Page 17 - GALIET PHYSICS BLOSSOMS III

(2) Spacetime, Mass, and Geometry
(i) We start by considering 2-dimensional geometries, which we can visualize ’from outside’ in our 3-dimensional world. Imagine that we compare (a) a triangle, and (b) a circle, in 2 different geometries, viz., in a flat Euclidean plane, and in a sphere of radius R.
Explain first of all how a being living only in one or other of these 2-d geometries would try to decide what geometry they were in, using measurements performed on the 2 geometrical objects. How would they define and measure angles and distances in their geometry?
A being confined to a 2D reality would have no idea of other higher dimensions, and could not decide or perceive with perfect clarity what geometry she is in. We can visualize a 2D’s curvature in a bent sheet of paper or the earth’s surface because they allow two way travel north-south (latitude-widthwise) and east-west (longitude-lengthwise). But it is impossible to visualize the curvature of a 3D or 4D spacetime (length, width, depth + time) unless we were at extra dimensions to visualize them. We can see the 2D’s curvature because we are seeing it from the 3D dimension (length, width, depth), but unfortunately, we cannot see any dimensions beyond the 3D. If we could see in the 4D, we could see time, or every event in someone’s life. However, we can find out whether space or spacetime is curved by figuring out what basic rules of geometry pertain to each environment, and there are 3 basic rules (flat geometry, spherical geometry, saddle-shaped geometry).
In the 2D world, she would observe the path of objects that follow the straightest possible paths between two points.13 She would discover that angles of a triangle don’t add up to 180 degrees; that squares don’t close up when we try to draw it by joining four equal lines with right angles; and that the circle’s area is not proportional to the radius’ square. She would not know that she dwells in a curved dimension. Only beings dwelling in a 3D reality would know that her 2-D world is curved. All she could perceive as a 2D being is that her 2D world’s internal geometry is non-Euclidean. Very small objects (little squares, triangles, circles, etc.) appear to be Euclidean, while all other larger objects appear to be deviated.14
Then show what they would find. You can show this using pictures, but you should also draw graphs of (i) how you think the total internal angle of the triangle (ie. the sum of the 3 internal angles) will vary with the size of the triangle, for both geometries, and (ii) how the area of a circle of radius r will vary as we vary r, for both geometries.
Triangle
In a flat Euclidean plane, the sum of the triangle’s angles is always 180 degrees.
On the sphere’s surface, the sum of the triangle’s angles is greater than 180 degrees.
Circle
13 Bennet, Donahue, Schneider, Voit. The Cosmic Perspective. 6th Edition. 436. 14 This information is derived from Dr. Stamp’s Slide 3.28.
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