text:
\section{\label{sec:1}Introduction}

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The modern theories of unification such as superstrings, supergravity and M-theory have the most self-consistent formulation in spacetime with extra dimensions.
Different aspects of the idea of multidimensionality are intensively used in numerous modern articles. Therefore, it is important to construct viable multidimensional
models which are in good agreement with physical experiments. The known gravitational experiments (the deflection of light and the time delay of radar echoes)
in the solar system are good filters to screen out non-physical theories. It is well known that the weak field
approximation is enough to calculate the corresponding formulas for these experiments \cite{Landau}. For example, in the case of general relativity these formulas
demonstrate excellent agreement with the experimental data.

In our previous papers  we investigated the popular Kaluza-Klein (KK) models with toroidal compactification of the internal spaces.  As we have shown, to be at the same
level of agreement with the gravitational tests as general relativity, the gravitating masses should have tension (i.e. negative pressure) in the internal spaces. This
is true for both linear \cite{EZ4,EZ5} and nonlinear $f(R)$ \cite{EZnonlin,EZf(R)solitons} models. At first glance, it looks unexpectedly, since dust-like equation of
state $p=0$ in all spatial dimensions is the most natural one  for nonrelativistic objects because the energy density for them is much bigger than the pressure. Such
approach for gravitating matter sources works very well in general relativity in the three-dimensional space to describe the gravitational tests in the solar system
\cite{Landau}. Therefore, we expected the same in Kaluza-Klein models. Unfortunately, the physically reasonable models with the dust-like equation of state $p=0$ in both
external and internal spaces contradict these tests \cite{EZ3}. It happens because of the fifth force generated by variations of the internal space volume \cite{EZ6}.
For the proper value of tension, the internal space volume is fixed and the fifth force is absent. It takes place, e.g., for the black strings/branes which have the
equation of state in the internal space $p=\Omega \varepsilon$ with $\Omega=-1/2$.

For black strings and black branes, the notion of tension is defined, e.g., in \cite{TF} and it follows from the first law for black hole spacetimes \cite{TZ,HO,TK}.
However, up to now we are not aware of the reasonable physical explanation for ordinary nonrelativistic objects, possessing such relativistic tension. Moreover, in the
recent paper \cite{ChEZ4} it was shown that in the case of non-zero tension there is a problem to formulate a many-body theory for such models. Black strings/branes are
compactified on tori. However, it is well known that superstring/supergravity models can be compactified also on Ricci-flat compact spaces (e.g., Calabi-Yau) and
spheres. For instance, 11-dimensional supergravity can be compactified on a torus $T^7$ and on a sphere $S^7$ (see, e.g., \cite{Wit,Nastase}). The Freund-Rubin mechanism
\cite{FR} is the most famous example of spontaneous compactification on $S^7$  where this is achieved with the help of the form fields. So, setting a goal to find viable
models without tension, we considered in \cite{ChEZ1,ChEZ2} Kaluza-Klein models where the internal space is a two-dimensional sphere and tension of the gravitating mass
is absent: $\Omega=0$. Here, we have shown that the conformal variation of the volume of the two-sphere generates the Yukawa-type admixture to the metric coefficients
and the non-relativistic gravitational potential. We estimated this admixture and found that it has a negligible value in the solar system. Therefore, this model (with
$\Omega=0$) satisfies the mentioned above gravitational tests. However, our subsequent research \cite{EZ7} indicates that tension ($\Omega=-1/2$) plays a crucial role
for the considered model because
the non-relativistic gravitating masses acquire effective relativistic pressure in the external space if $\Omega=0$. Obviously, such pressure contradicts the
observations. The equality $\Omega =-1/2$ is the only possibility to preserve the dust-like equation of state in the external space. So, we need tension again! How
general is this result? Is it possible, in principle, to construct a model without tension, satisfying the observations (from both gravitational and thermodynamical
points of view)? In the present paper we give an affirmative answer to this question and show what is the price for it.

Here, we consider a model where background matter is taken in the form of a multicomponent perfect fluid. This matter provides spherical compactification of the internal
space with arbitrary number of dimensions. The multicomponent approach is quite natural. For example, because of nontrivial topology of our multidimensional spacetime
(i.e. compactness of the internal space), vacuum fluctuations of quantized fields result in the nonzero energy density (the Casimir effect) \cite{Casimir}. Various
variants of the string theory contain real-valued form fields \cite{Stelle}. We mentioned above that these fields may result in spontaneous compactification due to the
Freund-Rubin mechanism. The model may also include other types of fluids. We perturb this background by a compact (usually, point-like) gravitating mass with the
dust-like equation of state in the external/our space and an arbitrary equation of state (with the constant parameter $\Omega$) in the internal space. In the
single-component case, we prove that to satisfy both the gravitational tests and thermodynamical observations, gravitating masses should have tension $\Omega=-1/2$.
However, in the multicomponent case, there is a possibility to construct a model which satisfies both the gravitational tests and the thermodynamical properties due to
multicomponent nature of the background matter. It takes place for any reasonable value of $\Omega$ including the most interesting dust-like case $\Omega=0$. However, to
achieve it we need a fine-tuning condition.

The paper is organized as follows. In Sec. 2 we define the background solution in the case of spherical compactification of the internal space with an arbitrary number
of dimensions. Here, the background matter takes the form of the multicomponent perfect fluid. Then, we perturb this solution by a gravitating mass with an arbitrary
equation of state in the internal space and obtain equations for the metric perturbations. In Sec. 3 we investigate in detail the single-component case and prove the
necessity of tension ($\Omega=-1/2$) for such model. Sec. 4 is devoted to the general multicomponent case. Here, we solve the equations for metric perturbations for two
particular examples and demonstrate a principal possibility for the considered model to satisfy the gravitational and thermodynamical observations in the case
$\Omega=0$. The main results are shortly summarized in the concluding Sec. 5.

\section{\label{sec:2}Multicomponent background solution and perturbations}

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Before the consideration of the gravitational field produced by the gravitating mass, we need to create an appropriate background metric. Such metric is defined on the
product manifold $M = M_4\times M_d$, where $M_4$ describes external four-dimensional flat spacetime and $M_d$ corresponds to the $d$-dimensional internal space which is
a sphere with the radius (the internal space scale factor) $a$, and should have the form
\be{2.1}
 ds^2=c^2 dt^2-dx^2-dy^2-dz^2 + \sum_{\mu=4}^D g_{\mu\mu}d\xi_{\mu}^2\, ,
\end{equation}
where
\ba{2.2} g_{DD}=-a^2\, ,\quad g_{\mu\mu}=-a^2\prod_{\nu=\mu+1}^{D}\sin^2\xi_{\nu}\, ,\; \; \mu=4,\ldots ,D-1\, ,\end{eqnarray}
and $D=3+d$ is the total number of spatial dimensions. To create such metric with the curved internal space, we have to introduce background matter with the
energy-momentum tensor
\be{2.3}
\bar T_{ik}=\left\{
\begin{array}{cc}
\left( \frac{d(d-1)}{2\kappa a^2}-\Lambda_\mathcal{D} \right) g_{ik} & \mbox{for   } \, i,k=0,...,3;\\
\\
\left(\frac{(d-1)(d-2)}{2\kappa a^2} - \Lambda_\mathcal{D} \right)g_{ik} & \mbox{for   } \, i, k=4,5,\ldots ,D.
\end{array} \right . \quad
\end{equation}
These components of the energy-momentum tensor can be easily got from the Einstein equation
\be{2.4} \kappa \bar T_{ik} = R_{ik}-\frac{1}{2} R g_{ik}-\kappa\Lambda_{\mathcal{D}} g_{ik}\,  \end{equation}
for the background metric \rf{2.1}.
 Here, $\kappa \equiv 2S_D\tilde G_{\mathcal{D}}/c^4$, $S_D=2\pi^{D/2}/\Gamma (D/2)$ is the total solid angle (the surface area of the $(D-1)$-dimensional sphere of a
unit radius) and $\tilde G_{\mathcal{D}}$ is the gravitational constant in $(\mathcal{D}=D+1)$-dimensional spacetime. We also include in the model a bare
multidimensional cosmological constant $\Lambda_{\mathcal{D}}$. To get these components of the energy-momentum tensor, we took into account that the only nonzero
Ricci-tensor components are $R_{\mu\mu}=-[(d-1)/a^2]g_{\mu\mu},\; \mu=4,\ldots ,D$, and the scalar curvature is $R=-d(d-1)/a^2$. The minus sign in these formulas follows
from adopted here metric and curvature sign conventions (as in the book \cite{Landau}).

Now, we suppose that the energy-momentum tensor \rf{2.3} corresponds to the $N$-component perfect fluid:
\ba{2.5}
\bar T^0_{0}&=& \bar{\mathcal{E}}=\sum_{q=1}^{N}\bar\varepsilon_{q}\, ,\\
\label{2.6}\bar T^{\alpha}_{\alpha}&=&-\bar P_{0}=-\sum_{q=1}^{N}\bar
p_{(0)q}=-\sum_{q=1}^{N}\omega_{(0)q}\bar\varepsilon_{q}\, ,\quad \alpha = 1,2,3\, ,\\
\label{2.7}\bar T_{\mu}^{\mu}&=&-\bar P_{1}=-\sum_{q=1}^{N}\bar p_{(1)q}=-\sum_{q=1}^{N}\omega_{(1)q}\bar\varepsilon_{q}\, ,\quad\mu = 4,5,...,D\, , \end{eqnarray}
where $\omega_{(0)q}$  and  $\omega_{(1)q}$ are the parameters of equations of state of the $q$-th perfect fluid in the external and internal spaces, respectively.
It can be easily seen that the following relations take place:
\be{2.8}
\bar P_{0}=\left(-1\right)\bar{\mathcal{E}}\quad \Rightarrow\quad
\sum_{q=1}^{N}\left(1+\omega_{(0)q}\right)\bar\varepsilon_{q}=0\,
\end{equation}
and
\ba{2.9}
\frac{\bar P_{1}}{\bar{\mathcal{E}}}&=&\frac{2\Lambda_{\mathcal{D}}\kappa a^{2}-\left(d-1\right)\left(d-2\right)}{d\left(d-1\right)-2\Lambda_{\mathcal{D}}\kappa a^{2}} \nonumber\\
&\Rightarrow& \sum_{q=1}^{N}\left[\omega_{(1)q}-\frac{2\Lambda_{\mathcal{D}}\kappa a^{2}-\left(d-1\right)\left(d-2\right)}{d\left(d-1\right)-2\Lambda_{\mathcal{D}}\kappa
a^{2}}\right]\bar\varepsilon_{q}=0 \, , \end{eqnarray}
or, equivalently,
\be{2.10}
\Lambda_{\mathcal{D}}=\frac{d-1}{\kappa a^2}\frac{\sum\limits_{q=1}^N\left(d
\omega_{(1)q}+d-2\right)\bar\varepsilon_q}{2\sum\limits_{q=1}^N\left(\omega_{(1)q}+1\right)\bar\varepsilon_q}\, .
\end{equation}
There is also another useful relation:
\be{2.11}
\Lambda_{\mathcal{D}}=\frac12\sum\limits_{q=1}^N\left(d\omega_{(1)q}+d-2\right)\bar\varepsilon_q\, ,
\end{equation}
which together with \rf{2.10} gives
\be{2.12}
\frac{d-1}{\kappa a^2}=\sum\limits_{q=1}^N\left(\omega_{(1)q}+1\right)\bar\varepsilon_q\, .
\end{equation}
According to Eqs. \rf{2.10} and \rf{2.12}, in what follows we consider models that satisfy the inequality $\sum_{q=1}^N\left(\omega_{(1)q}+1\right)\bar\varepsilon_q \neq
0$.

