7. Have a look at the derivation of the geodesic equation (see geodezyjna.pdf file).

OTW (zestaw 2 - środa 24.10.2018)

7. Have a look at the derivation of the geodesic equation (see geodezyjna.pdf file).

8. A vector field u

on R

has components u

= −y, u

= x, u

= 0 in the Cartesian basis.

Find the components of u

in the spherical coordinates.

9. Consider a curve in R

given by

x(t) = cos

t, y(t) = cos t sin t, z(t) = sin t .

(a) Find the components of the tangent vector to this curve in the Cartesian basis.

(b) Express the curve in the spherical coordinates and find the components of the tangent vector in the spherical basis.

(c) Show by a direct calculation that the results of (a) and (b) are related by the transfor- mation law for the vector.

10. Show that for any curve with a tangent vector that fulfills v

∇

v

= cv

, with c an arbitrary function on the curve, the parametrization of this curve can be changed to get v

∇

v

= 0 (this is called an affine parametrization). Show that different affine parametrizations are connected with a linear transform.

11. Consider the 2-dimensional metric ds

= f (u, v)du dv.

(a) Show that the curves u =const are geodesics. Are these timelike, spacelike, or null?

(b) Find the condition on f that v is an affine parameter along the u =const geodesics.

12. Solve the geodesic equation on two-sphere S

:

ds

= dθ

+ sin

θdφ

,

for geodesics with φ 6=const. Get the solution in the form θ = θ(φ).