Page 18 - GALIET PHYSICS BLOSSOMS III

In a flat Euclidean plane, the circle’s circumference is 2πr.
On a spherical surface, the circle’s circumference is less than 2πr.
As the circle’s radius increases, the area deviates more and more from the norm. It is not proportional to the square of the radius.
Conclusion
If a circle’s circumference in space is really 2πr, then we can affirm space has a flat geometry.
If a circle’s circumference in space is less than 2πr, then we can affirm space has a spherical geometry simply because these rules hold for a sphere’s surface.
Thus, there is a key difference between the rules of flat and spherical geometries. Indeed, in a flat plane, a straight line is the shortest distance between A and B, while in on a spherical surface, like the earth’s, the shortest and straightest distance is a great circle given that in our realm, a straight line does not exist.15
Fig. 116
(ii) In General Relativity, it is the geometry of 4-dimensional spacetime that one is interested in.
15 Bennet, Donahue, Schneider, Voit. The Cosmic Perspective. 6th Edition. 433-435.
16 Bennet, Donahue, Schneider, Voit. The Cosmic Perspective. 6th Edition. 435. Excerpt from figure S3.12.
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