$\Gamma[\phi]\equiv\dfrac{\Gamma_{\Pi}[\phi]}{\Gamma_{\Psi}[\phi]},$ (8.62)

which can be integrated straightforwardly

$\sqrt{A}\int\dfrac{\delta\Psi[\xi^{I}]}{\sqrt{{1-B(\Psi[\xi^{I}])^{2}}}}=\pm\Gamma[\phi]\xi^{I}+C,$
(8.63)

where $C$ is a constant of integration, with the result

$\sqrt{{\dfrac{A}{B}}}\arcsin\left\\{\sqrt{{\dfrac{B}{A}}}\Psi[\xi^{I}]\right\\}=\pm\Gamma[\phi]\xi^{I}+C,$
(8.64)

so that after elementary algebraic manipulations one obtains

$\Psi[\xi^{I}]=\sqrt{{\dfrac{A}{B}}}\sin\theta(\xi^{I},\phi),$ (8.65)

where $\theta(\xi^{I},\phi)$ is the phase

$\theta(\xi^{I},\phi)=\sqrt{{\dfrac{B}{A}}}\left(\pm\Gamma[\phi]\xi^{I}+C\right),$
(8.66)

Albeit, because of $\Psi[\xi^{I}]$ must be a functional of $\xi^{I}$ only,
must hold $\Gamma[\phi]=\Gamma_{0}$, where $\Gamma_{0}$ is a constant
independent on $\phi$ and $\xi^{I}$, for which the phase
$\theta(\xi^{I},\phi)$ is reduced to

$\theta(\xi^{I})=\sqrt{{\dfrac{B}{A}}}\left(\pm\Gamma_{0}\xi^{I}+C\right).$
(8.67)

Taking into account the relation (8.59) one obtains finally

$\displaystyle\Psi[\xi^{I}]$ $\displaystyle=$
$\displaystyle\sqrt{\dfrac{A}{B}}\sin\theta(\xi^{I}),$ (8.68)
$\displaystyle\Pi_{\Psi}[\xi^{I}]$ $\displaystyle=$
$\displaystyle\pm\sqrt{{\dfrac{1}{A}-\sin^{2}\theta(\xi^{I})}}.$ (8.69)

In the light of the equation (8.18), however, one of the relations

$\displaystyle\sin\theta(\xi^{I})$ $\displaystyle=$ $\displaystyle 0,$ (8.70)
$\displaystyle\sin\theta(\xi^{I})$ $\displaystyle=$
$\displaystyle\pm\sqrt{\dfrac{1}{A}},$ (8.71)

is always true. One sees that both these conditions define discrete values of
the initial data of the invariant global dimension $\xi_{I}$. Namely, the
first relation (8.70) leads to the solution

$\sqrt{{\dfrac{B}{A}}}\left(\pm\Gamma_{0}\xi^{I}+C\right)=k\pi,$ (8.72)

where $k\in\mathbb{Z}$ is an integer, what leads to

$\xi^{I}=\pm\dfrac{1}{\Gamma_{0}}\left(\sqrt{{\dfrac{A}{B}}}k\pi-C\right).$
(8.73)

Similarly the second relation (8.71) can be solved immediately

$\xi^{I}=\pm\dfrac{1}{\Gamma_{0}}\left(\pm\sqrt{{\dfrac{A}{B}}}\arcsin\sqrt{{\dfrac{1}{A}}}-C\right).$
(8.74)

For the first case one has

$\displaystyle\Psi[\xi^{I}]$ $\displaystyle=$ $\displaystyle 0,$ (8.75)
$\displaystyle\Pi_{\Psi}[\xi^{I}]$ $\displaystyle=$
$\displaystyle\pm\sqrt{{\dfrac{1}{A}}},$ (8.76)

and for the second one hold

$\displaystyle\Psi[\xi^{I}]$ $\displaystyle=$
$\displaystyle\pm\sqrt{{\dfrac{1}{B}}},$ (8.77)
$\displaystyle\Pi_{\Psi}[\xi^{I}]$ $\displaystyle=$ $\displaystyle 0.$ (8.78)

Finally one sees that the invariant global wave functional (8.41) is

$\Psi[\xi,\phi]=\pm\Gamma_{\Psi}[\phi]\Gamma_{0}\sqrt{{\dfrac{1}{A}}}\dfrac{\sin\left[\Omega(\xi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right]}{\sqrt{{\langle V\rangle(\xi,\phi)}}},$ (8.79)

in the first case (8.70), and

$\Psi[\xi,\phi]=\pm\Gamma_{\Psi}[\phi]\sqrt{{\dfrac{1}{B}}}\cos\left[\Omega(\xi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right],$ (8.80)

for the second one (8.71).

Applying the normalization condition (8.55) to the solutions (10.150) and
(10.157) one receives the equations

$\displaystyle|\Gamma_{\Psi}[\phi]\Gamma_{0}|^{2}$ $\displaystyle=$
$\displaystyle 1,$ (8.81) $\displaystyle\Gamma_{\Psi}[\phi]\Gamma_{0}$
$\displaystyle=$ $\displaystyle 1,$ (8.82)

which can be solved easy and lead to the relation

$\Gamma_{\Psi}[\phi]=\dfrac{1}{\Gamma_{0}}.$ (8.83)

Therefore, one obtains finally

$\displaystyle\Psi_{1}[\xi,\phi]$ $\displaystyle=$
$\displaystyle\pm\sqrt{{\dfrac{1}{|A|}}}\dfrac{\sin\left[\Omega(\xi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right]}{\sqrt{{\langle V\rangle(\xi,\phi)}}},$ (8.84)
$\displaystyle\Psi_{2}[\xi,\phi]$ $\displaystyle=$
$\displaystyle\pm\sqrt{{\dfrac{1}{|B|}}}\cos\left[\Omega(\xi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right],\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (8.85)

where now the constants $A$ and $B$ are equal to

$\displaystyle
A=\int_{\Sigma(\xi,\phi)}\left(\dfrac{\sin\left[\Omega(\xi^{\prime})\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}\right]}{\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}}\right)^{2}\delta\xi^{\prime}\delta\phi^{\prime},$
(8.86) $\displaystyle
B=\dfrac{1}{\Gamma_{0}}\int_{\Sigma(\xi,\phi)}\cos^{2}\left[\Omega(\xi^{\prime})\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}\right]\delta\xi^{\prime}\delta\phi^{\prime}.$
(8.87)

In this manner the general solutions of the theory of quantum gravity can be
now constructed straightforwardly by using of the solutions (10.150) and
(10.157), in which the integrals must be putted

$\displaystyle\Omega(\xi)$ $\displaystyle=$
$\displaystyle\int_{\Sigma(\xi)}\delta\xi^{\prime}=\xi,$ (8.88)
$\displaystyle\langle V\rangle(\xi,\phi)$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Omega(\xi)}\int_{\Sigma(\xi)}\delta\xi^{\prime}V[\xi^{\prime},\phi],$
(8.89)

where the $\xi$-measure is

$\displaystyle\delta\xi=\dfrac{1}{8\pi}\dfrac{\delta h}{\sqrt{6h}}.$ (8.90)

Because, however, the generalized gravitational potential $V[\xi,\phi]$ has
the form of an algebraic sum

$V[\xi,\phi]=-{{}^{(3)}\\!R}+2\Lambda+2\kappa\ell_{P}^{2}\varrho,$ (8.91)

one has very convenient separability

$\langle{V}\rangle(\xi,\phi)=-\dfrac{1}{\Omega(\xi)}\int_{\Sigma(\xi)}\delta\xi^{\prime}\leavevmode\nobreak\
{{}^{(3)}\\!R}+2\Lambda+\dfrac{2\kappa\ell_{P}^{2}}{\Omega(\xi)}\int_{\Sigma(\phi)}\delta\phi^{\prime}\int_{\Sigma(\xi)}\delta\xi^{\prime}\rho.$
(8.92)

Therefore, for a concretely given geometry of a three-dimensional space-like
embedded space one should estimate the functionally averaged three-dimensional
Ricci scalar

$\displaystyle\langle{{}^{(3)}\\!R}\rangle=\dfrac{1}{\Omega(\xi)}\int_{\Sigma(\xi)}\delta\xi^{\prime}\leavevmode\nobreak\
{{}^{(3)}\\!R},$ (8.93)

and the functionally averaged energy density of Matter fields

$\langle\rho\rangle=\dfrac{1}{\Omega(\xi)}\int_{\Sigma(\phi)}\delta\phi^{\prime}\int_{\Sigma(\xi)}\delta\xi^{\prime}\rho,$
(8.94)

and using these quantities construct the functionally averaged generalized
gravitational potential

$\langle
V\rangle(\xi,\phi)=-\langle{{}^{(3)}\\!R}\rangle+2\Lambda+2\kappa\ell_{P}^{2}\langle\rho\rangle.$
(8.95)

Applying this averaging method one can construct the solutions (10.150) and
(10.157) straightforwardly. If the Ricci scalar curvature ${{}^{(3)}\\!R}$ of
an embedded space and/or the energy density of Matter fields $\varrho$ are
functions of a space-time point $x^{\mu}=(x^{0},x^{1},x^{2},x^{3})$ or any one
space-time coordinate the integration procedure is understood as the procedure
of performing of the Lebesgue–Stieltjes integral. For instance in the case
when the integrand is a function of $x^{\mu}$ then the Lebesgue–Stieltjes
measure has the following form

$\delta\xi=\dfrac{\partial^{4}\xi}{\partial x_{0}\partial x_{1}\partial
x_{2}\partial x_{3}}d^{4}x,$ (8.96)

or in terms of determinant of induced metric $h$

$\delta\xi=\dfrac{1}{8\pi}\dfrac{\partial^{4}h}{\partial x_{0}\partial
x_{1}\partial x_{2}\partial x_{3}}\dfrac{d^{4}x}{\sqrt{6h}}.$ (8.97)

####  D Problem I: Inverted Transformation

There is the problem of inverted transformation

$\xi\rightarrow h_{ij},$ (8.98)

within the obtained solutions. In fact, this problem is strictly related to
the problem of finding of a induced metric if one knows its determinant. Such
a procedure can not be performed analytically, but there are functional
methods which enable performing of this step.

The solutions (10.150) and (10.157) expressed via $h_{ij}$ are

$\displaystyle\Psi_{1}[h_{ij},\phi]$ $\displaystyle=$
$\displaystyle\pm\sqrt{{\dfrac{1}{|A|}}}\dfrac{\sin\left[\Omega(h_{ij})\sqrt{{\langle
V\rangle(h_{ij},\phi)}}\right]}{\sqrt{{\langle V\rangle(h_{ij},\phi)}}},$
(8.99) $\displaystyle\Psi_{2}[h_{ij},\phi]$ $\displaystyle=$
$\displaystyle\pm\sqrt{{\dfrac{1}{|B|}}}\cos\left[\Omega(h_{ij})\sqrt{{\langle
V\rangle(h_{ij},\phi)}}\right],$ (8.100)

where now the constants $A$ and $B$ are equal to

$\displaystyle
A=\dfrac{1}{8\pi}\int_{\Sigma(h_{ij},\phi)}\left(\dfrac{\sin\left[\Omega(h_{ij}^{\prime})\sqrt{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}\right]}{\sqrt{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}}\right)^{2}\sqrt{{\dfrac{h^{\prime}}{6}}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime},$ (8.101) $\displaystyle
B=\dfrac{1}{8\pi\Gamma_{0}}\int_{\Sigma(h_{ij},\phi)}\cos^{2}\left[\Omega(h_{ij}^{\prime})\sqrt{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}\right]\sqrt{{\dfrac{h^{\prime}}{6}}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime},$ (8.102)

and assumed to be convergent and finite. The solutions (8.99) and (8.100) are
two independent states transformed to the standard quantum geometrodynamics,
but in general they are not solutions of the Wheeler–DeWitt equation.

Interestingly, in such a situation the normalization condition (8.55) becomes

$\dfrac{1}{8\pi}\int_{\Sigma(h_{ij},\phi)}|\Psi[h_{ij}^{\prime},\phi^{\prime}]|^{2}\sqrt{{\dfrac{h^{\prime}}{6}}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime}=1,$ (8.103)

and can be used to construct the $\pi$ number definition

$\pi=\dfrac{1}{8}\int_{\Sigma(h_{ij},\phi)}|\Psi[h_{ij}^{\prime},\phi^{\prime}]|^{2}\sqrt{{\dfrac{h^{\prime}}{6}}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime}.$ (8.104)

####  E Problem II: The Hilbert Space and Superposition

Because, however, both the equations of quantum geometrodynamics (6.101) and
(8.10) are linear in the field $\Psi$, in general the superposition

$\Psi=\sum_{i=1,2}\alpha_{i}\Psi_{i},$ (8.105)

where $\alpha_{i}$ are arbitrary constants, and $\Psi_{i}$ are (8.99) and
(8.100), should be also a solution. The normalization (8.103) of (8.105) gives

$|\alpha_{1}|^{2}+|\alpha_{2}|^{2}+(\alpha^{\star}_{1}\alpha_{2}+\alpha_{1}\alpha^{\star}_{2})I=1,$
(8.106)

where $I$ is the integral

$I=\dfrac{1}{\sqrt{|A||B|}}\dfrac{1}{8\pi}\int_{\Sigma(h_{ij},\phi)}\dfrac{\sin\left[2\Omega(h_{ij}^{\prime})\sqrt{{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}}\right]}{2\sqrt{{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}}}\sqrt{{\dfrac{h^{\prime}}{6}}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime}.$ (8.107)

For vanishing $I=0$ one obtains form (8.106) simply

$|\alpha_{2}|=\sqrt{{1-|\alpha_{1}|^{2}}}\quad,\quad|\alpha_{1}|\geqslant 1.$
(8.108)

The case of $I\neq 0$ is much more complicated. Note that the equation (8.106)
can be rewritten in form

$(\alpha_{1}+\alpha_{2}I)\alpha^{\star}_{1}+(\alpha_{2}+\alpha_{1}I)\alpha_{2}^{\star}=0,$
(8.109)

what leads to the result

$\dfrac{\alpha^{\star}_{1}}{\alpha_{2}^{\star}}=\dfrac{-\alpha_{1}I+\alpha_{2}}{\alpha_{1}+\alpha_{2}I},$
(8.110)

or equivalently

$\displaystyle C\alpha^{\star}_{1}$ $\displaystyle=$
$\displaystyle-\alpha_{1}I+\alpha_{2},$ (8.111) $\displaystyle
C\alpha_{2}^{\star}$ $\displaystyle=$ $\displaystyle\alpha_{1}+\alpha_{2}I,$
(8.112)

where $0\neq C\in\mathbb{R}$ is a constant. The relations (8.111)-(8.112) lead
to

$\displaystyle C|\alpha_{1}|^{2}$ $\displaystyle=$
$\displaystyle-\alpha^{2}_{1}I+\alpha_{2}\alpha_{1},$ (8.113) $\displaystyle
C|\alpha_{2}|^{2}$ $\displaystyle=$
$\displaystyle\alpha_{1}\alpha_{2}+\alpha_{2}^{2}I.$ (8.114)

Mutual addition and application of (8.106) yields

$CI[(\alpha^{\star}_{1}-\alpha_{2})\alpha_{2}+(\alpha_{2}^{\star}+\alpha_{1})\alpha_{1}]=\alpha_{1}\alpha_{2}+\alpha_{2}\alpha_{1},$
(8.115)

which generates the equations

$\displaystyle CI(\alpha^{\star}_{1}-\alpha_{2})$ $\displaystyle=$
$\displaystyle\alpha_{1},$ (8.116) $\displaystyle
CI(\alpha_{2}^{\star}+\alpha_{1})$ $\displaystyle=$ $\displaystyle\alpha_{2}.$
(8.117)

Complex decomposition of (8.116)-(8.117) leads to

$\displaystyle\Re\alpha_{2}$ $\displaystyle=$
$\displaystyle(CI-1)\Re\alpha_{1},$ (8.118) $\displaystyle\Im\alpha_{2}$
$\displaystyle=$ $\displaystyle(CI-1)\Im\alpha_{1},$ (8.119)

or equivalently

$\displaystyle\alpha_{2}$ $\displaystyle=$ $\displaystyle(CI-1)\alpha_{1},$
(8.120) $\displaystyle|\alpha_{2}|^{2}$ $\displaystyle=$
$\displaystyle(CI-1)^{2}|\alpha_{1}|^{2}.$ (8.121)

Employing (8.120)-(8.121) within the constraint (8.106) yields to

$\dfrac{1}{|\alpha_{1}|^{2}}=IC^{2}+(I^{2}-2I)C-I+2.$ (8.122)

Because of the natural condition $\dfrac{1}{|\alpha_{1}|^{2}}>0$ one has the
inequality for $C$

$IC^{2}+I(I-2)C-(I-2)>0,$ (8.123)

where $I\geqslant 0$. One sees that the case $I=0$ gives $2>0$ what is true.
The inequality (8.123) has two different solutions which are dependent on
$\mathrm{sgn}(I-2)$. When $0\leqslant I<2$ then the solution is

$C\in\left(C_{-},C_{+}\right).$ (8.124)

When $I>2$ the solution is somewhat different

$C\in(-\infty,C_{-})\cup(C_{+},\infty).$ (8.125)

Here are the constants

$C_{\mp}=\dfrac{2-I}{2}\mp\dfrac{1}{2}\sqrt{\dfrac{I^{3}-4I^{2}+8I-8}{I}}.$
(8.126)

For consistency the constants $C_{\mp}$ must be real numbers. For this it is
necessary and sufficient to satisfy the condition

$I^{3}-4I^{2}+8I-8\geqslant 0,$ (8.127)

what is satisfied for $I\geqslant 2$ or $I<0$. Because of $I>0$ the first case
is true, and therefore (8.125) is true region of validity of $C$. Applying the
definition of $I$, $A$, and $B$ one receives the following inequality

$\displaystyle\int_{\Sigma(h_{ij},\phi)}\dfrac{\sin\left[2\Omega(h_{ij}^{\prime})\sqrt{{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}}\right]}{2\sqrt{{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}}}\sqrt{h^{\prime}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime}\geqslant$
$\displaystyle\dfrac{2}{\sqrt{|\Gamma_{0}|}}\left(\int_{\Sigma(h_{ij},\phi)}\left(\dfrac{\sin\left[\Omega(h_{ij}^{\prime})\sqrt{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}\right]}{\sqrt{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}}\right)^{2}\sqrt{h^{\prime}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime}\right)^{1/2}\times$
$\displaystyle\left(\int_{\Sigma(h_{ij},\phi)}\cos^{2}\left[\Omega(h_{ij}^{\prime})\sqrt{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}\right]\sqrt{h^{\prime}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime}\right)^{1/2},$ (8.128)

or in terms of the invariant global dimension

$\displaystyle\int_{\Sigma(\xi,\phi)}\dfrac{\sin\left[2\Omega(\xi^{\prime})\sqrt{{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}}\right]}{2\sqrt{{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}}}\delta\xi^{\prime}\delta\phi^{\prime}\geqslant$
$\displaystyle\dfrac{2}{\sqrt{|\Gamma_{0}|}}\left(\int_{\Sigma(\xi,\phi)}\left(\dfrac{\sin\left[\Omega(\xi^{\prime})\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}\right]}{\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}}\right)^{2}\delta\xi^{\prime}\delta\phi^{\prime}\right)^{1/2}\times$
$\displaystyle\left(\int_{\Sigma(\xi,\phi)}\cos^{2}\left[\Omega(\xi^{\prime})\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}\right]\delta\xi^{\prime}\delta\phi^{\prime}\right)^{1/2}.$
(8.129)

Let us consider the concept of an inner product space, called also pre-Hilbert
space. Such a space is a vector space equipped with an additional structure
called an inner product

$\left<\bullet,\bullet\right>:V\times V\rightarrow\mathbb{F},$ (8.130)

where $V$ is a vector space over the field of scalars $\mathbb{F}$. A pre-
Hilbert space is called a Hilbert space if is complete as a normed space under
the induced norm $||\bullet||$. Every Hilbert space is the Banach space. The
structure of an inner product allows to build intuitive geometrical notions
such as length of a vector or the angle between two vectors, and deduce
orthogonality between vectors by vanishing of an inner product. Pre-Hilbert
spaces generalize Euclidean spaces to vector spaces of any, including
infinite, dimension and are the theme of functional analysis (For more
detailed discussion of functional analysis see e.g. the Ref. [603]). The
Lebesgue space $L^{2}\left(\Sigma,\mu\right)$ where $\Sigma$ is the
configurational space equipped with the Lebesgue measure $\mu$, called also
the space of square-integrable functions, which is the special case of
$L^{p}$-spaces (For more details see e.g. the Ref. [604]), and moreover is a
Hilbert space. This particular Hilbert space is the fundament of quantum
mechanics, in which wave functions belong to $L^{2}$.

