# A coincident stack of D3 branes vs a shell of them

• I would in general like to understand how to derive the low energy metrics to describe D-brane configurations. Any pedagogic reference which explains the method?

In particular I have in mind these two facts,

(1) For N coincident D3 branes in IIB it seems that the metric is,

$ds^2 = (1+ \frac{L^4}{y^4})^{-0.5}\eta_{ij}dx^idx^j + (1+ \frac{L^4}{y^4} )^{0.5}(dy^2+y^2d\Omega_5^2)$

where $L^4 = 4\pi g_s N (\alpha')^2$ ($L$ being called the radius of the Dbrane)

(2) A "shell" of D3 branes is apparently described by,

$ds^2 = h^{-1}(r)[-dt^2 + d\chi^2 + dx_1^2 + dx_2^2 ] + h(r)[dr^2 + r^2d\Omega_5^2 ]$

where $\chi$ is a compactified dimension of length $L$ and $h(r) = \frac{R^2}{r^2}$ for $r>r_0$ and $h(r) = \frac{R^2}{r_0^2}$ for $r<= r_0$

• How are these two related? What exactly is a "shell of Dbranes"? How are these two derived?

• What exactly is meant by "radius of the Dbrane" (L) in the first one? Is that the coordinate in the spacetime which was compactified and T-dualized and is that the same as the $L$ in the second metric?

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But the "shell of D3-branes" is meant to be spherical - literally a sphere that is "made out of" the D3-brane stuff. People use this shell in thought experiments where they allow them to collapse to create the black hole/brane. Inside, $r\lt r_0$, one has a flat 10-dimensional Minkowski space, and outside, we have $AdS_5\times S^5$. These two 10D geometries are "nonsmoothly" connected at $r=r_0$. The shell has this extra $r_0$ parameter, the radius. Naively, you could think that the limit $r_0\to 0$ of the shell gives you the stack because you eliminate the flat region inside but you actually make the branes infinitely curved so I think that you're getting away from the flat stack.
You have written all the formulae that define (and answer the question) what is meant by the "radius of the D-branes" $L$. $L$ is also the curvature radius of the AdS space, and the curvature radius (or ordinary radius) of the five-sphere. It is the length scale or a value of the $y$ coordinate at which the two components of the metric tensor (e.g. the redshift) change substantially. You have also written the radius $L$ in terms of $N,g_s,\alpha'$. The more branes you have or the stronger the string coupling is, the stronger the gravitational field is around them, and the larger $L$ is – the further the stack of D3-branes may affect the gravitational field.