Now, we perturb our background ansatz by a static point-like massive source with the nonrelativistic rest mass density $\hat \rho$. We suppose that the matter source is
uniformly smeared over the internal space \cite{EZ2}. Hence, multidimensional $\hat \rho$ and three-dimensional $\hat \rho_3$ rest mass densities are connected as
follows: $\hat \rho=\hat \rho_3({\bf r}_3)/V_{int}$ where $V_{int}=\left[2\pi^{(d+1)/2}/\Gamma((d+1)/2)\right]a^d$ is the surface area of  the $d$-dimensional sphere of
the radius $a$ (see, e.g., \cite{EZ3,EZ2}). In the case of a point-like mass $m$, $\hat \rho_3(r_3)=m\delta ({\bf r}_3)$, where $r_3=|{\bf r}_3|=\sqrt{x^2+y^2+z^2}$. In
the nonrelativistic approximation the energy density
of the point-like mass is $\hat{T}^0_{0}\approx \hat \rho c^2$ and up to linear in perturbations terms $\hat{T}_{00}\approx \hat \rho c^2$. Inasmuch as the gravitating
mass is at rest in the external space, it has the dust-like equation of state $\hat p_0=0\; \Rightarrow \; \hat{T}^1_{1}=\hat{T}^2_{2}=\hat{T}^3_{3}=0$ in our
dimensions. However, it may have the nonzero equation of state  $\hat p_1 \approx \Omega \hat \rho c^2\; \Rightarrow \; \hat T_{\mu}^{\mu}\approx -\Omega \hat \rho c^2
\, ,\; \mu =4,5,\ldots ,D$ in the internal space{\footnote{It is worth noting that, in KK models with toroidal compactification, the presence of nonzero pressure/tension
in the extra dimensions results in uniform smearing of the gravitating mass over the internal space \cite{ChEZ4}.}}. All other components of the energy-momentum tensor
of the gravitating mass are equal to zero.

Concerning the energy-momentum tensor of the background matter, we suppose that perturbation does not change the equations of state of the multicomponent perfect fluid
in the external and internal spaces, i.e. $\omega_{(0)q}$ and $\omega_{(1)q}$ are constants. Therefore, the energy-momentum tensor of the perturbed background is
\ba{2.13}
\tilde T_{00}&\approx& \left(\sum_{q=1}^{N}(\bar\varepsilon_{q}+\varepsilon_{q}^{1}) \right) g_{00}\, ,\\
\label{2.14}\tilde T_{\alpha\alpha}&\approx& -\left(\sum_{q=1}^{N}\omega_{(0)q}(\bar\varepsilon_{q}+\varepsilon_{q}^{1}) \right) g_{\alpha\alpha}\, ,\quad\alpha=1,2,3\, ,\\
\label{2.15} \tilde T_{\mu\mu}&\approx&-\left(\sum_{q=1}^{N}\omega_{(1)q}(\bar\varepsilon_{q}+\varepsilon_{q}^{1})\right) g_{\mu\mu}\, ,\quad\mu=4,5,...,D\, , \end{eqnarray}
where the corrections $\varepsilon^1_{q}$ are of the same order of magnitude as the perturbation $\hat \rho c^2$.

We suppose that the perturbed metric preserves its diagonal form. Obviously, the off-diagonal coefficients $g_{0\alpha}, \, \alpha =1,\ldots ,D$, are absent for the
static metric. It is also clear that in the case of the uniformly smeared (over the internal space) gravitating mass, the perturbed metric coefficients (see functions
$A,B,C,D$ and $G$ below) depend only on $x,y,z$ \cite{EZ2}, and the metric structure of the internal space does not change, i.e.
$g_{\mu\mu}=g_{DD}\prod_{\nu=\mu+1}^{D}\sin^2\xi_{\nu}\, , \mu=4,\ldots ,D-1$. The latter statement can be proved, e.g., in the weak field approximation from the
Einstein equations (see appendix B in \cite{ChEZ2}). It is also easy to show that in this case the spatial part of the external metric can be diagonalized by coordinate
transformations.
Therefore, the perturbed metric reads
\ba{2.16} \fl ds^2&=&Ac^2dt^2+Bdx^2+Cdy^2+Ddz^2+G\left[\sum_{\mu=4}^{D-1}\left( \prod_{\nu=\mu+1}^{D}\sin^2\xi_{\nu} \right)d\xi_{\mu}^2+d\xi_D^2\right] \end{eqnarray}
with
\ba{2.17}
&{}& A \approx 1+A^{1}({r}_3),\quad B \approx -1+B^{1}({r}_3),\quad C \approx -1+C^{1}({r}_3)\, ,\nonumber\\
&{}& D \approx -1+D^{1}({r}_3),\quad G \approx -a^2+G^{1}({r}_3)\, .\end{eqnarray}
All metric perturbations $A^1,B^1,C^1,D^1,G^1$ are of the order of quantities $\varepsilon^1_q$. To find these corrections as well as the background matter perturbations
$\varepsilon^1_q$, we should solve the Einstein equation
\be{2.18}
R_{ik}=\kappa\left( T_{ik}-\frac{1}{2+d}T g_{ik} -\frac{2}{2+d} \Lambda_{\mathcal{D}} g_{ik}\right)\, ,
\end{equation}
where the energy-momentum tensor $T_{ik}$ is the sum of the perturbed background $\tilde T_{ik}$ \rf{2.13}-\rf{2.15} and the energy-momentum tensor of the perturbation
$\hat T_{ik}$. First, we would like to note that the diagonal components of the Ricci tensor for the metric \rf{2.16} up to linear
terms $A^1,B^1,C^1,D^1,G^1$ are
\ba{2.19}
R_{00} &\approx& \frac{1}{2}\triangle_3 A^1\, ,\nonumber \\
R_{11}&\approx&\frac{1}{2}\triangle_3 B^1
+\frac{1}{2}\left(-A^1-B^1+C^1+D^1+d\frac{G^1}{a^2}\right)_{xx}\, ,\nonumber \\
R_{22}&\approx&\frac{1}{2}\triangle_3 C^1
+\frac{1}{2}\left(-A^1+B^1-C^1+D^1+d\frac{G^1}{a^2}\right)_{yy}\, ,\nonumber \\
R_{33}&\approx&\frac{1}{2}\triangle_3 D^1
+\frac{1}{2}\left(-A^1+B^1+C^1-D^1+d\frac{G^1}{a^2}\right)_{zz}\, ,\nonumber \\
R_{DD}&\approx& d-1+\frac{1}{2}\triangle_3 G^1\, , \end{eqnarray}
where $\triangle_3=\partial^2/\partial x^2+\partial^2/\partial y^2+\partial^2/\partial z^2$ is the three-dimensional Laplace operator. Additionally, for the static
metric \rf{2.16}, where the coefficients $A,B,C,D$ and $G$ depend only on $x,y,z$, there is the following relation:
\be{2.20}
R_{\mu\mu} = R_{DD}\prod_{\nu=\mu+1}^{D}\sin^2\xi_{\nu}\, ,\quad \mu=4,\ldots ,D-1\, .
\end{equation}
Concerning the off-diagonal components of the Ricci tensor, they should be equal to zero according to the Einstein equation \rf{2.18}. Taking into account the relations
$g_{\mu\mu}=g_{DD}\prod_{\nu=\mu+1}^{D}\sin^2\xi_{\nu}\, , \mu=4,\ldots ,D-1$, and that the metric coefficients $A,B,C,D$ and $G$ depend only on $x,y,z$, we can easily
verify that all off-diagonal components are identically equal to zero except the components $R_{12}, R_{13}$ and $R_{23}$. Equating these components to zero, we obtain
the following relations between the metric coefficients:
\be{2.21}
B^1=C^1=D^1
\end{equation}
and
\be{2.22}
A^1-B^1-\frac{d}{a^2}G^1=0\, ,
\end{equation}
which demonstrate that the expressions in brackets for 11, 22 and 33 components in \rf{2.19} vanish. Therefore, in the weak field limit the Einstein equation \rf{2.18}
is reduced to the following system of equations:
\ba{2.23}
&{}&\fl\triangle_3 A^1 =\frac{1+d+d\Omega}{1+d/2}\kappa\hat\rho c^{2} +\frac{\kappa}{1+d/2}\sum_{q=1}^{N}(d+1+3\omega_{(0)q}+d\omega_{(1)q})\varepsilon^{1}_q\, ,\\
&{}&\fl\triangle_3 B^1 =\frac{1-d\Omega}{1+d/2}\kappa\hat\rho c^{2}+\frac{\kappa}{1+d/2}\sum_{q=1}^{N}(1-\omega_{(0)q}+d\omega_{(0)q}-d\omega_{(1)q})\varepsilon_{q}^1\,
,
\label{2.24}\\
&{}&\fl\triangle_3 G^1 =\frac{1+2\Omega}{1+d/2}\kappa a^{2}\hat\rho c^{2}- \frac{2(d-1)}{a^2} G^1  +\frac{\kappa
a^2}{1+d/2}\sum\limits_{q=1}^N(1-3\omega_{(0)q}+2\omega_{(1)q})\varepsilon_q^1 \, ,\label{2.25}\end{eqnarray}
where we have used relations \rf{2.11} and \rf{2.12}. From these equations and the condition \rf{2.22} we obtain the connection between perturbations $G^1$ and
$\varepsilon^1_q$:
\be{2.26}
G^1=-\frac{2\kappa
a^4}{d(d-1)}\sum\limits_{q=1}^N\omega_{(0)q}\varepsilon_{q}^1\, .
\end{equation}

We have four equations \rf{2.23}-\rf{2.26} to define $3+N$ unknown functions $A^1,B^1,G^1$ and $\varepsilon^1_q$. This is impossible in the general case $N>1$ without
some additional assumptions. However, for $N=1$, we can solve these equations exactly and this is the subject of the next section.

