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Analyzing the radial geodesics of the Campanelli–Lousto solutions

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Abstract

When dealing with a spacetime, one usually searches for singularities, black holes, white holes and wormholes due to their importance to the motion of particles. There is a family of solution of the Brans–Dicke vacuum equations that has not been fully studied from this perspective. In this paper, I study some properties of this family and find the complete set of solutions that avoids singularity at the point where the metric diverges or degenerates. The possible changes in the metric signature when passing through this point is analyzed. In addition, I also study the radial geodesics and obtain the solutions of some particular cases.

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Notes

  1. It is defined as .

  2. The solution (4)–(6) is not exactly the same as that of Fisher because the scalar fields are different. In the case of Eq. (6), the scalar field becomes trivial for \(m=-n\).

  3. For the case \(r<r_0\) and m either an even integer or an irreducible fraction of the type even/odd, the situation is a little bit more involved because setting \(\dot{R}_i=0\) no longer means releasing the particle from rest at \(r_i\) (r is a time coordinate in this case). I shall not pursue this issue here.

  4. Notice that the expression \(1-n-(2-n)\frac{r_0}{2r}\) can be recast as \((1-n)(1-r_c/r)\), which clearly changes its sign.

  5. Using the plus sign in Eq. (43) one can show that the particle can come back to the universe \(r\in (r_m,\infty )\) if an appropriate initial velocity is given to the particle when it is at the other universe.

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  1. Centro de Ciências da Natureza, Universidade Estadual do Piauí, C. Postal 381, Teresina, Piauí, 64002-150, Brazil

    J. B. Formiga

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  1. J. B. Formiga
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Formiga, J.B. Analyzing the radial geodesics of the Campanelli–Lousto solutions. Gen Relativ Gravit 47, 146 (2015). https://doi.org/10.1007/s10714-015-1967-4

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  • DOI: https://doi.org/10.1007/s10714-015-1967-4

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