Weak-field limit of Kaluza–Klein models with spherically symmetric static scalar field: observational constraints

Abstract

Abstract In a multidimensional Kaluza–Klein model with Ricci-flat internal space, we study the gravitational field in the weak-field limit. This field is created by two coupled sources. First, this is a point-like massive body which has a dust-like equation of state in the external space and an arbitrary parameter $$\varOmega $$ Ω of equation of state in the internal space. The second source is a static spherically symmetric massive scalar field centered at the origin where the point-like massive body is. The found perturbed metric coefficients are used to calculate the parameterized post-Newtonian (PPN) parameter $$\gamma $$ γ . We define under which conditions $$\gamma $$ γ can be very close to unity in accordance with the relativistic gravitational tests in the solar system. This can take place for both massive or massless scalar fields. For example, to have $$\gamma \approx 1$$ γ ≈ 1 in the solar system, the mass of scalar field should be $$\mu \gtrsim 5.05\times 10^{-49}$$ μ ≳ 5.05 × 10 - 49 g $$\sim 2.83\times 10^{-16}$$ ∼ 2.83 × 10 - 16 eV. In all cases, we arrive at the same conclusion that to be in agreement with the relativistic gravitational tests, the gravitating mass should have tension: $$\varOmega = -\,1/2$$ Ω = - 1 / 2 .