Question

Consider the 2-sphere with the metric $ds^2 = g_{ab}dx^a dx^b = a^2 \left( \frac{dr^2}{1 - r^2} + r^2 d\theta^2 \right)$ (1) where $a$ is a constant with length dimensions, $0 < r < 1$ and $0 \le \theta < 2\pi$. a. Make a drawing of each of the coordinate curves b. Show that with this coordinate system it is not possible to cover the entire surface of the 2-sphere.

          Consider the 2-sphere with the metric
$ds^2 = g_{ab}dx^a dx^b = a^2 \left( \frac{dr^2}{1 - r^2} + r^2 d\theta^2 \right)$ (1)
where $a$ is a constant with length dimensions, $0 < r < 1$ and $0 \le \theta < 2\pi$.
a. Make a drawing of each of the coordinate curves b. Show that with this coordinate system it is not
possible to cover the entire surface of the 2-sphere.

Consider the 2-sphere with the metric
ds^2 = gabdx^a dx^b = a^2 ( (dr^2)/(1 - r^2) + r^2 dθ^2 ) (1)
where a is a constant with length dimensions, 0 < r < 1 and 0 ≤θ < 2π.
a. Make a drawing of each of the coordinate curves b. Show that with this coordinate system it is not
possible to cover the entire surface of the 2-sphere.

Added by Steven B.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Supreeta N

10/04/2023

Step 1

a. For constant r, the curve is a circle of radius r in the θ direction. b. For constant θ, the curve is a radial line from the origin in the r direction. Show more…

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Consider the 2-sphere with the metric ds^2 = gabdx^adx^b = a^2((dr^2)/(1-r^2) + r^2dθ^2) where a is a constant with length dimensions, 0 <= θ < 2π and 0 <= θ < 2π. a. Make a drawing of each of the coordinate curves b. Show that with this coordinate system it is not possible to cover the entire surface of the 2-sphere.