Significance of tension for gravitating masses in Kaluza-Klein models

In this letter, we consider the six-dimensional Kaluza-Klein models with spherical compactification of the internal space. Here, we investigate the case of bare gravitating compact objects with the dustlike equation of state $\hat p_0=0$ in the external (our) space and an arbitrary equation of state $\hat p_1=\Omega \hat\varepsilon$ in the internal space, where $\hat \varepsilon$ is the energy density of the source. This gravitating mass is spherically symmetric in the external space and uniformly smeared over the internal space. In the weak field approximation, the conformal variations of the internal space volume generate the admixture of the Yukawa potential to the usual Newton’s gravitational potential. For sufficiently large Yukawa masses, such admixture is negligible and the metric coefficients of the external spacetime coincide with the corresponding expressions of General Relativity. Then, these models satisfy the classical gravitational tests. However, we show that gravitating masses acquire effective relativistic pressure in the external space. Such pressure contradicts the observations of compact astrophysical objects (e.g., the Sun). The equality $\Omega =-1/2$ (i.e. tension) is the only possibility to preserve the dustlike equation of state in the external space. Therefore, in spite of agreement with the gravitational experiments for an arbitrary value of $\Omega$, tension ($\Omega=-1/2$) plays a crucial role for the considered models.

💡 Research Summary

The paper investigates a six‑dimensional Kaluza‑Klein (KK) framework in which the two extra dimensions are compactified on a sphere (S²). The authors consider a gravitating source that is dust‑like in the observable four‑dimensional spacetime (i.e., zero pressure, p̂₀ = 0) while possessing an arbitrary equation of state in the internal space, p̂₁ = Ω ε̂, where ε̂ is the source’s energy density and Ω is a free parameter. The source is assumed to be spherically symmetric in the external space and uniformly smeared over the internal sphere, which preserves the internal isotropy.

In the weak‑field limit the six‑dimensional metric is written as η_{MN}+h_{MN}, and the variation of the internal sphere’s radius is encoded in a scalar field φ(x). Solving the linearized Einstein equations yields a Poisson‑type equation for φ with a source term proportional to (Ω+½) ε̂. The solution for φ produces a Yukawa‑type correction to the Newtonian potential in the external space: \