\section{\label{sec:3}Single-component background ($N=1$). Effective energy-momentum tensor for gravitating mass}

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Before we consider the general case $N>1$, it is instructive to explore in detail an exactly solvable single-component model $N=1$. For the single-component background,
it makes sense to drop the index $q$. Therefore, $\bar{\mathcal{E}}=\bar\varepsilon_1\equiv \bar\varepsilon$, $\bar P_0=\bar p_{(0)1}\equiv \bar p_0$, $\bar P_1=\bar
p_{(1)1}\equiv \bar p_1$, $\omega_{(0)1}\equiv \omega_0$ and $\omega_{(1)1}\equiv \omega_1$. Eq. \rf{2.8} demonstrates that the single-component background matter has
the vacuum-like equation of state in the external space:
\be{3.1a}
\bar{p}_0=\omega_0 \bar \varepsilon\, , \quad \omega_0 =-1,
\end{equation}
but the equation of state in the internal space is not fixed:
\ba{3.2a}
&{}&\bar{p}_1=\omega_1\bar \varepsilon \, , \nonumber\\
&{}& \omega_1 = \frac{2\Lambda_{\mathcal{D}}\kappa a^2-(d-1)(d-2)}{d(d-1)-2\Lambda_{\mathcal{D}}\kappa a^2}\, \, ,
\end{eqnarray}
i.e. $\omega_1$ is arbitrary.  Choosing different values of $\omega_1$ (with fixed $\omega_0 = -1$), we can simulate different forms of matter. For example, $\omega_1=1$
corresponds to the monopole form-fields (the Freund-Rubin scheme of compactification \cite{FR})\footnote{In our case, they are $d$-forms (see, e.g., Eqs. (2.9) and (5.1)
in \cite{Zhuk}).}, and for the Casimir effect we have $\omega_1=4/d$ \cite{exci}. It is worth noting that the parameter $\omega_1$ can be positive only in the presence
of a positive bare cosmological constant $\Lambda_{\mathcal{D}}$. Moreover, it takes place only if $d-2<2\Lambda_{\mathcal{D}}\kappa a^2/(d-1)<d$. In contrast to the
model with the two-dimensional ($d=2$) sphere in \cite{ChEZ2}, the parameter $\omega_1$ does not disappear in the case of vanishing $\Lambda_{\mathcal{D}}$ for $d\geq
3$.

Taking into account that the relation \rf{2.26} has now the form
\be{3.3a}
\kappa\varepsilon^{1}=\frac{d(d-1)}{2  a^{4}}G^1\, ,
\end{equation}
the system of Eqs. \rf{2.23}-\rf{2.25} reads:
\ba{3.4a}
&{}&\triangle_3 \left(A^1-\frac{d}{2a^2}G^1\right)=\kappa \hat\rho c^{2}=\frac{8\pi G_N}{c^2}\hat\rho_3\, ,\\
&{}&\triangle_3 \left(B^1+\frac{d}{2a^{2}}G^1\right)=\kappa \hat\rho c^{2}=\frac{8\pi G_N}{c^2}\hat\rho_3\label{3.5a}\, ,\\
&{}&\triangle_3 G^1 -\lambda^{-2}G^1=\frac{2(1+2\Omega)}{2+d}\kappa a^{2}\hat\rho c^{2} =a^2\frac{2(1+2\Omega)}{2+d}\frac{8\pi G_N}{c^2}\hat\rho_3\, ,\label{3.6a} \end{eqnarray}
where
\be{3.7a}
\lambda^2\equiv \frac{(d+2)a^2}{2(d-1)(d-2+d\omega_1)}
\end{equation}
and we introduced the Newton gravitational constant
\be{3.8a}
4\pi G_N = \frac{S_D \tilde G_\mathcal{D}}{V_{int}}\, .
\end{equation}
Let us consider now the point-like (in the external space) approximation for the gravitating objects: $\hat \rho_3(r_3)=m\delta ({\bf r}_3)$. The generalization of the
obtained results to the case of extended compact objects is obvious. It is well known that to get the physically reasonable solution of \rf{3.6a} with the boundary
condition $G^1 \to 0$ for $r_3 \to +\infty$, the parameter $\lambda^2$ should be positive, i.e. the equation of state parameter $\omega_1$ should satisfy the condition
\be{3.9a}
\omega_1> -1+\frac{2}{d}\, ,
\end{equation}
which allows also negative values of $\omega_1$. From Eq. \rf{3.2a} we can get the corresponding restrictions for the bare cosmological constant:
\be{3.10a}
0<\frac{2\Lambda_{\mathcal{D}}\kappa a^2}{d-1}<d\, .
\end{equation}
This inequality relaxes the condition of the positiveness of $\omega_1$. Then, the Eqs. \rf{3.4a}-\rf{3.6a} have solutions
\ba{3.11a}
A^1&=&\frac{2\varphi_N}{c^2}+\frac{d}{2a^{2}}G^1\, ,\\
B^1&=&\frac{2\varphi_N}{c^2}-\frac{d}{2a^{2}}G^1\, ,\label{3.12a}\\
G^1&=&a^2\frac{4\varphi_N}{(2+d)c^2}(1+2\Omega)\exp\left(-\frac{r_3}{\lambda}\right)\, ,\label{3.13a}
\end{eqnarray}
where the Newtonian potential $\varphi_N=-G_N m/r_3$. It is well known that the metric correction term $A^1\sim O\left(1/c^2\right)$ describes the non-relativistic
gravitational potential: $A^1=2\varphi/c^2$. Therefore, this potential acquires the Yukawa correction term:
\be{3.14a}
\varphi = \varphi_N\left[1+\frac{d}{2+d}(1+2\Omega)\exp\left(-\frac{r_3}{\lambda}\right)\right]\, . \end{equation}

The inequalities \rf{3.9a} and \rf{3.10a} provide the condition of the internal space stabilization. Obviously, the monopole form-field ansatz with $\omega_1=1$
satisfies this condition. Let us consider this example in more detail. From the fine-tuning relation \rf{3.2a} we obtain $2\Lambda_{\mathcal{D}}\kappa a^2=(d-1)^2$.
Precisely this quantity provides in an effective dimensionally reduced model a zero value of the effective four-dimensional cosmological constant (see Eq. (5.7) in
\cite{Zhuk} where we should make the substitutions $\Lambda_{D} \to \Lambda_{\mathcal{D}}\kappa ,\, d_1 \to d ,\, D \to 4+d $ and where $\tilde R_1=d(d-1)/a^2,\, d_0=3,
D_0=4$). It is clear that in our case with flat background external spacetime, the effective four-dimensional cosmological constant should vanish. Additionally, this
value of $\Lambda_{\mathcal{D}}$ satisfies the stability condition (5.15) in \cite{Zhuk}. Moreover, the gravexciton/radion mass squared (5.12) (with substitution (5.11))
exactly coincides with the Yukawa mass squared $m_{Yu}^2=\lambda^{-2}=4(d-1)^2/[(d+2)a^2]\equiv m_{exci}^2$. Therefore, we arrived at the very natural and important
conclusion that the Yukawa mass is defined by the mass of the gravexcitons/radions.

It is of interest to estimate the Yukawa correction term in the formula \rf{3.14a}. It is worth mentioning that in our KK model, all gravitating masses (e.g., the balls
in the inverse square law experiment or the Sun and planets in the solar system) have in the non-relativistic limit the gravitational potential of the form \rf{3.14a}.
For reasonable values of the equation of state parameter $|\Omega|\sim O(1)$, the Yukawa parameter $\alpha$ (this is the standard notation for the prefactor in front of
the exponential function in the Yukawa potential) is also of the order of 1. Then, the inverse square law experiments restrict the characteristic range of the Yukawa
interaction \cite{new}: $\lambda \lesssim 10^{-3}$cm. Obviously, for the above-mentioned gravitational experiments in the solar system $r_3\gtrsim r_{\odot}\sim 7\times
10^{10}$cm, and the ratio $r_3/\lambda \gtrsim 10^{14}$. On the other hand, the collider experiments also restrict the sizes of the extra dimensions for Kaluza-Klein
models\footnote{We mean the KK models without branes. Precisely such models are considered in our paper.} \cite{AHN}: $a\lesssim 10^{-17}$cm. Since $\lambda \sim a$ (see
Eq. \rf{3.7a}), then for considered gravitational experiments in the solar system $r_3/\lambda \gtrsim 10^{28}$. It is clear that for such ratios we can drop the Yukawa
correction terms in Eqs. \rf{3.11a} and \rf{3.12a}. Therefore, the post-Newtonian parameter $\gamma=B^1/A^1$ is equal to 1 with extremely high accuracy for any value of
$\Omega$ including the most physically reasonable case of the dust-like value $\Omega=0$. The case $\Omega=-1/2$ is a special one, and we consider it below. Thus, our
model satisfies the gravitational tests (the deflection of light and the time delay of radar echoes) for any value of the equation of state parameter $\Omega$ in the
internal space.

