Perfect fluid equations with nonrelativistic conformal symmetry: Exact solutions

Abstract

The group-theoretic approach is used to construct exact solutions to perfect fluid equations invariant under the Schrodinger group, or the l-conformal Galilei group, or the Lifshitz group. In each respective case, the velocity vector field looks similar to the Bjorken flow. It is shown that one can reach an arbitrarily high density (and hence pressure) for a short period of time by adjusting the value of l and other free parameters available.

Figures (5)

• Figure 1: The graph of $z=\rho(t,x)$ for $\ell=\frac{1}{2}$ (orange), $\ell=\frac{5}{2}$ (blue), $\ell=\frac{9}{2}$ (green), $\ell=\frac{13}{2}$ (red) with $t \in [2,6]$, $x \in [-10,10]$, $c=0.1$, $a=0.5$.
• Figure 2: A flow generated by the vector field $\upsilon_i=\frac{\ell x_i}{t}$ in two spatial dimensions for $x_1 \in [-1,1]$, $x_2 \in [-1,1]$ at $\ell=1$ and $t=10$.
• Figure 3: The dependence of the mass of a disk of unit radius (centered at the origin of the coordinate system) upon time for $\ell=\frac{1}{2}$ (blue) and $\ell=\frac{5}{2}$ (orange) in two spatial dimensions with $c=0.1$, $a=0.5$, and $t \in [0.8,3]$.
• Figure 4: An example of a flow obtained by applying a specific acceleration transformation to the velocity vector field $\upsilon_i=\frac{\ell x_i}{t}$ for $\ell=1$ and $d=2$.
• Figure 5: The dependence of the mass of a disk of unit radius centered at the origin of the coordinate system upon time for $z=0.6$ (blue), $z=0.7$ (orange), $z=0.8$ (green) in two spatial dimensions with $c=0.1$, $a=0.5$, and $t \in [0.1,10]$.