Complexity growth of massive black hole with a probe string

We consider $(n+2)$-dimensional action of massive gravityVegh:2013sk ; Cai:2014znn ,

where $R$ is the scalar curvature, $l$ is the AdS radius and $f$ is a fixed symmetric tensor. In addition, $c_{i}$ is constant and $\mathscr{U}_{i}$ is the symmetric polynomial of the eigenvalue of the $(n+2)\times(n+2)$ matrix $\mathscr{K}_{\nu}^{\mu}=\sqrt{g^{\mu\alpha}f_{\alpha\nu}}$ which can be written as follows

where $[\mathscr{K}]=\mathscr{K}_{\mu}^{\mu}$ and the square root in $\mathscr{K}$ can be interpreted as $(\sqrt{\mathscr{K}})^{\mu}_{~{}\nu}(\sqrt{\mathscr{K}})^{\nu}_{~{}\lambda}=\mathscr{K}^{\mu}_{~{}\lambda}$. As known that the presence of impurities in realistic materials leads that the momentum is not conserved, so that the system gives finite DC conductivity. Modeling systems via translationally invariant quantum field theories always comes across problems unless the effects of momentum dissipation is incorporated. In holographic framework, several models have been proposed to involve momentum dissipation, which brings in finite DC conductivity. Massive gravity is a completive candidate in which the momentum dissipation is involved, and it is an effective bulk theory that does not conserve momentum without borrowing additional fields. In the action (1), the last terms represent massive potentials associated with the graviton mass which breaks the diffeomorphism invariance in the bulk, which produces momentum relaxation in the dual boundary theory.

Variating the action, we can obtain the equations of motion

where

The static black hole solution of the above action yields

where $h_{ij}dx^{i}dx^{j}$ is a line element of Einstein space with constant curvature $n(n-1)k$ and $k=1,0,-1$ corresponds to a spherical, Ricci flat, and hyperbolic horizon for black hole. We follow the ansatz in Cai:2014znn and set the reference metric $f_{\mu\nu}=diag(0,0,c_{0}^{2}h_{ij})$. Then $\mathscr{U}_{i}$ can be computed as

Putting them back to the Einstein equation, we have the metric function $f(r)$

where the integral constant $m_{0}$ is the black hole mass parameter.

Following the approach in Nagasaki:2017kqe ; Nagasaki:2018csh , we consider a Wilson line operator in the spacetime by inserting a fundamental string in the massive gravity. This corresponds to a test particle moving on the boundary gauge theory, which is described by a non-local operator. Inserting the Wilson loop is described by adding a Nambu-Goto (NG) term, so the total action consists of the Einstein-Hilbert term, the Nambu-Goto term and the boundary term. Since the contribution of Einstein-Hilbert action and boundary term to the complexity growth in massive black hole has been studied in Pan:2016ecg ; Guo:2017rul , here we shall investigate the effect of the Nambu-Goto action.

Setup To this end, we assume that a probe string moves in a subspace with different topologies. Then the induced metric was $h_{ij}dx^{i}dx^{j}=d\phi^{2}$. We take the worldsheet parameter as

where $v$ denotes the velocity of a string relative to the black hole. For simplicity, we will set $l$=1, then the induced metric is

The NG action is achieved by integrating over the WDW patch,

where the horizon $r_{h}$ is determined by $f(r_{h})=0$, and $T_{s}$ is the tension of fundamental string.

Varying the above ‘action’, we obtain the equation of motion for $\xi$

from which we could define the constant $c_{\xi}$ as

Subsequently, $\xi^{\prime}$ can be solved by

This expression must give real values, so the denominator and numerator in the square root should coincidentally be positive,negative or zero. It is noted that we shall set $c_{0}=c_{1}=c_{2}=c_{3}=c_{4}=1$ without loss of generality. Then regarding the numerator, $f(\sigma)-v^{2}\sigma^{2}$, as a function of $\sigma$, we find it is a monotonically increasing function and has a negative value for $\sigma=0$. Therefore, there is only one solution $\sigma=\sigma_{H}$ to $f(\sigma)-v^{2}\sigma^{2}=0$. Subsequently, the constant $c_{\xi}$ is determined by $\sigma_{H}^{2}f(\sigma_{H})-c_{\xi}^{2}=0$ as

Inserting the expressions (9),(13) and (14) into (10), we could rewrite the NG action as

In the following we shall study the effect of horizon curvature and the graviton mass on the NG action in different dimensions.