Let us construct the Lebesgue space $L^{2}$ for the theory of quantum gravity
_in accordance_ with the superposition principle. In such a situation the
configurational space is the midisuperspace $\Sigma(\xi,\phi)$ with the
measure $\mu=\delta\xi\delta\phi$. The Lebesgue space
$L^{2}\left(\Sigma(\xi,\phi),\delta\xi\delta\phi\right)$ is the Hilbert space
with the inner product

$\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle=\int_{\Sigma(\xi,\phi)}f^{\star}(\xi^{\prime},\phi^{\prime})g(\xi^{\prime},\phi^{\prime})\delta\xi^{\prime}\delta\phi^{\prime},$
(8.131)

satisfying the conditions

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle$
$\displaystyle=$
$\displaystyle\overline{\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle},$
(8.132)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\langle{af(\xi,\phi),g(\xi,\phi)}\right\rangle$
$\displaystyle=$ $\displaystyle
a\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle,$ (8.133)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\langle{f(\xi,\phi)+h(\xi,\phi),g(\xi,\phi)}\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle+\left\langle{h(\xi,\phi),g(\xi,\phi)}\right\rangle.$
(8.134)

The induced norm

$\left|\left|f(\xi,\phi)\right|\right|=\sqrt{\left\langle{f(\xi,\phi),f(\xi,\phi)}\right\rangle}=\left(\int_{\Sigma(\xi,\phi)}|f(\xi^{\prime},\phi^{\prime})|^{2}\delta\xi^{\prime}\delta\phi^{\prime}\right)^{1/2}<\infty,$
(8.135)

is homogeneous

$\left|\left|af(\xi,\phi)\right|\right|=|a|\left|\left|f(\xi,\phi)\right|\right|,$
(8.136)

and satisfies the triangle inequality

$\left|\left|f(\xi,\phi)+g(\xi,\phi)\right|\right|\leqslant\left|\left|f(\xi,\phi)\right|\right|+\left|\left|g(\xi,\phi)\right|\right|.$
(8.137)

For the orthogonal situation

$\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle=0,$ (8.138)

the Pythagoras theorem holds

$\left|\left|f(\xi,\phi)+g(\xi,\phi)\right|\right|^{2}=\left|\left|f(\xi,\phi)\right|\right|^{2}+\left|\left|g(\xi,\phi)\right|\right|^{2}.$
(8.139)

The parallelogram law

$\left|\left|f(\xi,\phi)+g(\xi,\phi)\right|\right|^{2}+\left|\left|f(\xi,\phi)-g(\xi,\phi)\right|\right|^{2}=2\left|\left|f(\xi,\phi)\right|\right|^{2}+2\left|\left|g(\xi,\phi)\right|\right|^{2},$
(8.140)

is a necessary and sufficient condition for the existence of a scalar product
corresponding to a given norm. This scalar product follows from the
polarization identity

$\left|\left|f(\xi,\phi)+g(\xi,\phi)\right|\right|^{2}=\left|\left|f(\xi,\phi)\right|\right|^{2}+\left|\left|g(\xi,\phi)\right|\right|^{2}+\Re\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle,$
(8.141)

called also the law of cosinuses, and equals to

$\displaystyle\left(f(\xi,\phi),g(\xi,\phi)\right)$ $\displaystyle=$
$\displaystyle\dfrac{\left|\left|f(\xi,\phi)+g(\xi,\phi)\right|\right|^{2}-\left|\left|f(\xi,\phi)-g(\xi,\phi)\right|\right|^{2}}{2}=$
(8.142) $\displaystyle=$
$\displaystyle\Re\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle.$

When $f_{i}(\xi,\phi)$, $i=1,\ldots,N$, are orthogonal vectors then holds the
relation

$\sum_{i=1}^{N}\left|\left|f_{i}(\xi,\phi)\right|\right|=\left|\left|\sum_{i=1}^{N}f_{i}(\xi,\phi)\right|\right|.$
(8.143)

When $V$ is a complete pre-Hilbert space, i.e. is a Hilbert space, then the
equation (8.143) becomes the Parseval identity

$\sum_{i=1}^{\infty}\left|\left|f_{i}(\xi,\phi)\right|\right|=\left|\left|\sum_{i=1}^{\infty}f_{i}(\xi,\phi)\right|\right|,$
(8.144)

provided that the infinite series on the LHS is convergent. Recall that
completeness of the space is necessary for convergence of the sequence of
partial sums on the space

$s_{k}=\sum_{i=1}^{k}\left|\left|f_{i}(\xi,\phi)\right|\right|,$ (8.145)

which is a Cauchy sequence.

The inner product and the norm can be used for definition of the angle
$\alpha$ between two vectors $f(\xi,\phi)$ and $g(\xi,\phi)$

$\alpha\left(f(\xi,\phi),g(\xi,\phi)\right)=\arccos\dfrac{\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle}{\left|\left|f(\xi,\phi)\right|\right|\left|\left|g(\xi,\phi)\right|\right|},$
(8.146)

and it can be shown by straightforward computation that the
Cauchy–Bunyakovsky–Schwarz inequality holds

$\left|\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle\right|\leqslant\left|\left|f(\xi,\phi)\right|\right|\left|\left|g(\xi,\phi)\right|\right|.$
(8.147)

Taking into account

$\displaystyle f(\xi,\phi)$ $\displaystyle=$
$\displaystyle\dfrac{\sin\left[\Omega(\xi)\sqrt{\langle
V\rangle(\xi,\phi)}\right]}{\sqrt{\langle V\rangle(\xi,\phi)}},$ (8.148)
$\displaystyle g(\xi,\phi)$ $\displaystyle=$
$\displaystyle\cos\left[\Omega(\xi)\sqrt{\langle V\rangle(\xi,\phi)}\right],$
(8.149)

one obtains

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle$
$\displaystyle=$
$\displaystyle\int_{\Sigma(\xi,\phi)}\dfrac{\sin\left[2\Omega(\xi^{\prime})\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}\right]}{2\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}}\delta\xi^{\prime}\delta\phi^{\prime},$
(8.150)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left|\left|f(\xi,\phi)\right|\right|^{2}$
$\displaystyle=$
$\displaystyle\int_{\Sigma(\xi,\phi)}\left(\dfrac{\sin\left[\Omega(\xi^{\prime})\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}\right]}{\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}}\right)^{2}\delta\xi^{\prime}\delta\phi^{\prime},$
(8.151)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left|\left|g(\xi,\phi)\right|\right|^{2}$
$\displaystyle=$
$\displaystyle\int_{\Sigma(\xi,\phi)}\cos^{2}\left[\Omega(\xi)\sqrt{\langle
V\rangle(\xi,\phi)}\right]\delta\xi^{\prime}\delta\phi^{\prime}.$ (8.152)

By this reason one can rewrite the integral (8.107) as

$I=\dfrac{\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle}{\left|\left|f(\xi,\phi)\right|\right|\left|\left|g(\xi,\phi)\right|\right|},$
(8.153)

or with using of the definition (8.146)

$I=\cos\alpha\left(f(\xi,\phi),g(\xi,\phi)\right).$ (8.154)

In this manner the inequality (8.129) can be rewritten as

$\cos\alpha\left(f(\xi,\phi),g(\xi,\phi)\right)\geqslant\dfrac{2}{\sqrt{|\Gamma_{0}|}}$
(8.155)

whereas the Cauchy–Bunyakovsky–Schwarz inequality (8.147) says that

$\left|\cos\alpha\left(f(\xi,\phi),g(\xi,\phi)\right)\right|\leqslant 1.$
(8.156)

Hence the superposition principle requires

$\dfrac{2}{\sqrt{|\Gamma_{0}|}}\leqslant\left|\cos\alpha\left(f(\xi,\phi),g(\xi,\phi)\right)\right|\leqslant
1,$ (8.157)

or equivalently

$\alpha\left(f(\xi,\phi),g(\xi,\phi)\right)\in\left[\arccos\dfrac{2}{\sqrt{|\Gamma_{0}|}}+2k\pi,2\pi+2k\pi\right],\quad
k\in\mathbb{Z},$ (8.158)

what is consistent if and only if

$\sqrt{|\Gamma_{0}|}\geqslant 2.$ (8.159)

Let us consider the wave functions (8.99) and (8.100) expressed via the
invariant global dimension

$\displaystyle\Psi_{1}[\xi,\phi]$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{\sin\left[\Omega(\xi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right]}{\sqrt{{\langle
V\rangle(\xi,\phi)}}}=\dfrac{1}{\sqrt{|A|}}f(\xi,\phi),$ (8.160)
$\displaystyle\Psi_{2}[\xi,\phi]$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cos\left[\Omega(\xi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right]=\dfrac{1}{\sqrt{|B|}}g(\xi,\phi),$ (8.161)

with the constants $A$ and $B$ given by

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!A$
$\displaystyle=$
$\displaystyle\int_{\Sigma(\xi,\phi)}\left(\dfrac{\sin\left[\Omega(\xi^{\prime})\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}\right]}{\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}}\right)^{2}\delta\xi^{\prime}\delta\phi^{\prime}\equiv\left|\left|f(\xi,\phi)\right|\right|^{2},$
(8.162)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!B$
$\displaystyle=$
$\displaystyle\dfrac{1}{|\Gamma_{0}|}\int_{\Sigma(\xi,\phi)}\cos^{2}\left[\Omega(\xi^{\prime})\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}\right]\delta\xi^{\prime}\delta\phi^{\prime}\equiv\dfrac{1}{|\Gamma_{0}|}\left|\left|g(\xi,\phi)\right|\right|^{2}.$
(8.163)

In other words

$\displaystyle\Psi_{1}[\xi,\phi]$ $\displaystyle=$
$\displaystyle\dfrac{f(\xi,\phi)}{\left|\left|f(\xi,\phi)\right|\right|},$
(8.164) $\displaystyle\Psi_{2}[\xi,\phi]$ $\displaystyle=$
$\displaystyle\sqrt{|\Gamma_{0}|}\dfrac{g(\xi,\phi)}{\left|\left|g(\xi,\phi)\right|\right|},$
(8.165) $\displaystyle I$ $\displaystyle=$
$\displaystyle\cos\alpha\left(\Psi_{1}[\xi,\phi],\Psi_{2}[\xi,\phi]\right).$
(8.166)

The wave functions (8.164) and (8.165) are elements of the Lebesgue space
$L^{2}\left(\Sigma(\xi,\phi),\delta\xi\delta\phi\right)$ with the scalar
product

$\left({\Psi_{1}(\xi,\phi),\Psi_{2}(\xi,\phi)}\right)=\Re\left\langle{\Psi_{1}(\xi,\phi),\Psi_{2}(\xi,\phi)}\right\rangle,$
(8.167)

where the inner product is

$\left\langle{\Psi_{1}(\xi,\phi),\Psi_{2}(\xi,\phi)}\right\rangle=\int_{\Sigma(\xi,\phi)}\Psi_{1}^{\star}(\xi^{\prime},\phi^{\prime})\Psi_{2}(\xi^{\prime},\phi^{\prime})\delta\xi^{\prime}\delta\phi^{\prime},$
(8.168)

and the induced norm

$\left|\left|\Psi_{i}(\xi,\phi)\right|\right|=\left(\int_{\Sigma(\xi,\phi)}|\Psi_{i}(\xi^{\prime},\phi^{\prime})|^{2}\delta\xi^{\prime}\delta\phi^{\prime}\right)^{1/2},$
(8.169)

where $i=1,2$. It is easy to see that

$\left({\Psi_{1}(\xi,\phi),\Psi_{2}(\xi,\phi)}\right)=\sqrt{|\Gamma_{0}|}I,$
(8.170)

and because of $I\geqslant 2$ the scalar product satisfies the inequality

$\left({\Psi_{1}(\xi,\phi),\Psi_{2}(\xi,\phi)}\right)\geqslant
2\sqrt{|\Gamma_{0}|}.$ (8.171)

In the light of the inequality (8.159) one has

$\left({\Psi_{1}(\xi,\phi),\Psi_{2}(\xi,\phi)}\right)\geqslant 4.$ (8.172)

It can be seen also that

$\displaystyle||\Psi_{1}||$ $\displaystyle=$ $\displaystyle 1,$ (8.173)
$\displaystyle||\Psi_{2}||$ $\displaystyle=$
$\displaystyle\sqrt{|\Gamma_{0}|}\geqslant 2,$ (8.174)

i.e. the state $\Psi_{1}$ is a ray whereas the state $\Psi_{2}$ is not a ray
in the Hilbert space. Because, however the superposition is given by

$\Psi=\alpha_{1}\Psi_{1}+\alpha_{2}\Psi_{2},$ (8.175)

one can selected the constants $\alpha_{1}$ and $\alpha_{2}$ in such a way
that the wave function $\Psi_{2}$ will be normalized to unity. The choice is
easy to deduce. Because of $\Psi_{1}$ is normalized to unity the coefficient
$\alpha_{1}$ is an arbitrary constant, while the coefficient $\alpha_{2}$
should be exchanged on

$\alpha_{2}^{\prime}=\dfrac{1}{\sqrt{|\Gamma_{0}|}}\alpha_{2},$ (8.176)

where $\alpha_{2}$ is an arbitrary constant. Still, however, in general the
problem of normalization of the state $\Psi_{2}$ is unsolved. Redefinition of
the scalar product

$\left({\Psi_{1}(\xi,\phi),\Psi_{2}(\xi,\phi)}\right)=\dfrac{\left\langle{\Psi_{1}(\xi,\phi),\Psi_{2}(\xi,\phi)}\right\rangle}{\left|\left|\Psi_{1}(\xi,\phi)\right|\right|\left|\left|\Psi_{2}(\xi,\phi)\right|\right|}=I\geqslant
2,$ (8.177)

does not work consistently in the light of (8.154). The only solution is to
apply _ad hoc_ the scaling

$\Psi_{2}^{\prime}=\dfrac{1}{\sqrt{|\Gamma_{0}|}}\Psi_{2}\leqslant\dfrac{1}{\sqrt{2}}\Psi_{2},$
(8.178)

which, however, is consistent with the choice (8.176).

Another alternative can be constructed as follows. Let us assume that the
superposition state

$\Psi^{\prime}=\alpha_{1}^{\prime}\Psi_{1}^{\prime}+\alpha_{2}^{\prime}\Psi_{2}^{\prime},$
(8.179)

is such that the states $\Psi_{1}^{\prime}$ and $\Psi_{2}^{\prime}$ are
orthonormal, i.e. are orthogonal rays in the Hilbert space

$\displaystyle||\Psi_{1}^{\prime}||$ $\displaystyle=$ $\displaystyle 1,$
(8.180) $\displaystyle||\Psi_{2}^{\prime}||$ $\displaystyle=$ $\displaystyle
1,$ (8.181) $\displaystyle\left(\Psi_{1}^{\prime},\Psi_{2}^{\prime}\right)$
$\displaystyle=$ $\displaystyle 0.$ (8.182)

The problem is to construct such constants $\alpha_{1}^{\prime}$,
$\alpha_{2}^{\prime}$, and states $\Psi_{1}^{\prime}$ and $\Psi_{2}^{\prime}$
which satisfy these requirements. This problem can be solved with help of the
Gram–Schmidt algorithm, in which the orthonormal states $\Psi_{i}^{\prime}$
(here $i=1,2$) are expressed via the non-orthonormal states $\Psi_{i}$ via
using of the Gram determinants

$\displaystyle G_{0}$ $\displaystyle=$ $\displaystyle 1,$ (8.183)
$\displaystyle G_{1}$ $\displaystyle=$
$\displaystyle\left|(\Psi_{1},\Psi_{1})\right|,$ (8.184) $\displaystyle G_{2}$
$\displaystyle=$
$\displaystyle\det\left[\begin{array}[]{cc}(\Psi_{1},\Psi_{1})&(\Psi_{1},\Psi_{2})\\\
(\Psi_{1},\Psi_{2})&(\Psi_{2},\Psi_{2})\end{array}\right],$ (8.187)

as follows

$\displaystyle\Psi_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{G_{0}G_{1}}}\Psi_{1},$ (8.188)
$\displaystyle\Psi_{2}^{\prime}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{G_{1}G_{2}}}\det\left[\begin{array}[]{cc}(\Psi_{1},\Psi_{1})&(\Psi_{1},\Psi_{2})\\\
\Psi_{1}&\Psi_{2}\end{array}\right].$ (8.191)

It can be seen by straightforward computation that

$\displaystyle G_{1}$ $\displaystyle=$ $\displaystyle 1,$ (8.192)
$\displaystyle G_{2}$ $\displaystyle=$
$\displaystyle|\Gamma_{0}|\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right),$
(8.193)

and consequently

$\displaystyle\Psi_{1}^{\prime}$ $\displaystyle=$ $\displaystyle\Psi_{1},$
(8.194) $\displaystyle\Psi_{2}^{\prime}$ $\displaystyle=$
$\displaystyle\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{-1/2}\left(-I\Psi_{1}+\dfrac{1}{\sqrt{|\Gamma_{0}|}}\Psi_{2}\right).$
(8.195)

One sees now that the second term in (8.195) converges with the proposed
scaled state (8.178). In this manner the superposition state (8.179) can be
rewritten as

$\Psi^{\prime}=\alpha_{1}^{\prime}\Psi_{1}+\alpha_{2}^{\prime}\Psi_{2}^{\prime},$
(8.196)

where now $\Psi_{2}^{\prime}=\dfrac{1}{\sqrt{|\Gamma_{0}|}}\Psi_{2}$ is the
scaled state (8.178) and

$\displaystyle\alpha_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\alpha_{1}-\alpha_{2}I\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{-1/2},$
(8.197) $\displaystyle\alpha_{2}^{\prime}$ $\displaystyle=$
$\displaystyle\alpha_{2}\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{-1/2}.$
(8.198)

The states $\Psi_{1}$ and $\Psi_{2}$ can be expressed via the orthonormal
states $\Psi_{1}^{\prime}$ and $\Psi_{2}^{\prime}$

$\displaystyle\Psi_{1}$ $\displaystyle=$ $\displaystyle\Psi_{1}^{\prime},$
(8.199) $\displaystyle\Psi_{2}$ $\displaystyle=$
$\displaystyle\sqrt{|\Gamma_{0}|}\left(\Psi_{2}^{\prime}+I\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{1/2}\Psi_{1}^{\prime}\right),$
(8.200)

and by this reason

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(\Psi_{1},\Psi_{2})$
$\displaystyle=$
$\displaystyle\left(\Psi_{1}^{\prime},\sqrt{|\Gamma_{0}|}\left(\Psi_{2}^{\prime}+I\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{1/2}\Psi_{1}^{\prime}\right)\right)=$
(8.201) $\displaystyle=$
$\displaystyle\sqrt{|\Gamma_{0}|}\left[(\Psi_{1}^{\prime},\Psi_{2}^{\prime})+I\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{1/2}(\Psi_{1}^{\prime},\Psi_{1}^{\prime})\right]=$
$\displaystyle=$
$\displaystyle\sqrt{|\Gamma_{0}|}I\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{1/2}.$

In the light of the relation (8.170) one has

$\sqrt{|\Gamma_{0}|}I\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{1/2}=\sqrt{|\Gamma_{0}|}I,$
(8.202)

which, because of $I\geqslant 2$, gives uniquely and unambiguously

$\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{1/2}=1,$ (8.203)

or equivalently

$I^{2}=\dfrac{1}{\sqrt{|\Gamma_{0}|}}-1.$ (8.204)

In the light of the fact $I\geqslant 2$ one has

$\sqrt{|\Gamma_{0}|}\leqslant\dfrac{1}{5},$ (8.205)

what however is inconsistent in the light of the fact (8.159). In this manner
the problem of superposition in the theory of quantum gravity can not be
solved by application of the Gram–Schmidt algorithm of orthonormalization to
the Hilbert space constructed in this subsection.

####  F Problem III: The Problem of Time

It is easy to see that within the invariant global one-dimensional quantum
gravity the problem of time is naturally solved. Namely the role of physical
time plays the invariant global dimension

$\xi=\dfrac{1}{4\pi}\sqrt{\dfrac{h}{6}}\equiv\dfrac{t_{\textrm{physical}}}{\tau},$
(8.206)

where $\tau$ is a reference constant, which can be taken _ad hoc_ as the
Planck time, i.e. $\tau=t_{P}$. Such an identification is justified by the
fact that $\xi$ is invariant with respect to action of the spatial
diffeomorphisms group, because of the volume form $\sqrt{h}$ is such a
diffeoinvariant. In this manner, in the sense of Kuchař $\xi$ is an
observable.

Such a situation is analogous to the quantum cosmology presented in the
chapter 5, in which the role of time plays the cosmic scale factor parameter
$a$. This parameter is a function of the conformal or cosmological time. By
this reason the proposed model of quantum gravity can be rewritten in more
fashionable form

$\left(\dfrac{d^{2}}{dt^{2}}+V[t]\right)\Psi(t,\phi)=0.$ (8.207)

This global one-dimensional evolutionary equation defines in itself non-
trivial dynamics. Namely, this is the time-dependent $0+1$ Schrödinger wave
equation.

###  Chapter 9 Examples of Invariant 1D Wave Functions

Let us present with no detailed computations the wave functionals associated
to the three classical solutions of the Einstein field equations.

