하틀 일반상대성이론 2.11 풀이

[역대급천재(valuable6233)]

James Burkett Hartle, Gravity: An Introduction to Einstein's General Relativity, 2nd edition

Chapter 2. Geometry as Physics 제2장. 물리학으로서의 기하학

문제(Problem) 11 및 해설(Solution)

Conical Projections

Conical projections map points on the globe into polar coordinates (r, ψ) in the plane of the map. (We use ψ to avoid confusion with the coordinate φ on the sphere.) Thus, in general, r = r(λ, φ) and ψ = ψ(λ, φ). A particularly simple class of conical projections uses the North Pole as the origin of the polar coordinates and has r = r(λ) and ψ = φ.

(a) For this simple class, express the line element on the sphere in terms of r and ψ.

(b) Find the function r(λ) that makes this an equal-area projection in which there is a constant proportionality between each area on map and the corresponding area on the sphere. (Hint: See the hint for Problem 10.)

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