The Principle of Relativity · Albert Einstein
III
By the world-postulate a similar treatment of the four determining quantities _x_, _y_, _z_, _t_, of a world-point is possible. Thereby the forms under which the physical laws come forth, gain in intelligibility, as I shall presently show. Above all, the idea of acceleration becomes much more striking and clear. I shall again use the geometrical method of expression. Let us call any world-point O as a “Space-time-null-point.” The cone _c²__t²_ - _x²_ - _y²_ - _z²_ = O consists of two parts with O as apex, one part having _t_ < 0, the other having _t_ > 0. The first, which we may call _the fore-cone_ consists of all those points which send light towards O, the second, which we may call _the aft-cone_, consists of all those points which receive their light from O. The region bounded by the fore-cone may be called the fore-side of O, and the region bounded by the aft-cone may be called the aft-side of O. (_Vide_ fig. 2). On the aft-side of O we have the already considered hyperboloidal shell F = _c²__t²_ - _x²_ - _y²_ - _z²_ = 1, _t_ > 0. The region inside the two cones will be occupied by the hyperboloid of one sheet -F = _x²_ + _y²_ + _z²_ - _c²__t²_ = _k²_, where _k²_ 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_ = -∞ and _t_ = ∞, asymptotically approaches the velocity of light _c_.
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