Revisiting diagonal tetrads: new Black Hole solutions in f(T) gravity

Abstract

We study various forms of diagonal tetrads that accommodate Black Hole solutions in f(T) gravity with certain symmetries. As is well-known, vacuum spherically symmetric diagonal tetrads lead to rather boring cases of constant torsion scalars. We extend this statement to other possible horizon topologies, namely, spherical, hyperbolic and planar horizons. All such cases are forced to have constant torsion scalars to satisfy the anti-symmetric part of the field equations. We give a full classification of possible vacuum static solutions of this sort. Furthermore, we discuss addition of time-dependence in all the above cases. We also show that if all the components of a diagonal tetrad depend only on one coordinate, then the anti-symmetric part of the field equations is automatically satisfied. This result applies to the flat horizon case with Cartesian coordinates. For solutions with a planar symmetry (or a flat horizon), one can naturally use Cartesian coordinates on the horizon. In this case, we show that the presence of matter is required for existence of non-trivial solutions. This is a novel and very interesting feature of these constructions. We present two new exact solutions, the first is a magnetic Black Hole which is the magnetic dual of a known electrically charged Black Hole in literature. The second is a dyonic Black Hole with electric and magnetic charges. We present some features of these Black holes, namely, extremality conditions, mass, behavior of torsion and curvature scalars near the singularity.

Static spherically symmetric solutions in new general relativity

“…In a sense, the singular horizons might be taken similarly to the popular firewalls. Note, at the same time, that even the usual Schwarzschild horizon is anyway singular in terms of the tetrad Ansatz (20), or its corresponding torsion [9], as was also mentioned, for example, in the [3]. And it is very natural indeed.…”

“…A lesson to take is that in search for such solutions one should attempt using different possible radial variables. In particular, we probably need to try other options in our long, but largely unsuccessful, go for constructing these configurations in f(T) models [3,4].…”

“…And even apart from the foundational issues, many modified models of teleparallel type appear to be much more complicated than it has ever been expected. For instance, we have not been able to find non-trivial exact Black Hole solutions in f(T) gravity, unless we accept topological flat horizons [3] or a complex tetrad [4] which is problematic since the tetrad is the dynamical variable in teleparallel gravity. Therefore it definitely makes sense to better study the physical properties of the simplest models.…”

Static Spherically Symmetric Perfect Fluid Solutions in Teleparallel F(T) Gravity

“…There are a number of possible approaches for solutions to the FEs described by Equations (14a), (19a), (19b) and (21) added by the conservation law solution described by Equation (18). The main goal is to find several possible F(T) solutions from these previous equations.…”

“…A proper frame is not invariantly defined since it is defined in terms of the connection, which is not a tensorial quantity, which leads to a number of problems when using such a frame to determine symmetries. There are several papers in the literature about static and non-static spherically symmetric solutions in teleparallel F(T) gravity [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. There are several perturbative solutions in TEGR (Teleparallel Equivalent of General Relativity) and there are some power-law F(T) solutions with power-law frame components (see [14][15][16][17] and references within).…”