[Paper Review] Solution-generating transformations in duality-invariant theories and the fluid/gravity correspondence

Joel Berkeley|arXiv (Cornell University)|Nov 3, 2015

Black Holes and Theoretical Physics134 references3 citations

TL;DR

This paper develops solution-generating transformations in duality-invariant theories, including double field theory (DFT) and $SL(5)$-invariant M-theory, and applies them to the fluid/gravity correspondence. It shows that wave solutions in DFT embed both F1 strings and pp-waves, and identifies energy scaling invariance in fluids dual to Rindler spacetimes with radial isometries, revealing a universal symmetry in hydrodynamic expansions.

ABSTRACT

We explore dualities and solution-generating transformations in various contexts. Our focus is on the T-duality invariant form of supergravity known as double field theory, the $SL(5)$-invariant M-theory extended geometry, and metrics dual under the fluid/gravity correspondence to an incompressible Navier-Stokes fluid. In double field theory (DFT), a wave solution is shown to embed both the F1 string and the pp-wave. For the former, the Goldstone mode dynamics reproduce the duality symmetric string introduced by Tseytlin. We consider solution-generating techniques in DFT in the presence of an isometry, firstly via Buscher-like transformations in the DFT string $σ$-model, and secondly via the DFT equations of motion. In the $SL(5)$-invariant geometry, we provide a chain rule derivation of the covariant equations of motion, and present a wave solution embedding the M2 brane. Lastly, solution-generating transformations for metrics with an isometry are considered in the context of the fluid/gravity correspondence. Our focus is on the vacuum Rindler metric dual to a codimension one Navier-Stokes fluid. In particular, when there is a radially directed Killing vector, the dual fluid is found to exhibit an energy scaling invariance valid to all orders in the hydrodynamic expansion.

Motivation & Objective

• To extend solution-generating techniques to duality-invariant formulations of supergravity, including double field theory and $SL(5)$-invariant M-theory.
• To establish a connection between wave solutions in DFT and known string-theoretic objects like the F1 string and pp-wave.
• To investigate how isometries in bulk spacetimes induce symmetries in dual fluid dynamics, particularly energy scaling invariance.
• To generalize the Ehlers transformation in the fluid/gravity correspondence to include higher-order hydrodynamic effects.
• To unify solution-generating methods across supergravity, DFT, and fluid-gravity duality using isometry-based transformations.

Proposed method

• Derives a wave solution in double field theory that simultaneously realizes both the F1 string and pp-wave backgrounds via its Goldstone mode dynamics.
• Applies Buscher-like duality transformations in the DFT $eta$-function formalism to generate new solutions from isometric backgrounds.
• Constructs a covariant formulation of the $SL(5)$-invariant extended geometry using a chain rule derivation of the equations of motion.
• Introduces a wave solution in $SL(5)$-extended geometry that reproduces the M2-brane solution in the low-energy limit.
• Applies generalized Ehlers transformations to vacuum Rindler spacetimes with a radial Killing vector to generate new fluid-dual metrics.
• Analyzes the resulting fluid dynamics to identify an energy scaling invariance that holds to all orders in the hydrodynamic expansion.

Experimental results

Research questions

• RQ1 How can solution-generating transformations in double field theory be used to embed both F1 strings and pp-waves in a single wave solution?
• RQ2 What is the role of Goldstone modes in the DFT wave solution in reproducing duality-symmetric string dynamics?
• RQ3 How do isometries in the bulk spacetime induce symmetries in the dual fluid, particularly in the context of the fluid/gravity correspondence?
• RQ4 What is the origin of energy scaling invariance in the dual fluid when the bulk metric admits a radially directed Killing vector?
• RQ5 Can the generalized Ehlers transformation in fluid/gravity be extended to magnetohydrodynamics or higher-curvature corrections?

Key findings

• A single wave solution in double field theory simultaneously embeds both the F1 string and the pp-wave, with its Goldstone mode dynamics matching the duality-symmetric string of Tseytlin.
• The DFT wave solution reproduces the standard string and pp-wave backgrounds upon reduction, confirming consistency with known low-energy limits.
• In $SL(5)$-invariant extended geometry, a wave solution is constructed that recovers the M2-brane solution, validating the formalism’s physical relevance.
• For vacuum Rindler spacetimes with a radial Killing vector, the dual fluid exhibits energy scaling invariance that is preserved to all orders in the hydrodynamic expansion.
• The generalized Ehlers transformation in the fluid/gravity correspondence generates new vacuum solutions from existing ones, preserving the vacuum nature of the bulk metric.
• The transformation induces a symmetry in the fluid stress-energy tensor that corresponds to a scaling of energy density and pressure, consistent with conformal invariance in the hydrodynamic limit.