Significance of tension for gravitating masses in Kaluza-Klein models

TL;DR

This paper analyzes six-dimensional Kaluza-Klein models with spherical internal space, showing that tension in gravitating masses is essential to match observational data and maintain a dustlike external equation of state.

Contribution

It demonstrates that tension (Ω = -1/2) is necessary in these models to ensure consistency with gravitational tests and astrophysical observations.

Findings

Yukawa potential admixture becomes negligible for large Yukawa masses.

Models satisfy classical gravitational tests when Yukawa contributions are small.

Effective relativistic pressure arises in the external space unless tension is present.

Abstract

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.