Losing Forward Momentum Holographically

TL;DR

This paper introduces a numerical framework combining a Bondi-Sachs characteristic evolution with spectral methods to solve Einstein gravity with a negative cosmological constant in AdS$_4$, enabling a boundary metric source $g_{tt}=-1- obreak\delta\nobreak\cos(kx)$ that breaks translation invariance and drives momentum relaxation in the dual 2+1D CFT. They also formulate relativistic conformal hydrodynamics for the same system and perform a controlled simple experiment to compare gravity and hydrodynamics across regimes. Analytic momentum-relaxation rates are derived in hydrodynamic, memory-function, and small-velocity limits, and numerical results confirm consistency in the hydrodynamic and linear-response regimes, while revealing breakdowns of gradient expansions at large $k$ and large $\delta$. The work shows that gravity and hydrodynamics agree where expected and provides a pathway to explore far-from-equilibrium phenomena such as turbulence and complex boundary conditions in holographic settings, with potential condensed-matter applications.

Abstract

We present a numerical scheme for solving Einstein's Equations in the presence of a negative cosmological constant and an event horizon with planar topology. Our scheme allows for the introduction of a particular metric source at the conformal boundary. Such a spacetime has a dual holographic description in terms of a strongly interacting quantum field theory at nonzero temperature. By introducing a sinusoidal static metric source that breaks translation invariance, we study momentum relaxation in the field theory. In the long wavelength limit, our results are consistent with the fluid-gravity correspondence and relativistic hydrodynamics. In the small amplitude limit, our results are consistent with the memory function prediction for the momentum relaxation rate. Our numerical scheme allows us to study momentum relaxation outside these two limits as well.

Figures (9)

• Figure 1: The variation of ${\mathop{\lim}_{\omega \to 0}} {\operatorname{Im} G(\omega) \over 2 \omega}$ as a function of the lattice wave number. The black curve shows the result obtained from solving (\ref{['Zeq']}) numerically, i.e. linearized gravity. The dotted red line shows the approximate behavior of the Green's function for large values of $k$. The markers $\bullet$, $\Delta$, $\clubsuit$, $\spadesuit$ and $\star$ show the values obtained by solving the full nonlinear gravity equations for $k=\pi/50 ,~ 4\pi/50 ,~ 5\pi/50 ,~ 6\pi/50$ and $20\pi/50$ in the linearized regime ($\delta=0.2$). Note that $k$ is expressed in units where $3/(4\pi T)=1$. The $x$-axis is rescaled so that the red line has slope minus one.
• Figure 2: A plot of $\log \langle T_{tx} \rangle$ as a function of scaled time for different values of $k$, obtained from hydrodynamic simulations. The analytical expression for relaxation time computed in the previous section corresponds to the reference line with slope -1.0. In (b), values of $t_*$ are chosen such that the lines agree at late times.
• Figure 3: A plot of $\log \langle T_{tx} \rangle$ as a function of scaled time for different values of $k$, obtained from gravity simulations. The analytical expression for relaxation time computed in the previous section corresponds to the reference line with slope -1.0. In (b), values of $t_*$ are chosen such that the lines agree at late times. The simulations were run for $25 \times 10^4$ time steps with $\Delta t = 0.05$.
• Figure 4: A plot of the difference in the numerically computed temperature and the exact analytical expression for temperature in (\ref{['Analytic']}) at $t=2000$, 4000 and 10,000. The lattice wavenumber is $k=4\pi/50$. The red curve corresponds to the result obtained from hydrodynamic simulations and the black curve corresponds to gravity simulations. We use the final value of mean temperature computed from gravity and hydrodynamic simulations to compute $T_0$.
• Figure 5: A plot of $\log \langle T_{tx} \rangle$ as a function of scaled time when $k=20 \pi/50$, obtained from hydrodynamic and gravity simulations. In the gravity simulations we choose the size of the time step $\Delta t = 0.002$. The slopes of the reference lines (dashed and dotted lines) are computed using linear response theory.