####  A The Minkowski Space-time

Let us consider first empty space with no cosmological constant, i.e. the
Minkowski space-time. In such a case, in the Cartesian coordinates, the
spatial metric coincides with the metric of the Euclidean space

$h_{ij}=\left[\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&1\end{array}\right],$
(9.1)

and therefore $h=1$ so that the invariant global dimension is

$\xi=\dfrac{1}{4\pi}\dfrac{1}{\sqrt{6}}.$ (9.2)

The Minkowski space-time is characterized by

${{}^{(3)}}R=0\quad,\quad\varrho=0\quad,\quad\Lambda=0,$ (9.3)

and by this reason the averaged generalized gravitational potential is

$\langle{V}\rangle(\xi,\phi)=0.$ (9.4)

In this manner the wave functions for this case are

$\displaystyle\Psi_{1}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\xi,$ (9.5) $\displaystyle\Psi_{2}$
$\displaystyle=$ $\displaystyle\dfrac{1}{\sqrt{|B|}},$ (9.6)

where the integration constants $A$ and $B$ are

$\displaystyle A$ $\displaystyle=$
$\displaystyle\dfrac{1}{18(4\pi)^{3}\sqrt{6}},$ (9.7) $\displaystyle B$
$\displaystyle=$ $\displaystyle\dfrac{1}{4\Gamma_{0}\pi\sqrt{6}}.$ (9.8)

In other words the wave functions

$\displaystyle\Psi_{1}$ $\displaystyle=$ $\displaystyle\sqrt{12\pi\sqrt{6}},$
(9.9) $\displaystyle\Psi_{2}$ $\displaystyle=$
$\displaystyle\sqrt{4|\Gamma_{0}|\pi\sqrt{6}},$ (9.10)

are constant on the midisuperspace. This case is singular because of constant
wave functions can not be normalized to unity separately. However, taking the
superposed state

$\Psi=\alpha_{1}\Psi_{1}+\alpha_{2}\Psi_{2},$ (9.11)

one obtains the normalization condition

$3|\alpha_{1}|^{2}+|\Gamma_{0}||\alpha_{2}|^{2}+\left(\alpha_{1}^{\star}\alpha_{2}+\alpha_{1}\alpha_{2}^{\star}\right)\sqrt{3|\Gamma_{0}|}=1,$
(9.12)

having the following solutions

$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle\dfrac{\beta_{1}}{\sqrt{3}}\exp(i\alpha),$ (9.13)
$\displaystyle\alpha_{2}$ $\displaystyle=$
$\displaystyle\dfrac{\beta_{2}}{\sqrt{|\Gamma_{0}|}}\exp(i\alpha),$ (9.14)

where $\alpha$ is an arbitrary real phase, and the constants $\beta_{1}$ and
$\beta_{2}$ are constrained by

$\beta_{2}=\pm 1-\beta_{1}.$ (9.15)

In other words the superposed states

$\Psi_{\pm}^{M}=\exp(i\alpha)\left(\dfrac{\beta}{\sqrt{3}}\Psi_{1}+\dfrac{\pm
1-\beta}{\sqrt{|\Gamma_{0}|}}\Psi_{2}\right)=\pm\exp(i\alpha)\sqrt{4\pi\sqrt{6}},$
(9.16)

are consistent wave functions of the Minkowski space-time. Because of the
Minkowski vacuum is nonlinearly stable, the wave functions (9.16) are the
reference states for another $\pi$ number definition

$\pi:=\dfrac{\left|\Psi_{\pm}^{M}\right|^{2}}{4\sqrt{6}}.$ (9.17)

####  B The Kasner Space-time

Let us consider the simple solution of the Einstein field equations describing
an anisotropic universe without matter, called the Kasner metric. The spatial
part of this metric is

$h_{ij}=\left[\begin{array}[]{ccc}t^{2p_{1}}&0&0\\\ 0&t^{2p_{2}}&0\\\
0&0&t^{2p_{3}}\end{array}\right],$ (9.18)

where $t$ is time coordinate, and

$\displaystyle\sum_{i}p_{i}$ $\displaystyle=$ $\displaystyle 1,$ (9.19)
$\displaystyle\sum_{i}p_{i}^{2}$ $\displaystyle=$ $\displaystyle 1.$ (9.20)

In this case the global dimension is

$h=t^{2},$ (9.21)

while the invariant global dimension is

$\xi=\dfrac{t}{4\pi\sqrt{6}}.$ (9.22)

Because of the cosmological constant and the Matter fields are absent here,
and the three-dimensional Ricci scalar curvature vanishes one has
$\langle{V}\rangle=0$. In this manner the wave functions of the Kasner space-
time are

$\displaystyle\Psi_{1}^{K}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{t}{4\pi\sqrt{6}},$ (9.23)
$\displaystyle\Psi_{2}^{K}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}},$ (9.24)

where the constants of integration are

$\displaystyle A$ $\displaystyle=$
$\displaystyle\dfrac{1}{3}\dfrac{T^{3}}{6(4\pi)^{3}\sqrt{6}},$ (9.25)
$\displaystyle B$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Gamma_{0}}\dfrac{T}{4\pi\sqrt{6}},$ (9.26)

where $T$ is some reference value of time $t$.

####  C The Schwarzschild Space-time

The second situation which we shall present here is the Schwarzschild space-
time. This metric is a spherically symmetric vacuum solution of the Einstein
field equations with no cosmological constant, i.e. a situation
${{}^{(3)}\\!R[h]}=0$, $\Lambda=0$, $\varrho=0$. It means that in this case
also $\langle V\rangle(h_{ij},\phi)=0$. The spatial part of the metric has the
form

$h_{ij}=\left[\begin{array}[]{ccc}\left(1-\dfrac{r_{S}}{r}\right)^{-1}&0&0\\\
0&r^{2}&0\\\ 0&0&r^{2}\sin^{2}\theta\end{array}\right],$ (9.27)

where $r_{S}$ is the Schwarzschild radius of the spherically symmetric non-
rotating object of mass $M$

$r_{S}=\dfrac{2GM}{c^{2}}=\dfrac{\kappa}{4\pi}Mc^{2}.$ (9.28)

Therefore in this case the global dimension is

$h=\dfrac{r^{4}\sin^{2}\theta}{1-\dfrac{r_{S}}{r}},$ (9.29)

so that the invariant global dimension is

$\xi=\dfrac{1}{4\pi}\dfrac{r^{2}\sin\theta}{\sqrt{6\left(1-\dfrac{r_{S}}{r}\right)}},$
(9.30)

and the total volume of the Schwarzschild midisuperspace is $\Omega(\xi)=\xi$.
The integration constants $A$ and $B$ can be established easy

$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\Omega^{2}(\xi)\delta\xi=\dfrac{\Xi^{3}}{3}=\dfrac{1}{(4\pi)^{3}}\dfrac{R^{6}\sin^{3}\Theta}{\left(6\left(1-\dfrac{r_{S}}{R}\right)\right)^{3/2}},$
(9.31) $\displaystyle B$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Gamma_{0}}\int\delta\xi=\dfrac{\Xi}{\Gamma_{0}}=\dfrac{1}{4\Gamma_{0}\pi}\dfrac{R^{2}\sin\Theta}{\sqrt{6\left(1-\dfrac{r_{S}}{R}\right)}},$
(9.32)

where $\Xi=\xi(R,\Theta)$ is

$\Xi=\dfrac{1}{4\pi}\dfrac{R^{2}\sin\Theta}{\sqrt{6\left(1-\dfrac{r_{S}}{R}\right)}}.$
(9.33)

In this manner the wave functions of the Schwarzschild space-time are

$\displaystyle\Psi_{1}^{S}$ $\displaystyle=$
$\displaystyle\sqrt{4\pi\sqrt{6}}\sqrt{\dfrac{1}{R^{2}\sin\Theta}\sqrt{1-\dfrac{r_{S}}{R}}}\sqrt{\dfrac{1-\dfrac{r_{S}}{R}}{1-\dfrac{r_{S}}{r}}}\dfrac{r^{2}\sin\theta}{R^{2}\sin\Theta},$
(9.34) $\displaystyle\Psi_{2}^{S}$ $\displaystyle=$
$\displaystyle\sqrt{4\pi|\Gamma_{0}|\sqrt{6}}\sqrt{\dfrac{1}{R^{2}\sin\Theta}\sqrt{1-\dfrac{r_{S}}{R}}}.$
(9.35)

####  D The (Anti-) De Sitter Space-time

Let us consider now the spherically symmetric solution of vacuum Einstein’s
field equations in presence of the cosmological constant, i.e. lambdavacuum
Einstein’s field equations

$R_{\mu\nu}=\Lambda g_{\mu\nu},$ (9.36)

called the (Anti-) De Sitter space-time. Such a space-time is a submanifold of
the Minkowski space-time of one higher dimension described by the hyperboloid
of one sheet

$-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=\pm\alpha^{2},$ (9.37)

where the sign plus describes De Sitter space-time, whereas the sign minus is
related to the Anti-De Sitter space. The (Anti-) De Sitter space-time is an
Einstein manifold

$R_{\mu\nu}=\pm\dfrac{3}{\alpha^{2}}g_{\mu\nu},$ (9.38)

what in the light of the lambdavacuum Einstein field equations (9.36) the
cosmological constant is

$\Lambda=\pm\dfrac{3}{\alpha^{2}}.$ (9.39)

The three-dimensional Ricci scalar curvature of the (Anti-) De Sitter space-
time can be established easy

${{}^{(3)}}R=h^{ij}R_{ij}=3\Lambda,$ (9.40)

and because of there is no Matter fields $\varrho=0$, so that the averaged
generalized gravitational potential has the value

$\langle{V}\rangle(\xi,\phi)=-3\Lambda+2\Lambda=-\Lambda.$ (9.41)

In other words for the De Sitter space-time

$\langle{V}\rangle(\xi,\phi)=-\dfrac{3}{\alpha^{2}},$ (9.42)

while for the Anti-De Sitter space-time

$\langle{V}\rangle(\xi,\phi)=\dfrac{3}{\alpha^{2}}.$ (9.43)

The spatial part of the (Anti-) De Sitter metric is

$h_{ij}=\left[\begin{array}[]{ccc}\left(1-\dfrac{\Lambda}{3}r^{2}\right)^{-1}&0&0\\\
0&r^{2}&0\\\ 0&0&r^{2}\sin^{2}\theta\end{array}\right],$ (9.44)

and by this reason the global dimension is

$h=\dfrac{r^{4}\sin^{2}\theta}{1-\dfrac{\Lambda}{3}r^{2}},$ (9.45)

whereas the invariant global dimension is

$\xi=\dfrac{1}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{\Lambda}{3}r^{2}}},$
(9.46)

and of course the volume of the related midisuperspace is $\Omega(\xi)=\xi$.

Let us consider first the case of the De Sitter space-time, i.e.
$\Lambda=\dfrac{3}{\alpha^{2}}$. The constants of integration can be evaluated
as follows

$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\left(\dfrac{\sinh\left(\sqrt{\Lambda}\xi\right)}{\sqrt{\Lambda}}\right)^{2}\delta\xi=\dfrac{1}{\Lambda}\left(-\dfrac{\Xi}{2}+\dfrac{\sinh\left(2\sqrt{\Lambda}\Xi\right)}{4\sqrt{\Lambda}}\right),$
(9.47) $\displaystyle B$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Gamma_{0}}\int\cosh^{2}\left(\sqrt{\Lambda}\xi\right)\delta\xi=\dfrac{1}{\Gamma_{0}}\left(\dfrac{\Xi}{2}+\dfrac{\cosh\left(2\sqrt{\Lambda}\Xi\right)}{4\sqrt{\Lambda}}\right),$
(9.48)

where $\Xi$ is the reference constant

$\Xi=\dfrac{1}{4\pi\sqrt{6}}\dfrac{R^{2}\sin\Theta}{\sqrt{1-\dfrac{\Lambda}{3}R^{2}}}.$
(9.49)

By this reason the wave functions of the De Sitter space-time are

$\displaystyle\Psi_{1}^{DS}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{1}{\sqrt{\Lambda}}\sinh\left(\dfrac{\sqrt{\Lambda}}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{\Lambda}{3}r^{2}}}\right),$
(9.50) $\displaystyle\Psi_{2}^{DS}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cosh\left(\dfrac{\sqrt{\Lambda}}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{\Lambda}{3}r^{2}}}\right).$
(9.51)

Similarly for the case of Anti-De Sitter space-time one obtains the following
wave functions

$\displaystyle\Psi_{1}^{ADS}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{1}{\sqrt{|\Lambda|}}\sin\left(\dfrac{\sqrt{|\Lambda|}}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1+\dfrac{|\Lambda|}{3}r^{2}}}\right),$
(9.52) $\displaystyle\Psi_{2}^{ADS}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cos\left(\dfrac{\sqrt{|\Lambda|}}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1+\dfrac{|\Lambda|}{3}r^{2}}}\right),$
(9.53)

where $|\Lambda|=\dfrac{3}{\alpha^{2}}$, and the constants of integration $A$
and $B$ are

$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\left(\dfrac{\sin\left(\sqrt{|\Lambda|}\xi\right)}{\sqrt{|\Lambda|}}\right)^{2}\delta\xi=\dfrac{1}{|\Lambda|}\left(\dfrac{\Xi}{2}-\dfrac{\sin\left(2\sqrt{|\Lambda|}\Xi\right)}{4\sqrt{|\Lambda|}}\right),$
(9.54) $\displaystyle B$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Gamma_{0}}\int\cos^{2}\left(\sqrt{|\Lambda|}\xi\right)\delta\xi=\dfrac{1}{\Gamma_{0}}\left(\dfrac{\Xi}{2}+\dfrac{\cos\left(2\sqrt{|\Lambda|}\Xi\right)}{4\sqrt{|\Lambda|}}\right),$
(9.55)

where $\Xi$ is the reference constant

$\Xi=\dfrac{1}{4\pi\sqrt{6}}\dfrac{R^{2}\sin\Theta}{\sqrt{1+\dfrac{|\Lambda|}{3}R^{2}}}.$
(9.56)

####  E The (Anti-) De Sitter–Schwarzschild Space-time

The (Anti-) De Sitter–Schwarzschild space-time is the case jointing the
(Anti-) De Sitter and Schwarzschild space-times. This is the spherically
symmetric solution of lambdavacuum Einstein’s field equations with suitable
boundary conditions. The spatial part of space-time metric has the form

$h_{ij}=\left[\begin{array}[]{ccc}\left(1-\dfrac{r_{S}}{r}-\dfrac{\Lambda}{3}r^{2}\right)^{-1}&0&0\\\
0&r^{2}&0\\\ 0&0&r^{2}\sin^{2}\theta\end{array}\right].$ (9.57)

In this manner the global dimension is

$h=\dfrac{r^{4}\sin^{2}\theta}{1-\dfrac{r_{S}}{r}-\dfrac{\Lambda}{3}r^{2}},$
(9.58)

while the invariant global dimension is

$\xi=\dfrac{1}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{r_{S}}{r}-\dfrac{\Lambda}{3}r^{2}}},$
(9.59)

and of course the volume of the related midisuperspace is $\Omega(\xi)=\xi$.
The wave functions of the De Sitter–Schwarzschild space-time, called also the
Kottler space-time, can be written in the form

$\displaystyle\Psi_{1}^{DS-S}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{1}{\sqrt{\Lambda}}\sinh\left(\dfrac{\sqrt{\Lambda}}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{r_{S}}{r}-\dfrac{\Lambda}{3}r^{2}}}\right),$
(9.60) $\displaystyle\Psi_{2}^{DS-S}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cosh\left(\dfrac{\sqrt{\Lambda}}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{r_{S}}{r}-\dfrac{\Lambda}{3}r^{2}}}\right).$
(9.61)

where the constants of integration $A$ and $B$ are

$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\left(\dfrac{\sinh\left(\sqrt{\Lambda}\xi\right)}{\sqrt{\Lambda}}\right)^{2}\delta\xi=\dfrac{1}{\Lambda}\left(-\dfrac{\Xi}{2}+\dfrac{\sinh\left(2\sqrt{\Lambda}\Xi\right)}{4\sqrt{\Lambda}}\right),$
(9.62) $\displaystyle B$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Gamma_{0}}\int\cosh^{2}\left(\sqrt{\Lambda}\xi\right)\delta\xi=\dfrac{1}{\Gamma_{0}}\left(\dfrac{\Xi}{2}+\dfrac{\cosh\left(2\sqrt{\Lambda}\Xi\right)}{4\sqrt{\Lambda}}\right),$
(9.63)

where $\Xi$ is the reference constant

$\Xi=\dfrac{1}{4\pi\sqrt{6}}\dfrac{R^{2}\sin\Theta}{\sqrt{1-\dfrac{r_{S}}{R}-\dfrac{\Lambda}{3}R^{2}}}.$
(9.64)

Similarly, the wave functions of the Anti-De Sitter–Schwarzschild space-time
are

$\displaystyle\Psi_{1}^{ADS-S}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{1}{\sqrt{|\Lambda|}}\sin\left(\dfrac{\sqrt{|\Lambda|}}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{r_{S}}{r}+\dfrac{|\Lambda|}{3}r^{2}}}\right),$
(9.65) $\displaystyle\Psi_{2}^{ADS-S}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cos\left(\dfrac{\sqrt{|\Lambda|}}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{r_{S}}{r}+\dfrac{|\Lambda|}{3}r^{2}}}\right),$
(9.66)

where $|\Lambda|=\dfrac{3}{\alpha^{2}}$, and the constants of integration $A$
and $B$ are

$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\left(\dfrac{\sin\left(\sqrt{|\Lambda|}\xi\right)}{\sqrt{|\Lambda|}}\right)^{2}\delta\xi=\dfrac{1}{|\Lambda|}\left(\dfrac{\Xi}{2}-\dfrac{\sin\left(2\sqrt{|\Lambda|}\Xi\right)}{4\sqrt{|\Lambda|}}\right),$
(9.67) $\displaystyle B$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Gamma_{0}}\int\cos^{2}\left(\sqrt{|\Lambda|}\xi\right)\delta\xi=\dfrac{1}{\Gamma_{0}}\left(\dfrac{\Xi}{2}+\dfrac{\cos\left(2\sqrt{|\Lambda|}\Xi\right)}{4\sqrt{|\Lambda|}}\right),$
(9.68)

where $\Xi$ is the reference constant

$\Xi=\dfrac{1}{4\pi\sqrt{6}}\dfrac{R^{2}\sin\Theta}{\sqrt{1-\dfrac{r_{S}}{R}+\dfrac{|\Lambda|}{3}R^{2}}}.$
(9.69)

####  F The Kerr Space-Time

Kerr’s space-time is a solution of vacuum Einstein’s field equations for
spherical body rotating with the angular momentum $J$. Because of the spatial
Ricci scalar curvature is zero, cosmological constant is not present, and
Matter fields are absent, in such a situation the averaged generalized
gravitational potential identically vanishes. The spatial part of the Kerr
metric has the form

$h_{ij}=\left[\begin{array}[]{ccc}\dfrac{r^{2}+\alpha^{2}\cos^{2}\theta}{r^{2}-r_{s}r+\alpha^{2}}&0&0\\\
0&r^{2}+\alpha^{2}\cos^{2}\theta&0\\\
0&0&r^{2}+\alpha^{2}+\dfrac{\alpha^{2}r_{S}r\sin^{2}\theta}{r^{2}+\alpha^{2}\cos^{2}\theta}\end{array}\right],$
(9.70)

where $\alpha$ is the coefficient related to the angular momentum and the mass
of the rotating object

$\alpha=\dfrac{J}{Mc}.$ (9.71)

In this manner the global dimension is

$h=\dfrac{\left(r^{2}+\alpha^{2}\cos^{2}\theta\right)^{2}}{r^{2}-r_{s}r+\alpha^{2}}\left(r^{2}+\alpha^{2}+\dfrac{\alpha^{2}r_{S}r\sin^{2}\theta}{r^{2}+\alpha^{2}\cos^{2}\theta}\right),$
(9.72)

while the invariant global dimension is

$\xi=\dfrac{1}{4\pi\sqrt{6}}\left(r^{2}+\alpha^{2}\cos^{2}\theta\right)\sqrt{\dfrac{\left(r^{2}+\alpha^{2}\right)^{2}-\alpha^{2}\left(r^{2}-r_{S}r+\alpha^{2}\right)\sin^{2}\theta}{\left(r^{2}-r_{s}r+\alpha^{2}\right)\left(r^{2}+\alpha^{2}\cos^{2}\theta\right)}},$
(9.73)

and of course the volume of the Kerr midisuperspace is $\Omega(\xi)=\xi$.
Introducing the reference parameter

