# Tension and horizon radius of brane (String theory and M-theory by Katrin Becker)

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.”

• Is the $r$ in the equation supposed to be the $r_2$ that you mentioned? – G. Smith Sep 29 '20 at 17:59
• 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 $r^2 = \sum_{\mu=1}^2x_2^2$. – KoKo_physmath Sep 30 '20 at 0:05