Reference: Einstein’s 1920 Book
Section XXXI (Part 3)
The Possibility of a “Finite” and Yet “Unbounded” Universe
Please see Section XXXI at the link above.
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Summary
A universe can be without bounds (infinite) from the perspective of its dimensions, but from a higher dimension it can be seen to be finite.
If the ratio of the circumference of a circle to its diameter is determined very accurately, and it is less than π, then that circle is drawn on a spherical surface.
Riemann discovered a three-dimensional analogy of a two-dimensional spherical surface. That means an object moving in a straight line in any direction will ultimately reach its starting point. Such a space is finite but it has no bounds.
Thus, closed spaces without limits are conceivable. From amongst these, the spherical space (and the elliptical) excels in its simplicity, since all points on it are equivalent. This is the approach taken by the general theory of relativity.
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Final Comments
Our space has the property of consistency beyond just having a location. Because of this consistency the space has a curvature. The smaller is the consistency, the larger is the curvature. Even light does not travel in straight lines but has a curvature on a cosmic scale; therefore, it will never escape this universe. This universe can therefore be viewed as unbound but finite in nature.
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