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I am trying to find the geometry of a near-extremal D3 brane. I have been told that this geometry is the same as the 5D analog to the Schwarzschild metric with a negative cosmological constant. Trying to mimic Schutz (Ch 10) I tried plugging the metric

[tex]ds^2=e^{2\eta}dq^2-e^{2\zeta}dt^2+e^{2\xi}dr^2+r^{2}d\Omega^2[/tex]

into the Einstein field equations with [tex]T_{a b}=0[/tex], to solve for the functions [tex]\eta,\zeta,\xi[/tex], which I am assuming are functions of r only. The coordinate q is for the extra dimension. The system of DEs I get from this are terrible.

Anyway, I'm not expecting anyone to solve this problem for me. I am almost certain that this calculation has been done before, so I am wondering if anyone knows of any references that might help me out, or if someone could maybe tell me if I am going about this problem the right way. Please keep in mind that my main goal is to get a handle on the near-extremal D3 brane geometry.

Note: I don't really know anything about string theory, so please try not to use to much of its language, or else please define any string theory jargon that you may use. Also, I have to go shovel some snow, so I may not be online to reply for an hour or so.

Thank you.

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# Schwarzschild in 5D

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