$\Xi=\dfrac{1}{4\pi\sqrt{6}}\left(R^{2}+\alpha^{2}\cos^{2}\Theta\right)\sqrt{\dfrac{\left(R^{2}+\alpha^{2}\right)^{2}-\alpha^{2}\left(R^{2}-r_{S}R+\alpha^{2}\right)\sin^{2}\Theta}{\left(R^{2}-r_{s}R+\alpha^{2}\right)\left(R^{2}+\alpha^{2}\cos^{2}\Theta\right)}},$
(9.74)

one can write out the constants of integration $A$ and $B$

$\displaystyle A$ $\displaystyle=$ $\displaystyle\dfrac{\Xi^{3}}{3},$ (9.75)
$\displaystyle B$ $\displaystyle=$ $\displaystyle\dfrac{1}{\Gamma_{0}}\Xi,$
(9.76)

so that the wave functions of the Kerr space-time have the form

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Psi_{1}^{Kerr}$
$\displaystyle=$
$\displaystyle\dfrac{r^{2}+\alpha^{2}\cos^{2}\theta}{4\pi\sqrt{|A|}\sqrt{6}}\sqrt{\dfrac{\left(r^{2}+\alpha^{2}\right)^{2}-\alpha^{2}\left(r^{2}-r_{S}r+\alpha^{2}\right)\sin^{2}\theta}{\left(r^{2}-r_{s}r+\alpha^{2}\right)\left(r^{2}+\alpha^{2}\cos^{2}\theta\right)}},$
(9.77)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Psi_{2}^{Kerr}$
$\displaystyle=$ $\displaystyle\dfrac{1}{\sqrt{|B|}}.$ (9.78)

####  G The Kerr–Newman Space-time

The Kerr–Newman space-time is a solution of the electrovacuum Einstein field
equations called also Einstein–Maxwell equations, which are combination of the
Einstein field equations with the stress-energy tensor of electromagnetic
field and no cosmological constant, and the source-free Maxwell equations

$\displaystyle R_{\mu\nu}=\kappa\ell_{P}^{2}{T}_{\mu\nu}^{em},$ (9.79)
$\displaystyle F_{\mu\nu;\kappa}+F_{\nu\kappa;\mu}+F_{\kappa\mu;\nu}=0,$
(9.80) $\displaystyle F^{\mu\nu}_{;\nu}=0,$ (9.81)

where $T_{\mu\nu}^{em}$ is the stress-energy tensor of electromagnetic field,
which can be given _ad hoc_ in analogy with the case of flat space-time

$T_{\mu\nu}^{em}=\dfrac{1}{\mu_{0}}\left(F^{\alpha}_{\mu}{F}_{\alpha\nu}-\dfrac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right),$
(9.82)

and $F_{\mu\nu}$ is the electromagnetic field tensor

$\displaystyle F_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}0&\dfrac{E_{x}}{c}&\dfrac{E_{y}}{c}&\dfrac{E_{z}}{c}\\\
-\dfrac{E_{x}}{c}&0&-B_{z}&-B_{y}\\\ -\dfrac{E_{y}}{c}&B_{z}&0&-B_{x}\\\
-\dfrac{E_{z}}{c}&-B_{y}&B_{x}&0\end{array}\right],$ (9.87) $\displaystyle
F^{\mu\nu}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}0&-\dfrac{E_{x}}{c}&-\dfrac{E_{y}}{c}&-\dfrac{E_{z}}{c}\\\
\dfrac{E_{x}}{c}&0&-B_{z}&-B_{y}\\\ \dfrac{E_{y}}{c}&B_{z}&0&-B_{x}\\\
\dfrac{E_{z}}{c}&-B_{y}&B_{x}&0\end{array}\right],$ (9.92)

where $\vec{E}=[E_{x},E_{y},E_{z}]$ and $\vec{B}=[B_{x},B_{y},B_{z}]$ are
three-vectors of the electric and the magnetic field, respectively. The speed
of light $c$ can be expressed in terms of the vacuum permeability $\mu_{0}$
and the vacuum permittivity $\epsilon_{0}$ via the formula
$\epsilon_{0}\mu_{0}=1/c^{2}$.

There is the question how to derive the stress-energy tensor of
electromagnetic field (9.82) via using of the definition

$T_{\mu\nu}=-\dfrac{2}{\sqrt{-g}}\dfrac{\delta(\sqrt{-g}\mathcal{L})}{\delta
g^{\mu\nu}},$ (9.93)

following from the Hilbert–Palatini action principle, and describing Matter
fields characterized by Lagrangian $\mathcal{L}$ in a four-dimensional
Riemannian space-time with a metric $g^{\mu\nu}$. The Lagrangian of the
Maxwell electromagnetic field has the form (For more general approach see e.g.
the Ref. [605])

$\mathcal{L}=-\dfrac{1}{4\mu_{0}}F_{\alpha\beta}F^{\alpha\beta}.$ (9.94)

With using of the Jacobi formula
$\delta\sqrt{-g}=-\dfrac{1}{2}\sqrt{-g}g_{\mu\nu}\delta{g}^{\mu\nu}$ the
stress-energy tensor (9.93) can be rewritten in more convenient form

$T_{\mu\nu}=-2\dfrac{\delta\mathcal{L}}{\delta
g^{\mu\nu}}+g_{\mu\nu}\mathcal{L},$ (9.95)

so that the problem is the establish the functional derivative

$\dfrac{\delta\mathcal{L}}{\delta
g^{\mu\nu}}=-\dfrac{1}{4\mu_{0}}\dfrac{\delta}{\delta
g^{\mu\nu}}\left(F_{\alpha\beta}F^{\alpha\beta}\right)=-\dfrac{1}{4\mu_{0}}\left(\dfrac{\delta{F}_{\alpha\beta}}{\delta
g^{\mu\nu}}F^{\alpha\beta}+F_{\alpha\beta}\dfrac{\delta{F}^{\alpha\beta}}{\delta
g^{\mu\nu}}\right).$ (9.96)

Let us notice that the tensor $F_{\alpha\beta}$ can be presented in more
complex form

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!F_{\alpha\beta}$
$\displaystyle=$ $\displaystyle
g^{\gamma}_{\alpha}{g}^{\delta}_{\beta}{F}_{\gamma\delta}=\left(g^{\gamma}_{\mu}{g}^{\mu\nu}g_{\nu\alpha}\right)\left(g^{\delta}_{\nu}{g}^{\nu\mu}g_{\mu\beta}\right){F}_{\gamma\delta}=$
(9.97) $\displaystyle=$
$\displaystyle\left(g^{\gamma}_{\mu}{g}^{\mu\nu}g_{\nu\alpha}\right)\left(g^{\delta}_{\nu}{g}^{\mu\nu}g_{\mu\beta}\right){F}_{\gamma\delta}={g}^{\mu\nu}{g}^{\mu\nu}\left(g^{\gamma}_{\mu}{g}_{\nu\alpha}g^{\delta}_{\nu}{g}_{\mu\beta}\right){F}_{\gamma\delta}.$

In this manner one has

$\displaystyle\dfrac{\delta{F}_{\alpha\beta}}{\delta{g}^{\mu\nu}}$
$\displaystyle=$ $\displaystyle
2{g}^{\mu\nu}\left(g^{\gamma}_{\mu}{g}_{\nu\alpha}g^{\delta}_{\nu}{g}_{\mu\beta}\right){F}_{\gamma\delta}=2g^{\gamma}_{\mu}{g}^{\delta}_{\nu}\left({g}^{\mu\nu}g_{\nu\alpha}\right){g}_{\mu\beta}{F}_{\gamma\delta}=$
(9.98) $\displaystyle=$ $\displaystyle
2\left(g^{\gamma}_{\mu}{g}^{\delta}_{\nu}{g}^{\mu}_{\alpha}{g}_{\mu\beta}\right){F}_{\gamma\delta}=2\left(g^{\gamma}_{\mu}{g}^{\mu}_{\alpha}\right){g}^{\delta}_{\nu}{g}_{\mu\beta}{F}_{\gamma\delta}=$
$\displaystyle=$ $\displaystyle
2\left(g^{\gamma}_{\alpha}{g}^{\delta}_{\nu}{g}_{\mu\beta}\right){F}_{\gamma\delta}=2\left({g}^{\delta}_{\nu}{g}_{\mu\beta}\right)g^{\gamma}_{\alpha}{F}_{\gamma\delta}=2\left({g}^{\delta}_{\nu}{g}_{\mu\beta}\right){F}_{\alpha\delta}=$
$\displaystyle=$
$\displaystyle-2{g}_{\mu\beta}\left({g}^{\delta}_{\nu}{F}_{\delta\alpha}\right)=-2{g}_{\mu\beta}F_{\nu\alpha},$

and consequently

$\dfrac{\delta{F}_{\alpha\beta}}{\delta
g^{\mu\nu}}F^{\alpha\beta}=-2{g}_{\mu\beta}F_{\nu\alpha}F^{\alpha\beta}=2F_{\nu\alpha}\left({g}_{\mu\beta}F^{\beta\alpha}\right)=2F_{\nu\alpha}F_{\mu}^{\alpha}=2F_{\mu}^{\alpha}{F}_{\alpha\nu}.$
(9.99)

Similarly, one can rewrite ${F}^{\alpha\beta}$ in the form

$F^{\alpha\beta}=g^{\alpha\kappa}g^{\beta\lambda}F_{\kappa\lambda},$ (9.100)

and consequently one has

$\displaystyle\dfrac{\delta{F}^{\alpha\beta}}{\delta{g}^{\mu\nu}}$
$\displaystyle=$
$\displaystyle\dfrac{\delta{g}^{\alpha\kappa}}{\delta{g}^{\mu\nu}}g^{\beta\lambda}F_{\kappa\lambda}+{g}^{\alpha\kappa}\dfrac{\delta{g}^{\beta\lambda}}{\delta{g}^{\mu\nu}}F_{\kappa\lambda}+{g}^{\alpha\kappa}{g}^{\beta\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
(9.101) $\displaystyle=$ $\displaystyle
g^{\alpha}_{\mu}{g}^{\kappa}_{\nu}{g}^{\beta\lambda}F_{\kappa\lambda}+{g}^{\alpha\kappa}g^{\beta}_{\mu}{g}^{\lambda}_{\nu}{F}_{\kappa\lambda}+{g}^{\alpha\kappa}{g}^{\beta\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$
$\displaystyle-g^{\alpha}_{\mu}{g}^{\kappa}_{\nu}{g}^{\beta\lambda}{F}_{\lambda\kappa}-g^{\alpha\kappa}g^{\beta}_{\mu}{g}^{\lambda}_{\nu}{F}_{\lambda\kappa}+{g}^{\alpha\kappa}{g}^{\beta\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$
$\displaystyle-g^{\alpha}_{\mu}{g}^{\kappa}_{\nu}{F}^{\beta}_{\kappa}-g^{\alpha\kappa}g^{\beta}_{\mu}{F}_{\nu\kappa}+{g}^{\alpha\kappa}{g}^{\beta\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$
$\displaystyle-g^{\alpha}_{\mu}{F}^{\beta}_{\nu}-g^{\beta}_{\mu}{g}^{\alpha\kappa}{F}_{\nu\kappa}+{g}^{\alpha\kappa}{g}^{\beta\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$ $\displaystyle
g^{\alpha}_{\mu}{F}^{\beta}_{\nu}+g^{\beta}_{\mu}{g}^{\alpha\kappa}{F}_{\kappa\nu}+{g}^{\alpha\kappa}{g}^{\beta\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$ $\displaystyle
g^{\alpha}_{\mu}{F}^{\beta}_{\nu}+g^{\beta}_{\mu}{F}^{\alpha}_{\nu}+{g}^{\alpha\kappa}{g}^{\beta\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}.$

By this reason one obtains

$\displaystyle
F_{\alpha\beta}\dfrac{\delta{F}^{\alpha\beta}}{\delta{g}^{\mu\nu}}$
$\displaystyle=$ $\displaystyle
F_{\alpha\beta}\left[g^{\alpha}_{\mu}{F}^{\beta}_{\nu}+g^{\beta}_{\mu}{F}^{\alpha}_{\nu}+{g}^{\alpha\kappa}{g}^{\beta\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}\right]=$
(9.102) $\displaystyle=$
$\displaystyle\left(F_{\alpha\beta}g^{\alpha}_{\mu}\right){F}^{\beta}_{\nu}+\left(F_{\alpha\beta}g^{\beta}_{\mu}\right){F}^{\alpha}_{\nu}+\left(F_{\alpha\beta}{g}^{\alpha\kappa}{g}^{\beta\lambda}\right)\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$
$\displaystyle\left(-F_{\beta\alpha}g^{\alpha}_{\mu}\right){F}^{\beta}_{\nu}-F_{\alpha\mu}{F}^{\alpha}_{\nu}-\left(F_{\beta\alpha}{g}^{\alpha\kappa}{g}^{\beta\lambda}\right)\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$ $\displaystyle-
F_{\beta\mu}{F}^{\beta}_{\nu}+{F}^{\alpha}_{\nu}{F}_{\alpha\mu}-\left(F_{\beta}^{\kappa}{g}^{\beta\lambda}\right)\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$
$\displaystyle{F}^{\beta}_{\nu}{F}_{\beta\mu}+{F}^{\alpha}_{\nu}{F}_{\alpha\mu}-F^{\kappa\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$ $\displaystyle
2{F}^{\alpha}_{\nu}{F}_{\alpha\mu}-2F_{\mu}^{\alpha}{F}_{\alpha\nu}=0,$

where we have applied the identity

${F}^{\alpha}_{\nu}{F}_{\alpha\mu}=g^{\alpha\beta}{F}_{\beta\nu}{F}_{\alpha\mu}=g^{\beta\alpha}{F}_{\alpha\mu}{F}_{\beta\nu}=F_{\mu}^{\beta}{F}_{\beta\nu}=F_{\mu}^{\alpha}{F}_{\alpha\nu}.$
(9.103)

Taking into account (9.99) and (9.102) one receives

$\dfrac{\delta{F}_{\alpha\beta}}{\delta{g}^{\mu\nu}}F^{\alpha\beta}+F_{\alpha\beta}\dfrac{\delta{F}^{\alpha\beta}}{\delta{g}^{\mu\nu}}=-2{F}^{\alpha}_{\mu}{F}_{\alpha\nu}.$
(9.104)

By this reason one obtains

$\dfrac{\delta\mathcal{L}}{\delta
g^{\mu\nu}}=-\dfrac{1}{4\mu_{0}}\left(-2{F}^{\alpha}_{\mu}{F}_{\alpha\nu}\right)=\dfrac{1}{2\mu_{0}}{F}^{\alpha}_{\mu}{F}_{\alpha\nu}=-\dfrac{1}{2\mu_{0}}{F}^{\alpha}_{\mu}{F}_{\alpha\nu},$
(9.105)

where in the last step we have changed order of indexes in
${F}^{\alpha}_{\mu}$, which is invisible by the only inconvenient notation. In
this manner one receives finally

$T_{\mu\nu}^{em}=-2\dfrac{\delta\mathcal{L}}{\delta
g^{\mu\nu}}+g_{\mu\nu}\mathcal{L}=\dfrac{1}{\mu_{0}}\left({F}^{\alpha}_{\mu}{F}_{\alpha\nu}-\dfrac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right),$
(9.106)

what coincides with the stress-energy tensor of electromagnetic field (9.82)
given _ad hoc_ in analogy with the case of flat space-time.

For the case of the Kerr–Newman metric in general the spatial Ricci scalar
curvature and energy density of Matter fields are manifestly non zero. Let us
calculate them straightforwardly. The spatial Ricci curvature tensor can be
taken from (9.79) as $R_{ij}=\kappa{T}^{em}_{ij}$, where

$T_{ij}^{em}=\dfrac{1}{\mu_{0}}\left(F^{\alpha}_{i}{F}_{\alpha{j}}-\dfrac{1}{4}h_{ij}F_{\alpha\beta}F^{\alpha\beta}\right),$
(9.107)

is the spatial part of the stress-energy tensor for electromagnetic field.
When the space-time metric is the Minkowski metric $\eta_{\mu\nu}$ then the
spatial part of the stress-energy tensor of electromagnetic field is

$\left.T_{ij}^{em}\right|_{g_{\mu\nu}=\eta_{\mu\nu}}=-\sigma^{f\/lat}_{ij},$
(9.108)

where $\sigma_{ij}^{f\/lat}$ is the Maxwell stress tensor of electromagnetic
field

$\sigma_{ij}^{f\/lat}=\epsilon_{0}E_{i}E_{j}+\dfrac{1}{\mu_{0}}B_{i}B_{j}-\dfrac{1}{2}\left(\epsilon_{0}\vec{E}^{2}+\dfrac{1}{\mu_{0}}\vec{B}^{2}\right)\delta_{ij}.$
(9.109)

In this manner the natural generalization of the Maxwell stress tensor of
electromagnetic field to the case of non-flat space-time is the tensor

$\sigma_{ij}=-\dfrac{1}{\mu_{0}}\left(F^{\alpha}_{i}{F}_{\alpha{j}}-\dfrac{1}{4}h_{ij}F_{\alpha\beta}F^{\alpha\beta}\right),$
(9.110)

which we shall call the the curved-space Maxwell stress tensor of
electromagnetic field. Therefore the three-dimensional Ricci scalar curvature
${{}^{(3)}}R=h^{ij}R_{ij}$ up to the minus sign becomes the curved-space
Maxwell stress tensor of electromagnetic field projected onto the induced
metric

$\displaystyle{{}^{(3)}}R$ $\displaystyle=$
$\displaystyle-\kappa\ell_{P}^{2}{h}^{ij}\sigma_{ij}=\kappa\ell_{P}^{2}{h}^{ij}\dfrac{1}{\mu_{0}}\left(F^{\alpha}_{i}{F}_{\alpha{j}}-\dfrac{1}{4}h_{ij}F_{\alpha\beta}F^{\alpha\beta}\right)=$
(9.111) $\displaystyle=$
$\displaystyle\dfrac{\kappa\ell_{P}^{2}}{\mu_{0}}\left({h}^{ij}F^{\alpha}_{i}{F}_{\alpha{j}}-\dfrac{3}{4}F_{\alpha\beta}F^{\alpha\beta}\right).$

where we have used the identity $h^{ij}h_{ij}=3$. Using of the transformation

${h}^{ij}F^{\alpha}_{i}{F}_{\alpha{j}}={h}^{ij}h_{ik}F^{k\alpha}{F}_{\alpha{j}}=F^{j\alpha}{F}_{\alpha{j}}=F^{\alpha{j}}{F}_{{j}\alpha},$
(9.112)

and the definition

$F_{\alpha\beta}F^{\alpha\beta}=F^{\alpha{j}}{F}_{\alpha{j}}+F^{\alpha{0}}{F}_{\alpha{0}}=-F^{\alpha{j}}{F}_{{j}\alpha}-F^{\alpha{0}}{F}_{{0}\alpha},$
(9.113)

together with the properties of the electromagnetic field tensor

$\displaystyle{F}_{\alpha\beta}F^{\alpha\beta}$ $\displaystyle=$
$\displaystyle 2\left(\vec{B}^{2}-\dfrac{\vec{E}^{2}}{c^{2}}\right),$ (9.114)
$\displaystyle{F}_{\alpha 0}F^{0\alpha}$ $\displaystyle=$
$\displaystyle\dfrac{\vec{E}^{2}}{c^{2}},$ (9.115)

one obtains

${h}^{ij}F^{\alpha}_{i}{F}_{\alpha{j}}=-F_{\alpha\beta}F^{\alpha\beta}-F^{\alpha{0}}{F}_{{0}\alpha}=-2\left(\vec{B}^{2}-\dfrac{\vec{E}^{2}}{c^{2}}\right)-\dfrac{\vec{E}^{2}}{c^{2}}=2\vec{B}^{2}+\dfrac{\vec{E}^{2}}{c^{2}}.$
(9.116)

Collecting all together one receives finally

${{}^{(3)}}R=\dfrac{\kappa\ell_{P}^{2}}{\mu_{0}}\left[2\vec{B}^{2}+\dfrac{\vec{E}^{2}}{c^{2}}-\dfrac{3}{2}\left(\vec{B}^{2}-\dfrac{\vec{E}^{2}}{c^{2}}\right)\right]=\dfrac{\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\vec{B}^{2}+5\epsilon_{0}\vec{E}^{2}\right).$
(9.117)

Similarly one can calculate the energy density of electromagnetic field.
Applying the definition $n^{\mu}={n}^{\beta}{g}^{\mu}_{\beta}$, and the
identity $g_{\mu\nu}n^{\mu}{n}^{\nu}=n^{\mu}{n}_{\mu}=-1$ one obtains

$\displaystyle\varrho$ $\displaystyle=$ $\displaystyle
T_{\mu\nu}n^{\mu}{n}^{\nu}=\dfrac{1}{\mu_{0}}\left(F_{\mu\alpha}g^{\alpha\beta}F_{\nu\beta}-\dfrac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right)n^{\mu}{n}^{\nu}=$
(9.118) $\displaystyle=$
$\displaystyle\dfrac{1}{\mu_{0}}\left({F}_{\mu\alpha}({g}^{\alpha\beta}n^{\mu}{n}^{\nu})F_{\nu\beta}-\dfrac{1}{4}g_{\mu\nu}n^{\mu}{n}^{\nu}{F}_{\alpha\beta}F^{\alpha\beta}\right).$

The second term in the formula (9.118) can be easy transformed with using of
the identity $g_{\mu\nu}n_{\mu}{n}_{\nu}=n^{\mu}{n}_{\mu}=-1$. The first term,
however, is not so easy to transform. Let us notice that

