AdS Black Holes with a Bouncing Interior

We thank both of the referees for taking the time to look at the paper and their positive comments.

Referee 2 has recommended publication in its current form.

Referee 1 has raised some minor points which we have addressed as described below.

We have also corrected the two typos found by Referee 1.

1. The referee correctly recalled there are two possible quantizations of the scalar field when m2=-2. We have therefore added “, using standard quantization” to the sentence below equation (3). As the referee has noted the falloff is given explicitly in equation (4).

2. We have pasted below Mathematica code that shows that (16) is correct as written in the paper. The steps are literally those given in the paper and outlined by the referee, so we do not believe that a clarification in the text is necessary. Most likely the referee has made a typo writing down the equation.

(* this is equation (12) *)

In[1]:= eq=4(2 H[\[Phi]]-v[\[Phi]])v[\[Phi]]v''[\[Phi]]/v'[\[Phi]]== (8 H[\[Phi]]^2 v[\[Phi]]- 2H[\[Phi]](3 v[\[Phi]]^2+4(3+v'[\[Phi]]))+v[\[Phi]](12 + v[\[Phi]]^2 + 4D[2 H[\[Phi]]-v[\[Phi]],\[Phi]]));

In[2]:= vv[\[Phi]_] = vo + \[Delta] dv[\[Phi]];

(* this is the linearized equation *)

In[3]:= eq1=Series[eq /. v -> vv,{\[Delta],0,0}] // Normal// FullSimplify

Out[3]= 6 (4+vo^2) H[\[Phi]]==vo (12+vo^2+8 H[\[Phi]]^2+8 (H^\[Prime])[\[Phi]]+(4 (vo-2 H[\[Phi]]) (dv^\[Prime]\[Prime])[\[Phi]])/(dv^\[Prime])[\[Phi]])

(* this is the expression written in (16), directly for the derivative*)

In[4]:= dvp[\[Phi]_] = a Exp[- (3 + vo^2/4)\[Phi]/vo](V[\[Phi]]- 2/vo V'[\[Phi]]);

(* here we verify that the expression (16) solves the linearized equation *)

In[5]:= eq1 /. dv'[\[Phi]] -> dvp[\[Phi]] /. dv''[\[Phi]] -> dvp'[\[Phi]] /. V''[\[Phi]] -> D[H[\[Phi]] V[\[Phi]],\[Phi]] /. V'[\[Phi]] -> H[\[Phi]] V[\[Phi]]// FullSimplify

Out[5]= True

3. The referee is concerned that Fig 3 is insufficient to demonstrate that |H| >> |v| over the extended bounce regime because it only covers a narrow window of rho. However, if we look at figure 4, which is shown for the same bounce, we see that Kasner behavior is recovered on either side over an even narrower window of rho. Therefore Fig 3 and Fig 4 combined indeed show that |H| >> |v| holds over a regime that extends from within one Kasner regime to within the next. The overlap with the Kasner regimes is the crucial point. There is not much to be gained from extending the plot range of Fig 3 as suggested by the referee. However, we do understand the referee’s concern and have therefore added “, see Fig. 4 below,” in the sentence underneath equation (17).

4. As we had already stated in the text, both statements that we made at end of the paragraph below (38) follow from equation (24). However, we do agree that the explanation here was a little compressed. We have added the clarifying comment “(from (24), 1/|k_n Delta v_n| is the width of the bounce)” and “(also from (24), phi-dot changes sign between rho >> rho_n and rho << rho_n when |k_n Delta v_n| is large)”.