Getting confused with different metrics for AdS black hole

Question

I'm getting confused with the convention for the metric that describes a (planar) AdS black hole in $1+d$ dimensions ($1$ timelike, $d$ spacelike).

The most common definition seems to be the one as in https://arxiv.org/abs/2210.09647 Eq (2.6):

$$ds^2 = -f(r)\, dt^2 + \frac{1}{f(r)}dr^2 + r^2d\vec{x}^2 \qquad\text{with}\qquad f(r) = \frac{r^2}{L^2}\left(1-\frac{r_h^d}{r^d}\right)$$

where $r = r_h$ is the location of the event horizon.

Now I want to compare this to the typical definition in 2D ($d=1$) as in https://arxiv.org/abs/2110.05522 Eq (4.8):

$$ds^2 = -f(r)\,dt^2 + \frac{1}{f(r)}dr^2 \qquad\text{with}\qquad f(r) = r^2-1$$

but I don't see how to get there from the previous equation. I know it is common to absorb $r_h$ and $L$ by rescaling $r\to \frac{r}{r_h}$ and $f(r)\to \frac{L^2}{r_h^2}f(r)$, however, the scaling with $d$ is incorrect (it looks as if one has to use $d=2$ to get to this result but that would be a 3D black hole which it is not).

I'm suspecting that the geometry has to do with it, more specifically the fact that in 2D there is no curvature singularity at $r=0$, but I don't see how.

Any help would be greatly appreciated (in particular if there is a reference to a paper that treats this in more detail, because I don't find any).

Answer 1

I have an idea regarding how to obtain the $d=1$ dimensional version of the AdS$_d$ black hole metric (not 100% sure if this is the way to go, but here goes). According to the papers you cite, Eq. (2.6) from https://arxiv.org/pdf/2210.09647.pdf, the AdS$_{d+1}$ black-hole metric is $$ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\vec{x}^2$$

In the other paper, however, namely https://arxiv.org/pdf/2110.05522.pdf, the authors state clearly that they set both the black hole horizon and the AdS radius of curvature to be 1. Following the same prescription, and setting $d=1$, Eq. (2.6) reduces to $$ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}$$ with $f(r)=r^2(1-\frac{1}{r})=r(r-1),\ r\ne0$. I can always rescale and shift the radial coordinate, since the AdS spacetime is invariant under such transformations. Let us perform the consecutive transformations

1. $r\rightarrow r+\frac{1}{2} \Rightarrow f(r)=r(r-1)\rightarrow r^2-\frac{1}{4}$

2. $r\rightarrow \frac{r}{2},\ t\rightarrow2t\Rightarrow f(r)=r^2-\frac{1}{4}\rightarrow \frac{1}{4} \tilde{f}(r)\equiv \frac{1}{4} (r^2-1)$

And, hence, the line element takes the form $$ds^2=-\tilde{f}(r)dt^2+\frac{dr^2}{\tilde{f}(r)}$$ where $\tilde{f}(r)$ is the blackening factor appearing in the other paper...

If there are any questions, please let me know.

Answer 2

The convention for the metric that you are using is not the most common one for the planar AdS black hole in 1+d dimensions. The more common convention is to use the following metric:

$$ds^2 = -f(r)dt^2 + \frac{dr^2}{f(r)} + r^2 d\vec{x}^2$$

where

$$f(r) = r^2 - \frac{M}{r^{d-2}}$$

and $M$ is a constant related to the mass of the black hole. This metric is obtained by taking the limit $a \to 0$ of the Kerr-AdS black hole metric, which is a rotating AdS black hole. The parameter $a$ is related to the angular momentum of the black hole.

The metric that you are using is obtained by rescaling $r \to r/L$ and $M \to M/L^{d-1}$, where $L$ is the AdS radius of curvature. This gives:

$$ds^2 = -f(r)dt^2 + \frac{dr^2}{f(r)} + r^2 d\vec{x}^2$$

where

$$f(r) = r^2 (1 - \frac{C}{r^{d-1}})$$

and $C$ is a constant related to $M$. This metric has the same asymptotic behavior as the previous one, but it has a different horizon radius and temperature.

To see this, note that the horizon radius $r_h$ is given by solving $f(r_h) = 0$. In the first convention, this gives:

$$r_h^{d-1} = M$$

while in the second convention, this gives:

$$r_h^{d-1} = C$$

The temperature of the black hole is given by:

$$T = \frac{f'(r_h)}{4\pi}$$

where $f'(r)$ is the derivative of $f(r)$ with respect to $r$. In the first convention, this gives:

$$T = \frac{d-1}{4\pi L} r_h$$

while in the second convention, this gives:

$$T = \frac{d-1}{4\pi} r_h (1 - \frac{C}{(d-1)r_h^{d-1}})$$

You can see that these two expressions are not equal unless $C = 0$, which corresponds to the pure AdS case with no black hole.

Therefore, if you want to compare the metrics in different dimensions, you should either use the same convention for both cases, or convert between them using the appropriate rescaling factors.

I hope this helps you understand the difference between the two conventions for the AdS black hole metric.

(1) ads cft - AdS Black holes - Physics Stack Exchange. AdS Black holes.

(2) AdS black hole - Wikipedia. https://en.wikipedia.org/wiki/AdS_black_hole.

(3) [1711.02744] Schwarzschild Black Hole in Anti-De Sitter Space - arXiv.org. https://arxiv.org/abs/1711.02744.