Tachyboloid

In special relativity, the tachyboloid of a linear spacetime is the set of time-like events one temporal unit into the future. The points of the tachyboloid represent the subluminal velocities for a constant motion through the origin of the spacetime. As special relativity is studied with a progression of one, two, and three spatial dimensions, completing the temporal one, the tachyboloid is a hypersurface of one, two, or three dimensions.
The simplest case starts with one space dimension and R representing events (t,x) of spacetime, say with t measured in nanoseconds and x measured in lengths of 30 centimeters. Then <math>t^2 = x^2</math> represents the light cone, and <math> t^2 - x^2 = 1, \quad t > 0</math> represents the tachyboloid T. In this case T is the branch of a hyperbola, which is parameterized by rapidity, an instance of hyperbolic angle.
For the two spatial dimension case, events are (t,x,y) ∈ R . The hyperboloid <math>t^2 - x^2 - y^2 = 1 </math> has one sheet in the past and one sheet in the future, which forms the tachyboloid. See Rhodes & Semon (2004). They say, "we restrict ourselves to two spatial dimensions because this is sufficient for understanding the most common cases of Thomas rotation and precession." They use the Poincare disk model as a conventional reference.
Minkowski space is a universe with three spatial dimensions containing events (t,x,y,z) ∈ R , and which has light cone given by <math>x^2 + y^2 +z^2 = t^2</math>. The tachyboloid is the future part of the hyperboloid <math>t^2 - x^2 - y^2 - z^2 = 1</math>. Each event on the tachyboloid corresponds to a Lorentz transformation. The combination of two Lorentz transformations in different directions results in a Thomas precession. It is this kinematic phenomena that requires hyperbolic geometry for looking at the tachyboloid as a metric space. Minkowski was aware of this fact, as recounted by Scott Walter. An early writer on the hyperbolic geometry of the tachyboloid was . See P.L. Galison (1979) for an image of Minkowski's annotations on the tachyboloid.
Hyperplane of simultaneity
Every event t on the tachyboloid corresponds to a hyperplane of space-like events that are hyperbolic-orthogonal to t. This hyperplane is the simultaneous hyperplane of t, or t ’s hyperplane of simultaneity. It is a reduced sense of classical simultaneity, since absolute time and space were dissolved by relativity. This "Minkowski simultaneity" evidences the relativity of simultaneity. One can consider an inertial frame of reference as composed of a proper time directed by the tachyboloid and its simultaneous hyperplane.
Suppose <math>p = (t,x,y,z)</math> is an event on the tachyboloid. Then the hyperplane of simultaneity of p is given by
:<math>S(p) = \lbrace q = (s,u,v,w) \ : \ ts - ux - vy - wz = 0 \rbrace</math>.
Using the bilinear form η of Minkowski, q ∈ S(p) when η(p,q) = 0 .

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