Reference: Einstein’s 1920 Book
Section XXIV (Part 2)
Euclidean and Non-Euclidean Continuum
Please see Section XXIV at the link above.
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Summary
A surface is continuous. It can be divided into square units. The property of flatness of the surface is quite special. It supports straight lines. This property underlies the Cartesian co-ordinate system. This is the Euclidean continuum.
A non-flat surface shall still form a continuum, but it would not support straight-line units of the same size throughout. It would no longer be a Euclidean continuum and it would not support the Cartesian co-ordinates directly. So we shall use a more flexible co-ordinate system that would comply with the inertial requirements of the gravitational field.
The Euclidean geometry and the Cartesian system represents a totally rigid medium with no flexibility. A gravitational field requires non-Euclidean geometry and a co-ordinate system that can take into account some flexibility in the medium. The mathematics, which takes this flexibility into account is the method of Riemann of treating multi-dimensional, non-Euclidean continua based on the principles outlined by Gauss.
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Final Comments
The Euclidean geometry is basically considering the surface of a sphere of infinite radius. Such a surface appears flat and it supports straight lines. These are boundary conditions that represent a substance of total flexibility. Within this boundary is substance that varies in the flexibility of its structure, such that the farthest point from the boundary is totally rigid.
The Euclidean geometry arbitrarily assumes a substance of totally rigid structure at the boundary and within that boundary throughout. The non-Euclidean geometry introduces the required flexibility, and the methods of Riemann accounts for the variations in that flexibility.
This flexibility represents the variations in the inertia of matter or, more generally, the variations in the consistency of substance filling the space-time.
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