### Abstract

The theory of spacelike congruences in general relativity is briefly reviewed and the physical interpretation of the rotation tensorR_{ a b }, the expansion E, and the shear tensorS_{ a b }, of the curves is discussed. It is proved that if the unit tangent vector to any curve of the congruence is everywhere orthogonal to the 4‐velocity field *u* ^{ a } of a self‐gravitating fluid, then observers comoving with the fluid can be employed along a curve of the congruence if and only if the curves are material curves in the fluid. A congruence of vortex lines is studied in detail. Starting from the Ricci identity for *u* ^{ a } and using Einstein’s equations, general expressions in terms of the kinematic quantities and fluid variables are derived for R_{ a b }, C, and S_{ a b } for a vortex congruence. It is found that E and S_{ a b } depend explicitly on the gravitational field only through the magnetic part of the Weyl tensor, and R_{ a b } only through a term proportional to the total energy flux *q* ^{ a } derived from Einstein’s equations. With the aid of Maxwell’sequations, properties of congruences of magnetic field lines, electric field lines, and a certain combination of vortex and magnetic field lines are determined. For a congruence of magnetic field lines in an electrically conducting fluid and assuming the magnetohydrodynamic approximation of vanishing electric field, it is proved that for a comoving observer, R_{ a b }=0 if and only if the conduction current in the fluid is orthogonal to the magnetic field. The propagation equations for R_{ a b }, E, and S_{ a b } along a curve of a spacelike congruence are considered. These equations are developed in full for the special case of a congruence of material curves in a fluid. The divergence, or constraint, equation for the rotation vector is also derived. Where appropriate, corresponding results in Newtonian gravitation theory are given for comparison.

Commenting has been disabled for this content