I'm trying to visualize the vector field on $\mathbb{R}^3$ given by the matrix $$\begin{pmatrix} 1 & 0 & 0\\ 0 & \cosh(t) & \sinh(t)\\ 0 & \sinh(t) & \cosh(t) \end{pmatrix},$$ where $t$ is a real number, when restricted to the upper sheet of the hyperboloid $x^2+y^2-z^2=-1$. Here are my two attempts:
VectorPlot3D[{x, y Cosh[t]+z Sinh[t], y Sinh[t]+z Cosh[t]},
{x, -5, 5}, {y, -5, 5}, {z, 0, 5},
RegionFunction -> Function[{a, b, c}, a^2 + b^2 - c^1 == -1]]
and
VectorPlot3D[{x, y Cosh[t]+z Sinh[t], y Sinh[t]+z Cosh[t]},
{x, -5, 5}, {y, -5, 5}, {z, 0, 5},
RegionFunction -> ((#1^2+#2^2-#3^2==-1) &)]
These are both based off of examples I saw in the VectorPlot3D documentation, but they both return empty graphs. What am I doing wrong? Thanks!