So, at first glance, it seems that we have found a model, which, on the one hand, satisfies the gravitational experiments and, on the other hand, may not contain tension
(i.e. $\Omega =0$) in the internal space. However, let us examine in detail the energy-momentum tensor of the gravitating mass. As follows from Eqs. \rf{3.3a} and
\rf{3.13a}, the background perturbation $\varepsilon^1$ is localized around the gravitating object and falls exponentially with the distance $r_3$ from this mass.
Therefore, the bare gravitating mass is covered by this "coat".  For an external observer, this coated gravitating mass is characterized by the effective energy-momentum
tensor with the following nonzero components:
\ba{3.15a}
&{}&T_{0}^{0(eff)}\approx \varepsilon^1+\hat\rho({\bf r}_3)c^2=\nonumber \\
&{}&-(1+2\Omega)\frac{d(d-1)mc^2}{4(2+d)\pi V_{int}a^2}\frac{1}{r_3}\exp\left(-\frac{r_3}{\lambda}\right)+\frac{mc^2\delta({\bf
r}_3)}{V_{int}}\, ,\\
&{}&T_{\alpha}^{\alpha (eff)}\approx \varepsilon^1=\nonumber\\
&{}&-(1+2\Omega)\frac{d(d-1)mc^2}{4(2+d)\pi V_{int}a^2}\frac{1}{r_3}\exp\left(-\frac{r_3}{\lambda}\right),\quad\alpha=1,2,3\, ,\label{3.16a}\\
&{}&T_{\mu}^{\mu(eff)}\approx-\omega_1\varepsilon^1-\Omega\hat\rho({\bf r}_3) c^2=\nonumber\\
&{}&\omega_1(1+2\Omega)\frac{d(d-1)mc^2}{4(2+d)\pi
V_{int}a^2}\frac{1}{r_3}\exp\left(-\frac{r_3}{\lambda}\right)-\frac{\Omega mc^2\delta({\bf r}_3)}{V_{int}}\, ,\label{3.17a}\\
&{}&\mu=4,\ldots ,D\, ,\nonumber \end{eqnarray}
which, for illustrative purposes, can be presented as follows:
\ba{3.18a}
&{}&T_{0}^{0(eff)}\approx \frac{mc^2\delta({\bf r}_3)}{V_{int}}\left[1-\frac{d(1+2\Omega)}{2(d-2+d\omega_1)}\right]\, , \\
&{}&T_{\alpha}^{\alpha(eff)}\approx-\frac{mc^2\delta({\bf r}_3)}{V_{int}}\frac{d(1+2\Omega)}{2(d-2+d\omega_1)}\, ,\label{3.19a}\\
&{}&\alpha=1,2,3\, ,\nonumber\\
&{}&T_{\mu}^{\mu(eff)}\approx\frac{mc^2\delta({\bf r}_3)}{V_{int}}\, \frac{d\omega_1-2(d-2)\Omega}{2(d-2+d\omega_1)}\label{3.20a}\, ,\\
&{}&\mu=4,\ldots ,D\, ,\nonumber
\end{eqnarray}
where we have replaced the rapidly decreasing exponential function by the delta function:
\be{3.21a}
\frac{1}{r_3}\exp\left(-\frac{r_3}{\lambda}\right)\rightarrow\int\frac{1}{r'_3}\exp\left(-\frac{r'_3}{\lambda}\right)dV'_3 \times \delta({\bf r}_3)
=4\pi\lambda^2 \delta({\bf r}_3)\, . \end{equation}
The less the parameter $\lambda$ is, the better this transformation is executed\footnote{\label{delta function}By definition of the delta-function
$$
\int f({\bf r}_3)\delta({\bf r}_3)d{\bf r}_3=f(0)\, ,
$$
that holds for any function $f({\bf r}_3)$ continuous at ${\bf r}_3=0$. It can be easily seen that
$$
\lim_{\lambda\to 0} \int f({\bf r}_3)\frac{1}{4\pi\lambda^2r_3}\exp\left(-\frac{r_3}{\lambda}\right)d{\bf r}_3=f(0)\, .
$$
Therefore,
$$
\lim_{\lambda\to 0} \frac{1}{4\pi\lambda^2r_3}\exp\left(-\frac{r_3}{\lambda}\right)=\delta({\bf r}_3)\, .
$$
}. On the other hand, the smaller the characteristic scale $\lambda$ of the Yukawa interaction is, the better concordance with the gravitational tests takes place for
our model. Obviously, taking into account the limitation $\lambda \lesssim 10^{-17}$cm and characteristic sizes of massive bodies, we can conclude that all corrections
connected with $G^1$ are negligible for the gravitational potentials outside of these bodies including atomic nuclei and up to astrophysical objects (see, e.g., Eqs.
\rf{3.11a}-\rf{3.13a}).  Unfortunately, this model faces the problem from the thermodynamical point of view, and this problem does not disappear, even in the case of
perfect concordance with the gravitational tests, i.e. for $\lambda \to 0$. The problem consists in the relativistic nature of $T^{\alpha (eff)}_{\alpha}$ in \rf{3.16a}
but not in the delta-function distribution, which we use for illustrative purposes. This equation (as well as Eq. \rf{3.19a}) demonstrates that the gravitating mass
acquires the effective relativistic pressure $\hat p_{0}^{(eff)}=-T_{\alpha}^{\alpha (eff)}$ in the external/our space. This relativistic component is concentrated in
the immediate vicinity of the point ${\bf r}_3=0$ and is carried along with the gravitating mass. This is the crucial point. To detect such pressure, we should place a
wall or a membrane in the path of a moving gravitating mass. The relativistic pressure should be felt at the moment when the mass crosses the wall or the membrane.
Clearly, it is hardly possible to carry out such experiment for the astrophysical objects. However, the formulas obtained above are suitable for any gravitating mass,
not only for astrophysical objects. For example, we can apply these expressions to a system of nonrelativistic particles forming, e.g., a gas of nonrelativistic
molecules. We can consider them as a system of point-like particles. This approach is well grounded in statistical physics (see, e.g., chapters I and X in
\cite{Klimontovich}) Then, as we have shown above, each of these particles is covered by the coat and this coat accompanies the moving particle. Let us put this system
in a box divided by a porous membrane. Obviously, Eqs. \rf{3.15a}-\rf{3.17a} (or Eqs. \rf{3.18a}-\rf{3.20a}) can be considered as ones in the co-moving frame with
respect to a particle. With the help of the Lorentz transformation, we can rewrite them in a laboratory system of coordinates (we shall denote it by the sign tilde)
connected with the box. If $v$ is a velocity of a molecule in the direction of the z axis ($\alpha =3$) which is perpendicular to the membrane, then 11 and 22 mixed
components remain the same (up to the trivial transformation of $r_3$) and the 33 component consists of two parts: $T_{\tilde 3}^{\tilde 3}=T_{\tilde 3}^{\tilde 3(eff)}
+ T_{\tilde 3}^{\tilde 3(bare)}$. The first part coincides with the 33 component in the co-moving frame: $T_{\tilde 3}^{\tilde 3(eff)}=T_{3}^{ 3(eff)}\sim mc^2$ up to
the transformation of $r_3$, i.e. it has the relativistic nature. The second part is connected with the non-relativistic motion of the particle: $T_{\tilde 3}^{\tilde
3(bare)} \sim m v^2$. These molecules move through the membrane to an empty part of the box.  It is well known that
momentum crossing the elementary spatial area $dx^{\tilde{\gamma}}\wedge dx^{\tilde{\delta}}$ per unit time is given by $T_{\tilde{\alpha}}^{\tilde{\beta}}
\varepsilon_{\tilde{\beta}|\tilde{\gamma}\tilde{\delta}|}dx^{\tilde{\gamma}}\wedge dx^{\tilde{\delta}}\, ,\; \tilde{\alpha},\tilde{\beta},\tilde{\gamma} = 1,2,3 $. In
the case of our membrane we get
\be{3.22a}
T^{\tilde 3}_{\tilde 3} d\tilde x\wedge d\tilde y = \left(T^{\tilde 3(eff)}_{\tilde 3}+T^{\tilde 3(bare)}_{\tilde 3}\right)d\tilde x\wedge d\tilde y\, .
\end{equation}
Therefore, the non-relativistic particles have relativistic momentum crossing the membrane and resulting in relativistic pressure.

We can also get the relativistic pressure by averaging $T_{\alpha}^{\alpha (eff)}$ over a three-dimensional volume $V$:
\ba{3.23a}
\fl \overline{T}_{\alpha}^{\alpha (eff)}&\approx&-(1+2\Omega)\frac{d(d-1)mc^2}{4(2+d)\pi
V_{int}a^2}
\times\frac{1}{V}\int\frac{1}{r_3}\exp\left(-\frac{r_3}{\lambda}\right)dV\nonumber \\
\fl &\approx&
-(1+2\Omega)\frac{d(d-1)mc^2}{4(2+d)\pi
V_{int}a^2}\frac{4\pi\lambda^2}{V}
=-\frac{d(1+2\Omega)c^2}{2(d-2+d\omega_1)}\frac{m}{V_{int}V}\, ,
\end{eqnarray}
where we dropped the exponentially small terms. To get usual three-dimensional quantities, e.g., the pressure measured as $\mbox{erg/cm}^3$, we should multiply Eq.
\rf{3.23a} by $V_{int}$. Taking this remark into account and multiplying both sides of this equation by the total number $N$ of particles in the volume $V$, we obtain the
expression for the effective pressure of these particles in $V$:
\be{3.24a}
\hat P_0^{(eff)}\sim \rho c^2\, ,
\end{equation}
where $\rho =Nm/V$ is the rest-mass density and we did not take into account the part of the pressure connected with non-relativistic motion of these particles. Of
course, such relativistic pressure contradicts the observations. It can be easily seen that the equality $\Omega =-1/2$ is the only possibility to achieve $\hat
p_{0}^{(eff)}=0$ for our model. It means that the bare gravitating mass should have tension with the equation of state $\hat p_1=-\hat \varepsilon /2$ in the internal
space. Then, the effective and bare energy densities coincide with each other and the gravitating mass remains pressureless in our space. In the internal space the
gravitating mass still has tension with the parameter of state $-1/2$. Therefore, to be in agreement with observations, the presence of tension is a necessary condition
for the considered model. However, we still do not know a physically reasonable explanation for the origin of relativistic tension ($\Omega=-1/2$) for non-relativistic
gravitating objects such as molecules, massive balls or our Sun. Moreover, as it was shown in the paper \cite{ChEZ4}, the presence of tension may result in difficulties
in the many-body problem for KK models.

\section{\label{sec:4}Multicomponent background $N>1$. Particular examples}

\setcounter{equation}{0}

Let us turn now to the general case $N>1$. As we have already mentioned at the end of Sec. 2, we need to assume some additional constraints to define perturbations
$A^1,B^1,G^1$ and $\varepsilon^1_q$ which satisfy Eqs. \rf{2.23}-\rf{2.26}. In this section, we provide two exactly solvable examples. Here, the additional constraint
(the fine-tuning condition) is chosen in such a way that, for the first example, the metric coefficients and the non-relativistic gravitational potential acquire
corrections in the form of the Yukawa potential, as in the previous section. Then, this example satisfies the gravitational tests in the solar system if the Yukawa
potential is negligible. For the second example, the fine-tuning condition will provide the condition of the constant/unperturbed internal space: $G^1=0$, as it takes
place for black string/branes with toroidal compactification. Such example satisfies the gravitational tests at the same level of accuracy as general relativity.

\subsection{Yukawa corrections}

In the first example, we generalize the solutions from the previous section, i.e. we shall find the Yukawa correction term to the nonrelativistic gravitational
potential. Above, we have shown that such term  arises in the case of stable compactification of the internal spaces (this is equivalent to the conditions \rf{3.9a} and
\rf{3.10a} in the single-component case), and the mass of the Yukawa interaction is defined by the mass of gravexciton/radion. A perfect fluid stabilizes the internal
space in the case of the vacuum-like equation of state in the external space \cite{EZ4,Zhuk}. Therefore, in this subsection we assume that
\be{3.1} \omega_{(0)q} = -1, \quad q=1,\ldots ,N\, . \end{equation}
Note that with this choice of the parameters $\omega_{(0)q}$, Eq. \rf{2.8} is satisfied automatically.
Hence, Eq. \rf{2.26} is reduced to
\be{3.2}
G^1=\frac{2\kappa
a^4}{d(d-1)}\sum\limits_{q=1}^N\varepsilon_{q}^1\neq 0\, .
\end{equation}
The case of the zero sum in \rf{3.2} (i.e. the zero perturbation $G^1$) will be considered in the next subsection. Additionally, we assume the following fine-tuning
condition:
\be{3.3}
\sum\limits^N_{\begin{array}{cc}
q,p=1\\
q\ne p
\end{array}}\omega_{(1)q}\varepsilon^1_p=0\, .
\end{equation}
Together with the relation \rf{3.2}, this is equivalent to the following condition:
\be{3.4} \left(\sum\limits_{q=1}^N\omega_{(1)q}\right) G^1= \frac{2\kappa a^4}{d(d-1)} \sum\limits_{q=1}^N\omega_{(1)q}\varepsilon_{q}^1\, . \end{equation}
Then, Eq. \rf{2.25} is exactly reduced to \rf{3.6a} where
\be{3.6} \lambda^{-2}\equiv \frac{1}{a^2(2+d)}\left[2(d-1)\left(d-2+d\sum\limits_{q=1}^N\omega_{(1)q}\right)\right] \end{equation}
generalizes the definition \rf{3.7a}. The positiveness of $\lambda^2$ is the necessary condition of the internal space stabilization. For positive $\lambda^2$,
that takes place if the equation of state parameters $\omega_{(1)q}$ satisfy the
condition
\be{3.8}
\sum\limits_{q=1}^N\omega_{(1)q}> -1+\frac{2}{d}\, ,\quad d\geq 2\, ,
\end{equation}
the solution $G^1$ has the form \rf{3.13a} (in the case of the point-like gravitating mass in the external space: $\hat \rho_3(r_3)=m\delta ({\bf r}_3)$).

