Abstract
We consider the Complexity=Action (CA) proposal in Einstein gravity and investigate new counterterms which are able to remove all the UV divergences of holographic complexity. We first show that the two different methods for regularizing the gravitational on-shell action proposed in ref. [1] are completely equivalent, provided that one considers the Gibbons-Hawking-York term as well as new counterterms inspired from holographic renormalization on timelike boundaries of the WDW patch. Next, we introduce new counterterms on the null boundaries of the WDW patch for four and five dimensional asymptotically AdS spacetimes which are able to remove all the UV divergences of the on-shell action. Moreover, they are covariant and do not change the equations of motion. At the end, by applying the null counterterms, we calculate the holographic complexity of an AdS-Schwarzschild black hole as well as the complexity of formation. We show that the null counterterms do not change the complexity of formation.
Highlights
We first show that the two different methods for regularizing the gravitational on-shell action proposed in ref. [1] are completely equivalent, provided that one considers the Gibbons-Hawking-York term as well as new counterterms inspired from holographic renormalization on timelike boundaries of the Wheeler-De Witt (WDW) patch
In the second regularization shown in the right panel of figure 1, the spacetime is cut at r = δ and the null boundaries of the WDW patch start from r = δ
One might write some types of counterterms on the timelike boundaries the of WDW patch, which are similar to those applied in holographic renormalization [47,48,49]
Summary
We will study the UV divergences of the holographic complexity of an eternal two-sided AdS-Schwarzschild black hole in Einstein gravity by applying the CA proposal for two different regularizations shown in figure 1. Inspired by holographic renormalization, one might consider the following counterterms on the timelike boundaries of the WDW patch in the left panel of figure 1. From eq (3.11), (3.13) and eq (3.23), one can conclude that the divergent parts of the total action in the two regularizations are equal to each other, Ireg.1|div = Ireg.2|div In both regularizations the structure and coefficients of the UV divergences of holographic complexity are exactly the same, provided that one takes into account all surface terms including the counterterms inspired by holographic renormalization, i.e. eq (3.15). We used them on a small time interval of the AdS boundary, which is one of the boundaries of the WDW patch It seems that in the calculation of the on-shell action in any region of spacetime, it is necessary to consider the role of counterterms on all boundaries of that region.. It seems that in the calculation of the on-shell action in any region of spacetime, it is necessary to consider the role of counterterms on all boundaries of that region. Since, the calculation of holographic complexity in the second regularization is easier, in the rest of the paper, we apply it
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