Virtual black hole

!Proof

{{Equation box 1

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|equation=$G\_\{\backslash mu\backslash nu\}\; +\; \backslash Lambda\; g\_\{\backslash mu\backslash nu\}\; =\; \{8\; \backslash pi\; G\; \backslash over\; c^4\}\; T\_\{\backslash mu\backslash nu\}$

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where $G\_\{\backslash mu\backslash nu\}\; =\; R\_\{\backslash mu\backslash nu\}\; -\; \{R\; \backslash over\; 2\}\; g\_\{\backslash mu\backslash nu\}$ is the Einstein tensor, which combines the Ricci tensor, the scalar curvature and the metric tensor; $\backslash Lambda$ is the cosmological constant; а $T\_\{\backslash mu\backslash nu\}$ is the energy-momentum tensor of matter; $\backslash pi$ is the mathematical constant pi; $c$ is the speed of light; and $G$ 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.{{harvnb|Dirac|1975|p=9}}

For any tensor field $N\_\{\backslash mu\backslash nu...\}$, we may call $N\_\{\backslash mu\backslash nu...\}\backslash sqrt\{-g\}$ a tensor density, where $g$ is the determinant of the metric tensor $g\_\{\backslash mu\backslash nu\}$. The integral $\backslash int\; N\_\{\backslash mu\backslash nu...\}\backslash sqrt\{-g\}\backslash ,d^4x$ 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.{{harvnb|Dirac|1975|p=37}} Here we consider only small domains. This is also true for the integration over the three-dimensional hypersurface $S^\{\backslash nu\}$.

Thus, the Einstein field equations for a small spacetime domain can be integrated by the three-dimensional hypersurface $S^\{\backslash nu\}$. Have{{Cite journal |last=Klimets |first=Alexander |date=2017 |title=On the fundamental role of massless form of matter in physics. Quantum gravity. |url=https://philpapers.org/archive/ALXOTF.pdf |journal=Fizika B |issue=9 |pages=23–42}}

: $\backslash frac\{1\}\{4\backslash pi\}\backslash int\backslash left\; (G\_\{\backslash mu\backslash nu\}\; +\; \backslash Lambda\; g\_\{\backslash mu\backslash nu\}\backslash right\; )\backslash sqrt\{-g\}\backslash ,dS^\{\backslash nu\}\; =\; \{2G\; \backslash over\; c^4\}\; \backslash int\; T\_\{\backslash mu\backslash nu\}\backslash sqrt\{-g\}\backslash ,dS^\{\backslash nu\}$

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

{{Equation box 1

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|equation=$R\_\{\backslash mu\}=\backslash frac\{2G\}\{c^3\}P\_\{\backslash mu\}$

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where $P\_\{\backslash mu\}=\backslash frac\{1\}\{c\}\backslash int\; T\_\{\backslash mu\backslash nu\}\backslash sqrt\{-g\}\backslash ,dS^\{\backslash nu\}$ is the component of the 4-momentum of matter, $R\_\{\backslash mu\}=\backslash frac\{1\}\{4\backslash pi\}\backslash int\backslash left\; (G\_\{\backslash mu\backslash nu\}\; +\; \backslash Lambda\; g\_\{\backslash mu\backslash nu\}\backslash right\; )\backslash sqrt\{-g\}\backslash ,dS^\{\backslash nu\}$ is the component of the radius of curvature small domain.

The resulting tensor equation can be rewritten in another form. Since $P\_\{\backslash mu\}=mc\backslash ,U\_\{\backslash mu\}$ then

:$R\_\{\backslash mu\}=\backslash frac\{2G\}\{c^3\}mc\backslash ,U\_\{\backslash mu\}=r\_s\backslash ,U\_\{\backslash mu\}$

where $r\_s$ is the Schwarzschild radius, $U\_\{\backslash mu\}$ is the 4-speed, $m$ is the gravitational mass. This record reveals the physical meaning of the $R\_\{\backslash mu\}$ values as components of the gravitational radius $r\_\backslash text\{s\}$.

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

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

or

{{Equation box 1

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|title=The basic equation of quantum gravity

|equation=$-2i\backslash ell^2\_\{P\}\backslash frac\{\backslash partial\}\{\backslash partial\; x^\{\backslash mu\}\}|\backslash Psi(x\_\{\backslash mu\})\backslash rangle=\backslash hat\; R\_\{\backslash mu\}|\backslash Psi(x\_\{\backslash mu\})\backslash rangle$

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

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

From here follow the specified uncertainty relations

{{Equation box 1

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|equation=$\backslash Delta\; R\_\{\backslash mu\}\backslash Delta\; x\_\{\backslash mu\}\backslash ge\backslash ell^2\_\{P\}$

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Substituting the values of $R\_\{\backslash mu\}=\backslash frac\{2G\}\{c^3\}m\backslash ,c\backslash ,U\_\{\backslash mu\}$ and $\backslash ell^2\_\{P\}=\backslash frac\{\backslash hbar\backslash ,G\}\{c^3\}$

and reducing identical constants from two sides, we get Heisenberg's uncertainty principle

:$\backslash Delta\; P\_\{\backslash mu\}\backslash Delta\; x\_\{\backslash mu\}=\backslash Delta\; (mc\backslash ,U\_\{\backslash mu\})\backslash Delta\; x\_\{\backslash mu\}\backslash ge\backslash frac\{\backslash hbar\}\{2\}$

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

:$\backslash Delta\; R\_\{0\}\backslash Delta\; x\_\{0\}=\backslash Delta\; r\_\backslash text\{s\}\backslash Delta\; r\backslash ge\backslash ell^2\_\{P\}$

where $r\_\backslash text\{s\}$ is the Schwarzschild radius, $r$ is the radial coordinate. Here $R\_0=r\_\backslash text\{s\}$ and $x\_0=c\backslash ,t=r$, 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 $dS$ в in the Schwarzschild solution has the form

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

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

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

It is seen that at the Planck scale $r=\backslash ell\_P$ 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.). 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.