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

• Pacopag
In summary, the conversation revolved around finding the geometry of a near-extremal D3 brane by using a 5D analog to the Schwarzschild metric with a negative cosmological constant. The individual seeking help had tried solving for the functions \eta, \zeta, and \xi, which are assumed to be functions of only r, but encountered difficulties. They were looking for references or advice on how to approach the problem and mentioned not being familiar with string theory. Another individual suggested looking into Wesson's work on the arxiv for potential solutions. The use of software to solve the equations was also mentioned.
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.

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

Cheers.

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

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.

## What is the "geometry of near-extremal D3 brane"?

The "geometry of near-extremal D3 brane" refers to the mathematical description of the shape and structure of a D3 brane, which is a type of fundamental object in string theory. In this context, "near-extremal" means that the brane is close to being in a state of maximum energy, known as the extremal limit.

## What is the Schwarzschild metric in 5 dimensions?

The Schwarzschild metric in 5 dimensions is a mathematical description of the curvature of space-time in the vicinity of a massive object, such as a black hole. It is a solution to Einstein's equations of general relativity and is characterized by a parameter known as the mass of the object.

## Why is it important to study the geometry of near-extremal D3 branes and Schwarzschild in 5D?

Studying the geometry of near-extremal D3 branes and Schwarzschild in 5D is important for understanding the behavior of fundamental objects in string theory and the nature of space-time in the presence of massive objects. It also has implications for understanding the physics of black holes and the connections between string theory and general relativity.

## What are some practical applications of this research?

While the study of near-extremal D3 branes and Schwarzschild in 5D is primarily theoretical, it has potential applications in areas such as cosmology, high-energy physics, and quantum gravity. For example, it may help in developing a more complete understanding of the behavior of matter and energy in the early universe or in the extreme conditions near the event horizon of a black hole.

## What are some current research topics related to this area of study?

Current research topics related to the geometry of near-extremal D3 branes and Schwarzschild in 5D include investigating the behavior of black holes in higher dimensions, exploring the connections between string theory and other areas of physics, and studying the implications of this research for the holographic principle and the nature of space-time.

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