Download normal and two other geometries.

C= 2  R, A =  R 2
C < 2  R, A <  R 2
C > 2  R, A >  R 2
Comparison of the regions within a fixed distance of a point (as could be determined by a taught string in that space) in
example spaces: Euclidean (plane), Riemann (sphere) and Lobachevsky (hyperbolic saddle). The region is ‘cut out’ and
flattened into Euclidean plane. Ref. Wolfgang Rindler, Essential Relativity, p 107.
Euclid:
1. Parallel postulate: Through a point P outside a line  there is one and only one line parallel to .
2. A straight line segment is the shortest distance between two points.
Pythagorean theorem r 2 = x 2 + y 2 (right angle at local point).
3. The sum of the angles of any triangle is equal to 180. Curvature is zero or neutral.
Riemann:
1. Substitute postulate: Through a point P outside a line  there is no line parallel to it; that is, every
pair of lines in a plane must intersect. = > construction of a non-Euclidean geometry.
2. On a sphere, the shortest route is along a great circle. The mathematical term for the shortest route
along a surface is geodesic.
1. restated: Through a given point P outside a geodesic  there is no geodesic which does not meet .
3. The sum of the angles of any triangle is more than 180. Curvature is positive.
The two-dimensional surface of a sphere can be embedded in a larger three-dimensional Euclidean
space: Spherical coordinates are then given by (R, , ):
Radius R of the sphere:
R2=x2+y2 +z2 .
Colatitude (measured from north pole, like zenith angle)  :
tan  = z / R,
(latitude = 90°-  )
Longitude (or azimuth) angle  :
tan  = y / x .
Lobachevsky:
1. Substitute postulate: In a plane, through a point outside a line there are an infinite number of lines
which do not intersect . = > construction of a non-Euclidean geometry.
Restated: Through a given point P outside a geodesic  there are an infinite number of geodesics
which do not intersect .
2. Shortest routes are curves.
3. The sum of the angles of any triangle is less than 180. Curvature is negative.
Einstein: General theory says that the presence of matter distorts space-time in its neighborhood.
First test: observed amount of perturbation in the orbit of Mercury predicted better by Einsteinian
theory than Newtonian theory.
Second test: more impressive because it was a matter of complete prediction from the theory of facts
never before observed. If the particle theory of light is accepted, then as light from a distant star passes
near the sun, it must, according to Newton, be bent toward the sun, because of the sun’s gravitational
attraction. The amount of this deflection computed by Einstein’s general theoy was double that figured
from Newtonian theory. This cannot be checked ordinarily, since the sun’s brilliance makes it
impossible to see the star rays passing near, but total eclipses furnish an opportunity.
British expeditions to observe such eclipses in Brazil and West Africa in 1919, as well as observations
of late eclipses, confirmed Einstein’s prediction beyond a doubt.
3rd test: vibration of an atom is slower near sun than a similar atom on earth—redshifted
…
regions that are relatively small portions of any curved surface may be considered as approximately
flat.