Finding Geometry of Near-Extremal D3 Brane | Schwarzschild in 5D

[Pacopag]

Hi;
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
$$ds^2=e^{2\eta}dq^2-e^{2\zeta}dt^2+e^{2\xi}dr^2+r^{2}d\Omega^2$$
into the Einstein field equations with $$T_{a b}=0$$, to solve for the functions $$\eta,\zeta,\xi$$, 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.

 

[George Jones]

I'm not sure if it will help, but try looking up Wesson's work on the arxiv. He does stuff in 5d,

For example,

http://arxiv.org/abs/gr-qc/0611137

 

[Pacopag]

Great. Thanks George. That looks like the track that I want to get on.

Cheers.

 

[robousy]

Do you have software to solve the equations? That can make things easier.
Also, wouldn't the variables be a function of t?

 

[Pacopag]

I have maple with grtensor. I asked maple to solve the system, but it doesn't really give me anything useful. As for the functions of t: that may very well be the case, but I am looking for the static solution, so I am just naively assuming that the metric components are t-independent.