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In this letter, we consider the six-dimensional Kaluza-Klein models withspherical compactification of the internal space. Here, we investigate the caseof bare gravitating compact objects with the dustlike equation of state $\hatp_0=0$ in the external (our) space and an arbitrary equation of state $\hatp_1=\Omega \hat\varepsilon$ in the internal space, where $\hat \varepsilon$ isthe energy density of the source. This gravitating mass is sphericallysymmetric in the external space and uniformly smeared over the internal space.In the weak field approximation, the conformal variations of the internal spacevolume generate the admixture of the Yukawa potential to the usual Newton'sgravitational potential. For sufficiently large Yukawa masses, such admixtureis negligible and the metric coefficients of the external spacetime coincidewith the corresponding expressions of General Relativity. Then, these modelssatisfy the classical gravitational tests. However, we show that gravitatingmasses acquire effective relativistic pressure in the external space. Suchpressure contradicts the observations of compact astrophysical objects (e.g.,the Sun). The equality $\Omega =-1/2$ (i.e. tension) is the only possibility topreserve the dustlike equation of state in the external space. Therefore, inspite of agreement with the gravitational experiments for an arbitrary value of$\Omega$, tension ($\Omega=-1/2$) plays a crucial role for the consideredmodels.

Significance of tension for gravitating masses in Kaluza-Klein models