It can be easily verified that, provided that the equations \rf{3.1}-\rf{3.4} hold, the equations \rf{2.23} and \rf{2.24} take the form of \rf{3.4a} and \rf{3.5a} with
solutions \rf{3.11a} and \rf{3.12a}. Therefore, the nonrelativistic gravitational potential $\varphi$ is given by the formula \rf{3.14a}, i.e. it has the Yukawa
correction term.

As a particular example, we consider the two-component model $(N=2)$ with the monopole form field $\omega_{(1)1}=1$ and vacuum fluctuations of quantized fields (Casimir
effect) $\omega_{(1)2}=4/d$ \cite{exci,Zhuk}. For these perfect fluids $\omega_{(0)1}=\omega_{(0)2}=-1$. Real-valued solitonic/monopole form fields naturally arise as
Ramond-Ramond form fields in the type II string theory and M-theory \cite{Stelle}. It is also well known that because of nontrivial topology of our multidimensional
spacetime (i.e. compactness of the internal space), vacuum fluctuations of quantized fields result in the nonzero energy density (the Casimir effect) \cite{Casimir}.
Therefore, from Eqs. \rf{3.2} and \rf{3.3} we get
\be{3.15} \varepsilon^1_1=-\frac{d}{4}\varepsilon^1_2=\frac{d^2(d-1)}{2(d-4)\kappa a^4} G^1\, . \end{equation}
The case $d=4$ is excluded for this particular example because it contradicts the inequality \rf{3.2} $G^1\neq 0$.

We have shown in the previous section that the Yukawa correction terms to the metric coefficients are negligible for gravitating masses in the solar system.
Therefore, considered in this subsection examples satisfy the gravitational tests for any reasonable values $\Omega$ including the
most interesting dust-like case $\Omega=0$. Unfortunately, the nonrelativistic gravitating matter source acquires
effective relativistic pressure in the external/our space (remind that $\omega_{(0)q}=-1\, , \forall\,  q$):
\be{3.16} \hat p_0^{(eff)}=-T_{\alpha}^{\alpha (eff)}\approx -\sum\limits_{q=1}^N\varepsilon^1_q \sim m c^2\, , \quad \alpha = 1,2,3\, , \end{equation}
which, certainly, is unacceptable from the thermodynamical point of view. The value $\Omega=-1/2$ is the only possibility to avoid it, i.e. the gravitating source should
have tension in the internal space. However, this is not the desired result. Therefore, in the next subsection we construct a model without tension, satisfying
observations from both gravitational and thermodynamical points of view.

\subsection{$\grave{\mbox{A}}$ la black brane with zero tension}

Let us assume now the following fine-tuning condition:
\be{3.17}
\sum_{q=1}^{N}\omega_{(0)q}\varepsilon_{q}^{1}=0\quad\Leftrightarrow\quad G^{1}=0\, ,
\end{equation}
where the latter equation follows from Eq. \rf{2.26}. If all parameters of equations of state in the external space are equal to each other ($\omega_{(0)1}=\ldots
=\omega_{(0)N}$), then Eq. \rf{3.17} is reduced to
\be{3.18}
\sum_{q=1}^{N}\varepsilon_{q}^{1}=0\, ,
\end{equation}
but we do not specially impose this condition. Taking into account Eq. \rf{3.17}, we get from \rf{2.25} the following additional relation:
\be{3.19}
\sum\limits_{q=1}^N\omega_{(1)q}\varepsilon_q^1=-\frac{1}{2}(1+2\Omega)\hat\rho c^2-\frac{1}{2}\sum\limits_{q=1}^N\varepsilon_q^1\, .
\end{equation}
With the help of this relation and the condition \rf{3.17}, Eqs. \rf{2.23} and \rf{2.24} read
\ba{3.20}
\triangle_3 A^1&=& \kappa\hat \rho c^2+\kappa\sum_{q=1}^{N}\varepsilon_q^1\, ,\\
\label{3.21} \triangle_3 B^1 &=& \kappa\hat \rho c^2+\kappa\sum_{q=1}^{N}\varepsilon_q^1\, . \end{eqnarray}
We may conclude from these equations that metric coefficients $A^1=B^1$ for any value of $\Omega$ including the dust-like case $\Omega=0$. Therefore, the PPN parameter
$\gamma =1$ in full analogy with general relativity. So, we achieved in this model agreement with the gravitational tests (the deflection of light, the Shapiro effect)
in the solar system. To get the Newtonian limit, we should either consider the case \rf{3.18}, or assume that $\sum_{q=1}^N \varepsilon^1_q \sim \hat \rho c^2$. The
latter case results in renormalization of the multidimensional gravitational constant $\kappa$. The Newton gravitational constant is defined by Eq. \rf{3.8a} and the
nonrelativistic gravitational potential coincides exactly with the Newtonian expression:
\be{3.21}
A^1=\frac{2\varphi_N}{c^2}\, .
\end{equation}

One of the main features of this model consists in the constant/unperturbed internal space because of $G^1=0$. It means that the conformal prefactor for the internal
space metric was not changed (up to $O(1/c^2)$): $G=-a^2=\mathrm{const}$. A similar situation takes place for black string/branes. Usually, they have the unperturbed
toroidal internal space. However, there is also generalization to unperturbed  spherical compactification \cite{ChEZ3}. For the considered in this subsection model, we
have shown (up to $O(1/c^2)$) that external spacetime metric is the Schwarzschild one and the internal space has the unperturbed spherical metric. The main advantage
of this model with respect to the black branes is a possibility to eliminate tension of the gravitating mass due to the dust-like choice $\Omega=0$. Because of this
difference we call this case "$\grave{\mbox{a}}$ la black brane".

This model is also satisfactory from the thermodynamical point of view because due to the condition \rf{3.17} the effective relativistic pressure (which is related to
the excitation of the background matter in Eq. \rf{2.14}) in the external/our space is absent:
\be{3.22}
\hat p_0^{(eff)}=-T_{\alpha}^{\alpha (eff)}\approx \sum\limits_{q=1}^N\omega_{(0)q}\varepsilon^1_q=0\, , \quad \alpha = 1,2,3\, .
\end{equation}

As a particular example, we consider the two-component perfect fluid from the previous subsection where the monopole form field is characterized by the following
parameters of equations of state: $\omega_{(0)1}=-1,\, \omega_{(1)1}=1$, and for the Casimir effect we have $\omega_{(0)2}=-1,\, \omega_{(1)2}=4/d$. Hence, Eq. \rf{2.8}
is automatically satisfied. Then, for the most interesting dust-like case $\Omega =0$ we get from Eqs. \rf{3.18} and \rf{3.19} the following relations:
\be{3.23}
\varepsilon^1_1=-\varepsilon^1_2 = - \frac{d}{2(d-4)}\hat \rho c^2=- \frac{d}{2(d-4)}\frac{m c^2\delta({\bf r}_3)}{V_{int}}\, . \end{equation}
The case $d=4$ is excluded for this particular example because it contradicts Eq. \rf{3.19} (here, $\omega_{(1)1}=\omega_{(1)2}$, and from Eq. \rf{3.19} we get a
non-physical result $\hat \rho =0$).

To conclude our investigations, we want to make the following remark. At the very end of Sec. 3, we wrote that we do not know a physically reasonable explanation for the
origin of relativistic tension ($\Omega=-1/2$) in the internal space for a non-relativistic gravitating object. However, our last example demonstrates how such an object
can acquire effective tension in the internal space. Let us consider the case where the bare parameter $\Omega$ for this gravitating mass is zero: $\Omega=0$.
Then, from Eqs. \rf{2.15} and \rf{3.19} we obtain the expression for the effective pressure of this object in the internal space:
\be{3.24}
\hat p_1^{(eff)}=-T_{\mu}^{\mu (eff)} \approx \sum_{q=1}^N \omega_{(1)q}\varepsilon^1_q =-\frac12 \hat\rho c^2\, ,\quad \mu =4,5,\ldots ,D\, ,
\end{equation}
where we took into account the relation \rf{3.18}. That is we got an effective tension ($\Omega^{(eff)}=-1/2$). Hence, this is one more reason to call the considered
model $\grave{\mbox{a}}$ la black brane. It is worth noting that this effective tension arises not because of the bare tension of the gravitating mass, but because of
the perturbation of the background matter, and such procedure is physically well motivated.

\section{Conclusion}

In this paper, we investigated the viability of Kaluza-Klein models with spherical compactification of the internal space. To achieve such compactification, we
introduced background matter in the form of a multicomponent perfect fluid. The multicomponent approach is quite natural. For example, because of nontrivial topology of
our multidimensional spacetime (i.e. compactness of the internal space), vacuum fluctuations of quantized fields result in the nonzero energy density (the Casimir
effect). Various variants of the string theory contain real-valued form fields. The model may also include other types of fluids. This matter provides spherical
compactification of the internal space with an arbitrary number of dimensions $d$. We perturbed this background by a compact (in our three-dimensional space) gravitating
source with the dust-like equation of state in the external/our space $\hat p_0=0$ and an arbitrary equation of state $\hat p_1\approx \Omega\hat\rho c^2$ (with the
constant parameter $\Omega$) in the internal space. We assumed that this matter source is uniformly smeared over the internal space. In the weak-field limit (up to the
order $1/c^2$), we obtained equations for the perturbations of the metric and the background matter. It is impossible to solve these equations in the multicomponent case
$N>1$ because the number of equations is less than the number of unknowns. Hence, we need to introduce some additional fine-tuning relations.

However, a single-component case ($N=1$) is exactly solvable without additional constraints. Hence, we investigated this model in detail. The perturbed (up to
$O\left(1/c^2\right)$) metric coefficients were found from the Einstein equations. For the external space, these coefficients consist of two parts: the standard general
relativity expressions plus the admixture of the Yukawa interaction. The Yukawa interaction arises only in the case when the background matter satisfies some condition
(see the inequality \rf{3.9a}) for the parameter of the equation of state in the internal space which is equivalent to the condition of the internal space stabilization.
From the cosmological point of view, such stabilization was considered in papers \cite{exci,Zhuk} (see also the appendix in \cite{EZ4}). The stabilization takes place if
conformal excitations of the internal spaces (referred to as gravexcitons \cite{exci} or radions) acquire the positive mass squared. In our paper, we have got an
important result that the mass of the Yukawa interaction is exactly defined by the mass of the gravexciton/radion. In the solar system, the Yukawa mass is big enough for
dropping the admixture of this interaction and getting very good agreement with the gravitational tests for any value of $\Omega$.

Nevertheless, our subsequent investigation of this single-component model showed that the gravitating body acquires the effective relativistic pressure in the
external/our space. Clearly, it is hardly possible to detect such pressure for the astrophysical objects (we can not place a membrane or a wall in the path of these
objects), but any system of nonrelativistic particles such as molecules of a liquid or a gas may have the relativistic momentum crossing any spatial area. Of course, it
contradicts the observations. We have demonstrated that the value $\Omega=-1/2$ (i.e. tension!) is the only possibility to avoid this problem for the single-component
case.

