From ##D3-Brane## to ##AdS_5-Schwarzschild##

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In summary, the conversation discusses the difficulty in understanding the near-horizon limit of the metric for a near-extremal D3-brane, specifically the transition from equation (1) to equation (2). It is suggested that there may be a typo in equation (1) and that the correct form should include a constant radius for the five-sphere. The author then explains that in the near-horizon limit, the S^5 component of the metric is dropped as it no longer plays a role.
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I'm trying to read this paper. Of course there are lots of things about it that are beyond me but for now its only the calculations in the main body of the paper that I'm trying to, more or less, understand. So I'm trying to go through it and find out the things that I don't get.

For now, I have problem with the beginning of the section 2:
The metric of the near-extremal D3-brane is
## ds_{10}^2=H^{-\frac 1 2} (-h dt^2+d\vec x ^2)+H^{\frac 1 2} (\frac{dr^2}{h}+d\Omega_5^2) \ \ \ \ \ \ \ \left( H=1+(\frac{L}{r})^4 \ \ , \ \ h=1-(\frac{r_H}{r})^4 \right) \ \ \ \ \ (1)##
Here ##\vec x = (x, y, z) ## are the spatial coordinates along which the D3-brane is extended and ##dΩ^2_5## is the standard metric on the five-sphere ##S^5## with unit radius. The near-horizon limit consists of “dropping the 1” from H. Then the metric is ##AdS_5-Schwarzschild##,
##ds_5^2=(\frac{r}{L})^2 (-hdt^2+d\vec x^2)+(\frac{L}{r})^2 \frac{dr^2}{h} \ \ \ \ \ \ \ \ \ \ (2)##
times the metric for an ##S^5## of constant radius L.

I don't understand how he got equation (2) from equation (1) by just "dropping the 1 from H"! Because when I do that, I get:
## (\frac r L)^2 ( -h dt^2+ d \vec x ^2)+(\frac L r)^2 ( \frac{dr^2}{h}+d\Omega_5^2) ##
And this is really a mystery to me how that ## d\Omega_5^2 ## vanishes and reappears as an overall multiplicative factor! This really strange because in equation (1), the "radius" of the ##S^5## is ## H^{\frac 1 2} ## but in the near-horizon limit, it involves other coordinates too, actually their differentials which is non-sense because terms like ## dt^2 d\Omega_5^2 ## will appear in the metric. What is the author doing here?
Thanks
 
  • #3
I think there might be a typo in (1) and it should be

$$ds^2 = H^{-1/2} (-h dt^2 + d \vec x^2) + H^{1/2} (\frac{dr^2}{h} + r^2 d\Omega^2)$$
Then you get Schwarzschild-AdS x S^5 as the near-horizon limit. You should see that the S^5 now gets a constant radius equal to the AdS radius.

In (2), Gubser also drops the S^5 part of the metric, because it will no longer play a role. (There are fancier situations where it does play a role...if you allow your fields to depend on the S^5 coordinates then things can quickly become a mess.)
 
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1. What is a D3-brane?

A D3-brane is a type of brane, or extended object, in string theory that has three spatial dimensions. It is a fundamental object in the theory and plays a key role in various string theoretic constructions.

2. What is AdS5-Schwarzschild?

AdS5-Schwarzschild is a black hole solution in five-dimensional anti-de Sitter space (AdS5) that is often used in string theory to study the AdS/CFT correspondence. It is a combination of the AdS5 space and the Schwarzschild black hole solution.

3. How are D3-branes related to AdS5-Schwarzschild?

In the AdS/CFT correspondence, D3-branes are related to AdS5-Schwarzschild through the holographic principle. This principle states that a quantum system in a higher-dimensional space can be described by a classical system in a lower-dimensional space. In this case, the D3-branes in the higher-dimensional AdS5 space are dual to the black hole in the lower-dimensional AdS5-Schwarzschild space.

4. What applications does the study of From D3-Brane to AdS5-Schwarzschild have?

The study of From D3-Brane to AdS5-Schwarzschild has many applications in string theory and theoretical physics. It allows for a better understanding of the AdS/CFT correspondence and the holographic principle, and also has implications for quantum gravity and black hole thermodynamics.

5. What are the current developments in the study of From D3-Brane to AdS5-Schwarzschild?

Currently, there is ongoing research in this area to deepen our understanding of the AdS/CFT correspondence and to explore its applications in other areas of physics, such as condensed matter systems and quantum information theory. There is also interest in extending this correspondence to other types of branes and higher dimensional spacetimes.

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