The separation ds of adjacent events is defined by ds^2 = -dx^2 - dy^2 - dz^2 + (c^2)dt^2. It is an extension of the notion of distance and this is the new invariant. Now if x, y, z, t are transformed to any set of new variables x_{1}, x_{2}, x_{3}, x_{4}, we shall get a quadratic expression for ds^2 = (g_{1 1}x_{1})^2 + 2g_{1 2}x_{1}x_{2}+. . . = [sum]g_{i j}, x_{i},x_{j}, where the g's are functions of x_{1}, x_{2}, x_{3}, x_{4} depending on the transformation.
The special properties of space and time in any region are defined by these g's which are themselves determined by the actual distribution of matter in the locality. Thus from the Newtonian point of view, these g's represent the gravitational effect of matter while from the Relativity stand-point, these merely define the non-Newtonian (and incidentally non-Euclidean) space in the neighbourhood of matter.
We have seen that Einstein's theory requires local curvature of space-time in the neighbourhood of matter. Such altered characteristics of space and time give a satisfactory explanation of an outstanding discrepancy in the observed advance of perihelion of Mercury. The large discordance is almost completely removed by Einstein's theory.
Again, in an intense gravitational field, a beam of light will be affected by the local curvature of space, so that to an observer who is referring all phenomena to a Newtonian system, the beam of light will appear to deviate from its path along an Euclidean straight line.
This famous prediction of Einstein about the deflection of a beam of light by the sun's gravitational field was tested during the total solar eclipse of May, 1919. The observed deflection is decisively in favour of the Generalised Theory of Relativity.
