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We investigate the transverse modes of the gravitational and velocity fieldsin $\Lambda$CDM, based on a high-resolution simulation performed using theadaptive-mesh refinement general-relativistic $N$-body code GRAMSES. We studythe generation of vorticity in the dark matter velocity field at low redshift,providing fits to the shape and evolution of its power spectrum over a range ofscales. By analysing the gravitomagnetic vector potential, which is absent inNewtonian simulations, in dark matter haloes with masses ranging from$\sim10^{12.5}~h^{-1}{M}_{\odot}$ to $\sim10^{15}~h^{-1}{M}_{\odot}$, we findthat its magnitude correlates with the halo mass, peaking in the inner regions.Nevertheless, on average, its ratio against the scalar gravitational potentialremains fairly constant, below percent level, decreasing roughly linearly withredshift and showing a weak dependence on halo mass. Furthermore, we show thatthe gravitomagnetic acceleration in haloes peaks towards the core and reachesalmost $10^{-10}$ $h$ cm/s$^2$ in the most massive halo of the simulation.However, regardless of the halo mass, the ratio between the magnitudes of thegravitomagnetic force and the standard gravitational force is typically ataround the $10^{-5}$ level inside the haloes, again without significant radiusdependence. This result confirms that the gravitomagnetic effects have anegligible impact on structure formation, even for the most massive structures,although its behaviour in low density regions remains to be explored. Likewise,the impact on observations remains to be understood in the future.

Vector modes in $\Lambda$CDM: the gravitomagnetic potential in dark matter haloes from relativistic $N$-body simulations