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The initial data of effective field theories of relativistic viscous fluids and gravity

Lorenzo Gavassino, Áron D. Kovács, Harvey S. Reall

Abstract

There has been recent progress in developing well-posed theories of relativistic viscous hydrodynamics and of gravitational effective field theories. These have in common the feature that they introduce unphysical degrees of freedom. We address the problem of how these should be treated. We propose a ''reduction of order'' approach which is applied not at the level of equations of motion but only to initial data. This specifies uniquely the data for the unphysical modes in terms of the data for the physical modes. We argue that the apparent breaking of Lorentz invariance associated with this approach is not a problem provided one restricts to Lorentz frames for which the assumptions of effective field theory are manifestly valid.

The initial data of effective field theories of relativistic viscous fluids and gravity

Abstract

There has been recent progress in developing well-posed theories of relativistic viscous hydrodynamics and of gravitational effective field theories. These have in common the feature that they introduce unphysical degrees of freedom. We address the problem of how these should be treated. We propose a ''reduction of order'' approach which is applied not at the level of equations of motion but only to initial data. This specifies uniquely the data for the unphysical modes in terms of the data for the physical modes. We argue that the apparent breaking of Lorentz invariance associated with this approach is not a problem provided one restricts to Lorentz frames for which the assumptions of effective field theory are manifestly valid.
Paper Structure (18 sections, 3 theorems, 49 equations, 2 figures)

This paper contains 18 sections, 3 theorems, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Two toy models
  3. Causal heat conduction in a moving rod
  4. Dispersive sound in ultradense matter
  5. General considerations
  6. The extra degrees of freedom are always fast
  7. Reducing multiple orders
  8. Consistency of the order reduction
  9. Lorentz and general covariance
  10. Preservation of Lorentz covariance
  11. Preservation of general covariance
  12. Applications
  13. Causal heat conduction in rotating stars
  14. BDNK
  15. Regularised Einstein-Gauss-Bonnet gravity
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\phi:\textup{"Minkowski"}\rightarrow \mathbb{C}$ be a classical scalar field, governed by the linear partial differential equation where $\mathcal{N}(a,b_j)$ and $\mathcal{M}(a,b_j)$ are polynomials in $a$ and $b_j$ with constant coefficients. Call $\mathfrak{n}$ and $\mathfrak{m}$ the degrees of respectively $\mathcal{N}(a,b_j)$ and $\mathcal{M}(a,b_j)$ in the variable $a$ (and assume that

Figures (2)

  • Figure 1: Comparison between various solutions of \ref{['heatBDNK']} (solid lines), under the ansatz $T(t,x)=T(t)e^{i k x}$, with $\lambda k=0.05$ and $v=0$. The red curves represent solutions initialized with the same $T(0)$, but various choices of $\partial_t T(0)\sim \lambda k^2$. These solutions exhibit an initial transient relaxation on a timescale $t \sim \lambda$, after which they converge to solutions of \ref{['dt1heatBDNK']} (dashed) with a different$T(0)$. The only trajectory that does not display any transient behavior is the one arising from the Cauchy problem \ref{['correctCauchy']} (blue curve), which perfectly overlaps the one solution of \ref{['dt1heatBDNK']} having the same $T(0)$. Note that, in general, the order-reduced equation may not be well-posed. Here, this property happens to hold, so we can actually contrast solutions of \ref{['heatBDNK']} and \ref{['dt1heatBDNK']}.
  • Figure 2: Comparison between solutions of \ref{['dispersive']} for various initial data (solid lines), under the ansatz $\phi(t,x)=\phi(t)e^{i k x}$, with $\lambda k=0.2$ and $c_s=1/2$. As initial conditions, we fix $\phi(0)$, and set $\partial_t \phi(0)=0$ in all cases. The blue curves show the solution of \ref{['cauchysound']}. The red curves show solutions of \ref{['dispersive']} obtained by varying the higher-order derivatives in the initial data. Specifically, in the left panel we vary $\partial_t^2\phi(0)\sim c_s^2 k^2$ while keeping $\partial_t^3 \phi(0)=0$, whereas in the right panel we fix $\partial_t^2\phi(0)$ according to \ref{['cauchysound']} and vary $\partial_t^3 \phi(0)\sim c_s^2 k^2/\lambda$ (we just need $\lambda^3 \partial^3_t\ll1$ for validity of EFT, and $c_s^3k^3$ would be too small to be visible). All such solutions develop additional fast oscillations with frequency of order $\lambda^{-1}$, superimposed on the main infrared sound mode. In the right panel, changes of initial data also result in a global shift of phase of the underlying infrared mode. The only trajectory free from fast oscillations is the one corresponding to the Cauchy problem \ref{['cauchysound']} (blue curve), which exactly overlaps with the solution of \ref{['dispersiveacausal']} initialized with the same $\phi(0)$ and $\partial_t \phi(0)$ (black dashed). For reference, we also include the solution of $(\partial_t^2 - c_s^2\partial_x^2)\phi = 0$ with identical initial data (dotted gray line).

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof