Hi everyone,
I was reading an article about quasilocal formalism and calculations in a Kerr BH ( this:<https://arxiv.org/pdf/hep-th/0102001.pdf>). I was trying to reproduce the results obtained on it, but I found an expresion that I can't understand: (eq. 8).
Qij=√−γ16π∂Lct∂γij
where γij is the metric induced in a surface of r=constant (with spacetime Kerr metric), γ is the determinant of γij and Ict is:
Ict=2√2∫∂Md3x√−γ√R(γ)
The resulting Qij obtained in the paper for a Kerr BH is the following (eq. 23 and eq. 24):
Qij=Qij2+Qij3
Qij2=√−γ16π√2R(Rij−Rγij)
Qij3=√−γ16π1√2(∇a(∇aR−1/2)γij−12∇(i(∇j)R−1/2))
My understanding in partial derivatives is that a partial derivative of a function of several variables is its derivative with respect to one of those variables. Then my problem is that I can't understand what's the operation ∂Lct∂γij and I can't obtain the Qij2 and Qij3. I was wondering if anyone could explain me what's the meaning of this equation or how can I compute this?
Thanks for everything!