${g}^{\alpha\beta}n^{\mu}{n}^{\nu}=g^{\alpha\mu}g^{\nu\beta}(g_{\mu\nu}n^{\mu}{n}^{\nu})=-g^{\alpha\mu}g^{\nu\beta},$
(9.119)

where we have applied $g_{\mu\nu}n^{\mu}{n}^{\nu}=n_{\nu}n^{\nu}=-1$. In this
manner

$\displaystyle{F}_{\mu\alpha}({g}^{\alpha\beta}n^{\mu}{n}^{\nu})F_{\nu\beta}$
$\displaystyle=$
$\displaystyle-{F}_{\mu\alpha}(g^{\alpha\mu}g^{\nu\beta})F_{\nu\beta}={F}_{\mu\alpha}(g^{\alpha\mu}g^{\nu\beta})F_{\beta\nu}=$
(9.120) $\displaystyle=$
$\displaystyle{F}^{\mu}_{\mu}{F}^{\nu}_{\nu}=F_{\mu\nu}(g^{\nu\mu}g_{\nu\mu})F^{\mu\nu}=$
$\displaystyle=$ $\displaystyle
F_{\mu\nu}(g^{\nu\mu}g_{\mu\nu})F^{\mu\nu}=4F_{\mu\nu}F^{\mu\nu},$

where we have applied the identities $F^{\mu}_{\mu}=F_{\mu\nu}g^{\nu\mu}$,
$F^{\nu}_{\nu}=g_{\nu\mu}F^{\mu\nu}$, and $g^{\nu\mu}g_{\mu\nu}=4$. In this
manner one receives finally the energy density

$\displaystyle\varrho$ $\displaystyle=$
$\displaystyle\dfrac{1}{\mu_{0}}\left(4{F}_{\alpha\beta}{F}^{\alpha\beta}+\dfrac{1}{4}{F}_{\alpha\beta}F^{\alpha\beta}\right)=\dfrac{17}{4\mu_{0}}{F}_{\alpha\beta}F^{\alpha\beta}=$
(9.121) $\displaystyle=$
$\displaystyle\dfrac{17}{2\mu_{0}}\left(\vec{B}^{2}-\dfrac{\vec{E}^{2}}{c^{2}}\right)=\dfrac{17}{2}\left(\dfrac{1}{\mu_{0}}\vec{B}^{2}-\epsilon_{0}\vec{E}^{2}\right),$

where we have applied the property (9.114) of electromagnetic field tensor.
Consequently the generalized gravitational potential is

$\displaystyle
V=-{{}^{(3)}}R+2\kappa\ell_{P}^{2}\varrho=-\dfrac{\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\vec{B}^{2}+5\epsilon_{0}\vec{E}^{2}\right)+17\kappa\ell_{P}^{2}\left(\dfrac{1}{\mu_{0}}\vec{B}^{2}-\epsilon_{0}\vec{E}^{2}\right)=$
$\displaystyle=\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\vec{B}^{2}-\dfrac{13}{11}\epsilon_{0}\vec{E}^{2}\right).$
(9.122)

Now it is easy to see that $V$ averaged on midisuperspace is

$\langle{V}\rangle=\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right),$
(9.123)

where $\langle\vec{B}^{2}\rangle$ and $\langle\vec{E}^{2}\rangle$ are the
midisuperspace means of the squared fields $\vec{B}$ and $\vec{E}$,
respectively. One has explicitly

$\displaystyle\langle\vec{B}^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Omega(\xi)}\int\vec{B}^{2}\delta\xi,$ (9.124)
$\displaystyle\langle\vec{E}^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Omega(\xi)}\int\vec{E}^{2}\delta\xi.$ (9.125)

Because of the fields $\vec{B}$ and $\vec{E}$ are in general functions on
space-time, it is evident that both the averages (9.124) and (9.125) must be
treated as functions on space-time, of course after performing in all the
functional integrals the suitable transformation from midisuperspace to space-
time $\xi\rightarrow\xi(x)$. In other words one has

$\langle{V}\rangle(\xi,\phi)=\langle{V}\rangle(x).$ (9.126)

Because of the considerations presented above are independent on the concrete
form of the electrovacuum solution, they can be applied to any solution of the
Einstein–Maxwell equations. Interestingly, one can _ad hoc_ add the
cosmological constant

$\langle{V}\rangle=\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)+2\Lambda,$
(9.127)

so that the averaged generalized gravitational potential can be ordered by two
equivalent ways

$\displaystyle\langle{V}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\epsilon_{0}\langle\vec{E}^{2}\rangle\right)+2\Lambda-3\kappa\ell_{P}^{2}\epsilon_{0}\langle\vec{E}^{2}\rangle=$
(9.128) $\displaystyle=$
$\displaystyle\dfrac{39\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\epsilon_{0}\langle\vec{E}^{2}\rangle\right)+2\Lambda-\dfrac{3\kappa\ell_{P}^{2}}{\mu_{0}}\langle\vec{B}^{2}\rangle.$
(9.129)

If the averages $\langle\vec{B}^{2}\rangle=B_{\Lambda}^{2}$ and
$\langle\vec{E}^{2}\rangle=E_{\Lambda}^{2}$ are constant then one can take

$2\Lambda=3\kappa\ell_{P}^{2}\epsilon_{0}E^{2}_{\Lambda}=\dfrac{3\kappa\ell_{P}^{2}}{\mu_{0}}B^{2}_{\Lambda},$
(9.130)

and one the averaged generalized gravitational potential vanishes

$\langle{V}\rangle=0.$ (9.131)

The condition (9.130, however, leads to the relation

$E_{\Lambda}=cB_{\Lambda},$ (9.132)

which can be used to define the speed of light

$c:=\dfrac{E_{\Lambda}}{B_{\Lambda}},$ (9.133)

if one knows $E_{\Lambda}$ and $B_{\Lambda}$.

Let us return to the Kerr–Newman space-time. The spatial part of the
Kerr–Newman metric has the form

$h_{ij}=\left[\begin{array}[]{ccc}\dfrac{r^{2}+\alpha^{2}\cos^{2}\theta}{\Delta}&0&0\\\
0&r^{2}+\alpha^{2}\cos^{2}\theta&0\\\
0&0&\dfrac{(r^{2}+\alpha^{2})^{2}-\alpha^{2}\Delta\sin^{2}\theta}{r^{2}+\alpha^{2}\cos^{2}\theta}\sin^{2}\theta\end{array}\right],$
(9.134)

where $\Delta=r^{2}\alpha^{2}-r_{S}r+r_{Q}^{2}$, and by this reason the global
dimension can be deduced easy

$h=\left(r^{2}+\alpha^{2}\cos^{2}\theta\right)\left[r^{2}+\alpha^{2}\cos^{2}\theta+\dfrac{(r^{2}+\alpha^{2})(r_{S}r-r_{Q}^{2})}{r^{2}+\alpha^{2}-r_{S}r+r_{Q}^{2}}\right]\sin^{2}\theta,$
(9.135)

so that the invariant global dimension is

$\xi=\dfrac{1}{4\pi\sqrt{6}}\sqrt{\left(r^{2}+\alpha^{2}\cos^{2}\theta\right)\left[r^{2}+\alpha^{2}\cos^{2}\theta+\dfrac{(r^{2}+\alpha^{2})(r_{S}r-r_{Q}^{2})}{r^{2}+\alpha^{2}-r_{S}r+r_{Q}^{2}}\right]}\sin\theta.$
(9.136)

The volume of the Kerr-Newman midisuperspace is $\Omega(\xi)=\xi$. Here
$r_{S}$ is the Schwarzschild radius and $r_{Q}$ is a length-scale
corresponding to the electric charge Q of the mass

$r_{Q}^{2}=\dfrac{Q^{2}G}{4\pi\epsilon_{0}c^{4}}=\dfrac{Q^{2}}{4\pi\epsilon_{0}}\dfrac{\kappa}{8\pi}.$
(9.137)

There are three possible situations. Namely, when the averaged generalized
gravitational potential (9.123) is $1^{\circ}$ vanishing, $2^{\circ}$
positive, $3^{\circ}$ negative. The first case is rather trivial.
$\langle{V}\rangle$ vanishes if and only if

$\langle\vec{B}^{2}\rangle=\dfrac{13}{11}\dfrac{\langle\vec{E}^{2}\rangle}{c^{2}},$
(9.138)

what in fact means that

$|\vec{B}|=\sqrt{\dfrac{13}{11}}\dfrac{|\vec{E}|}{c}\approx
1.087\dfrac{|\vec{E}|}{c}.$ (9.139)

In such a situation one can construct straightforwardly and easy the wave
functions of the Kerr–Newman space-time

$\Psi_{1}^{KN}=\dfrac{\sin\theta}{4\pi\sqrt{|A|}\sqrt{6}}\sqrt{\left(r^{2}+\alpha^{2}\cos^{2}\theta\right)\left[r^{2}+\alpha^{2}\cos^{2}\theta+\dfrac{(r^{2}+\alpha^{2})(r_{S}r-r_{Q}^{2})}{r^{2}+\alpha^{2}-r_{s}r+r_{Q}^{2}}\right]},$
(9.140) $\Psi_{2}^{KN}=\dfrac{1}{\sqrt{|B|}},$ (9.141)

where the constants of integration $A$ and $B$ are

$\displaystyle A$ $\displaystyle=$ $\displaystyle\dfrac{\Xi^{3}}{3},$ (9.142)
$\displaystyle B$ $\displaystyle=$ $\displaystyle\dfrac{1}{\Gamma_{0}}\Xi,$
(9.143)

where $\Xi$ is the reference constant

$\Xi=\dfrac{1}{4\pi\sqrt{6}}\sqrt{\left(R^{2}+\alpha^{2}\cos^{2}\Theta\right)\left[R^{2}+\alpha^{2}\cos^{2}\Theta+\dfrac{(R^{2}+\alpha^{2})(r_{S}R-r_{Q}^{2})}{R^{2}+\alpha^{2}-r_{s}R+r_{Q}^{2}}\right]}\sin\Theta.$
(9.144)

The second situation, i.e. positive $\langle{V}\rangle$, is defined for

$\langle\vec{B}^{2}\rangle>\dfrac{13}{11}\dfrac{\langle\vec{E}^{2}\rangle}{c^{2}},$
(9.145)

what can be presented in the form

$|\vec{B}|>\sqrt{\dfrac{13}{11}}\dfrac{|\vec{E}|}{c}.$ (9.146)

In such a situation the constants of integration $A$ and $B$ are not easy to
calculate for general fields $\vec{B}$ and $\vec{E}$, but can be presented in
the compact form

$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\left[\dfrac{\sin\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}\right)}{\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}}\right]^{2}\delta\xi,$
(9.147) $\displaystyle B$ $\displaystyle=$
$\displaystyle\int\cos^{2}\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}\right)\delta\xi,$
(9.148)

and the wave functions of the Kerr–Newman space-time are

$\displaystyle\Psi_{1}^{KN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{\sin\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}\right)}{\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}},$
(9.149) $\displaystyle\Psi_{2}^{KN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cos\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}\right).$
(9.150)

The case of negative averaged generalized gravitational potential can be
considered analogously. In this case

$|\vec{B}|<\sqrt{\dfrac{13}{11}}\dfrac{|\vec{E}|}{c}.$ (9.151)

So that the wave functions of the Kerr–Newman space-time are

$\displaystyle\Psi_{1}^{KN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{\sinh\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}\right)}{\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}},$
(9.152) $\displaystyle\Psi_{2}^{KN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cosh\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}\right).$
(9.153)

where the integration constants $A$ and $B$ are

$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\left[\dfrac{\sinh\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}\right)}{\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}}\right]^{2}\delta\xi,$
(9.154) $\displaystyle B$ $\displaystyle=$
$\displaystyle\int\cosh^{2}\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}\right)\delta\xi,$
(9.155)

####  H The Reissner–Nordström Space-time

Another solution of the electrovacuum Einstein field equations is the
Reissner–Nordström metric describing spherically symmetric static massive
charged object. Therefore in this case one has also

$\langle{V}\rangle=\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right),$
(9.156)

and the only change is the metric. The spatial part of Reissner–Nordström
space-time metric is

$h_{ij}=\left[\begin{array}[]{ccc}\left(1-\dfrac{r_{S}}{r}+\dfrac{r_{Q}^{2}}{r^{2}}\right)^{-1}&0&0\\\
0&r^{2}&0\\\ 0&0&r^{2}\sin^{2}\theta\end{array}\right],$ (9.157)

so that the global dimension is

$h=\dfrac{r^{4}\sin^{2}\theta}{1-\dfrac{r_{S}}{r}+\dfrac{r_{Q}^{2}}{r^{2}}}$
(9.158)

while the invariant global dimension has the form

$\xi=\dfrac{1}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{r_{S}}{r}+\dfrac{r_{Q}^{2}}{r^{2}}}}$
(9.159)

and of the volume of the Reissner–Nordström midisuperspace is
$\Omega(\xi)=\xi$. There are three possible situations defined by absolute
values of the electric and magnetic fields. For the case of negative
$\langle{V}\rangle$ the wave functions of the Reissner–Nordström space-time
are

$\displaystyle\Psi_{1}^{RN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{\sinh\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}\right)}{\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}},$
(9.160) $\displaystyle\Psi_{2}^{RN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cosh\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}\right).$
(9.161)

where the integration constants $A$ and $B$ are

$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\left[\dfrac{\sinh\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}\right)}{\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}}\right]^{2}\delta\xi,$
(9.162) $\displaystyle B$ $\displaystyle=$
$\displaystyle\int\cosh^{2}\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}\right)\delta\xi.$
(9.163)

When $\langle{V}\rangle$ is positive then

$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\left[\dfrac{\sin\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}\right)}{\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}}\right]^{2}\delta\xi,$
(9.164) $\displaystyle B$ $\displaystyle=$
$\displaystyle\int\cos^{2}\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}\right)\delta\xi,$
(9.165)

and the wave functions of the Reissner–Nordström space-time are

$\displaystyle\Psi_{1}^{RN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{\sin\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}\right)}{\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}},$
(9.166) $\displaystyle\Psi_{2}^{RN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cos\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}\right).$
(9.167)

For vanishing $\langle{V}\rangle$ one has

$\displaystyle\Psi_{1}^{RN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4\pi\sqrt{|A|}\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{r_{S}}{r}+\dfrac{r_{Q}^{2}}{r^{2}}}},$
(9.168) $\displaystyle\Psi_{2}^{RN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}},$ (9.169)

where the constants of integration $A$ and $B$ are

$\displaystyle A$ $\displaystyle=$ $\displaystyle\dfrac{\Xi^{3}}{3},$ (9.170)
$\displaystyle B$ $\displaystyle=$ $\displaystyle\dfrac{1}{\Gamma_{0}}\Xi,$
(9.171)

where $\Xi$ is the reference constant

$\Xi=\dfrac{1}{4\pi\sqrt{6}}\dfrac{R^{2}\sin\Theta}{\sqrt{1-\dfrac{r_{S}}{R}+\dfrac{r_{Q}^{2}}{R^{2}}}}.$
(9.172)

####  I The Gödel Space-time

Let us consider the solution of the Einstein field equations with presence of
the cosmological constant and the stress-energy tensor of dust

$\displaystyle
R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}{{}^{(4)}}R+\Lambda{g}_{\mu\nu}=\kappa\ell_{P}^{2}{T}_{\mu\nu},$
(9.173) $\displaystyle T_{\mu\nu}=\varepsilon{u}_{\mu}{u}_{\nu},$ (9.174)

which is given by the Gödel space-time

$\displaystyle g_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}-a^{2}&0&0&-a^{2}e^{x}\\\
0&a^{2}&0&0\\\ 0&0&a^{2}&0\\\
-a^{2}e^{x}&0&0&-\dfrac{1}{2}a^{2}e^{2x}\end{array}\right],$ (9.179)
$\displaystyle g^{\mu\nu}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}\dfrac{1}{a^{2}}&0&0&-\dfrac{2e^{-x}}{a^{2}}\\\
0&\dfrac{1}{a^{2}}&0&0\\\ 0&0&\dfrac{1}{a^{2}}&0\\\
-\dfrac{2e^{-x}}{a^{2}}&0&0&\dfrac{2e^{-2x}}{a^{2}}\end{array}\right].$
(9.184)

having the following spatial part

$h_{ij}=\left[\begin{array}[]{ccc}a^{2}&0&0\\\ 0&a^{2}&0\\\
0&0&-\dfrac{1}{2}a^{2}e^{2x}\end{array}\right]\quad,\quad
h^{ij}=\left[\begin{array}[]{ccc}\dfrac{1}{a^{2}}&0&0\\\
0&\dfrac{1}{a^{2}}&0\\\ 0&0&\dfrac{2e^{-2x}}{a^{2}}\end{array}\right],$
(9.185)

where $a=a(x,y,z)$. In this case the global dimension is

$|h|=\dfrac{a^{6}}{2}e^{2x},$ (9.186)

so the invariant global dimension has the form

$\xi=\dfrac{1}{8\pi\sqrt{3}}a^{3}e^{x},$ (9.187)

and therefore the volume of the Gödel midisuperspace is $\Omega(\xi)=\xi$. The
Gödel metric satisfies the Einstein field equations for which the cosmological
constant is related to the parameter $a$ by the relation

$\Lambda=-\dfrac{1}{2a^{2}}.$ (9.188)

The energy density $\epsilon$ of the dust is

$\epsilon=\dfrac{1}{\kappa\ell_{P}^{2}a^{2}}.$ (9.189)

The Gödel metric can be decomposed in the ADM $3+1$ form. It is easy to see
that such an _ad hoc_ decomposition generates the equations

$\displaystyle-N^{2}+N_{i}N^{i}$ $\displaystyle=$ $\displaystyle-a^{2},$
(9.190) $\displaystyle N_{i}$ $\displaystyle=$
$\displaystyle\left[0,0,-a^{2}e^{x}\right].$ (9.191)

Taking into account the spatial metric (9.185) one receives

$N^{i}=h^{ij}N_{j}=\left[0,0,-2e^{-x}\right],$ (9.192)

what gives $N_{i}N^{i}=2a^{2}$ and by this reason

$N=\sqrt{3}a.$ (9.193)

In this manner one can establish the normal unit vector field

$\displaystyle n^{\mu}$ $\displaystyle=$
$\displaystyle\left[\dfrac{1}{N},-\dfrac{N^{i}}{N}\right]=\dfrac{1}{\sqrt{3}}\left[\dfrac{1}{a},0,0,\dfrac{2e^{-x}}{a}\right],$
(9.194) $\displaystyle n_{\mu}$ $\displaystyle=$
$\displaystyle\left[-N,0_{i}\right]=\sqrt{3}\left[-a,0,0,0\right].$ (9.195)

The problem is to choose the velocity vector $u_{\mu}$ according to the
general rules

$\displaystyle u_{\mu}{u}^{\mu}$ $\displaystyle=$ $\displaystyle-1,$ (9.196)
$\displaystyle u_{\mu}{n}^{\mu}$ $\displaystyle=$ $\displaystyle-1,$ (9.197)
$\displaystyle u_{\mu}{n}^{\mu}{u}_{\nu}{n}^{\nu}$ $\displaystyle=$
$\displaystyle 1.$ (9.198)

However, it can be seen by straightforward calculation that the following
choice

$\displaystyle u^{\mu}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{3}}\left[\dfrac{1}{a},0,0,0\right],$ (9.199)
$\displaystyle u_{\mu}$ $\displaystyle=$ $\displaystyle
g_{\mu\nu}u^{\nu}=\sqrt{3}\left[-a,0,0,-ae^{x}\right],$ (9.200)

satisfies the conditions (9.196)-(9.198). The contraction of the Einstein
field equations (9.173)-(9.174) with metric $g^{\mu\nu}$ leads to

${{}^{(4)}}R=4\Lambda+\kappa\ell_{P}^{2}\epsilon,$ (9.201)

so that the equations (9.173)-(9.174) can be presented in the form

$R_{\mu\nu}=\Lambda{g}_{\mu\nu}+\kappa\ell_{P}^{2}\epsilon\left({u}_{\mu}{u}_{\nu}+\dfrac{1}{2}g_{\mu\nu}\right),$
(9.202)

what after including the fact that for the Gödel Universe one has

$\kappa\ell_{P}^{2}\epsilon=-2\Lambda,$ (9.203)

one receives the four-dimensional Ricci curvature tensor

$R_{\mu\nu}=\kappa\ell_{P}^{2}\epsilon{u}_{\mu}{u}_{\nu}=-2\Lambda{u}_{\mu}{u}_{\nu}.$
(9.204)

In this manner the three-dimensional Ricci curvature tensor has the form

$R_{ij}=-2\Lambda{u}_{i}{u}_{j},$ (9.205)

and consequently the three-dimensional Ricci scalar curvature is

${{}^{(3)}}R=h^{ij}R_{ij}=-2\Lambda{u}_{i}{u}^{i}=0.$ (9.206)

Similarly one can establish the energy density of Matter fields

$\varrho=T_{\mu\nu}n^{\mu}{n}^{\nu}=\epsilon{u}_{\mu}{n}^{\mu}{u}_{\nu}{n}^{\nu}=\epsilon=-2\dfrac{\Lambda}{\kappa\ell_{P}^{2}}.$
(9.207)

In this manner the generalized gravitational potential for the Gödel Universe
has the following form

$V=-{{}^{(3)}}R+2\Lambda+2\kappa\ell_{P}^{2}\varrho=-2\Lambda=\kappa\ell_{P}^{2}\epsilon=\dfrac{1}{a^{2}}>0,$
(9.208)

so that

$\displaystyle\langle{V}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{1}{a^{3}e^{x}}\int\dfrac{d({a}^{3}e^{x})}{{a}^{2}}=\dfrac{1}{a^{3}e^{x}}\int\dfrac{3a^{2}dae^{x}+a^{3}e^{x}dx}{{a}^{2}}=$
(9.209) $\displaystyle=$
$\displaystyle\dfrac{1}{a^{3}e^{x}}\int\left(3e^{x}da+ae^{x}dx\right),$
(9.210)

and by this reason

$\sqrt{\langle{V}\rangle}\left(\xi,\phi\right)=\dfrac{1}{a^{3/2}e^{x/2}}\sqrt{\int\left(3e^{x}da+ae^{x}dx\right)}.$
(9.211)

In this manner the wave functions of the Gödel dust Universe are

$\displaystyle\Psi_{1}^{G}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{\sin\left(\dfrac{a^{3/2}e^{x/2}}{8\pi\sqrt{3}}\sqrt{\int\left(3e^{x}da+ae^{x}dx\right)}\right)}{\dfrac{1}{a^{3/2}e^{x/2}}\sqrt{\int\left(3e^{x}da+ae^{x}dx\right)}},$
(9.212) $\displaystyle\Psi_{2}^{G}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cos\left(\dfrac{a^{3/2}e^{x/2}}{8\pi\sqrt{3}}\sqrt{\int\left(3e^{x}da+ae^{x}dx\right)}\right),$
(9.213)

where $\mathrm{sgn}(a)\neq 0$, and the constants of integration $A$ and $B$
are

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!A$
$\displaystyle=$
$\displaystyle\dfrac{1}{2\sqrt{3}S_{P}}\int\left(\dfrac{\sin\left(\dfrac{a^{3/2}e^{x/2}}{8\pi\sqrt{3}}\sqrt{\int\left(3e^{x}da+ae^{x}dx\right)}\right)}{\dfrac{1}{a^{3/2}e^{x/2}}\sqrt{\int\left(3e^{x}da+ae^{x}dx\right)}}\right)^{2}d(a^{3}e^{x}),$
(9.214)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!B$
$\displaystyle=$
$\displaystyle\dfrac{1}{\Gamma_{0}}\dfrac{1}{2\sqrt{3}S_{P}}\int\cos^{2}\left(\dfrac{a^{3/2}e^{x/2}}{8\pi\sqrt{3}}\sqrt{\int\left(3e^{x}da+ae^{x}dx\right)}\right)d(a^{3}e^{x}).$
(9.215)

Of course when $a=constans$ the situation is much more simpler, but we will
not derive these particular results.

