My main question is, is it possible to manually add Newton's constant $G$ to the metric function of a black hole? Is there such a possibility for Black Brane? How to add? Should it be added to the main parameters of the black hole? That is, to parameters such as mass and charge...
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2$\begingroup$ $G$ is already in the Schwarzschild metric. Most practicing physicists use units where $G = 1$, but if you want to see it in SI units look at the version on Wikipedia. $\endgroup$– Michael SeifertCommented Jun 29, 2023 at 19:37
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2$\begingroup$ Use dimensional analysis to determine where factors of $G$ and $c$ and $1/4\pi\epsilon_0$ should be. $\endgroup$– GhosterCommented Jun 30, 2023 at 5:56
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The Schwarzschild metric (for a black hole with mass $M$) already contains the gravitational constant $G$, when spelling out the Schwarzschild radius $r_s=\frac{2GM}{c^2}$: $$\begin{align} ds^2 &= \left(1-\frac{2GM}{c^2r}\right)c^2dt^2 \\ &-\left(1-\frac{2GM}{c^2r}\right)^{-1}dr^2 \\ &-r^2d\theta^2-r^2\sin^2\theta\ d\phi^2 \end{align}$$
And in a similar way the Reissner-Nordström metric (for a black hole with mass $M$ and charge $Q$) already contains the gravitational constant $G$ and the electric constant $\epsilon_0$: $$\begin{align} ds^2 &= \left(1-\frac{2GM}{c^2r}+\frac{GQ^2} {4\pi\epsilon_0c^4r^2}\right)c^2dt^2 \\ &-\left(1-\frac{2GM}{c^2r}+\frac{GQ^2}{4\pi\epsilon_0c^4r^2}\right)^{-1}dr^2 \\ &-r^2d\theta^2-r^2\sin^2\theta\ d\phi^2 \end{align}$$
For a black brane you can do it in the same way by spelling out the Schwarzschild radius $r_s$.
answered Jul 2, 2023 at 22:25
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$\begingroup$ Thank you very much for the clear and detailed explanations. I tried to apply such a process to equation (14) from (arxiv.org/pdf/1810.09242.pdf), but I encountered some problems. Do you think this equation has a dimensional problem? How can we add this parameter without causing dimensional problems? $\endgroup$– SaberCommented Jul 3, 2023 at 13:01
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$\begingroup$ @Saber This article seems to use so-called natural units all over the place, i.e. $G=1$ and $c=1$. For example in equation (5) $G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi T_{\mu\nu}$, which actually is meant to be $G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^4} T_{\mu\nu}$. So you need to add the missing factors of $G$ and $c$ in many places to make them dimensionally consistent. $\endgroup$ Commented Jul 3, 2023 at 14:36
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$\begingroup$ With respect to the above arXiv, we have parameters with dimensions as follows, $$[r] = L$$ $$[m] = L^{-1}$$ $$[e] = [C] = 1$$ $$[b] = L$$ $$[c_0] = L$$ $$[c_1] = [c_2] = [c_3] = 1$$ The $[G]$ for this 5d black brane has, $$[G] = L^3$$ So we need to add one term such as $G^{2/3}$ in the third term to solve the dimension problem with added Newton constant is this true? I didn't see in the other papers such as $G^{2/3}$ in the metric function $\endgroup$– SaberCommented Jul 8, 2023 at 13:21