Killing Vectors of a 2-Sphere

[Airsteve0]

Homework Statement

Find all killing vectors of the 2-sphere.

Homework Equations

$ds^2=dθ^2+(sinθ)^2d\phi^2$

The Attempt at a Solution

The goal of this problem is pretty self-explanatory; however, I didn't feel that the topic of killing vectors was covered very well in my course. I was just wondering if someone knew of a site which explained how to approach this sort of problem and the steps which need to be taken. Thanks in advance!

 

[mighty2000]

Hello! I understand your concern about not feeling confident in your understanding of killing vectors. Here are some steps you can follow to approach this problem:

1. Understand the concept of a killing vector: A killing vector is a vector field that preserves the metric of a manifold. In other words, it is a vector field that preserves distances and angles on a manifold.

2. Familiarize yourself with the 2-sphere: The 2-sphere is a two-dimensional surface embedded in three-dimensional space. It is defined by the equation x^2 + y^2 + z^2 = 1. It is important to understand the geometry of the 2-sphere in order to find its killing vectors.

3. Use the metric to find the infinitesimal transformation: The metric of the 2-sphere can be written as ds^2 = dθ^2 + (sinθ)^2d\phi^2. This metric allows us to find the infinitesimal transformation, which is a vector field that preserves the metric. This transformation can be written as ξ = ξ^θ ∂/∂θ + ξ^φ ∂/∂φ, where ξ^θ and ξ^φ are functions of θ and φ, respectively.

4. Apply the killing equation: The killing equation is a differential equation that must be satisfied by the components of the infinitesimal transformation. In this case, it is given by ξ^θ,_φ + ξ^φ,_θ = 0, where the comma denotes partial differentiation. Solving this equation will give you the components of the killing vector.

5. Repeat for all possible combinations of ξ^θ and ξ^φ: There are four possible combinations of ξ^θ and ξ^φ that can satisfy the killing equation. These combinations are: ξ^θ = 0, ξ^φ = constant, ξ^θ = constant, ξ^φ = constant, ξ^θ = sinφ, ξ^φ = cosφ, and ξ^θ = cosφ, ξ^φ = -sinφ. Each of these combinations will give you a different killing vector.

I hope this helps you in understanding how to approach this problem. If you need more help, there are many online resources that explain the concept of killing vectors in more detail. Good luck!