####  J The Einstein–Rosen Gravitational Waves

The example of vacuum solution are also the Einstein–Rosen cylindrical
gravitational waves. The spatial part of the metric has the form

$h_{ij}=\left[\begin{array}[]{ccc}e^{2\gamma-2\psi}&0&0\\\ 0&e^{2\psi}&0\\\
0&0&r^{2}e^{-2\psi}\end{array}\right],$ (9.216)

where $r$ is radial distance from the $z$ axis, and $\psi=\psi(t,r)$ and
$\gamma=\gamma(t,r)$ are functions satisfying the equations

$\displaystyle\ddot{\psi}$ $\displaystyle=$
$\displaystyle\dfrac{\psi^{\prime}}{r}+\psi^{\prime\prime},$ (9.217)
$\displaystyle\dfrac{\gamma^{\prime}}{r}$ $\displaystyle=$
$\displaystyle\psi^{\prime 2}+\dot{\psi}^{2},$ (9.218)
$\displaystyle\dfrac{\dot{\gamma}}{r}$ $\displaystyle=$ $\displaystyle
2\dot{\psi}\psi^{\prime}.$ (9.219)

In other words when one solves the equation (9.217) then the function $\gamma$
can be established by

$\gamma=\int\left[r\left(\psi^{\prime
2}+\dot{\psi}^{2}\right)dr+2r\dot{\psi}\psi^{\prime}dt\right].$ (9.220)

In this manner the global dimension for the Einstein–Rosen waves is

$h=r^{2}e^{2\gamma-2\psi},$ (9.221)

while the invariant global dimension has the form

$\xi=\dfrac{1}{\sqrt{6}S_{P}}re^{\gamma-\psi}.$ (9.222)

The generalized gravitational potential vanishes in this case, so that the
wave functions are

$\displaystyle\Psi_{1}^{ER}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{1}{4\pi\sqrt{6}}re^{\gamma-\psi},$
(9.223) $\displaystyle\Psi_{2}^{ER}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}},$ (9.224)

where the constants of integration $A$ and $B$ are determined by the relations

$\displaystyle A$ $\displaystyle=$ $\displaystyle\dfrac{\Xi^{3}}{3},$ (9.225)
$\displaystyle B$ $\displaystyle=$ $\displaystyle\dfrac{1}{\Gamma_{0}}\Xi,$
(9.226)

where $\Xi$ is the reference constant

$\Xi=\dfrac{1}{4\pi\sqrt{6}}Re^{\gamma(T,R)-\psi(T,R)},$ (9.227)

where $T$ and $R$ are reference values of $t$ and $R$, respectively.

####  K The Taub–Newman–Unti–Tamburino Space-time

Let us consider the generalized axisymmetric solution of the vacuum Einstein
field equations presented in the Weyl canonical coordinates

$h_{ij}=\left[\begin{array}[]{ccc}e^{2\gamma-2\psi}&0&0\\\
0&e^{2\gamma-2\psi}&0\\\
0&0&r^{2}e^{-2\psi}-A^{2}e^{2\gamma}\end{array}\right],$ (9.228)

for which the field equations are

$\displaystyle\bigtriangleup\psi$ $\displaystyle=$ $\displaystyle 0,$ (9.229)
$\displaystyle\dfrac{\gamma^{\prime}}{r}$ $\displaystyle=$
$\displaystyle\psi^{\prime 2}+\dot{\psi}^{2},$ (9.230)
$\displaystyle\dfrac{\gamma^{\prime}_{z}}{r}$ $\displaystyle=$ $\displaystyle
2\dot{\psi}\psi^{\prime}.$ (9.231)

Here $r$ is the radial distance from the axis of symmetry $z$. In such a
situation the global dimension is

$h=e^{4\gamma-4\psi}\left(r^{2}e^{-2\psi}-A^{2}e^{2\gamma}\right),$ (9.232)

so that the invariant global dimension is

$\xi=\dfrac{1}{4\pi\sqrt{6}}e^{2\gamma-2\psi}\sqrt{r^{2}e^{-2\psi}-A^{2}e^{2\gamma}}.$
(9.233)

For this case the cosmological constant as well as Matter fields are absent.
Therefore the three-dimensional Ricci scalar curvature, and the generalized
gravitational potential are trivial. The wave functions of any static
axisymmetric vacuum solution can be established as

$\displaystyle\Psi_{1}^{SAVS}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{1}{4\pi\sqrt{6}}e^{2\gamma-2\psi}\sqrt{r^{2}e^{-2\psi}-A^{2}e^{2\gamma}},$
(9.234) $\displaystyle\Psi_{2}^{SAVS}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}},$ (9.235)

where the constants of integration $A$ and $B$ are

$\displaystyle A$ $\displaystyle=$ $\displaystyle\dfrac{\Xi^{3}}{3},$ (9.236)
$\displaystyle B$ $\displaystyle=$ $\displaystyle\dfrac{1}{\Gamma_{0}}\Xi,$
(9.237)

where $\Xi$ is the reference constant

$\Xi=\dfrac{1}{4\pi\sqrt{6}}e^{2\gamma(R,Z)-2\psi(R,Z)}\sqrt{R^{2}e^{-2\psi(R,Z)}-A^{2}(R,Z)e^{2\gamma(R,Z)}},$
(9.238)

where $R$ and $Z$ are reference values of $r$ and $z$. Let us see the wave
functions of two particular cases.

The Taub–Newman–Unti–Tamburino space-time is the particular case the static
axisymmetric vacuum space-time characterized by

$\displaystyle e^{2\psi}$ $\displaystyle=$
$\displaystyle\dfrac{(r_{+}+r_{-})^{2}-(r_{S}^{2}+4l^{2})}{(r_{+}+r_{-}+r_{S})^{2}+4l^{2}},$
(9.239) $\displaystyle e^{2\gamma}$ $\displaystyle=$
$\displaystyle\dfrac{(r_{+}+r_{-})^{2}-(r_{S}^{2}+4l^{2})}{4r_{+}r_{-}},$
(9.240) $\displaystyle A$ $\displaystyle=$
$\displaystyle\dfrac{2l(r_{+}-r_{-})}{\sqrt{r_{S}^{2}+4l^{2}}},$ (9.241)
$\displaystyle r_{\pm}^{2}$ $\displaystyle=$ $\displaystyle
r^{2}+\left(z\pm\dfrac{1}{2}\sqrt{r_{S}^{2}+4l^{2}}\right)^{2}.$ (9.242)

###  Chapter 10 The Functional Objective Geometry

####  A Effective Scalar Curvature

Let us assume that the concrete form of the gravitational potential $V_{eff}$
(8.2) is fixed _ad hoc_ as functional or function of the global dimension $h$.
In such a situation one can express the Ricci scalar of a three-dimensional
embedded space as follows

${{}^{(3)}}R=2\Lambda+2\kappa\ell_{P}^{2}\varrho-6(8\pi)^{2}hV_{eff},$ (10.1)

whereas the global one-dimensional quantum gravity is given by the
evolutionary equation

$\left(\dfrac{\delta^{2}}{\delta h^{2}}+V_{eff}\right)\Psi[h]=0.$ (10.2)

In this manner the quantum gravity is the system describing geometry (10.1),
and quantum mechanics (10.2) of an embedded space.

The quantum gravity (10.1)-(10.2) is in itself non trivial. In fact, this can
be expressed in more general notation

$\displaystyle{{}^{(3)}}R$ $\displaystyle=$ $\displaystyle f[h_{ij}],$ (10.3)
$\displaystyle\dfrac{\delta^{2}\Psi[h]}{\delta h^{2}}$ $\displaystyle=$
$\displaystyle-V_{eff}[h_{ij}]\Psi[h],$ (10.4)

where both $f[h_{ij}]$ as well as $V_{eff}[h_{ij}]$ are scalar-valued
functionals of an induced three-dimensional metric $h_{ij}$, i.e. are
_objective functionals_

$\displaystyle f[h_{ij}]$ $\displaystyle=$ $\displaystyle
f[I_{\mathbf{h}},II_{\mathbf{h}},III_{\mathbf{h}}],$ (10.5)
$\displaystyle\Psi[h_{ij}]$ $\displaystyle=$
$\displaystyle\Psi[I_{\mathbf{h}},II_{\mathbf{h}},III_{\mathbf{h}}],$ (10.6)

where $I_{\mathbf{h}}$, $II_{\mathbf{h}}$, and $III_{\mathbf{h}}$ are the
$3\times 3$ matrix invariants of an induced metric $h_{ij}$

$I_{\mathbf{h}}=\mathrm{Tr}\mathbf{h}\quad,\quad{II}_{\mathbf{h}}=\dfrac{\left(\mathrm{Tr}\mathbf{h}\right)^{2}-\mathrm{Tr}\mathbf{h}^{2}}{2}\quad,\quad{III}_{\mathbf{h}}=\det\mathbf{h},$
(10.7)

which according to the Cayley–Hamilton theorem are the coefficients of the
characteristic polynomial of the matrix $h_{ij}$

$\mathbf{h}^{3}-I_{\mathbf{h}}\mathbf{h}^{2}+II_{\mathbf{h}}\mathbf{h}-III_{\mathbf{h}}\mathbf{I}_{3\times
3}=0.$ (10.8)

In this manner we shall call _the functional objective geometry_ the quantum
gravity given by system of equations (10.3)- (10.4).

We shall call the Ricci scalar curvature (10.1) describing the three-geometry
of an embedded space _the effective scalar curvature_ and study its meaning in
this section. For convenience let us present $V_{eff}$ as an algebraic sum of
three elementary energetic constituents

$V_{eff}=V_{G}+V_{C}+V_{M},$ (10.9)

where $V_{G}$, $V_{C}$, and $V_{M}$ are the geometric and the cosmological,
and the material contributions

$\displaystyle V_{G}$ $\displaystyle=$
$\displaystyle-\dfrac{1}{6(8\pi)^{2}}\dfrac{{{}^{(3)}\\!R}}{h},$ (10.10)
$\displaystyle V_{C}$ $\displaystyle=$
$\displaystyle\dfrac{1}{6(8\pi)^{2}}\dfrac{2\Lambda}{h},$ (10.11)
$\displaystyle V_{M}$ $\displaystyle=$
$\displaystyle\dfrac{1}{6(8\pi)^{2}}\dfrac{2\kappa}{h}\varrho.$ (10.12)

One can list several examples of physical scenarios within the global one-
dimensional quantum gravity, with respect to the choice of the form of the
potential $V_{eff}$.

  1. 1.

The case of constant non vanishing effective gravitational potential
$V_{eff}=V_{c}\neq 0$. In such a situation the Ricci scalar curvature of an
embedded space and the global one-dimensional quantum gravity are

$\displaystyle{{}^{(3)}}R=2\Lambda+2\kappa\ell_{P}^{2}\varrho-6(8\pi)^{2}hV_{c},$
(10.13)
$\displaystyle\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+V_{c}\right)\Psi_{c}[h]=0,$
(10.14)

where $\Psi_{c}[h]$ is a wave functional related to $V_{eff}=V_{c}$.

  2. 2.

The case of trivial effective gravitational potential $V_{eff}=0$. In such a
situation the three-dimensional Ricci scalar curvature and the global one-
dimensional quantum gravity are

$\displaystyle{{}^{(3)}\\!R}=2\Lambda+2\kappa\ell_{P}^{2}\varrho,$ (10.15)
$\displaystyle\dfrac{\delta^{2}}{\delta{h^{2}}}\Psi_{0}[h]=0,$ (10.16)

where $\Psi_{0}$ is a ”free” wave functional related to $V_{eff}=0$.

  3. 3.

The case when a sum of geometric and cosmological contributions is trivial
$V_{G}+V_{C}=0$, but the effective potential does not vanish identically
$V_{eff}\neq 0$. In such a situation the three-dimensional Ricci scalar
curvature and the global one-dimensional quantum gravity are

$\displaystyle{{}^{(3)}}R=2\Lambda,$ (10.17)
$\displaystyle\left(\dfrac{\delta^{2}}{\delta{h^{2}}}-\dfrac{1}{6(8\pi)^{2}}\dfrac{2\kappa\ell_{P}^{2}}{h}\varrho[h]\right)\Psi_{M}[h]=0,$
(10.18)

where $\Psi_{M}$ is a ”material” wave functional related to $V_{M}\neq 0$.

  4. 4.

The case when a sum of geometric and material contributions vanishes
$V_{G}+V_{M}=0$, but the gravitational potential is in general non trivial
$V_{eff}\neq 0$. In such a situation the Ricci scalar curvature of an embedded
space and the global one-dimensional quantum gravity are

$\displaystyle{{}^{(3)}\\!R}=2\kappa\ell_{P}^{2}\varrho,$ (10.19)
$\displaystyle\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+\dfrac{1}{6(8\pi)^{2}}\dfrac{2\Lambda}{h}\right)\Psi_{C}[h]=0.$
(10.20)

Here $\Psi_{C}$ is the ”cosmological” wave functional related to $V_{C}\neq
0$.

  5. 5.

The case when a sum of cosmological and material contributions is trivial
$V_{C}+V_{M}=0$, but the effective gravitational potential is non zero
$V_{eff}\neq 0$. In such a situation the energy density of Matter fields and
the global one-dimensional quantum gravity are

$\displaystyle\varrho=-\dfrac{\Lambda}{\kappa\ell_{P}^{2}},$ (10.21)
$\displaystyle\left(\dfrac{\delta^{2}}{\delta{h^{2}}}-\dfrac{1}{6(8\pi)^{2}}\dfrac{{{}^{(3)}\\!R}}{h}\right)\Psi_{G}[h]=0,$
(10.22)

where $\Psi_{G}$ is the ”geometric” wave functional related to $V_{G}\neq 0$.

  6. 6.

More general approach can be based on complex analysis (For basics advances
see e.g. [606]). Let us pu _ad hoc_ the functional Laurent series expansion in
the global dimension $h$ of the effective gravitational potential $V_{eff}[h]$
in an infinitesimal neighborhood, i.e. a 1-sphere (circle) of a radius
$h_{\epsilon}$, of any fixed initial value $h_{0}$

$V_{eff}[h]=\sum_{-\infty}^{\infty}a_{n}\left(h-h_{0}\right)^{n}\quad\mathrm{in}\quad
C(h_{\epsilon})=\left\\{h:|h-h_{0}|<h_{\epsilon}\right\\},$ (10.23)

where $a_{n}$ are the series coefficients given by the classical functional
integral

$a_{n}=\dfrac{1}{2\pi
i}\int_{C(h_{\epsilon})}\dfrac{V_{eff}[h]}{\left(h-h_{0}\right)^{n+1}}\delta
h,$ (10.24)

which is the Cauchy integral with the Radon/Lebesgue–Stieltjes measure $\delta
h$. Let us take into considerations $h_{0}=0$. Then the Ricci scalar curvature
of a three-dimensional embedded space is

${{}^{(3)}\\!R}=2\Lambda+2\kappa\ell_{P}^{2}\varrho-6(8\pi)^{2}\sum_{-\infty}^{\infty}b_{n}(h-h_{0})^{n},$
(10.25)

where $b_{n}$ is the series coefficient

$b_{n}=a_{n-1}+h_{0}a_{n}=\dfrac{1}{2\pi
i}\int_{C(h_{\epsilon})}\dfrac{h}{\left(h-h_{0}\right)^{n+1}}V_{eff}[h]\delta
h,$ (10.26)

and the global one-dimensional quantum gravity yields

$\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+\sum_{-\infty}^{\infty}a_{n}(h-h_{0})^{n}\right)\Psi[h]=0.$
(10.27)

By the triangle inequality one has

$|b_{n}|\leqslant|a_{n-1}|+|h_{0}||a_{n}|$ (10.28)

so it is easy to see that

$\dfrac{|b_{n}|}{|a_{n}|}\leqslant\dfrac{|a_{n-1}|}{|a_{n}|}+|h_{0}|.$ (10.29)

Applying the inequality

$\left|\int{f}\right|\leqslant\int|f|,$ (10.30)

where $f$ is considered as Riemann-integrable function and integral is
defined, to the coefficients $a_{n}$ and $b_{n}$ one has

$\displaystyle|a_{n}|$ $\displaystyle\leqslant$
$\displaystyle\dfrac{1}{h_{\epsilon}^{n+1}}\dfrac{1}{2\pi}\int_{C(h_{\epsilon})}\left|V_{eff}\right|\delta{h}\leqslant\dfrac{1}{h_{\epsilon}^{n+1}}|a_{-1}|,$
(10.31)

where $a_{-1}$ is the residue of the effective gravitational potential in the
point $h=h_{0}$ given by the Cauchy integral formula

$a_{-1}=\mathrm{Res}(V_{eff},h_{0})=\dfrac{1}{2\pi
i}\int_{C(h_{\epsilon})}{V}_{eff}\delta{h},$ (10.32)

where $C(h_{\epsilon})$ traces out a circle around $h_{0}$ in a
counterclockwise manner on the punctured disk
$D=\left\\{z:0<|h-h_{0}|<R\right\\}$. If the point $h=h_{0}$ is a pole of
order $n$, then

$\mathrm{Res}(V_{eff},h_{0})=\dfrac{1}{\Gamma(n)}\lim_{h\rightarrow
h_{0}}\dfrac{\delta^{n-1}}{\delta{h}^{n-1}}\left((h-h_{0})V_{eff}\right).$
(10.33)

It can be seen straightforwardly that

$\dfrac{|a_{n-1}|}{|a_{n}|}\geqslant{h}_{\epsilon},$ (10.34)

and hence the inequality (10.29) gives

$\dfrac{|b_{n}|}{|a_{n}|}\geqslant{h}_{\epsilon}+|h_{0}|.$ (10.35)

