March 6, 2005

Small Blackholes

I recently wrote about Greg Moore and collaborators’ work computing the entropy of a certain class of $N=2$ blackholes which arise in the compactification of Type IIA on a Calabi-Yau.

Alex Maloney visited us this past week, and gave us a beautiful talk on his work on the geometry of the blackholes in question. The situation is really quite striking. Classically, these blackholes have vanishing horizon area. They have a null singularity, a pathology which, says Alex, is characterized as “naked with the lights off” in the GR literature.

Umh, whatever

One-loop corrections in the string theory induce certain curvature-squared corrections, which can be written as integrals over half of $N=2$ superspace (a holomorphic function of the Kähler moduli times square of the self-dual part of the Weyl curvature, plus Hermitian conjugate).

These corrections have a dramatic effect. They make the singularity time-like and cloak it behind a horizon. In the corrected geometry, one recovers a formula in which the blackhole entropy is proportional to the area. But, instead of the Bekenstein-Hawking formula, $S=\frac{1}{4}A$, you find $S=\frac{1}{2}A$!

Now, at first you might be suspicious: if the one-loop corrections radically change the answer, how about two-loops …? But that’s the wrong way to think about it. For these blackholes, the tree-level answers (for the area and entropy) happen to *vanish*. The one-loop answer is the *leading* nonvanishing contribution. Higher corrections are still controllably small compared to the leading term.

In a little more detail, consider the (tree plus) one-loop Topological String free energy, which I called ${F}_{\text{pert}}$ in the previous post,

(1)$F\left({X}^{A},{T}^{2}\right)=\frac{1}{6}{D}_{abc}\frac{{X}^{a}{X}^{b}{X}^{c}}{{X}^{0}}+\frac{1}{2}{A}_{ab}{X}^{a}{X}^{b}+\frac{1}{24}{c}_{2a}\frac{{X}^{a}}{{X}^{0}}\frac{{T}^{2}}{64}$

The (generalized) attractor equations are

(2)${C}^{2}{T}^{2}=256,\phantom{\rule{1em}{0ex}}\mathrm{Re}\left(C{X}^{A}\right)={p}^{A},\phantom{\rule{1em}{0ex}}\mathrm{Re}\left(C{F}_{A}\right)={q}_{A}$

Let’s solve them in this one-loop approximation, assuming the electric graviphoton charge, ${p}^{0}=0$. The computation is fairly easy;

(3)$\begin{array}{rl}C{X}^{0}& =i\sqrt{\frac{D+{c}_{2a}{p}^{a}}{6{\stackrel{^}{q}}_{0}}}\\ C{X}^{a}& ={p}^{a}+i\sqrt{\frac{D+{c}_{2d}{p}^{d}}{6{\stackrel{^}{q}}_{0}}}{D}^{ab}\left({q}_{b}-{A}_{bc}{p}^{c}\right)\end{array}$

where

(4)$\begin{array}{rl}{\stackrel{^}{q}}_{0}& ={q}_{0}+\frac{1}{2}{D}^{ab}\left({q}_{a}-{A}_{ac}{p}^{c}\right)\left({q}_{b}-{A}_{bd}{p}^{d}\right)\\ {D}_{ab}& ={D}_{abc}{p}^{c},\phantom{\rule{1em}{0ex}}{D}^{ab}{D}_{bc}={\delta }_{c}^{a},\phantom{\rule{1em}{0ex}}D={D}_{abc}{p}^{a}{p}^{b}{p}^{c}\end{array}$

Plugging into the formula for the horizon area,

(5)$\begin{array}{rl}A& =\pi i\mid C{\mid }^{2}\left({X}^{A}{\overline{F}}_{A}-{\overline{X}}^{A}{F}_{A}\right)\\ & =2\pi \sqrt{\frac{{\stackrel{^}{q}}_{0}}{6\left(D+{c}_{2a}{p}^{a}\right)}}\left(D+\frac{1}{2}{c}_{2a}{p}^{a}\right)\end{array}$

and the (Wald-corrected) formula for the entropy,

(6)$\begin{array}{rl}S& =\frac{1}{4}A+128\pi i\left(\frac{\partial \overline{F}}{\partial {\overline{T}}^{2}}-\frac{\partial F}{\partial {T}^{2}}\right)\\ & =\frac{\pi }{2}\sqrt{{\stackrel{^}{q}}_{0}\left(D+{c}_{2a}{p}^{a}\right)/6}\end{array}$

Now, in “conventional” four-(or more)-charged blackholes, such that $D={D}_{abc}{p}^{a}{p}^{b}{p}^{c}$ and ${\stackrel{^}{q}}_{0}$ are both large, this just gives

(7)$A=2\pi \sqrt{{\stackrel{^}{q}}_{0}D/6}$

and

(8)$S=\frac{1}{4}A\phantom{\rule{thinmathspace}{0ex}}\left(1+\frac{1}{2}{c}_{2a}{p}^{a}/D\right)$

the Bekenstein-Hawking result plus a small one-loop correction. However, for these two-charged blackholes, $D=0$, and we get

(9)$A=\pi \sqrt{{\stackrel{^}{q}}_{0}{c}_{2a}{p}^{a}/6}$

and $S=\frac{1}{2}A$.

It’s really quite nice that we are seeing essentially stringy effects rescuing cosmic censorship, while generating substantial corrections to the Bekenstein-Hawking formula.

Posted by distler at March 6, 2005 12:43 AM

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