauchy problems e. See Hawking and Ellis, p. The solvability features of a wave equation as opposed to, say, a heat equation actually depend on the fact that the "hyperbolic" wave equation is a PDE whose symbol is a function with certain properties - the same properties that one would need to define a riemannian metric of negative curvature.
However, they are completely different. Show 1 more comment. Active Oldest Votes. Improve this answer. Carlo Beenakker Carlo Beenakker k 12 12 gold badges silver badges bronze badges. Thanks for pointing out this connection! Add a comment.
The spacetime of Special Relativity is flatnot curved like hyperbolic geometry. So, the geometry of Special Relativity satisfies the parallel postulate.
Triangle sides are straight. However, since the "angles" between timelike vectors is based on the unit hyperbola playing the role of a circle in Minkowski spacetime geometry , Special Relativity uses hyperbolic trigonometry.
So, you don't need to learn hyperbolic geometry to understand the fundamentals of Special Relativity. The "space of rapidities which is related to velocities " is a hyperbolic space. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.
Create a free Team What is Teams? Learn more. Is spacetime in special relativity hyperbolic? Ask Question. Asked 2 years, 4 months ago. Active 2 years, 4 months ago. Viewed times. Do I have to learn hyperbolic geometry to understand it? Improve this question. Ryder Rude Ryder Rude 2, 7 7 silver badges 18 18 bronze badges.
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