Because by the triangle inequality

$|b_{n+1}|=|a_{n}+h_{0}a_{n+1}|\leqslant|a_{n}|+|h_{0}||a_{n+1}|,$ (10.36)

and

$\dfrac{|a_{n+1}|}{|a_{n}|}\leqslant\dfrac{1}{h_{\epsilon}},$ (10.37)

one obtains

$\dfrac{|b_{n+1}|}{|a_{n}|}\leqslant 1+\dfrac{|h_{0}|}{h_{\epsilon}}.$ (10.38)

applying the inequality (10.35) in the equivalent form

$\dfrac{|a_{n}|}{|b_{n}|}\leqslant\dfrac{1}{{h}_{\epsilon}+|h_{0}|}.$ (10.39)

one receives the upper bound

$\dfrac{|b_{n+1}|}{|a_{n}|}\dfrac{|a_{n}|}{|b_{n}|}=\dfrac{|b_{n+1}|}{|b_{n}|}\leqslant\dfrac{1}{{h}_{\epsilon}+|h_{0}|}\left(1+\dfrac{|h_{0}|}{h_{\epsilon}}\right).$
(10.40)

Another bound for $\dfrac{|b_{n+1}|}{|b_{n}|}$ can be obtained as follows.
Because of

$a_{n}=\dfrac{b_{n}-a_{n-1}}{h_{0}},$ (10.41)

one has

$b_{n+1}=\dfrac{b_{n}-a_{n-1}}{h_{0}}+h_{0}a_{n+1},$ (10.42)

or after small algebraic manipulations

$h_{0}b_{n+1}+a_{n-1}=h_{0}a_{n+1}+b_{n}.$ (10.43)

This equation can be rewritten in the form

$1=\left|\dfrac{h_{0}b_{n+1}}{h_{0}a_{n+1}+b_{n}}+\dfrac{a_{n-1}}{h_{0}a_{n+1}+b_{n}}\right|,$
(10.44)

which after taking into account the triangle inequality

$\left|\dfrac{h_{0}b_{n+1}}{h_{0}a_{n+1}+b_{n}}+\dfrac{a_{n-1}}{h_{0}a_{n+1}+b_{n}}\right|\leqslant\left|\dfrac{h_{0}b_{n+1}}{h_{0}a_{n+1}+b_{n}}\right|+\left|\dfrac{a_{n-1}}{h_{0}a_{n+1}+b_{n}}\right|,$
(10.45)

leads to

$|h_{0}a_{n+1}+b_{n}|\leqslant|h_{0}||b_{n+1}|+|a_{n-1}|.$ (10.46)

Applying again the triangle inequality

$|h_{0}a_{n+1}+b_{n}|\leqslant|h_{0}||a_{n+1}|+|b_{n}|,$ (10.47)

one receives

$|h_{0}||b_{n+1}|-|b_{n}|\leqslant|h_{0}||a_{n+1}|-|a_{n-1}|.$ (10.48)

This inequality can be rewritten as

$|h_{0}|\dfrac{|b_{n+1}|}{|a_{n}|}-\dfrac{|b_{n}|}{|a_{n}|}\leqslant|h_{0}|\dfrac{|a_{n+1}|}{|a_{n}|}-\dfrac{|a_{n-1}|}{|a_{n}|},$
(10.49)

or equivalently

$|h_{0}|\dfrac{|b_{n+1}|}{|b_{n}|}-1\leqslant\dfrac{|a_{n}|}{|b_{n}|}\left(|h_{0}|\dfrac{|a_{n+1}|}{|a_{n}|}-\dfrac{|a_{n-1}|}{|a_{n}|}\right).$
(10.50)

In the light of the inequality (10.39) and the relation

$|h_{0}|\dfrac{|a_{n+1}|}{|a_{n}|}-\dfrac{|a_{n-1}|}{|a_{n}|}\leqslant\dfrac{|h_{0}|}{{h}_{\epsilon}}-{h}_{\epsilon},$
(10.51)

one obtains the bound

$\dfrac{|b_{n+1}|}{|b_{n}|}\leqslant\dfrac{1}{|h_{0}|}\left(1+\dfrac{1}{h_{\epsilon}+|h_{0}|}\left(\dfrac{|h_{0}|}{h_{\epsilon}}-h_{\epsilon}\right)\right).$
(10.52)

Comparing this bound to the previous one (10.40) one receives

$|h_{0}|(|h_{0}|-1)\geqslant 0,$ (10.53)

what gives the condition for $h_{0}$

$|h_{0}|\in\\{0\\}\cup[1,\infty).$ (10.54)

Naturally, there is many other opportunities for selection of a form of the
effective gravitational potential $V_{eff}[h]$. However, in this section we
shall discuss the only a particular case.

####  B The Newton–Coulomb Potential

Let us consider _the residual approximation_ in which the series coefficient
of the effective gravitational potential are

$a_{n}=\left\\{\begin{array}[]{cc}a_{-1}=const&\mathrm{for}\leavevmode\nobreak\
n=-1\\\ 0&\mathrm{for}\leavevmode\nobreak\ n\neq-1\end{array}\right.,$ (10.55)

i.e. the effective gravitational potential (8.2) has the form of the
Newton–Coulomb potential

$V_{eff}=\dfrac{a_{-1}}{h-h_{0}}.$ (10.56)

In such a situation the coefficients $b_{n}$ are

$b_{n}=\left\\{\begin{array}[]{cc}b_{-1}=h_{0}a_{-1}&\mathrm{for}\leavevmode\nobreak\
n=-1\\\ b_{0}=a_{-1}&\mathrm{for}\leavevmode\nobreak\ n=0\\\
0&\mathrm{for}\leavevmode\nobreak\ n\neq-1,0\end{array}\right.,$ (10.57)

so that the Ricci scalar curvature of a three-dimensional space is

${{}^{(3)}\\!R}=2\Lambda+2\kappa\ell_{P}^{2}\varrho-6(8\pi)^{2}a_{-1}\left(1+\dfrac{h_{0}}{h-h_{0}}\right),$
(10.58)

whereas the Klein–Gordon equation (8.1) is

$\left(\dfrac{\delta^{2}}{\delta h^{2}}+\dfrac{a_{-1}}{h-h_{0}}\right)\Psi=0.$
(10.59)

The equation (10.64) defines some three-geometries, but even in the vacuum
situation, i.e. $\varrho=0$ and $\Lambda=0$, it is difficult to establish an
induced geometry which Ricci scalar curvature behaves like

${{}^{(3)}\\!R}\sim 1+\dfrac{h_{0}}{h-h_{0}}.$ (10.60)

Interestingly, in general the residue of the three-dimensional Ricci scalar
curvature in the point $h_{0}$ is

$\mathrm{Res}({{}^{(3)}}R,h_{0})=2\kappa\ell_{P}^{2}\mathrm{Res}(\varrho,h_{0})-6(8\pi)^{2}a_{-1}h_{0},$
(10.61)

i.e. it can be taken equal to zero if and only if the residue of energy
density of Matter fields is

$\mathrm{Res}(\varrho,h_{0})=\dfrac{3(8\pi)^{2}}{\kappa\ell_{P}^{2}}a_{-1}h_{0}.$
(10.62)

If one takes _ad hoc_ the relation

$a_{-1}=\dfrac{\Lambda}{3(8\pi)^{2}},$ (10.63)

then the Ricci scalar curvature of induced three-geometry has the form

${{}^{(3)}\\!R}=2\kappa\ell_{P}^{2}\varrho-\dfrac{2\Lambda{h_{0}}}{h-h_{0}},$
(10.64)

and its residue

$\mathrm{Res}({{}^{(3)}}R,h_{0})=2\kappa\ell_{P}^{2}\mathrm{Res}(\varrho,h_{0})-2\Lambda{h_{0}},$
(10.65)

vanishes if and only if

$\mathrm{Res}(\varrho,h_{0})=\dfrac{\Lambda}{\kappa\ell_{P}^{2}}h_{0}.$
(10.66)

Then also the geometry of an embedded three-manifold is Ricci-flat if and only
if the energy density of Matter fields has the following form

$\varrho=\dfrac{\Lambda}{\kappa\ell_{P}^{2}}\dfrac{{h_{0}}}{h-h_{0}}.$ (10.67)

Another possible Ricci-flat three-manifold is obtained for $\Lambda=0$ and

$\displaystyle\varrho=\dfrac{3(8\pi)^{2}}{\kappa\ell_{P}^{2}}a_{-1}\left(1+\dfrac{h_{0}}{h-h_{0}}\right).$
(10.68)

In general three-spaces having induced metrics satisfying the Ricci scalar
curvature (10.60) are not known yet. However, it is evidently seen that in the
particular case $h_{0}=0$, which is in full accordance with the general
condition (10.54), the situation is much more simpler, i.e.

$\displaystyle{{}^{(3)}\\!R}=2\Lambda+2\kappa\ell_{P}^{2}\varrho-6(8\pi)^{2}a_{-1},$
(10.69) $\displaystyle\left(\dfrac{\delta^{2}}{\delta
h^{2}}+\dfrac{a_{-1}}{h}\right)\Psi=0.$ (10.70)

Let us consider this particular case as the basic situation. We shall call the
global one-dimensional quantum gravity described by the system of equations
(10.69)-(10.70) _the Newton–Coulomb quantum gravity_.

As the example we shall consider vanishing of the energy density

$\varrho\equiv 0,$ (10.71)

i.e. stationarity of Matter fields. We shall call this case _the
Newton–Coulomb stationary quantum gravity_. In such a situation the Ricci
scalar curvature of three-dimensional embedded space becomes

${{}^{(3)}}R=2\Lambda-6(8\pi)^{2}a_{-1}=constant.$ (10.72)

Therefore the Ricci curvature tensor of the three-dimensional manifolds
describes the three-dimensional Einstein manifolds [597]

$R_{ij}=\lambda h_{ij},$ (10.73)

where the sign $\lambda$ of the Einstein manifolds is defined by the
Newton–Coulomb stationary quantum gravity as follows

$\dfrac{2}{3}\Lambda-2(8\pi)^{2}a_{-1}=\lambda.$ (10.74)

Because, however, energy density of Matter fields vanishes therefore the
Einstein manifolds described by the sign (10.74) possess _maximal symmetry_.
By this reason the Newton–Coulomb stationary quantum gravity geometrically
corresponds to three-dimensional _the maximally symmetric Einstein manifolds_.

In this manner in general one can consider the classification of the three-
dimensional spaces, which are maximally symmetric three-dimensional Einstein
manifolds (10.73), with respect to the value of the sign $\lambda$ (10.74) of
a manifold. The following conclusion can be deduced straightforwardly.

######  Conclusion.

The Newton–Coulomb stationary quantum gravity, defined by the effective
gravitational potential $V_{eff}[h]=\dfrac{a_{-1}}{h}$, determines the three-
dimensional embedded spaces which are the maximally symmetric three-
dimensional Einstein manifolds, characterized by the sign (10.74). There are
particular situations:

  1. 1.

When the sign is non zero $\lambda\neq 0$ and the residue of the effective
gravitational potential is negative $a_{-1}=-|\alpha|$, then the effective
potential $V_{eff}[h]$ becomes the Newtonian attractive potential energy

$V_{eff}=-\dfrac{|\alpha|}{h}=-\dfrac{Gm_{1}m_{2}}{\ell_{P}h}.$ (10.75)

When the cosmological constant is positive $\Lambda=+|\Lambda|$ then the
maximally symmetric Einstein three-manifolds are characterized by positive
Ricci scalar curvature

${{}^{(3)}}R=\dfrac{2}{3}|\Lambda|+2(8\pi)^{2}|\alpha|.$ (10.76)

When the cosmological constant is negative $\Lambda=-|\Lambda|$ the maximally
symmetric Einstein three-manifolds are characterized by the Ricci scalar
curvature

${{}^{(3)}}R=-\dfrac{2}{3}|\Lambda|+2(8\pi)^{2}|\alpha|,$ (10.77)

which is negative if $|\Lambda|>3(8\pi)^{2}|\alpha|$, and positive if
$|\Lambda|<3(8\pi)^{2}|\alpha|$.

  2. 2.

When the sign is non zero $\lambda\neq 0$ and the residue of the effective
gravitational potential is positive $a_{-1}=+|\alpha|$, then the effective
potential $V_{eff}[h]$ becomes the Coulomb repulsive potential energy

$V_{eff}=\dfrac{|\alpha|}{h}=\dfrac{q_{1}q_{2}}{4\pi\epsilon_{0}\ell_{P}h}.$
(10.78)

When the cosmological constant is negative $\Lambda=-|\Lambda|$ then the
maximally symmetric Einstein three-manifolds are characterized by negative
Ricci scalar curvature

${{}^{(3)}}R=-\dfrac{2}{3}|\Lambda|-2(8\pi)^{2}|\alpha|.$ (10.79)

When the cosmological constant is positive $\Lambda=+|\Lambda|$ then the
maximally symmetric Einstein three-manifolds are characterized by the Ricci
scalar curvature

${{}^{(3)}}R=\dfrac{2}{3}|\Lambda|-2(8\pi)^{2}|\alpha|,$ (10.80)

which is negative if $|\Lambda|<3(8\pi)^{2}|\alpha|$, an positive if
$|\Lambda|>3(8\pi)^{2}|\alpha|$.

  3. 3.

When the sign is vanishing $\lambda=0$, i.e. the maximally symmetric Einstein
three-manifolds are Ricci-flat, one determines uniquely the value of the
reside of the effective gravitational potential as

$a_{-1}=\pm\dfrac{|\Lambda|}{3(8\pi)^{2}},$ (10.81)

In such a case one obtains the values of cosmological constant

$|\Lambda|=\left\\{\begin{array}[]{rl}\dfrac{3(8\pi)^{3}E_{P}}{4\ell_{P}^{2}}r_{g}(m_{1})r_{g}(m_{2})&\mathrm{for\leavevmode\nobreak\
the\leavevmode\nobreak\ Newton\leavevmode\nobreak\ law}\vspace*{10pt}\\\
\dfrac{3(8\pi)^{3}E_{P}}{\ell_{P}^{2}}r_{e}(q_{1})r_{e}(q_{2})&\mathrm{for\leavevmode\nobreak\
the\leavevmode\nobreak\ Coulomb\leavevmode\nobreak\ law}\end{array}\right.$
(10.82)

where $m$ is mass of a body generating Newtonian gravitational field in vacuum
and $r_{g}(m)=\dfrac{2Gm}{c^{2}}=\kappa{c^{2}}\dfrac{m}{4\pi}$ is its
gravitational radius, $q$ is charge generating Coulombic electrical field in
vacuum and
$r_{e}(q)=\dfrac{q}{\sqrt{4\pi\epsilon_{0}}}\sqrt{\dfrac{G}{c^{4}}}=\sqrt{\dfrac{\kappa}{2\epsilon_{0}}}\dfrac{q}{4\pi}$
is its electrical radius.

Note that in fact, by assuming the relation for the series coefficients
(10.24), the residue $a_{-1}$ is the Cauchy integral of the effective
potential $V_{eff}$ in the fixed point $h_{0}=0$

$a_{-1}=\textrm{Res}\left[\dfrac{1}{6(8\pi)^{2}}\dfrac{1}{h}\left(-{{}^{(3)}\\!R}+2\Lambda+2\kappa\ell_{P}^{2}\varrho\right),h=0\right],$
(10.83)

and its value can be straightforwardly established as

$a_{-1}=\dfrac{1}{6(8\pi)^{2}}\left.\left(-{{}^{(3)}\\!R}+2\Lambda+2\kappa\ell_{P}^{2}\varrho\right)\right|_{h=0}=-\dfrac{{{}^{(3)}}R_{0}}{6(8\pi)^{2}}+\dfrac{\Lambda}{3(8\pi)^{2}}+\dfrac{\kappa\ell_{P}^{2}\varrho_{0}}{3(8\pi)^{2}}\quad,$
(10.84)

where subscript ”$0$” on the LHS means value of a quantity in $h=0$.

When one associates the effective gravitational potential
$V_{eff}=\dfrac{a_{-1}}{h}$ the Newton or the Coulomb potential energy, then
the global dimension becomes a spatial distance $r=\sqrt{x^{2}+y^{2}+z^{2}}$

$h\equiv r\quad,$ (10.85)

so that the evolution (10.59) becomes radial wave equation

$\left(\dfrac{d^{2}}{d{r^{2}}}+\dfrac{\mp|\alpha|}{r}\right)\Psi(r)=0,$
(10.86)

describing a quantum Kepler problem. There is a lot of metrics $h_{ij}$
possessing the same determinant, for instance

$h_{ij}=r^{1/3}r_{ij},$ (10.87)

where $r_{ij}$ is $SO(3)$ group rotation matrix, which can be expressed via
the Euler angles $(\theta,\varphi,\phi)$

$r_{ij}(\theta,\varphi,\phi)\equiv
r_{il}^{(3)}(\theta)r_{lk}^{(2)}(\varphi)r_{kj}^{(3)}(\phi)\quad,$ (10.88)

where $r_{ij}^{(p)}(\vartheta)$ are rotation matrices around a $p$-axis

$\displaystyle r_{ij}^{(3)}(\vartheta)$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}\cos\vartheta&-\sin\vartheta&0\\\
\sin\vartheta&\cos\vartheta&0\\\ 0&0&1\end{array}\right],$ (10.92)
$\displaystyle r_{ij}^{(2)}(\vartheta)$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}\cos\vartheta&0&\sin\vartheta\\\
0&1&0\\\ -\sin\vartheta&0&\cos\vartheta\end{array}\right].$ (10.96)

Interestingly, one can connect the radial wave equation (10.86) with the
radial Schrödinger equation for a particle equipped with energy $E$, mass $m$,
and potential energy $V$ (For more detailed discussion of the radial
Schrödinger equation see e.g. the Ref. [607]). Applying the substitution
$\Psi(r)=rR(r)$ where $R(r)$ is the radial part of the wave function of a
particle, one can present the radial Schrödinger equation in the following
form

$\left[\dfrac{d^{2}}{dr^{2}}-\left(\dfrac{2m}{\hslash^{2}}V+\dfrac{l(l+1)}{r^{2}}\right)\right]\Psi(r)=\dfrac{2mE}{\hslash^{2}}\psi(r),$
(10.97)

where $l$ is one of subscripts of the spherical harmonics
$Y_{l}^{m}(\theta,\phi)$ of degree $l$ and order $m$

$Y_{l}^{m}(\theta,\phi)=\sqrt{\dfrac{2l+1}{4\pi}\dfrac{(l-m)!}{(l+m)!}}P_{lm}(\cos\theta)e^{im\phi},$
(10.98)

which are the angular part of the wave function of a particle. Here
$P_{lm}(x)$ are associated Legendre polynomials

$P_{lm}(x)=\dfrac{1}{2^{l}l!}(1-x^{2})^{m/2}\dfrac{d^{l+m}}{dx^{l+m}}(x^{2}-1)^{l}.$
(10.99)

Putting _ad hoc_ the values $E=0$, $l=0$ and
$-\dfrac{2m}{\hslash^{2}}V=V_{eff}$ one receives excellent coincidence with
the equation (10.86). This nice property allows to construct the physical
interpretation of the wave functional $\Psi(r)$. Namely, it is strictly
related to the wave function $\psi(r,\theta,\phi)=R(r)Y_{0}^{m}(\theta,\phi)$
of a particle equipped with mass $m$ and having the total energy $E=0$ and the
potential energy $V=-\dfrac{\hslash^{2}}{2m}\dfrac{\pm|\alpha|}{r}$. The
identification is as follows

$\Psi(r)=r\dfrac{\psi(r,\theta,\phi)}{Y_{0}^{0}(\theta,\phi)},$ (10.100)

where $Y_{0}^{0}(\theta,\phi)=\dfrac{1}{2\sqrt{\pi}}$ is the only non trivial
value of $Y_{0}^{m}(\theta,\phi)$.

