Need to calculate Christoffel connection from a given metrics

[chinared]

Hi all,
I am trying to find the Christoffel connections of this metric:

ds²= -(1+2∅)dt² +(1-2∅)[dx²+dy²+dz²]
where ∅ is a general function of x,y,z,t.

I tried to solve this through the least action principle, but some of my results(t-related terms) were different from the answer with a minus sign. So, I guess it's a problem about the part of t of the action.

I regarded this part as -1/2(1+2∅)$\dot{t}$², should I remove the minus sign to get the correct answer?

$\dot{t}$: the derivative of t regard to the affine parameter λ

Thanks for your help!

 

[HallsofIvy]

That's pretty straight forward isn't it? The Christoffel symbols (of the first kind) are given by
$$[ij, k]= \frac{1}{2}\left(\frac{\partial g_{ij}}{\partial x^k}+ \frac{\partial g_{ik}}{\partial x^j}- \frac{\partial g_{jk}}{\partial x^i}\right)$$

Here, if we take $x^1= x$, $x^2= y$, $x^3= z$, and $x^4= t$, then $g_{11}= g_{22}= g_{33}= 1- 2\phi$, $g_44= -(1+ 2\phi)$. Of course, the result will depend upon the partial derivatives of $\phi$. If $\phi$ can be any function of the variables, then the Christoffel symbols can be just about anything!