First we set the graviton mass as $m=0$ and study the effect of BTZ black hole mass on the NG action. The explicit behaviors with different parameters are plotted in Fig.1. In the left plot we show the relation between action growth and string velocity with different black hole masses. We see that the complexity growth takes the maximum value when the string is stationary, and as the string moves faster, the action growth becomes smaller. Moreover, the peak value of action growth is larger for larger black hole mass. These features are similar as the observe for static black holes Nagasaki:2017kqe and Nagasaki:2018csh . Additionally, the action growth will vanish as the velocity approaches to light speed, which was also observed in the rotating BTZ black holes Nagasaki:2018csh even though the peak is shifted.

The right plot of Fig.1 shows the relation between action growth and black hole mass with different string velocities. In this case, the faster string gives the smaller action growth, which is consistent with that in the left plot. It behaves as a monotonically increasing function of the black hole mass, which is similar to the observes in rotating BTZ black hole Nagasaki:2017kqe . A possible interpretation is that a lager system may have larger information and so it is more complex.

The above numerical results can be analytically confirmed. With tricks, the growth of NG action (16) could be analytically integrated as

which depends on the black hole mass and string velocity in a simple way. It indicates that with fixed $m_{0}$, the action growth is maximum when the string is stationary with $v=0$ and the maximal value is $\sqrt{m_{0}}$. As $\lvert v\rvert$ increases, it decreases and could vanish when the speed approaches to the light speed with $\lvert v\rvert=1$. Moreover, the action growth could increase monotonously as $m_{0}$ when $v$ is fixed.

Then we turn on the graviton mass and study its effect on the NG action growth. So we fix the black hole mass as $m_{0}=2$. The results are shown in Fig.2.

In left plot of Fig.2, we show the relation between action growth and string velocity for massive black hole. As usual, the maximal complexity growth appears when the string velocity is equal to zero for different $m$. The interesting point is that the maximal value in massive BTZ black hole is smaller than that in BTZ black hole, and as the graviton mass increases, the maximal value becomes smaller. For all chosen $m$, as the string moves faster, the action growth becomes smaller. Moreover, comparing to the BTZ case, the action growth will not vanish when the string moves with light speed. In addition, It is also obvious that the effect of $m$ on the action growth depends on the velocity.

Then in the right plot of Fig.2, we present the relation between action growth and the graviton mass. We focus on the vicinity of the light speed. In all cases, as $m$ increases, the action growth increases into a peak and then decreases, i.e., it is not a monotonically increasing function. Moreover, the velocity dependence is significant for small graviton mass, but it seems to be slight for large $m$, which is also explicit in the inserted plot. Thus, on the vicinity of the light speed, the graviton mass suppresses the velocity dependence of the action growth, which is very different from the effect of black hole mass that enhances it in all cases as studied in Nagasaki:2017kqe ; Nagasaki:2018csh .

It is noted that in this case, it is difficult to integrate the action (16) analytically. However, we could estimate the behavior in some limit regimes to further check the numerical analysis. In the limit $v\rightarrow 0$, we can integrate (16) as

which is a monotone decreasing function of positive $m$. So the value of the action growth at $v=0$ is smaller as the graviton mass increases, which is consistent with that shown in the left plot of Fig.2.

In the speed light limit with $v\rightarrow 1$, a semi-analytic integration is obtained as

which is shown in Fig.3. It shows that in this limit, as $m$ increases, the action growth increases into a peak and then decreases, i.e., it is not a monotonic function, which matches the numerical study shown in the right plot of Fig.2.

We first study the effect of horizon curvature. So we focus on the AdS black hole by setting the graviton mass as $m=0$.

The velocity dependence and black hole mass dependence in four dimensional case are shown in Fig.4. In the left plot with the black hole mass $m_{0}=2$, the action growth for $k=1$ reproduces the curve with black hole mass $M=1$ in figure 1 of Nagasaki:2018csh . In all cases, the maximal value appears when the string does not move, and as the velocity increases, the action growth decreases. It is obvious that the maximal values in the case with $k=-1$ is the largest while it is larger in $k=0$ than in $k=1$ case. This phenomena can be understood in an analytical way. In the limit $v\to 0$, the integration (15) with four dimension $(n=2)$ is

Then recalling the event horizon satisfying $f(r_{h})=0$, which gives us

It is obvious that for AdS case with $m=0$ and $m_{0}=2$, $k$ decreases as $r_{h}$ increases, which indicates that $k=-1$ corresponds to largest $r_{h}$, so does the largest action growth while $k=1$ corresponds to the smallest action growth. It is easy to see that the conclusion is also hold in massive gravity with $m\neq 0$, which will match the numerical results as we will see soon. Noted that similar analysis could also be done in higher dimension cases.