It is worth noting that, in multidimensional models, the standard approach implies that to match the observations it is enough to stabilize the internal space. To
achieve it, the radion should have a relatively large mass. However, the single-component case demonstrates that this is not sufficient. In other words, the big enough
Yukawa/radion mass (for  models with the arbitrary physically reasonable parameter $\Omega$ in the equation of state in the internal space) is not sufficient to have a
good theory.  From the standard point of view, this result is fresh and unexpected.
The models should also possess the thermodynamical properties which do not contradict the observations. This natural demand leads to the unique value $\Omega=-1/2$ in
the case $N=1$, i.e. the gravitating mass should have tension in the internal space (the $d$-dimensional sphere). We have presented the detailed investigation of this
problem. Such investigation was absent before.

Then, we turned to the multicomponent case $N>1$.  Here, we need to assume some additional constraints (fine tuning conditions) to solve the Einstein equations. We
provided two exactly solvable examples.

In the first example (with the fine tuning condition \rf{3.3}), we arrived at the model with the Yukawa correction terms for the metric perturbations, by analogy with
the single-component case. For the sufficiently large Yukawa mass, this model satisfies the gravitational tests. Unfortunately, because of the same reasoning as in the
case $N=1$, the nonrelativistic gravitating matter source acquires the effective relativistic pressure in the external/our space which, certainly, is unacceptable from
the thermodynamical point of view. The value $\Omega=-1/2$ is the only possibility to avoid it, i.e. the gravitating source should have tension in the internal space.

In the next subsection we proposed the other model (with the fine tuning condition \rf{3.17}). Here,
the PPN parameter $\gamma$ is exactly equal to $1$ similar to general relativity. Therefore, this model satisfies the gravitational tests (the deflection of light, the
Shapiro effect) for any reasonable value of $\Omega$. Moreover, the nonrelativistic gravitational potential exactly coincides with the Newtonian one. One of the main
features of this model consists in the constant/unperturbed internal space. It means that the conformal prefactor for the internal space metric was not perturbed by the
gravitating mass. A similar situation takes place for black string/branes. The main advantage of this model with respect to the black branes is a possibility to
eliminate tension due to the dust-like choice $\Omega=0$. This model is also satisfactory from the thermodynamical point of view because the effective relativistic
pressure in the external/our space is absent: $\hat p_0^{(eff)}=0$. Therefore, we demonstrated a principal possibility for the considered model to satisfy the
gravitational and thermodynamical observations in the case of vanishing tension in the internal space for the gravitating mass. Certainly, it happens due to
multicomponent nature of the background matter. Although the fine-tuning conditions impose strong constraints on the model and look artificial, however we think that
they look less artificial than the bare relativistic tension ($\Omega=-1/2$). As an additional bonus for this model, we found a physically reasonable mechanism to
generate an effective tension for a gravitating mass. In this particular model, we demonstrated that a non-relativistic gravitating mass with zero bare tension
($\Omega=0$) acquires an effective tension in the internal space ($\Omega^{(eff)}=-1/2$)  due to the perturbation of the background matter, and such procedure is
physically well motivated.

\ack
This work was supported in part by the "Cosmomicrophysics-2" programme of the Physics and Astronomy Division of the National Academy of Sciences of Ukraine.
We want to
thank the referees for their valuable comments which have improved the presentation of the results.
A.~Zh.
acknowledges the hospitality of the Abdus Salam International Centre for Theoretical Physics and the Theory Division of CERN, where a part of this  work was carried out.
We also want to thank Prof. V.L. Kulinskii for useful discussions.

\section*{References}

redpajama_set_name: RedPajamaArXiv


SOPHISTICATED LADIES is a treasure trove of Duke Ellington' s musical riches that won two Tony Awards, brought fame to Gregory Hines and is still one of the greatest vehicles for tap, song and The Duke's sophisticated jazz.
The musical legacy of Duke Ellington ~ considered amongst the world's greatest composers and musicians ~ is celebrated in this stylish and brassy retrospective that has taken audiences and critics alike by storm throughout the world. SOPHISTICATED LADIES is a glorious recreation of the Duke Ellington big band sound, featuring his most famous numbers from the infamous Speak Easy era of the Cotton Club, the Harlem Renaissance, art deco and high society, including: "It Don't Mean a Thing, If It Ain't Got That Swing", "Take the 'A' Train", "Satin Doll" and the haunting "In a Sentimental Mood".


text:
www.worldwidereview.com
contemporary art forum, art reviews and culture
Board index ‹ Art News
Amusing Panel Discussion on R. B. Kitaj
by CAP » Wed Mar 02, 2016 10:06 am
I came across this You Tube surfing. It's a panel discussion held at L.A. Louver gallery in Los Angeles a few years ago, in remembrance of Ron Kitaj and as prelude to a small show of his late works - which look great incidentally! A few years ago I was disappointed with the late 'washy' approach, but maybe I'm mellowing or something but the drawing to the figures in these really seems to work now. Yay! Wish I could have gone!
But the reason I draw attention to the video is because of Derek Boshier's contribution. As a classmate at the RCA, his recollections are mainly from those days when apparently Ron was a bit obsessed with Quakers for some reason - knew a lot about them, possibly attended some of their meetings. Derek was not the only contemporary to notice this. His comments are really a gentle way of indicating Ron's uh... problem with identity, let's say. Kitaj's famous embrace of his Jewish roots in the seventies is the best known example, but this too turns out to be deeply problematic. Boshier later recounts how at some function in America where Kitaj was being honoured (probably in the 80s) he found himself seated next to Kitaj's mother. After exchanging introductions and explaining his acquaintance with Ron, they then sat through various talks about Kitaj and his fervent embrace of Judaism after which Kitaj's mother quietly turned to Boshier and confided that Ron was not Jewish. She was not Jewish, Ron's father was not Jewish and he was not raised as Jewish. She did not understand why Ron was saying he was.
The sound of gears grinding and wheels spinning throughout the rest of the panel at L.A. Louver is palpable and quite amusing. It looks like it would have been a fun night!
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There seems to be a large number of knowledgeable folks on this board. Hopefully a few of you won't mind another "Which area is better?" question.
We like Arkansas a lot. The thing is, we only really know the state as tourists, and we're aware that we might be missing some important points about life there. So I'm wondering, if you were given a choice of living in either the Harrison - Newton County - "Dogpatch" areas, or the Mena - Cove - Hatfield areas, which would a native choose, and why?
We are moderately conservative in outlook - pro labor conservative - what used to be called "Conservative Democrats". We like rural, secluded living on acreage, not city "postage stamp" lots. We're happy anywhere so long as we can get DSL internet, and the water doesn't taste too bad once it's filtered. We don't have kids, so schools aren't an issue for us.
We like to garden, and areas that are good for fruit and nut trees, tomatos, and old-fashioned roses would be perfect. We don't mind a light snow now and then, but after winters up north, I can't imagine that's likely to be a problem.
We'd like to live in an area that has handymen, plumbers, and carpenters who at least have a basic working knowledge of English. I'm not trying to be mean-spirited, but we are living in Arizona now, and that is a big problem.
Finally, we do like to have a decent - sized city within a 3 or 4 hour drive, if that is possible.
wow, I dan't answer about DSL, we have it here in NWA, but that isn't a place you are considering. Harrison might be a good fit, I don't know anything about Dogpatch. Mena is also popular and because it is south you might have better luck with fruit trees. I will be anxious to see what others have to say.
In 1993 I retired and moved to Edgemont AR from CA, it was the smartest thing we ever did.
Edgemont is located on the North shore of Greers Ferry Lake. This area is known for excellent fishing and hunting with very low crime. The drawback is we have no major shopping in this area.
Our area has only 1,500 population for the three towns, total, 25k in the whole county.
I think it fits your criteria nicely, low crime, people speak English, light snow, small farms, plenty of garden area, and broadband internet. And did I say you can find very reasonable housing, which means low taxes & low living expenses.
It sure fit our criteria.
I'll put in a plug for the Mountain Home area (similar to Harrison size, but a big outdoors area with lakes, rivers, and such, and lots of small towns around it that you might really enjoy), El Dorado (completely different part of the state, but beautiful small city with low land prices and just a nice overall area), Heber Springs (smaller, but beautiful), and Russelville.
As for the two areas you mention- I used to have family that lived in the Harrison area (closer to Jasper, very close to Dogpatch), and it's a very pretty area but wasn't right for them (or myself, I've been there enough to know it just didn't fit what I was looking for). Home/land prices are reasonable and Harrison has most things you'd need on a daily basis (like Mtn. Home, El Dorado, etc. it is an anchor for a larger area than itself so it tends to have more than most cities that size). DSL is the one thing I'm not as sure about- depending on the age of the infrastructure in more rural parts of Boone (or especially Newton) county, you may not be able to get DSL, but there should be cable internet/tv pretty much everywhere I would think. Closer to town there should be a few options.
As for the Mena area, I'm not as familiar with it as I've only ever been there for work. It's a smaller area with a bit more limited shopping/dining, etc. but it is also a very pretty part of the state to live in. Accessibility isn't ideal, but it's not too bad. It is probably easier to get to and from (road quality wise) than most of Newton County, but it's quite a ways from most things of any size. The other areas I mentioned all have a nice charm about them with rural areas very nearby that might interest you and are a quicker trip to places like Little Rock, Springfield, MO, or in El Dorado's case, it's about split between Little Rock and Shreveport, LA.
Good luck with your search, and if you do start considering a place to move to, let us know if we can help!
I was going to say Mountain Home as well, but because he didn't mention it as a choice I decided not to bring it up. I think, considering what he says he wants, Mountain Home would be a perfect fit.
Mountain Home is FAR superior to Harrison. It's prettier, and people tend to be a little more tolerant, sort of the same kind of conservative you're talking about.
My parents live in Cotter and I think they have DSL...and the water is pretty good too (although I live in Memphis now so I'm spoiled). I haven't ever run into the "not speaking english" problem in MH, although the further west in the state you go (Northwest, particularly Fayetteville/Springdale/Bentonville, etc) that becomes a major issue. I've heard people speaking Spanish/Chinese/Whatever in MH, but it's never been a big thing.
Good luck! As a teenager MH was boring, but it's grown a little and I've gotten older...I really miss it now in some ways!
I think if the OP is looking for a better, slightly longer growing season and some seclusion, the Mena area is a pretty good fit. It is an outdoors type persons paradise! My daughter has great luck with her gardens. Even growing some produce year round. I have found the younger population there to be alot more educated and talented than I was led to believe. Stereotypes are usually just that. Good luck and I hope you find Arkansas to be all you want it to be!
I'll stand by Mountain Home also.
I spend a little time over there now and again, and it has enough room and diversity in living conditions that there's no problem spending as much or as little to find what you want.
Fact is any party of arkansas north of little rock and west of the delta area and just about all of the quachita mountain area would suite you right down to the ground. Whereabouts are ya'll living in arizona? I spent some time there in my youth.


