Time-dependent $N{AdS}_2$ holography with applications

We develop a method for obtaining exact time-dependent solutions in Jackiw-Teitelboim gravity coupled to nonconformal matter and study consequences for $N{AdS}_2$ holography. We study holographic quenches in which we find that the black hole mass increases. A semiholographic model composed of an infrared $N{AdS}_2$ holographic sector representing the mutual strong interactions of trapped impurities confined at a spatial point is proposed. The holographic sector couples to the position of a displaced impurity acting as a self-consistent boundary source. This effective $0+1$-dimensional description has a total conserved energy. Irrespective of the initial velocity of the particle, the black hole mass initially increases, but after the horizon runs away to infinity in the physical patch, the mass vanishes in the long run. The total energy is completely transferred to the kinetic energy or the self-consistent confining potential energy of the impurity. For initial velocities below a critical value determined by the mutual coupling, the black hole mass changes sign in finite time. Above this critical velocity, the initial condition of the particle can be retrieved from the $SL(2,R)$ invariant exponent that governs the exponential growth of the bulk gravitational $SL(2,R)$ charges at late time.

Figure 1

Sources and responses: As expected, the responses die down at late time once the sources vanish. (a) Plot of $J(u)$. (b) Plot of $O(u)$. (c) Plot of $J_{th}(\tau (u))$. (d) Plot of $O_{th}(\tau (u))$.

Figure 2

The time dependence of the black hole mass and the $SL(2,R)$ charges. (a) Plot of $H_{\mathrm{sch}}=-1/2M(u)$ This plot is very similar to the case of quenches in higher dimensional holographic systems where $M(u)$ grows but not monotonically. (b) Plot of the $SL(2,R)$ charges as a function of time: Note that the final $SL(2,R)$ frame is different since $Q^{\pm }$ are non-vanishing.

Figure 3

The plot of $\tau (u)$, which maps the time of the physical state to that of the fixed temperature state, and its derivatives. The generic result is that $\tau (u)$ saturates to a constant and its derivatives vanish.

Figure 4

The dilaton grows on the black hole event horizon monotonically within numerical accuracy interpolating between the thermal limits at early and late times.

Figure 5

The plots for energies and $SL(2,R)$ charges for $v_0=2.0$ and $\lambda =0.4$. (a) Plot of energies as function of time: Note that the total energy $H_{\mathrm{tot}}=H_{\mathrm{kin}}+H_{\mathrm{int}}+H_{\text{Sch}}$ is conserved after the initial impulse and is finally transferred to Hkin, the particle kinetic energy. The mass of the black hole ${M=-2H}_{\mathrm{sch}}$ remains positive and decays to zero eventually. (b) Plot of the $SL(2,R)$ charges as a function of time in the first phase. Although all of them diverge at late time, their Casimir (and thus the black hole mass) vanishes.

Figure 6

The plots of sources and responses for $v_0=2$ and $\lambda =0.4$. (a) Plot of $X(u)$ and $X_{th}(\tau (u))$ as functions of time. Note $X(u)$ eventually reaches linear growth regime implying that the particle reaches a terminal velocity. $X_{th}(\tau (u))$, the source conformally mapped to a black hole of unit mass, saturates to a constant. (b) Plot of $O(u)$ as a function of time. The eventual rapid decay of $O(u)$ ensures that $H_{\mathrm{int}}\alpha X(u)O(u)$vanishes at long time. (c) Plot of $O_{th}(\tau (u))$: Note as $X_{th}(\tau (u))$ saturates, $O_{th}(\tau (u))$ grows with time. However, the rapid decay of 0 ensures that $H_{\mathrm{int}}\alpha {\tau }^{\prime }(u)X_{th}(\tau (u))O_{th}(\tau (u))$ also decays in this frame.

Figure 7

Here $v_0=2.0$ and $\lambda =0.4$. (a) Plots of $\tau ,{\tau }^{\prime },{\tau }^{\prime \prime }$ and ${\tau }^{\prime \prime \prime }$ vs $u$. Due to our choice of initial $SL(2,R)$ frame, ${\tau }^{\prime }=1$ (so $\tau$ is linear in $u$) for $u<0$ (before the kick), while ${\tau }^{\prime \prime }={\tau }^{\prime \prime \prime }=0$. Note $\tau$ saturates at late time, while all its derivatives vanish. (b) ${{\tau }^{\prime }}^2/{\tau }^{\prime \prime }$ as a function of time. Clearly ${{\tau }^{\prime }}^2$ decays faster than ${\tau }^{\prime \prime }$ with time. Also ${{\tau }^{\prime }}^2/{\tau }^{\prime \prime }$ is always negative since ${\tau }^{\prime \prime }$ is so. (c) $G(u)$ as a function of time. The dilaton singularity at $r=\infty$ maps to $z=G(u)$ (see text). This implies that the singularity is far from the Poincare horizon $z=\infty$ except at initial time.

Figure 8

$a$, the exponent for late-time growth of $SL(2,R)$ charges as a function of $v_0$ for $\lambda =0.4$ and fixed unit initial mass of the black hole. Note that $a$ grows monotonically with $v_0$.

Figure 9

The plots for energies and $SL(2,R)$ charges for $v_0=0.9$ and $\lambda =0.4$. (a) Plot of energies as function of time: Note that the total energy $H_{\text{tot}}$ is conserved after initial kick and is finally transferred to $H_{\mathrm{int}}$, the confining potential energy. The mass of the black hole $M=-2H_{\mathrm{sch}}$ becomes negative after finite time and then eventually vanishes at long time. (b) Plot of $SL(2,R)$ charges as a function of time. Note that they saturate to finite values.

Figure 10

The plots of sources and responses for $v_0=1.4$ and $\lambda =0.4$. (a) Plot of $X(u)$ and $X_{th}(\tau (u))$: Both of them saturate to constant values at late time. The particle stops at a finite distance from the origin. (b) Plot of $O(u)$: $O(u)$ saturates to a constant value at late time so that $H_{\mathrm{int}}\alpha X(u)O(u)$ also saturates to a constant. (c) Plot of $O_{th}(\tau (u))$: Note $O_{th}(\tau (u))$ diverges at late time since $X_{th}(\tau (u))$ saturates to a constant value. However, $H_{\mathrm{int}}\alpha {\tau }^{\prime }(u)X_{th}(\tau (u))O_{th}(\tau (u))$ also saturates to a constant in this frame because the decay of ${\tau }^{\prime }$ compensates for the growth of $O_{th}(\tau (u))$.

Figure 11

The four energies in the case of $v_0=1.1$ and $\lambda =0.5$. The double crossing of $H_{\mathrm{sch}}$ about zero is hard to discern here, so one can refer to the inset plot in Fig. 12. The final transfer of energy goes to the kinetic energy of the particle.

Figure 12

Different phases for $\lambda =0.5$ as can be seen from the behavior of $M(u)=-2H_{\mathrm{sch}}(u)$ for various choices of initial velocities. The inset plot shows that multiple crossings of zero is possible for $M(u)$ when the total conserved energy is positive.

Figure 13

The behavior of the horizon $r(u)$ given by (140) for $\lambda =0.5$, and different initial velocities of the particle, namely 2.0 and 0.8. In the former, the system is in the first phase and in the latter, it is in the other phase. Note that the initial horizons coincide because we start with the same black hole mass. The horizon runs away to infinity irrespective of whether the black hole mass becomes negative or not. Also it happens roughly when the black hole mass stops increasing monotonically as evident from comparison with Fig. 12. After this the horizon emerges on the other (unphysical) sheet (see text).