But the action growth decays fastest when $k=-1$. As the string velocity increases, there exists a intersection and the effect of $k$ becomes opposite to that in small velocity. Another notable feature is that for $k=0$ and $k=-1$, the action growth with light speed goes to zero, namely the contribution from NG action disappears such that the effect of string is zero in these cases. This phenomena is in contrast to the nonzero value for $k=1$. As we increase the dimension(see Fig.5 for 5D,6D and 10D cases.), the action growth from left to right becomes gentler and gentler as usual, and the intersection occurs at larger velocity. Moreover, similar as $k=1$, the action growth near light speed for $k=0$ also increases to be nonzero for higher dimensions, while for $k=-1$, it keeps zero.

In the right plot of Fig.4, we show how the action growth depends on the black hole mass near the light speed. Again the blue curve for $k=1$ reproduces the curve with the same velocity in figure 1 of Nagasaki:2018csh , and there exists a peak at small black hole mass. In a contrast, it behaves as a monotonically increasing function of the black hole mass for $k=-1$ and $k=0$. Moreover, it shows that in the vicinity of light speed, the action growth for $k=1$ is the largest, which is consistent with that in the left plot.

As the dimension increases (see Fig.6 for $5D,6D$ and $10D$ cases), the peak for $k=1$ could disappear and the black hole mass dependence becomes monotonically increasing function, which reproduces the outcome of Nagasaki:2018csh . Moreover, it also shows that as the dimension increases, the gap between $k=0$ and $k=-1$ is wider while the gap between $k=0$ and $k=-1$ is narrower. Unfortunately, it is difficult to find analytical insight of the features in the limit $v\to 1$ even in four dimensional case.

Then, we turn on $m$ to study the effect of the graviton mass and horizon curvature.

We first study in detail the effect of graviton mass on the string velocity dependence as well as graviton mass dependence in the vicinity of light speed. We show the results for four dimensional case in Fig.7 and Fig.8, respectively. In all plots of Fig.7, larger $m$ gives the lower maximal values of action growth. This feature could be expected from (20) in $v\to 0$ limit if we reform (21) as $m^{2}=(m_{0}/r_{h}-r^{2}_{h}-k)/(r_{h}/2+1)$, which indicates that with fixed $m_{0}$ and $k$, larger $m$ corresponds to smaller $r_{h}$. There exists a intersection where the effect of $m$ on the string velocity dependence will change. This observe is consistent with that in three dimensional case (see Fig.2), even though here the curves are more gentler.

Fig.8 shows the $m$ dependence of action growth in the vicinity of light speed. In all cases, the smaller string velocity corresponds to larger action growth when the graviton mass is small. As the increase of $m$, it approaches to a maximal value and then decreases. The $m$ dependence is slightly affected by the string velocity for large enough $m$. Though the behavior is similar to that for 3D case studied in previous section, we see that the location of peak tends to smaller $m$.

Then we fix $m=0.5$ and study the effect of horizon curvature. The results in four dimensional massive gravity are shown in Fig.9. Comparing the left plots in Fig.9 and Fig.4, we see that in massive gravity, the effect of horizon curvature on the velocity dependent action growth is similar to that in Einstein gravity. As in the AdS case, the maximal value for stationary string with $k=-1$ is the largest, the reason of which has been stated below equation (21). There also exists a intersection. As the dimension increases (see Fig.10), the velocity dependent action growth is more gentler. One new feature in massive gravity is that in all cases, the action growth in light speed limit becomes non-vanished. Comparing with that in AdS case, the intersection in massive case for different curvatures could separate in lower dimension.

In the right plot of Fig.9, we show the effect of $k$ on the graviton mass dependence, which is more significant for small $m$ than for large $m$. It also shows a peak at certain critical $m_{c}$ as in three dimensional case. Here the location value of $m$ for the peak decreases as $k$ increases. Furthermore, in Fig.11 we show the graviton mass dependence for higher dimensions. As the dimension increases, $m$ for the location of peak decreases. Then for $k=1$, the peak shows at $m=0$, meaning the action grow monotonically decreases as $m$ increases(see the middle plot for six dimension), which is also explicit in the ten dimensional case. Further increasing the dimension, the action grow could also become monotonically decreasing function of $m$ for the massive black hole with $k=-1$ and $k=0$. This is a new phenomena.