Another linkage to the radial Schrödinger equation is also possible. Let the
wave functional $\Psi(r)$ is the radial part $R(r)$ of the wave function of a
particle, i.e. $\psi(r,\theta,\phi)=\Psi(r)Y_{l}^{m}(\theta,\phi)$, equipped
with mass $m$ and total energy $E$, and moving in _arbitrary_ potential
$V(r)$. Then $\Psi(r)$ satisfies the radial Schrödinger equation

$-\dfrac{1}{r^{2}}\dfrac{d}{dr}\left(r\dfrac{d\Psi}{dr}\right)+\left(\dfrac{2m}{\hslash^{2}}V+\dfrac{l(l+1)}{r^{2}}\right)\Psi=\dfrac{2mE}{\hslash^{2}}\Psi,$
(10.101)

or equivalently

$\dfrac{d^{2}\Psi}{dr^{2}}+\dfrac{1}{r}\dfrac{d\Psi}{dr}-\left(\dfrac{2m}{\hslash^{2}}rV+\dfrac{l(l+1)}{r}\right)\Psi=-\dfrac{2mE}{\hslash^{2}}r\Psi.$
(10.102)

This equations can be separated on the system of two equations

$\displaystyle\dfrac{d^{2}\Psi}{dr^{2}}-\dfrac{2m}{\hslash^{2}}rV\Psi$
$\displaystyle=$ $\displaystyle 0,$ (10.103)
$\displaystyle\dfrac{d\Psi}{dr}-l(l+1)\Psi+\dfrac{2mE}{\hslash^{2}}r^{2}\Psi$
$\displaystyle=$ $\displaystyle 0.$ (10.104)

The equation (10.103) is exactly the wave equation (10.86) if and only if the
equation holds

$V(r)=-\dfrac{\hslash^{2}}{2mr}V_{eff}.$ (10.105)

The equation (10.104) must be solved. It is easy to see that this equation can
be integrated straightforwardly

$\int_{\Psi(r_{0})}^{\Psi(r)}\dfrac{d\Psi}{\Psi}=\int_{r_{0}}^{r}\left(l(l+1)-\dfrac{2mE}{\hslash^{2}}r^{2}\right)dr,$
(10.106)

where $r_{0}$ is some reference value of $r$, what gives

$\ln\left|\dfrac{\Psi(r)}{\Psi(r_{0})}\right|=l(l+1)(r-r_{0})-\dfrac{2mE}{\hslash^{2}}\dfrac{r^{3}-r_{0}^{3}}{3},$
(10.107)

and results in the solution

$\Psi(r)=\Psi(r_{0})\exp\left\\{-l(l+1)r_{0}+\dfrac{2mE}{3\hslash^{2}}r_{0}^{3}\right\\}\exp\left\\{l(l+1)r-\dfrac{2mE}{3\hslash^{2}}r^{3}\right\\}.$
(10.108)

Now one can differentiate the equation (10.104) with respect to $r$ and
obtains

$\displaystyle\dfrac{d^{2}\Psi}{dr^{2}}$ $\displaystyle=$ $\displaystyle
l(l+1)\dfrac{d\Psi}{dr}-\dfrac{2mE}{\hslash^{2}}\left(2r\Psi+r^{2}\dfrac{d\Psi}{dr}\right)=$
(10.109) $\displaystyle=$
$\displaystyle\left[l(l+1)-\dfrac{2mE}{\hslash^{2}}r^{2}\right]\dfrac{d\Psi}{dr}-\dfrac{4mE}{\hslash^{2}}r\Psi=$
$\displaystyle=$
$\displaystyle\left\\{\left[l(l+1)-\dfrac{2mE}{\hslash^{2}}r^{2}\right]^{2}-\dfrac{4mE}{\hslash^{2}}r\right\\}\Psi.$

Application of the equation (10.103) to the result (10.109) leads to

$\dfrac{2m}{\hslash^{2}}rV=\left\\{\left[l(l+1)-\dfrac{2mE}{\hslash^{2}}r^{2}\right]^{2}-\dfrac{4mE}{\hslash^{2}}r\right\\},$
(10.110)

what allows to establish the potential $V$

$V(r)=\dfrac{\hslash^{2}}{2m}\dfrac{l^{2}(l+1)^{2}}{r}-2E-2El(l+1)r+\dfrac{2mE^{2}}{\hslash^{2}}r^{3}.$
(10.111)

Using of the identification (10.105) and the explicit form of
$V_{eff}=\dfrac{\pm|\alpha|}{r}$, one receives the equation for $r$

$\left(\dfrac{2mE}{\hslash^{2}}\right)^{2}r^{5}-\dfrac{4mE}{\hslash^{2}}l(l+1)r^{3}-\dfrac{4mE}{\hslash^{2}}r^{2}+l^{2}(l+1)^{2}r\pm|\alpha|=0,$
(10.112)

which is very difficult to solve, but simplifies for suggested value $l=0$

$\left(\dfrac{2mE}{\hslash^{2}}\right)^{2}r^{5}-\dfrac{4mE}{\hslash^{2}}r^{2}\pm|\alpha|=0.$
(10.113)

The equation (10.113) can be solved easy. Let us focus on the real solution
which for the Coulombic case is

$\displaystyle r$ $\displaystyle=$
$\displaystyle\Bigg{[}\dfrac{\hslash^{2}}{2mE}\left(\dfrac{mE}{\hslash^{2}}|\alpha|\right)^{1/3}\left(\sqrt{1-\dfrac{32}{27}\dfrac{2mE}{\hslash^{2}|\alpha|^{2}}}-1\right)^{1/3}+$
(10.114) $\displaystyle+$
$\displaystyle\dfrac{2}{3}\left(\dfrac{mE}{\hslash^{2}}|\alpha|\right)^{-1/3}\left(\sqrt{1-\dfrac{32}{27}\dfrac{2mE}{\hslash^{2}|\alpha|^{2}}}-1\right)^{-1/3}\Bigg{]}^{1/2},$

while for the Newtonian case one obtains

$\displaystyle r$ $\displaystyle=$
$\displaystyle\Bigg{[}\dfrac{\hslash^{2}}{2mE}\left(\dfrac{mE}{\hslash^{2}}|\alpha|\right)^{1/3}\left(\sqrt{1-\dfrac{32}{27}\dfrac{2mE}{\hslash^{2}|\alpha|^{2}}}+1\right)^{1/3}+$
(10.115) $\displaystyle+$
$\displaystyle\dfrac{2}{3}\left(\dfrac{mE}{\hslash^{2}}|\alpha|\right)^{-1/3}\left(\sqrt{1-\dfrac{32}{27}\dfrac{2mE}{\hslash^{2}|\alpha|^{2}}}+1\right)^{-1/3}\Bigg{]}^{1/2},$

Recall that in our theory $h=r$, i.e. in fact the results (10.114)-(10.115)
fix value of $h$. The distance must be real number. For this must be

$E\leqslant\dfrac{27}{32}\dfrac{\hslash^{2}}{2m}|\alpha|^{2}.$ (10.116)

In general solutions of the algebraic equation (10.112) can be found by
methods of the Galois group. Despite the real and positive solution is
difficult to extract, some simpler solution can be constructed. Let us
substitute to the equation (10.112) as the unknown $r=x+iy$, where the
imaginary part $y$ will be $y\rightarrow 0$ in a certain stage of the
construction. In such a situation the equation (10.112) is equivalent to the
statement that its both real and imaginary part vanish. Such a method
generates the system of equations

$\displaystyle\left(\dfrac{2mE}{\hslash^{2}}\right)^{2}y^{4}+2\left(\dfrac{2mE}{\hslash^{2}}\right)\left(l(l+1)-5\left(\dfrac{2mE}{\hslash^{2}}\right)x^{2}\right)y^{2}-$
$\displaystyle-\left(\dfrac{2mE}{\hslash^{2}}\right)x\left(4+6l(l+1)x-5\left(\dfrac{2mE}{\hslash^{2}}\right)x^{3}\right)+l^{2}(l+1)^{2}=0,$
(10.117) $\displaystyle
5\left(\dfrac{2mE}{\hslash^{2}}\right)^{2}xy^{4}+2\left(\dfrac{2mE}{\hslash^{2}}\right)\left(1+3l(l+1)x-5\left(\dfrac{2mE}{\hslash^{2}}\right)x^{3}\right)y^{2}+$
$\displaystyle+x\left(-2\left(\dfrac{2mE}{\hslash^{2}}\right)x+\left(l(l+1)-\left(\dfrac{2mE}{\hslash^{2}}\right)^{2}\right)^{2}\right)\pm|\alpha|=0,$
(10.118)

where we introduced the shortened notation $L=l(l+1)$. Now one can put $y=0$,
so that $x=r$, and obtain the system of equations

$\displaystyle-\left(\dfrac{2mE}{\hslash^{2}}\right)r\left(4+6l(l+1)r-5\left(\dfrac{2mE}{\hslash^{2}}\right)r^{3}\right)+l^{2}(l+1)^{2}=0,$
(10.119) $\displaystyle
l^{2}(l+1)^{2}r-\left(\dfrac{2mE}{\hslash^{2}}\right)r^{2}\left(2+2l(l+1)r-\left(\dfrac{2mE}{\hslash^{2}}\right)r^{3}\right)\pm|\alpha|=0,$
(10.120)

which can be presented in the form

$\displaystyle\left(\dfrac{2mE}{\hslash^{2}}\right)r^{2}$ $\displaystyle=$
$\displaystyle\dfrac{l^{2}(l+1)^{2}r}{\left(4+6l(l+1)r-5\left(\dfrac{2mE}{\hslash^{2}}\right)r^{3}\right)},$
(10.121) $\displaystyle\left(\dfrac{2mE}{\hslash^{2}}\right)r^{2}$
$\displaystyle=$
$\displaystyle\dfrac{l^{2}(l+1)^{2}r\pm|\alpha|}{\left(2+2l(l+1)r-\left(\dfrac{2mE}{\hslash^{2}}\right)r^{3}\right)},$
(10.122)

and lead to the relation

$\dfrac{l^{2}(l+1)^{2}r}{\left(4+6l(l+1)r-5\left(\dfrac{2mE}{\hslash^{2}}\right)r^{3}\right)}=\dfrac{l^{2}(l+1)^{2}r\pm|\alpha|}{\left(2+2l^{2}(l+1)^{2}r-\left(\dfrac{2mE}{\hslash^{2}}\right)r^{3}\right)}.$
(10.123)

The relation (10.123), however, can be rewritten in much more convenient form
of the 4th order algebraic equation

$\displaystyle 4\left(\dfrac{2mE}{\hslash^{2}}\right)l^{2}(l+1)^{2}r^{4}\pm
5\left(\dfrac{2mE}{\hslash^{2}}\right)|\alpha|r^{3}-4l^{3}(l+1)^{3}r^{2}-$
$\displaystyle 2l(l+1)\left(l(l+1)\pm 3|\alpha|\right)r-(\pm 4|\alpha|)=0.$
(10.124)

In other words the 5th order algebraic equation very difficult to solve has
been reduced to the 4th order algebraic equation which in general can solved
straightforwardly. In our interest is the real and positive solution of the
equation (B) which is given by the formula

$\displaystyle r=-\dfrac{\pm
5|\alpha|}{16l^{2}(l+1)^{2}}+\dfrac{1}{2}\Bigg{(}\dfrac{1}{2^{2/3}3A_{l}}\left(45\dfrac{|\alpha|^{2}}{l(l+1)}-81(\pm|\alpha|)+\dfrac{4\hslash^{2}l^{4}(l+1)^{4}}{mE}\right)+$
$\displaystyle\dfrac{25|\alpha|^{2}}{64l^{4}(l+1)^{4}}+\dfrac{\hslash^{2}l(l+1)}{3mE}+\dfrac{\hslash^{2}A_{l}}{{2}^{10/3}3mEl^{2}(l+1)^{2}}\Bigg{)}^{1/2}+\dfrac{1}{2}\Bigg{(}\dfrac{25|\alpha|^{2}}{32l^{4}(l+1)^{4}}+$
$\displaystyle\dfrac{2\hslash^{2}l(l+1)}{3mE}-\dfrac{1}{2^{2/3}3A_{l}}\left(45\dfrac{|\alpha|^{2}}{l(l+1)}-81(\pm|\alpha|)+\dfrac{4\hslash^{2}l^{4}(l+1)^{4}}{mE}\right)-$
$\displaystyle\dfrac{\hslash^{2}A_{l}}{2^{10/3}3mEl^{2}(l+1)^{2}}+\dfrac{1}{4l(l+1)}\left(\dfrac{2\hslash^{2}l(l+1)}{mE}-\dfrac{\pm
125|\alpha|^{3}}{64l^{5}(l+1)^{5}}+\dfrac{\pm
7\hslash^{2}|\alpha|}{2mE}\right)\times$
$\displaystyle\Bigg{(}\dfrac{25|\alpha|^{2}}{64l^{4}(l+1)^{4}}+\dfrac{\hslash^{2}l(l+1)}{3mE}+\dfrac{\hslash^{2}A_{l}}{{2}^{10/3}3mEl^{2}(l+1)^{2}}+$
$\displaystyle\dfrac{1}{2^{2/3}3A_{l}}\left(45\dfrac{|\alpha|^{2}}{l(l+1)}-81(\pm|\alpha|)+\dfrac{4\hslash^{2}l^{4}(l+1)^{4}}{mE}\right)\Bigg{)}^{-1/2}\Bigg{)}^{1/2},$
(10.125)

where

$\displaystyle A_{l}=\dfrac{2mE}{\hslash^{2}}\Bigg{(}-\dfrac{\pm
675\hslash^{2}|\alpha|^{3}}{2mE}+\dfrac{\hslash^{4}l^{6}(l+1)^{6}}{2(mE)^{2}}\bigg{(}\dfrac{351|\alpha|^{2}}{l(l+1)}-\dfrac{\pm
297|\alpha|}{l(l+1)}-$
$\displaystyle\dfrac{8\hslash^{2}l^{3}(l+1)^{3}}{mE}+54\bigg{)}+\dfrac{\hslash^{2}}{2mE}\Bigg{(}-\dfrac{\hslash^{2}l^{6}(l+1)^{6}}{mE}\bigg{(}\dfrac{45|\alpha|^{2}}{l(l+1)}+\dfrac{4\hslash^{2}l^{4}(l+1)^{4}}{mE}-$
$\displaystyle 81(\pm|\alpha|)\bigg{)}^{3}+\bigg{(}\pm
675|\alpha|^{3}-\dfrac{\hslash^{2}l^{8}(l+1)^{8}}{mE}\bigg{(}\dfrac{351|\alpha|^{2}}{l(l+1)}-\dfrac{\pm
297|\alpha|}{l(l+1)}-$
$\displaystyle\dfrac{8\hslash^{2}l^{3}(l+1)^{3}}{mE}\bigg{)}\bigg{)}^{2}+54\Bigg{)}^{1/2}\Bigg{)}^{1/3}.$
(10.126)

Another separation of the radial Schrödinger equation (10.102) is possible.
Namely, one can rewrite this equation as the system

$\displaystyle\dfrac{d^{2}\Psi}{dr^{2}}-\left(\dfrac{2m}{\hslash^{2}}rV+\dfrac{l(l+1)}{r}\right)\Psi$
$\displaystyle=$ $\displaystyle 0,$ (10.127)
$\displaystyle\dfrac{d\Psi}{dr}+\dfrac{2mE}{\hslash^{2}}r^{2}\Psi$
$\displaystyle=$ $\displaystyle 0.$ (10.128)

The procedure analogous to the previous separation gives the solution

$\Psi(r)=\Psi(r_{0})\exp\left\\{\dfrac{2mE}{3\hslash^{2}}r_{0}^{3}\right\\}\exp\left\\{-\dfrac{2mE}{3\hslash^{2}}r^{3}\right\\},$
(10.129)

the potential

$V(r)=-\dfrac{\hslash^{2}}{2m}\dfrac{l(l+1)}{r^{2}}-2E+\dfrac{2mE^{2}}{\hslash^{2}}r^{3},$
(10.130)

and the equation for $r$

$\left(\dfrac{2mE}{\hslash^{2}}\right)^{2}r^{4}-\dfrac{4mE}{\hslash^{2}}r-\dfrac{\hslash^{2}}{2m}l^{2}(l+1)^{2}\pm|\alpha|=0.$
(10.131)

This is equation can be solved for arbitrary $l$ with no problems. There are
real and positive solutions if and only if

$\pm|\alpha|\leqslant\dfrac{\hslash^{2}}{2m}l^{2}(l+1)^{2}.$ (10.132)

Then there is the only one real and positive solution given by

$\displaystyle
r=\dfrac{1}{2}\Bigg{\\{}\dfrac{2\hslash^{2}}{mE}\Bigg{[}\dfrac{2^{-1/3}}{(mE/\hslash^{2})^{2/3}}\beta_{l}^{1/3}+\dfrac{2^{2/3}\alpha_{l}}{3(mE/\hslash^{2})^{4/3}}\beta_{l}^{-1/3}\Bigg{]}^{-1/2}$
$\displaystyle-\dfrac{2^{-1/3}}{(mE/\hslash^{2})^{2/3}}\beta_{l}^{1/3}-\dfrac{2^{2/3}\alpha_{l}}{3(mE/\hslash^{2})^{4/3}}\beta_{l}^{-1/3}\Bigg{\\}}^{1/2}$
$\displaystyle+\dfrac{1}{2}\left(\dfrac{2^{-1/3}}{(mE/\hslash^{2})^{2/3}}\beta_{l}^{1/3}+\dfrac{2^{2/3}\alpha_{l}}{3(mE/\hslash^{2})^{4/3}}\beta_{l}^{-1/3}\right)^{1/2}$
(10.133)

where we have introduced the shortened notation

$\displaystyle\alpha_{l}$ $\displaystyle=$
$\displaystyle-\dfrac{\hslash^{2}}{2m}l^{2}(l+1)^{2}\pm|\alpha|,$ (10.134)
$\displaystyle\beta_{l}$ $\displaystyle=$ $\displaystyle
1+\sqrt{1-\dfrac{4}{27}\dfrac{\hslash^{4}}{m^{2}E^{2}}\alpha_{l}^{3}}.$
(10.135)

It is easy to deduce the solution for $l=0$.

####  C Boundary Conditions for The Wave Functionals

In this subsection we shall consider certain solutions of the global one-
dimensional quantum gravity (8.1) for the approximation of the effective
gravitational potential $V_{eff}$ discussed in the previous subsection. For
the considered situation the $h$-evolution

$\left(\dfrac{\delta^{2}}{\delta{h^{2}}}\mp\dfrac{|\alpha|}{h}\right)\Psi^{\mp}[h]=0,$
(10.136)

is solved by two type of wave functions $\Psi^{\mp}$ where the attractive wave
functions $\Psi_{G}^{-}[h]$ are associated with the the Newton-like effective
gravitational potential, and the repulsive ones $\Psi^{+}[h]$ are associated
with the Coulomb-like effective gravitational potential. Because of manifest
one-dimensionality of the functional evolutionary equation (10.136) one can
solve this equation in frames of the theory of ordinary differential equations
by treatment of the functional derivative as the ordinary one, i.e.
$\dfrac{\delta}{\delta h}=\dfrac{d}{dh}$, and the functional and a function
$\Psi[h]=\Psi(h)$ with no loss of the generality.

In this manner we shall consider here the equation

$\left(\dfrac{d^{2}}{dh^{2}}\mp\dfrac{|\alpha|}{h}\right)\Psi^{\mp}(h)=0.$
(10.137)

The general solution of this differential equation can be constructed
straightforwardly by application of the Bessel functions $J_{n}$ and $Y_{n}$
for the case of the attractive potential

$\Psi^{-}[h]=\sqrt{|\alpha|h}\left[C_{1}^{-}J_{1}\left(2\sqrt{|\alpha|h}\right)+2iC_{2}^{-}Y_{1}\left(2\sqrt{|\alpha|h}\right)\right],$
(10.138)

and in terms of the modified Bessel $I_{n}$ and $K_{n}$ for the case of the
repulsive potential

$\Psi^{+}[h]=-\sqrt{|\alpha|h}\left[C_{1}^{+}I_{1}\left(2\sqrt{|\alpha|h}\right)+2C_{2}^{+}K_{1}\left(2\sqrt{|\alpha|h}\right)\right],$
(10.139)

where $C_{1}^{\pm}$ and $C_{2}^{\pm}$ are constants of integration. In
concrete calculations one can take the standard definitions of the Bessel
functions of first and second kind [602], $J_{\alpha}(x)$ and $Y_{\alpha}(x)$,
which are

$\displaystyle J_{\alpha}(x)$ $\displaystyle=$
$\displaystyle\dfrac{1}{\pi}\int_{0}^{\pi}dt\cos\left(x\cos t-\alpha
t\right),$ (10.140) $\displaystyle Y_{\alpha}(x)$ $\displaystyle=$
$\displaystyle\dfrac{J_{\alpha}(x)\cos\left(\alpha\pi\right)-J_{-\alpha}(x)}{\sin\left(\alpha\pi\right)},$
(10.141)

as well as the modified Bessel functions of the first and the second kind,
$I_{\alpha}(x)$ and $K_{\alpha}(x)$, which are

$\displaystyle I_{\alpha}(x)$ $\displaystyle=$
$\displaystyle\dfrac{1}{\pi}\int_{0}^{\pi}dt\exp\left(x\cos
t\right)\cos\left(\alpha t\right),$ (10.142) $\displaystyle K_{\alpha}(x)$
$\displaystyle=$
$\displaystyle\dfrac{\pi}{2}\dfrac{I_{-\alpha}(x)-I_{\alpha}(x)}{\sin\left(\alpha\pi\right)}.$
(10.143)

Recall that standardly the values of the Bessel functions of second kind and
the modified Bessel functions of second kind for any integers $n$ can be
received by application of the limiting procedure

$\displaystyle Y_{n}(x)$ $\displaystyle=$
$\displaystyle\lim_{\alpha\rightarrow n}Y_{\alpha}(x),$ (10.144)
$\displaystyle{K}_{n}(x)$ $\displaystyle=$
$\displaystyle\lim_{\alpha\rightarrow n}K_{\alpha}(x).$ (10.145)

In further part of this section we shall present solutions to the one-
dimensional quantum mechanics (10.137) with respect to several selected
boundary conditions for the general solutions (10.138) and (10.139).

###### Boundary Conditions I

Let us consider the global one-dimensional quantum mechanics (8.1) with the
boundary conditions for some selected initial value of the dimension
$h=h_{I}$:

$\displaystyle\Psi[h_{I}]$ $\displaystyle=$ $\displaystyle\Psi_{I},$ (10.146)
$\displaystyle\dfrac{\delta\Psi}{\delta h}[h_{I}]$ $\displaystyle=$
$\displaystyle\Psi^{\prime}_{I}.$ (10.147)