How to define a relativistic irrotational flow?

I have been studying the article https://arxiv.org/abs/1001.1310 in order to understand relativistic fluid dynamics and analogue models of gravity using the same. I am having difficulty visualising the way a relativistic irrotational flow is defined there. I am describing the important features. Let the spacetime in which the fluid is flowing be described by the general metric g. There are no restrictions on this metric. Following the paper referred to above, let $V$ denote the four-velocity (vector) field of the fluid. Now, using the metric, it is possible to construct the 1-form field $v$ corresponsing to the vector filed V.
The 1-form $v$, the 2-form $omega_2$ and the 3-form fields $t_3$ are defined by,
begin{align}
v=g(V), , , omega_2 = dv, , , t_3 = v wedge omega_2 = v wedge dv~.
end{align}
The author says, “In a relativistic context, setting the 2-form $ω_2 = 0$ is too strong a condition, setting the 3-form $t_3 = 0$ is just right”. This statement I do not understand. Furthermore, I am not able to visualise how $t_3 = v wedge dv$ looks. I have tried visualising forms with the help of Dan Piponi’s article `On the Visualisation of Differential Forms’. Basically, a 1-form is picturised as a codimension 1 surface, a 2-form as a co-dimension 2 surface etc. and their wedge products as intersections of the (hyper)surfaces.

It would be great if you could help me see why setting $t_3=v wedge dv =0$ would imply an irrotational relativistic flow.