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 = ∞.