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Local Lorentz transformation and exact spherically symmetric vacuum solutions in f(T) gravity theories

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Abstract

In this paper a non-diagonal, spherically symmetric, tetrad field that contains an arbitrary function, S(r), which corresponds to a local Lorentz transformation, is applied to the field equations of f(T) gravity theories. Analytic vacuum solutions with integration constants are derived. These constants are studied by calculating the total conserved charge associated with each solution. The study shows that the obtained solutions represent the Schwarzschild–Ads spacetime.

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Notes

  1. Spacetime indices μ,ν,… and SO(3,1) indices a,b,… run from 0 to 3. Time and space indices are indicated by μ=0,i, and a=(0),(i).

  2. (Λ i j ) g is the general local Lorentz transformation matrix to a general three-dimensional rotation R parametrized by its three Euler angles θ, ϕ and ψ such that we can write

    where the R x , R y , R z are the rotation matrices about the Cartesian coordinate axis with angles ϕ, ϑ, ψ, respectively. These well-known matrices are given by

    In our study we consider the following values for the three Euler angles:

    $$\phi=\delta(r), \qquad\vartheta=\theta+\frac{\pi}{2}, \qquad \psi=\phi, $$

    where δ(r) is taken to be a general function of r, and when \(\delta(r)=\frac{\pi}{2}\), for spherical symmetric case, we get the local Lorentz transformation (Λ i j ) presented in Eq. (9).

  3. The asymptote behavior of Eq. (17), after using Eq. (18), is like \(S(r) \sim \mathrm{constant}+O (\frac{1}{r^{2}} )\) which has no effect on the calculations of the physical quantities, i.e., the conserved quantities related to output II [34, 41].

  4. Equations (11)–(13) and Eq. (20) are four differential equations in four unknown functions f(T)=f(r), A(r), B(r) and S(r), but those equations are not in consistence with each other. This inconsistency mainly comes from the appearance of f T and f TT terms. If these terms are vanishing one returns to the linear case of f(T) and can find exact solution to these system, otherwise it will not be easy. Therefore, we put some constraint on the unknown functions to be able to find some especial solutions.

  5. The calculations were checked using Maple software 15.

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Authors and Affiliations

  1. Centre for Theoretical Physics, The British University in Egypt, P.O. Box 43, Sherouk City, 11837, Egypt

    Gamal G. L. Nashed

  2. Mathematics Department, Faculty of Science, Ain Shams University, Cairo, 11566, Egypt

    Gamal G. L. Nashed

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Nashed, G.G.L. Local Lorentz transformation and exact spherically symmetric vacuum solutions in f(T) gravity theories. Eur. Phys. J. C 73, 2394 (2013). https://doi.org/10.1140/epjc/s10052-013-2394-x

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