Black holes in f(R) theory of gravity with compact extra dimensions

We study static, spherically symmetric solutions in $f(R)$ gravity within a $D = 4+n$-dimensional spacetime, where the extra dimensions form a compact $n$-sphere of constant radius. We derive an exact solution in which the four-dimensional part of the metric corresponds to the Schwarzschild - de Sitter metric, while the extra dimensions are stabilized at a constant radius $L_0$. A consistency condition relates the size of the internal space to the effective four-dimensional cosmological constant $\Lambda_4 = (n-1)/L_0^2$, which is generally too large to be compatible with observations. To overcome this issue, we include the vacuum polarization effects of quantized matter fields nonminimally coupled to curvature. Using the semiclassical approach, we obtain an asymptotically flat four-dimensional Schwarzschild solution as a limiting case, where the size of the extra sphere is determined by the vacuum expectation values of the quantum fields. Finally, we discuss the dependence of the effective four-dimensional Planck mass and the extra-dimensional radius on the radial coordinate.