Page:SahaSpaceTime.djvu/13

be called the Velocity-vector, and the Acceleration-vector of the substantial point at P. Now we have

$\left.{\begin{array}{l}c^{2}{\dot {t}}^{2}-{\dot {x}}^{2}-{\dot {y}}^{2}-{\dot {z}}^{2}=c^{2}\\c^{2}{\dot {t}}{\ddot {t}}-{\dot {x}}{\ddot {x}}-{\dot {y}}{\ddot {y}}-{\dot {z}}{\ddot {z}}=0\end{array}}\right\}$

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 ${\frac {c^{2}}{\rho }}$ in the direction of MP.

If ${\ddot {x}},\ {\ddot {y}},\ {\ddot {z}},\ {\ddot {t}}$ are nil, then the hyperbola of curvature at P reduces to the straight line touching the world-line at P, and ρ = ∞.

IV

In order to demonstrate that the assumption of the group G_c for the physical laws does not possibly lead to any contradiction, it is unnecessary to undertake a revision of the whole of physics on the basis of the assumptions underlying this group. The revision has already been successfully made in the case of