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Schwarzschild-like exteriors for stars in Kaluza-Klein gravity

In this work we examine the effective four-dimensional world that emanates from a general class of static spherical Ricci-flat solutions in Kaluza-Klein gravity in $D$-dimensions. By means of dimensional reduction we obtain a family of asymptotically flat Schwarzschild-like metrics for which all the components of the Ricci tensor, except for $R_{11}$, are zero. Although the reduced spacetime is not empty, it is similar to vacuum in the sense that the effective matter satisfies an equation of state which is the generalization of $(ρ+ 3p) = 0$ for "nongravitating matter" in 4D. In Kaluza-Klein gravity these Schwarzschild-like metrics describe the exterior of a spherical star without rotation. In this framework, we generalize the well-known Buchdahl's theorem for perfect fluid spheres whose mass density does not increase outward. Without any additional assumptions, we develop the most general expression for the compactness limit of a star. We provide some numerical values for it, which in principle are observationally testable and allow us to compare and contrast different theories and exteriors. We find that in Kaluza-Klein gravity the compactness limit of a star can be larger than 1/2, without being a black hole: the general-relativistic upper limit $M/R < 4/9$ is increased as we go away from the Schwarzschild vacuum exterior. We show how this limit depends on the number of dimensions of spacetime, and demonstrate that the effects of gravity are stronger in 4D than in any other number of dimensions.