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Diving into a holographic superconductor
by Sean A. Hartnoll, Gary T. Horowitz, Jorrit Kruthoff, Jorge E. Santos
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Submission summary
Authors (as registered SciPost users):  Sean Hartnoll · Jorrit Kruthoff 
Submission information  

Preprint Link:  scipost_202011_00016v1 (pdf) 
Date accepted:  20210107 
Date submitted:  20201124 19:29 
Submitted by:  Kruthoff, Jorrit 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Charged black holes in antide Sitter space become unstable to forming charged scalar hair at low temperatures $T < T_\text{c}$. This phenomenon is a holographic realization of superconductivity. We look inside the horizon of these holographic superconductors and find intricate dynamical behavior. The spacetime ends at a spacelike Kasner singularity, and there is no Cauchy horizon. Before reaching the singularity, there are several intermediate regimes which we study both analytically and numerically. These include strong Josephson oscillations in the condensate and possible 'Kasner inversions' in which after many efolds of expansion, the EinsteinRosen bridge contracts towards the singularity. Due to the Josephson oscillations, the number of Kasner inversions depends very sensitively on $T$, and diverges at a discrete set of temperatures $\{T_n\}$ that accumulate at $T_c$. Near these $T_n$, the final Kasner exponent exhibits fractallike behavior.
Author comments upon resubmission
We wish to thank the referees for their positive comments about our manuscript. Both referees 2 and 3 ask for further clarification about our analytic approximations. In response, we have added a paragraph to the introduction in which we explain our methodology more explicitly: the terms we drop in each epoch are motivated by the numerical solutions, but the approximation of dropping them can be verified to be selfconsistent. We repeat this statement in footnote 2 on page 8, where the first approximations are introduced. Our argument is certainly not ``circular" since the analytic expressions we obtain explain complicated behavior in terms of a few parameters.
We offer the following further responses to the referees comments:
{\bf Referee 1} Our results are not contained in previous analysis of BKL behavior near spacelike singularities. The papers that the referee mentions analyze gravity coupled to a neutral scalar field which behaves quite differently from the charged scalar that we analyze. We have added a footnote in the introduction about this, as well as a new sentence in the discussion section.
{\bf Referee 2} The logic in section 4.2 very closely follows our reference [9], as we state several times. In that reference some of the approximations are explained more slowly, as is the justification for the name `collapse of EinsteinRosen Bridge'. Nonetheless, the information needed to answer most of the referee's questions is also contained quite explicitly in section 4.2 of the current paper as written. To respond to the referee's questions:
1) If we include the $\delta z$ dependence in the equations, it would only add a small correction to the solutions we find since $z$ changes very little during this period. The functions do however change a lot over this short range, but not due to the explicit $z$ dependence in the equation.
2) At $z=z_0$, $g_{tt}$ is given by solving $c_1^2 \log(g_{tt}) + g_{tt} = 0$. This follows immediately from (17).
3) The statement we are making in the paragraph above equation (18) is that $\Phi$ can be set to a constant in equations (13) and (14). This is not the same as saying that the derivative is small everywhere. The variation of $\Phi$ must be small compared to its initial value. However, we thank the referee for pointing to this paragraph, as the argument it contained was not correct. We have fixed it in the revised version and hopefully also made it clearer.
4) Since $t$ is a spacelike coordinate inside the horizon, the length of a spacelike surface going from one horizon to the other in a twosided black hole depends on $g_{tt}$. If this shrinks rapidly, this length ``collapses".
{\bf Referee 3} Firstly, there was indeed a typo in the definition of $p_\phi$ in equation (36) that we corrected. We thank the referee for bringing this to our attention.
Regarding the referee's concerns about dropping terms. We hope that our new paragraph in the introduction and the new footnote 2 clarifies the methodology. It is not circular. The referee takes as an example the right hand side of equation (7). The terms to be compared are $m^2/z^2$ and $e^{\chi}\Phi^2/f$. Consider for example the collapse of the EinsteinRosen bridge. What happens there (as we show in section 4.2) is that $\Phi$ is constant while $g_{tt} \sim e^{\chi} f$ becomes exponentially small. The second term above then becomes exponentially large and we see that, indeed, the mass term is negligible in this regime. One can do a similar analysis for all the other cases.
In the new footnote 2 we have also clarified explicitly that the approximations can also hold away from $T_c$ when the nonlinear dynamics becomes strong. In particular, at fast Kasner inversions.
We hope the manuscript is now suitable for publication.
Published as SciPost Phys. 10, 009 (2021)