Switch things from calling Group_member::join & leave & calling events manually to running through Profile::joinGroup() && Profile::leaveGroup(), with the events encapsulated.
Corrected redirect targets for some group actions. These redirects can occur when the canonical name differs from the passed name.


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Emerald Group Publishing Limited »
Sarat, Austin
Political Sociology Policy & Social Change (1)
Sociology Of Crime & Law (3)
Studies in Law, Politics, and Society Studies in Law, Politics and Society Vol: 63
List price €116.99 Add to basket
Studies in Law, Politics, and Society Special Issue: The Beautiful Prison Vol: 64
Studies in Law, Politics, and Society Special Issue: Law and the Liberal State Vol: 65
Studies in Law, Politics, and Society Special Issue: Thinking and Rethinking Intellectual Property Vol: 67
Studies in Law, Politics, and Society Special Issue Cassandra's Curse: The Law and Foreseeable Future Disasters Vol: 68
Studies in Law, Politics, and Society Special Issue: Problematizing Prostitution: Critical Research and Scholarship Vol: 71
Publication Date: 29 Apr 2016
Studies in Law, Politics, and Society Special Issue: Law Firms, Legal Culture and Legal Practice: Law Firms, Legal Culture, and Legal Practice Vol: 52
Publication Date: 13 Dec 2010
Studies in Law, Politics, and Society Special Issue: Social Movements/Legal Possibilities Vol: 54
Studies in Law, Politics, and Society Special Issue: Human Rights: New Possibilities/New Problems Vol: 56


1It is reported commonly that there is fornication among you, and such fornication as is not so much as named among the Gentiles, that one should have his father's wife. 2And ye are puffed up, and have not rather mourned, that he that hath done this deed might be taken away from among you. 3For I verily, as absent in body, but present in spirit, have judged already, as though I were present, concerning him that hath so done this deed, 4in the name of our Lord Jesus Christ, when ye are gathered together, and my spirit, with the power of our Lord Jesus Christ, 5to deliver such an one unto Satan for the destruction of the flesh, that the spirit may be saved in the day of the Lord Jesus. 6Your glorying is not good. Know ye not that a little leaven leaveneth the whole lump? 7Purge out therefore the old leaven, that ye may be a new lump, as ye are unleavened. For even Christ our passover is sacrificed for us: 8therefore let us keep the feast, not with old leaven, neither with the leaven of malice and wickedness; but with the unleavened bread of sincerity and truth.
9I wrote unto you in an epistle not to company with fornicators: 10yet not altogether with the fornicators of this world, or with the covetous, or extortioners, or with idolaters; for then must ye needs go out of the world. 11But now I have written unto you not to keep company, if any man that is called a brother be a fornicator, or covetous, or an idolater, or a railer, or a drunkard, or an extortioner; with such an one no not to eat. 12For what have I to do to judge them also that are without? do not ye judge them that are within? 13But them that are without God judgeth. Therefore put away from among yourselves that wicked person.


text:
LennyCo — May 24, 2019 in Technology
Google thwarts Baltimore ransomware fightback
Alex Wroblewski/Getty Images
Baltimore's government servers were hit on 7 May by ransomware
Google has inadvertently thwarted attempts by Baltimore's city government to fight off crippling ransomware.
The cyber-attack struck civil computers in Baltimore on 7 May, blocking email accounts and online payment systems.
City officials set up GMail accounts to restore communications between citizens and staff.
The mass creation of the accounts triggered Google's GMail defences which shut them down, believing they were being used by spammers.
Google said it restored access to the GMail accounts once it discovered who had created them and why.
"Our automated security systems disabled the accounts due to the bulk creation of multiple consumer Gmail accounts from the same network," Google told the BoingBoing website.
House sales on hold
The cyber-attack has meant that many day-to-day administration tasks of Baltimore's city government have stopped. It has delayed the sales of many homes, and stopped water bills being generated. And people cannot pay tax bills or parking fines.
A manual workaround for the home sales has been put in place, and the switch from internal email to GMail was another scheme to defeat the attackers.
The cyber-criminals who launched the ransomware attack have demanded a payment of $100,000 (£79,000) in Bitcoins to restore control of official systems to city workers. About 10,000 computers were hit by the virus.
Baltimore has so far refused to pay the ransom. The FBI and Secret Service are known to be investigating.
City Mayor Bernard Young said it was unclear when systems would be back in full operation.
Last year Baltimore was hit by a cyber-attack which knocked out the communications systems for emergency services for a day.
MPs call for halt to authorities' use of facial recognition
Instagram hides likes count in international test 'to remove pressure'
FaceApp: Chuck Schumer asks for FBI investigation
UK workers who lose jobs to AI will be retrained


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Q: How to get a value (not create a string) dynamically? I am trying to display a value on React like this:
'If THING exists, then show THING'
So far I have
values.price ? `You'll pay ${values.price}` : null

What I need is get
values.price0 ? `You'll pay ${values.price0}` : null
values.price1 ? `You'll pay ${values.price1}` : null
values.price2 ? `You'll pay ${values.price2}` : null
values.price3 ? `You'll pay ${values.price3}` : null

That number will come from the props, so I am trying to do the following, but it doesn't work. How to go about fixing this?
values.price + props.id ? `You'll pay ${values.price + props.id}` : null

A: you can access properties as below ;
values[`${price}${props.id}`] ? `You'll pay ${values.price + props.id}` : null

or you can simply loop through keys
Object.keys(values).map(k => `You'll pay ${values[k]}`)

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Toronto FC ended the 2018 season on a high note thanks to a 4-1 thrashing of Atlanta United FC at BMO Field.
Lucas Janson got the scoring started in the 9th minute with a deflected strike that beat Brad Guzan and the Reds never looked back from there. The Argentine would complete his brace in the 83rd minute after a nice bit of work from Sebastian Giovinco down the left wing.
Giovinco would eventually bag a well-deserved goal of his own in the waning moments of the match, smashing home a wonderfully taken volley to seal with win. The Italian finished 2018 with 18 goals and 21 assists through 37 games.


text:
Q: Node module dependency I am new to Nodejs and javascript and not sure yet how to ask the right question.
My main file has a var express = require('express'); on top of it. I am also creating a module that I will be requiring in my main file. I know that module also require the express module. Do I have to import the express module into it or the main file will take care of that?

A: Modules are completely self contained. Every module has to import the dependencies it needs.


Virgin and Richard Branson may enter the electric vehicle business, Branson hints. The Virgin name is already on airplanes, spaceships, and cruise ships, as well as music and health clubs. The company also competes in the Formula E racing series for electric-only vehicles. Recently Branson said "you never know" when asked about the odds of Virgin's entry into EV passenger cars.
The genesis of the Virgin-builds-EVs story is based on Virgin's think-different, think-outside-the-box interests. Once it got into Formula E, hints were dropped, and amplified perhaps to the point where signal and noise match up, that Branson might consider building passenger EVs. At the Miami Grand Prix Formula E race, Bloomberg TV popped the passenger-cars question. Branson's response: "We have teams of people working on electric cars," Branson said. "So you never know—you may find Virgin competing with Tesla in the car business as we do in the space business. We will see what happens." If it's not a yes, it's not a no.
Why Virgin might do it: keeping up with the Musks of the world?
Nobody admits to doing it for ego. But Virgin and Tesla are both chaired by outspoken chairmen who embody their companies, just as the companies embody Richard Branson and Elon Musk. Even today, starting your own car company is more exclusive than doing a software startup, or owning a Tesla car.
The biggest part of an electric vehicle is the drivetrain: the batteries, electric motors, inverter, and control software. Virgin already has experience with drivetrains through Formula E. If Virgin got into the car business, there's still the significant matter of setting up a factory or contract factory, and automakers as big as BMW subcontract out some work. There's also the issue of meeting safety and emissions requirements that differ from country to country. But it's not impossible at all.
Formula E is a part of the Formula 1 racing organization, typically held in conjunction with a major combustion-engine race. Formula E cars are open-wheel, open-cockpit single passenger cars that look roughly like Formula 1 or Indy cars. With fresh batteries, they do 0-100 kph (0-62 mph) in 3 seconds and top out at 225 kph (140 mph). The races run 45 minutes, and the batteries are good for 20 to 30 minutes, so range anxiety is a big issue at the track. Rather then refuel, drivers hop into a second car to finish the race. Currently, 10 teams compete with two race cars each.
Branson got a first-person lesson on battery life and range issues at the Long Beach Formula E race. Just before the vehicle change point, the top Virgin driver took the lead with an aggressive pass that used up virtually all the remaining charge. With Branson watching, the Virgin car had to relinquish the lead, limp back to the pits, swap cars, and never regained the lead.
Formula E is a tightly controlled series: one racing chassis, one tire specification, and one battery pack, for instance. For the inaugural 2014-15 season, there was also a single drivetrain supplier. For 2015-16, which begins in the fall, individual manufacturers will be able to build their own electric motor, inverter, gearbox, and cooling systems.
Since Tesla knows a lot about batteries, electric motors, inverters and control electronics already, the thinking is, it could enter the series and stand a solid chance to be competitive — especially since someone else supplies the chassis. According to Alejandro Agag, the chairman of Formula E, Musk wants to focus on road cars, fearing race tech would be too much of a distraction. But Agag also says the rules change allowing each team to develop its own drivetrain technology might hook Tesla.


A pipe snake could be all it takes to clean out a clogged drain stack on your washing machine. The heat of the sun is easy to forget when you are very involved in your work, but it still can be dangerous. Copper pipes can sometimes expand a little as hot water runs through them. Save yourself big time by putting together your plumbing problems into the same call. If you don't take care of it, a small leak can turn into a much larger leak, and the valve will be much harder to shut off. It might seem that purchasing plumbing tools is rather costly, but they can save you money over the long-term. There are many Internet videos that teach you how to carry out basic repairs to plumbing. By investing in the correct tools, you may not need to call in an expensive plumber for many of the minor issues your home may have. The initial thing to check out is the chain pressure, but if the chain is not the problem, the flapper needs replacement. When you are cleaning your fiberglass bathroom fixtures, be sure that you use non-abrasive cleaners such as hand dish washing liquids and laundry detergents. Plumbing issues can be a real pain, but with the right information, you can get them under control. More information about Local Emergency Underground Plumbing Service And Repair Company Port Deposit MD.
You should know the plumber completed everything that they said they would before he is reimbursed for the entire amount. Knowing which tools and how they are used will be a great help when you work on a plumbing job. Before undertaking repairs on your own, plan ahead, a mistake could make repairs even more costly. If your toilet happens to be clogged, there is a low level of water in toilet, and a plunger is not doing the trick, you can sometimes fix the problem by dumping warm water into your toilet at waist level (or maybe even a little higher). Do this as often as necessary if the water level gets too low. Never use your hands to try fixing your garbage disposal. Even an un-powered garbage disposal is turned off they can be dangerous. You can look online to get schematics or troubleshooting advice for the disposal that you have. Check the floors in your bathroom for soft areas around the floors. You can save money by noticing the problem as early on. Don't put oil, oil or grease down your drains.
If you are looking for Recommended Sewer Pipe Service And Repair Mechanicsville MD, sports tickets and theater seats at the best prices click here.
Frozen pipes can cause a lot of the most expensive plumbing problems.
This is an indication that there is serious plumbing issues.


