Abstract
On the holographic complexity dual to the bulk action, we investigate the action growth for a shock wave geometry in a massive gravity theory within the Wheeler-De Witt (WDW) patch at the late time limit. For a global shock wave, the graviton mass does not affect the action growth in the bulk, i.e. the complexity on the boundary, showing that the action growth (complexity) is the same for both the Einstein gravity and the massive gravity. Nevertheless, for a local shock wave that depends on transverse coordinates, the action growth (complexity) is proportional to the butterfly velocity for the two gravity theories, but the butterfly velocity of the massive gravity theory is smaller than that of the Einstein gravity theory, indicating that the action growth (complexity) of the massive gravity is depressed by the graviton mass. In addition, we extend the black hole thermodynamics of the massive gravity and obtain the right Smarr formula.
Highlights
The holographic principle shows [1] that the bulk dynamical evolution can be coded in the boundary field theory without gravity
For a local shock wave that depends on transverse coordinates, the action growth caused by the boundary disturbance is proportional to the butterfly velocity for the two gravity theories, but the butterfly velocity of the massive gravity theory is smaller than that of the Einstein gravity theory, indicating that the action growth of the massive gravity is depressed by the graviton mass
We find that the action growth of the massive gravity in the case of the global shock wave is equal to that of the Einstein gravity because the effect of the global shock wave shifts the Kruskal coordinate v only a transverse-coordinate-independent quantity which does not depend on the graviton mass
Summary
The holographic principle shows [1] that the bulk dynamical evolution can be coded in the boundary field theory without gravity. In the recent studies [10,11,12,13,14,15] on the holographic complexity dual to the bulk action in different gravity theories, the spacetime with a shock wave reflects more characteristics of the boundary complexity, such as the criterion of existence of firewalls. We shall investigated the action growth in the bulk, i.e. the complexity on the boundary, for the shock wave geometry in the massive gravity [18] which contains only a massive mode.. We find that the action growth (complexity) of the massive gravity in the case of the global shock wave is equal to that of the Einstein gravity because the effect of the global shock wave shifts the Kruskal coordinate v only a transverse-coordinate-independent quantity which does not depend on the graviton mass.
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