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A static axisymmetric exact solution of $f(R)$-gravity · Around

2012

A static axisymmetric exact solution of $f(R)$-gravity

Gutierrez-Pineres, Antonio C., Lopez-Monsalvo, Cesar S.

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We present an exact, axially symmetric, static, vacuum solution for $f(R)$ gravity in Weyl's canonical coordinates.

We obtain a general explicit expression for the dependence of $df(R)/dR$ upon the $r$ and $z$ coordinates and then the corresponding explicit form of $f(R)$, which must be consistent with the field equations.
We analyze in detail the modified Schwarzschild solution in prolate spheroidal coordinates.
Finally, we study the curvature invariants and show that, in the case of $f(R)\neq R$, this solution corresponds to a naked singularity.

Built on

M. Carmeli, Group theory and general relativity: representations of the Lorentz group and their applications to the gravitational field, Imperial College Press, 1977
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H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt, Exact solutions of Einstein’s field equations, Cambridge University Press, 2003
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T. Multamäki, I. Vilja, Spherically symmetric solutions of modified field equations in f ⁡ ( r ) f(r) theories of gravity, Physical Review D 74 (6) (2006) 064022
2006
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