- #1

- 19

- 0

## Homework Statement

It's not exactly a homework question. I can find Christoffel Symbols using general definition of Christoffel symbol. But, when I try to find Christoffel Symbols using variational principle, I end up getting zero.

I have started with the space-time metric in a weak gravitational field (with the assumption of low velocity):

ds

^{2}=−(1+2ϕ)dt

^{2}+(1−2ϕ)(dx

^{2}+dy

^{2}+dz

^{2})

Where

ϕ<<1 is the gravitational potential.

## Homework Equations

Euler Lagrange(EL) Equation: [itex]\frac{d}{d\tau}[/itex]([itex]\frac{dL}{d\dot{x^{a}}}[/itex]) = [itex]\frac{dL}{dx}[/itex]

## The Attempt at a Solution

Lagrangian:

L = −(1+2ϕ)[itex]\dot{t}[/itex]

^{2}+(1−2ϕ)([itex]\dot{x}[/itex]

^{2}+[itex]\dot{y}[/itex]

^{2}+[itex]\dot{z}[/itex]

^{2})

using EL:[itex]\frac{d}{d\tau}[/itex]([itex]\frac{dL}{d\dot{t}}[/itex]) = [itex]\frac{dL}{dx}[/itex]

−(1+2ϕ)[itex]\ddot{t}[/itex]= 0

I repeated the similar process for x, y, and z and I got zero for all. Can someone please help me?

Answer should be:

Γ

^{t}

_{ti}=ϕ,

_{i}

Γ

^{i}

_{00}=ϕ,

_{i},Γ

^{i}

_{jk}=δjkϕ,i−δijϕ,k−δikϕ,j

Last edited: