Then find the formula for the Doppler effect in the case s = αk

GR problem set 02

9. Let n be the photon wave (four-)vector (n · n = 0), and u, u⁰ (four-)velocities of inertial observers O and O⁰ (u · u = −1 = u⁰ · u⁰). The photon frequency as measured by O and O⁰ reads ω = −n · u and ω⁰ = −n · u⁰ respectively. Projecting: n = ωu + k and u⁰ = γu + s, with γ = −u · u⁰ show that u · k = 0 = u · s and ω⁰ = γω − s · k (Doppler’s effect). Then find the formula for the Doppler effect in the case s = αk.

10. Show that two coordinate systems are enough to cover an S² sphere. Hint: use stereographic projection from the north and south poles of the sphere.

11. Consider a curve in ordinary 3-dimensional space given in terms of Cartesian coordinates x(t) = cos²t, y(t) = cos t sin t, z(t) = sin t .

(a) Calculate the components of the tangent to this curve at parameter t in the Cartesian coordinate basis.

(b) Express the curve in terms of spherical polar coordinates, i.e., give r(t), θ(t), φ(t), where these coordinates are defined by

x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ.

(c) Explicitly show that the results of (a) are related to the results of (b) by the vector transformation law.

12. Find the components g^αβ and gαβ of Minkowski metric ds² = −dt² + dx² + dy² + dz² in a rotating frame

t⁰ = t x⁰ = (x²+ y²)^1/2cos(φ − ωt), y⁰ = (x²+ y²)^1/2sin(φ − ωt), z⁰ = z, where tan φ = y/x.

13. Having two vector fields on R²: u = (xy, x²+ y²) i v = (y³, x³) in Cartesian coordinates, find their commutator [u, v].

14. Varying the action

S =

B

Z

A

r

−g_αβ(x(λ))dx^α(λ) dλ

dx^β(λ) dλ dλ ,

find the geodesic equation for the geodesic going from A to B, parametrized with the length along the geodesic s i.e. ds =

q

−g_αβ(x(λ))^dx_dλ^α(λ)dx_dλ^β(λ)dλ.

15. Let g_αβ be a nondegenerate metric on a manifold M (in x^α coordinates).

(a) Show that in a proximity of of any point P ∈ M one can find coordinates x^α0, such that gα⁰β⁰(P ) (the metric in x^α0 coordinates) is diagonal with diagonal elements ±1.

(b) Show that one can further specialize the choice of the coordinates from (a) in such a way

that ∂g_α⁰_β⁰

∂x^γ0 (P ) = 0.

Hint: Without loosing generality we can assume that x^α(p) = 0 = x^α0(p), then consider a coordinate transformations

x^α0(x^µ) = b^α_µ⁰x^µ+ c^α_µν⁰x^µx^ν + O(x³), x^µ(x^α0) = ˜b^µ_α0x^α0 + ˜c^µ_α0β⁰x^α0x^β0 + O(x⁰³).

A. Rostworowski http://th.if.uj.edu.pl/ arostwor/