Latent solitons, black strings, black branes, and equations of state in Kaluza-Klein models

A particular soliton solution was also found in Ref. [ 13 ] . In the case of four external spacetime coordinates and D ′ D^{\prime} internal spatial coordinates, the parameters in the corresponding solution (25) in Ref. [ 13 ] are: q = 2 , p = D ′ q=2,\,p=D^{\prime} and Δ 2 = 2 ​ ( D ′ + 1 ) / ( D ′ + 2 ) \Delta^{2}=2(D^{\prime}+1)/(D^{\prime}+2) . If we rewrite this solution in the isotropic coordinates, then we immediately get the connection with our parameters, namely: γ i = − 1 , i = 1 , … , N , τ = − D ′ , σ = D ′ \gamma_{i}=-1,\,i=1,\ldots,N,\,\tau=-D^{\prime},\,\sigma=D^{\prime} and θ = 2 / [ ( D ′ + 1 ) ​ ( D ′ + 2 ) ] \theta=\sqrt{2/[(D^{\prime}+1)(D^{\prime}+2)]}

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It is worth noting the singular nature of τ = 2 \tau=2 , which follows from Eq. ( 20

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In the case of one d d -dimensional internal space ( N = 1 N=1 ), we can rewrite this bound with respect to the parameter k k defined in [ 11 ] as | k | ≥ d × 2.3 × 10 4 |k|\geq d\times 2.3\times 10^{4} . This inequality shows that the increase of the number of internal space dimensions d d imposes stronger restrictions on k k

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It can be easily seen from Eqs. ( 10

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If we rewrite equations of state in the form p i = ( α i − 1 ) ε , i = 0 , … , N p_{i}=(\alpha_{i}-1)\varepsilon\,,\;i=0,\ldots,N , then for the latent solitons we have α 0 = 1 , α i = ( 1 + γ i ) / 2 , i = 1 , … , N \alpha_{0}=1,\,\alpha_{i}=(1+\gamma_{i})/2\,,\;i=1,\ldots,N . For these values of α 0 \alpha_{0} and α i \alpha_{i} , we get on the right-hand side of Eq. (A15) in Ref. [ 1 ] the terms ( γ i ​ d i / 2 ) ​ κ N ​ ρ 3 (\gamma_{i}d_{i}/2)\kappa_{N}\rho_{3} . These terms are dynamical functions because of the dynamical behavior of the energy density ρ 3 \rho_{3} . This results in a violation of the necessary condition for the internal space stabilization

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