Virtual black hole

where ${\displaystyle G_{\mu \nu }=R_{\mu \nu }-{R \over 2}g_{\mu \nu }}$ is the Einstein tensor, which combines the Ricci tensor, the scalar curvature and the metric tensor; ${\displaystyle \Lambda }$ is the cosmological constant; а ${\displaystyle T_{\mu \nu }}$ is the energy-momentum tensor of matter; ${\displaystyle \pi }$ is the mathematical constant pi; ${\displaystyle c}$ is the speed of light; and ${\displaystyle G}$ is the Newtonian constant of gravitation.

Einstein suggested that physical space is Riemannian, i.e. curved and therefore put Riemannian geometry at the basis of the theory of gravity. A small region of Riemannian space is close to flat space.^[5]

For any tensor field ${\displaystyle N_{\mu \nu ...}}$, we may call ${\displaystyle N_{\mu \nu ...}{\sqrt {-g}}}$ a tensor density, where ${\displaystyle g}$ is the determinant of the metric tensor ${\displaystyle g_{\mu \nu }}$. The integral ${\displaystyle \int N_{\mu \nu ...}{\sqrt {-g}}\,d^{4}x}$ is a tensor if the domain of integration is small. It is not a tensor if the domain of integration is not small, because it then consists of a sum of tensors located at different points and it does not transform in any simple way under a transformation of coordinates.^[6] Here we consider only small domains. This is also true for the integration over the three-dimensional hypersurface ${\displaystyle S^{\nu }}$.

Thus, the Einstein field equations for a small spacetime domain can be integrated by the three-dimensional hypersurface ${\displaystyle S^{\nu }}$. Have^[4][7]

${\displaystyle {\frac {1}{4\pi }}\int \left(G_{\mu \nu }+\Lambda g_{\mu \nu }\right){\sqrt {-g}}\,dS^{\nu }={2G \over c^{4}}\int T_{\mu \nu }{\sqrt {-g}}\,dS^{\nu }}$

Since integrable space-time domain is small, we obtain the tensor equation

where ${\displaystyle P_{\mu }={\frac {1}{c}}\int T_{\mu \nu }{\sqrt {-g}}\,dS^{\nu }}$ is the component of the 4-momentum of matter, ${\displaystyle R_{\mu }={\frac {1}{4\pi }}\int \left(G_{\mu \nu }+\Lambda g_{\mu \nu }\right){\sqrt {-g}}\,dS^{\nu }}$ is the component of the radius of curvature small domain.

The resulting tensor equation can be rewritten in another form. Since ${\displaystyle P_{\mu }=mc\,U_{\mu }}$ then

${\displaystyle R_{\mu }={\frac {2G}{c^{3}}}mc\,U_{\mu }=r_{s}\,U_{\mu }}$

where ${\displaystyle r_{s}}$ is the Schwarzschild radius, ${\displaystyle U_{\mu }}$ is the 4-speed, ${\displaystyle m}$ is the gravitational mass. This record reveals the physical meaning of the ${\displaystyle R_{\mu }}$ values as components of the gravitational radius ${\displaystyle r_{\text{s}}}$.

In a small area of space-time is almost flat and this equation can be written in the operator form

${\displaystyle {\hat {R}}_{\mu }={\frac {2G}{c^{3}}}{\hat {P}}_{\mu }={\frac {2G}{c^{3}}}(-i\hbar ){\frac {\partial }{\partial \,x^{\mu }}}=-2i\,\ell _{P}^{2}{\frac {\partial }{\partial \,x^{\mu }}}}$

or

Then the commutator of operators ${\displaystyle {\hat {R}}_{\mu }}$ and ${\displaystyle {\hat {x}}_{\mu }}$ is

${\displaystyle [{\hat {R}}_{\mu },{\hat {x}}_{\mu }]=-2i\ell _{P}^{2}}$

From here follow the specified uncertainty relations

Substituting the values of ${\displaystyle R_{\mu }={\frac {2G}{c^{3}}}m\,c\,U_{\mu }}$ and ${\displaystyle \ell _{P}^{2}={\frac {\hbar \,G}{c^{3}}}}$ and reducing identical constants from two sides, we get Heisenberg's uncertainty principle

${\displaystyle \Delta P_{\mu }\Delta x_{\mu }=\Delta (mc\,U_{\mu })\Delta x_{\mu }\geq {\frac {\hbar }{2}}}$

In the particular case of a static spherically symmetric field and static distribution of matter ${\displaystyle U_{0}=1,U_{i}=0\,(i=1,2,3)}$ and have remained

${\displaystyle \Delta R_{0}\Delta x_{0}=\Delta r_{\text{s}}\Delta r\geq \ell _{P}^{2}}$

where ${\displaystyle r_{\text{s}}}$ is the Schwarzschild radius, ${\displaystyle r}$ is the radial coordinate. Here ${\displaystyle R_{0}=r_{\text{s}}}$ and ${\displaystyle x_{0}=c\,t=r}$, since the matter moves with velocity of light in the Planck scale.

Last uncertainty relation allows make us some estimates of the equations of general relativity at the Planck scale. For example, the equation for the invariant interval ${\displaystyle dS}$ в in the Schwarzschild solution has the form

${\displaystyle dS^{2}=\left(1-{\frac {r_{\text{s}}}{r}}\right)c^{2}dt^{2}-{\frac {dr^{2}}{1-{r_{\text{s}}}/{r}}}-r^{2}(d\Omega ^{2}+\sin ^{2}\Omega d\varphi ^{2})}$

Substitute according to the uncertainty relations ${\displaystyle r_{\text{s}}\approx \ell _{P}^{2}/r}$. We obtain

${\displaystyle dS^{2}\approx \left(1-{\frac {\ell _{P}^{2}}{r^{2}}}\right)c^{2}dt^{2}-{\frac {dr^{2}}{1-{\ell _{P}^{2}}/{r^{2}}}}-r^{2}(d\Omega ^{2}+\sin ^{2}\Omega d\varphi ^{2})}$

It is seen that at the Planck scale ${\displaystyle r=\ell _{P}}$ space-time metric is bounded below by the Planck length (division by zero appears), and on this scale, there are real and virtual Planckian black holes.

Similar estimates can be made in other equations of general relativity. For example, analysis of the Hamilton–Jacobi equation for a centrally symmetric gravitational field in spaces of different dimensions (with help of the resulting uncertainty relation) indicates a preference (energy profitability) for three-dimensional space for the emergence of virtual black holes (quantum foam, the basis of the "fabric" of the Universe.).^[4][7] This may have predetermined the three-dimensionality of the observed space.

Prescribed above uncertainty relation valid for strong gravitational fields, as in any sufficiently small domain of a strong field space-time is essentially flat.