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In exercise 12.1 tells the relation between $N_2$, the number of M2-brane, and $r_2$, the horizon radius.

The following relation is derived $$-g_{00} = -1 + \frac{16\pi G_{11}N_2T_\text{M2}}{9\Omega_7}r^{-6}. $$ I dont understand how to get it. The book explain “ignore the directions along the brane and the result is a tension.”

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  • $\begingroup$ Is the $r$ in the equation supposed to be the $r_2$ that you mentioned? $\endgroup$ – G. Smith Sep 29 '20 at 17:59
  • $\begingroup$ No $r_2$ is included in the metric as $ds^2 = H^{-2/3}dx\cdot dx + H^{1/3}dy\cdot dy$, where $x$ are the coordinates along the brane, $y$ are the coordinates perpendicular to the brane and $H(r) = 1+(r_2/r)^6$. $r$ is the distance from the brane, that is $r^2 = \sum_{\mu=1}^2x_2^2$. $\endgroup$ – KoKo_physmath Sep 30 '20 at 0:05

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