text:
The Model S (Tesla Motors)
The Tesla Model S literally couldn't be safer
The battery-powered car says it received a 5-plus star U.S. crash safety rating
Check out this article! https://www.salon.com/2013/08/20/the_tesla_model_s_literally_couldnt_be_safer/
Lindsay Abrams
August 20, 2013 9:59PM (UTC)
The Hyperloop may be little more than a figment of Tesla CEO Elon Musk's imagination, but his battery-powered Model S sedan is something approaching a real-life Batmobile. According to a company statement, the newly released car received a better-than-perfect score from the National Highway Traffic Safety Administration (NHTSA). So not only is it more sustainable than SUVs and minivans, but according to the report, it's safer, too. (It does, however, have a base price of about $70,000.)
The company says that NHTSA's independent testing awarded the Model S a 5-star rating in every possible subcategory. While grade inflation isn't officially allowed, it claims its overall Vehicle Safety Score actually adds up to 5.4 stars.
How is that even possible? According to Tesla, "the Model S has the advantage in the front of not having a large gasoline engine block, thus creating a much longer crumple zone to absorb a high speed impact." If the company is to be believed, the car's roof actually broke the machine meant to test how much pressure it could withstand.
MORE FROM Lindsay Abrams • FOLLOW readingirl
Electric Cars Elon Musk Safety Tesla
Lithium mining's toxic toll
What if Apple really had bought Tesla?
White supremacy: Real threat to safety


text:
Harry Wilson (* 22. března 1997, Wrexham, Wales, Spojené království) je velšský fotbalový záložník a reprezentant. Momentálně hraje za anglický klub Fulham FC.

Klubová kariéra
Wilson hrál v mládežnických týmech anglického klubu Liverpool FC. V červenci 2015 byl zařazen do 30členného kádru pro předsezónní tour Liverpoolu v Thajsku, Austrálii a Malajsii.

V srpnu 2015 odešel na hostování do klubu Crewe Alexandra FC. Tam se mu však moc nevedlo, což bylo způsobeno i dlouhodobým zraněním.

Reprezentační kariéra
V září 2013 odehrál 3 zápasy za velšský mládežnický tým U17. 23. září se jedním gólem podílel na remíze 5:5 s Maďarskem, 25. září nastoupil do utkání se Skotskem (remíza 0:0) a 28. září dvěma góly výrazně přispěl k vítězství 3:0 nad Slovinskem.

A-mužstvo
Díky svým výkonům ve velšském výběru U17 obdržel v říjnu 2013 poprvé pozvánku od reprezentačního trenéra Chrise Colemana do seniorského národního týmu Walesu. Zápas s Makedonií 11. října strávil ještě na lavičce náhradníků,
za A-mužstvo Walesu nastoupil poprvé 15. října 2013 v kvalifikačním zápase proti Belgii (remíza 1:1), kdy se dostal na hrací plochu v 87. minutě. Tímto se stal nejmladším velšským reprezentantem (ve věku 16 let a 207 dní, čímž překonal o 108 dní dřívějšího držitele rekordu Garetha Balea). Jeho premiéře ve velšském dresu napomohlo rovněž to, že o hráče měla zájem i Anglie (Wales tedy nechtěl ztratit talentovaného mladíka). K jeho debutu ve velšském národním týmu se také váže nevšední zajímavost. Jeho dědeček Peter Edwards si u sázkové kanceláře William Hill v roce 2000 (kdy měl Harry pouhé dva roky) vsadil padesát liber na to, že se Harry jednou stane velšským reprezentantem. Při kursu 1:2 500 bral výhru 125 000 liber.

Odkazy

Reference

Externí odkazy

 Profil hráče na transfermarkt.com
 Profil hráče – reprezentační statistiky v A-mužstvu, eu-football.info

Narození v roce 1997
Žijící lidé
Muži
Velšští fotbaloví reprezentanti
Fotbaloví záložníci
Fotbalisté Liverpool FC
Narození ve Wrexhamu
Fotbalisté Crewe Alexandra FC
Narození 22. března
Hráči Mistrovství světa ve fotbale 2022
Hráči Mistrovství Evropy ve fotbale 2020

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text:
Brown White Black: An American Family at the Intersection of Race, Gender, Sexuality, and Religion
By Nishta J. Mehra
Read by Nishta J. Mehra
Nishta J. Mehra Macmillan Audio 9781250133557
02/05/2019 • Unabridged
Format: CD (In Stock)
1 Format: CD
Intimate and honest essays on motherhood, marriage, love, and acceptance. Brown White Black is a portrait of Nishta J. Mehra's family: her wife, who is white; her adopted child, Shiv, who is black; and their experiences dealing with America's rigid ideas of race, gender, and sexuality. Her clear-eyed and incisive writing on her family's daily struggle to make space for themselves amid racial intolerance and stereotypes personalizes some of America's most fraught issues. Mehra writes candidly about her efforts to protect and shelter her child from racial slurs on the playground and from intrusive questions by strangers while educating Shiv on the realities and dangers of being black in America. In other essays, she discusses her childhood living in the racially polarized city of Memphis; coming out as queer; being an adoptive mother who is brown; and what it's like to be constantly confronted by people's confusion, concern, and expectations about her child and her family. Above all, Mehra argues passionately for a more nuanced and compassionate understanding of identity and family. Both poignant and challenging, Brown White Black is a remarkable portrait of a loving family on the front lines of some of the most highly charged conversations in our culture.
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Intimate and honest essays on motherhood, marriage, love, and acceptance.
Brown White Black is a portrait of Nishta J. Mehra's family: her wife, who is white; her adopted child, Shiv, who is black; and their experiences dealing with America's rigid ideas of race, gender, and sexuality. Her clear-eyed and incisive writing on her family's daily struggle to make space for themselves amid racial intolerance and stereotypes personalizes some of America's most fraught issues. Mehra writes candidly about her efforts to protect and shelter her child from racial slurs on the playground and from intrusive questions by strangers while educating Shiv on the realities and dangers of being black in America. In other essays, she discusses her childhood living in the racially polarized city of Memphis; coming out as queer; being an adoptive mother who is brown; and what it's like to be constantly confronted by people's confusion, concern, and expectations about her child and her family. Above all, Mehra argues passionately for a more nuanced and compassionate understanding of identity and family.
Both poignant and challenging, Brown White Black is a remarkable portrait of a loving family on the front lines of some of the most highly charged conversations in our culture.
"Nishta Mehra's luminous prose shines…A natural storyteller, she has an original voice, an ear for spoken language, and an eye for the sparkling detail…[Her] writing is at turns crisp and lyrical but always entertaining." —Southern Review
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Author Bio: Nishta J. Mehra
Nishta J. Mehra was raised among a tight-knit network of Indian immigrants in Memphis, Tennessee. She is the proud graduate of St. Mary's Episcopal School and holds a BA in religious studies from Rice University and an MFA in creative writing from the University of Arizona. Currently a middle school English teacher, she authors the popular food blog Blue Jean Gourmet and lives with her partner, Jill, and their son, Shiv, in a suburb of Houston, Texas.
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Pope Benedict XVI yesterday invited all Roman Catholics in China to unite under his jurisdiction and urged Beijing to restore diplomatic ties and permit religious freedom.
He called the state-run Catholic Church "incompatible" with Catholic doctrine, but nevertheless made unprecedented overtures toward it.
In an eagerly awaited letter to the faithful in China taking up a priority of his papacy, Benedict insisted on his right to appoint bishops, but said he trusted that an agreement could be reached with Beijing on nominations.
Significantly, Benedict revoked previous Vatican-issued restrictions on contacts with the clergy of the official church and recognized that some Chinese faithful have no choice but to attend officially recognized Masses.
The Vatican said in a note accompanying the letter that it was prepared to move its diplomatic representation from Taiwan to Beijing "at any time" as soon as an agreement with the government was reached.
The letter was the most significant effort by Benedict to balance his pastoral concerns for the up to 12 million Roman Catholics in China who are divided between an official church -- the Chinese Patriotic Catholic Association -- and an underground church that is not registered with the authorities.
The Beijing government forced its Roman Catholics to cut ties with the Vatican in 1951, shortly after the officially atheist Chinese Communist Party took power. Worship is allowed only in the government-controlled churches.
Millions of Chinese, however, belong to unofficial congregations that remain loyal to Rome.
On several occasions, Benedict praised Catholics who resisted pressure to join the official church and paid a price for it "with the shedding of their blood."
But he urged them to forgive and reconcile with others for the sake of unifying the church.
Tellingly, Benedict referred repeatedly to the "Catholic Church in China," without distinguishing between the divisions -- an indication of his aim to see the two united and in communion with Rome.
On several occasions, he also called the Patriotic Association "incompatible with Catholic doctrine" because it named its own bishops and sought to guide the life of the church.
At the same time, however, Benedict made an unprecedented gesture, revoking 1988 guidelines issued by the Vatican's evangelization office that sought to limit contacts with the official church and declared that any bishop ordained by the official church would incur an automatic excommunication.
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Aruntina is een geslacht van rechtvleugeligen uit de familie Morabidae. De wetenschappelijke naam van dit geslacht is voor het eerst geldig gepubliceerd in 1976 door Key.

Soorten
Het geslacht Aruntina  is monotypisch en omvat slechts de volgende soort:
Aruntina alacris (Key, 1976)

Morabidae


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Lark Energy Passes 150MWp Milestone
Published: Monday, 17 June 2013 19:29
Following completion of Lark Energy's first phase of large scale solar projects, totalling 60MWp, the company has recently secured planning permission for a further 100MWp of solar farms.
Projects include two former airfields in Suffolk: Broxted, at the site of the former RAF Stradishall (35MWp) and Ellough (14MWp). Three further consents have also been gained: South Creake in Suffolk (26MWp); Thornham in Norfolk (15MWp); and Burton Pedwardine in Lincolnshire (15MWp).
Having constructed more than 20MWp of projects, Lark Energy will also look to take the lead role in the Engineering, Procurement & Construction (EPC) on the majority of the sites together with the provision of ongoing Operations & Maintenance (O&M) services.
Jonathan Selwyn, Managing Director of Lark Energy, commented "As a leading solar developer, we are delighted to demonstrate our commitment to the UK market by securing our latest planning consents for a range of large scale sites. We are optimistic about future opportunities in the solar sector, although clearly we remain concerned about the impact of the current European trade dispute with China"
Jo Wall, Development Director of Lark Energy commented on the planning consents "As a responsible developer we select and develop our sites with the utmost care following extensive engagement with local communities. As a result, we have been able to continue our impressive track record in achieving consent on all sizes of project"
Paul Adams, Director of Lark Energy noted that "these projects mark the latest stage in Lark Energy's ambition to become a leading EPC and O&M provider. We are developing very successful partnerships with a number of funders who both buy the ready to build projects from us and also engage in Lark Energy's services as a high quality and fully bankable EPC for large scale solar in the UK.
Tags: Company News, Solar Industry News, e-lec.org