Non-Euclidean geometry and the Robertson-Walker metric

More documents

Recommendations

Info

• On a flat Euclidean plane, a circle has circumference C = 2πr. • On a sphere of radius R, the radius of the circle r = Rθ, but the circumference is C = 2π(R sin θ) (1) ( ) sin θ = 2πr (2) θ < 2πr (3) So, an observer on the surface of a sphere could measure the radius and circumference of a circle, and deduce that they were living in a positively curved world. R sin θ r θ R Figure 2: A circle on the surface of a sphere. • Similarly, in a negatively curved space, C > 2πr. 2 Measuring distances and the Robertson-Walker metric These differences in “big” distances must reflect differences in the infinitesimal distances from which they are made up: ∫ L = dl (4) In Euclidean space, the distance between two points is L 2 = x 2 + y 2 (5) or equivalently in polar coordinates, We express infinitesimal separations similarly: L 2 = r 2 . (6) (dl) 2 = (dx) 2 + (dy) 2 (7) 2
y dy dl dx r r dɸ dl dr dɸ x Figure 3: Infinitesimal distances. or in polar coordinates (dl) 2 = (dr) 2 + (r dφ) 2 (8) We call the rule for how distances are measured in a space the metric, e.g. dl 2 = dr 2 + r 2 dφ 2 is the metric for a flat Euclidean space. We can define a different metric for a different geometry. For example, the distance between any two points on the surface of a sphere of radius R is given by (dl) 2 = (R dθ) 2 + (r dφ) 2 , (9) with radial coordinate r and angular coordinates θ and φ. Since r = R sin θ, dr = R cos θ dθ, and R dθ = dr cos θ = R dr √ R 2 − r 2 = dr √ 1 − r 2 /R 2 (10) (since r = R sin θ and sin 2 θ + cos 2 θ = 1). So our metric for distance on the surface of the sphere is ( ) 2 (dl) 2 dr = √ + (r dφ) 2 . (11) 1 − r 2 /R 2 For a general two-dimensional surface with curvature K ≡ 1 R 2 , (12) the metric is ( ) (dl) 2 dr 2 = √ + (r dφ) 2 . (13) 1 − Kr 2 We generalize this to three dimensions: ( ) (dl) 2 dr 2 = √ + (r dθ) 2 + (r sin θ dφ) 2 . (14) 1 − Kr 2 K is now the curvature of the 3-d space. Note that as K → 0 or r → 0, the space acts Euclidean. 3
Page 1: Astronomy 401 Lecture 28 Non-Euclid

Non-Euclidean geometry and the Robertson-Walker metric

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?