Master field treatment of metric perturbations sourced by the trailing string

TL;DR

This work presents a fully diffeomorphism-invariant master-field formalism for linearized Einstein equations with a trailing-string source in $AdS_5$-Schwarzschild, enabling decoupled and separable equations for tensor, vector, and scalar sectors. By constructing master fields $igl( extPhi_T^{ m even/odd}, extPhi_V^{ m even/odd}, extPhi_Sigr)$, the authors derive explicit master equations and connect bulk perturbations to the holographic boundary stress tensor, including the identification of conserved and drag contributions to $iglra T_{mn}igrra$. They analyze both small-$k$ and large-$k$ regimes, obtaining analytic expressions for the Fourier components of the boundary stress and the near-field energy density, and they compute the drag force on the moving quark, confirming consistency with known results. The approach provides a versatile framework for extending holographic stress-tensor calculations to more general backgrounds and nonuniform quark motion, with potential numerical-analytic hybrids to cover all scales. The findings illuminate how energy and momentum are distributed in the finite-temperature plasma, including a notable near-field front enhancement at high velocities.

Abstract

We present decoupled, separable forms of the linearized Einstein equations sourced by a string trailing behind an external quark moving through a thermal state of N=4 super-Yang-Mills theory. We solve these equations in the approximations of large and small wave-numbers.

Figures (3)

• Figure 1: Comparisons between the large $K$ analytic approximations found in \ref{['QSHighK']} (plots a,b), (\ref{['QVHighK']}) (plots c,d), and (\ref{['QTHighK']}) (plots e,f) and corresponding numerical solutions of the linearized Einstein equations in the form studied in Friess:2006fk. The unbroken colored lines come from numerical solutions at the values of $K=k/\pi T$ indicated, while the dashed lines come from evaluations of (\ref{['QSHighK']}), suitably transformed into evaluations of $Q_E$, $Q_D$, and $Q_A$ as described in the text. The values of $K$ used in plots c,e are the same as in a, and those in d,f are the same as in b. For values of $K$ larger than the largest one shown in each plot, the numerical evaluations are almost indistinguishable from the $K\to\infty$ approximation. The Mach angle is indicated, but there is no reason to expect structure near this angle because large $K$ is the opposite limit of where hydrodynamics is expected to apply.
• Figure 2: (a) A contour plot of the near-field energy density as computed from (\ref{['T00Position']}) at time $t=0$. The red regions in the left side of the plot represent negative $\langle T_{00} \rangle$---more precisely, an energy deficit compared to the average energy density of the thermal bath. The deficit arises because the ${\cal O}(T^2)$ dominates over the ${\cal O}(1)$ term, which leads us to question the accuracy of the asymptotics in these regions. (b) The ${\cal O}(T^2)$ correction $\epsilon^{(2)}$ as a function of $\Delta\phi = \pi - \arctan x_\perp/x_1$ for several different velocities $v$. $\epsilon^{(2)} \propto 1/x^2$, and a different value of $x = \sqrt{x_1^2+x_\perp^2}$ was chosen for each velocity in order to make the qualitative effects easy to discern.
• Figure 3: The near-field Poynting vector to the approximation shown in (\ref{['PoyntingPosition']}) in the plane $\pi T x_3 = 0.02$. The arrows show the direction of the projection of $\vec{S}$ into the plane, and the color corresponds to the magnitude of of $\vec{S}$, where red means large and blue means small.