General Relativity-Variational approach

[QED_81]

Homework Statement

Hi,
I am given the following action S:
S=1/2∫〖g^ab ((∇_a Ф ∇_b Ф)+(1/6 R_ab Ф^2) ) √(-g) dx^4 〗

Where R_ab is the Ricci tensor, Ф is a scalar field and g^ab is the metric.

I am asked to find the equation of motion for the field.

Homework Equations

I know I have to substitute my Lagrangian: L= 1/2 g^ab ((∇_a Ф ∇_b Ф)+(1/6 R_ab Ф^2) )

into the Euler-Lagrange equations, but I don't know in which EL equation exactly!

The Attempt at a Solution

I mean should I use the EL equation where Ф and ∇_b Ф are my independent components? Or should I use the 2nd order EL equations where g_ab; g_ab,c & g_ab,cd are my independent components (as in the Einstein Hilbert action)?


Thanks :)

 

[mark726]

Dear student,

Thank you for sharing your question on the forum. I can provide you with some guidance on how to approach this problem.

Firstly, it is important to understand the concept of the Euler-Lagrange equations. These equations are used to find the equations of motion for a given Lagrangian, which in turn describes the dynamics of a system. In this case, your Lagrangian is given as L= 1/2 g^ab ((∇_a Ф ∇_b Ф)+(1/6 R_ab Ф^2) ), and you are asked to find the equation of motion for the scalar field Ф.

To do this, you will need to use the Euler-Lagrange equations for the scalar field Ф, which are given as:

∂L/∂Ф - ∂/∂x^μ (∂L/∂(∂x^μ Ф)) = 0

where L is your Lagrangian and x^μ represents the independent variables (in this case, the coordinates x, y, z, and t).

Next, you will need to substitute your Lagrangian into the above equation and solve for the equation of motion for Ф. This will involve taking derivatives with respect to Ф and its derivatives (∂Ф/∂x, ∂Ф/∂y, ∂Ф/∂z, and ∂Ф/∂t).

I hope this helps you in finding the equation of motion for the scalar field Ф. If you need further clarification or assistance, please do not hesitate to ask. Good luck!