# Quasilocal Formalism and Computations in Kerr Black Hole

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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).

$$Q^{ij}= \frac{\sqrt{-\gamma}}{16\pi}\frac{\partial \mathcal{L}_{ct}}{\partial \gamma_{ij}}$$

where $\gamma_{ij}$ is the metric induced in a surface of $r=constant$ (with spacetime Kerr metric), $\gamma$ is the determinant of $\gamma_{ij}$  and $I_{ct}$ is:

$$I_{ct}=2\sqrt{2}\int_{\partial \mathcal{M}}d^{3}x\sqrt{-\gamma}\sqrt{\mathcal{R}(\gamma)}$$

The resulting $Q^{ij}$ obtained in the paper for a Kerr BH is the following (eq. 23 and eq. 24):

$$Q^{ij}=Q^{ij}_{2}+Q^{ij}_{3}$$

$$Q^{ij}_{2}= \frac{\sqrt{-\gamma}}{16\pi} \sqrt{\frac{2}{\mathcal{R}}}(R^{ij}-\mathcal{R}\gamma^{ij})$$

$$Q^{ij}_{3}=\frac{\sqrt{-\gamma}}{16\pi}\frac{1}{\sqrt{2}}\left( \nabla_{a}(\nabla^{a}\mathcal{R}^{-1/2})\gamma^{ij} - \frac{1}{2}\nabla^{(i}(\nabla^{j)}\mathcal{R}^{-1/2})\right)$$

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 $\frac{\partial \mathcal{L}_{ct}}{\partial \gamma_{ij}}$ and I can't obtain the $Q^{ij}_{2}$ and $Q^{ij}_{3}$. I was wondering if anyone could explain me what's the meaning of this equation or how can I compute this?

Thanks for everything! asked Mar 8, 2021
recategorized Mar 11, 2021

The term is reminiscent of a stress-energy tensor, or maybe it's just a simple variation. Could you elaborate on what part you don't understand?

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