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One of the solutions to these equations is the following:. Substituting 15 into 5 we get a vanishing value of the scalar torsion. To understand the physical meaning of the integration constant that appears in solution 15 we discuss the physics related to this solution. First, the metric associated with the tetrad field given by 8 after using solution 15 in 2 has the form.
We rewrite the tetrad field 8 after using 15 in a way to understand its structure. To do this we use the fact that any tetrad field can be rewritten as. Using 18 to rewrite the tetrad field 8 after using 15 , we get. The matrices 21 and 22 give local Lorentz transformations; that is, they satisfy We have considered a solution in the vacuum case. Nine nonlinear differential equations have been obtained.
To solve these differential equations, we have imposed two constraints related to the derivative of the scalar torsion T. Then the number of the nonlinear differential equations reduced to four nonlinear differential equations. Wehavederived a solution which contains one constant of integration. To understand the physical meaning of the constant of integration, which exists in the solution, we have calculated its associated metric. To understand the properties of the derived solution, we have rewritten it as three matrices.
The first local Lorentz transformation local Lorentz transformation means that it satisfies A. By the use of 5 and 8 , we obtain the torsion scalar and its derivatives in terms of r and 0 as. The author declares that there is no conflict of interests regarding the publication of this paper.
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From A. Equations A. If this term is vanishing, then A. Hehl, J. McCrea, E. Mielke, and Y. Ne'eman, "Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance," Physics Reports, vol. Wanas and H. Wanas, "The other side of gravity and geometry: anti-gravity and anticurvature," Advances in High Energy Physics, vol. Wanas, N. Youssef, and A. Youssef and A. Sid-Ahmed, "Linear connections and curvature tensors in the geometry of parallelizable manifolds," Reports on Mathematical Physics, vol.
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