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
c²t² - 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 = c²t² - 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 - c²t² = 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.
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 = ∞ 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 at the point R of the surface, then the vectors shall be called normal to each other. Accordingly
c²tt₁ - xx₁ - yy₁ - zz₁ = 0,
which is the condition that the vectors with the components (x, y, z, t) and (x₁ y₁ z₁ t₁) are normal to each other.
For the measurement of vectors in different directions, the unit measuring rod is to be fixed in the following manner;—a space-like vector from 0 to -F = I is always to have the measure unity, and a time-like vector from O to +F = 1, t > 0 is always to have the measure 1/c.
Let us now fix our attention upon the world-line of a substantive point running through the world-point (x, y, z, t); then as we follow the progress of the line, the quantity
dτ = (1/c) √(c²dt² - dx² - dy² - dz²),
corresponds to the time-like vector-element (dx, dy, dz, dt).
The integral τ = ∫dτ, taken over the world-line from any fixed initial point P₀ to any variable final point P, may be called the “Proper-time” of the substantial point at P₀ upon the world-line. We may regard (x, y, z, t), i.e., the components of the vector OP, as functions of the “proper-time” τ; let ([.x], [.y], [.z], [.t]) denote the first differential-quotients, and ([..x], [..y], [..z], [..t]) the second differential quotients of (x, y, z, t) with regard to τ, then these may respectively be called the Velocity-vector, and the Acceleration-vector of the substantial point at P. Now we have
c² [.t²] - [.x²] - [.y²] - [.z²] = c²
c² [.t][..t] - [.x][..x] - [.y][..y] - [.z][..z] = 0
i.e., the ‘Velocity-vector’ is the time-like vector of unit measure in the direction of the world-line at P, the ‘Acceleration-vector’ at P is normal to the velocity-vector at P, and is in any case, a space-like vector.
Now there is, as can be easily seen, a certain hyperbola, which has three infinitely contiguous points in common with the world-line at P, and of which the asymptotes are the generators of a ‘fore-cone’ and an ‘aft-cone.’ This hyperbola may be called the “hyperbola of curvature” at P (vide fig. 3). If M be the centre of this hyperbola, then we have to deal here with an ‘Inter-hyperbola’ with centre M. Let P = measure of the vector MP, then we easily perceive that the acceleration-vector at P is a vector of magnitude c²/ρ in the direction of MP.
If [..x], [..y], [..z], [..t] are nil, then the hyperbola of curvature at P reduces to the straight line touching the world-line at P, and ρ = ∞.