Page:The principle of relativity (1920).djvu/154

 The region inside the two cones will be occupied by the hyperboloid of one sheet

-F = x^2 + y^2 + z^2 - c^2t^2 = k^2,

where k^2 can have all possible positive values. The hyperbolas which lie upon this figure with O as centre, are important for us. For the sake of clearness the individual branches of this hyperbola will be called the "Inter-*hyperbola with centre O." Such a hyperbolic branch, when thought of as a world-line, would represent a motion which for t = -[infinity] and t = [infinity], asymptotically approaches the velocity of light c.

If, by way of analogy to the idea of vectors in space, we call any directed length in the manifoldness x, y, z, t a vector, then we have to distinguish between a time-vector directed from O towards the sheet ±F = 1, t > 0 and a space-vector directed from O towards the sheet -F = 1. The time-axis can be parallel to any vector of the first kind. Any world-point between the fore and aft cones of O, may by means of the system of reference be regarded either as synchronous with O, as well as later or earlier than O. Every world-point on the fore-side of O is necessarily always earlier, every point on the aft side of O, later than O. The limit c = [infinity] corresponds to a complete folding up of the wedge-shaped cross-section between the fore and aft cones in the manifoldness t = 0. In the figure drawn, this cross-section has been intentionally drawn with a different breadth.

Let us decompose a vector drawn from O towards (x, y, z, t) into its components. If the directions of the two vectors are respectively the directions of the radius vector OR to one of the surfaces ±F = 1, and of a tangent RS