Calculating Homothetic Vector Fields: A Step-by-Step Guide

[axlsaml1]

can anyone please give me an example of how to calculate the homothetic vector field of
say

a Bianchi Type I exact solution of the Einstein field equation

(refer to dynamical systems in cosmology by Wainwright and Ellis chapter 9)


note : I know already how to calculate the killing vector field of a homogeneous, isotropic line element

 

[smallphi]

Don't you have to take the given metric, write explicitly the defining differential equations for a homotetic vector field and try to solve these? Is there another way?

 

[Entropia]

Sure, I would be happy to provide an example of calculating the homothetic vector field for a Bianchi Type I exact solution of the Einstein field equation. Before we begin, let's first define what a homothetic vector field is.

A homothetic vector field is a vector field that satisfies the condition that its Lie derivative with respect to the metric tensor is proportional to the metric tensor itself. In other words, the vector field scales the metric tensor by a constant factor at each point. This means that the vector field is a symmetry of the metric, and its integral curves are geodesics.

Now, let's look at the specific example of a Bianchi Type I exact solution of the Einstein field equation. This type of solution is characterized by having three commuting, homothetic vector fields. We will use the notation from Wainwright and Ellis' book, where the three homothetic vector fields are denoted by X, Y, and Z.

To calculate the homothetic vector fields, we will use the Killing equation, which is a special case of the Lie derivative condition for homothetic vector fields. The Killing equation is given by:

L_X g = 0

where L_X denotes the Lie derivative with respect to the vector field X and g is the metric tensor.

In this case, we are looking for three vector fields X, Y, and Z that satisfy the Killing equation. We can start by choosing a specific Bianchi Type I metric, such as the Kasner metric:

ds^2 = -dt^2 + t^{2p_1}dx^2 + t^{2p_2}dy^2 + t^{2p_3}dz^2

where p_1, p_2, and p_3 are constants.

Now, we can substitute this metric into the Killing equation and solve for the three vector fields. This will give us the following equations:

L_X g = -2p_1t^{2p_1-1}dt^2 + 2p_2t^{2p_2-1}dx^2 + 2p_3t^{2p_3-1}dy^2 + 2p_4t^{2p_4-1}dz^2 = 0

L_Y g = 2p_1t^{2p_1-1}dt^2 - 2p_2t^{