Effective spacetime from multidimensional gravity

We study the effective spacetimes in lower dimensions that can be extracted from a multidimensional generalization of the Schwarzschild-Tangherlini spacetimes derived by Fadeev, Ivashchuk, and Melnikov (Phys. Lett. A 161, 98 (1991)). The higher-dimensional spacetime has D = (4 + n + m) dimensions, where n and m are the number of “internal” and “external” extra dimensions, respectively. We analyze the effective (4 + n) spacetime obtained after dimensional reduction of the m external dimensions. We find that when the m extra dimensions are compact (i) the physics in lower dimensions is independent of m and the character of singularities in higher dimensions, and (ii) the total gravitational mass M of the effective matter distribution is less than the Schwarzschild mass. In contrast, when the m extra dimensions are large, this is not so; the physics in (4 + n) does explicitly depend on m as well as on the nature of singularities in high dimensions, and the mass of the effective matter distribution (with the exception of wormhole-like distributions) is larger than the Schwarzschild mass. These results may be relevant to observations for an experimental/observational test of the theory.