Strings in Gödel CTC and in hairy axionic black holes with Kerr Nieh–Yan charges in teleparallelism

In this paper, we present three types of teleparallel frames with cylindrical and spherical symmetry. The first example is the closed time-like curves (CTC) in cosmic strings in teleparallel T₄ T 4 gravity, which shows they are forbidden. Here, we show that in T₄ T 4 , in order to allow for CTC, we are also led to a lower bound on the angular momentum of the cosmic string. This result matches the interior T₄ T 4 solution to a General Relativity (GR) vacuum solution. One of the main differences between the present report and the one by Soleng is that, here, the interior symmetric solution does not have polarized spins but only Cartan torsion in the spirit of teleparallelism. Torsion flux is computed to show that a cylindrical central singularity corresponds to a $$(2+1)$$ ( 2 + 1 ) -spacetime rotating point particle in T₄ T 4 . Therefore, the possibility of building time machines seems to be strongly constrained compared with Einstein–Cartan (EC) gravity. We present a spherically symmetric black hole with an open string along a particular direction of the black hole (BH). In this case, it is shown that when the teleparallel frame used is the one where all spin coefficients vanish, a torsion flux cutoff appears along one of the directions. In the stringy BH case, choosing a restricted teleparallel frame, a torsion-axion-like hair appears in the form of an axion scalar. A Kerr Nieh–Yan (NY) invariant black hole is obtained. Coley et al. previously obtained BH solutions in teleparallel spacetime, which present horizon singularities but not torsion singularities.